Picking and Smoothing of Seismic Events and CRS Attributes ...

Picking and Smoothing of Seismic
Events and CRS Attributes,
Application for Inversion
Picken und Glätten von seismischen
Ereignissen und CRS Attributen,
Anwendung bei der Inversion
Diplomarbeit
von
Ingo Koglin
Geophysikalisches Institut
der
Universität Karlsruhe
Das Thema wurde von Prof. Dr. P. Hubral gestellt.
Korreferent: Prof. Dr. F. Wenzel
Karlsruhe, November 2001

EHRENWÖRTLICHE ERKLÄRUNG:
Hiermit versichere ich, dass ich die vorliegende Arbeit selbständig und nur mit
den angegebenen Hilfsmitteln verfasst habe.
Karlsruhe, November 2001 Unterschrift

Zusammenfassung
Englische Fachbegriffe
Im Sprachgebrauch der Geophysik werden viele eigenständige Begriffe mit ihrem
englischen Namen verwendet, da entweder für sie keine adäquate deutsche
Übersetzung gefunden werden konnte oder sie inzwischen allgemein üblich sind.
Diese Wörter werden im Verlauf dieser Zusammenfassung, mit Ausnahme ihrer
großgeschriebenen Abkürzungen, kursiv geschrieben.
Einleitung
Wissenschaftler der Geophysik wollen ihr Wissen über die Struktur der Erde erweitern,
um Kohlenwasserstoffreservoirs aufzufinden, um Erdbebenrisiken oder
das Gefährdungspotential eines Vulkan abschätzen zu können. Jedoch ist es nicht
möglich die Erde zweizuteilen, um die Strukturen direkt vor Augen zu haben.
Daher beschreibt die vorliegende Arbeit Verarbeitungsmethoden, die genutzt werden
können, um schliesslich ein leichter interpretierbares Abbild des Untergrundes
aus reflexionsseismischen Daten zu erhalten.
Die Erfassung seismischer Daten in der zweidimensionalen Reflexionsseismik
wird hauptsächlich mit common-shot (CS) Anordnungen durchgeführt (siehe Abbildung
1.1(a)). Dabei wird die “Antwort” des Untergrundes auf seismische Impulse
aufgenommen. Die aufgezeichneten seismischen Spuren, die in einer CS-
Sektion dargestellt werden, gehören zu einem Ereignis z. B. einer Explosion, die
eine sich im Untergrund ausbreitende seismische Welle auslöst. Ein Seismogramm
enthält Spuren, die abhängig von ihrem offset beziehungsweise half-offset sortiert
werden. Der half-offset ist die halbe und der offset die ganze Distanz zwischen
Quell- und Empfängerposition für jedes Schuss-Empfänger-Paar. Die midpoint-
Koordinate ist der Mittelpunkt zwischen der Quell- und Empfängerposition. Die
erzeugte Welle wird an Diskontinuitäten im Untergrund refraktiert oder reflektiert.
Die Teile der reflektierten oder refraktierten Welle, die die Empfängern erreichen,
werden anhand der vergangenen Zeit relativ zur Auslösung der Quelle,
I

Zusammefassung
d. h. der Laufzeit, aufgezeichnet. Die CS-Anordnung wird dann im Falle zweidimensionaler
Messungen entlang einer geradlinigen seismischen Linie verschoben,
um viele CS-Sektionen zu erhalten, die Reflexionen von jeweils gleichen
Punkten im beleuchteten Untergrund enthalten. Solch eine Redundanz in dem
gesamten aufgezeichneten Datensatz ergibt den sogenannten multi-coverage Datensatz,
da er Tiefenpunkte mehrfach abdeckt.
Verschiedene Umsortierungen der aufgenommenen Spuren können vorgenommen
werden, um andere Sektionen zu bilden, die einen Schritt auf dem Weg
zur Interpretation der aufgenommenen Daten repräsentieren. Eine Möglichkeit
ist die Spuren in einer common-offset (CO) Sektion oder einer common-midpoint
(CMP) Sektion zusammenzufassen. Eine CO-Sektion beinhaltet alle Spuren mit
einem bestimmten konstanten offset, die nach ihrer midpoint-Koordinate sortiert
sind (siehe Abbildung 1.1(b)).
Eine spezielle CO-Sektion ist die zero-offset-Sektion. Hier ist der (half-) offset Null,
das heisst, dass die Positionen eines Quell-Empfänger-Paares zusammenfallen
(siehe Abbildung 1.1(c)). Jedoch kann diese ZO-Anordnung bei der reflexionsseismischen
Datenerfassung nicht realisiert werden, da die Quelle möglicherweise
den koinzidenten Empfänger zerstört. Die ZO-Sektion muss üblicherweise von
Stapelungsmethoden simuliert werden.
Die CMP-Sektion kombiniert alle Spuren des gleichen midpoint sortiert nach aufsteigenden
(half-) offsets (siehe Abbildung 1.1(d)). Die vier Darstellungen in Abbildung
1.1 verdeutlichen die Strahlenwege der meist verbreiteten Anordnungen
am Beispiel eines horizontalen Reflektors mit einem Überbau konstanter
Geschwindigkeit. Die CMP-Sektion wird manchmal auch als common-depth-point
(CDP) Darstellung bezeichnet, was aber nur für den Fall von ebenen horizontalen
Schichten richtig ist. Dort ist die x-Koordinate des CMP und des CDP gleich.
Sobald der Reflektor aber nicht mehr horizontal ist (siehe Abbildung 1.2), unterscheiden
sich die x-Werte des CMP und des CDP. Daher erreichen die Strahlen für
eine CMP-Konfiguration den ebenen geneigten Reflektor in einem verschmierten
Bereich.
Innerhalb dieser Diplomarbeit werden alle aufgenommenen Spuren in einem
dreidimensionalen Datenraum plaziert. Dieser Datenraum ist durch die folgenden
Achsen bestimmt: die xm-Achse bezeichnet die midpoint-Koordinate, die h-
Achse steht für den half-offset und die t-Achse gibt die seit dem Schuss relativ
vergangene Zeit, die Laufzeit, an. Die vier in Abbildung 1.1 erwähnten Sektionen
sind in diesem Datenraum enthalten. Abbildung 1.3 zeigt die zu CS-, COund
CMP-Sektionen gehörigen Ebenen. Rot eingefärbte Ebenen bezeichnen CS-
Sektionen. Jede CS-Sektion (xm const.) schliesst mit jeder Ebene const.
oder const. einen Winkel von 45 Grad ein. Grün eingefärbte Ebenen sind Bei-
xm¡ h¡ h¡
spiele für CMP-Sektionen. Innerhalb einer CMP-Sektion ist der midpoint konstant
const.), d. h. gleich (engl. common) für alle enthaltenen Spuren. Die blau
eingefärbten Ebenen stellen einige CO-Sektionen (h¡ const.) dar. Hierbei ist der
(xm¡
II

half-offset beziehungsweise offset konstant für alle Spuren einer CO-Sektion. Der
Spezialfall einer ZO-Sektion ist die vorderste Ebene dieses Datenraumes, d. h.,
die Ebene 0.
Der beschriebene Datenraum wird für die common-reflection-surface (CRS) Stapelung
(engl. stack) verwendet. Stapeln bedeutet hierbei, die Summation aller Amplituden
einer Sektion entlang einer Laufzeitkurve oder Laufzeitfläche in den Daten.
Falls die Laufzeitkurve der realen Kurve des Reflexionsereignisses in den
Daten entspricht, dann werden die kohärenten Amplituden konstruktiv aufsummiert.
Das Ergebnis wird im entsprechenden Punkt der ZO-Sektion plaziert. Der
Nutzen ist ein höheres Signal-zu-Rauschen (engl. signal-to-noise (S/N)) Verhält-
h¡
nis, welches die Identifizierung von Reflexionsereignissen erleichtert. Das S/N-
Verhältnis ist definiert als das Verhältnis der maximalen Amplitude aller Reflexionsereignisse
eines Datensatzes zur root mean square Amplitude des Rauschens.
Ein S/N-Verhältnis kleiner als eins bedeutet, dass die Signale eines Reflexionsereignisses
meist nicht visuell erfasst werden können, da ihre Amplituden kleiner
sind als die des Rauschens. Daher wird ein hoher S/N-Wert angestrebt. Die sogenannte
CMP-Stapelung ist ein Beispiel für die Summation von Amplituden
in einer CMP-Sektion entlang einer Laufzeitkurve. Im Gegensatz zu der CMP-
Stapelung legt die CRS-Stapelungsmethode eine Stapelfläche innerhalb des 3D
Datenraumes fest. Das Ziel ist wesentlich mehr kohärente Amplituden aufzusummieren.
Somit steigt das S/N-Verhältnis weiter an.
Die simulierte ZO-Sektion ist die Basis für viele Inversionsverfahren, um
ein leichter interpretierbares Abbild des Untergrundes zu erhalten. In
der ZO-Sektion werden möglichst viele leicht erkennbare Reflexionsereignisse
identifiziert. Die Zweiweg-ZO-Laufzeit und einige andere Attribute,
im Folgenden weiter erklärt, werden verwendet, um ein Iso-Schichtgeschwindigkeitenmodell
oder Makro-Geschwindigkeitsmodell des Untergrundes
zu erstellen. Hierbei ist der Unterschied zwischen einem Iso-Schichtgeschwindigkeitenmodell
und einem Makro-Geschwindigkeitsmodell, dass das
letztere keine Schichtgrenzen mit scharfen Geschwindigkeitssprüngen enthält.
Das Iso-Schichtgeschwindigkeitenmodell wird aufgebaut aus Schichten mit
Geschwindigkeitsverteilungen (hier: mit konstanter Geschwindigkeit) unterteilt
durch Schichtgrenzen. Solche Iso-Schichtgeschwindigkeitenmodelle können
durch Glättung in Makro-Geschwindigkeitsmodelle umgeformt werden. Die Berechnung
dieser Geschwindigkeitsmodelle wird Inversion genannt, da sie das
inverse Problem löst. Das bedeutet, ein Geschwindigkeitsmodell aus den Informationen
gegeben durch die Reflexions-Antworten des Untergrundes zu bestimmen.
Der letzte Schritt ist die Tiefenmigration. Die post-stack Tiefenmigration transformiert
alle Punkte der simulierten ZO-Sektion aus dem Zeitbereich in den Tiefenbereich.
Diese Transformation benötigt das Makro-Geschwindigkeitsmodell des
Untergrundes, damit die Punkte mit ihren “wahren” Tiefen plaziert werden.
III

Zusammefassung
Diese Diplomarbeit behandelt die Lösung des inversen Problems, d. h. den Inversionsprozess.
Die Inversion ist ein wichtiger Schritt, um am Ende ein optimales
Abbild des Untergrundes zu erhalten. Sie reagiert sehr empfindlich auf Veränderungen
der verwendeten Attribute (z. B. Laufzeit, Stapelgeschwindigkeit, usw.).
Hier kommt die Glättung ins Spiel, die einen Hauptanteil der vorliegenden Diplomarbeit
ausmacht. Ich habe das Glätten der Attributkurven mit den folgenden
sechs statistischen Methoden, die einen Mittelwert errechnen, getestet:
der
der ¢
Eine
der
¢ ¢
¢
arithmetische Mittelwert
Median
Kombination aus arithmetischen Mittelwert und Median mit einem
Grenzwert, die im Folgenden als mean difference cut bezeichnet wird
gewichtete arithmetische Mittelwert
Polynomregression
robust locally weighted (RLW) regression (Cleveland, 1979)
die
die
Es würde den Rahmen dieser Arbeit sprengen, wenn alle Ergebnisse gezeigt werden
würden, so dass ich mich auf die Darstellung der Ergebnisse der zwei Glättungsalgorithmen,
die sich am stärksten voneinander unterscheiden und brauchbare
Resultate für eine nachfolgende Inversion ergeben, beschränkt habe. Der
andere Hauptteil dieser Arbeit befasst sich mit der Erstellung eines 2D Iso-Ge-
¢
schwindigkeitsmodells aus den geglätteten Ergebnissen der durch die CRS-Stapelung
direkt aus den Daten erhaltenen
¢
Attribute.
Überblick und Zusammenfassung der Arbeit
Kapitel 2 behandelt den theoretischen Hintergrund einiger Standardmethoden
und der Methoden, die in dieser Arbeit verwendet wurden. Ich beginne mit der
Einführung einiger konventioneller Bearbeitungsmethoden wie migration to ZO
(MZO), CMP-Stapelung und pre-stack depth migration (PreSDM). Danach wird die
CRS-Stapelung erklärt (Abschnitt 2.2), und es wird beschrieben, wie man die
CRS-Attribute erhält. Bevor ich die in dieser Arbeit verwendeten Inversionsmethoden
erkläre, ist es notwendig, geglättete CRS-Attribute entlang der zu invertierenden
(Primär-) Reflexionsereignisse zu haben. Folglich werden die eingesetzte
Pickmethode und die verwendeten Glättungsmethoden kurz beschrieben.
Das nachfolgende Kapitel 3 umfasst zwei synthetische Beispiele, anhand derer
die Zuverlässigkeit der Inversionsalgorithmen getestet wird und die Auswirkungen
der Glättungsalgorithmen auf die Inversion veranschaulicht werden. Das erste
Modell enthält vier Schichtgrenzen mit geringer Komplexität, d. h. mit nicht
IV

zu stark gekrümmten Schichtgrenzen. Mit diesem Modell (Abbildung 3.2) werden
die Auswirkungen der Glättung der Eingangsdaten auf alle vier beschriebene
Inversionsalgorithmen überprüft. Das zweite Modell (Abbildung 3.42) ist
etwas komplexer. Es hat eine synklinale Struktur, die in den Seismogrammen eine
Triplikation hervorruft. Dieses Modell dient dazu, festzustellen, welcher Inversionsalgorithmus
in der Lage ist, die Schleife der Triplikation aufzulösen und
auch Strukturen unterhalb der Triplikation invertieren zu können. Dabei ergab
sich, dass es nur der Horizont-Inversion bei beiden synthetischen Modellen möglich
war, ein vernünftiges Ergebnis ähnlich dem synthetischen Modell zu erzielen
(siehe Abbildungen 3.37 und 3.74). Desweiteren wird auch der Einfluss der
Glättung deutlich. Die RLW-Regression zeigte im Gegensatz zu der Glättung mit
dem arithmetischen Mittelwert eine signifikante Stabilisierung der Inversionsergebnisse
(siehe z. B. Abbildungen 3.20 und 3.21).
Die Anwendung der Inversionsalgorithmen auf Realdaten wird in Kapitel 4 präsentiert.
Hierbei habe ich drei verschiedene Zielbereiche (Abbildung 4.2) festgelegt,
von denen angenommen wird, dass sie verschieden komplexe Strukturen
enthalten. Der erste Bereich wurde gewählt, um zu sehen, ob die Inversionsalgorithmen
überhaupt mit realen Daten umgehen können. Der zweite Zielbereich
enthält sich überschneidende Reflexionsereignisse, die im Konflikt zu den Annahmen
der derzeitigen Implementierung der Inversionsalgorithmen steht. Zielbereich
drei überdeckt die Dom-ähnliche Struktur, die aus der Form der Reflexionsereignisse
zu erwarten ist und wird verwendet, um die Grenzen der Anwendbarkeit
der vier Inversionsalgorithmen zu erkennen. Hierbei kann wieder
festgestellt werden, dass die Horizont-Inversion im Vergleich zu den anderen Inversionsmethoden
vorzuziehen ist (vergleiche Abbildungen 4.5, 4.7, 4.9 und 4.11,
usw.). Auch die RLW-Regression führte zu glatteren Resultaten verglichen mit
dem Median (siehe Abbildungen 4.18 und 4.17).
In Kapitel 5 verwende ich eines der invertierten Geschindigkeitsmodelle und sein
entsprechendes Pendant des mit dem Realdatensatz zur Verfügung gestellten
Geschwindigkeitsmodells als Eingangsdaten für eine Kirchhoff Tiefenmigration.
Die sich ergebenden Tiefenabbilder (Abbildungen 5.5 und 5.6) des Untergrundes
werden von der Erdöl- und Erdgasindustrie verwendet, wobei es in der Verantwortung
der interpretierenden Experten liegt, zu entscheiden, welches Abbild
die Realität besser beschreibt. Eine Anwendung dieser Abbilder ist, zu entscheiden,
wo ein Bohrloch zum Ausbeuten eines Kohlenwasserstoffreservoirs abgeteuft
werden soll. Das Abbild muss dazu so exakt sein, wie es aus den aufgenommenen
Sektionen möglich ist, so dass ein Bohrloch nicht “trocken” ist, das heisst,
dass kein Kohlenwasserstoffreservoir gefunden wurde. Ist dies der Fall, bedeutet
das einen finanziellen Verlust im Bereich von Millionen Dollar. Ich kann nur betonen,
dass ich bei der Erstellung des Geschindigkeitsmodells mehr Informationen
aus dem multi-coverage Datensatz verwendet habe als das bei dem Geschwindigkeitsmodell,
das aus CMP-Stapelungen erstellt wurde, der Fall war.
V

Contents
1 Introduction 1
1.1 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Theory 7
2.1 Conventional processing methods . . . . . . . . . . . . . . . . . . . 7
2.1.1 Migration to ZO and normal moveout/dip moveout/stack 8
2.1.2 Common-midpoint stack . . . . . . . . . . . . . . . . . . . . 11
2.1.3 Pre-stack depth migration . . . . . . . . . . . . . . . . . . . . 12
2.2 Common-reflection-surface stack . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Projected first Fresnel zone . . . . . . . . . . . . . . . . . . . 16
2.2.2 CRS attributes search strategy . . . . . . . . . . . . . . . . . . 19
2.3 Picking of reflection events . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Smoothing by means of statistical methods . . . . . . . . . . . . . . 21
2.4.1 The arithmetic mean . . . . . . . . . . . . . . . . . . . . . . . 22
2.4.2 The median . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4.3 The mean difference cut . . . . . . . . . . . . . . . . . . . . . 23
2.4.4 The weighted arithmetic mean . . . . . . . . . . . . . . . . . 25
2.4.5 Robust locally weighted regression . . . . . . . . . . . . . . . 26
2.4.6 Spline interpolation/approximation . . . . . . . . . . . . . . 28
2.5 Inversion methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5.1 Dix inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.5.2 Plane inversion . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5.3 Circular inversion . . . . . . . . . . . . . . . . . . . . . . . . 37
2.5.4 Horizon inversion . . . . . . . . . . . . . . . . . . . . . . . . 37
3 Synthetic data examples 41
3.1 Model A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.1 Dix inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.1.2 Plane inversion . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.1.3 Circular inversion . . . . . . . . . . . . . . . . . . . . . . . . 62
3.1.4 Horizon inversion . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2 Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
i

Contents
3.2.1 Dix inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.2.2 Plane inversion . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.2.3 Circular inversion . . . . . . . . . . . . . . . . . . . . . . . . 85
3.2.4 Horizon inversion . . . . . . . . . . . . . . . . . . . . . . . . 89
4 Real data examples 95
4.1 Target range 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.1.1 Dix inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.1.2 Plane inversion . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.1.3 Circular inversion . . . . . . . . . . . . . . . . . . . . . . . . 101
4.1.4 Horizon inversion . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2 Target range 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.2.1 Dix inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.2.2 Plane inversion . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.2.3 Circular inversion . . . . . . . . . . . . . . . . . . . . . . . . 105
4.2.4 Horizon inversion . . . . . . . . . . . . . . . . . . . . . . . . 109
4.3 Target range 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.3.1 Dix inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.3.2 Plane inversion . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.3.3 Circular inversion . . . . . . . . . . . . . . . . . . . . . . . . 114
4.3.4 Horizon inversion . . . . . . . . . . . . . . . . . . . . . . . . 115
5 Application examples 117
6 Conclusions 123
A 2D ZO CRS attribute£ 125
B Correction of CRS attributes 127
C Used hard- and software 131
List of Figures 133
References 137
Danksagung 139
ii

Chapter 1
Introduction
Geophysicists want to gain knowledge of the structure of the earth to find hydrocarbon
reservoirs, to estimate the risk of earth quakes or of volcanic hazards,
and so on. However, it is not possible to cut the earth into halves to see the structure
directly. Therefore, the thesis on hand describes processing methods that
can be used to finally obtain an easier interpretable image of the subsurface out
of reflection seismic data.
Seismic data acquisition in 2D reflection seismics is mainly performed with a
common-shot (CS) configuration (see Figure 1.1(a)) to obtain the subsurface reflection
response on seismic impulses. The recorded seismic traces plotted in a CS
section or CS gather belong to one experiment, where, e. g., an explosion event
initiates a seismic wave that propagates through the subsurface. A seismogram
contains traces that are sorted with increasing offset or half-offset, respectively.
The half-offset is half the distance and the offset is the whole distance between
the source location and the receiver location for every shot-receiver pair. The
midpoint coordinate is the center between the source and the receiver location.
The generated wave is refracted and reflected at discontinuities in the subsurface.
Those parts of the reflected or refracted wave that emerge at the receivers
are recorded with respect to the elapsed time relatively to the start of the source
emission, i. e., the traveltime. For 2D measurements, the CS configuration is then
moved along a straight seismic line to obtain many CS sections that contain reflections
of the same reflector points in the illuminated subsurface. The whole
recorded data set forms a so-called multi-coverage data set and contains a certain
redundance as it covers depth points multiple times.
Several rearrangements of the recorded traces can be performed to form other
sections that represent one step on the way to interpret the recorded data. One
way is to resort the traces into a common-offset (CO) gather or into a commonmidpoint
(CMP) gather. A CO gather contains all traces with a certain constant
offset that are sorted by their midpoint coordinate (see Figure 1.1(b)).
A special CO gather is the zero-offset (ZO) gather. Here, the (half-) offset is zero,
1

Chapter 1. Introduction
z
depth
z
depth
z
depth
z
depth
shotpoint
receivers
(a) All rays start from the same shotpoint.
shotpoints
receivers
(b) Every shot-receiver-distance is the same.
shotpoints and receivers coincide
(c) Here, all shot-receiver-distances are zero.
shotpoints
common-
midpoint
receivers
common-depth-point
location along the
seismic line
x
location along the
seismic line
x
location along the
seismic line
x
location along the
seismic line
x
(d) The shot-receiver-distances are chosen to have the
same midpoint.
Figure 1.1: Ray paths of four different seismic data arrays that form: (a) a
common-shot (CS) gather, mainly used for data acquisition, (b) a common-offset
(CO) gather, (c) a zero-offset (ZO) gather, that is a special CO gather and (d) a
common-midpoint (CMP) gather with the common-depth-point (CDP).
2

z
depth
shotpoints
common-
midpoint
receivers
dip angle
location along the
seismic line
x
reflector
smeared area of depth points
Figure 1.2: Ray paths of a CMP gather for a plane dipping reflector. Here, a CDP
could not be assigned exactly anymore but the smeared area of the depth points
is taken as CDP as long as some assumptions are satisfied.
i. e., the location for one source and receiver pair coincides (see Figure 1.1(c)).
However, this ZO configuration cannot be realized for reflection seismic acquisitions
as the source would possibly destroy the coincidental receiver. The ZO
section usually has to be simulated by stacking methods.
The CMP gather combines all traces with the same midpoint and sorts them with
increasing (half-) offsets (see Figure 1.1(d)). The four parts of Figure 1.1 illustrate
the ray paths belonging to the most commonly used configurations for the example
of a horizontal reflector with a constant velocity overburden. The CMP gather
sometimes is also called common-depth-point (CDP) gather which is only correct
for the case of plane horizontal layers. There, the x-coordinate of the CMP and
the CDP are the same. But as soon as the reflector is not horizontal anymore (see
Figure 1.2), the x-coordinate of the CMP and the CDP differ. Thus, the rays for a
CMP configuration emerge at the plane dipping reflector in a smeared area.
For this thesis, all recorded traces are placed in a three dimensional data volume
with the following axes: the xm-axis denotes the midpoint coordinate, the h-axis
stands for the half-offset and the t-axis is the relatively elapsed time since the
shot, i. e., the traveltime. The four gathers mentioned in Figure 1.1 are contained
in this 3D data volume. Figure 1.3 shows the planes belonging to CS, CO and
CMP gathers. Red colored planes denote CS gathers. Every CS gather (xm
const. const. h¡
const.) builds with every plane h¡
or xm¡
an angle of 45 degree.
Green colored planes are examples for CMP gathers. Within one CMP gather,
the midpoint is (xm¡ constant const.), i. e., common for all contained traces. The
blue colored planes depict some CO (h¡ gathers const.). There, the half-offset
respectively the offset is constant for all traces within one CO gather. The special
case of a ZO gather is the front plane of this data volume, i. e., the plane 0.
The described data volume is used by the common-reflection-surface (CRS) stack.
Stacking means to sum up all amplitudes in a section along a traveltime curve or
surface in the data. If the traveltime curve matches the true curve of the reflection
event then the coherent amplitudes are summed up constructively. The result is
h¡
3

Chapter 1. Introduction
t [s]
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 1 2 3 4 5 6 7 8 9
x [km]
m
0
1
2
3
4
5
6
7
h [km]
Figure 1.3: Three dimensional data volume with axes xm for midpoint coordinate,
h for half-offset and t for time. The red planes are CS gathers, the green planes
are CMP gathers and the blue planes are CO gathers. The special case of 0
denotes the ZO section which is parallel to the blue planes and belongs to the
front plane of the 3D data volume.
h¡
assigned to the corresponding point of the ZO section. The benefit is a higher
signal-to-noise (S/N) ratio that makes it easier to identify reflection events. The
S/N ratio is defined as the quotient of the maximum amplitude of all reflection
events within one data set over the root-mean-square amplitude of the noise. A
S/N ratio smaller than one means that the signals of the reflection events mostly
cannot be distinguished visually from the noise because their amplitudes are
smaller than those of the noise. Thus, a high S/N value is desired. The so-called
CMP stack is one example for summing up amplitudes in CMP gathers along a
traveltime curve. In contrast to the CMP stack, the CRS stacking method determines
a stacking surface within the 3D data volume. The aim is to sum up even
more coherent amplitudes. Thus, the S/N ratio increases even more.
4
8
9

The simulated ZO section is the basis for many inversion procedures to obtain
a clearer image of the subsurface. Within the ZO section, many easy visible
reflection events are picked. The two-way ZO traveltime and some other attributes,
further explained below, are used to obtain an iso-velocity layer model or
a macro-velocity model of the subsurface. Here, the difference of an iso-velocity
layer model and a macro-velocity model is that the latter does not contain interfaces
with sharp velocity steps. The iso-velocity model is build up by layers of velocity
distributions (here, of constant velocity) separated by interfaces. Those isovelocity
models can be transformed to macro-velocity models by smoothing. The
computation of such velocity models is called inversion as it solves the inverseproblem.
This means to determine a velocity model out of the information given
by the reflection response of the subsurface.
The last step is the depth migration. The post-stack depth migration transforms
all points of the simulated ZO section from the time domain into the depth domain.
This transformation requires a macro-velocity model of the subsurface in
order to place the points at their “true” depth locations.
This diploma thesis aims at solving the inverse-problem, i. e., the inversion process.
The inversion is an important step to receive an optimal image of the subsurface
in the end. It is very sensitive on variations of the used attributes (e. g.,
traveltime, stacking velocity,¤¥¤¥¤). There, the smoothing takes place which is one
of the main tasks of the thesis on hand. I have tested the smoothing of the attribute
curves with the following six different statistical methods that calculate a
mean value:
¢ arithmetic
mean
combination of arithmetic mean and median with a threshold, referred to
as mean difference cut in the following
arithmetic mean
median
a ¢ ¢
regression
locally weighted regression (Cleveland, 1979)
weighted
polynomial ¢
robust
It is beyond the frame of this thesis to show all their results, so that I restrict the
displayed results to the two smoothing algorithms that differ the most and yield
useful results for a subsequent inversion. The other main task of this thesis is to
obtain a 2D iso-velocity model of the subsurface out of the smoothed results from
¢
data-derived attributes obtained by the CRS stack.
¢
5

Chapter 1. Introduction
1.1 Structure of the thesis
Chapter 2 provides the theoretical background of some standard methods and the
methods discussed in this thesis. I start with an introduction to conventionally
processing methods as migration to ZO (MZO), CMP stack, and pre-stack depth
migration (PreSDM). Afterwards, the common-reflection-surface (CRS) stacking
method (Section 2.2) is explained and it is described how the CRS attributes
are obtained. Before I describe the inversion methods used in this thesis, it is
necessary to have smoothed CRS attributes along identified (primary) reflection
events. Therefore, the used picking algorithm and the used smoothing algorithms
are shortly explained.
Chapter 3 contains two synthetical examples that are used to check the reliability
of the inversion algorithms and to observe the effect of the smoothing algorithms
on the inversion process. The first model contains four interfaces of low complexity,
i. e., with not too strongly curved interfaces. From this model (Figure 3.2), the
effect of smoothing the input is investigated for all four described inversion algorithms.
The second model (Figure 3.42) is a little bit more complex as it includes
a synclinal structure that produces a triplication in the seismograms. This model
was used to see which inversion algorithm is able to unwrap the bow tie of the
triplication and to invert structures also beneath the triplication.
In Chapter 4, the inversion algorithms are applied to real data and the results are
presented. I have chosen three different target areas (Figure 4.2) that are expected
to contain structures of different levels of complexity. The first range is chosen
to observe if the inversion algorithms can handle real data at all. The second
target range contains intersecting events which is in conflict with the assumptions
made for the current implementation of the inversion algorithms. Target range
three covers the dome-like structure expected from the shape of reflection events
within the simulated ZO section and is used to find the limits of applicability of
the four inversion methods.
In Chapter 5, I used one of the inverted velocity models and its corresponding
part of the velocity model provided with the real data set as input for a Kirchhoff
depth migration. The resulting depth images of the subsurface are used by the
gas and oil industry to decide where to drill a bore-hole to exploit hydrocarbon
reservoirs. This image has to be as exact as it is possible from the recorded sections,
so that the bore-hole is not “dry”, i. e., no hydrocarbon reservoir was found.
If this is the case then it means a financial loss in the range of millions of dollars.
6

Chapter 2
Theory
The data from most 2D seismic reflection experiments either onshore or offshore
are acquired using the common-shot (CS) configuration and can be rearranged
in many different ways for subsequent interpretation processes. The data can be
sorted with respect to their half-offset coordinate h and their midpoint coordinate
xm. The half-offset h is half the distance between the location of the source xs and
the receiver xg assuming all shots and receivers are located on a straight line:
xg
xg
h¡
xs¦ xm¡
xs
2
The midpoint coordinate xm is the midpoint between xs and xg:
(2.1)
xg h (2.2)
2
Then, the recorded seismic trace of each receiver is plotted along the time axis at
its corresponding xm and h coordinates. This rearrangement forms, with the time
t as the third axis, the three dimensional data volume described in Chapter 1.
In the following, I will shortly introduce several processing methods that are
steps within different approaches of seismic imaging to obtain a depth migrated
image of the subsurface. To describe some of those methods, I make use of
common-reflection-point (CRP) trajectories. A CRP trajectory defines the
¡ xs¦ h¡
location
of the primary reflection event in the 3D data volume that pertains to one
and the same reflection point of a reflector. I refer to such a point as reflection
point, where—in contrast to a “diffraction” point—the dip of the reflector (at this
point) has to be considered. The resulting CRP traveltime curve is, in general, not
parallel to a CMP, CO, or CS gather.
2.1 Conventional processing methods
Conventional processing chains are mostly analyzing CMP gathers, represented
by the green planes in Figure 1.3, to subsequently obtain 2D velocity models of
7

Chapter 2. Theory
the subsurface. Many different velocity analysis methods (for further details see
Yilmaz, 1987) are used to receive a stacking velocity. By varying this stacking velocity,
a CMP traveltime curve fitting best to reflection events can be found along
which the amplitudes are summed up, i. e., stacked. The aim of stacking is to
improve the signal-to-noise (S/N) ratio by up to a theoretical factor of§N (Yilmaz,
1987), where N is the number of contributing traces, e. g., in a CMP gather.
In reality, this factor is smaller than§N: the signal is not always coherent, the
fitted curve does not match exactly the true reflection traveltime curve, and noise
(amplitudes caused by wind, traffic,¤¥¤¥¤) does not always interfere destructively
during the stack.
In the following sections, I introduce the idea of some standard processing methods
for simulating a 2D ZO section. The so-called migration to ZO (MZO), the
normal moveout/dip moveout/stack (NMO/DMO/stack), or the CMP stack are
pre-stack processing methods. They simulate a ZO section by stacking along traveltime
surfaces or curves. From the by-product, i. e., the NMO or stacking velocity
a velocity model is computed which is needed for the subsequent migration process.
In contrast to these methods, also a short insight into one pre-stack depth
migration (PreSDM) method is provided that transforms reflection events from
the time domain into the depth domain directly in one step.
2.1.1 Migration to ZO and normal moveout/dip moveout/stack
One conventional processing method, the migration to ZO (MZO), aims to correct
the offset dependency of reflection events in CMP and CO gathers in order to
produce a stacked ZO section. Therefore, the MZO sums up all amplitudes along
the CRP trajectories belonging to all reflection points on the ZO isochron defined
by the recorded traveltime t0. The stacked signal is placed into point P0. The fanshaped
MZO operator is illustrated in the upper part of Figure 2.1 for the case of
one reflector point R with a constant velocity overburden (v0). For this case, the
ZO isochron is the lower semicircle with the center at x0 and radius v0t0¨2. The
MZO is decomposed into three separate steps: normal moveout corrections, dip
moveout corrections, and the stacking.
The traveltime curve of a single horizontal interface with a homogeneous overburden
in a CMP configuration has the shape of a hyperbola (Yilmaz, 1987):
t 2 (h)¡ t 2 0¦ 4h
2
v 2 NMO¤ (2.3)
with the ZO traveltime t0. Here, the normal moveout (NMO) velocity vNMO is
identical with the constant velocity in the overburden. In the case of n horizontal
iso-velocity layers, the traveltime curve can still be approximated by a hyperbola
as long as h stays small against the depth of the illuminated interface point. The
8

Depth [m] Time [s]
0.6
0.4
0.2
0
-200
-400
-600
-1000
t
h
-500
P0
X0
MZO operator
R
0
Midpoint [m]
ZO isochron
2.1 Conventional processing methods
500
x
1000
0
300
400
200
Half-offset [m]
100
Figure 2.1: The fan-shaped MZO operator is the reflection response of the ZO
isochron. In the lower part of this figure, the ZO isochron is displayed that is
circular for this particular case of one constant velocity layer.
traveltimes for ray paths from the sources through all layers down to the reflecting
interface and back to the receivers within a CMP configuration are given by
(Taner and Koehler, 1969)
t 2 (h)¡ C0¦
C1h
2¦ C2h
4¦ C3h
6¦ (2.4) ¤¥¤¥¤�©
with t C0¡ 2 0 , C1¡ 2 RMS . C1, C2, C3,¤¥¤¥¤depend on the thicknesses and interval
velocities vi of the layers above the current reflecting interface. vRMS denotes the
root mean square (RMS) velocity which is a special average velocity. It replaces
all layers above the reflecting interface by one hypothetical layer with constant
velocity defined as
4¨v
v 2 RMS¡ 1
t0
n
�i�1
v 2 i Δti¤ (2.5)
The two-way traveltime Δti is twice the traveltime along a vertical ray segment
in the ith layer. Taking into account the small-spread approximation (Taner and
Koehler, 1969), i. e., small-offset approximation, the higher orders of h 2 can be
neglected which yields the second order approximation of the traveltime curve
9

Chapter 2. Theory
given by
t 2 (h)¡ t 2 0¦ 4h
2
v 2 RMS¤ (2.6)
It is obvious from comparing Equations (2.3) and (2.6) that v 2 RMS equals v2 NMO ,
provided the small-spread approximation holds. The offset dependent part of
the traveltime hyperbola is removed by a correct estimation of the NMO velocity
vNMO. Thus, the reflection event is flattened in the CMP gather. Now, all
signals located at t0 within the N contributing traces are horizontally stacked
which yields the simulated ZO signal at the current CMP and ZO traveltime
(P(x0©t0)¡ P0).
Considering plane reflectors with small dips, the dip moveout (DMO) tries to
correct for the reflection point dispersal (see Figure 1.2) occurring from the dip of
the reflector. The traveltime for a single dipping reflector is (Levin, 1971)
t 2 (h)¡ t 2 0¦ 4h
2 cos2� v2 NMO
(2.7)
with the dip angle�of the reflector. The second term can be separated into a
NMO and a DMO part,
t 2 (h)¡ t 2 0¦ 4h
2
v2 4h
NMO 2 sin 2� v2 NMO ¤
(2.8)
The above equation implies that the NMO/DMO correction can be divided into
two steps. Firstly, the NMO correction within the CMP gather is applied to give
an initial estimation of the velocity. Secondly, one of several different methods
(Deregowski, 1986; Hale, 1991) performs the DMO correction while assuming not
too large interface dips. The DMO correction investigates CO gathers to eliminate
the dip dependency of vNMO. With the small-dip approximation (Hubral and Krey,
1980), Equation (2.8) reduces to Equation (2.3) because 0. Thus, the
lim���0 last term of Equation (2.8) vanishes. Finally, the ZO section is obtained by stacking
the signals located at the corrected traveltimes which is also called normal
moveout/dip moveout/stack (NMO/DMO/stack).
The aim of MZO or NMO/DMO/stack is to provide a simulated ZO section with
a high S/N ratio. With an improved S/N ratio, it is easier to identify reflection
events because they more prevail the noise. Their amplitudes are easier to distinguish
from the noise. The MZO operator is the fan-shaped surface shown in
Figure 2.1 but it fits not very well to the true traveltime surface of the
sin���
illustrated
reflection event. Only along the light green line the operator sums up the amplitudes
of the reflection event which is the CRP trajectory of R in the displayed
case. The remaining part of the stack surface adds noise to the stacked result.
Thus, it deteriorates the stacking result because the noise does not always interfere
destructively during the stack.
10

X1
vst1
vst 2
vst3
v st4
Stacked ZO traces
X 2
Seismic line
X3
t
2.1 Conventional processing methods
Offset h
Figure 2.2: The red lines are the best-fit hyperbolae for certain traveltimes and
CMP gathers obtained by coherency analysis. From one picked traveltime to the
next in the same trace, the 1D stacking velocity functions are filled up constantly
with the velocities belonging to the next reflection event. A 2D stacking-velocity
model is build up by, e. g., interpolating or approximating between the 1D velocity
functions of chosen CMP gathers.
2.1.2 Common-midpoint stack
For the common-midpoint (CMP) stack, Equation 2.7 is reformulated as:
t 2 stack(h)¡ t 2 0�stack¦ 4h 2
v 2 stack¤ (2.9)
The stacking velocities v stack are provided by coherency analyses (Yilmaz, 1987).
One whole CMP gather is corrected many times with different constant velocities.
Then, the coherency value is plotted for all traveltimes and stacking velocities
which forms the velocity spectrum. The maximum coherency value indicates
the stacking velocity for the best-fit hyperbola at a certain traveltime. Conventionally,
the maxima within the velocity spectrum of this CMP gather are interactively
picked. This velocity analysis yields the final stacked ZO trace by correcting
the whole CMP gather with a one dimensional velocity function. The one
dimensional velocity function is a function with up and down steps of constant
stacking velocities between the picked stacking velocities.
A 2D velocity model can be obtained by performing the CMP stack for certain
CMP gathers and afterwards, e. g., interpolating or approximating between the
one dimensional interval velocity functions computed out of the stacking velocity
functions. Figure 2.2 depicts a short example showing an interpolated 2D
stacking-velocity model in the front panel which also indicates the ZO section.
11

Chapter 2. Theory
With the knowledge of the traveltimes for the picked stacking velocities and assuming
horizontal interfaces, the interface velocities can be calculated by recursively
solving Equation (2.5) as the stacking velocities are interpreted as RMS
velocities.
It must be emphasized that the NMO velocity vNMO and the stacking velocity vstack
are two different velocities. Their difference becomes obvious for large offsets,
i. e., large spread lengths. While the NMO velocity considers the small-spread
approximation to fit a hyperbola to the reflection event, the CMP stack varies the
stacking velocity as long as it finds the best-fit hyperbola for the whole spread
length. The difference between the NMO velocity vNMO and the stacking velocity
vstack is called spread-length bias (Al-Chalabi, 1973; Hubral and Krey, 1980). It is
obvious from Equations (2.3) and (2.9) that the smaller the offset, the closer the
optimum stacking hyperbola to the small-spread hyperbola, hence the smaller
the difference between vNMO and vstack.
Nevertheless, the spread-length bias is generally neglected in practice assuming
locally horizontal stratified media or areas with gently dipping, plane reflectors in
the vicinity of the illuminated depth point. However, the velocity analysis must
be redone by iterative interaction in order to improve the resolution of primary
reflection events which implies that this iterative process yields a better stacking
result. The resolution of the subsequent migration can also be improved by taking
more CMP gathers into account for the construction of the 2D stacking-velocity
model. Migration can also benefit from the iterative velocity analysis which gives
more accurate one dimensional velocity functions for the selected CMP gathers.
A post-stack depth migration can be used to obtain an image of the subsurface
out of these simulated ZO sections and inverted velocity models.
2.1.3 Pre-stack depth migration
Other well-known methods to receive an interpretable image of the subsurface
are pre-stack depth migration (PreSDM) methods. For a Kirchhoff PreSDM, the
reflector is build up by the kinematic response of diffractors according to Huygens’
principle. Thus, the reflected wavefront is the envelope of all reflection
curves from the diffractors representing the reflector. The summation surface can
then be regarded as a collection of Huygens traveltime curves. This is called
a Kirchhoff summation (for further details on Kirchhoff summation see Yilmaz,
1987).
The PreSDM operator in Figure 2.3 fits in a larger area to the reflection traveltime
curve than the MZO operator because it uses the CRP trajectories of point
R with all possible dips (green lines) to simulate a diffraction point at R. But the
PreSDM operator strongly depends on an a priori known velocity model which
is usually not available. An initial velocity model can be obtained from the CMP
or NMO/DMO/stack but these velocity models have to be refined iteratively to
12

Depth [m] Time [s]
0.6
0.4
0.2
0
-200
-400
-600
-1000
t
h
-500
P0
X0
R
0
Midpoint [m]
PreSDM operator
500
2.2 Common-reflection-surface stack
x
1000
0
300
400
200
Half-offset [m]
100
Figure 2.3: The red Huygens’ surface in the time domain shows the PreSDM operator
which is the kinematic reflection response of the point diffractor at R. The
diffraction rays are shown in the lower part of the 3D data volume and correspond
to the red diffraction traveltime curve shown in the ZO plane of the upper
part.
improve the result of the PreSDM.
A further aspect of improvement in contrast to post-stack migration methods is
the process itself, it provides the depth migrated section in one step. However, to
enhance the result of the PreSDM, the interpreter can “only” intervene by changing
the velocity model. Thus, the PreSDM operator fits in a larger area to the true
traveltime surface but still much noise is collected along the stacking surface of
the PreSDM operator.
2.2 Common-reflection-surface stack
The common-reflection-surface (CRS) stack (Höcht, 1998; Jäger, 1999) is another
stacking method that produces a simulated ZO section, e. g., for migration purposes.
The CRS stack fits a surface to the true reflection traveltime surface within
the 3D data volume considering the CRP trajectories of reflection points on an
arc segment with the local radius of curvature CR of the reflector at the investigated
reflection point R (see Figure 2.5). In contrast to the NMO/DMO/stack
13

Chapter 2. Theory
which provides only one parameter, i. e., the NMO velocity, this surface is parameterized
by three attributes. The parameters are two radii of curvature and
the emergence angle of the ZO rays emerging at the surface from the corresponding
reflector point. These CRS attributes are obtained by coherency analyses and
will be further explained below.
The aim of the CRS stack is not only to obtain a simulated 2D ZO section with
an improved S/N ratio as obtained by the NMO/DMO/stack. It will also provide
additional information about the subsurface without the knowledge of a
velocity model. Therefore, it is a data-driven and a model independent stacking
procedure that only makes use of information directly provided by the input data
rearranged to the 3D data volume (Figure 1.3).
The CRS stack is based on the paraxial ray theory. In paraxial ray theory (Bortfeld,
1989; Červený, 2001), it is assumed that a ray in the vicinity of a central ray,
i. e., the paraxial ray can be described by two parameters. These parameters at
any point of a paraxial ray should be linearly dependent on those at its initial
point. Bortfeld (1989) introduced the 4�4 surface-to-surface propagator matrix
in order to describe the changes of these parameters from the anterior surface
(surface with sources) to the posterior surface (surface with receivers). With the
propagator matrix, the traveltime for transmitted rays can be expressed with the
parabolic traveltime approximation using Hamilton’s equation. In the case of 2D,
the 4�4 propagator matrix reduces to the 2�2 propagator matrix.
It is known for many years that the parabolic traveltime approximation for a simple
layered media and near vertical reflections yields not as good results as the
hyperbolic traveltime approximation. Therefore, Schleicher et al. (1993) squared
the parabolic traveltime equation and retained only terms up to second-order in
(x x0) and h. Both traveltime equations (parabolic and hyperbolic) can be expressed
in terms of midpoint and half-offset coordinates while the propagator
matrix elements are formulated in terms of wavefront curvatures as proposed by
Hubral (1983). The parabolic traveltime approximation is given by
2
(x x0) sin£¦ cos
tpar(x©h)¡
v0 t0¦ 2£ �(x x0)
v0 2 h
RN ¦ 2
(2.10)
and the hyperbolic traveltime
RNIP�
equation by
t 2 hyp (x©h)¡ 2
(x sin£�2¦ 2
x0) t0 cos
v0
v0 �t0¦ 2£ �(x x0) 2 h
RN ¦ 2
(2.11)
RNIP�¤
It is assumed that the near-surface velocity v0 is a priori known.
The three parameters£, RNIP, and RN are the CRS attributes. To explain their
physical meaning, Hubral (1983) introduced two theoretical experiments. These
two experiments are so-called eigenwave experiments, which means that the respective
wavefronts before and after the reflection at the point of interest are the
14

z
Normal wave
NIP wave
NIP
v 2
2.2 Common-reflection-surface stack
Figure 2.4: Illustration of the two eigenwaves, viz., the NIP-wave and the normal
wave at several instants in time.
same except their direction of propagation. The term “eigenwave” is further explained
by Hubral (1983). In the following, I will only describe the eigenwave
experiments by upgoing wavefronts.
One of the eigenwave experiments is the normal incidence point (NIP) wave experiment.
This experiment can be interpreted as a point source exploding at the
endpoint of the normal incidence ray in the subsurface (blue NIP in Figure 2.4).
The CRS attribute “emergence angle of the normal ray”,£, is measured between
the normal incidence ray shown in blue in Figure 2.4 and the surface normal at x0,
i. e., at the surface. The local curvature of the NIP wavefront at x0 is the CRS attribute
“radius of curvature of the NIP wavefront”, RNIP. Local curvature because
the wavefronts are, in general, not circular when they impinge at the surface due
to refraction at arbitrary curved interfaces during their propagation.
The other eigenwave experiment is the normal wave experiment. An interpretation
of this experiment can be an exploding reflector including the illuminated
reflector point NIP. The resulting wavefront is per definition perpendicular to all
normal rays belonging to a certain reflector in the subsurface while it propagates
upwards. The angle of emergence,£, is again measured between the surface normal
and the central (normal) ray at the surface point x0. The radius of curvature
of the normal wave at x0 is the CRS attribute “radius of curvature of the normal
wavefront”, RN. The brown lines in Figure 2.4 are normal rays with respect to the
local arc segment of the reflector which yields a radius of wavefront curvature of
RN at x0.
The angle of emergence,£, is identical for both experiments as observable in
Figure 2.4. Thus, there are only three different CRS attributes in the case of a
2D subsurface which have to be determined. Consider one has found the CRS
x 0
α
v 0
v 1
x
15

Chapter 2. Theory
Depth [m] Time [s]
0.6
0.4
0.2
0
-200
-400
-600
-1000
t
h
-500
P0
x 0
CRS stack surface
R
CR
0
Midpoint [m]
500
x
1000
0
300
400
200
Half-offset [m]
100
Figure 2.5: The green surface is the CRS stacking surface, i. e., the primary reflection
response for all zero-offset and finite-offset reflections associated with the
red arc segment. The stacked result is assigned to the ZO point P0.
attributes for a reflector, then the CRS stack operator is the green surface in Figure
2.5. This operator fits better to the reflection event than the MZO operator or the
PreSDM operator.
2.2.1 Projected first Fresnel zone
The CRS stacking operator needs to be limited, otherwise it would sum up noise
as it is done by the PreSDM operator. A user-defined ZO aperture size was first
implemented to constrain the size of the CRS stacking operator. Vieth (2001) introduced
the projected first Fresnel zone as an adequate and automatic value for
the ZO aperture size.
The first Fresnel zone is a measure of lateral resolution depending on the frequency,
the velocity of the medium, and the traveltime. This emphasizes the
importance of the first Fresnel zone for imaging.
The extension of the first Fresnel zone at the target reflector, i. e., the first interface
Fresnel zone between points M1 and M2 is determined by a traveltime difference
of two rays due to their different paths (see Figure 2.6). One ray is the reflected ray
(SMRG) and the other ray is the diffracted ray (SM1G) or (SM2G), respectively.
16

t
z
S G
M
M
1
R
M2
2.2 Common-reflection-surface stack
Figure 2.6: The difference of the traveltime t0 of the reflected ray (SMRG) and
the traveltime td of the two diffracted rays (SM1G) and (SM2G) defines the (first)
Fresnel zone, i. e., td T¨2. The first interface Fresnel zone is the reflector
between points M1 and M2. t0�
The traveltime difference between the reflected and one diffracted ray equals half
the period, T¨2, of a mono-frequent wave.
In the following, I omit the word ’first’ as no ambiguities occur. The projected
Fresnel zone (Hubral et al., 1993) is defined by the upper endpoints of the bundle
of normal, paraxial rays (brown lines in Figure 2.7) that are perpendicular
to the considered reflector within the interface Fresnel zone. The straight bold
lines in Figure 2.7 depict the projected Fresnel zones corresponding to x0 where
x0 does not have to be the midpoint between x1 and x2. x0 is the center of the
projected Fresnel zone only in case of horizontally stratified media. Otherwise,
its location within the projected Fresnel zone depends on the ray paths of the
central and paraxial rays. Figure 2.7(b) shows the primary ZO paraxial reflection
1©2.
rays (xiMixi) and the corresponding diffraction rays (xiMRxi) with i¡
T/2
t d
t 0
x
The
boundary condition of the projected Fresnel zone is matched if the traveltime difference
pertaining to the reflected and diffracted ray equals T¨2. For real data, T
is the period of the dominant frequency because the source wavelet is a combination
of many frequencies in all cases as it is assumed to be causal. The definition
of the first interface Fresnel zone is illustrated by Figure 2.7(c). A point Mi belongs
to the first interface Fresnel zone if and only if the traveltime difference of
the ZO central ray (x0MRx0) and a Fresnel zone ray (x0Mix0) is not larger than
T¨2.
Please refer to Vieth (2001) for further explanation regarding the calculation and
implementation of projected Fresnel zones. In Figure 2.5, the red arc segment
17

Chapter 2. Theory
t
τ R
τD
Paraxial rays
M 1
M R
N
R
x1 x0
x2
1 0 2
M 2
Central ray
x
Diffraction
rays
M 1
Fresnel zone
rays
M 1
M R
M R
x
x
1 0 2
z (a) (c)
Figure 2.7: Projection of the interface Fresnel zone to the surface. The reflection
traveltime curve�R and the diffraction traveltime curve�D are tangent at NR and
differ up to T¨2 between x1 and x2. The bold lines represent the projected and
interface Fresnel zone, respectively.
denotes the interface Fresnel zone. Therefore, the CRS stacking surface sums up
the amplitudes only along the shown green surface and places the result in the
corresponding ZO point P(x0©t0)¡ P0. The resulting simulated 2D ZO section
has a better S/N ratio than even PreSDM. Vieth (2001) also emphasizes two improvements
of the CRS Fresnel stack against older CRS stack implementations.
On the one hand, an improved resolution of reflection events is obtained and
on the other hand the continuity of reflection events is enhanced both because
of the smaller aperture size. However, the CRS stack—as any other data driven
method—does not provide directly a velocity model, but it gives more information
about the subsurface through the ZO section itself and the CRS attributes.
With little additional effort, a stacking-velocity model as obtained by the CMP
stack can be achieved by picking several reflection events in the simulated ZO
section and using their corresponding CRS attributes£ and RNIP to calculate the
stacking velocities by means of Equation (2.13) below. In the following, I will
18
M 2
M 2
x
x
x
x
(b)

2.2 Common-reflection-surface stack
shortly explain the application of CRS attributes for inversion to obtain an isovelocity
layer model and finally a macro-velocity model.
2.2.2 CRS attributes search strategy
How to find the CRS attribute triplet which builds up the best traveltime surface
to simulate the point P0? The search for this CRS attribute triplet can be
done in one step as a three-parameter search. However, this requires very much
computational performance which is very expensive even on the nowadays highperformance
computer systems. Therefore, Müller (1999) and Jäger (1999) suggested
to perform three subsequent one-parameter search steps. This search strategy
does not require an initial guess to start from, but the range to search for each
attribute is user-defined. To be on the safe side, this range should be chosen rather
too large than too small. Optionally, a local optimization can be performed in the
three-dimensional attribute domain where the initial triplet determines the starting
point and the optimized values are obtained in one step. In the following, I
will shortly explain how a ZO section is simulated from a multi-coverage data set
by means of the CRS stack. Note, that the following steps are described for the
hyperbolic traveltime Equation (2.11). They are, nevertheless, the same for the
use of the parabolic traveltime Equation (2.10).
¢ First
¢ Second
step: A one-parameter search for a combined parameter vstack is performed
within the CMP gather, thus, and Equation (2.11) reads
x¡
x0
t 2 hyp(x©h)�(x�x t 0)¡ 2 2 0¦ t0
cos
v0 2£ RNIP¤
h
(2.12)
Compared with Equation (2.3), it is obvious that the stacking velocity can
be expressed by means of£and RNIP (Hubral and Krey, 1980):
v 2 stack¡ 2v0RNIP
t0 cos 2£
(2.13)
This step is called Automatic CMP Stack (Mann et al., 1999) and represents a
non-interactive velocity analysis as described for the CMP stack above.
step: The Automatic CMP Stack provides a ZO section in which
Equation (2.11) can be reduced to
2
thyp(x©h)�(h�0�R (x x0) (2.14)
v0
where the second order term in (x x0) has been neglected. This firstorder
approximation can be regarded as a plane wave approximation with
N���)¡
. From this step, called Plane Wave Stack, the emergence
t0¦
is obtained. Inserting this angle into Equation (2.13), a solution for RNIP is
found.
sin£�©
angle£
RN¡��
2
19

Chapter 2. Theory
¢ Third
¢ Fourth
step: While£and RNIP are already known, the third parameter RN is
searched for by means of
t 2 hyp(x©h)�(h�0)¡ �t0¦
2
(x sin£�2¦ 2
x0) t0 cos
v0
v0
2£(x x0) 2
(2.15)
RN ¤
RN associated with the maximum coherency is chosen to simulate the corresponding
ZO point in step four.
step: After all three parameters are found for a certain ZO point,
they can be used to calculate the traveltime with Equation (2.11). The subsequent
stack along the traveltime surface, i. e., the Hyperbolic ZO Stack is
called Initial CRS Stack. The word ’initial’ is used to emphasize that the
achieved parameters with this stack serve as initial values for the optimization
process which provides the Optimized CRS Stack.
2.3 Picking of reflection events
Picking reflection events and extracting their corresponding CRS attributes is relevant
for the inversion. All inversion methods presented in Section 2.5 are only
able to invert structures set up by discrete interfaces. Thus, they require the CRS
attributes along reflection events. They cannot use the whole attribute section because
the CRS attributes obtained for the noise have no physical meaning. These
values stem from the coherency analysis of the CRS stack which does not distinguish
between reflection events and noise.
The picking of reflection events in the current implementation depends only on
the amplitudes. The interpreter starts the semi-automatic picking algorithm at
the maximum amplitude belonging to the searched event at its left border. Then,
the algorithm uses the traveltime of the maximum amplitude as starting value
for searching the maximum amplitude in a user-defined symmetrical window in
the next trace to the right. The found maximum amplitude is marked and its
traveltime is the new start value for searching in the following trace. Problems
may occur in noisy areas where the amplitudes of the noise within the searching
window can be larger than the amplitudes of the reflection event. There, the
user has to help the algorithm to bypass those areas. Another problem can be
that two events intersect or just come close to each other, so that their maximum
amplitudes are within the same search window. Then, the user has to identify
which local amplitude maximum is of interest for him otherwise the algorithm
takes the global maximum that might belong to the wrong event and follow this
one from now on.
A future task could be to implement—as presented by Liebhardt (1997)—the attributes
of the instantaneous envelope R(t), the instantaneous phase�(t) and the
20

2.4 Smoothing by means of statistical methods
instantaneous frequency f (t) into the detection algorithm for the picking process
(for formulae refer to Yilmaz, 1987). The motivation behind this is to automatically
pick (primary) reflection events that the inversion can be automatically performed
after the CRS stack. Thus, the processing chain stack-inversion-migration
would be fully automated and data-driven. Also an interpolation of the traces
can be considered to pick the “true” maximum amplitude instead of picking only
the maximum sampled value to decrease the error bound of the subsequent inversion.
2.4 Smoothing by means of statistical methods
The attributes of the CRS stack (£, RNIP, RN) after the picking are one dimensional
series of data along the identified reflection events, i. e., values belonging
to picked traveltimes and their corresponding locations along the seismic line by
means of CMP coordinates. These functions are not smooth from the very beginning.
Especially, RNIP or RN contain high frequency fluctuations from one trace
to the next or, even worse, outliers. Thus, a smoothing filter has to be applied
to get the extracted samples sufficiently smooth along the picked reflection event
and/or to exclude the outliers for the subsequent inversion. This is necessary, because
the inversion procedure is very sensitive to the variation of the CRS stack
attributes. Therefore, I have tested several different filters used in statistics to find
a mean value. The filters for smoothing the input, i. e., the CRS attributes along
the identified reflection events, for the inversion are
the
the ¢
a
the
¢ ¢
¢
arithmetic mean,
median,
combination of the arithmetic mean and the median named the mean difference
cut,
weighted arithmetic mean, and
robust locally weighted regression (Cleveland, 1979).
the
For all these filters, one has to define among several parameters at least one window
length for smoothing over the traces, i. e., along the reflection which I refer
to as x-direction in the following. Those window parameters specify how many
samples n are added in positive and negative direction to the picked one in timeand/or
x-direction separately to be used during the calculation of the filtering
algorithms. The window length itself, e. g., for the time-direction is then calculated
as
¢
1 and the window is centered around the picked time value. These
values are used to initiate the smoothing with one of the filters explained in the
following subsections.
2n¦
21

Chapter 2. Theory
As result from these filters, only one value, namely the “smoothed” one, is written
back into the position of the original picked sample and to be used for the
inversion. Smoothed value means smooth with respect to all values along the
picked reflection event.
2.4.1 The arithmetic mean
The simplest algorithm to find a mean value is the arithmetic mean. All samples�i
of the pre-defined window are summed up and divided by the number of
samples afterwards. Here, the index starts at negative values to emphasize the
symmetry of the window to its center, i. e., the picked value (i¡ 0). Then, the
arithmetic mean is given by
1
(2.16)
1
For the calculation of the arithmetic mean, it is assumed that the values are nor-
2n¦
mal distributed. However, all�i are arbitrarily distributed values. This computation
strongly depends on all values within the window because all values are
equally weighted. If there is at least one outlier even at an end of the window then
the arithmetic mean calculates the same smoothed value as if the outlier was the
�
picked value. Thus, it cannot perform a good smoothing. Nevertheless, it
�¡
is still
better than keeping the outliers.
i���n�i¤
The implementation of the arithmetic mean filter firstly smooths all picked values
by taking a window in time-direction into account. Afterwards, the smoothed
values are again filtered but now in x-direction, i. e., along the reflection event.
The first smoothing in time-direction is used to enlarge the numbers of used values
for the smoothing. The window length in time-direction depends on the
dominant frequency and the sampling rate and is used to provide a few more
values for statistics. But it has to be chosen rather too small than too large that
values associated with noise are not considered, i. e., noise-related values do not
affect the smoothing. The window length in x-direction is not restricted but the
more samples are taken, the lower is the dominant frequency of the output which
can suppress information for high resolution results. Using the time-direction
smoothing does not always enhance the result as shown in Chapter 3.
2.4.2 The median
All samples within the pre-defined window are sorted from smallest to largest
values. The sample left of the middle for even window length or the sample of
the middle for odd window length is the median.
22
n

2.4 Smoothing by means of statistical methods
This filter does not depend on outliers as strongly as the arithmetic mean. But if
there are more than n outliers, then the median will take one of those outliers as
output. This can happen if the chosen window length is too large: then at both
window ends more outliers are considered than values belonging to the actual
reflection event because of the noise within the CRS attributes.
The median filter implemented in the smoothing algorithm is also able to use
neighboring samples in time-direction firstly to eliminate outliers. The values for
all samples belonging to the wavelet of the picked reflection event are assumed to
be identical or at least close to each other. But the CRS stack uses coherency analysis
to find the attribute triplet for every trace and traveltime. Thus, the values of
the reflection event can vary especially at the zero crossings of the wavelet. Using
the hyperbolic traveltime approximation to fit the CRS operator to the reflection
event accounts for the change of the stacking velocity which slightly increases
from the beginning of the wavelet to its end. The variation of the CRS attributes
belonging to a reflection event can be observed in Figures 3.6 and 3.45. There, in
the vicinity of the picked traveltime the CRS attributes vary slightly in contrast to
the variation of CRS attributes resulting from the noise.
2.4.3 The mean difference cut
I combined the arithmetic mean and the median to the mean difference cut (MDC).
With this combination, I tried to combine the advantages of both methods. The
MDC is calculated as follows:
1. the arithmetic mean according to Equation (2.16) for the defined window is
calculated,
2. the values are checked whether they exceed a given threshold (see below).
If yes, they are excluded from further calculations by subtracting them from
the sum and the window length is decreased by one,
3. the arithmetic mean is calculated again with the new sum and window
length (see Figure 2.8(b)) and
4. if all�i exceed the threshold, the median of the original window is taken as
output (see Figure 2.8(d)).
Thus, the MDC does not only have the good properties of the arithmetic mean
and the median, it also gives a little more control to the user by means of the
threshold.
The threshold is a user-defined percentage of the arithmetic mean after the first
step which represents a centered window around this temporary arithmetic mean.
I used a percentage and not a fixed value to account for the value of the initial
mean. This percentage is added to the first calculated arithmetic mean and also
23

Chapter 2. Theory
value
value
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
−6 −4 −2 0
sample
2 4 6
30
25
20
15
10
5
(a) triangle weight function
0
−6 −4 −2 0
sample
2 4 6
(c) example 2
value
value
18
16
14
12
10
8
6
4
2
0
−6 −4 −2 0
sample
2 4 6
20
18
16
14
12
10
8
6
4
2
(b) example 1
0
−6 −4 −2 0
sample
2 4 6
(d) example 3
Figure 2.8: The threshold is set to 10 percent and the window length in timedirection
is 9 samples. (a) The blue line depicts the weight function for the
weighted arithmetic mean. Green dots represent the samples used for the calculation.
At the samples marked with red dots, the weight function is zero. For
the following examples, the red dashed lines are the arithmetic mean and the red
dotted lines denote (the borders of) the thresholds. The median results are shown
as blue dashed lines and the weighted arithmetic mean are the green dashed lines.
The final calculated output of the MDC is shown as dashed line in magenta. (b)
Here, the threshold of 10 percent was too small to use more than just three values
for the second calculation of the arithmetic mean. (c) This example shows what
happens when no values are in the threshold. The median is taken as output.
(d) A disadvantage can be that only one value closer to the outliers is inside the
threshold which then defines the final output.
24

2.4 Smoothing by means of statistical methods
subtracted from it that the centered interval is obtained. Only the values within
this interval are used for further computations.
If there is an outlier in the original window, the threshold must be a large percentage
to include some values. Otherwise the median will be taken and this can be
the better choice. The threshold should be chosen carefully. If a large threshold is
considered, outliers can be used again for the second calculation of the arithmetic
mean and therefore the smoothed value does not change strongly.
2.4.4 The weighted arithmetic mean
An improvement of the standard arithmetic mean is to use a weighted arithmetic
mean. This is the sum over all samples of the pre-defined window that are multiplied
with a weight function wi (here: triangular):
���
�
���
wi¡
(n�1)�i
n � �W¡
wi�i
i���n
(2.17a)
(n�1) 2 n� i� for 0
(n�1)�i
(n�1) 2 �i��� 0� i�
�
for n
(2.17b)
0 for n
The weight function (Figure 2.8(a)) must comply with the prerequisite that the
sum over all values is one:
1 i�����wi¡ �
(2.17c)
The result of this filter becomes the better the further the outliers deviate from the
originally picked sample. It is obvious that the arithmetic mean itself can also be
seen as an weighted arithmetic mean with a rectangular weight function (boxcar).
Then all samples are treated equally.
Equation (2.17c) has to be fulfilled so that the weight function wi does not change
the scale of the original input. If the result of the sum is not equal to one then it
can be considered as a scaling factor.
The weighted arithmetic mean is also able to smooth in time-direction like the
other filters above. I only implemented a triangular weight function because for
the smoothing in time-direction where the algorithm can make use of just a few
values it does not make sense to have a more complex weight function. The current
implementation of this filter supports different window lengths for filtering
in time-direction and smoothing in x-direction. In future, the user can additionally
change the weight function.
As a new task, it might be interesting to test also other mean value calculation
formulae as listed in Bronstein and Semendjajew (1996) that also assume a normal
distribution as the arithmetic mean and the median:
25

Chapter 2. Theory
the ¢
the ¢
¢ and
geometric mean
harmonic mean �H¡
the quadratic mean
�G¡��i���n�i� 2n�1© �
1
n
(2.18)
���
�Q¡
� 2n¦
1 �
2.4.5 Robust locally weighted regression
�i© 1
n
1 i���n
(2.19)
2n¦ 1
n
(2.20)
i���n�2 i¤ �
All these formulae above have in common that they assume a normal distribution
of the values. This may hold for the values belonging to one reflection event in
each trace, i. e., for smoothing in time-direction. However, this can not hold for
the values along the reflection event, i. e., for smoothing in x-direction. The values
along the picked event may represent any arbitrary curve. Therefore, I want to
use a kind of regression to find an adequate approximation of the extracted CRS
attributes that is smooth enough for the inversion. The linear regression over
all values extracted from one reflection event does not make sense as it is clear
that the CRS attributes will, in general, not increase or decrease with a constant
gradient for complex subsurface structures. Therefore, I tried to use a polynomial
regression for the whole reflection event, but the polynomial regression with low
orders can not account for fast changes. Using higher orders introduces too much
fluctuations in areas with small variations after a fast change. A solution for
this problem can be to consider only small windows. Thus, one has to define
conditions for the borders of the windows that the resulting curve is continuous
in the first derivative.
Cleveland (1979) introduced a method using windows for every point called robust
locally weighted (RLW) regression that is based on a polynomial fit within
the windows but with weighted least squares. This method iteratively improves
the weight function for every point to obtain a fitted curve for the whole reflection
event.
For the first calculation, a weight function W(x) is needed, where x is normalized
to the half window length (see below). The center of this initial weight function
26

2.4 Smoothing by means of statistical methods
0) is related with each x-coordinate of the current investigated reflection
event xi, so that every picked reflection signal has its own but same initial weight
function. The initial weight function W must comply with the following proper-
W(x¡
ties:
1. W(x)� 0 for�x�� 1,
2. W( x)¡ W(x) ,
3. W(x) is a monotonously decreasing function for 0, x�
4. W(x)¡ 0 for�x�� 1 .
and
Weight functions with such properties are, e. g., a boxcar, a triangle, a cosine
(within the boundaries from to¦�), a bisquare, or a tricube function scaled
1 .
Consider a set of values�i from an arbitrary picked reflection event with n traces
belonging to the locations xi along the seismic line. Thus, the RLW regression
has to smooth the values�i
��
for xi from 1©¥¤¥¤¥¤�©n. The degree of smoothing is
defined by
to�x��
the parameter 1. Let f n be rounded to the nearest integer.
Using the weight function W, the weight function wk(xi) is assigned to each xi
with 1©¥¤¥¤¥¤�©n . The weight function W is centered at xi and scaled such that
the first point at which W becomes zero is situated at the rth nearest neighbor of
xi, where r is the nearest integer to f n.
With these weight functions wk(xi), an initial fit is performed for every xi. The
fitted value
i¡ 0� f� r¡ k¡ ˆ �i is the result of a dth degree polynomial fit to all points of the input
data using weighted least squares obtained from the weights wk(xi). This calculation
of the initial fitted values is referred to as the locally weighted regression.
The next step of the iterative procedure is to calculate a new set of weight functions.
Therefore, a different set of weights�i is defined for each (xi,�i) based on
the size of the residuals�i �i. ˆ The larger the residuals, the smaller the weights
of�i. Now, the initial weights wk(xi) are replaced by the new weights�iwk(xi) and
the fitted values are calculated again with the new weights. This latter step can
be repeated iteratively. The entire procedure with the initial fit and all following
iterations is called the robust locally weighted (RLW) regression.
The RLW regression also assumes a normal distribution but not directly for the
data itself. The smoothing procedure has been designed to accommodate data
which comply to �i¡
(2.21)
where g is a smooth function of xi and the�i are random variables with mean 0
and constant scale. This means that the residuals calculated from the final fitted
values to the original values are normal distributed around the fitted values. The
g(xi)¦��i©
27

Chapter 2. Theory
smoothness of the resulting curve is controlled by the parameter f . If f is large
then more neighboring points are taken into account for the calculation of one
fitted value. For a decreasing weight function W like the above mentioned triangular
or cosine function, the values contained in the window affect the more the
fitted value the closer they are to the center.
The benefits and the disadvantages of the RLW regression in contrast to the other
described statistical smoothing methods can be observed in the following chapter.
2.4.6 Spline interpolation/approximation
Another kind of polynomial fitting procedures are the spline interpolation and
the spline approximation. The difference between interpolation and approximation
is that the interpolated curve includes all given points while the approximation
contains the given points in a certain error bound.
The cubic spline interpolation uses a third-order polynomial to obtain a curve
segment that is smooth in the first derivative and continuous in the second derivative.
In a special case, if the first derivatives mi at the given points are provided
from outside, then the interpolation requires only two neighboring points (xi©��i)
and (xi�1©��i�1) to determine the spline segment which is given by
a1(x a2(x
xi)¦
xi) 2¦ a3(x
xi) 3¤ (2.22)
The solution for the coefficients has to satisfy the linear equation system
a0¦ �(x)¡
with
1 ����
0 0 0
0 1 0 0
1 xi�1 xi (xi�1 xi) 2 (xi�1 xi) 3
0 1 2(xi�1 xi) 3(xi�1 xi) 2
�x dx�i
��
d� mi¡
and mi�1¡
� ���
�
a0
a1
a2
a3
�
���
d�
� ���¡
���
�
�x �i�1¤
� dx�
mi
�i�1
mi�1
� �i
(2.23) ����
�
For the considered inversion problem, the first derivatives mi for all interface
points are given by the direction of the ZO rays. The ZO rays are normal rays
and end in the NIP. Therefore, the first derivatives have to be perpendicular to
the direction of the ZO rays at the corresponding NIP.
The spline approximation is based on the same method as the interpolation but
with error intervals around the given interface points (for an application example
see Figure 2.14). For further details, please refer to Nürnberger (1989).
28

2.5 Inversion methods
2.5 Inversion methods
To understand the structure of the subsurface, geophysicists need a subsurface
model representing the physical properties. The knowledge of these properties
allows predictions which can be compared with the measured data whether the
assumptions made during the computation of the subsurface model are close
enough to reality. Thus, analyzing the effects computed on these models leads
to a better understanding of the real data. In my case, I use the data-derived
CRS attributes to directly obtain an adequate velocity model, i. e., without any
iteration.
The inversion methods presented in the following subsections yield a 2D isovelocity
layer model that is used to obtain the macro-velocity model necessary
for the subsequent depth migration. A depth image of the subsurface is obtained
from pre- or post-stack migration procedures (e. g., see PreSDM in Section 2.1.3).
These depth images are used, e. g., in oil and gas exploration to identify possible
locations of hydrocarbon reservoirs.
The computed velocity model is subject to several restrictions due to the current
implementation (some used terms right below will be further explained later on):
The ¢
The ¢
Only ¢
The ¢
layers are bounded by arbitrarily curved interfaces that are spread over
the whole range of the traces for the inversion methods described in Sections
2.5.1 – 2.5.3. The interfaces obtained from the inversion method explained
in the subsection 2.5.4 are spread between the extension of depth
points of the outermost back-propagated rays.
interfaces must not intersect each other that the ascending order of interfaces
is not changed.
the P-wave velocity is considered. Thus, no conversion of rays due to
reflection or refraction is taken into account. In other words, the inversion
methods provide acoustic models which contain only P-wave velocities.
high-frequency approximation (Červený, 2001) must be fulfilled, so
that the ray theory can be applied. That means, that in order to describe
high-frequency seismic wave propagation in inhomogeneous media by a
ray-theoretical approach, it is required that the material parameters of the
medium do not vary strongly over the distance of the order of the prevailing
wavelength. This signifies for the 2D layered model that the radii of
curvature of reflecting interfaces are assumed to be much larger than the
wavelengths of waves that impinge on them. Then, an interface will act
predominantly as a reflector rather than a diffractor. Furthermore, the thickness
of a layer has to be larger than the dominant wavelength of the signal.
29

Chapter 2. Theory
The ¢
The ¢
layer velocity is constant for the temporary and final iso-velocity layer
model. This yields straight lines for the ray segments within the layers due
to Fermat’s principle.
construction of the interfaces agrees with Huygens’ principle which assumes
that wavefronts can be regarded as the envelope of diffractor wavefronts
at a certain moment in time. The propagation of a wavefront can also
be expressed with the propagation of rays as used in geometrical optics.
The rays within isotropic media are perpendicular to the local tangent of
the wavefront at any point (in time) or space.
For the model that satisfies the limitations above, some laws and conventions
concerning the propagation of rays and wavefront curvatures are presented in
the following. Hubral and Krey (1980) introduced analytical expressions for the
following laws treating the changes of wavefront curvature radii:
Refraction
30
¢
¢ Propagation
¢ Reflection
law:
RP RP 2¡
vΔt 1¦
(2.24)
The propagation law describes the change of a wavefront’s radius of curvature
as it is propagating along a straight ray path through a layer with
constant parameters. Thus, the radius of the wavefront curvature RP 2 in
point P2 at a later moment in time (Δt� 0) directly depends on the wavefront
curvature radius RP 1 at point P1 and the time Δt that the wavefront
requires to travel the distance between both points. v is the constant velocity
of the surrounding medium. The spread of the wavefront curvature is
called geometrical spreading.
law:
1 vR RR¡
cos2�I vIRI cos2�R¦ 1
RF cos2�R�vR cos�I cos�R� (2.25)
vI
The radius of the reflected wavefront curvature RR is build up by a sum
of two terms. The first term stems from the change depending on a planar
interface, i. e., the radius of the interface curvature . The second
term accounts for the influence of the interface’s curvature. Index I denotes
the variables before the reflection, index R after the reflection. The angles
�I and�R are measured between the ray ending or starting at the reflection
RF¡��
point and the interface normal.
law:
1 vT RT¡
cos2�I vIRI cos2�T¦ 1
RF cos2�T�vT cos�I cos�T� (2.26)
vI

2.5 Inversion methods
The refraction law has the same structure as the reflection law above: one
term for the refraction at plane interfaces and one term describing the change
due to arbitrarily curved interfaces represented at the point of refraction by
the local radius of interface curvature. Here, index T denotes the variables
after the refraction.
Whether the radius of wavefront curvature for a propagating ray is positive or
negative is a matter of definition. I will stick to the same definition as used by
Hubral and Krey (1980). Figure 2.9(a) depicts that a radius of wavefront curvature
lagging behind (red arc) its local tangent (green line) is defined as positive. If the
wavefront lies ahead of its tangential plane the radius of curvature is negative
(see blue arc in Figure 2.9(a)). The following definition describes the interface
curvature radius. If the interface at the point of intersection with an incident ray
appears convex for this incident ray then RF is positive, illustrated by the red
ray and the red dashed arc in Figure 2.9(b). The green ray and green dashed arc
depict the concave case.
Another law determines the direction of a ray before and after a refraction or
reflection by defining the sign of angles. This is Snell’s law given by
sin�I
vI ¡
vT ¡
sin�T
sin�R
vR ¤
The visualization of Snell’s law is shown in Figure 2.10. At the intersection of a
ray with an interface the area around this point is divided into four quadrants
separated by the interface normal and the local interface tangent. The yellow
quadrants depict negative angles and the red ones positive angles. Thus, the
incidence angle�I is negative, the reflected angle�R is positive and the refracted
angle�T is negative for the case depicted in Figure 2.10. The same signs are
obtained if the incident ray starts in the fourth quadrant. For the case of the
incident ray beginning in the third or first quadrant, the signs are changed.
In my case, the S-wave velocity is set to zero, i. e., the acoustic case is considered
which implies no wave-type conversion. Thus, the velocity vR for the medium
around the reflected ray is equal to the medium velocity of the incident ray vI
which yields�R¡ �I. With this special case also the reflection law (2.25) simplifies
to
1 1 RR¡
RI
or RR¡
RI
(2.27)
as cos�R¡ cos ( �I). However, the reflection law is not considered during all
inversion procedures presented below because multiples are not taken into account
for the construction of the interfaces. For the back propagation of primary
reflection events, only the propagation law and the refraction law is required.
31

Chapter 2. Theory
direction of ray propagating through homogeneous medium
local tangent
R0
(a)
V1
V 2
R >0
F, 1
R

t 0
z
0,1
Z
X 0
x
x
x
x
x
x
x
x
x
v 1
v 2
v 3
x
x
x
Seismic line
2.5 Inversion methods
Figure 2.11: Dix inversion. Upper part: stacked ZO section with identified reflection
events of different reflectors. Lower part: Iso-velocity layer model with
linearly interpolated interfaces and layers with constant average vi from all interval
velocities belonging to one layer.
2.5.1 Dix inversion
The conventional Dix inversion algorithm (Dix, 1955) determines an iso-velocity
layer model referring to the assumptions made for NMO corrections. Thus, the
subsurface consists of horizontal or at least gently dipping plane layers of constant
velocities during the inversion of reflection events from one trace. Here,
gently dipping means an interface dip less than five degrees which implies that
the small-dip approximation holds. After all traces are processed, the velocities
belonging to one layer are averaged to obtain the constant P-wave layer velocity.
Figure 2.11 shows the construction of the iso-velocity layer model. The first
picked reflection event (green amplitudes, 1) of the red (i¡ trace 0) is inverted
using the corresponding interval velocity vi�0�j�1 which equals the NMO
velocity but only in case of the first interface. The thickness of the first layer, i. e.,
the depth of the first interface is calculated by v0�1t0�1¨2. For the subsequent
events 2©3©¥¤¥¤¥¤),
j¡
the formula for the interval velocities from small to
(j¡ zi�0�j�1¡
33

Chapter 2. Theory
large traveltimes t0�j provided by Dix (1955) has to be used:
2 NMO�i�j ti�j v 2 NMO�i�j�1 ti�j�1
v
(2.28)
ti�j ti�j�1
where index j counts the identified reflection events and index i accounts for the
traces. In Figure 2.11, the red trace belongs to 0 and the black ones to 1©2©¥¤¥¤¥¤.
©
The NMO velocities and the traveltimes are assumed to be known. For
the traveltimes, this is fulfilled by simply picking the identified primary reflection
events in the ZO section. The NMO velocities are provided by the CRS attribute
RNIP according to Equation (2.13)
vi�j¡
0
i¡
which implies horizontal interfaces.
Here, the CRS attribute RN is not considered but set to infinity which implies
planar interfaces. Thus, Equation (2.28) simplifies to
i¡
¡ with£
RNIP�i�j RNIP�i�j�1
2v0
ti�j ti�j�1 © vi�j¡
(2.29)
with the known near-surface velocity v0 of the CRS stack. The calculation of
the corresponding layer thicknesses makes use of the interval velocities and is
obtained recursively
dzi�j¡ dzi�j�1¦
by
1
2 vi�j(ti�j (2.30)
where dz0�j�1 denotes the depth of the previous, i. e., the (i 1)th calculated temporary
interface. This yields the dark blue horizontal straight lines in Figure 2.11
for the first trace. Going on to the next traces, the blue crosses are obtained but
with different interval velocities. Hence, to construct the iso-velocity layer model
the temporary interval velocities of each layer
ti�j�1)©
const¤) are averaged and assigned
to the whole layer (colored layers in Figure 2.11).
As the inversion procedure firstly inverts all identified reflection events along
(j¡
one trace and then continues with the next trace, the Dix inversion is a trace-bytrace
inversion process. The Dix inversion is very robust concerning the computation,
but the range of application is very limited as the formulae are based on
the assumptions made for NMO corrections. The method is not valid for more
complex structures because it strictly considers all depth points to be located on
the vertical depth lines for each trace and does not account for interface dips and
curvatures. Dix (1955) did not give a solution for negative values for the quotient
under the square root in Equation (2.28), which is the case for intersecting events
as they are not part of the assumptions made for NMO corrections.
2.5.2 Plane inversion
The plane dipping interface inversion (shortly: plane inversion) is a generalization
of the Dix inversion where the dips of the interfaces have an effect on the
34

Depth [km]
0.5
1.5
2
2.5
1
3
0
v i,1
int.1
v i,2
int.2
v i,3
int.3
v i,4
int.4
x
Ri,2 S i,1
z
z
z
z
X
i
i,1
x
i,2
x
i,3
x
i,4
x
x
Ri,1 2.8 3 3.2 3.4 3.6 3.8 4 4.2
Distance [km]
2.5 Inversion methods
Seismic line
Figure 2.12: Construction of the temporary iso-velocity model for each picked
reflection event within one trace. The normal incidence points Ri�j determine
the location (red crosses) and the local dip of the interfaces. Assuming plane
interfaces, the depth points zi�j(blue crosses) are obtained by the intersection with
the vertical depth line. The interval velocities vi�j are assigned to the zi�j.
x
Ri,3 x
R i,4
35

Chapter 2. Theory
interface point in depth. This approach is based on the more general formulae by
Dürbaum (1953) but still uses plane interfaces. I will only explain the incitement
of this method, for a deeper insight into the formulae, refer to Majer (2000).
The construction of the temporary iso-velocity layer model for each trace starts
in almost the same manner as the Dix inversion with the first identified reflection
event. To find the depth point, now the CRS attribute of the emergence angle
must be given in advance to the Dix inversion. With the traveltime ti�0�j�1,
the CRS attribute RNIP�i�0�j�1 and the CRS attribute£i�0�j�1, the NIP wavefront
is propagated along its corresponding normal ray in the direction of£i�j until
RNIP�i�j¡ 0, i. e., referred as back-propagating the ray in the following. This defines
the first reflector point Ri�j in Figure 2.12 by
RNIP�i�j sin£i�j©RNIP�i�j cos£i�j) T¤ (2.31)
(xi
Consider plane interfaces which is done by setting the CRS attribute RN�i�j
for all interfaces 1©2©¥¤¥¤¥¤)
Ri�j¡
and all traces 0©1©¥¤¥¤¥¤). (i¡ The plane interface
must be perpendicular to the ray ending in Ri�0�j�1 as it is the NIP. Then, the intersection
with the vertical depth line is obtained by following the plane interface
(j¡
through Ri�0�j�1 with
:¡��
slope tan£i�0�j�1 . This yields the depth point
zi�0�j�1.
mi�0�j�1¡
During the back-propagation of the next reflection event, not only the propagation
law has to be used, but also the refraction law and Snell’s law must be
applied at the intersection of the new ray with the temporary interface of the first
reflection event (point Si�0�1, see Figure 2.12). This is done in the same way for
all reflection events belonging to one trace, and afterwards the same procedure is
performed for the next traces. In the end, the iso-velocity model is, as for the Dix
inversion, constructed by linear interpolation of all depth points of each reflection
event within the current implementation. In future, they can be interpolated by
splines which can be used to find reflector points where no picking of reflection
signals was possible that yield continuous reflectors. The constant layer velocity
is taken as the average of all interval velocities belonging to one interface. Thus,
the plane inversion is also a trace-by-trace inversion procedure which includes
the interface dip into the calculation. The range of application compared to the
Dix inversion can now be extended for plane dipping interfaces. However, the
method might fail for steep dipping reflection events because of intersections of
the temporary interfaces of one trace with each other before they intersect the
vertical depth line. This changes the ascending order of interface points on the
vertical depth line. Therefore, only a not too large spread area of the ZO rays
around the vertical depth line is considered which implies to take only gentle
dips into account.
36

2.5.3 Circular inversion
2.5 Inversion methods
The circular curved interface inversion method (shortly: circular inversion) makes
use of the same back-propagation process as the plane inversion, but now takes
also the curvature of the interfaces into account to find a more reliable interface
point in depth. The interface curvature is related to the CRS attribute RN that is
constant for each interface point during the construction of the temporary isovelocity
model.
The depth points zi�j (see blue crosses in Figure 2.13) are also situated at an intersection
with the vertical depth line, but now—in contrast to the plane inversion—
with the circular interface segment. The back-propagation with£and RNIP yields
the interface points Ri�j (green crosses) and the slope of the temporary plane interface.
The CRS attribute RN also has to be propagated back until RNIP¡ 0 to
obtain the true interface curvature radius R�N . To find the depth point zi�j, the center
of the circle with radius R�N has to be calculated. Therefore, we make use of
the knowledge that the temporary plane is tangent to the temporary circle. Now,
the intersection of the circle and the vertical depth line can be calculated.
The circular inversion uses all three attributes of the CRS stack extracted for the
identified (picked) reflection events. It fits the interface with circular segments
which is one step closer to reality than the plane inversion. However, as it still is
a trace-by-trace inversion method, the same limitation as for the plane inversion
for the ZO rays, that they are in a “small” area around the vertical depth line, has
to be considered.
2.5.4 Horizon inversion
In contrast to the three inversion methods described above, the horizon inversion
generates the whole interfaces one after the other. Thus, the horizon inversion is a
so-called layer-stripping inversion method. To perform this recursive calculation
of the interfaces from top to bottom, the model is assumed to consist of constant
velocity layers.
Picking identified reflection events and extracting their corresponding CRS attributes
is necessary to obtain the information needed to construct the jth interface
of the 2D iso-velocity layer model. The current implementation of the
horizon inversion algorithm makes use of the following data:
¢ The
The
The ¢ ¢
location of the ZO ray within the seismic line, here, xi�j with index j
for the interface number and index 1©¥¤¥¤¥¤�©n j counting the picked values
along one reflection event. Thus, the total number of picked values for each
reflection event n j does not have to be the same for all interfaces.
i¡
picked two-way traveltimes ti�j.
associated emergence angle£i�j.
37

Chapter 2. Theory
Depth [km]
0.5
1.5
2
2.5
1
3
0
int.1
int.2
int.3
int.4
x
R i,2
x
R i,4
Xi
z i,1
xx
R i,1
z i,2
x
z i,3
x
z i,4
x
x
R i,3
1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2
Distance [km]
Seismic line
Figure 2.13: Construction of the temporary iso-velocity model for each picked reflection
event within one trace. The normal incidence points Ri�j determine the
location (green crosses), the local dip, and the local curvature of the interfaces.
Assuming circular interfaces, the depth points zi�j (blue crosses) are obtained by
the intersection with the vertical depth line. The interval velocities vi�j are assigned
to the zi�j.
38
v i,1
v i,2
v i,3
v i,4

1.2
Depth z [km]
1.4
1
x
x
x x
x
1.4 1.6 1.8 2 2.2
Distance x [km]
x
x
x
x
2.5 Inversion methods
Figure 2.14: Interface construction for the horizon inversion. The black crosses
are the back-propagated interface points that are fitted by a spline interpolation
(green line) or by a spline approximation (red line). The dark blue lines denote
the refracted ray segments from the last known interface (black line).
corresponding radius of curvature of the NIP wavefront RNIP.
The
For the jth interface, the rays are traced back through the (j 1) known layers.
Applying recursively the propagation law (see Equation (2.24)) and the refraction
law (see Equation (2.26)) until the NIP wavefront shrinks to zero, yields as shown
for the plane and circular inversion the interface points. These interface points are
depicted as black crosses in Figure 2.14. In contrast to the trace-by-trace inversion
methods, no intersection with the vertical depth line is necessary, but the backpropagated
points belonging to one reflection event are assumed to be at their
¢
true location as we have used the CRS attributes to find these points. Then, the
final jth interface is constructed by applying the spline interpolation (green line)
or the spline approximation (red line).
The spline interpolation is not the optimum choice for the construction of the
final interface because it strongly depends on the calculated positions as it has to
cross all these points. The positions can fluctuate that much from one point to
the next that the back-propagation for a subsequent time horizon can fail because
there a successive refraction was not possible. Another reason for the algorithm
to fail can be that the input still contains outliers. On the other hand, the spline
approximation may smooth the interface too much such that complex structures
are lost. To find a compromise between smoothing and interpolating, the error
bound for the spline approximation has to be chosen interactively by the user.
39

Chapter 2. Theory
The jth interface is obtained by means of spline approximation of the ray end
points back-propagated through the (j 1) known layers in the current implementation.
A future task can be to test the RLW regression for the interface construction
to account more for small variations along an interface due to complex
structures. The CRS attribute could also be used to constrain the spline approximation.
The final result of the 2D iso-velocity layer model fits closer to complex structures
of real data than the other presented inversion methods. The horizon inversion
can follow the rays even outside the boundaries of the input locations along the
seismic line. This means, that the x-location spread is not essentially the same as
of the input, i. e., it depends on the x-location of the outermost back-propagated
ray of each reflection event. The algorithm does not need the same number of input
points for all interfaces as it constructs the whole interface using approximating
functions before going on to the next interface. In future implementations, the
limitations made for the model can be revoked to get closer to reality. Accounting
for velocity gradients within the layers can be one of these enhancements.
40

Chapter 3
Synthetic data examples
In the following sections, two acoustic 2D iso-velocity models of different complexity
are discussed. The corresponding results of the methods involved in the
inversion process are described. The two synthetic multi-coverage data sets are
generated in order to demonstrate the ability of the inversion methods, described
in Section 2.5, to adequately reconstruct the structures and the P-wave velocities
of the used models. The results of the inversion algorithms are compared with
the original model to determine how stable the different algorithms are with respect
to variations of the CRS attributes. For this purpose, the generated synthetic
data sets consist only of primary P-wave reflections of continuous interfaces with
a S/N ratio of 10. The source wavelet is a zero-phase Ricker wavelet with a dominant
frequency of 15 Hz shown as blue line in Figure 3.1. The red dots in Figure
3.1 are examples of amplitudes of a recorded seismogram without noise. Here,
the red dot of the maximum amplitude does not coincide with the real maximum
because of the sampling interval of 4 ms.
Picking the maximum amplitude of identified primary reflection events within
the simulated ZO section yields nearly the correct ZO two-way traveltime for the
case of the used synthetic data. This means, the ZO two-way traveltime is obtained
by picking the maximum amplitude of the source wavelet which belongs
0¤0
to times around t¡
s in Figure 3.1. The maximum error regarding to time for
the picking does not exceed half the sampling interval of the seismograms without
noise. For seismograms with noise, the error can be greater depending on the
dominant frequency of the source wavelet and the S/N ratio. The picked time
error leads to a depth error and can be accumulated for the following interfaces,
but it can also cumulate destructively.
All four inversion algorithms have one problem in common, they are not able to
calculate layer velocities beneath the last interface. It is the responsibility of the
interpreter to define an adequate velocity distribution for the half plane beneath
the last interface. In my case, this filling velocity is constant, but in future it
could be made up by blocks with constant velocities or even smoothly varying
41

Chapter 3. Synthetic data examples
amplitude normalized to 1
1
0.5
0
−0.5
−0.06 −0.04 −0.02 0
time t in [s]
0.02 0.04 0.06
Figure 3.1: The zero-phase Ricker wavelet with a dominant frequency of 15 Hz.
Red dots simulate a seismogram without noise and with a sampling rate of 4 ms.
Here, the maximum is not at 0 s. t¡
velocities in horizontal or also in vertical direction. However, as long as I try to
reconstruct models with homogeneous velocity layers, it makes no sense to fill
the half plane with an inhomogeneous velocity distribution. The velocity of the
underlying half plane is important for the reflection coefficient which is required
to calculate the reflected amplitudes. As long as the inversion algorithms are
not designed to account for “true amplitudes” but only for traveltimes and CRS
attributes during the computation of velocity models, the problem is at this stage
of development negligible.
3.1 Model A
Figure 3.2 shows the initial 2D iso-velocity layer model with four curved interfaces
spread over the entire profile. The black circles depict the given points for
constructing the whole interfaces by means of spline interpolation (thin and thick
black lines through the circles). The interpolated splines of the interfaces are used
to calculate the local dips shown in Figure 3.3. The interfaces separate five layers
with constant velocities. The different constant P-wave velocities for the first four
layers are 1¤5 km/s, 1¤8 km/s, 2¤0 km/s, and 2¤2 km/s (see
also red numbers in the left part of Figure 3.2). The velocity for the half plane,
2¤5 v4¡ v3¡ v2¡ v1¡
i. e., the fifth layer, is set to v5¡
km/s, all velocities are also indicated by the
different colors. The curvature of the interfaces, the thickness of the layers, and
their corresponding P-wave velocities are chosen to yield “simple” primary Pwave
reflections. That means, that the reflection events within the simulated ZO
section do not intersect each other and have, e. g., no triplications. This choice of
42

z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
1.5
1.8
2.0
2.2
2.5
−4
0 1 2 3 4 5
x [km]
6 7 8 9 10
3.1 Model A
Figure 3.2: Model A: 2D iso-velocity layer model with smooth interfaces. The red
numbers denote the homogeneous P-wave velocities in [km/s] and belong to the
colored layers. The colorbar depicts the velocities for all following models in this
section.
the original model will make it easy for the half-automatic picker to follow the
reflection events in one step. Thus, the picker is able to follow the maximum amplitude
from the trace at the left border to the trace at the right border without
losing track of the reflection event because of small gaps or intersecting events.
To obtain the synthetic multi-coverage primary reflection data set, the ray method
is used to calculate the two-way traveltimes of CMP gathers with half offsets from
0 1 h¡
km to h¡
km with increments of Δh¡ 0¤02 km at midpoint intervals of
0¤01 km. In this way, a multi-coverage data volume is obtained. Also, the
ZO (h¡ section 0 km) is part of the generated multi-coverage data set, but the
aim of this thesis is to make use of data-driven attributes obtained from the
Δxm¡
2D
ZO CRS stacking method for inversion algorithms and not only of ZO two-way
traveltimes. The synthetic ZO section can be used to evaluate the accuracy and
the improvement of the S/N ratio of the CRS stacking procedure against the CMP
stack. This is explained in detail in Jäger (1999) and Höcht (1998).
From this data set, the simulated ZO section with a S/N ratio of 10, shown in
Figure 3.4, is the result of the initial CRS stacking procedure using projected Fresnel
zones as ZO aperture size. The small window in the right part of Figure 3.4,
where the amplitudes are plotted as excursions to the left and right and not as
colors, is called wiggle plot and illustrates the S/N ratio of 10. The CRS stacking
method with projected Fresnel zones is shortly described in Section 2.2.1. For
further explanations, please refer to Vieth (2001). The ZO section is cut off at
midpoint coordinates 1 km and 8¤5km. At lower and higher midpoint
coordinates, the coverage of the data set becomes too low for the CRS stacking
xm¡ xm¡
43

Chapter 3. Synthetic data examples
theta [degree]
8
6
4
2
0
−2
−4
−6
−8
−10
−12
1 2 3 4 5
x [km]
6 7 8
Figure 3.3: Local dips of synthetic velocity model A. Red: 1st interface, green:
2nd interface, blue: 3rd interface and yellow: 4th interface.
procedure to yield reliable results.
The emergence angle£ of only the first interface represents the local dip (see
Appendix A). Here, the red line of£ in Figure 3.12 and 3.13 differ from the
synthetic dips in Figure 3.3 not only because of the smoothing, also the extracted
original angles in Figure 3.8 do not coincide with the synthetic dips. The CRS
stack was intentionally performed with a near-surface velocity of 1.7 km/s which
is not the same velocity as of the first layer. This difference leads to wrong CRS
attributes. However, the attributes can be corrected by adding a virtual layer
above the first layer with infinitesimal thickness (see Appendix B).
The CRS stack produces the simulated ZO section and three additional sections.
Each of these three sections contains one attribute for each time sample: one for all
emergence angles£(see Figure 3.5), one for all radii of curvature of the NIP wavefronts
(see Figure 3.6), and one for all radii of curvature of the normal wavefronts
(see Figure 3.7). Because of the coherence analysis which is used to determine the
attribute triplet for each time sample, I obtain attributes also for the noise. But
these attributes should not be used for the inversion methods which are based on
reflection events. Thus, it is necessary to extract the CRS attributes along the reflection
events. The two-way traveltimes are picked at the maximum amplitudes
of the identified primary P-wave reflections. Their corresponding CRS attributes
can now be extracted with these picked times directly from the attribute sections
obtained from the CRS stack. The picked traveltimes are depicted as green lines
44

time [s]
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
1 2 3 4
Distance [km]
5 6 7 8
time [s]
1.20
1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
1.75
1.80
3.1 Model A
7.1
Distance [km]
7.2 7.3 7.4
Figure 3.4: Simulated ZO section with S/N ratio of 10 from the initial Fresnel
CRS stack. Negative amplitudes are blue, positive amplitudes are red and picked
traveltimes are displayed as green lines. The S/N ratio can be observed in the
small window to the right.
time [s]
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
1 2 3 4
Distance [km]
5 6 7 8
Figure 3.5: Section of CRS attribute£obtained from the CRS stacking procedure.
The colorbar depicts£[�]
45
30
20
10
0
-10
-20
-30

Chapter 3. Synthetic data examples
46
time [s]
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
1 2 3 4
Distance [km]
5 6 7 8
Figure 3.6: Section of CRS attribute RNIP. The colorbar depicts RNIP [km].
time [s]
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
1 2 3 4
Distance [km]
5 6 7 8
Figure 3.7: Section of CRS attribute RN. The colorbar depicts RN [km].
6
4
2
0
-2
-4
-6
50
40
30
20
10
0
-10
-20
-30
-40
-50

angle [degree]
8
6
4
2
0
-2
-4
-6
-8
-10
1 2 3 4 5
Distance [km]
6 7 8
3.1 Model A
Figure 3.8: The extracted original CRS attributes of the emergence angle£.
Rnip [km]
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
1 2 3 4 5 6 7 8
Distance [km]
Figure 3.9: The extracted original CRS radii of curvature of the NIP wavefront
RNIP.
47

Chapter 3. Synthetic data examples
Rn [km]
600
500
400
300
200
100
0
-100
-200
-300
-400
-500
-600
1 2 3 4 5
Distance [km]
6 7 8
Figure 3.10: The extracted original CRS radii of curvature of the normal wavefront
RN.
1/Rn [1/km]
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
1 2 3 4 5 6 7 8
Distance [km]
Figure 3.11: The reciprocal value of the extracted original CRS radii of curvature
of the normal wavefront RN.
48

3.1 Model A
in Figure 3.4. The extracted original CRS attributes are shown in Figures 3.5 – 3.7
also with green lines for the picked values. In the following figures, the attributes
of the first interface are displayed as red lines, green lines stand for attributes of
the second interface, attributes of the third interface are depicted as blue lines,
and fourth interface attributes as yellow lines. These colors for each attribute and
interface are used during the whole thesis.
In Figure 3.11 and in all following figures of the CRS attribute RN, RN is displayed
as its reciprocal value to better reveal the behavior of the reflectors curvature.
Roughly speaking, if a reflector changes its curvature from an anticlinal
to a synclinal structure or vice versa, respectively, the original CRS attribute RN
undergoes a sign change from plus to minus while passing�� or vice versa,
respectively. In Figure 3.10, the peaks of RN can be observed, but the remaining
behavior of the curve gets lost in the large scale of the Figure which was clipped
at 600 km, i. e., there were even greater values of RN. After calculating the reciprocal
values of RN, the sign change converts into a zero crossing as shown in
Figure 3.11. Another advantage of the reciprocal of RN is, that smoothing the
curve can be performed without any additional calculations. That means, while
smoothing the original attribute RN, the peaks of the sign changes must remain in
the smoothed curves. Therefore, the smoothing algorithm has to detect such sign
changes and has to leave them unchanged. In other words, the window length
for the smoothing has to be tapered to zero and has to start with zero again on
the other side of the sign change. If this is not taken into account, the smoothing
produces artefacts as shown in Section 2.4.1. But for the reciprocal value of RN,
zero crossings appear that can be smoothed without detecting them. To obtain
the smoothed RN itself, just the reciprocal value of the smoothed 1¨RN has to be
taken again.
The CRS attribute of the emergence angle£ in Figure 3.8 is, in contrast to the
radii, nearly a smooth curve and thus it is not necessary to smooth it even more
for further usage. However, the CRS attribute radius of curvature of the NIP
wavefront RNIP in Figure 3.9 fluctuates so much from trace to trace that it is not
rich in meaning to use these data directly for the inversion algorithms. The CRS
attribute RN in Figure 3.11 shows small peaks which can be interpreted as small
outliers that should be eliminated for inversion purposes. Therefore, it is necessary
to smooth at least the CRS attributes RNIP and RN. Although the angle is a
smooth attribute from the very beginning, I nevertheless applied several filters to
it. Sometimes these filters can corrupt the original data that much that they can
lead to false results. This can be observed in Figure 3.13, where the RLW regression
corrupted the original shape of the decreasing parts of the emergence angle
£. Thus, the filter has not improved the data at all.
In Section 2.4 several smoothing methods are presented, but to show all their results
is beyond the frame of this thesis. Thus, I only focus on the results of the
arithmetic mean in time- and x-direction (see Figures 3.12, 3.14 and 3.16) in com-
49

Chapter 3. Synthetic data examples
angle [degree]
8
6
4
2
0
-2
-4
-6
-8
-10
1 2 3 4 5
Distance [km]
6 7 8
Figure 3.12: CRS attribute£smoothed with arithmetic mean with seven samples
window length in time- and in x-direction.
angle [degree]
8
6
4
2
0
-2
-4
-6
-8
-10
1 2 3 4 5
Distance [km]
6 7 8
Figure 3.13: CRS attribute£ smoothed with RLW regression with f=0.1 which
represents 75 samples window length in x-direction.
50

Rnip [m]
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
1 2 3 4 5 6 7 8
Distance [km]
3.1 Model A
Figure 3.14: CRS attribute RNIP smoothed with arithmetic mean with seven samples
window length in time- and in x-direction.
Rnip [m]
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
1 2 3 4 5 6 7 8
Distance [km]
Figure 3.15: CRS attribute RNIP smoothed with RLW regression with f=0.1 which
represents 75 samples window length in x-direction.
51

Chapter 3. Synthetic data examples
1/Rn [1/km]
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
1 2 3 4 5 6 7 8
Distance [km]
Figure 3.16: CRS attribute 1¨RN smoothed with arithmetic mean with seven samples
window length in time- and in x-direction.
1/Rn [1/km]
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
1 2 3 4 5 6 7 8
Distance [km]
Figure 3.17: CRS attribute 1¨RN smoothed with RLW regression with f=0.1 which
represents 75 samples window length in x-direction.
52

z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
1.499
1.795
1.968
2.188
2.500
−4
1 2 3 4 5
x [km]
6 7 8
3.1 Model A
Figure 3.18: Velocity model from Dix inversion and attributes smoothed with
arithmetic mean. The constant velocities of the layers are depicted by the red
numbers within the layers in [km/s].
parison to the results of the RLW regression in x-direction (see Figures 3.13, 3.15
and 3.17). For the arithmetic mean, the filter length in time- and x-direction is
7 samples. Finding the filter length in time-direction is difficult because if the
length is too large, then the window is greater than the wavelet length and values
of the noise will affect the calculations. These noise values are often quite
different to the values of the reflection event, which will lead to a wrong result
of the smoothing. The dominant frequency of the reflection event can be used to
find a suitable value for the window length in time-direction. The parameter f
for the RLW regression is set to 0.1 which means a filter length of 75 samples in
x-direction. The parameter for the number of iterations nsteps was set to 2.
3.1.1 Dix inversion
The well-known Dix inversion method uses plane horizontal interfaces to find the
corresponding depth points as described in Section 2.5.1. This assumption means,
that the CRS attribute of the normal wavefront curvature RN is equal� which
implies a plane reflector and therefore RN will not be used any further. Also, the
angle£ emergence has no effect on the calculation of depth points because the
plane interfaces are assumed to be yields£ horizontal which 0 cos£ and 1.
Thus,£will also not be involved in the calculations. Those prerequisites might be
misleading in that the inverted result is a model with four plane and horizontal
interfaces in my case. However, I obtain models as shown in Figures 3.18 and
3.19 because the current implementation of the Dix inversion algorithm makes
use of the assumptions while it is calculating the depth points for one x-location
and not for the whole interface. Before the algorithm moves to the next trace, it is
¡ ¡
53

Chapter 3. Synthetic data examples
z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
1.500
1.793
1.947
2.177
2.500
−4
1 2 3 4 5
x [km]
6 7 8
Figure 3.19: Velocity model from Dix inversion and attributes smoothed with
RLW regression. The constant velocities of the layers are depicted by the red
numbers within the layers in [km/s].
calculating all depth points within the actual trace. This is called a trace-by-trace
inversion algorithm.
The resulting iso-velocity model should consist of homogeneous velocities within
the layers. To end up with the Figures 3.18 and 3.19, the constant layer velocity
has to be found. Therefore, I assume that the computed interval velocities are
variations around the searched for value with a normal distribution. Then, I get
the constant interval velocity for each layer by calculating the arithmetic mean of
all interval velocities obtained for one layer:
�i�1
n
vi�j
v (3.1)
n ¤ j¡
Here i is the index for each trace, n is the number of traces and j is the index for
each reflection event. The mean velocity v j is assigned to the layer above the j-th
interface that was constructed from the j-th reflection event.
The result of the Dix inversion should not differ very much from the original
synthetic depths as long as the interfaces are nearly plane and horizontal. This is
obvious in the middle part of Figure 3.18 and 3.19 around midpoint coordinate
5 xm¡
km. There, the interfaces of the original model have at all four depth
locations a nearly horizontal tangent, i. e., no dip. This fact can also be observed
in the depth difference plots of Figures 3.20 and 3.21 and in the velocity difference
plots of Figures 3.22 and 3.23. The differences in depth and velocity are calculated
by subtracting the synthetic values from the inverted ones. The mean difference
54

dz [m]
50
0
−50
−100
−150
1 2 3 4 5
x [km]
6 7 8
3.1 Model A
Figure 3.20: Differences in depth from Dix inversion result for each inverted
depth point and in the same colors as the attributes for each interface smoothed
with arithmetic mean.
dz [m]
40
20
0
−20
−40
−60
−80
−100
−120
−140
1 2 3 4 5
x [km]
6 7 8
Figure 3.21: Differences in depth from Dix inversion result for each inverted
depth point and in the same colors as the attributes for each interface smoothed
with RLW regression.
55

Chapter 3. Synthetic data examples
dv [m/s]
200
100
0
−100
−200
−300
−400
1 2 3 4 5
x [km]
6 7 8
Figure 3.22: Velocity differences from Dix inversion result for each interface, attributes
smoothed with arithmetic mean. The fluctuating lines are the velocity
differences for every inverted point while the horizontal lines of the same color
depict the mean velocity difference of the whole constant velocity layer: Δv1¡ 1
m/s, Δv2¡ 5 m/s, Δv3¡ 32 m/s, Δv4¡ 12 m/s.
dv [m/s]
100
50
0
−50
−100
−150
−200
−250
1 2 3 4 5
x [km]
6 7 8
Figure 3.23: Velocity differences from Dix inversion result for each interface, attributes
smoothed with RLW regression. The fluctuating lines are the velocity differences
for every inverted point while the horizontal lines of the same color depict
the mean velocity difference of the whole constant velocity layer: Δv1� 0
m/s, Δv2¡ 7 m/s, Δv3¡ 53 m/s, Δv4¡ 23 m/s.
56

of the velocities is then obtained by:
3.1 Model A
�i�1
n
Δvi�j
Δv (3.2)
n ¤ j¡
Negative differences in depth and/or velocity are obtained, if the inverted depth
points are above the synthetic depth points and/or if the velocities are too low.
Positive differences imply that the inverted depth points are placed too deep, i. e.,
beneath the synthetic depth points and/or that the velocities are too high.
Looking at the first reflection event and its attributes, especially the
angle£
emergence
which represents the dip (see appendix A) but only in the case of the
first reflector leads to the following expectation. According to the actual dips, I
should receive three regions 0), where the results of the Dix inversion should
be close to the synthetic values due to the assumption of plane horizontal layers.
2¤5©5¤0 (��
For these regions around xm¡
and 7¤5 km, the differences in depth (red
lines in Figures 3.20 and 3.21) are around zero and the rest of the curve shows a
similar behavior as the dip given by£. This means, if the dips of the interfaces
are not in the vicinity of zero degrees, then the Dix inversion result cannot be the
real depth. The differences in velocities for the first reflection event (red lines in
Figure 3.22 and 3.23) do not show a strong effect that can be correlated with the
dip or one of the attributes. The velocity is nearly constant which implies that
dip£ the plays only a small role and that the correction of the attributes was
successfull.
The second interface (green lines) has its maximum depth displacement to the
5
synthetic depth points around xm¡
km. There, the first and second interface are
both dipping with nearly the same maximum dip. This leads to an accumulation
of the error from the first and the second interface, because the emergence angle£
is ignored while calculating the temporary layer velocity. As the dip increases, the
factor cos 2£decreases which yields a too small NMO velocity as the factor cos 2£
is in the denominator (see Equation (2.13) and Figures 3.22 and 3.23). However,
RNIP can also be related with these errors because of the stronger smoothing for
deeper reflectors.
For the third (blue lines) and the fourth layer (yellow lines) in Figure 3.20 and
3.21, the depth differences from trace to trace are fluctuating more and more because
the errors of the layers above cumulate partly constructively and partly
destructively. The same happens for the velocity differences. However, the maximum
depth error is less than 7 percent and the mean differences of the layer
velocities remain smaller than 5 percent of the synthetic values.
Comparing the results of the two different smoothing algorithms applied to the
input data before the Dix inversion, it is obvious that the RLW regression yields
a more stable inversion process than the arithmetic mean smoothing. The differences
of the RLW regression and the synthetic values (Figures 3.21 and 3.23) can
57

Chapter 3. Synthetic data examples
z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
1.499
1.792
1.975
2.175
2.500
−4
1 2 3 4 5
x [km]
6 7 8
Figure 3.24: Velocity model from plane inversion and attributes smoothed with
arithmetic mean. The constant velocities of the layers are depicted by the red
numbers within the layers.
be regarded as the trend of the difference curves obtained by input data smoothed
with the arithmetic mean (Figures 3.20 and 3.22). That the mean velocity difference
does not decrease from the result of the arithmetic mean to the result of the
RLW regression is caused by the smaller variance of the reflector points differences
compared with the synthetic interface points.
3.1.2 Plane inversion
In contrast to the Dix inversion, the plane dipping interface inversion (or shortly:
plane inversion) takes the local dip of the reflectors into account to determine
their corresponding depth points as described in Section 2.5.2. For the first interface,
the dip is identical to the 2D ZO CRS attribute of the emergence angle£
as shown in Appendix A if the CRS was performed with the “true” near-surface
velocity. Otherwise, the CRS attributes have to be corrected first (see Appendix
B). For the second or further reflectors the attribute£may provide also information
about the local dip of that reflector. There, the model down to the currently
constructed interface has to be known, especially the layer velocities. Then, the
refraction law can be applied at all shallower interfaces in order to obtain the dip
from£ again. Unfortunately, the velocity is only a priori given in the vicinity
of the surface by the surface velocity of the CRS stack which is not always the
true first layer velocity. The CRS attributes obtained with a wrong near-surface
velocity v0 can be corrected (see Appendix B) in such a way that the inversion
algorithm uses the “true” CRS attributes.
The construction of the interfaces remains the same as for the Dix inversion. The
58

z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
1.500
1.791
1.952
2.166
2.500
−4
1 2 3 4 5
x [km]
6 7 8
3.1 Model A
Figure 3.25: Velocity model from plane inversion and attributes smoothed with
RLW regression. The constant velocities of the layers are depicted by the red
numbers within the layers.
picked reflection events of one trace are inverted from top to bottom, i. e., from
low to high two-way traveltimes. This is done for one trace after the other which
yields the 2D iso-velocity layer model in a trace-oriented manner. Thus, I do
not get a 2D velocity model with four interfaces of constant dip. The dips are
only constant for one interface point of each trace during the calculation of one
trace, i. e., within the temporary model of that trace. The local interface dips can
be calculated for this temporary model by£, but they have not to be the local
interface dips of the final model. The temporary model for the next trace is, in
general, different to the previous temporary model according to the layer velocity
and the location of the temporary constructed interfaces.
Accounting for the local interface dips, should enhance the result of the inversion
algorithm if the interfaces can be locally approximated with dipping plane segments.
Thus, the differences of the plane inversion results to the synthetic data
should be smaller as for the Dix inversion. This holds true for the first interface
as shown in Figure 3.26 and 3.27. The differences in depth are closer to zero than
the Dix inversion results. The velocity is not affected very much according to the
local dip because the dip is smaller than�8�which yields a maximum change
of the temporary inverted velocities of two percent. For the second time horizon,
the depths differences of the points are over all also closer to the synthetic
depths points which can be directly correlated with the accounting for the local
dips during the back-propagation of the rays. Around midpoint location 4
6 xm¡
km and xm¡
km the depth points are placed slightly lower than by the Dix
inversion. There, it happened that—in contrast to the case displayed in Figure
2.12—the intersection of the planar temporary interfaces with the vertical depth
59

Chapter 3. Synthetic data examples
dz [m]
150
100
50
0
−50
−100
−150
1 2 3 4 5
x [km]
6 7 8
Figure 3.26: Differences in depth from plane inversion result and attributes
smoothed with arithmetic mean for each inverted depth point and separately for
each interface.
dz [m]
100
50
0
−50
−100
−150
1 2 3 4 5
x [km]
6 7 8
Figure 3.27: Differences in depth from plane inversion result and attributes
smoothed with RLW regression for each inverted depth point and separately for
each interface.
60

dv [m/s]
200
100
0
−100
−200
−300
−400
1 2 3 4 5
x [km]
6 7 8
3.1 Model A
Figure 3.28: Velocity differences from plane inversion result and attributes
smoothed with arithmetic mean separately for each interface. The blue lines are
the velocity differences for every inverted point while the red lines depict the
mean velocity difference of the whole constant velocity layer: Δv1¡ 1 m/s,
Δv2¡ 8 m/s, Δv3¡ 25 m/s, Δv4¡ 25 m/s.
dv [m/s]
100
50
0
−50
−100
−150
−200
−250
1 2 3 4 5
x [km]
6 7 8
Figure 3.29: Velocity differences from plane inversion result and attributes
smoothed with RLW regression separately for each interface. The blue lines are
the velocity differences for every inverted point while the red lines depict the
mean velocity difference of the whole constant velocity layer: Δv1� 0 m/s,
Δv2¡ 9 m/s, Δv3¡ 48 m/s, Δv4¡ 34 m/s.
61

Chapter 3. Synthetic data examples
line provides a smaller depth than the depth of the back-propagated reflection
point. The layer velocity which is provided by RNIP according to Equation (2.13)
increases due to considering the local dip, but the provided layer velocities can
5
also decrease if shallower layers are thicker. Around xm¡
km, the maximum
difference of the temporary velocity does not change as the dips of the first and
second interface are close to zero, i. e., the thickness of the layers stays the same
as for the Dix inversion.
The third inverted reflector points show the depth point dependence with respect
to the local dip along the whole interface. Compared with the Dix inverted layer
velocity, the change of the layer velocity can be neglected. Thus, the down shift
of the inverted points, i. e., the up shift of the depth differences to the synthetic
values is directly correlated with the interface dip.
At the left border of the fourth inverted interface, it happened that the interface
points are placed significantly deeper than by the Dix inversion despite a smaller
layer velocity. This is explained with a back-propagated reflection point far away
from the vertical depth line, so that the intersection of its temporary plane with
the vertical depth line provides an extremly larger depth value than the Dix inversion
did or even larger than the original depth in the model is.
Comparing the results of the different smoothing algorithms, it is obvious that
the RLW regression provides a more stable inversion result than the inversion
with input data smoothed by the arithmetic mean algorithm.
3.1.3 Circular inversion
The circular curved interface inversion (shortly: circular inversion) goes one step
further in using the information provided by the CRS attributes. It calculates
the interface points in the same way as the Dix and plane inversion, i. e., traceby-trace
and along the vertical depth line. But it also uses the CRS attribute RN
which is related to the radius of curvature of the local reflector segment to find
the intersection point at the vertical depth line as describe in Section 2.5.3.
The depth differences for the first inverted interface are slightly more fluctuating
than those from the plane inversion. The temporary layer velocities are not
affected of this stronger fluctuations. Comparing the second inverted interface
with the Dix inversion result, it is obvious that the depth differences and the temporary
layer velocities now also depend on the refraction at the first interface,
where the radius of the interface curvature RF is no longer� . Thus, the second
term of Equation (2.26) has to be considered as it does not vanish. This yields the
larger temporary layer velocities obtained from the circular inversion from midpoint
coordinates 1 km to 4 km and around 7 km. In the vicinity
of 5 km, the latter mentioned effect resulted in lower layer velocities. The
changes of the depth differences compared to the results of the Dix inversion are
mainly dependent on the larger layer velocity differences.
xm¡ xm¡ xm¡ xm¡
62

z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
1.499
1.793
1.989
2.177
2.500
−4
1 2 3 4 5
x [km]
6 7 8
3.1 Model A
Figure 3.30: Velocity model from circular inversion and attributes smoothed with
arithmetic mean. The constant velocities of the layers are depicted by the red
numbers within the layers.
z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
1.500
1.792
1.968
2.165
2.500
−4
1 2 3 4 5
x [km]
6 7 8
Figure 3.31: Velocity model from circular inversion and attributes smoothed with
RLW regression. The constant velocities of the layers are depicted by the red
numbers within the layers.
The maximum error of the layer velocity for the third interface around 5
km is again an effect of the refraction but now at the second interface, where the
radius of interface curvature is negative, so that the second term of the refraction
law is negative. Thus, the refracted radius of wavefront curvature is larger which
yields a larger layer velocity. With this layer velocity, the depth point should be
placed deeper, but with the too small depth of the second interface, also the third
interface is still at a too small depth.
For the fourth inverted interface, the effects of the above layers according to the
xm¡
layer velocity cumulate destructively. This leads to an over all smaller depth
63

Chapter 3. Synthetic data examples
dz [m]
60
40
20
0
−20
−40
−60
−80
1 2 3 4 5
x [km]
6 7 8
Figure 3.32: Differences in depth from circular inversion result and attributes
smoothed with arithmetic mean for each inverted depth point and separately for
each interface.
dz [m]
30
20
10
0
−10
−20
−30
−40
−50
−60
1 2 3 4 5
x [km]
6 7 8
Figure 3.33: Differences in depth from circular inversion result and attributes
smoothed with RLW regression for each inverted depth point and separately for
each interface.
64

dv [m/s]
300
200
100
0
−100
−200
1 2 3 4 5
x [km]
6 7 8
3.1 Model A
Figure 3.34: Velocity differences from circular inversion result and attributes
smoothed with arithmetic mean separately for each interface. The blue lines are
the velocity differences for every inverted point while the red lines depict the
mean velocity difference of the whole constant velocity layer: Δv1¡ 1 m/s,
Δv2¡ 7 m/s, Δv3¡ 11 m/s, Δv4¡ 23 m/s.
dv [m/s]
150
100
50
0
−50
−100
−150
1 2 3 4 5
x [km]
6 7 8
Figure 3.35: Velocity differences from circular inversion result and attributes
smoothed with RLW regression separately for each interface. The blue lines are
the velocity differences for every inverted point while the red lines depict the
mean velocity difference of the whole constant velocity layer: Δv1� 0 m/s,
Δv2¡ 8 m/s, Δv3¡ 32 m/s, Δv4¡ 35 m/s.
65

Chapter 3. Synthetic data examples
z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
1.499
1.795
1.964
2.189
2.500
−4
1 2 3 4 5 6 7 8
x [km]
Figure 3.36: Horizon inversion result shown as thick black line. Attributes
smoothed with arithmetic mean.
difference compared to the Dix and plane inversion result. I shortly mention
again that the RLW regression stabilized the inversion procedure also in case of
the circular inversion.
3.1.4 Horizon inversion
The so-called horizon inversion differs from the three inversion algorithms above
in three points: (i) the end points of back-propagated rays are not projected to
the vertical depth line, (ii) the interfaces are constructed one after the other which
means that all points of the first reflection event are inverted before the second reflection
event is considered (layer-stripping method) and (iii) to obtain a smooth
interface for inverting following reflectors the interfaces are approximated by
splines with arbitrary user-defined error bounds.
As I mentioned above, the interface is constructed directly from the back-propagated
reflection points which yields an interface stretched between the left most
and right most reflection points belonging to one inverted reflection event. These
approximated interface points can have larger x-coordinates at the right border or
smaller x-coordinates at the left border than the input (first and second interface
in Figures 3.36 and 3.37) or just the other way round (third and fourth interface)
depending on the ray paths. If the interfaces do not stretch over the same x-range
then I constantly extrapolated the border values to the borders of the largest xrange
of the inverted interface results (see dotted parts of the third and fourth
interfaces in Figures 3.36 and 3.37). The differences to the synthetic values are
calculated only within the x-range of the used input x-range. At the borders,
where the inverted x-ranges are smaller than the input x-range, I marked the
borders of the inverted x-range with vertical dashed lines in the same color as the
66

z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
1.500
1.792
1.944
2.184
2.500
−4
1 2 3 4 5 6 7 8
x [km]
3.1 Model A
Figure 3.37: Horizon inversion result shown as thick black line. Attributes
smoothed with RLW regression.
interface is displayed (see Figures 3.38 – 3.41).
The back-propagation process of the horizon inversion is based on the same formulae
as the plane or circular inversion, but it takes only£and RNIP into account
from the input. The radius of interface curvature is obtained from the approximated
spline function after all reflection points of one reflection event are computed.
Thus, it can happen that a ray of the next reflection event does not impinge
at the last approximated interface due to its CRS attributes and its refraction(s) at
shallower interfaces. Then, this ray drops out for the calculation of the current
interface.
The depth differences in Figures 3.38 and 3.39 of all inverted interfaces are at most
points the larger the deeper the interface is located. This is due to the larger fluctuations
of the local layer velocities that are obtained for every ray path segment.
As I use these velocities to obtain the whole layer velocity by their arithmetic
mean, the errors in depth and velocity mostly cumulate constructively (see also
Figures 3.40 and 3.41). The stabilizing effect of the RLW regression compared to
the arithmetic mean is clearly visible within the velocity difference plots. Within
the depth difference plots, this effect is most visible for the first inverted interface.
For the other interfaces, the effect is lost because of the approximation of the
interfaces.
In the end, the investigation of this model showed that the inversion algorithm
results are, despite some minor exceptions, the better the more information are
accounted for while computing the depth points and the smoother the input was.
Thus, the RLW regression should be preferred instead of the arithmetic mean
smoothing. The order among the inversion methods is from low preference to
high preference: Dix – circular – plane – horizon inversion.
67

Chapter 3. Synthetic data examples
dz [m]
80
60
40
20
0
−20
−40
−60
1 2 3 4 5
x [km]
6 7 8
Figure 3.38: Differences in depth from horizon inversion result and attributes
smoothed with arithmetic mean for each inverted depth point and separately for
each interface.
dz [m]
60
40
20
0
−20
−40
−60
1 2 3 4 5
x [km]
6 7 8
Figure 3.39: Differences in depth from horizon inversion result and attributes
smoothed with RLW regression for each inverted depth point and separately for
each interface.
68

dv [m/s]
600
500
400
300
200
100
0
−100
−200
−300
−400
1 2 3 4 5
x [km]
6 7 8
3.1 Model A
Figure 3.40: Velocity differences from horizon inversion result and attributes
smoothed with arithmetic mean separately for each interface. The blue lines are
the velocity differences for every inverted point while the red lines depict the
mean velocity difference of the whole constant velocity layer: Δv1¡ 1 m/s,
Δv2¡ 5 m/s, Δv3¡ 36 m/s, Δv4¡ 11 m/s.
dv [m/s]
200
150
100
50
0
−50
−100
−150
−200
−250
−300
1 2 3 4 5
x [km]
6 7 8
Figure 3.41: Velocity differences from horizon inversion result and attributes
smoothed with RLW regression separately for each interface. The blue lines are
the velocity differences for every inverted point while the red lines depict the
mean velocity difference of the whole constant velocity layer: Δv1� 0 m/s,
Δv2¡ 8 m/s, Δv3¡ 56 m/s, Δv4¡ 16 m/s.
69

Chapter 3. Synthetic data examples
z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
1.5
1.8
2.0
2.5
3.0
−3.5
0 1 2 3 4 5 6 7
x [km]
Figure 3.42: Model B: 2D iso-velocity layer model with interfaces of stronger curvature
than in model A. The red numbers denote the homogeneous P-wave velocities
[km/s] and belong to the colored layers. The colorbar depicts the velocities
for all following models in this section.
3.2 Model B
Now, in contrast to model A, the curvatures of the interfaces in Figure 3.42 are
larger. This yields a reflection event with a triplication for interface number 2.
The generation of the multi-coverage data set is the same as for model A. The
interfaces which separate the five layers are constructed by means of spline interpolation
through the given points depicted as black circles in Figure 3.42. From
the spline interpolation, the local dips shown in Figure 3.43 are obtained. The
P-wave velocities for the first four constant velocity layers are 1¤5 km/s,
1¤8 km/s, 2¤0 km/s, and 2¤5 km/s. The half plane beneath the
3¤0 v4¡ v3¡ v2¡ v1¡
fourth layer is filled with the fifth layer velocity v5¡
km/s. The chosen layer
velocities and depth of the interfaces yield primary reflections that may intersect
themselves (see interface two) but not each other as shown in the simulated ZO
section in Figure 3.44. The triplication at the right border of the second primary
reflection event is too “small”, so that it is neglected, i. e., the bow tie like structure
is not picked there. Thus, the picker has to be started at least three times to
obtain all segments of the triplication.
The multi-coverage primary reflection event is again obtained by means of ray
theory to calculate the two-way traveltime from the above described original
model. This is done with a half-offset range from 0 km to 1 km with
increments of Δh¡ 12¤5m. As displayed in Figure 3.42, the midpoint coordinates
h¡ h¡
70

theta [degree]
30
20
10
0
−10
−20
−30
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
x [km]
3.2 Model B
Figure 3.43: Local dips of synthetic velocity model B. Red: 1st interface, green:
2nd interface, blue: 3rd interface and yellow: 4th interface.
are spread km� between 0 7 km with midpoint distances Δxm¡ of 6¤25 m.
The source wavelet is a zero-phase Ricker wavelet with a dominant frequency of
30 Hz. Its maximum peak is placed at the calculated two-way traveltime which
yields an anti-causal source signal. Noise is added to obtain a S/N ratio of 10
within the simulated 2D ZO section of the Initial Fresnel CRS stack (see Figure
3.44). The wiggle plot to the right is a zoomed part of the simulated ZO section
which shows the triplication. The CRS stack was performed with a near-surface
velocity of 1¤5 km/s which is the velocity of the first layer, so that no correction
of the CRS attributes is necessary. Figure 3.45 shows the CRS attribute
xm�
section
of RNIP. I omitted to display the other attribute sections (as they have not to be
visualized necessarily).
The simulated ZO section is cut off at 0¤5 km and 6 km because the
redundance of the synthetic multi-coverage is too low for the CRS stacking procedure
to yield reliable results out of the whole region. The original extracted
CRS attributes along the picked primary reflection events (green lines in Figure
3.44) are displayed in Figures 3.46 (£), 3.47 (RNIP), and 3.48 (1¨RN) with the same
colors for the different interfaces as in the previous section.
xm¡ xm¡
Figures 3.49 and 3.50 depict the results of the smoothing with the arithmetic mean
and the RLW regression, respectively. Here, the RLW regression which cannot
optimally handle too steeply decreasing functions has corrupted the angle values
71

Chapter 3. Synthetic data examples
time [s]
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
1 2
Distance [km]
3 4 5 6
time [s]
2.0
2.1
2.2
2.3
2.4
2.5
2.6
Distance [km]
2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5
Figure 3.44: Simulated ZO section with S/N ratio of 10 from the initial Fresnel
CRS stack. Negative amplitudes are blue, positive amplitudes are red and picked
traveltimes are displayed as green lines. The S/N ratio can be observed in the
small window to the right.
72
time [s]
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
1 2
Distance [km]
3 4 5 6
Figure 3.45: Section of CRS attribute RNIP. The colorbar depicts RNIP [km].
6
4
2
0
-2
-4
-6

angle [degree]
30
20
10
0
-10
-20
-30
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
Distance [km]
3.2 Model B
Figure 3.46: The extracted original CRS attributes of the emergence angle£.
Rnip [km]
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
Distance [km]
Figure 3.47: The extracted original CRS radii of curvature of the NIP wavefront
RNIP.
73

Chapter 3. Synthetic data examples
1/Rn [1/km]
6
4
2
0
-2
-4
-6
-8
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
Distance [km]
Figure 3.48: The reciprocal value of the extracted original CRS radii of curvature
of the normal wavefront RN.
of the third interface. The next two Figures 3.51 and 3.52 are the smoothing results
of RNIP. Both smoothing algorithms were not able to totally remove all outliers
especially from the triplication. The two Figures 3.53 and 3.54 show the reciprocal
value of the result for smoothing RN. There, the arithmetic mean inserted more
outliers than it has removed due to first smoothing in time-direction.
In the following, I will emphasize the differences between the four inversion algorithms
in handling the values belonging to the triplication. Here, I focus on the
inversion of the triplication and structures beneath the triplication.
74

angle [degree]
30
20
10
0
-10
-20
-30
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
Distance [km]
3.2 Model B
Figure 3.49: CRS attribute£smoothed with arithmetic mean with seven samples
window length in time- and in x-direction.
angle [degree]
30
20
10
0
-10
-20
-30
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
Distance [km]
Figure 3.50: CRS attribute£ smoothed with RLW regression with f=0.2 which
represents 35 samples window length in x-direction.
75

Chapter 3. Synthetic data examples
Rnip [m]
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
Distance [km]
Figure 3.51: CRS attribute RNIP smoothed with arithmetic mean with seven samples
window length in time- and in x-direction.
Rnip [m]
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
Distance [km]
Figure 3.52: CRS attribute RNIP smoothed with RLW regression with f=0.2 which
represents 35 samples window length in x-direction.
76

1/Rn [1/km]
6
4
2
0
-2
-4
-6
-8
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
Distance [km]
3.2 Model B
Figure 3.53: CRS attribute 1¨RN smoothed with arithmetic mean with seven samples
window length in time- and in x-direction.
1/Rn [1/km]
6
4
2
0
-2
-4
-6
-8
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
Distance [km]
Figure 3.54: CRS attribute 1¨RN smoothed with RLW regression with f=0.2 which
represents 35 samples window length in x-direction.
77

Chapter 3. Synthetic data examples
z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
1.491
1.775
1.978
2.741
3.000
1
2 2
3
−3.5
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
x [km]
Figure 3.55: Velocity model from Dix inversion and attributes smoothed with
arithmetic mean. The constant velocities of the layers are depicted by the red
numbers within the layers in [km/s].
3.2.1 Dix inversion
The Dix inversion is a primitive transform of traveltimes into depth points. It only
uses one velocity, i. e., the NMO velocity obtained by Equation (2.13) with£¡0 to
find the corresponding depth points with respect to Equation (2.30) and interval
velocities by Equation (2.28). Therefore, it does not even try to unwrap the bow
tie structure of the triplication. The Dix inversion treats the additional values of
the second interface with the same x-location as new reflection events. Thus, the
bow tie structure is imaged to the depth section of the iso-velocity model. This
is emphasized by the green numbers in Figures 3.55 and 3.56, where ’1’ denotes
the “first” interface part, i. e., the lowest two-way traveltimes of the second reflection
event with the constant layer velocity depicted to the left. ’2’ is the second
inverted part of the interface with a different layer velocity obtained by the arithmetic
mean of values from the second part of the second inverted reflection event.
The last part is marked by a ’3’ and the layer velocity is again the arithmetic mean
of values belonging only to this part.
The velocity differences of the third part of the second inverted interface are that
large because they depend on the differences of the second part. The great angles
of rays which impinge at the flanks of the syncline structure and belong to the
second part are not considered for the calculation which yields too high velocities.
This also results in too deeply placed depth points of the third part.
Looking at the depth differences of the third and fourth inverted interface be-
78

z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
1.491
1.777
1.969
2.751
3.000
1
2 2
3
−3.5
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
x [km]
3.2 Model B
Figure 3.56: Velocity model from Dix inversion and attributes smoothed with
RLW regression. The constant velocities of the layers are depicted by the red
numbers within the layers in [km/s].
neath the triplication (Figures 3.57 and 3.58), the RLW regression has resulted in a
slightly better depth positioning than the results with arithmetic mean smoothed
input. For the velocity differences of the third interface, also the RLW regression
should be preferred because of its smaller differences. Observing the parts besides
the triplication, no significant difference according to the two smoothing
algorithms occurred as their results of smoothing the input attribute RNIP show
only slight differences.
3.2.2 Plane inversion
The plane inversion starts with unwrapping the bow tie structure of the triplication
because it back-propagates the rays down to their “true” end points. However,
it still assumes planar interfaces and projects these end points to the vertical
depth lines as the final depth points of the inverted interfaces. The current implementation
does not distinguish if an input value belongs to the same reflection
event and has the same x-values but different traveltimes. Thus, those values are
handled as new reflection events which are back-propagated through the current
temporary model.
In Figures 3.61 and 3.62, only two parts of the triplication are displayed. This is
caused by limiting the algorithm to the restriction of the assumptions made for
the model, i. e., if the angle within the current calculated layer is larger than 45�
then this back-propagated ray is neglected. Thus, all rays of the second part of
79

Chapter 3. Synthetic data examples
dz [m]
300
200
100
0
−100
−200
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
x [km]
Figure 3.57: Differences in depth from Dix inversion result for each inverted
depth point and in the same colors as the attributes for each interface smoothed
with arithmetic mean.
dz [m]
300
200
100
0
−100
−200
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
x [km]
Figure 3.58: Differences in depth from Dix inversion result for each inverted
depth point and in the same colors as the attributes for each interface smoothed
with RLW regression.
80

dv [m/s]
2000
1500
1000
500
0
−500
−1000
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
x [km]
3.2 Model B
Figure 3.59: Velocity differences from Dix inversion result for each interface, attributes
smoothed with arithmetic mean. The fluctuating lines are the velocity
differences for every inverted point while the horizontal lines of the same color
depict the mean velocity difference of the whole constant velocity layer: Δv1¡ 9
m/s, Δv2¡ 25 m/s, Δv3¡ 22 m/s, Δv4¡ 241 m/s.
dv [m/s]
2000
1500
1000
500
0
−500
−1000
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
x [km]
Figure 3.60: Velocity differences from Dix inversion result for each interface, attributes
smoothed with RLW regression. The fluctuating lines are the velocity
differences for every inverted point while the horizontal lines of the same color
depict the mean velocity difference of the whole constant velocity layer: Δv1¡ 9
m/s, Δv2¡ 23 m/s, Δv3¡ 31 m/s, Δv4¡ 251 m/s.
81

Chapter 3. Synthetic data examples
z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
1.491
1.771
1.979
2.616
3.000
1
2 2
−4
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
x [km]
Figure 3.61: Velocity model from plane inversion and attributes smoothed with
arithmetic mean. The constant velocities of the layers are depicted by the red
numbers within the layers.
z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
1.491
1.773
1.971
2.591
3.000
1
2 2
−4
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
x [km]
Figure 3.62: Velocity model from plane inversion and attributes smoothed with
RLW regression. The constant velocities of the layers are depicted by the red
numbers within the layers.
82

dz [m]
800
600
400
200
0
−200
−400
−600
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
x [km]
3.2 Model B
Figure 3.63: Differences in depth from plane inversion result and attributes
smoothed with arithmetic mean for each inverted depth point and separately for
each interface.
dz [m]
600
400
200
0
−200
−400
−600
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
x [km]
Figure 3.64: Differences in depth from plane inversion result and attributes
smoothed with RLW regression for each inverted depth point and separately for
each interface.
83

Chapter 3. Synthetic data examples
dv [m/s]
2500
2000
1500
1000
500
0
−500
−1000
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
x [km]
Figure 3.65: Velocity differences from plane inversion result and attributes
smoothed with arithmetic mean separately for each interface. The blue lines are
the velocity differences for every inverted point while the red lines depict the
mean velocity difference of the whole constant velocity layer: Δv1¡ 9 m/s,
Δv2¡ 29 m/s, Δv3¡ 21 m/s, Δv4¡ 116 m/s.
dv [m/s]
2500
2000
1500
1000
500
0
−500
−1000
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
x [km]
Figure 3.66: Velocity differences from plane inversion result and attributes
smoothed with RLW regression separately for each interface. The blue lines are
the velocity differences for every inverted point while the red lines depict the
mean velocity difference of the whole constant velocity layer: Δv1¡ 9 m/s,
Δv2¡ 27 m/s, Δv3¡ 29 m/s, Δv4¡ 91 m/s.
84

z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
1.491
1.773
1.941
2.433
3.000
−4
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
x [km]
3.2 Model B
Figure 3.67: Velocity model from circular inversion and attributes smoothed with
arithmetic mean. The constant velocities of the layers are depicted by the red
numbers within the layers.
the second reflection event are omitted and the third part is here denoted with
the green ’2’.
The difference curves of the layer velocities of Figures 3.59 and 3.60 contain some
small constant parts, i. e., parts parallel to the x-axis. There, the algorithm has
again omitted rays because their angle in the current layer is larger than 45�or
these rays do not even reach the previous inverted interface due to the backpropagation
of their attributes. To be able to calculate the differences, I simply
inserted the same value of the left end of those gaps which is certainly not the best
choice. Interpolation can be a better choice but it cannot hide that the algorithm
cannot fulfill the model assumptions in these areas. Hence, the plane inversion is
in the current implementation not applicable to solve triplications.
The restriction that the temporary planar interfaces should not intersect before it
intersect their associated vertical depth lines is also not fulfilled for the third and
2¤8
fourth reflection event near xm¡
km. There, the points of the fourth interface
are placed at lower depths than those of the third interface which is depicted by
points within a constantly colored area (see Figures 3.55 and 3.56).
3.2.3 Circular inversion
Like the plane inversion, the circular inversion projects the inverted end points to
the vertical depth line but with the use of RN which describes the local curvature
85

Chapter 3. Synthetic data examples
z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
1.491
1.769
1.947
2.564
3.000
−4
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
x [km]
Figure 3.68: Velocity model from circular inversion and attributes smoothed with
RLW regression. The constant velocities of the layers are depicted by the red
numbers within the layers.
of the interface. This does not imply that an interface point is always obtained.
There are two conditions that have to be fulfilled: (i) If the horizontal distance
from the center of the local interface circle related to the current investigated ray
end point to the vertical depth line is larger than the radius of this circle then no
intersection with the vertical depth line exists. Thus, these attribute triplets are
omitted for further calculations within the temporary velocity model. (ii) Consider
the local circular interface intersects the vertical depth line. This yields, in
general, two depth points. Then, the desired depth point must be at larger depth
than the depth point of the previous inverted intersection point. Thus, the order
of reflection events along the vertical depth line is not permuted. Otherwise this
attribute triplet is also not taken into account. If both intersection points are beneath
the previous inverted intersection point, then it depends on the sign of the
back-propagated R�N . If R�N is positive, the intersection point at larger depth has to
be used as the desired depth point and for further calculations of the temporary
velocity model. If R�N is negative, the shallower intersection point is the desired
depth point.
Theoretically, the curvature of a turning point is zero (Bronstein and Semendjajew,
1996). Thus, the reciprocal value of the curvature has to be equal to� . Each
(the left and the right) flank of the syncline has a turning point which implies that
the interface curvature has a zero crossing and the radius of interface curvature
86

dz [m]
600
500
400
300
200
100
0
−100
−200
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
x [km]
3.2 Model B
Figure 3.69: Differences in depth from circular inversion result and attributes
smoothed with arithmetic mean for each inverted depth point and separately for
each interface.
dz [m]
600
400
200
0
−200
−400
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
x [km]
Figure 3.70: Differences in depth from circular inversion result and attributes
smoothed with RLW regression for each inverted depth point and separately for
each interface.
87

Chapter 3. Synthetic data examples
dv [m/s]
2000
1500
1000
500
0
−500
−1000
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
x [km]
Figure 3.71: Velocity differences from circular inversion result and attributes
smoothed with arithmetic mean separately for each interface. The blue lines are
the velocity differences for every inverted point while the red lines depict the
mean velocity difference of the whole constant velocity layer: Δv1¡ 9 m/s,
Δv2¡ 27 m/s, Δv3¡ 59 m/s, Δv4¡ 67 m/s.
dv [m/s]
2000
1500
1000
500
0
−500
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
x [km]
Figure 3.72: Velocity differences from circular inversion result and attributes
smoothed with RLW regression separately for each interface. The blue lines are
the velocity differences for every inverted point while the red lines depict the
mean velocity difference of the whole constant velocity layer: Δv1¡ 9 m/s,
Δv2¡ 31 m/s, Δv3¡ 53 m/s, Δv4¡ 64 m/s.
88

z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
1.491
1.774
2.043
2.467
3.000
1 2 3 4 5 6
x [km]
3.2 Model B
Figure 3.73: Horizon inversion result shown as thick black line. Attributes
smoothed with arithmetic mean.
has to change its sign passing� by at such a point.
In contrast to the theory, it is obvious for the second inverted interface in Figures
3.67 and 3.68 that the circular inversion was not able to obtain intersection points
at the vertical depth line for nearly all values related to the triplication. The in-
,
verted flanks of the syncline tend more and more to �
that they exceed the
threshold of the inversion algorithm before the turning point is reached. Thus,
the inversion algorithm omitted those values and investigates the next reflection
event. For the values of the fourth reflection event, the Inversion algorithm also
failed in a larger area around the triplication. The obtained iso-velocity layer
models from both kinds of smoothing of the input are not very close to the original
model. Here, the RLW regression does not show a significant enhancement
of the final result.
3.2.4 Horizon inversion
The horizon inversion does not have to distinguish in which order the parts of
the triplication have to be back-propagated. As it is a layer-stripping method, the
horizon inversion uses directly the end points of followed up rays to construct the
inverted interface. Thus, the obtained positions are distributed along the “true”
interface which is assumed to have only one depth value for every x-location, i. e.,
a unique function of the midpoint coordinate xm.
The parts of the second reflection event are not treated as new reflections as the
other three inversion algorithms did. Their attributes£and RNIP are used to back-
89

Chapter 3. Synthetic data examples
z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
1.491
1.780
2.022
2.495
3.000
1 2 3
x [km]
4 5 6
Figure 3.74: Horizon inversion result shown as thick black line. Attributes
smoothed with RLW regression.
propagate all corresponding rays even when their starting point is the same for
some rays. Thus, the bow tie structure is unwrapped and yields the illuminated
points of the syncline structure (see Figures 3.73 and 3.74).
The gaps, where no ZO ray has emerged from the inverted interfaces, are closed
by the approximation of the whole interface. Especially beneath the syncline
structure, the subsequently inverted reflection points yield an approximated interface
close to those of the original model. But at the right end of the second
interface inverted from the input smoothed with the RLW regression (see Figure
3.74), many rays are omitted due to their attributes. Thus, also the rays of the
third and fourth reflection event that do not intersect the previous approximated
interface are also omitted even if they could be used to construct the next interface.
Despite that the arithmetic mean smoothed input yields more inverted depth
points, also their differences in depth and velocity to the synthetic values are
at many points smaller than the differences of the inversion of RLW regression
smoothed input. Thus, the smoothing has to be applied carefully, so that in the
end the inversion is able to optimally reconstruct the original model.
The depth differences and velocity differences are smaller compared to the differences
of the Dix inversion. I will not compare them with the results of the plane
or circular inversion because these algorithms broke down for the inversion of
the triplication.
90

dz [m]
100
80
60
40
20
0
−20
−40
−60
−80
−100
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
x [km]
3.2 Model B
Figure 3.75: Differences in depth from horizon inversion result and attributes
smoothed with arithmetic mean for each inverted depth point and separately for
each interface.
dz [m]
100
50
0
−50
−100
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
x [km]
Figure 3.76: Differences in depth from horizon inversion result and attributes
smoothed with RLW regression for each inverted depth point and separately for
each interface.
91

Chapter 3. Synthetic data examples
dv [m/s]
600
400
200
0
−200
−400
−600
−800
−1000
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
x [km]
Figure 3.77: Velocity differences from horizon inversion result and attributes
smoothed with arithmetic mean separately for each interface. The blue lines are
the velocity differences for every inverted point while the red lines depict the
mean velocity difference of the whole constant velocity layer: Δv1¡ 9 m/s,
Δv2¡ 26 m/s, Δv3¡ 43 m/s, Δv4¡ 33 m/s.
dv [m/s]
1000
500
0
−500
−1000
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
x [km]
Figure 3.78: Velocity differences from horizon inversion result and attributes
smoothed with RLW regression separately for each interface. The blue lines are
the velocity differences for every inverted point while the red lines depict the
mean velocity difference of the whole constant velocity layer: Δv1¡ 9 m/s,
Δv2¡ 20 m/s, Δv3¡ 22 m/s, Δv4¡ 5 m/s.
92

3.2 Model B
To summarize the results for this model, it has to be emphasized that the plane
and circular inversion algorithms in their current implementation are only applicable
for reflection events without triplications. The Dix inversion should also
not be used to invert triplications because it handles those parts as new reflection
events. Nevertheless, the Dix inversion yields a more or less usable iso-velocity
model for a subsequent migration. Only the horizon inversion was able to unwrap
the bow tie structure of the reflection event due to the syncline structure
of the second interface of the original model. Therefore, the horizon inversion
should be preferred to obtain an iso-velocity model from events with triplications.
The choice of which algorithm should be used to smooth the input can be
made dependent on the area of interest, i. e., the target zone.
93

Chapter 4
Real data examples
To compare the different inversion methods with each other not only synthetic
data sets of more or less complexity are used. In this chapter, a real data set,
kindly provided by the oil and gas company BEB, is the main target for the inversion
processes.
In the introduction of Chapter 3, I mentioned that picking the ZO two-way traveltimes
at the maximum amplitudes yields an error that will shift the inverted
depth points. For the synthetic data, this error depends mainly on the sampling
rate if the source wavelet is known. But for real data, the source wavelet is in
most cases unknown (e. g., source wavelet of an explosion).
The “true” ZO two-way traveltime of real data will not be found at the maximum
absolute amplitude in most cases, but it should be situated at or before the
absolute maximum of the source wavelet which means it could be found at times
50 t�
ms in Figure 4.1(a). The source wavelet is assumed to be causal which
means that it should be a minimum-phase wavelet and not a zero-phase wavelet
as for the synthetic data. Therefore, picking traveltimes at the maximum amplitude
of one reflection event implies a small time shift not only depending on the
sampling rate but also on the shape and on the length of the source wavelet. This
time shift results in a depth shift while constructing the interfaces and could be
calculated if we once have the velocity model, as the depth shift depends only on
the velocity along the ray path and the time shift. To avoid such a depth shift,
the zero crossings before the identified maximum amplitude should be picked
which is much more difficult due to the noise and afterwards a one-sided window
should be used. The shape of the source wavelet is assumed to be symmetrical
to its maximum, so that I can use symmetrical windows in time-direction for
smoothing the attributes. This choice is a kind of limitation, but it seems to be satisfied
in most cases, also for real data (see small wiggle plot part in Figure 4.1(b)).
Even if the source wavelet is not symmetrical to its maximum amplitude, it will
have an area around the maximum within that the CRS attributes are close to
each other and different to the values belonging to the noise. The window length
95

Chapter 4. Real data examples
amplitude normalized to 1
time t in [ms]
(a)
time [s]
1.65
1.70
1.75
1.80
1.85
1.90
1.95
6.80 6.85
Distance [km]
6.90 6.95
Figure 4.1: (a) The minimum-phase Ricker wavelet with a dominant frequency
of 15 Hz. It is the minimum-phase equivalent to Figure 3.1. The red dots denote
amplitude values without noise for a sampling rate of 4 ms. (b) Small window of
the simulated ZO section obtained from the real data set shown as wiggle plot.
The green line denotes a picked event.
in time-direction is chosen by accounting for the dominant frequency of the real
data set and its sampling rate. It should be rather too small than too large, so that
attributes of the surrounding noise will not affect the smoothing.
I divided the whole data set into three parts that are displayed in Figure 4.2. In the
following, I will describe the obtained results from the four inversion methods using
input smoothed with the median and the RLW regression, respectively. Here,
I used the median instead of the arithmetic mean because it yields a smoother
result even with a window length smaller or equal to the median. With the arithmetic
mean smoothed attributes, a stable inversion was not possible due to their
strong fluctuations.
4.1 Target range 1
This rather simple region at the left part of the whole real data set is chosen firstly
to see if the inversion methods are able to handle real data at all. There, the observable
reflection events imply that the expected subsurface structure contains
not too steeply dipping and not too strongly curved interfaces.
I picked ten (most likely primary) reflection events between midpoint coordinates
2¤8 km and 9¤0 km and two-way traveltimes 0¤2 s and 3¤3 s to t¡ t¡ xm¡ xm¡
96
(b)

target range 1 target range 3 target range 2
Distance [km]
28
26
24
22
20
18
16
14
12
10
8
6
4
2
4.1 Target range 1
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
time [s]
Figure 4.2: Simulated ZO section from the initial Fresnel CRS stack. Negative
amplitudes are blue, positive amplitudes are red and picked traveltimes are displayed
as green lines. Target range 1 is the part located between the red lines,
target range 2 between the blue lines and target range 3 between the yellow lines.
97

Chapter 4. Real data examples
z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
−4
−4.5
−5
3 4 5 6 7 8 9
x [km]
Figure 4.3: Provided interval velocity model within the boundaries of the inverted
velocity models. The dotted lines depict iso-velocity lines, i. e., lines of
the same velocity. The velocity difference between two neighboring iso-velocity
lines is 50 m/s.
obtain a high resolution for the inverted iso-velocity model. In this area, all identified
events spread over the whole horizontal target range, so that there should
be no gaps along the inverted interfaces. To determine whether the identified reflection
events are primary reflections or not, I make use of the stacking velocity
section obtained from the automatic CMP stack during the CRS stack procedure.
If I identified a multiple then the stacking velocity should be smaller than for
neighboring events.
An iso-velocity model (Figure 4.3) is also provided by BEB based on the CMP
stack method (see Section 2.1.2). This model is obtained by calculating interval
velocities from several picked stacking velocities and interpolating between
them.
4.1.1 Dix inversion
The Dix inversion of these ten identified primary reflection events results in Figures
4.4 and 4.5. Compared with the provided velocity model, the Dix inversion
has computed nearly the same velocity model. It is not exactly the same
because the Dix inversion computes constant velocity layers out of the inverted
temporary velocity models, so that it does account for the lateral changes only
with a mean velocity for the whole layer within the inverted x-range. To obtain a
98

z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
−4
−4.5
−5
1.837
2.289
2.599
2.601
2.535
2.687
3.687
3.391
4.000
1.728
2.483
3 4 5 6
x [km]
7 8 9
4.1 Target range 1
Figure 4.4: Dix inversion result of target range 1 from CRS attributes smoothed
with the median and a total window length of 21 samples.
model with laterally varying velocities, the layer velocity can be calculated within
smaller windows in the future. Or to use the current implementation of the Dix
algorithm, the velocity model can be calculated for parts of the whole identified
reflection event and afterwards merged together. However, this procedure of dividing
the input and merging the results can only work for parts with the same
picked reflection events. Otherwise, if the number of picked events varies then
the continuity of reflection events that belong to the same reflector is not maintained.
The stabilizing effect of the RLW regression for the inversion process becomes
again obvious if I compare Figures 4.4 and 4.5: The fluctuations along the interfaces
in the result of the RLW regression smoothed input are smaller than those
in the result obtained by the input smoothed with the median.
4.1.2 Plane inversion
The plane inversion fails to invert more than the first six identified reflection
events because the angles of ray segments within deeper layers to the right side of
this target exceed the “small-dip” assumption. Also for the input smoothed with
the RLW regression, the inverted interfaces do not fluctuate much from trace to
trace, but the gaps remain the same (see Figures 4.6 and 4.7).
Thus, the plane inversion is over all only applicable within a small depth zone
of this target range. The RLW regression can only help to find smoother inverted
99

Chapter 4. Real data examples
z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
−4
−4.5
−5
1.858
2.282
2.593
2.582
2.589
2.673
3.711
3.387
4.000
1.660
2.559
3 4 5 6
x [km]
7 8 9
Figure 4.5: Dix inversion result of target range 1 from CRS attributes smoothed
with RLW regression with 0¤2 and four iterations.
f¡
z [km]
0
−1
−2
−3
−4
−5
−6
1.837
2.287
2.583
2.532
2.515
2.761
3.612
3.308
1.728
2.454
4.000
3 4 5 6
x [km]
7 8 9
Figure 4.6: Plane inversion result of target range 1 from CRS attributes smoothed
with the median and a total window length of 21 samples.
interfaces. The obtained layer velocities in the upper part are close to those of the
Dix inversion or those of the provided interval velocity model computed from
stacking velocities obtained by several CMP stacks.
100

z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
−4
−4.5
−5
1.858
2.281
2.574
2.542
2.552
2.720
2.647
4.000
1.660
2.533
3.313
3 4 5 6
x [km]
7 8 9
4.1 Target range 1
Figure 4.7: Plane inversion result of target range 1 from CRS attributes smoothed
with RLW regression with 0¤2and four iterations.
f¡
4.1.3 Circular inversion
The circular inversion is able to provide an inverted iso-velocity model for all
given reflection events. In contrast to the plane inversion, the construction of the
shallower interfaces yield smaller angles for the ray segments within subsequent
layers, so that these rays still comply with the model assumptions.
Figure 4.8 shows the obtained result from the median smoothed input. The strong
fluctuations to the right are mainly depending on the angles and the RN-values.
With the RLW regression, these fluctuations are decreased, so that an adequate
iso-velocity model is obtained. The velocity and depth variations compared with
the result of the Dix inversion are small.
4.1.4 Horizon inversion
Looking at the results of the horizon inversion (Figures 4.10 and 4.11), it is obvious
that the obtained velocity model is prolonged to the right due to the interfaces
dipping to the left. The interfaces from the median smoothed input show in
the middle stronger variation than the interfaces from the RLW regression which
depends on the approximation of the computed depth points to construct the
interface for the subsequent interfaces.
That the horizon inversion approximates the interfaces around the “true” depth
points, i. e., where the rays are back-propagated to, yields gaps at the borders of
the input target range. To obtain an iso-velocity model for at least the whole target
101

Chapter 4. Real data examples
z [km]
0
−1
−2
−3
−4
−5
−6
−7
−8
−9
3 4 5 6
x [km]
7 8 9
Figure 4.8: Circular inversion result of target range 1 from CRS attributes
smoothed with the median and a total window length of 21 samples.
z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
−4
−4.5
−5
1.858
2.290
2.617
2.626
2.662
2.785
3.903
3.537
4.000
1.660
2.561
3 4 5 6
x [km]
7 8 9
Figure 4.9: Circular inversion result of target range 1 from CRS attributes
smoothed with RLW regression with 0¤2 and four iterations.
f¡
102

z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
−4
−4.5
−5
1.725
2.154
2.403
2.432
2.353
2.477
3.412
2.974
4.000
1.650
2.380
3 4 5 6 7 8 9
x [km]
4.1 Target range 1
Figure 4.10: Horizon inversion result of target range 1 from CRS attributes
smoothed with the median and a total window length of 21 samples.
z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
−4
−4.5
−5
1.745
2.154
2.405
2.415
2.410
2.486
3.426
3.186
4.000
1.585
2.425
3 4 5 6
x [km]
7 8 9
Figure 4.11: Horizon inversion result of target range 1 from CRS attributes
smoothed with RLW regression with 0¤2 and four iterations.
f¡
range or even for the covered x-range, I prolonged the interfaces and velocities
constantly to the borders. To omit this simple filling of the iso-velocity model with
possibly wrong velocities, one should define a smaller target zone than the input
target range which is done for the subsequent migration in most cases (targetoriented
migration).
103

Chapter 4. Real data examples
z [km]
0
−1
−2
−3
−4
−5
−6
20 21 22 23
x [km]
24 25 26
Figure 4.12: Provided interval velocity model within the boundaries of the inverted
velocity models. The dotted lines depict iso-velocity lines, i. e., lines of the
same velocity. The velocity difference between two neighboring iso-velocity lines
is 50 m/s.
4.2 Target range 2
Within this target range, an event partly overlapping another event is contained
to test the ability of the current implementation of the inversion algorithms to
handle decreasing layer thicknesses despite the restriction that the identified reflection
events should not intersect each other. But the inversion algorithms can
not handle both parts of the picked overlapping events. Thus, I omitted the left
reflection event and used only the picked events that spread till the right border
of the target range. The first picked event with the smallest two-way traveltimes
is also not taken into account because its x-range does not cover the intersection
close to the left border of this target range.
Figure 4.12 shows the provided CMP stack iso-velocity model for this target
range. The lower part contains lower velocities than the following inverted results.
But to be able to compare them, I used the same colorbar as for the inverted
velocity models displayed in the following subsections.
104

4.2.1 Dix inversion
4.2 Target range 2
Here, the Dix inversion failed for the first time because it uses only the RNIPvalues
which are here smoothed that much by the RLW regression in contrast to
the median that the eighth layer intersects with the seventh layer. But the assumptions
to apply Dix formulae exclude intersecting events. Where the subsequent
reflection events yield points above the previous computed interface
points, the algorithm fails. Thus, also for the points of all subsequent reflection
events (here, only some points of the ninth reflection event) can not be inverted
beneath these points because they directly depend on the previously calculated
interface.
Comparing both inverted velocity models with the provided model yields that
the lower part of the inverted results contain higher velocities than in the provided
model partly due to the chosen filling velocity for the half space beneath
the last inverted time horizon. But the trend of dipping to the right is also contained
in both inversion results.
Some of the picked reflection events do not spread over the whole x-range of this
target zone which can be observed in Figures 4.13 and 4.14. The results are again
constantly prolonged to the borders of the target range. This yields an error at
the left border of the fifth interface for the construction of the final iso-velocity
model. There, it is not checked whether subsequent inverted interfaces do not
intersect with shallower interfaces but can have lower depth values. Thus, the
Dix inversion is not applicable for all identified reflection events. Nevertheless,
the results are useful for a subsequent migration.
4.2.2 Plane inversion
In the middle of this target range, the angles of ray segments after the fifth interface
exceed again the assumptions made for models to which the plane inversion
is applicable. As for target range 1, only the upper part is nearly the same as
for the Dix inversion according to the velocities and the depth of the inverted
interfaces.
The RLW regression smoothed the input in such a way that here some more
points could be inverted and resulted in a better visible ninth layer (see Figure
4.16) compared to the input smoothed by the median (see Figure 4.15). But the
gaps still remain, so that in the end also only the upper part of the inverted velocity
model could be used for migration purposes.
4.2.3 Circular inversion
The result of the circular inversion with the median smoothed input (Figure 4.17)
is able to invert the values where the plane inversion failed but with very large
105

Chapter 4. Real data examples
z [km]
0
−1
−2
−3
−4
−5
−6
20 21 22 23
x [km]
24 25 26
1.850
1.840
1.995 2.738
Figure 4.13: Dix inversion result of target range 2 from CRS attributes smoothed
with median and a total window length of 41 samples.
z [km]
0
−1
−2
−3
−4
−5
−6
2.961
20 21 22 23
x [km]
24 25 26
2.479
2.897
3.329
5.091
6.000
1.864
1.821
1.982 2.676
Figure 4.14: Dix inversion result of target range 2 from CRS attributes smoothed
with RLW regression with 0¤2 and two iterations.
f¡
106
2.977
2.477
2.924
3.399
4.894
6.000

z [km]
0
−1
−2
−3
−4
−5
−6
3.851
20 21 22 23
x [km]
24 25 26
1.850
1.840
1.994 2.706
2.424
2.997
3.100
2.777
6.000
4.2 Target range 2
Figure 4.15: Plane inversion result of target range 2 from CRS attributes smoothed
with median and a total window length of 41 samples.
z [km]
0
−1
−2
−3
−4
−5
−6
1.864
1.821
1.982 2.647
2.354
3.494
2.430
3.022
3.110
20 21 22 23
x [km]
24 25
6.000
26
Figure 4.16: Plane inversion result of target range 2 from CRS attributes smoothed
with RLW regression with 0¤2and two iterations.
f¡
107

Chapter 4. Real data examples
z [km]
0
−1
−2
−3
−4
−5
−6
−7
−8
−9
−10
−11
20 21 22 23
x [km]
24 25 26
Figure 4.17: Circular inversion result of target range 2 from CRS attributes
smoothed with median and a total window length of 41 samples.
fluctuations. With the RLW regression these fluctuations decrease that much,
that the effect of the different numbers of identified reflection events per trace
becomes obvious. The seventh layer in Figure 4.18 contains steps that can not be
related with steps in the input attributes. They are at the x-positions where above
the seventh layer the number of previous inverted interfaces increased.
This effect intensifies for the inversion of subsequent reflection events, so that
here also the results of the circular inversion are not applicable for migration pro-
108

z [km]
0
−1
−2
−3
−4
−5
−6
20 21 22 23
x [km]
24 25 26
1.864
1.821
1.987 2.710
3.037
4.661
2.479
2.951
3.442
6.000
4.2 Target range 2
Figure 4.18: Circular inversion result of target range 2 from CRS attributes
smoothed with RLW regression with 0¤2 and two iterations.
f¡
cesses at depths beneath the sixth inverted interface. Compared with the Dix
inversion, the layer velocities and interface depths differ the more the deeper the
interfaces are.
4.2.4 Horizon inversion
As mentioned in Subsection 3.2.4, the horizon inversion only accounts for backpropagated
rays that intersect all shallower interfaces. Thus, interface points at
the left part of the fifth reflection event are omitted because there the inverted
ray segments do not intersect with the fourth obtained reflection event. The same
happens for a part at the left border of the seventh reflection event. Therefore, the
target zone for a subsequent migration process should be chosen smaller than
from the Dix inversion result if the filled up parts should be taken into account.
The velocity for filling the half-space beneath the last inverted interface is maybe
chosen too high but it should proceed with the increase of layer velocities with
increasing depths.
Despite of the parts of some reflection events omitted by the horizon inversion,
the results of the horizon inversion should be preferred for a subsequent migration
as they yield the most continuous results at large depths within this target
109

Chapter 4. Real data examples
z [km]
0
−1
−2
−3
−4
−5
−6
1.721
1.904
20 21 22 23
x [km]
24 25 26
Figure 4.19: Horizon inversion result of target range 2 from CRS attributes
smoothed with median and a total window length of 41 samples.
z [km]
0
−1
−2
−3
−4
−5
−6
20 21 22 23
x [km]
24 25 26
Figure 4.20: Horizon inversion result of target range 2 from CRS attributes
smoothed with RLW regression with 0¤2 and two iterations.
f¡
110
1.697
1.896
1.738
2.548
2.623
2.672
2.953
3.495
5.026
6.000
1.751
2.497
2.692
2.703
2.852
3.479
5.096
6.000

z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
12 13 14 15
x [km]
16 17 18
4.3 Target range 3
Figure 4.21: Provided velocity model within the boundaries of the inverted velocity
models. The dotted lines depict iso-velocity lines, i. e., lines of the same
velocity. The velocity difference between two neighboring iso-velocity lines is 50
m/s.
range.
4.3 Target range 3
Target range three was used to get an iso-velocity layer model for a dome-like
structure, i. e., to see if the inversion algorithms can handle focusing structures.
With the third target range, a velocity model for the whole real data set can be
obtained in future. Then, the identified reflection events that are contained in all
three or at least two target ranges should be interpolated between the target range
borders. This model could also be used to compare it with the whole provided
interval velocity model obtained by CMP stacks and not only with small parts
(Figure 4.21).
Here, I can make use of only seven of the picked reflection events because the
others could not be distinguished from the noise over a long part of the specified
target range. The noise prevented to determine whether the parts with the lowest
two-way traveltimes within this range belong to one and the same reflection
event or not. Therefore, I assume that they do not belong to one event, so that
I omitted the left part. The two picked events very close to each other are too
closely together to separate their attributes.
111

Chapter 4. Real data examples
z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
2.169
2.404
1.859 1.828
3.319
3.092
2.664
3.500
12 13 14 15
x [km]
16 17 18
Figure 4.22: Dix inversion result of target range 3 from CRS attributes smoothed
with median and a total window length of 21 samples.
z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
2.128
2.531
1.952 1.711
3.299
2.745
3.500
2.911
12 13 14 15
x [km]
16 17 18
Figure 4.23: Dix inversion result of target range 3 from CRS attributes smoothed
with RLW regression with 0¤2 and four iterations.
f¡
4.3.1 Dix inversion
The Dix inversion results (Figures 4.22 and 4.23) are close to the provided CMP
stack results (Figure 4.21) because they contain the high-velocity layer around
depths of 1 km. The median smoothed input yields the same result as the RLW
regression but with some stronger fluctuations along the inverted events. The
RLW regression just inserted an obvious error at the left end of the sixth interface
which can not be that close to the fifth interface according to the traveltime
difference of the picked reflection events.
112

z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
2.152
2.393
1.859 1.827
3.294
2.795
2.527
3.500
12 13 14 15
x [km]
16 17 18
4.3 Target range 3
Figure 4.24: Plane inversion result of target range 3 from CRS attributes smoothed
with median and a total window length of 21 samples.
z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
2.127 2.522
1.952 1.711
3.272
3.036
3.500
2.594
12 13 14 15
x [km]
16 17 18
Figure 4.25: Plane inversion result of target range 3 from CRS attributes smoothed
with RLW regression with 0¤2and four iterations.
f¡
4.3.2 Plane inversion
The same error due to the RLW regression mentioned above occurred at the plane
inversion (Figure 4.25). However, the angles for the last inverted interface exceed
the model assumptions, so that the plane inversion is only reliable above the sixth
inverted interface. There, the obtained velocity models (Figures 4.24 and 4.25) are
close to results of the Dix inversion with respect to layer velocities and interface
depths.
113

Chapter 4. Real data examples
z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
2.212
2.406
1.861 1.809
3.308
3.500
3.227
2.824
12 13 14 15
x [km]
16 17 18
Figure 4.26: Circular inversion result of target range 3 from CRS attributes
smoothed with median and a total window length of 21 samples.
z [km]
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
2.128
2.538
1.952 1.712
3.328
2.873
3.500
3.023
12 13 14 15
x [km]
16 17 18
Figure 4.27: Circular inversion result of target range 3 from CRS attributes
smoothed with RLW regression with 0¤2 and four iterations.
f¡
4.3.3 Circular inversion
The circular inversion can not invert the median smoothed input without strong
fluctuations along the interfaces (Figure 4.26). But it was able to obtain the depth
points for the seventh interface in contrast to the plane inversion.
The described error of the inversion with the RLW regression smoothed sixth interface
is still contained in the velocity model (Figure 4.27) but the fluctuations
along the interfaces are eliminated, so that the obtained iso-velocity model is
closer to the provided velocity model than those from the Dix or plane inversion
despite the high half-space velocity.
114

z [km]
0
−0.5
−1
−1.5
−2
−2.5
2.060 2.224
1.746 1.670
3.090
3.011
2.578
3.500
−3
13.5 14 14.5 15 15.5 16
x [km]
16.5 17 17.5 18 18.5
4.3 Target range 3
Figure 4.28: Horizon inversion result of target range 3 from CRS attributes
smoothed with median and a total window length of 21 samples.
z [km]
0
−0.5
−1
−1.5
−2
−2.5
2.019 2.275
1.834 1.549
3.084
2.787
2.637
3.500
−3
13.5 14 14.5 15 15.5 16
x [km]
16.5 17 17.5 18 18.5
Figure 4.29: Horizon inversion result of target range 3 from CRS attributes
smoothed with RLW regression with 0¤2 and four iterations.
f¡
4.3.4 Horizon inversion
The horizon inversion limited the range of the inverted interfaces to the x-range
of the first identified reflection event. Thus, the focusing effect of the expected
structure is clearly visualized only at the right border (see Figures 4.28 and 4.29).
For the sixth interface, the error of the RLW regression as observed from the other
inversion method is decreased due to the approximation of the interface points.
Looking at the seventh inverted interface, the RLW regression smoothing yields
not as large depths variations as that from the median smoothing. Compared
with the provided interval velocity model, the velocity differences are smaller
except for the half plane than those of the other inverted velocity models to the
provided interval velocity model.
115

Chapter 4. Real data examples
To summarize the results obtained from the application of CRS attributes obtained
from real data for the inversion of identified primary reflection events, the
same order of preference as in Chapter 3 should be applied. This means that the
plane and circular inversion algorithms should better not be preferred or even
used as they are not always able to yield a reliable inverted iso-velocity model
or even a continuous inverted interface. The Dix inversion is simple and stable
but it cannot handle intersecting events which is one main interest according to
real data. The horizon inversion in its current implementation can also not handle
intersecting events but it has the potential to solve this problem in future.
According to the smoothing algorithm, one still should be careful with the chosen
parameters as even the RLW regression cannot eliminate all errors. Thus, an
interaction with the user is supposed to yield an optimum enhanced inversion
result.
A future task can be to merge the inverted velocity models of the three target
ranges to finally obtain a macro-velocity model for the whole real data set and to
perform a subsequent migration with this model.
116

Chapter 5
Application examples
What can the results of the inversion be used for? Firstly, the inverted iso-velocity
layer models have to be transformed into macro-velocity models. This transformation
is usually performed by a Backus averaging (Backus, 1962). The Backus
averaging means to smooth the squared slowness instead of the velocity itself.
The slowness p is defined as the reciprocal value of the velocity. The smoothing
of the squared slowness is performed as a convolution with a triangle in
x-direction and z-direction independently from each other. Then, the smoothed
squared slowness section is transformed back to obtain the smoothed iso-velocity
model, i. e., the macro-velocity model. In Figure 5.1, I plotted the simulated ZO
section of the Initial CRS Fresnel stack with amplitudes leveled by an automatic
gain control (AGC) with a window length of 1 s. In this figure, the amplitudes
at high two-way traveltimes are exaggerated and the amplitudes at low two-way
traveltimes are understated. The green lines are the picked reflection events that
are used to compute the iso-velocity model of Figure 5.2. To obtain this velocity
model, the input for the inversion was smoothed by the RLW regression with
0¤2
parameter f¡
and two iterations. For the subsequent Backus averaging, I
used a window length of 41 samples in both (x- and z-) directions. The resulting
macro-velocity is shown in Figure 5.3 and the provided macro-velocity model of
this part is shown in Figure 5.4.
Secondly, a post-stack depth migration is applied to the data set based on those
macro-velocity models. For this purpose, I used a Kirchhoff depth migration that
obtains the amplitudes for each depth point of the chosen depth target zone by
stacking all amplitudes along the Huygens curve in the ZO section. The Huygens
curves are calculated with the help of Greens function tables (GFT) which
are obtained by raytracing within the macro-velocity model. After the migration
process several changes are obvious. One is that the diffraction traveltime curve
in the time domain shrinks to a point in the depth domain which is unfortunately
not contained in this example. Another fact is, that the reflection events of the
time domain are not only placed to their corresponding vertical depth, they are
117

Chapter 5. Application examples
time [s]
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
20 21 22
Distance [km]
23 24 25 26
Figure 5.1: Simulated ZO section of target range 2 plotted after an AGC with an
1 s window length. The green and grey lines denote picked events, where the
green are used for the inversion.
0 1 2 3 4 5 6
z [km]
x [km]
20 21 22 23 24 25 26
Figure 5.2: The iso-velocity model obtained by the horizon inversion. The picked
reflection events are smoothed by means of the RLW regression.
118
1.5 2.5 3.5 4.5 5.5
v [km/s]

0 1 2 3 4 5 6
z [km]
x [km]
20 21 22 23 24 25 26
Figure 5.3: The macro-velocity model which is the iso-velocity model smoothed
by means of Backus averaging with a window length of 41 samples in x- and
z-direction.
0 1 2 3 4 5 6
z [km]
x [km]
20 21 22 23 24 25 26
Figure 5.4: The macro-velocity model subset of the provided interval velocity
model for target range 2.
119

Chapter 5. Application examples
1 2 3 4 5
z [km]
x [km]
20 21 22 23 24 25 26
Figure 5.5: Result of a Kirchhoff-type migration based on the macro-velocity
model obtained by the horizon inversion.
1 2 3 4 5
z [km]
x [km]
20 21 22 23 24 25 26
Figure 5.6: Result of the same Kirchhoff-type migration but now with the provided
macro-velocity model.
also laterally moved to their “true” x-coordinates.
Looking at the bottom of both migrated images, it is obvious that the reflection
event around 3¤7 s in Figure 5.1 is in Figure 5.5 placed deeper than in Figure
5.6 because of the higher velocities at the bottom of the corresponding macrovelocity
model. The second effect of the migration mentioned above becomes
obvious for the reflection event at the left border around 3¤2 s in Figure 5.1.
Its image is moved to larger x-locations as in the time domain.
t¡
t¡
120

The depth image is easier to interpret than the simulated ZO sections because
diffraction curves are not contained anymore and cannot be misinterpreted. Diffractors
and reflection events are now at their “true” depth locations. E. g. geologists
can use this image to tell oil companies where they can find hydrocarbon reservoirs.
This is the main target of reflection seismics, to provide an easily interpretable
image of the subsurface. However, I cannot decide which depth image
is closer to the true subsurface structure. I can only emphasize that I used more
information out of the multi-coverage data set than for the velocity model obtained
by CMP stacks to finally yield the shown depth image. Thus, it is in the
responsibility of the interpreter to decide whether the presented inversion methods
yield improved velocity models in contrast to conventional methods. Therefore,
the velocity models can be compared with existing velocity models from
well logs if there exists at least one bore-hole in the vicinity of the seismic line.
Those well logs can also be used to constrain the inversion results during their
computation that I propose for future improvements of the inversion methods.
The informations obtained from other geophysical methods as, e. g., seismological
tomography can provide additional insight to the subsurface. The resulting
depth images or velocity models can be used to compare them with the presented
results.
121

Chapter 6
Conclusions
I compared the results of four different inversion algorithms and their inputs
smoothed with two different algorithms with the synthetically generated velocity
models. The different inversion algorithms from the Dix inversion to the horizon
inversion make use of more input data and less restrictive assumptions step by
step, i. e., more attributes obtained by the CRS stack and less projections of the
“true” depth points. The Dix inversion only used the CRS attribute “radius of
NIP wavefront curvature”, RNIP which can be related to the NMO velocity (see
Equation with£ (2.13) 0) and placed the corresponding depth points along
the vertical depth line assuming temporary horizontal plane layers. The plane
inversion goes one step further and uses also the CRS attribute£. With this information,
the rays can be back-propagated to their “true” location. However, the
plane inversion still calculates the intersection of planar interfaces with the vertical
depth line. Thus, the complexity of invertable structures can be expanded
from temporary horizontal interfaces for the Dix inversion to temporary plane
dipping interfaces. With the use of the CRS attribute RN, the circular inversion
¡
is able to invert temporary circular interfaces. But as for the plane inversion,
the intersection of these circular interfaces with the vertical depth line is still obtained
as output. The horizon inversion defines the last step and uses the “true”
back-propagated ray end points for calculating the inverted interface by means
of spline approximation. In the current implementation RN is not used but can be
used to check the interface curvature or to constrain the spline approximation in
future.
The different effects of the smoothing algorithms applied to the input of the inversion
becomes obvious for both presented synthetic velocity models. The RLW
regression has shown that it can eliminate trace-to-trace fluctuations better than
the arithmetic mean or other smoothing algorithms discussed in Section 2.4. The
smoother input attributes after the RLW regression resulted in a more stable inversion
process according to the depth and velocity fluctuations from trace to
trace. For the second presented synthetic model, the plane and circular inver-
123

Chapter 6. Conclusions
sion algorithms failed due to the complexity of the model. The result of the Dix
inversion is also not applicable for a subsequent migration because the bow tie
structure is still contained in the inverted velocity model. Only the horizon inversion
was able to obtain a velocity model that is close to the synthetic model. Thus,
I recommend to use the horizon inversion with the RLW regression to produce
iso-velocity models useful for migration purposes.
The inversion of real data subsets in Chapter 4 tested if the inversion is also applicable
on real data. Here, the plane, the circular, and the Dix inversion showed
their dependence on the complexity of the illuminated structures. It must be
emphasized that they also strongly depend on the degree of smoothing the input
to obtain a physical meaningful velocity model, especially for the circular
inversion. Despite those trace-by-trace inversion algorithms, the layer-stripping
horizon inversion resulted even for the median smoothed input in a useful isovelocity
model for a subsequent migration. To improve the results of the horizon
inversion, I propose to divide the inverted constant velocity layers into blocks
of constant velocity or to use slight vertical or horizontal gradients to account
more for lateral changes and for changes due to structures between the picked
reflection events that are not used for the inversion.
At last, migration results of a part of the provided velocity model and of one inverted
iso-velocity model are shown in Chapter 5 in comparison to each other.
However, it is from there on in the responsibility of the interpreter to decide
which obtained depth image fits better to the reality.
124

Appendix A
2D ZO attribute�
CRS
The 2D ZO CRS attribute of the emergence angle£represents in case of constant
velocity layers the local dip of the very first interface at R. In the uppermost layer,
the ray is a straight line due to Fermat’s principle. In the case of ZO, the ray must
emerge at the interface perpendicular to the interface itself (for planar interfaces
see Figure A.1(a) and for arbitrary curved interfaces see A.1(b)) because all back
propagated rays are normal rays. Thus, a source generated wavefront downward
propagates along the same ray as the reflected wavefront propagates upward.
The normal ray impinges at the interface in reflection point R (Figure A.1). This
yields an angle�I¡ incidence 0�which implies according to Snell’s law (Equation
2.27) even with wave-type (vI�
conversion vR). The ray related to the re- ¡
flected wavefront is identical to the ray paths along that the wavefront traveled
down to the investigated reflector to meet the ZO condition at the surface.
The CRS attribute triplets describe local interface segments that fulfill the normal
ray condition. For rays belonging to the first interface, only the transmission law,
the reflection law, and Snell’s law have to be considered. The rays reflected at
subsequent interface must fulfill also the refraction law at shallower interfaces.
Thus, the CRS attribute£�of subsequent interfaces changes along the ray path
due to the change of the P-wave velocity indicated by Snell’s law. As all CRS
attributes are measured at the surface, the local dips are only equal to£in case of
the first interface.
For deeper reflectors, the dips can also be calculated from£�but only if the velocities
of all layers passed through are a priori known, so that the effect of the
refractions at previous interfaces can be corrected. Therefore, the dip is obtained
after at least one calculation and not directly represented by a part of the input
data itself.
125

Chapter A. 2D ZO CRS attribute£
v 1
v2
v 1
v2
α’
α’
R
α
(a) plane dipping interface
R
α
(b) arbitrary curved interface
δ
δ
surface
1. reflector
surface
local tangent of
1. reflector
1. reflector
Figure A.1: CRS attribute£ of the red rays represents (a) the interface dip or (b)
the local interface dip, both for the 2D ZO case and only for the first reflector.£��
is the emergence angle for a ZO ray of a following reflector.
126

Appendix B
Correction of CRS attributes
The CRS attributes are a representation of the subsurface as described in Section
2.2. They are obtained by a search strategy that depends on a user-defined nearsurface
velocity v0. If v0 is chosen wrongly then also the found attribute triplets
are not correct. However, they are not totally wrong, but represent the same
reflection event for a different near-surface velocity v0.
The obtained CRS attributes of a reflection event which is considered to be the primary
reflection of the uppermost interface allow immediately to verify whether
the user-defined near-surface velocity was correct or not. Therefore, the following
difference Δv is analyzed given by
2RNIP�i�j�1
(B.1)
t0�i�j�1
with index i for each trace and index 1 for the considered first interface. Equa-
Δv¡
tion (B.1) is based on the fact that the thickness of the first homogeneous layer is
represented by the radius of curvature of the NIP wavefront. This is true, because
the NIP wavefront has to focus into a point per definition and this point is
the center of the circle with radius RNIP. Thus, the velocity for the first layer is
the first term on the right side of Equation (B.1) with the ZO two-way traveltime
v0©
t0�i�j�1 of the first identified
j¡
reflection event.
Δv� If 0 then the first layer velocity v1 is equal to the user-defined near-surface
velocity Δv� v0. For 0, the kinematic wavefield attributes of the CRS stack must
be corrected to obtain the “true” depth points from the inversion methods.
For this correction, I consider a hypothetical layer above the seismic line with
layer velocity v0. In this case, the layer velocity v0 is not equal to the velocity
of the first subsurface layer v1,
¡
because 0. Thus, an additional refraction at
the seismic line occurs, i. e., an additional calculation has to be performed. The
refraction law is given by Equation (2.26) which simplifies to ¡ Δv�
1
RT¡ vT
cos2�I vIRI cos2�T© (B.2a)
127

Chapter B. Correction of CRS attributes
with because the interface, i. e., the seismic line, is planar. Furthermore, I
make use of Snell’s law
RF¡��
vT sin�I
vI
and the relation of sine and cosine
sin�T¡
cos�T¡��1 sin
(B.2b)
2�T¤ (B.2c)
The next equation is used to obtain the velocity of the layer into which the ray is
refracted:
t0
d1
t2¦ t3¦ ¤¥¤¥¤¡ t1¦ 2¡
(B.2d)
with the one-way traveltime t0¨2, which is the sum of the one-way traveltimes ¤¥¤¥¤ v�3¦
within each layer. The traveltimes within one layer t j are the quotient of the
distance traveled in one layer d j and the layer velocity v�j . In case of the correction,
only two layers are considered, (j¡ i. e., the hypothetical 1, vI) and the
first (j¡ subsurface layer 2, vT). Thus, Equation (B.2d) simplifies and
is solved for the velocity of the first subsurface layer:
v1¡ v�2¡ v0¡ v�1¡
vT
d2 d3 v�2¦ v�1¦
d2
(B.2e)
t0 d1 2
and d2 is the corrected RNIP because the thickness of the hypo-
v�1© v�2¡
where
thetical layer is set to zero, 0, so that the traveltimes remain unchanged. RI
and�I are substituted with the wrong CRS attributes R�NIP v�2¡ d1¡ and£��. Thus, comparing
the coefficients of Equations (B.2a), (B.2b), (B.2c), and (B.2e) yields a common
correction factor given by
fcorr ¡
RI
R�NIP ¡
(cos2�It0vI¦ 2
2
(cos2£�t0v0¦ 2
sin 2�IRI)RI
2
sin 2£�R�NIP )R�NIP¤ (B.3)
With this factor, the corrected CRS attributes£and RNIP are obtained by
128
fcorr
1
2
RNIP¡ fcorrt0v0¤ (B.4b)
sin£ ¡
(B.4a)
sin£�©

R’
Seismic line
R
Figure B.1: Illustration of wrong CRS attributes (green and primed) and the
“true” CRS attributes after the correction.
Figure B.1 is an illustration of what can happen if the uncorrected CRS attributes
£�and R�NIP are used to find the interface point, i. e., R�.£�and R�NIP are obtained
from the CRS stack with an improper near-surface velocity v0 and are shown in
green. The “true” attributes£ CRS and RNIP depicted in red lead to the “true”
reflection point R.
To correct the CRS attribute R�N to the “true” CRS attribute RN, the refraction law
of Equation (B.2a) has to be applied again. Thus, RN is given by
1 v1 RN¡
R’ NIP
with the corrected first subsurface layer velocity given by
v1¡ fcorrv0¤
X 0
α
α ’
v
0
R NIP
v 1
cos2£� v0 cos2£R�N© (B.5a)
(B.5b)
129

Appendix C
Used hard- and software
The computations were done on a SILICON GRAPHICS ORIGIN 3200 server with
six processors and on dual-processor Linux PCs (with S.u.S.E. Linux 6.2).
The half-automatic Picker of REFLEX was used on Win NT Workstations.
The smoothing algorithms and the inversion algorithms are implemented in C
and C++, respectively, and the latter algorithms make use of the NURBS++ library
(version 3.0.10) (public domain).
The synthetic data was produced by a ray tracing program which included forward-calculation
of the kinematic wavefield attributes. This software was kindly
provided by German Höcht. The synthetic data set was produced with the software
package NORSAR.
For analytical calculations, I used Maple V Release 5.1 (Waterloo Maple). For numerical
calculations and for visualization of the inversion results, I used Matlab
6.0.0.88 Release 12.
The 2-D figures of the seismic data sets and the picked data sets were generated
with the Seismic Unix package (Center of Wave Phenomena at Colorado School
of Mines).
This thesis was written on a PC (S.u.S.E. Linux 6.2) using the freely available word
processing package TEX, the macro package L ATEX 2�, and several extensions. The
bibliography was generated with BIBTEX. Figures were mainly constructed with
Xfig 3.2 (rev2).
131

List of Figures
1.1 Four seismic data array examples . . . . . . . . . . . . . . . . . . . . 2
1.2 CMP array with plane dipping reflector . . . . . . . . . . . . . . . . 3
1.3 3D data volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 MZO operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 CMP stack example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 PreSDM operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Illustration of eigenwaves . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 CRS stack operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.6 Determination of first Fresnel zone . . . . . . . . . . . . . . . . . . . 17
2.7 Projected first Fresnel zone . . . . . . . . . . . . . . . . . . . . . . . . 18
2.8 Mean difference cut examples and weight function . . . . . . . . . . 24
2.9 Sign conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.10 Snell’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.11 Temporary interface construction (Dix inversion) . . . . . . . . . . . 33
2.12 Temporary interface construction (Plane inversion) . . . . . . . . . 35
2.13 Temporary interface construction (Circular inversion) . . . . . . . . 38
2.14 Interface construction (Horizon inversion) . . . . . . . . . . . . . . . 39
3.1 Zero-phase Ricker wavelet . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2 Synthetic velocity model A . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3 Local dips of synthetic velocity model A . . . . . . . . . . . . . . . . 44
3.4 ZO section of model A . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.5 Section of CRS attribute£ . . . . . . . . . . . . . . . . . . . . . . . . 45
3.6 Section of CRS attribute RNIP . . . . . . . . . . . . . . . . . . . . . . 46
3.7 Section of CRS attribute RN . . . . . . . . . . . . . . . . . . . . . . . 46
3.8 Extracted original CRS attribute£ . . . . . . . . . . . . . . . . . . . 47
3.9 Extracted original CRS attribute RNIP . . . . . . . . . . . . . . . . . . 47
3.10 Extracted original CRS attribute RN . . . . . . . . . . . . . . . . . . . 48
3.11 Reciprocal value of extracted original CRS attribute RN . . . . . . . 48
3.12 Arithmetic mean smoothed CRS attribute£ . . . . . . . . . . . . . . 50
3.13 RLW regression smoothed CRS attribute£ . . . . . . . . . . . . . . 50
3.14 Arithmetic mean smoothed CRS attribute RNIP . . . . . . . . . . . . 51
133

List of Figures
134
3.15 RLW regression smoothed CRS attribute RNIP . . . . . . . . . . . . . 51
3.16 Arithmetic mean smoothed CRS attribute 1¨RN . . . . . . . . . . . 52
3.17 RLW regression smoothed CRS attribute 1¨RN . . . . . . . . . . . . 52
3.18 Velocity model of Dix inversion (arithmetic mean) . . . . . . . . . . 53
3.19 Velocity model of Dix inversion (RLW regression) . . . . . . . . . . 54
3.20 Dix inversion depth differences (arithmetic mean) . . . . . . . . . . 55
3.21 Dix inversion depth differences (RLW regression) . . . . . . . . . . 55
3.22 Dix inversion velocity differences (arithmetic mean) . . . . . . . . . 56
3.23 Dix inversion velocity differences (RLW regression) . . . . . . . . . 56
3.24 Velocity model of plane inversion (arithmetic mean) . . . . . . . . . 58
3.25 Velocity model of plane inversion (RLW regression) . . . . . . . . . 59
3.26 Plane inversion depth differences (arithmetic mean) . . . . . . . . . 60
3.27 Plane inversion depth differences (RLW regression) . . . . . . . . . 60
3.28 Plane inversion velocity differences (arithmetic mean) . . . . . . . . 61
3.29 Plane inversion velocity differences (RLW regression) . . . . . . . . 61
3.30 Velocity model of circular inversion (arithmetic mean) . . . . . . . . 63
3.31 Velocity model of circular inversion (RLW regression) . . . . . . . . 63
3.32 Circular inversion depth differences (arithmetic mean) . . . . . . . 64
3.33 Circular inversion depth differences (RLW regression) . . . . . . . . 64
3.34 Circular inversion velocity differences (arithmetic mean) . . . . . . 65
3.35 Circular inversion velocity differences (RLW regression) . . . . . . 65
3.36 Velocity model of horizon inversion (arithmetic mean) . . . . . . . 66
3.37 Velocity model of horizon inversion (RLW regression) . . . . . . . . 67
3.38 Horizon inversion depth differences (arithmetic mean) . . . . . . . 68
3.39 Horizon inversion depth differences (RLW regression) . . . . . . . . 68
3.40 Horizon inversion velocity differences (arithmetic mean) . . . . . . 69
3.41 Horizon inversion velocity differences (RLW regression) . . . . . . 69
3.42 Synthetic velocity model B . . . . . . . . . . . . . . . . . . . . . . . . 70
3.43 Local dips of synthetic velocity model B . . . . . . . . . . . . . . . . 71
3.44 ZO section of model B . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.45 Section of CRS attribute RNIP . . . . . . . . . . . . . . . . . . . . . . 72
3.46 Extracted original CRS attribute£ . . . . . . . . . . . . . . . . . . . 73
3.47 Extracted original CRS attribute RNIP . . . . . . . . . . . . . . . . . . 73
3.48 Reciprocal value of extracted original CRS attribute RN . . . . . . . 74
3.49 Arithmetic mean smoothed CRS attribute£ . . . . . . . . . . . . . . 75
3.50 RLW regression smoothed CRS attribute£ . . . . . . . . . . . . . . 75
3.51 Arithmetic mean smoothed CRS attribute RNIP . . . . . . . . . . . . 76
3.52 RLW regression smoothed CRS attribute RNIP . . . . . . . . . . . . . 76
3.53 Arithmetic mean smoothed CRS attribute 1¨RN . . . . . . . . . . . 77
3.54 RLW regression smoothed CRS attribute 1¨RN . . . . . . . . . . . . 77
3.55 Velocity model of Dix inversion (arithmetic mean) . . . . . . . . . . 78
3.56 Velocity model of Dix inversion (RLW regression) . . . . . . . . . . 79

List of Figures
3.57 Dix inversion depth differences (arithmetic mean) . . . . . . . . . . 80
3.58 Dix inversion depth differences (RLW regression) . . . . . . . . . . 80
3.59 Dix inversion velocity differences (arithmetic mean) . . . . . . . . . 81
3.60 Dix inversion velocity differences (RLW regression) . . . . . . . . . 81
3.61 Velocity model of plane inversion (arithmetic mean) . . . . . . . . . 82
3.62 Velocity model of plane inversion (RLW regression) . . . . . . . . . 82
3.63 Plane inversion depth differences (arithmetic mean) . . . . . . . . . 83
3.64 Plane inversion depth differences (RLW regression) . . . . . . . . . 83
3.65 Plane inversion velocity differences (arithmetic mean) . . . . . . . . 84
3.66 Plane inversion velocity differences (RLW regression) . . . . . . . . 84
3.67 Velocity model of circular inversion (arithmetic mean) . . . . . . . . 85
3.68 Velocity model of circular inversion (RLW regression) . . . . . . . . 86
3.69 Circular inversion depth differences (arithmetic mean) . . . . . . . 87
3.70 Circular inversion depth differences (RLW regression) . . . . . . . . 87
3.71 Circular inversion velocity differences (arithmetic mean) . . . . . . 88
3.72 Circular inversion velocity differences (RLW regression) . . . . . . 88
3.73 Velocity model of horizon inversion (arithmetic mean) . . . . . . . 89
3.74 Velocity model of horizon inversion (RLW regression) . . . . . . . . 90
3.75 Horizon inversion depth differences (arithmetic mean) . . . . . . . 91
3.76 Horizon inversion depth differences (RLW regression) . . . . . . . . 91
3.77 Horizon inversion velocity differences (arithmetic mean) . . . . . . 92
3.78 Horizon inversion velocity differences (RLW regression) . . . . . . 92
4.1 Synthetic minimum-phase Ricker wavelet and real data example . 96
4.2 ZO section of real data set . . . . . . . . . . . . . . . . . . . . . . . . 97
4.3 Provided interval velocity model (target range 1) . . . . . . . . . . . 98
4.4 Dix inversion result (median, target range 1) . . . . . . . . . . . . . 99
4.5 Dix inversion result (RLW regression, target range 1) . . . . . . . . 100
4.6 Plane inversion result (median, target range 1) . . . . . . . . . . . . 100
4.7 Plane inversion result (RLW regression, target range 1) . . . . . . . 101
4.8 Circular inversion result (median, target range 1) . . . . . . . . . . . 102
4.9 Circular inversion result (RLW regression, target range 1) . . . . . . 102
4.10 Horizon inversion result (median, target range 1) . . . . . . . . . . . 103
4.11 Horizon inversion result (RLW regression, target range 1) . . . . . . 103
4.12 Provided interval velocity model (target range 2) . . . . . . . . . . . 104
4.13 Dix inversion result (median, target range 2) . . . . . . . . . . . . . 106
4.14 Dix inversion result (RLW regression, target range 2) . . . . . . . . 106
4.15 Plane inversion result (median, target range 2) . . . . . . . . . . . . 107
4.16 Plane inversion result (RLW regression, target range 2) . . . . . . . 107
4.17 Circular inversion result (median, target range 2) . . . . . . . . . . . 108
4.18 Circular inversion result (RLW regression, target range 2) . . . . . . 109
4.19 Horizon inversion result (median, target range 2) . . . . . . . . . . . 110
135

List of Figures
136
4.20 Horizon inversion result (RLW regression, target range 2) . . . . . . 110
4.21 Provided velocity model (target range 3) . . . . . . . . . . . . . . . . 111
4.22 Dix inversion result (median, target range 3) . . . . . . . . . . . . . 112
4.23 Dix inversion result (RLW regression, target range 3) . . . . . . . . 112
4.24 Plane inversion result (median, target range 3) . . . . . . . . . . . . 113
4.25 Plane inversion result (RLW regression, target range 3) . . . . . . . 113
4.26 Circular inversion result (median, target range 3) . . . . . . . . . . . 114
4.27 Circular inversion result (RLW regression, target range 3) . . . . . . 114
4.28 Horizon inversion result (median, target range 3) . . . . . . . . . . . 115
4.29 Horizon inversion result (RLW regression, target range 3) . . . . . . 115
5.1 ZO section target range 2 . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.2 Iso-velocity model from the horizon inversion . . . . . . . . . . . . 118
5.3 Macro-velocity model (horizon inversion) . . . . . . . . . . . . . . . 119
5.4 Macro-velocity model (provided model) . . . . . . . . . . . . . . . . 119
5.5 Migrated image (horizon inversion) . . . . . . . . . . . . . . . . . . 120
5.6 Migrated image (provided model) . . . . . . . . . . . . . . . . . . . 120
A.1 2D ZO CRS attribute£and the interface dip� . . . . . . . . . . . . 126
B.1 CRS attribute correction . . . . . . . . . . . . . . . . . . . . . . . . . 129

References
Al-Chalabi, M., 1973, Series approximation in velocity and traveltime computations:
Geophysical Prospecting, 21, no. 4, 783–795.
Backus, G. E., 1962, Long wave elastic anisotropy produced by horizontal layering:
Journal of Geophysical Research, 67, 4427–4441.
Bortfeld, R., 1989, Geometrical ray theory: Rays and traveltimes in seismic systems
(second-order approximation of the traveltimes): Geophysics, 54, no. 3,
342–349.
Bronstein, I. N., and Semendjajew, K. A., 1996, Teubner-Taschenbuch der Mathematik:
B. G. Teubner Stuttgart - Leipzig.
Červený, V., 2001, Seismic Ray Theory: Cambridge University Press.
Cleveland, W. S., 1979, Robust locally weighted regression and smoothing scatterplots:
Journal of the American Statistical Association, 74, 829–836.
Deregowski, S. M., 1986, What is DMO?: First Break, 4, no. 7, 7–24.
Dix, C. H., 1955, Seimic velocity from surface measurements: Geophysics, 20, no.
1, 68–86.
Dürbaum, H., 1953, Possibilities of constructing true ray paths in reflection seismic
interpretation: Geophysical Prospecting, pages 125–139.
Hale, D., 1991, Dip moveout processing: Society of Exploration Geophysicists.
Höcht, G., 1998, Common Reflection Surface Stack: Master’s thesis, Universität
Karlsruhe, Germany.
Hubral, P., and Krey, T., 1980, Interval velocities from seismic reflection time measurements:
Society of Exploration Geophysicists.
Hubral, P., Schleicher, J., Tygel, M., and Hanitzsch, C., 1993, Determination of
Fresnel zones from traveltime measurements: Geophysics, 58, no. 5, 703–712.
137

References
Hubral, P., 1983, Computing true amplitude reflections in a laterally inhomogeneous
earth: Geophysics, 48, no. 8, 1051–1062.
Jäger, R., 1999, The Common Reflection Surface Stack - Theory and Application:
Master’s thesis, Universität Karlsruhe, Germany.
Levin, F. K., 1971, Apparent velocity from dipping interface reflections: Geophysics,
36, 510–516.
Liebhardt, G., 1997, Automatic detection of first-arrivals and inversion of refraction
seismic signals: Ph.D. thesis, Universität Karlsruhe, Germany.
Majer, P., 2000, Inversion of seismic parameters: Determination of the 2-D isovelocity
layer model: Master’s thesis, Universität Karlsruhe, Germany.
Mann, J., Hubral, P., Höcht, G., and Jäger, R., 1999, Common-Reflection-Surface
Stack - a real data example: Journal of Applied Geophysics, 43, no. 3,4, 301–318.
Müller, T., 1999, The Common Reflection Surface Stack Method - Seismic imaging
without explicit knowledge of the velocity model: Der Andere Verlag, Bad
Iburg.
Nürnberger, G., 1989, Approximation by Spline Functions: Springer Verlag
(Berlin).
Schleicher, J., Tygel, M., and Hubral, P., 1993, Parabolic and hyperbolic paraxial
two-point traveltimes in 3D media: Geophysical Prospecting, 41, no. 4, 495–
514.
Taner, M. T., and Koehler, F., 1969, Velocity spectra – Digital computer derivation
and applocations of velocity functions: Geophysics, 34, no. 6, 859–881.
Vieth, K.-U., 2001, Kinematic wavefield attributes in seismic imaging: Ph.D. thesis,
Universität Karlsruhe, Germany.
Yilmaz, O., 1987, Seismic data processing: Society of Exploration Geophysicists.
138

Danksagung
Ich danke Herrn Prof. Dr. Peter Hubral für die Übernahme des Hauptreferats und
auch für die Vergabe dieses Themas, das es mir ermöglichte einen tieferen Einblick
in die Geophysik zu bekommen.
Herrn Prof. Dr. Friedemann Wenzel danke ich für die Übernahme des Korreferats.
Dr. Kai-Uwe Vieth danke ich für die Betreuung bei der Erarbeitung dieser Diplomarbeit.
Er beantwortete immer alle Fragen und gab mir neue Anregungen
für die Lösung praktischer und theoretischer Probleme.
Für die Hilfe bei der Lösung programmiertechnischer Probleme danke ich German
Höcht und Jürgen Mann.
Für das Korrekturlesen und Verbessern meines nicht immer eindeutigen Englisch
danke ich: Jürgen Mann, German Höcht, Christoph Jäger, Steffen Bergler
und Thomas Hertweck.
Meinen Kommilitonen Christoph Jäger, Joachim Miksat, Christian Weidle sage
ich danke für die Sorge um meine Gesundheit (Mittagessen!). Bettina Bayer danke
ich für den sportlichen Anteil, das Tennisspielen war anstrengend aber auch
ein guter Ausgleich zur geistigen Arbeit.
Ausserdem danke ich meinen Eltern dafür, dass sie es mir ermöglichten, zum
einen so lange studieren zu können, und zum anderen, dass ich mich dabei auch
voll auf meine Arbeit konzentrieren konnte und mich nicht zu sehr um meinen
Unterhalt kümmern musste.
Mein Dank gilt auch sonst allen, die mich tatkräftig physisch wie auch psychisch
unterstützt haben und mit mir eine schöne und lehrreiche Zeit verbrachten (Messfahrten),
aber hier leider nicht namentlich erwähnt werden können.
139