Theory and application of residual static correction by means of CRS ...

Theory and application of residual static
correction by means of CRS attributes
Theorie und Anwendung von reststatischen
Korrekturen mit Hilfe der CRS Attribute
Diplomarbeit
von
Erik Ewig
Geophysikalisches Institut
der
Universität Karlsruhe
Das Thema wurde von Prof. Dr. P. Hubral gestellt.
Korreferent: Prof. Dr. F. Wenzel
Karlsruhe, Mai 2003

EHRENWÖRTLICHE ERKLÄRUNG:
Hiermit versichere ich, dass ich die vorliegende Arbeit selbständig und nur mit
den angegebenen Hilfsmitteln verfasst habe.
Karlsruhe, Mai 2003 Unterschrift

Zusammenfassung
Vorbemerkung
Diese Diplomarbeit ist bis auf diese Zusammenfassung in Englisch geschrieben.
Da auch in der deutschen Sprache einige englische Fachausdrücke gebräuchlich
sind, wurde bei diesen Ausdrücken auf eine Übersetzung verzichtet. Sie werden,
mit Ausnahme ihrer groß geschrieben Abkürzungen, kursiv dargestellt.
Zu Gunsten der Lesbarkeit habe ich in dieser Zusammenfassung weitgehend auf
die Angaben von Referenzen verzichtet. Diese Informationen sind im Hauptteil
der Arbeit zu finden.
Einleitung
Die Reflektionsseismik dient als ein Mittel um Abbilder des Untergrundes zu erstellen.
Es werden verschiedene Arten von Quellen (Explosionen, airguns, Hammerschlag,
usw.) benutzt, um akustische Wellen in das Gebiet zu senden, für welches
man sich interessiert. Diese Wellen werden an Impedanzkontrasten (schnelle
Änderung in der Geschwindigkeit und/oder der Dichte des Mediums) im Untergrund
reflektiert und propagieren danach zum Teil zurück an die Oberfläche, an
welcher mehrere Empfänger platziert sind. Im Fall einer 2D Akquisition befinden
sich Schüsse und Empfänger entlang einer Linie an der Erdoberfläche, in
Bohrlöchern oder als Hydrophone knapp unter der Wasseroberfläche. Für den
3D Fall wird eine flächenhafte Akquisition benötigt. Die Eigenschaften des Reflektors
(Tiefe, Neigung, Krümmung), an welchem die Welle reflektiert wird und
des Mediums (Geschwindigkeit, Dichte) durch welche die Welle propagiert, beeinflussen
die Laufzeit der Welle. Aufgrund dieser Beeinflussung ist es möglich,
aus den Laufzeiten Informationen über die Struktur des Untergrundes zu erhalten.
Wie bereits zuvor erwähnt, befinden sich Quelle und Empfänger entlang einer
Linie. Um nun einen mehrfach überdeckten Datensatz zu erhalten, müssen Quelle
und Empfänger entlang dieser so genannten seismischen Linie verschoben
werden. Dieser so erhaltene dreidimensionale Datensatz hängt sowohl von der
I

Zusammenfassung
Quell- und Empfängerposition als auch von der aufgezeichneten Laufzeit ab. Die
seismischen Daten durchlaufen mehrere Verarbeitungschritte (z.B. Dekonvolution,
Stapelung, Migration) bis als Endresultat ein Abbild des Untergrundes zur
Verfügung steht. In der seismischen Datenverarbeitung werden normalerweise
Mittelpunkt xm (Mittelpunkt zwischen Quelle und Empfänger) und half-offset h
(Distanz zwischen Mittelpunkt und Quelle bzw. Empfänger) als Koordinaten benutzt.
Somit wird der 3-D Datensatz normalerweise im xm-h-t Raum dargestellt.
Es bestehen verschiedene Möglichkeiten, Spuren für weitere Verarbeitungsschritte
in so genannten Sektionen oder gather zu gruppieren.
Von großer Bedeutung sind die common-midpoint (CMP) gather. Hierbei besitzen
alle Spuren, im Falle eines horizontalen Reflektor, denselben Reflektionspunkt. In
der Literatur wird oft der Term common-depth point (CDP) gather gleichbedeutend
zu CMP gather benutzt, aber der CMP und der CDP sind nur für horizontale Reflektoren
identisch. In dieser Arbeit wird deshalb nur der Term CMP gebraucht.
Jede Quelle stellt ein common-shot (CS) gather zur Verfügung, was bedeutet, dass
jede Spur in diesem gather den selben Quellpunkt besitzt. Die xm und h Koordinaten
sind zwar verschieden für jede Spur, aber sie stehen in einem linearen Zusammenhang.
Dementsprechend können die Spuren auch nach common-receiver
(CR) Lokationen sortiert werden.
Eine andere Möglichkeit ist es, alle Spuren mit dem selben offset zu gruppieren,
um ein common-offset (CO) gather zu erhalten. In dieser Konstellation haben alle
Spuren die selbe h Koordinate aber unterschiedliche xm Koordinaten. Ein Spezialfall
des CO gathers ist das zero-offset (ZO) gather. In diesem gather ist der offset
für jede Spur gleich Null und damit sind Quelle und Empfänger für jede Spur
koinzident.
Jedoch ist es in der Realität nicht möglich Quelle und Empfänger am selben Ort
zu platzieren, weswegen die ZO Sektion simuliert werden muss. Dies geschieht
durch Abbildungsverfahren wie die CMP Stapelung, die normal moveout/dip moveout
(NMO/DMO) Stapelung oder die Common-Reflection-Surface (CRS) Stapelung,
usw. Im Fall der CMP Stapelung werden die Amplituden entlang des ZO
Stapeloperators, einer Näherung der tatsächlichen Reflexionsantwort im xm-h-t
Raum in der Umgebung des ZO Punktes, aufsummiert. Das Ergebnis wird dem
jeweiligen ZO Punkt zugeordnet. Wird dies für jeden Punkt des ZO gather durchgeführt,
erhält man eine ZO gestapelte Sektion. Der Stapeloperator kann anhand
der normal moveout (NMO) Analyse, welche eine Hyperbel an die wahre Laufzeit
annähert, bestimmt werden. Dieses Analyse und die darauf folgende Stapelung
werden unter dem Begriff der CMP Stapelung zusammengefasst. Im allgemeinen
geht man bei der CMP Stapelung von horizontalen Reflektoren aus. Für geneigte
Reflektoren wird eine andere Methode, die so genannte NMO/DMO Stapelung,
benötigt. Die mit dieser Methode gewonnene ZO Sektion hat ein höheres Signal
zu Rauschen (S/N) Verhältnis, da bei diesem Verfahren im Gegensatz zur CMP
Stapelung in einem vielfach überdeckten Datenraum aufgestapelt wird. Der Be-
II

griff des Rauschens wird für alle nicht erwünschten Informationen in den Daten
verwendet. Rauschen kann in inkohärentes und in kohärentes Rauschen unterteilt
werden. Inkohärentes Rauschen ist zufälliges, unvorhersehbares Rauschen,
welches durch äußere Einflüsse wie zum Beispiel das Bewegen der Bäume im
Wind oder das Vorbeifahren eines Fahrzeugs verursacht wird.
Im Laufe der letzten Jahre haben sich neue Stapeltechniken etabliert, welche bessere
Stapelergebnisse erzielen als die oben erwähnten konventionellen Methoden.
Eine dieser Methoden ist die CRS Stapelmethode. Dies ist eine rein auf den
Daten basierende Methode, welche auch die lokale Krümmung des Reflektors
am Reflektionspunkt mit berücksichtigt. Das genaue Konzept der CRS Stapelung
wird in Kapitel 4 dieser Arbeit erklärt.
Die mit Hilfe der Reflektionseismik erhaltenen Abbilder des Untergrundes dienen
unter anderem dazu, Kohlenwasserstoffvorkommen zu finden. Heutzutage
wird es immer schwieriger, diese Vorkommen zu finden, da ein Großteil der Vorkommen
bereits erschlossen ist. Desweiteren ist es sehr teuer Bohrungen abzuteufen.
Dies führt zu einem erhöhten Interesse an besseren Abbildern des Untergrundes,
damit die Chance auf Erfolg so hoch wie möglich ist. Die nahe Oberfläche
bzw. die Verwitterungsschicht beeinflusst oftmals die Qualität von Landdaten
negativ. Methoden, um diesen Einfluss der Verwitterungsschicht zu beseitigen,
werden unter dem Begriff der statischen Korrekturen oder oft kurz statics
zusammengefasst (siehe Kapitel 2). Das Ziel der statischen Korrekturen ist, die
seismischen Daten so zu korrigieren, dass man Laufzeiten erhält, welche beobachtet
worden wären, wenn alle Messungen auf einer ebenen Fläche ohne das
Vorhandensein der Verwitterungsschicht oder einer Zone niedriger Geschwindigkeit
gemacht worden wären (vgl. Abbildung 1).
Die Durchführung der statischen Korrekturen geschieht folgendermaßen: Die
Geschwindigkeit und die Dicke der Verwitterungsschicht müssen durch geophysikalische
oder geologische Untersuchungen bekannt sein. Danach wird die
Laufzeit der Wellen durch die Verwitterungsschicht unterhalb der Quelle und
des Empfängers berechnet und die eigentliche Laufzeit der Welle um diesen Betrag
verringert. Die Auswirkungen der Topographie auf die Laufzeiten wird im
Prinzip durch diesen ersten Schritt, die so genannte Feldkorrektur beseitigt.
Es bleiben dennoch Störungen in der Reflektionslaufzeit zurück, welche durch
schnelle Variationen in der topographischen Höhe, der Geschwindigkeit der Verwitterungsschicht
oder der Dicke der Verwitterungsschicht verursacht werden.
Um diese verbleibenden Zeitverschiebungen, welche die Feldkorrektur nicht völlig
beseitigen konnte, zu bestimmen, ist ein weiterer Bearbeitungsschritt, die so
genannte reststatische Korrektur, notwendig. Hierbei wird jedem Schuss und
Empfänger eine zusätzliche statische Zeitverschiebung zugeordnet. Dieser Prozess
erzielt eine verbesserte Kontinuität der Reflektionsereignisse und ein besseres
Signal zu Rauschen Verhältnis (S/N) in der gestapelten Sektion. Nochmals
zusammengefasst besteht die statische Korrektur aus zwei Teilen, der Feldkorrek-
III

Zusammenfassung
S
R
S’ R’
Oberfläche
Unterkante der Verwitterungsschicht
datum line
Reflektor
Abbildung 1: Strahlweg von Quelle (S) durch die Verwitterungsschicht zu einem
horizontalen Reflektor und zurück zur Oberfläche zum Empfänger (R). S’ bzw.
R’ sind die hypothetischen Quell und Empfängerpunkte nach der Feldkorrektur.
tur und der reststatischen Korrektur. Diese Arbeit beschäftigt sich hauptsächlich
mit der reststatischen Korrektur.
Es müssen zwei Voraussetzungen erfüllt sein, damit man die statische Korrektur
auf seismische Daten anwenden kann:
• Oberflächenkonsistenz: Dies bedeutet, daß der Strahl nahezu senkrecht
durch die Verwitterungsschicht propagieren muss.
• Zeitkonsistenz: Die statische Verschiebung für eine Spur hängt nicht davon
ab, in welcher Tiefe der zugehörige Strahl reflektiert worden ist.
• Zusätzlich muss für die reststatische Korrektur gelten: Die Verwitterungsschicht
hat keinen Einfluß oder wenn doch, denselben Einfluß auf das wavelet
aller Spuren.
Ein Blick in die Geschichte zeigt, dass statische Korrekturen schon seit dem Beginn
der modernen seismischen Exploration, zu Beginn des zwanzigsten Jahrhunderts,
angewandt werden. Zu Anfang der digitalen Datenaufzeichnung wurde
die meiste Arbeit im Gebiet der statischen Korrekturen in die feldstatische
Korrekturen investiert. Heutzutage hat sich der Schwerpunkt mehr zu den restatischen
Korrekturen verlagert. Desweiteren werden heute auch zunehmend
Refraktionseinsätze benutzt, um die Genauigkeit der feldstatischen Korrektur zu
verbessern. Mittlerweile existieren verschiedenste Möglichkeiten (Überblick über
die konventionellen Methoden in Kapitel 3), um statische Laufzeitverschiebungen
zu beseitigen. Die Bestimmung der reststatischen Verschiebung geschieht in
IV

time [s]
0.8
0.9
1.0
1.1
1.2
trace number [#]
2 4
(a) synthetisches gather
time [s]
0.8
0.9
1.0
1.1
1.2
trace number [#]
1
(b) gestapelt ohne restatische
Korrektur
time [s]
0.8
0.9
1.0
1.1
1.2
trace number [#]
1
(c) gestapelt mit restatischer
Korrektur
Abbildung 2: Beispiel für die Verbesserung des Stapelergebnisses mit Hilfe der
restatischen Korrektur.
konventionellen Methoden folgendermaßen: Zuerst wird versucht, einen zeitlichen
Versatz zwischen den moveout korrigierten Spuren zu finden. Diese Versätze
werden dann in einen statischen Quellterm, einen statischen Empfängerterm
und andere Terme zerlegt. Es gibt verschiedene Methoden um diesen Versatz zu
bestimmen, von denen die meisten auf der Kreuzkorrelation zwischen zwei Spuren
basieren. Abbildung 2 zeigt mehrere gegeneinander zeitverschobene Spuren
nach der NMO Korrektur. Wenn diese Spuren ohne eine vorherige reststatische
Korrektur gestapelt werden, sind die Maxima des resultierenden wavelets nicht
mehr zwangsläufig am richtigen Ort und das wavelet ist deformiert (siehe Abbildung
2(b)). Die Stapelung der reststatisch korrigierten Spuren zeigt hingegegn
ein nicht deformiertes wavelet mit größerer Amplitude (siehe Abbildung (c)).
Die in dieser Arbeit vorgestellte Methode (Kapitel 5) macht Gebrauch von der
CRS Stapelmethode (Kapitel 4). Die aus der CRS Stapelmethode gewonnen kinematischen
Wellenfeldattribute, die sogenannten CRS Attribute, dienen als Basis
für die moveout Korrektur. Nach der moveout Korrektur erhält man für jede ZO
Spur ein CRS super gather, dieses gather beinhaltet alle moveout korrigierten Spuren,
welche nicht von einer Linie wie z. B. bei der CMP Stapelung, sondern aus einer
Fläche des Datenraumes stammen. Die mit der CRS Methode gestapelten ZO
Spuren dienen als Pilotspuren und werden mit jeder Spur des zugehörigen CRS
supergather kreuzkorreliert. Das Ergebnis dieser Kreuzkorrelation wird der entsprechenden
Quelle und dem entsprechenden Empfänger zugeordnet. Da jeder
Empfänger bzw. jede Quelle zu mehreren Spuren gehört, werden die Ergebnisse
der Kreuzkorrelationen für die selben Quellen oder Empfänger aufgestapelt.
V

Zusammenfassung
Dadurch ist die Lokation des Maximums in dieser Kreuzkorrelationsstapelung
die jeweilige reststatische Laufzeitverschiebungen des Empfänger bzw. der Quelle.
Diese Laufzeitverschiebungen können nun auf die Originaldaten angewendet
werden, und bei Bedarf kann eine weitere Iteration der reststatischen Korrektur
durchgeführt werden.
Um das Potential der neuen Methode zu demonstrieren, wurde diese an einem
synthetischen Datensatz, welcher künstlich mit Rauschen und statischen Verschiebungen
belegt wurde, getestet (Kapitel 6). Es zeigt sich, daß die neue Methode
mit nur zwei Iterationen in der Lage ist, die bekannten künstlichen Zeitverschiebungen
nahezu vollständig in den Daten zu beseitigen. Es wird jedoch
auch deutlich, dass eine Mindestanzahl an Beiträgen zur Korrelationstapelungen
nötig ist, um ein sinnvolles Ergebnis zu erzielen.
Desweiteren wurde die neue Methode an einem realen Datenbeispiel getestet,
welches zuvor schon einer konventionellen reststatischen Korrektur unterworfen
wurde (Kapitel 7). Hierbei zeigte sich, dass die reststatische Korrektur mit
Hilfe der CRS Attribute in der Lage war, weitere Zeitverschiebungen zu finden
und die Qualität der aus der CRS Stapelung resultierenden ZO Sektion zu verbessern.
In einem zweiten Schritt wurde dieser Realdatensatz künstlich mit statischen
Zeitverschiebungen belegt, um zu sehen, ob die neue Methode in der
Lage ist, auch diese Verschiebungen zu beseitigen. Es zeigte sich, dass in diesem
Beispiel mehr Iterationen benötigt wurden, um die Zeitverschiebungen zu beseitigen.
Auch wurde ein Problem dieser Methode deutlich. Die beste Abschätzung
der Zeitverschiebung war nach vier Iterationen erreicht, aber die gefunden Zeitverschiebungen
konvergierten erst nach der fünften Iteration gegen Null. Hierbei
muß beachtet werden, dass die künstlichen Zeitverschiebungen auf die Originaldaten
addiert worden sind, welche schon zu Beginn noch statische Laufzeitverschiebungen
aufwiesen. In Zukunft muss dieses Problem noch näher untersucht
werden. Zusammenfassend kann aber gesagt werden, dass die ersten Tests dieser
neuen Methode vielversprechende Resultate liefern. Allerdings muß sich die
reststatische Korrektur mit Hife der CRS Attribute erst an grösseren, vollständig
realen Daten bewähren.
VI

Contents
1 Introduction 1
2 Static correction 7
2.1 Weathering layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 CMP stack and the intention of static correction . . . . . . . . . . . 8
2.3 Assumptions for static correction . . . . . . . . . . . . . . . . . . . . 9
2.4 Field static correction . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 Residual static correction . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Conventional methods for residual static correction 17
3.1 Linear traveltime inversion . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Estimation of time shifts between traces . . . . . . . . . . . . . . . . 19
3.2.1 Searching for the maximum cross correlation . . . . . . . . . 20
3.3 Stack power maximization . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Non-linear traveltime inversion . . . . . . . . . . . . . . . . . . . . . 22
4 Common Reflection Surface stack 25
4.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2.1 Search for CRS attributes . . . . . . . . . . . . . . . . . . . . 27
4.2.2 CRS aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5 Residual static correction by means of CRS attributes 33
5.1 Moveout correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.2 Cross correlation and search for the estimated static time shift . . . 35
5.3 Iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6 Synthetic data example 39
6.1 Model and survey design . . . . . . . . . . . . . . . . . . . . . . . . 39
6.2 Residual static correction . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.2.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
i

Contents
7 Real data examples 63
7.1 The real dataset and the real dataset with artifical static time shifts . 63
7.2 Results of the new residual static correction approach . . . . . . . . 65
7.2.1 Original data . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
7.2.2 Original data with artifical static time shifts . . . . . . . . . . 68
8 Summary 81
A Refraction seismics in relation to the near-surface 83
B Cross Correlation 87
C Used hard- and software 89
List of Figures 91
List of Tables 95
References 97
Danksagung 99
ii

Chapter 1
Introduction
Reflection seismic serves as an method to construct images of the subsurface,
whereby different kinds of sources (explosion, airgun, sledgehammer, etc.) send
elastic or acoustic waves to the area of interest in the subsurface. These waves
are then reflected and transmitted at impedance contrasts (rapid changes in the
medium velocity and/or the density) and partly propagate back to the surface,
where a series of receivers is positioned. In case of 2-D data acquisition, the
source and receivers are usually disposed along a straight line on the surface,
in boreholes or as hydrophones shortly under the water surface, whereas in 3-D
acquisition, one needs, of course, an extensive acquisition. The properties of the
reflectors (depth, dip, curvature) and of the subsurface media (velocity, density)
affect the traveltimes of the waves. Therefore, it is possible to deduce information
about the subsurface structure from the observed traveltimes.
As mentioned before, the sources and receivers are placed on a straight line in
2-D acquisition. To acquire a multicoverage dataset, the source and receiver arrays
have to be moved along this so-called seismic line. The received three dimensional
dataset depends on shot and receiver locations as well as on the traveltime.
The data pass through several processing steps (e. g., deconvolution, stacking,
migration) untill an image of the subsurface is obtained as final product. Seismic
data processing is conventionally done in midpoint xm (point midway between
source and receiver) and half-offset h (half distance between source and receiver)
coordinates, thus the traces of the 3-D data volume are usually displayed in the
xm − h − t space (see Figure 1.1). However, different possibilities exist to group
the traces into so called gathers or sections for further processing.
Of major importance are the common-midpoint (CMP) gathers (see Figure 1.2(a)),
because for a horizontal reflector all rays that contribute to one CMP gather have
the same reflection point. Some examples for such CMP gathers are shown as
green planes in Figure 1.1. In literature the term common-depth-point (CDP)
gather is often used synonymously, but with Figure 1.2 in mind it is obvious that
the CMP and the CDP are only the same for horizontal reflectors. In this work,
1

Chapter 1. Introduction
t [s]
9
8
7
6
5
4
3
9
8
2
7
6
1
4
3
5
h [km]
2
0
1
0 1 2 3 4 5 6 7 8
0
9
x m[km]
Figure 1.1: Three dimensional data volume with axes xm for midpoint coordinate,
h for half-offset and t for traveltime. The red planes are CS gathers, the green
planes are CMP gathers and the blue planes are CO gathers. The special case
of h = 0 denotes the zero-offset section which is parallel to the blue planes and
represents the front plane of the 3D data volume.
only the term CMP is used.
Each shot provides a common-shot (CS) gather, which means that every trace in
this gather (red planes in Figure 1.1) refers to the same source point. The xm and
h coordinate are different for every trace but they linearly depend on each other.
Analogously, the traces can also be sorted with respect to common-receiver (CR)
locations.
Another possibility is to group all traces with the same offset together, to make
up a common-offset (CO) gather (blue planes in Figure 1.1). In this constellation,
all traces have the same h coordinate but different xm coordinates. A special case
of the CO section is the zero-offset (ZO) section, in which, as the name implies,
the offset is equal to zero and, thus, the sources and receivers coincide for every
trace. This section is very valuable for interpretation as it provides a time domain
image of the subsurface whereas the half reflection traveltime is exactly the time
which the wavefront needs to travel from the source to the reflection point for
every subsurface.
However, in reality, it is not possible to place shot and receiver at the same point.
Therefore, the ZO section must be simulated. This is done by imaging methods
2

z
depth
z
depth
shotpoints
shotpoints
common-
midpoint
receivers
common-depth-point
(a) horizontal reflector
common-
midpoint
(b) dipping reflector
receivers
dip angle
location along the
seismic line
x
location along the
seismic line
x
reflector
smeared area of depth points
Figure 1.2: Both figures illustrate a CMP acquisition. At (a) the figure with a
horizontal reflector, the CMP is identical with the CDP, but this is not the case at
(b) the figure with the dipping reflector.
like the CMP stack, the NMO/DMO/stack, or the CRS stack etc. In case of the
CMP stack, the amplitudes along the (ZO) stacking operator, an approximation
of the actual reflection traveltimes in the xm − h − t space in the vicinity of a ZO
point, are summed up. The result is assigned to the respective ZO point. Doing
the same for all points of the ZO gather yields the ZO stack section. The stacking
operator can be established by the normal moveout (NMO) analysis, which fits a
hyperbola to the reflection event within a CMP gather. This analysis and the subsequent
stack are consolidated to the term CMP stack. In general, the CMP stack
assumes horizontal reflectors. For dipping reflectors another method is needed,
the so-called normal moveout/dip moveout (NMO/DMO)/stack. The resulting
ZO section has a higher signal-to-noise ratio (S/N) because correlated events in
the multi-coverage data are summed up. The term noise is used for every undesired
information in the data. Noise is often divided into coherent noise and
incoherent noise. Multiple reflections are regarded as coherent noise, for example.
Incoherent noise or random noise is not predictable and is due to external
influences like wind shaking a receiver or a car driving near a receiver.
In the course of the last years, new stacking techniques have been established
3

Chapter 1. Introduction
which yield better stacking results than the conventional methods mentioned
above. One of this methods is the Common-Reflection-Surface (CRS) stack (see,
e. g., Mann, 2002). It is a data-driven method and takes also the local curvature
of the reflector at the reflection point into account. The NMO/DMO/stack take
only the dip of the reflector into account, whereas the CMP stack considers neither
the curvature of the reflector nor the dip of the reflector. The concept of the
CRS stack is explained in Chapter 4 of this thesis.
The images of the subsurface obtained by reflection seismics are of use for the
search for hydrocarbons. Today, the most of the deposits are made available.
Thus, it becomes more and more difficult to find new ones. Furthermore, drilling
for hydrocarbon deposits is very expensive. This causes an increased interest in
obtaining images of the subsurface of good quality. Land data are influenced by
the topography and irregularities in the near-surface, i. e., the weathering layer,
which deteriorate the data quality. Methods to remove the effects caused by the
weathering layer are combined by the term static correction, often shortened to
statics. To explain the idea of static correction, I refer to the definition of Sheriff
(2002): "Corrections applied to seismic data to compensate for effects of variations
in elevation, weathering thickness, weathering velocity, or reference to a datum.
The objective is to determine the reflection arrival times which would have been
observed if all measurement had been made on a (usually) flat plane with no
weathering or low velocity material present."
The principle behind static correction is shown in Figure 1.3, which displays
a simple field acquisition and shows the raypath (solid line) from a source S
through two layers down to the reflector and up to a receiver R. The reference
datum is shown as dotted line. The concept of static correction is to transform
the measured data, as if acquired with the new source S ′ and receiver point R ′
located vertically below the old points S and R and below the base of weathering.
The raypath (dashed line) of this hypothetical experiment is unaffected of
the weathered layer.
The topographic effect on the reflection times is significantly reduced by applying
field static correction (Section 2.4), which means the redatuming from S to S ′ and
from R to R ′ in Figure 1.3, respectively. However, rapid changes in elevation and
rapid changes in the near-surface velocity or the thickness of the weathering layer
still remain as reflection time distortions. To eliminate these remains which the
field static correction did not compensate, another processing step is necessary.
This step, which is called residual static correction (Section 2.5), assigns to every
shot and every receiver an additional static time shift. The time shifts of residual
static correction aim to enhance the continuity of reflection events and to improve
the signal-to-noise (S/N) ratio after stacking. In summary, one can say that the
static correction consists of two parts, the field static correction and the residual
static correction. In this work, I focus on the part of residual static correction.
A look into the historical background shows that static corrections are applied
4

S
R
S’ R’
surface
base of weathering
datum plane
reflector
Figure 1.3: Ray path through the weathering layer and reflection at a horizontal
reflector from a source (S) to a receiver (R) at the surface. S’ and R’ are the pseudosource
and pseudo-receiver location after the field correction, respectively
since the beginning of modern seismic exploration in the early years of the twentieth
century. However, since the introduction of digital recording and processing
the biggest effort has been undertaken to improve the so-called field static correction.
Nowadays, the main emphasis in the area of static correction is placed
on improving methods for computing residual static correction. Furthermore, refracted
arrivals are more and more used to improve the accuracy of the field static
correction.
Today, a lot of different methods exist to provide the residual static time shift. All
these methods use the assumption of one exclusive time shift for every source and
receiver, which implies the assumption of surface consistency (see Section 2.3). In
conventional methods the determination of the residual static time shifts works
in principle as follows: Firstly, one tries to estimate the misalignment of the
moveout corrected traces. Then, the misalignment of every single trace is decomposed
in a source static term, a receiver static term, and other terms (see
Section 2.5). As mentioned before, there are different methods to obtain these
static time shifts, but most of them are based on cross correlations of the traces of
the dataset (see Appendix B). Figure 1.4(a) shows a reflection event after NMO
correction distorted by residual static time shifts. Stacking the traces from (a)
without any corrections results in dislocated peaks for this reflection event and
in a deformed wavelet (see Figure 1.4(b)), while the stack with residual static correction
clearly shows an undeformed wavelet with correctly stacked amplitudes
(see Figure 1.4(c)).
The new approach presented in this work makes use of the CRS stack. So far,
5

Chapter 1. Introduction
time [s]
0.8
0.9
1.0
1.1
1.2
trace number [#]
2 4
(a) synthetic gather
time [s]
0.8
0.9
1.0
1.1
1.2
trace number [#]
1
(b) stacked ’as is’
time [s]
0.8
0.9
1.0
1.1
1.2
trace number [#]
1
(c) with residual static
correction
Figure 1.4: Example of residual static correction enhancement after an approximate
NMO correction was applied.
the 2D ZO CRS stack method does not account for residual static correction. The
CRS attributes serve as a basis for the moveout correction and the result, for every
ZO trace is a so-called CRS super gather which contains all CRS moveout corrected
prestack traces. This traces are not like in, for example, the CMP gather, only from
a plane of the data cube but from an entire subvolume of the data volume. Thus,
one of the advantages of this method is that it make use of more information
of the seismic multi-coverage reflection data. The estimate of the misalignment
inside the CRS super gather is also based on cross correlation (see Chapter 5).
6

Chapter 2
Static correction
2.1 Weathering layer
In many land data acquisition areas, the ground is covered with a relatively thin
layer of low seismic velocity. Geophysicists call this layer the weathering layer
which differs from the same term used by geologists. The seismic weathering
layer is a near-surface low velocity layer in which the portion of air filled pore
space of rocks is usually more than of water filled (Cox, 1974). The ’geological’
weathering layer is the result of a rock decomposition. But weathering is only
one of several processes that cause the phenomena of decomposition, that usually
affects the surface and continues downward. The main part is caused through the
physical or chemical properties of water, gas, and organisms in and outside the
rock.
In general, the thickness of the seismic weathering layer is between a few centimeters
and 50 meters or more, but the thickness of this layer can be extremely
irregular. Also, the velocity can vary rapidly in the lateral and vertical direction
and the elevation can vary. In most cases, the seismic weathering layer is thicker
than the geological one. The base of the seismic weathering layer is defined as the
depth where a change to a significant higher velocity occurs or where the velocity
stabilizes. It coincides sometimes with the water table and/or with the base
of the geological weathering layer.
The term low velocity layer (LVL) is often used for the seismic weathering layer.
The typical velocity for the weathering layer is between 500 m/s and 800 m/s
compared to subweathering velocities of 1500 m/s and up. Velocities in unconsolidated
materials typically depend on water saturation and are related to compaction
and thickness. Furthermore, the ratio between compressional (P-wave)
and shear wave (S-wave) velocities in this material has values from about 1.3 to
10.0. The reason for this deviation from the rule of thumb vP/vS ≈ √ 3 is that the
water saturation is a significant factor influencing the compressional wave velocity,
but has no effect on shear wave velocity. Usually, the thickness of such a
7

Chapter 2. Static correction
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surface
Figure 2.1: Some of the possible near-surface conditions which cause time distortions
in the observed seismic traces
seismic weathering layer is determined by refraction seismics (see Appendix A)
or by uphole-surveys. If an uphole survey is used, the information is obtained
only at discrete points along the seismic line. Between these points it is necessary
to interpolate because uphole survey locations and the source and receiver points
usually do not coincide. This interpolation can be based on one or more of the
following: reflection data, geologic data, simple numerical interpolation, or, of
course, refraction data.
Some parameters associated with the near-surface can change with the seasons or
even in smaller time scales because of several reasons, e. g., temperature changes,
rainfall, tidal effects, ice movements, wind, recent erosion, deposition, earthquakes
and human activities. The recognition and observation of these changes is
very important for monitoring or time-lapse surveys. Here, the data acquisition
is repeated after many months or even years.
Also very important for the static correction is the kind of the topography (see,
Cox, 1974), i. e., sand dunes, mountain front, youthful (characterized by active
vertical erosion) and mature topography (the surface profile gives no real indication
of the variation in the near surface). In Figure 2.1, one can see some surface
conditions which are frequently observed.
It is important to point out that the traveltime distortions do not occur because
of the presence of the LVL but due to the variation in thickness and velocity. The
so caused deterioration in the quality of land seismic data can be significant (see
Figure 2.5).
2.2 CMP stack and the intention of static correction
The raypath from a source to one receiver is shown in Figure 1.3. One can observe
that the traveltime along the raypath is influenced by the elevation of the
geophone and the shotpoint, by the velocity and thickness of the near-surface
8

Frequency [Hz]
240
220
200
180
160
140
120
100
80
60
40
20
notch
6dB
3dB
0
0 1 2 3 4 5 6
Static [ms]
7 8 9 10 11 12
2.3 Assumptions for static correction
Figure 2.2: A small static shift between two traces acts as a high-cut filter in the
amplitude spectrum. The frequency attenuation in the stacked trace is a function
of the static shift between the two traces which are summed up.
layer above the reference datum, by the depth and dip of the reflector itself, by
the distance separating the source and receiver, and finally by the average velocity
between the datum and the reflector. As described in Chapter 1, the NMO
corrected traces inside the CMP gather can be stacked, and so a ZO trace with enhanced
S/N ratio at the midpoint between the shots and the receivers is obtained.
This point leads to the intention of the static correction. When the traces are time
shifted because of the propagation through the LVL, the resulting stack cannot
reproduce the correct wavelet (Figure 1.4(b)). Therfore, it is necessary to correct
the traces, so that the data can be properly stacked. Time shifts can be applied
in two different ways: static or dynamic. Static correction means a constant time
shift is added to the whole data trace, whereas dynamic correction means time
shifts varying with traveltime. This work deals only with static correction. These
short-period static anomalies (see Figure 1.4) act as high-cut filters on the amplitude
spectrum and also cause a phase distortion of the wavelet. Figure 2.2 shows
the damping of the amplitude spectrum of the stack of two traces in relation to
the time shift between this two traces and there dominant frequency.
2.3 Assumptions for static correction
In principle, three assumptions have to be made to apply static correction to the
traces in the data:
• Surface consistency: The rays are assumed to propagate nearly vertically
9

Chapter 2. Static correction
v’
1
v’ 2
v’
3
S
R1
R2
R3
(a) model surface consistent
v 1
v
v
2
3
S
R1
(b) model not surface consistent
Figure 2.3: Rays in a surface consistent model and rays in a model which is not
surface consistent.
10
through the LVL layer. That this assumption is fundamental for static correction
becomes clear with Figure 2.3. The rays in the non-surface consistent
model (Figure 2.3(b)) take completely different paths through the LVL,
under the source point, from the source S to the receiver R1 as to the receiver
R2. Therefore, the time shifts are no unique property of the source
or receiver location, as it would be necessary for a static correction. The
lower the velocity in the near-surface layer in relation to the other layers, the
more the assumption of surface consistency is fulfilled. This fact is based on
Snell’s law which states that the ratio of the sines of the incident and refraction
angles is equal to the ratio of the velocities of the two layers (see
Equation A.1).
• Time consistency: This assumption follows from the surface consistency. It
stands for the fact that time shifts do not depend on the traveltimes of the
different reflection events. Surface consistency assumed, the ray reflected
on the deep reflector (blue line in Figure 2.3(a)) takes the same way in the
uppermost layer as the ray reflected on the shallow reflector (red line in
Figure 2.3(a)). Therefore, the time shift for every traveltime is a property of
the source or receiver location, only.
• In addition, residual static correction demands that the LVL has no or
at least the same influence on the wavelet of all emerging waves: if the
wavelets in all traces are influenced in the same way, this does not affect the
result of cross correlation. The effect of the LVL on the wavelet can only be
undone by a dynamic correction where every time point is shifted with a
different time shift.
R2
R3

Elevation
A’
A
tAW
t
Ae
A’’
tBW
t Be
B
B’’
B’
LVL
surface
C
base
of weathering t CW tCe 2.4 Field static correction
C’’
C’
reference
datum
Figure 2.4: Sketch illustrating the two components of datum correction: weathering
correction (the LVL is removed, so that the base of LVL (blue line) becomes
the new reference surface) and elevation correction (the datum (horizontal green
line) becomes the new reference surface)
2.4 Field static correction
The idea of field or datum static correction is to introduce a new horizontal plane
(datum plane) below the LVL, in order to place all sources and receivers on this
plane, and to simulate a new hypothetical experiment (see Figure 1.3). Note that
the term datum static correction and field static are in synonyms. The term field is
used because the field crew often performs this processing step. The new seismic
data should be free of the effects of rapid variation in elevation, weathering layer
thickness, or weathering layer velocity. To explain how this is applied in practice,
Figure 2.4 shows a simple near-surface model with a single LVL.
All preprocessing should have been done before as a logical first step. The datum
static correction consists of two parts: the weathering correction and the elevation
correction. In the first step the LVL is removed, so that the base of the LVL
becomes the new reference surface. That means, all points are projected from surface
to the base of the LVL, e. g., A to A ′ , B to B ′ , and C to C ′ (see Figure 2.4). Thus,
the surface-referenced traveltimes must be adjusted to give a new set of traveltimes
that would have been observed if the data had actually been recorded at the
base of the LVL (blue line). In Figure 2.4 the traveltime between A and A ′ , e. g.,
is shown as tAW, which depends on the thickness of the LVL at point A and the
(not necessarily constant) velocity vw of the LVL. This means in turn, that a ZO
trace for point A must be static corrected with 2tAW because the ray propagated
one time downwards and one time upwards along the same raypath through the
LVL. Before proceeding, a sign convention is required: correcting with negative
time shifts corresponds to positive elevations and reduces the reflection travel-
11

Chapter 2. Static correction
time.
In the next step the elevation correction is necessary to simulate data acquisition
on another surface, the reference datum plane. Back to Figure 2.4, the green line
indicates the reference datum, so again every point must be projected vertically
to this datum. It is to be mentioned that an error occurs in this step if the datum
plane is below the LVL, because surface consistency is fulfilled only within the
LVL. In the other layers the ray, in general, is not vertical. The effort to remove
this error will be investigated later on in Section 2.5. Neglecting this error, the
traveltime between A ′ and A ′′ is shown as tAe, which results from the distance
from point A ′ to A ′′ and the velocity ve, usually chosen to be constant. Similar
corrections are shown for locations B and C, respectively. At point A the datum
correction can be written as tA = −tAW − tAe. At the points B and C the signs must
be accommodated, according to the relative location of the considered datum
surface, i. e., below or a above the surface to which the point should be corrected
to. This means for point B tB = −tBW + tBe and for point C: tC = −tCW + tCe. In the
model depicted in Figure 2.4, the chosen datum is below or above the top surface
and/or the bottom of the weathering layer. The datum correction for the trace
with the source point at A and the receiver point at B, e. g., is according to the
previous considerations given by
tAB = − AA′
vw
− A′ A ′′
ve
− BB′
vw
+ B′ B ′′
ve
. (2.1)
To determine the traveltime for a datum correction, some parameters are required.
For the weathering correction, the elevation of source and receiver, the
thickness, and the velocity of the LVL are needed. For the elevation correction,
the elevation of the base of the LVL as well as the velocity between this base and
the datum plane are needed, too. The properties (velocity, thickness, elevation)
of the near-surface medium can be determined by uphole surveys and refraction
seismics (see Appendix A). At this point the difficulty of datum correction becomes
obvious, because, as described in Section 2.1, the parameters can change
not only slowly but also rapidly in horizontal and vertical direction.
Now, the question arises how to choose the reference datum. In the ideal case,
two conditions should be satisfied: Firstly, the datum surface should be a horizontal
surface with a constant velocity along the plane. Secondly, the static corrections
should be as small as possible. In some areas, the first of these two conditions
can only be satisfied if the datum is chosen at a significant depth below
the measurement surface. This might be necessary if the presence of a local topographic
feature produces a varying overburden, or load effect, which results in
a slightly higher acoustic impedance beneath the higher surface elevations due
to the increased pressure. The so introduced change in the acoustic impedance
has, in general, its maximum close to the feature and decreases with distance (see
Widess, 1946). Thus, choosing a deep datum is preferred which is in contradiction
12

2.5 Residual static correction
to the second condition mentioned above. The deeper the datum, the greater the
error between the vertical projection and the true raypath. As aforementioned,
the assumption of surface consistency is only fulfilled in the LVL. Another effect
of large field static corrections is that the corrected reflection traveltimes deviates
from a hyperbolic relationship for the actual NMO velocity (see Profeta et al.,
1995). From this, it follows that the stacking velocity determined by means of the
NMO correction differs from the correct one. This error is called residual NMO
(see Section 2.5).
To solve the problem of residual NMO, the concept of an intermediate so-called
floating datum has been developed. The floating datum is chosen, so that it is
close to the surface and the field static correction is kept small. There are several
possibilities to map a trace to an intermediate datum. For example, the mean
values of all field static corrections inside every CMP gather are calculated first.
Afterwards, every trace is corrected by the difference between the proper value of
the field static of the trace and the mean value of the corresponding CMP gather.
After application of the NMO correction, the data are converted to the reference
datum by the application of a further time correction, the mean datum static correction
for the CMP. This concept avoids distortions of the near-surface velocities,
caused among other things by shifting shallow events above zero time and avoids
distortion of the hyperbolic shape of the near-surface reflections prior to NMO.
However, the elevation correction can also be done with the CRS stack which
takes the surface topography into account (see Zhang, 2003). If this approach
is applied, the problem of the residual NMO does not exist, because the CRS
stack needs no stacking velocity. Nevertheless, the weathering correction is also
necessary and, consequently, the properties of the weathering layer have to be
known.
In general, field static correction is not necessary for marine datasets because an
LVL at the sea bottom is assumed to be non-existent and, of course, the water
layer has an almost constant velocity and a flat surface. Only in areas of rapid
varying water-bottom topography, or with variable low-velocity material below
the water-bottom, marine field static correction is computed in the same way as
for land data.
2.5 Residual static correction
As described in Section 2.4, errors have been made by the field static correction
which are mainly due to one factor, namely, the inaccuracies in the near-surface
model. The model is only a simplification of the geology. The field static correction
is only an approximation of a more complex problem. It could even happen
that the datum correction introduces misalignments in a CMP gather that are
larger than the misalignments in the data without datum correction. Thus, an
13

Chapter 2. Static correction
(a) without static correction (b) with static correction
Figure 2.5: CMP stack from a land survey (a) before and (b) after static correction
(taken from Yilmaz (1987)).
additional processing step is necessary to compensate these errors. This processing
step also serves to eliminate small variations of reflection traveltime caused
by rapid changes in elevation, the base of weathering, and weathering velocity.
Field static correction together with residual static correction form the static correction.
An example for static correction is shown in Figure 2.5, where (a) the
CMP stack of the ’original’ dataset and (b) the CMP stack after applying static
correction is displayed.
To achieve surface consistency, residual static correction techniques have to provide
one exclusive time shift for every source or receiver location. Residual static
correction is usually applied after datum correction but it is also possible to do
residual static correction without any preceding datum static correction. However
the residual correction without applying datum static correction is often unphysical
and mainly provides ’cosmetic’ improvements of the final stack section.
The time shifts that result from residual static analysis are basicly composed of
ti, j,h = ri + s j + Gk,h + MkhX 2 i, j
i + j
with k =
, 2 (2.2)
(see Taner et al., 1974; Wiggins et al., 1976; Fitch, 1981; Cox, 1974) where
14
• ri is the receiver static of the ith receiver location,
• s j is the source static of the jth source location,
• G k,h is the structural term, which is an arbitrary time shift for the kth CMP
gather along the hth horizon and depends on the subsurface structure. If

2.5 Residual static correction
the traces within a CMP gather are cross correlated (see Appendix B), then
Gk,h can be ignored, because it remains constant within this CMP gather.
• Mkh is the residual moveout term at the kth CMP gather for the hth horizon,
and Xi, j = s j − ri is the source-receiver distance or simply the offset. This
term depends on the error which has been made if the datum static correction
had been applied before the NMO correction was performed (see
Section 2.4).
Equation 2.2 assumes that the corrected arrival times are constant at a CMP location
index, meaning that the reflector in the depth for this CMP gather has
no cross-line dip. The dependency of the structural and of the residual moveout
terms from the particular horizon is often discarded in the conventional methods.
This also applies to the new approach which is presented in this work (Chapter 5).
Equation 2.2 can be modified for cross-line dipping reflectors in the following
way:
ti, j,h = ri + s j + Gk + Mk,hX 2 i + j
i, j + Dk,hYi, j with k = , (2.3)
2
where Dk,h is the cross-line dip coefficient at the kth CMP gather for the hth horizon
and Yi, j is the distance between the CMP and the reference line. In the next
chapter, residual static corrections are discussed in more detail and some conventional
methods are described.
15

Chapter 3
Conventional methods for residual
static correction
There are a lot of surface consistent techniques to estimate shot and receiver residual
static time shifts. But what most of them have in common is that they are
based on cross correlation (see Appendix B) to obtain the time shift τ between
two moveout corrected traces. The expressions for the static shifts of two traces
are
ti1, j = ri + s 1 1 j + Gk + Mk X 1 1 1 2 i1, j with k1 = 1 i1 + j1
, (3.1)
2
, (3.2)
2
where the first expression is Equation 2.2 without the dependency on the horizon.
The time shift τ is equal to the difference of the static time shifts of Equation 3.1
and 3.2 and reads
ti 2, j 2 = ri 2 + s j 2 + G k2 + M k2 X 2 i 2, j 2 with k2 = i2 + j2
τ = (ri 1 − ri 2 ) + (s j 1 − s j 2 ) + (Gk 1 − Gk 2 ) + (Mk 1 X 2 i 1, j 1 − Mk 2 X 2 i 2, j 2 )
with k1 = i1 + j1
2
and k2 = i2 + j2
.
2
(3.3)
usually, the time shifts are determined in certain subsets of the entire data volume.
Some conventional methods compare traces within CMP gathers. Then i2,
i1, j2 and j1 are linked by i2 = i1 + l and j2 = j1 + l. Furthermore, the two structural
and the two residual moveout terms are identical. Thus, the time shift between
two traces is independent of the subsurface structure and the expression for the
static shift of the traces simplifies to
τ = ri − ri 1 1−l + s j − s 1 j1+l + Mk (X 1 2 i,l − X 2 i−l, j+l) with k1 = i1 + j1
. (3.4)
2
17

Chapter 3. Conventional methods for residual static correction
In addition, two other categories exist in which the conventional methods could
be classified: the common-surface-location methods and the common-offsetbased
methods. In each of this methods, another term in Equation 3.3 vanishes,
e. g., for one of the common-surface-methods in which traces are compared which
belong to a common-source-plane, the source static term vanishes. Thus, every
method has advantages in certain cases, e. g., the common-surface-location methods
are usually applied for problematic datasets, such as data with large locationto-location
residual static corrections (see Cox, 1974). This chapter mainly deals
with common-midpoint based methods, as the majority of today’s processing
projects uses such kinds of methods. In this section, the general time shift between
two arbitrary traces was introduced.
3.1 Linear traveltime inversion
Every trace within the dataset, recorded above an LVL has a residual static time
shift after field static corrections have been applied. The time shift consists of the
terms presented in Equation 2.2. For convenience, Equation 2.2 is repeated here:
ti, j = ri + s j + Gk + MkX 2 i + j
i, j with k =
2 .
Let NS be the number of source locations, NR the number of receiver locations,
NG the number of CMP gathers, and NF the constant CMP fold, i. e., the number
of traces within a single CMP gather. The constant CMP fold is used for simplicity.
Generally, it depends on the particular CMP gather. The number of equations
in this system is equal to NGNF and the number of unknowns is NS + NR + 2NG.
Two times NG, because the structural term and the residual NMO term depend
only on the CMP location and are consequently constant within a CMP gather.
Typically, NGNF > NS + NR + NG + NG. Thus, there are more equations than unknowns,
but not all of them are independent. ”The linear simultaneous equations
that define the static problem are said to be overspecified (there are more equations
than unknowns) and underconstrained (they are deficient in the number
of independent equations available to solve for the unknowns).” (Wiggins et al.,
1976).
However, this kind of problem can be solved by least-square algorithms. Usually,
after the decomposition process, a time difference ɛi j between an observed time t ′ i j
and the actual value ti j of all calculated components exist. The sum of the squared
errors
Q = �
i, j,h
(ɛi jh) 2 = �
(ti jh − t ′ i jh )2
i, j,h
needs to be minimized. Inserting Equation 2.2 leads to
18
(3.5)

Q = �
3.2 Estimation of time shifts between traces
(ti jh − r
i, j,h
′ i − s ′ j − G ′ kh − M ′ khX′2 i j )2 . (3.6)
The minimum can be found by equating the partial derivatives of Equation 3.6
with respect to all variables with zero. This results in
∂Q/∂r ′ i = ∂Q/∂s ′ j = ∂Q/∂G ′ k,h = ∂Q/∂M ′ k,h = 0, (3.7)
which yields NS + NR + 2NG equations and that many unknowns. This equation
system has now a unique solution. After reducing the number of equations by
means of the least square algorithm, the number of equations and unknowns is
still large. There exist some solution techniques for such a system of equations,
e. g., the iterative Gauss-Seidel Method (Wiggins et al., 1976) where starting values
for all unknowns are needed. ti jh = ri = s j = Gkh = Mkh = 0 can be such a set
of initial values. The changes after every iteration can be examined and the computation
is stopped when this rate is below a specified threshold. These kinds of
methods are usually suited to obtain residual static corrections of up to half the
dominant period of the data. For larger static time shifts the non-linear traveltime
inversion methods (explained in Section 3.4) are more suitable.
3.2 Estimation of time shifts between traces
Before it is possible to decompose the traveltime equations, it is, of course, necessary
to determine the time shifts between the traces. Firstly, in the conventional
method two different kinds of traces exist which could be compared. The first
approach is to compare all traces within the CMP gather with each other. The
second method is to compare all traces within the CMP gather with a model trace
or a pilot trace. The determination of the time shift is performed by cross correlating
two traces. Then, the time associated with the maximum of the correlation
result (correlation coefficient over time shifts) is the searched for time shift.
Using a model trace or stacked pilot trace with increased S/N ratio should improve
the cross correlation result and should yield a more reliable time shift. The
simplest pilot trace is the stack of a CMP gather. The short-wavelength components
of the source and receiver residual static shifts will, in general, have an
average value close to zero. This means that the remaining time shift in the pilot
trace approximate the structural term, provided the residual moveout component
is small. Furthermore, a lot of methods exist to improve the quality of the pilot
trace, e. g., for low-coverage data the individual trace is subtracted from the pilot
trace because there may be an undesirable auto correlation component within the
cross correlation. Another method is to apply dip filters or velocity filters to some
areas of the prestack traces before the pilot trace is produced.
19

Chapter 3. Conventional methods for residual static correction
In this linear traveltime inversion method, the cross correlation is not performed
for the whole trace but only for a finite time window. The reason for this proceeding
is that this method accounts for the fact that not all reflectors have the
same structural dip. In the case of differently dipping reflectors, the data should
be divided into separate time windows so that the dip is almost constant within
each window. However, if the whole trace is used, the structural and residual
NMO term are averaged whereby in most cases no large errors are introduced.
Furthermore, the quality of the cross correlation is better using a long window
with many primary reflections. Furthermore, at shallow events and large offsets
the surface consistency is not fulfilled. Thus, small traveltimes should not be used
for the cross correlation. Time windows should be chosen in such a way that at
least several cycles of the dominant frequency associated with the main event are
inside the time window.
3.2.1 Searching for the maximum cross correlation
The time at the maximum peak of the correlation is the searched-for time shift.
This is the simplest approach, but yields only in some cases the correct solution
(see Figure 3.1 trace #1). There is a basic rule of thumb for picking: to pick the
peak closest to zero time unless there is another one 10 dB higher in magnitude.
The decibel difference N between the larger amplitude AY at point Y and smaller
amplitude AX at point X is given by N = 20 lg(AY/AX). This situation is illustrated
in Figure 3.1 trace #2, where peak X is the closest to zero but peak Y has a
magnitude 10dB greater. Another case is shown with trace #3 in Figure 3.1. There,
one of the input traces to the cross correlation is phase shifted, and the relevant
global extremum is not a maximum but a minimum. This minimum is the point
that should be picked in such cases.
The search for the true time shift can be simplified by defining a maximum possible
time shift that constrains the part over which the cross correlation is performed
and the maximum is searched for. Another reason to limit the maximum
possible time shift is the presence of short-period multiples, reverberation energy,
or data with narrow bandwidth or high noise level. There, the cross correlation
can yield a multitude of peaks with almost the same amplitude that can cause
uncertainty in the estimated time shifts (cycle skipping). On the one hand, the
maximum time shift must be larger than all expected combinations of shot and
receiver static time shifts. On the other hand, if this time shift is larger than the
dominant period of the data, then cycle skips might occur, especially in data with
low S/N ratio. Such cases must be taken into account during the picking part of
the processing.
The normalized cross correlation (see Appendix B) has a maximum amplitude of
1 if the two traces are identical. Thus, it is possible to use the value of the peak
(normalized correlation coefficient) to define a criterion for the reliability of a pick
20

Amplitude
Negative time shifts
0
X
Y
X
X
Y
Positive time shifts
3.3 Stack power maximization
trace #1
trace #2
trace #3
Figure 3.1: Cross Correlations to illustrate possible time picks
that serves as a weight factor for the resulting time shift equation (Equation 3.3).
Furthermore, the static time shifts could, in general, only be estimated as an integer
multiple of the sampling interval and if the sampling interval is, in general,
lower than 1 ms. However, the residual static correction is in the order of a few
ms. Therfore, an interpolation of the cross correlation trace should be considered.
3.3 Stack power maximization
In the linear traveltime inversion method, picking peaks of cross correlations (see
Section 3.2.1) is sensitive to errors in the presence of ambiguities or noise. That is
why the stack power maximization method was introduced by Ronen and Claerbout
(1985). The idea of this approach is that the static time shifts should be determined
to maximize the power, i. e., the sum of the squared amplitudes of the
stack. The power of the stacked section is a good measure of the quality, because
if all traces are aligned with no relative shift, the stack have the highest power.
In principle, one can try every, within reasonable limits, possible time shift δt of
a single trace inside a CMP gather, stack along the offset direction and sum the
squares of the stack S. Then, one chooses the time shift which gives the highest
power E given by (Cox, 1974)
E(Δt) = � S 2 (Δt) ! = max. (3.8)
What is actually done in the stack power method is equivalent but more efficient:
a super-trace built from all traces of the shot profile (trace F in Figure 3.2) is cross
correlated with another super-trace built of all the traces in the relevant part of
the stack without the contribution of that shot (trace G in Figure 3.2). Then, the
21

Chapter 3. Conventional methods for residual static correction
UNSTACKED
DATA
STACK
SUPER−TRACE
CROSS−CORRELATION
OFFSET
SHOT PROFILE
MID−POINT
* =
F G
Figure 3.2: Example of super-traces for one moveout corrected shot gather. Supertraces
F and G are cross correlated to determine the corresponding source static.
Figure taken from Ronen and Claerbout (1985)
global maximum of that cross correlation is picked, which yields the source static
of this shot. Afterwards, the stack is corrected and the process is analogously
repeated for each shot and receiver. Finally, a residual static corrected prestack
dataset is obtained and a new stack can be performed. It is possible to apply the
stack maximization method again to the corrected traces, which means to build
up new super-traces and to cross correlate them, or to consider the stack as the final
result. Usually, the process is repeated 5-20 times to achieve a convergence in
the results. If even more iterations are needed, the cost of the stack power method
is lower than the cost of traveltime picking. For the evidence of the equivalence
of maximizing the power of the sum and maximizing a cross correlation, see Ronen
and Claerbout (1985). Furthermore, an optional constraining routine may be
included in the procedure to remove linear trends and drop outs from the estimated
static time shifts and to avoid that the process selects a local maximum of
the stack power rather than the searched-for global maximum.
3.4 Non-linear traveltime inversion
In Section 3.1, the residual static corrections are estimated by linear inversion of
observed traveltime deviations. Rothman (1985) showed that the linearization of
equation 3.3 fails, if τ contains errors due to large static time shifts and noisy data.
This also applies to the stack power maximization method of Ronen and Claerbout
(1985). Therefore, a global optimization technique must be used for such
22

3.4 Non-linear traveltime inversion
Figure 3.3: Simulated annealing technique; the normalized stack power is plotted
against the number of iterations performed (Figure taken from Rothman, 1986).
problems. Rothman (1986) introduced a method in which, the cross correlation
is transformed into a probability distribution, instead of picking the peaks in the
cross correlation. From this probability distributions, random numbers are drawn
with the help of a simulated annealing algorithm and used to iteratively update
estimates of the static corrections until the stack converges to the maximum stack
power (Equation 3.8). The simulated annealing technique is based on a Monte
Carlo algorithm and has its origin in chemistry where the successful growth of
a crystal is tied to the global minimum of an optimization process (Kirkpatrick,
1983). Figure 3.3 shows the normalized stack power against the number of iterations.
It can be observed that the power initially decreases very quickly, but in
this example it starts around the 1000th iteration to converge to the final solution.
Usually, this approach needs a lot of iterations and according to this it needs a
lot of computer time and should therefore only be used for data with large residual
static time shifts. Another global optimization method is a genetic approach
based on biological processes that mimic biological evolution (see Wilson, 1994).
23

Chapter 4
Common Reflection Surface stack
4.1 Theory
The common-reflection-surface (CRS) stack (e. g., Mann, 2002) approximates
the reflection response of an unknown reflector segment in depth by means of
a second-order surface. This so-called stacking operator (green surface in Figure
4.2) is fitted to the true reflection traveltime surface (blue surface) in the (txm-h)
space, using coherence analysis. The CRS stacking surface is described by
t 2 �
hyp(xm, h) =
t0 +
2 sin α
v0
�2 (xm − x0) + 2t0 cos2 �
α (xm − x0)
v0
2
+
RN
h2
�
. (4.1)
RNIP
This traveltime is expressed in terms of midpoint xm and half-offset h coordinates.
The three parameters α, RNIP, and RN are the CRS attributes. Their physical
meaning is explained by means of two hypothetical experiments. These experiments
were introduced by Hubral (1983) and are called eigenwave experiments,
because the respective wavefronts before and after the reflection at the point of
interest are the same except for their direction of propagation. In the following,
only the upgoing wavefronts are considered. In the first of these two experiments,
a point source is placed at point SNIP, located on a reflector in the subsurface (see
Figure 4.1). The suffix of SNIP indicates that this is the normal incidence point
(NIP) of the central ZO ray. Figure 4.1 shows a wavefront with a point source in
SNIP at different instants of time.
The radius of local curvature of the NIP wavefront at x0 is the CRS attribute RNIP.
The term local is used because the wavefronts, in general, deviate from a circular
form, e. g., due to refraction at curved interfaces during their propagation. The
CRS attribute α is the emergence angle of the normal ray (blue line in Figure 4.1),
which is measured with respect to the surface normal.
The second experiment is performed in the same way, but now starting with a
wavefront that has the local curvature of the reflector at SNIP. Such an experi-
25

Chapter 4. Common Reflection Surface stack
z
Normal wave
NIP wave
SNIP
v 2
Figure 4.1: Illustration of the two eigenwaves, viz., the NIP wave and the normal
wave at several instants of time.
ment is called an exploding reflector experiment. All normal rays belong to a
certain reflector segment in the subsurface and are perpendicular to the reflector
segment. The emergence angle α is the same as the emergence angle from the NIP
wave and the CRS attribute RN is the radius of the generated normal wavefront
measured at x0. The brown lines in Figure 4.1 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 velocity v0 in Equation 4.1 is the near-surface velocity and is assumed to be
known a priori. If v0 is unknown, it can be calculated as the reciprocal slope of
the traveltime curve corresponding to the direct wave within the dataset. The
CRS stacking operator can be derived in different ways. Höcht (1998) presented
a geometrical approach which yields a parametric representation of the stacking
operator.
He proceeds as follows: firstly, the equation of the common-reflection-point
(CRP) trajectory for a homogeneous overburden is derived. A CRP trajectory defines
the location of all primary reflection events in the (t-xm-h) space that pertain
to the same reflection point on the reflector (brown line in Figure 4.2).
In the next step, the obtained representation for the CRP trajectory is expanded
to the case of inhomogeneous overburden by means of the concept of object and
image point, known from geometrical optics (see, e. g., Bergmann and Schaefer,
1993): an auxiliary model is created with constant velocity, which is equal
to the near-surface velocity. The corresponding image points for this model are
appointed, such that the two hypothetical experiments, now performed in the
auxiliary model, yield exactly the same wavefront curvatures and emergence angle
as in the actual model. The traveltimes of these two models are different but a
relationship can be found. Thus, the equation for a homogeneous overburden can
26
x 0
α
v 0
v 1
x

4.2 Implementation
be applied to the auxiliary model and with the relationship of the two different
traveltimes an approximated CRP trajectory for an inhomogeneous overburden
can be derived.
In order to achieve an approximation of the kinematic reflection response (green
surface in Figure 4.2) of an arbitrarily curved reflector segment (red arc segment
in Figure 4.2) in depth, the basic idea is to place a CRP trajectory at every ZO
point in the (t-xm-h) space corresponding to the reflector segment. Furthermore,
the time shift between the traveltimes of the auxiliary model and the actual model
is approximately constant for all CRP trajectories. Thus, one obtains an system
of equations which represents a family of CRP trajectories. Unfortunately, this
semi-parametric solution is very difficult to handle.
As an explicit function is much easier to implement in the CRS stack approach, a
second order Taylor-series expansion with respect to t 2 is more convenient. The
result of this expansion is the hyperbolic Equation 4.1. It is also possible to expand
with respect to t which yields a parabolic form. However, Ursin (1982)
concluded that a hyperbolic approximation is better suited than a parabolic one.
With Equation 4.1, a formula is at hand that is easy to handle and easy to implement
.
The advantage of the CRS stack is not only the simulation of a 2-D ZO section
with improved S/N ratio and better continuity of the events compared to the
conventional NMO/DMO/stack. It also provides additional information about
the subsurface in terms of the wavefield attributes α, RNIP, and RN, and does not
require the knowledge of a velocity model. Therefore, it is a data-driven stacking
procedure that makes use of information directly provided by the input data,
only.
4.2 Implementation
4.2.1 Search for CRS attributes
Starting from the hyperbolic Taylor expansion of the CRS reflection response
(Equation 4.1), the CRS attribute triplet must be determined that builds up a surface
that fits best an actual reflection event. This is a non-linear global optimization
problem. In principle, it is possible to solve it by a three parametric search,
however this direct solution is too time consuming. Therefore, Müller et al. (1998)
and Müller (1999) suggested to perform three subsequent one-parameter search
steps:
1. Automatic CMP stack:
In the CMP configuration (xm = x0), the CRS operator (Equation 4.1) depends
only on one (combined) parameter:
27

Chapter 4. Common Reflection Surface stack
Depth [m] Time [s]
0.6
0.4
0.2
0
−200
−400
−600
t
−1000
h
−500
P0
x 0
CRS stack surface
R
CRP trajectory
KR
0
Midpoint [m]
500
xm
1000
0
300
400
200
Half−offset [m]
100
Figure 4.2: The green surface is the CRS stacking surface, i. e., the primary approximate
reflection response for all zero-offset and finite-offset reflections associated
with the red arc segment. The blue lines represent the actual CO traveltime
curves. KR is the local curvature of the reflector at reflection point R. The CRP trajectory
for the emergence point x0 is shown in brown.
28
t 2 hyp (xm, h)|(xm=x 0) = t 2 0 + 2 t0
v0
cos 2 α h2
RNIP
In comparison to this equation, the CMP stack formula
. (4.2)
t 2 (h) = t 2 0 + 4h2
v2 , (4.3)
stack
shows that the stacking velocity vstack can be alternatively expressed in
terms of the CRS wavefield attributes:
v 2 stack = 2v0RNIP
t0 cos2 2v0
=
α t0q with q = cos2 α
RNIP
(4.4)
Hereby, the parameter q is the searched-for combined parameter. This parameter
is varied to fit a hyperbolic curve to the traveltime curve in the
CMP gather. The maximum of the coherence determines the best fitting

4.2 Implementation
curve. Note that the near-surface velocity v0 is constant within the paraxial
vicinity of the emergence location of the considered normal ray.
The name for this first step, automatic CMP stack, results from similarity
to the conventional CMP stack method and the fact that it can be applied
without any user interaction due to the coherence analysis.
2. Linear ZO stack: From step one, a ZO stacked section is obtained which
has a higher S/N ratio than the prestack data. With the assumption RN= ∞
and with h=0, i. e., within the ZO section, Equation 4.1 reduces to
t 2 hyp (xm,
�
h)| (h=0,RN=∞) =
t0 +
2 sin α
v0
�2 (xm − x0) . (4.5)
The assumption RN = ∞ is possible for small midpoint displacements.
From Equation 4.5, it is possible to determine the emergence angle α. In
turn, with α and Equation 4.4 RNIP can be calculated. The name of this step
based on the fact, that a linear stacking operator is used to determine α in
the ZO section.
3. Hyperbolic ZO stack:
After the two parameters RNIP and α are determined, the last attribute RN
is searched for again in the CMP stacked ZO section. For this case, Equation
4.1 is reduced to
t 2 hyp (x, h)|(h=0)
�
= t0 + 2
�2 (x − x0) sin α +
v0
2
t0 cos
v0
2 2 (x − x0)
α . (4.6)
RN
Again, the searched RN is obtained by determining the highest coherency
along the calculated traveltime curve in the prestack data. In this step RN,
is determined in the ZO section by using a hyperbolic stacking operator,
therefore this step is called hyperbolic ZO stack.
This search is performed for every grid point of the simulated ZO section and
an attribute triplet at each point is obtained. By means of these three attributes
and Equation 4.1, the CRS stacking operator is defined. Afterwards, the stack
along this operator is performed, and the result is assigned to the corresponding
ZO traveltime point. This method is applied for every ZO point, successively.
The result is the so-called initial CRS stack. The term initial is used to emphasize
that the parameter for this stack serve as initial values for a local optimization
process, where all three parameters are refined simultaneously within the entire
spatial CRS aperture. The result of the initial stack are not only the starting values
for the optimization process, but it also includes a preliminary stacked ZO
29

Chapter 4. Common Reflection Surface stack
Depth [m] Time [s]
0.6
0.4
0.2
0
−200
−400
−600
t
−1000
h
−500
P0
x 0
CRS stack surface
R
0
Midpoint [m]
500
xm
1000
0
300
400
200
Half−offset [m]
100
Figure 4.3: Same model like in Figure 4.2. Illustrated is the projection of the first
interface Fresnel zone (red arc segment) for ZO. The marginal normal rays are
shown as red solid lines and the resulting projected first Fresnel zone in the time
domain is signified with the red dotted lines.
section. The stack with the optimized values is called optimized CRS stack. Further
refinements are required in case of conflicting dip situations, corresponding
to the presence of relevant local coherence maxima.
As mentioned before, the search for the best fitting CRS stacking operator, which
is potentially related to a reflection event, is done by a coherence analysis. This
means that a multitude of different operators is calculated, the respective coherence
values are determined, and the attribute triplet which yields the highest
coherence value is selected. Different coherence measures might be used. In the
used implementation of the CRS stack the coherence measure semblance is used
(see Neidell and Taner, 1971; Mann, 2002).
4.2.2 CRS aperture
The CRS stacking operator deduced in Section 4.2.1 is an approximation of the
kinematic reflection response of a curved reflector segment in a paraxial vicinity
of the central ray. Therefore, it is necessary to limit this operator to a range where
the approximation is sufficiently accurate, otherwise it would sum up noise. The
projected first Fresnel zone serves as a suitable measure for the size of the ZO
30

Time [s]
3.5
3.0
2.5
2.0
1.5
1.0
0.5
ZO aperture
0
-0.8 -0.4 0 0.4 0.8
Midpoint displacement [km]
(a)
CMP aperture
0 0.4 0.8 1.2 1.6
Half-offset [km]
(b)
Half-offset [km]
1.5
1.0
0.5
4.2 Implementation
CRS aperture
0
-0.8 -0.4 0 0.4 0.8
Midpoint displacement [km]
Figure 4.4: a) ZO aperture for a dominant frequency of 30 Hz and an average
velocity ranging from 1.5 km/s to 3 km/s. The minimum aperture is set to 50 m,
the maximum to 750 m. b) CMP aperture given by the points (t0, h)=(0.5 s, 0.1 km)
and (3.5 s, 1.75 km). c) Spatial CRS aperture set up by the ZO and CMP apertures
at t0 = 3 s (taken from Mann, 2002).
aperture.
In this work, it is so far assumed that the seismic energy propagates along mathematical
rays within a virtually infinitesimal volume. But that does not accord
to the truth, because the wave propagation is affected by a finite volume in the
subsurface around the considered ray. The first Fresnel volume is the part of that
volume in which all rays contribute constructively at the emergence point on the
surface. The size of this area around the central ray depends on the frequency
content of the data. Furthermore, the first interface Fresnel zone is the intersection
of the Fresnel volume and the reflector and it is, thus, a measure for the
maximum achievable resolution in terms of reflector properties.
The first interface Fresnel zone stands for a depth domain area, but one wants to
limit the CRS stacking operator within the time domain. Hubral (1983) developed
a way to interlink these two domains. The emergence locations of all normal
rays reflected within the first interface Fresnel zone (Figure 4.3) define the above
mentioned projected first Fresnel zone for the ZO configuration.
To determine the projected first Fresnel zone in this way, a certain knowledge
of the subsurface is necessary. Hubral (1983) showed that an adequate approximation
exists which makes it possible to determine the projected first Fresnel
zone out of the data without additional information. The result is an equation for
the ZO aperture which depends on the three CRS attributes and the difference
between reflection and (hypothetical) diffraction traveltimes.
However, the implementation of the ZO aperture in the second step of the search
(c)
31

Chapter 4. Common Reflection Surface stack
for the CRS attributes, the linear ZO stack, is not possible because of two reasons:
At first, the full attribute triplet is necessary to define the aperture as noted below,
and at that time RN is not yet available. At second, the coherence analysis is very
sensitive to the number of contributing traces. However, an aperture is essential
and, therefore, an approximated projected first Fresnel zone is used which is calculated
for a horizontal interface (RN = ∞) with a homogeneous overburden. An
example for such an aperture is shown in Figure 4.4(a).
The CMP aperture as well as the full spatial aperture for the CRS super gather
could only be assigned empirically. In the current implementation, the CMP aperture
is a linear function of the ZO traveltime, which is defined by two user-given
points, see Figure 4.4(b). An elliptic form is used as a CRS super gather aperture
with the half-axes given by the ZO and the CMP apertures, respectively (Figure
4.4(c)). To get more information about the implementation and a lot of data
examples of the CRS stack, see Mann (2002).
32

Chapter 5
Residual static correction by means
of CRS attributes
In Chapter 3, a review of the conventional residual static methods was given.
Now, in this chapter a new approach will be presented which uses the CRS attributes
and the CRS stack for the determination of the residual static corrections.
It is a linear traveltime inversion method and it is in principle similar to the technique
of Ronen and Claerbout (1985). The improved S/N ratio of the CRS stack
compared to the NMO/DMO/stack method and the fact that the CRS stack is
purely data driven (see Mann, 2002; Trappe et al., 2001) are advantageous for the
new residual static correction method explained the in the following.
Figure 5.1 shows the basic steps of this method. The first step is to perform at least
the initial 2D ZO CRS stack to obtain the CRS attribute sections and the simulated
ZO section. Either the optimized or the initial 2D ZO CRS stack can be used for
the following steps. In Chapter 6, an analysis about the improvements, whether
the optimized or the initial CRS stack is used, is discussed. In the following sections,
the theory behind the particular steps is explained and in Chapters 6 and 7
the approach is applied to synthetic as well as to real data.
5.1 Moveout correction
As mentioned above, the CRS stack provides the CRS attributes (α, RN, RNIP).
With the knowledge of these attributes, the events which are approximated by the
CRS stack operator with the considered ZO traveltime t0, could be transformed
into a horizontal plane (black plane in Figure 5.2): every point of the prestack data
on the CRS stack operator is shifted by the corresponding moveout correction
tmoveout(x, h) = thyp(x, h) − t0 . (5.1)
The reflection response of each reflector segment is now independent of the half-
33

Chapter 5. Residual static correction by means of CRS attributes
method 1
method 2
CRS search and stack
CRS attributes pilot trace
CRS moveout correction of
the CRS super gather
cross correlation of pilot trace
and all CRS super gather traces
summing up correlation results for
common−source and common−receiver locations
correlation stack maxima
residual static values
correcting prestack traces
final stack
method 3
correcting and stacking
of CRS supergather
Figure 5.1: Flowchart for the residual static correction by means of CRS attributes.
An iterative process is possible in three different ways: 1) with a complete ’new’
CRS stack (dashed line) in each iteration; 2) moveout correction with the attributes
from the last CRS stack (dotted line), or 3) only residual static correction
of the CRS supergathers (dashed-dotted line). It is to be mentioned that Method 3
is not surface consistent.
offset h and the midpoint xm. This correction must be done for every time sample
of each simulated ZO trace. The result for one ZO trace is then a moveout corrected
CRS supergather, which contains all prestack traces lying inside the corresponding
CRS aperture. Thus, one prestack trace is contained in several CRS
supergathers but with different moveout corrections.
34

0.7
0.6
0.5
t [s]
0.4
0.3
0.2
5.2 Cross correlation and search for the estimated static time shift
CRS aperture
P 0
CRS stack surface
moveout corrected reflection event
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6
xm [km]
0
0.4
h [km]
0.2
Figure 5.2: The gray lines are the true CO reflection responses of a curved reflector.
The corresponding model is not displayed, it is similar to the model in
Figure 4.2 and 4.3 but not identical. The blue surface is the CRS stack surface for
one point P0 in the ZO section and the yellow line is the CRS aperture belonging
to this CRS surface. Furthermore, the black half ellipse is the moveout corrected
reflection event which is constrained by the CRS aperture.
5.2 Cross correlation and search for the estimated
static time shift
The new method is based on cross correlation (see Appendix B), namely, the cross
correlation between every single moveout corrected trace inside the CRS super
gather and a pilot trace. Here, the traces of the simulated ZO section serve as pilot
traces. This is one advantage of this new approach: as mentioned before, the CRS
stack yields a ZO stack section with a higher S/N ratio compared to conventional
stacking methods. Thus, in this approach, the quality of the pilot trace is much
better than that used in conventional residual static correction methods.
After the cross correlation is performed, all correlation results that belong to the
same source or receiver location, respectively, are summed up. Finally, the resid-
35

Chapter 5. Residual static correction by means of CRS attributes
ual static value is usually given by the time associated with the maximum of the
summed cross correlation results. Two assumptions are essential for this procedure:
structural and residual moveout terms are non-existent and the mean of the
receiver as well as the source and receiver residual static corrections are equal to
zero.
In some cases the searched time shifts do not correspond to the global maximum
but to a local maximum or a global/local minimum due to different reasons (see
Section 3.2.1), e. g., cycle skips. The discussed implementation of this residual
static method searches only for the global maximum and the corresponding time
shift. Nevertheless, to avoid picking the wrong time a limit for the maximum
correlation shift is implemented. The limit has to be defined by the user.
Problems might occur at the boundary of the dataset as only few correlation results
will contribute to a source or receiver location, respectively. In the future,
it is needful to implement a threshold which defines the minimum number of
traces in a supergather that are necessary to use this supergather for residual
static correction. From this follows that it is not possible to determine residual
static corrections for every shot and receiver location at the borders of the seismic
line.
Furthermore, the calculated coherence along the CRS stack surface can be used
to weight every time sample of the pilot trace as well as every time sample of the
moveout corrected traces of the CRS supergather before the cross correlation is
performed. The CRS stacking surfaces are only calculated for the ZO traces and,
thus, only a coherence section for the ZO traces is available. Therefore, the time
samples of the traces of the CRS supergather are weighted with the coherence of
the corresponding ZO trace time sample. In addition, the coherence is set to be
zero for every time sample which is outside the CRS aperture.
5.3 Iterations
After the residual static values are obtained from the cross correlation results, the
prestack traces are time shifted with the corresponding total time shifts. The total
time shift is simply the sum of the corresponding source and receiver static values
of each prestack trace. Now, there are several possibilities to proceed (see also
flowchart in Figure 5.1). If the quality of the CRS stack of the corrected prestack
traces is adequate, the corrected prestack traces and the stack, respectively, can
be used as the final result. If the result of the stack is not adequate, additional
iterations can be performed. The quality of the result must be evaluated by an
interpreter with the aid of the values of the static correction. If the most of the
values are zero it make no sense to perform another iteration. It might happen
that the static correction values do not converge. Consequently, the result of the
stack does not necessarily improve with more iterations and the quality of the
36

5.3 Iterations
stack should always be kept in view.
In principle, three different ways exist how the iterations can be realized. One
way is to repeat the whole process from the beginning, shown as method 1 in
Figure 5.1. That means to perform a complete new CRS stack with the corrected
prestack traces, moveout correction of the CRS supergather with new CRS attributes
and performing the cross correlation to estimate a new set of residual
static correction values. Finally, the prestack traces can be residual static corrected
again. As the CRS search is time consuming, this method requires the
most computing time.
The second way is to apply residual static corrections to the prestack traces and
to perform the moveout correction with the ’old’ attributes (method 2 in the
flowchart). This method saves a lot of time because the CRS search does not
have to be performed again. However, the attributes are applied to the wrong
time samples. In future research, it has to be analyzed how this error affects the
results and how this error relates to the saving of time.
The fastest method is the third one (Method 3 in the flowchart). Here, the moveout
corrected prestack traces within a CRS supergather are corrected with the
determined time shift. Afterwards, only the stack for every CRS supergather is
performed. However, this method violates the assumption of the surface consistency
as it is applied to the traces after a non-linear moveout correction. Shifting
the moveout corrected traces and performing an inverse moveout correction
shows that the resulting correction in the primary traces are different for diverse
times. Many conventional methods, e. g., the method from Ronen and Claerbout
(1985) do not perform a new NMO after applying the residual static correction
and, thus, also violate the assumption of surface consistency. It remains to examine,
considering data examples (Chapters 6 and 7), how strongly these violations
affect the results of the residual static correction.
Furthermore, various combinations of the three ways are possible, that means,
e. g., that the residual static time shifts are estimated several times only by means
of correcting the supergather and then, after a specific number of iterations, a
complete new CRS stack is performed. The new attributes can then, again, serve
as a basis for further iterations. One of them is applied to the real data in Chapter
7 and the advantages and drawbacks are analyzed. The used strategy has to
be chosen appropriately for each particular dataset and a cost-benefit calculation
should be performed.
37

Chapter 6
Synthetic data example
The new approach for static correction by means of CRS attributes is tested on a
synthetic data example. The results of this test are presented in this chapter. The
three different possibilities to realize the estimation of the static corrections, as
mentioned in Chapter 5, are compared and their advantages and drawbacks are
evaluated. Furthermore, different settings of the maximum allowable time shift
and/or the length of the cross correlation window are examined.
6.1 Model and survey design
The model, is used to generate the synthetic data set is shown in Figure 6.1(a).
There are four constant velocity layers, whereby the first reflector is a horizontal
interface, the second one has a dip of -7 ◦ , and the third one has a syncline
structure. The velocity increases with depth from layer to layer. The contrast in
velocity between the LVL and the next layer with a significantly higher velocity
is large. Most of the seismic energy is reflected at the first interface and deeper
reflectors can be hardly seen in the seismic data. Of course, this is also the case in
real data but in this first synthetic model the LVL was neglected to simplify matters.
Instead, the LVL is simulated by adding artificial surface consistent residual
static time shifts to the seismic data for this model.
The survey design is adapted from a marine data acquisition geometry. The first
source point is located at 2000 m. The maximum offset between source and receiver
is 2000 m and the minimum offset is 0 m. The receiver spacing is 25 m. Furthermore,
the source points have also a distance of 25 m and range to the maximum
source location of 10 000 m. Thus, the prestack dataset consists of 26001
traces and 721 CMP gathers with 321 different source locations and 401 different
receiver locations. The used wavelet is a zero-phase Ricker wavelet with a dominant
frequency of 30 Hz. The traces of the multicoverage dataset have a time
sampling interval of 4 ms.
39

Chapter 6. Synthetic data example
Depth [km]
Distance [km]
0 2 4 6 8 10
0
v 1=
2.2 km/s
1
2
3
4
5
6
(a) 2D model
v = 2.5 km/s
2
v = 3.0 km/s
3
v = 3.5 km/s
4
time [s]
1.0
1.5
2.0
2.5
3.0
3.5
4.0
CMP location [km]
2 4 6 8 10
(b) CRS stacked ZO section
Figure 6.1: (a) Model used to simulate a synthetic prestack dataset, (b) result
of the optimized CRS stack from prestack traces without any synthetic noise or
artifical static time shifts added.
The result of the optimized CRS stack for this synthetic dataset without artificial
noise and without artifical static time shifts is shown in Figure 6.1(b). The used
processing parameters for the ZO simulation are given in Table 6.1. The reason for
the lack of data in the ZO traces of the first 40 CMPs is due to the fact that the first
CMPs are not fully covered due to the CS configuration and due to the chosen
CMP aperture. Only traces with larger offsets are arranged in this CMPs and
Figure 4.4 shows that traces with larger offsets only contribute for larger times in
the CRS stack. However, the three reflection events can clearly be recognized in
the whole section.
Synthetic noise with a S/N ratio of 10 is now added to the prestack traces and
a new CRS stack is performed with the new prestack dataset. The result of this
stack is shown in Figure 6.2(a). The noise does not influence the shape of the
wavelet at the events in the ZO section, it can only be seen at the borders of
the events and between the events. Subsets (second reflector, 1.7 - 2.6 s) of the
coherence section and the three CRS attribute sections for these data are shown
in Figures 6.16 to 6.19 subfigures (a).
40

6.1 Model and survey design
Context Processing parameter Setting
Dominant frequency 30 Hz
General Coherence measure Semblance
parameters Data used for coherence analysis Original traces
Temporal width of coherence band 28 ms
Velocity and Near surface velocity 2200 m/s
constraints Tested stacking velocities 2000 . . .6000 m/s
Simulated ZO traveltimes 0 . . .4.5 s
Target Simulated temporal sampling interval 4 ms
zone Number of simulated ZO traces 721
Spacing of simulated ZO traces 12.5 m
Minimum ZO aperture 50 m @ 1 s
Aperture Maximum ZO aperture 567 m @ 2.5 s
and Minimum CMP aperture 1000 m @ 1 s
taper Maximum CMP aperture 3000 m @ 2.5 s
Relative taper size 30 %
Automatic Initial moveout increment for largest offset 4 ms
CMP stack Number of refinement iterations 3
Linear Tested emergence angles −60 . . .60◦ ZO Initial emergence angle increment 1◦ stack Number of refinement iterations 3
Hyperbolic Initial moveout increment for largest ZO distance 4 ms
ZO stack Number of refinement iterations 3
Hyperbolic Initial moveout increment for largest offset 4 ms
CS/CR stack Number of refinement iterations 3
Coherence threshold for smallest traveltime 0.1
Coherence threshold for largest traveltime 0.1
Maximum number of iterations 100
Local Maximum relative deviation to stop 10−4 optimization Initial variation of emergence angles 6◦ Initial variation of RNIP
5 %
Initial variation of transformed RN
6◦ Transformation radius for RN
100 m
Table 6.1: Synthetic data example: processing parameters used for the ZO simulation
by means of the CRS stack.
The next step is to add artifical residual static time shifts to the noisy data, namely
surface consistent source and receiver static time shifts, respectively, between
-10 ms and 10 ms. This yields total static time shifts between -20 ms and 20 ms
for every trace. The result of the optimized CRS stack of this data is shown in
Figure 6.2(b). In this figure, it is obvious that the wavelet is blurred; it is almost
impossible to recognize the shape of the wavelet, and the positions of the reflection
events are also not accurately detectable. The reason for the blurred wavelet
is easy to explain with Figure 1.4. All traces of the CRS supergather have different
41

Chapter 6. Synthetic data example
time [s]
1.0
1.5
2.0
2.5
3.0
3.5
4.0
CMP location [km]
2 4 6 8 10
(a) CRS stack section with noise
time [s]
1.0
1.5
2.0
2.5
3.0
3.5
4.0
CMP location [km]
2 4 6 8 10
(b) CRS stack section with noise and artifical
static time shifts added
Figure 6.2: The result of the optimized CRS stack from prestack traces with noise
(a) and with noise and static time shifts between ± 20 ms (b)
static time shifts. If all this traces are summed up to the simulated ZO trace, the
resulting wavelet has not the shape of the original wavelet (see Figure 1.4(b)). In
contrast, the stack of the original traces or corrected traces, respectively, reproduces
the wavelet, even with a higher amplitude than a single trace. The noise
is, of course, unaffected by the static time shifts because of its incoherent nature.
Subsets (second reflector, 1.7 - 2.6 ms) of the coherency section and the three CRS
attribute sections for this stack are shown in Figures 6.16 to 6.19 subfigures (b).
The coherence value of the original dataset along the event (Figure 6.16 (a)) is 0.7
or above whereas the maximum value at this event in the coherence section with
artifical static (Figure (b)) time shifts is 0.2. The explanation for this behavior is, in
principle, the same as for the blurred wavelet of the stack. The implemented coherence
measure in the CRS stack is “semblance”, which provides a normalized
ratio of the correlated and uncorrelated seismic energy in data subvolume around
the CRS operator. As the trace are now time shifted, the coherence analysis is not
able to sum up the same value of seismic energy as it is the case if the trace are
not shifted. Due to the used coherence band and the temporal extension of the
42

offset [m]
1000
750
500
250
0
I
II
III
IV
4100 4500 4900 5300
midpoint [m]
(a) Spatial CRS aperture at t0 = 5 s
V
time [s]
0
1
2
3
4
5
6.1 Model and survey design
I II III III III IV V
(b) Subsets of the CRS supergather
Figure 6.3: A schematic example for the CRS supergather. (a) The spatial CRS
aperture at t0 = 5 s, where the green and blue lines indicate the selected CMP
gathers which are shown in Figure (b).
wavelet, the searched attributes are not only determined along the CRS operator
but within a temporal window around it.
The artifical static time shift also affects the sections of the emergence angle α
(Figure 6.17) and radius of curvature of the NIP - wavefront RNIP (Figure 6.18
(a)) in the same way. Directly along the events these two attributes are nearly
the same as without static time shifts, but the width of the band with consistent
values is smaller and the borders of the band are not as sharp. The reason for
this is again the coherence. In the border areas no distinguished CRS operators
exist; the operator which sums up most energy can change from sample to sample
and, accordingly, the CRS attributes, too. Also on the left side, at the CMPs with
lower coverage, the band with consistent values breaks off earlier than in the
attributes sections without static time shifts. For supergathers with many traces
the coherence measurement is less affected by time shifts.
In the sections of the curvatures of the normal wavefronts KN = 1/RN (Figure 6.19
(a) and Figure 6.19 (b)), the value directly along the events are the same in both
sections, namely zero. The second reflector is planar. Consequently, its curvature
is zero. However, another characteristic appears at the border of the band in the
KN section: the section with artifical static time shift shows higher positive or negative
values, respectively. Higher positive curvatures of the normal wavefront at
the upper part of the band create stronger curved CRS operators which intersect
the event under a larger angle. Consequently, the curvature at the lower border
is negative. It is not investigated why this behavior of the CRS operator yields a
higher coherency value at the static time shifted traces.
43

Chapter 6. Synthetic data example
correlation stacks [#]
8000
7000
6000
5000
4000
3000
2000
1000
0
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
receiver/source position [m]
Figure 6.4: Number of traces contributing to the cross correlation stack for receivers
(blue line) and sources (red line), respectively.
6.2 Residual static correction
In the following, a residual static correction with the new approach based on the
CRS stack and the CRS attributes is applied to the data with artifical static time
shifts. An advantage of this procedure is that the resulting time shifts can directly
be compared with the previously applied artifical static time shifts. Figure 6.3
shows an example for a CRS supergather, where part (a) shows the schematic
shape of a supergather in the xm-h plane (compare with Figure 5.2) and part (b)
shows some CMP gathers from the CRS supergather influenced by the CRS aperture
(compare with Figure 4.4). The displayed CMP gathers are indicated in green
and blue in part (a). Furthermore, the moveout corrected events are visible in part
(b) at a traveltime of 0.8 s and around 3.5 s and especially the second event around
2 s. In case of full coverage of the CMPs, the supergather consists of about 35000
traces in this example.
The three different methods which are applied to obtain the residual static correction
by mean of CRS attributes should be mentioned here again because the
different methods are labeled as Method 1, 2 and 3 in the following (see also Section
5.3):
44
• Method 1: every new iteration step starts with a complete application of the
CRS stack.
• Method 2: the prestack traces are corrected with the residual static time
shifts, but the moveout correction is always performed with the attributes
from the last CRS stack.
• Method 3: the residual static time shifts are only applied to traces in the
moveout corrected CRS supergathers.

6.2 Residual static correction
Method number of temporal max. allowable mean deviation mean deviation
iteration window [s] time shift [ms] receiver [ms] source [ms]
1 1 1 - 4.2 24 2.17 1.98
1 1 0.6 - 1.1 24 1.84 1.73
1 1 1.7 - 2.6 24 1.97 1.93
1 1 2.6 - 4.2 24 3.62 3.56
1 1 0.6 - 4.2 56 1.84 1.73
1 2 0.6 - 1.1 24 1.29 0.95
2 2 0.6 - 1.1 24 1.46 1.18
3 2 0.6 - 1.1 24 1.36 1.00
1 3 0.6 - 1.1 24 1.28 0.95
1 INI 2 0.6 - 1.1 24 1.73 1.50
Table 6.2: Mean deviation of estimated static corrections from the artifical static
time shifts for the different methods, time windows and/or maximum time shifts.
INI in the last row stands for Initial CRS stack used for the processing instead of
the optimized CRS stack.
In the Figures 6.5 - 6.8, 6.13, and 6.14, subsets of the results of the different methods
with different settings regarding the cross correlation windows and/or different
maximum allowable time shifts are displayed. Problems occur in areas
with low coverage of the receivers. In Figure 6.4, the number of cross correlation
results which contribute to the cross correlation stack for receivers (blue line)
and sources (red line) are shown. The number of contributions depends on the
coverage of the particular CMP and on the chosen aperture. The most of these
problematic receiver values are equal to zero. Thus, they do not affect the following
processing. Anyway, in further investigations the problematic receiver, also,
are treated to see what happens with these static corrections. For the estimation
of the mean deviation of the result of the different methods, explained later in
this chapter, only the fully covered sources and receivers are accounted for. For
the source statics, no problem occurs because they are all fully covered. In the
future, it is necessary to implement a coverage threshold for source and receiver
locations at which the static corrections are to be considered. This threshold must
be carefully defined for every dataset according to its characteristic. Figures 6.5 -
6.8, 6.13 and 6.14 show subsets of the results.
6.2.1 Results
Figures 6.5(a) and 6.6(a) show a comparison of the original static time shifts (red
line) and the static time shifts extracted with Method 1 after the first iteration
(blue line). The shape of the static corrections is obtained very well but not always
with the exact amplitude. The mean deviation of the estimated corrections
from the original ones is about 2.17 ms for the receiver statics and about 1.98 ms
45

Chapter 6. Synthetic data example
statics [ms]
statics [ms]
statics [ms]
15
10
5
0
−5
−10
−15
4000 4200 4400 4600 4800 5000
source position [m]
5200 5400 5600 5800 6000
(a) red line: original static time shift; blue line: estimated statics with Method 1 after 1 iteration.
15
10
5
0
−5
−10
−15
4000 4200 4400 4600 4800 5000
source position [m]
5200 5400 5600 5800 6000
(b) red line: original static time shift; green line: estimated statics with Method 1 after 2
iterations.
15
10
5
0
−5
−10
−15
4000 4200 4400 4600 4800 5000
source position [m]
5200 5400 5600 5800 6000
(c) red line: original static time shift; blue line: estimated statics with method 1 after 1 iteration;
green line: estimated statics with Method 1 after 2 iterations.
Figure 6.5: Subsets of source statics with a maximum allowable time shift of 24 ms
and a cross correlation time window of 0.6 s - 4.4 s (I).
46

statics [ms]
statics [ms]
statics [ms]
15
10
5
0
−5
−10
6.2 Residual static correction
−15
4000 4200 4400 4600 4800 5000
receiver position [m]
5200 5400 5600 5800 6000
(a) red line: original static time shift; blue line: estimated static time shift with Method 1 after
1 iteration
15
10
5
0
−5
−10
−15
4000 4200 4400 4600 4800 5000
receiver position [m]
5200 5400 5600 5800 6000
(b) red line: original statics; green line: estimated static time shift with Method 1 after 2
iterations
15
10
5
0
−5
−10
−15
4000 4200 4400 4600 4800 5000
receiver position [m]
5200 5400 5600 5800 6000
(c) red line: original statics; blue line: estimated statics with Method 1 after 1 iteration; green
line: estimated statics with Method 1 after 2 iterations
Figure 6.6: Subsets of receiver statics with maximum allowable time shift of 24
ms and a cross correlation time window of 0.6 s - 4.4 s (I).
47

Chapter 6. Synthetic data example
statics [ms]
statics [ms]
statics [ms]
15
10
5
0
−5
−10
−15
4000 4200 4400 4600 4800 5000
source position [m]
5200 5400 5600 5800 6000
(a) red line: original static time shift; green line: estimated statics with Method 1 after 2
iterations; yellow line: estimated statics with Method 3 after 2 iterations
15
10
5
0
−5
−10
−15
4000 4200 4400 4600 4800 5000
source position [m]
5200 5400 5600 5800 6000
(b) red line: original static time shift; green line: estimated statics with Method 1 after 2 iterations;
black line: estimated statics with Method 2 after 2 iterations
15
10
5
0
−5
−10
−15
4000 4200 4400 4600 4800 5000
source position [m]
5200 5400 5600 5800 6000
(c) red line: original static time shift; yellow line: estimated statics with Method 3 after 2
iterations; black line: estimated statics shift with Method 2 after 2 iterations
Figure 6.7: Subsets of source static time shifts with a maximum time shift of 24 ms
and a cross correlation time window of 0.6 s - 4.4 s (II).
48

statics [ms]
statics [ms]
statics [ms]
15
10
5
0
−5
−10
6.2 Residual static correction
−15
4000 4200 4400 4600 4800 5000
receiver position [m]
5200 5400 5600 5800 6000
(a) red line: original statics time shift; green line: estimated statics with Method 1 after 2
iterations yellow line: estimated statics with Method 3 after 2 iterations
15
10
5
0
−5
−10
−15
4000 4200 4400 4600 4800 5000
receiver position [m]
5200 5400 5600 5800 6000
(b) red line: original static time shift; green line: estimated statics with Method 1 after 2
iterations; black line: estimated statics with Method 2 after 2 iterations
15
10
5
0
−5
−10
−15
4000 4200 4400 4600 4800 5000
receiver position [m]
5200 5400 5600 5800 6000
(c) red line: original static time shift; yellow line: estimated statics with Method 3 after 2
iterations; black line: estimated statics with Method 2 after 2 iterations
Figure 6.8: Subsets of receiver statics with a maximum time shift of 24 ms and a
cross correlation time window of 0.6 s - 4.4 s (II).
49

Chapter 6. Synthetic data example
statics [ms]
20
10
0
−10
−20
1200 1210 1220 1230 1240 1250
trace number [#]
1260 1270 1280 1290 1300
Figure 6.9: Subset of the total statics after two iterations of Method 1. Red line:
original total static time shift; green line: estimated total statics with Method 1
after 2 iterations
for the source statics (see Table 6.2). After the second iteration with Method 1, the
mean deviation of the obtained statics from iteration 1 and 2 decrease to a value
of 1.84 ms for the receiver statics or to 1.73 ms for the source statics, respectively.
This improvement can also be seen in Figures 6.5(b) and 6.6(b). There are still
parts of the subsets of the static corrections where the estimated static time shifts
do not follow the distribution of the original values, e. g., the source statics between
4075 m and 4125 m. However, they are close to the original ones. In this
part, the original static time shifts have a constant value of -10 ms and the estimated
static time shifts vary in a range of ± 1 ms around this value. However,
other parts exist where the shape is better estimated but with higher amplitude
deviation, e. g., for the sources between 4575 m and 4625 m. In the Figures 6.5(c)
or 6.6(c), the first iteration (blue line) and the second iteration (green line) are
directly comparable. Many improvements can be seen, but, e. g., at the source location
5775 m, 5950 m, or 5975 m, the second iteration deteriorates the estimated
static corrections.
A small subset of the total static corrections is shown in Figure 6.9. At some points
the original static time shifts are obtained very well (trace number 1230 - 1240).
Other points (traces number 1255 - 1260) are not well estimated but at least have
the correct order of magnitude.
A third iteration yields no significant improvements, the mean deviation after the
third iteration differs by less than 1% from the mean deviation after the second
iteration. The variation at some points is ±1 ms, but most of the estimated time
shifts are equal to zero. Thus, the methods almost converged after the second
iteration for this synthetic dataset. As the resulting static corrections of the third
iteration do not differ very much from the results of the second iteration they are
not displayed.
Method 2 and Method 3 are only applicable after the first iteration of Method 1,
50

6.2 Residual static correction
because the CRS stack must be performed at least once to obtain the CRS attributes
and the moveout-corrected CRS supergathers. Therefore, the results of
the different methods can only be compared starting with the second iteration.
The yellow lines in Figure 6.7(a) and 6.8(a) represent the source or receiver static
corrections for Method 3 after the second iteration. In comparison with the the
statics from Method 1 after the second iteration (green line), Method 3 yields less
accurate results, but in comparison with the first iteration of Method 1 (blue lines
Figure 6.5(a)) one can see that the statics after the second iteration of Method 3
are closer to the original ones. At some points Method 3 leads to a better result
especially at the points where the second iteration of Method 1 yields a deterioration
in comparison to the first iteration, e. g., at the source locations 5950 m and
5975 m. However, also independent improvements are visible, e. g., at receiver locations
4575 m - 4625 m. The mean deviation of the second iteration of Method 3
is about 0.17 ms for sources and about 0.23 ms for receivers and, thus, larger than
the mean deviation of the second iteration of Method 1.
The value of the mean deviation of Method 2 is between the value of Method 1
and Method 3. It is for the sources about 1.00 ms and for the receivers 1.36 ms
and there exist also points where this method is better than the others (see Figures
6.8(b) and 6.8(c)).
The advantages of Method 2 and Method 3 are definitely their superior computational
efficient. The complete CRS stack (Automatic CMP stack, Zero-offset
stacks, Initial stack , Local optimization) needs 4.8 h on a MIPS R 1200/400 MHz
processor, whereas the real residual static correction only takes 2.0 h. The CPU
time for the static correction can be reduced in future implementations because
at the moment all supergathers are generated with self-contained programs and
are written to disc. Thus, it is possible to extend the static correction program
such that the supergathers are generated ’on-the-fly’ without buffering on disc.
A realistic comparison of the required CPU time for Method 2 and Method 3 is
not possible at this stage of the implementation. Nevertheless it is possible to say
that Method 3 requires the least CPU time because no moveout correction is necessary.
Instead, only the time shifts have to applied to the previously obtained
supergathers.
So far, all the residual static corrections are estimated by means of the optimized
CRS attributes. The optimization process usually takes more than the half of the
CPU time of the whole CRS stacking approach (see Mann, 2002). The result of
the moveout correction and the following cross correlation performed with the
initial CRS attribute sections and the initial stack section is shown in Figure 6.12.
The direct comparison of this static time shift values with the values estimated
by Method 1 using the optimized attributes shows that the optimized attributes
yield much better results. This is also confirmed by the mean deviation of estimated
static corrections from the original ones (see Table 6.2). The obtained static
corrections are the worst of all different methods after the second iteration. The
51

Chapter 6. Synthetic data example
time [s]
time [s]
1.0
1.5
2.0
2.5
3.0
3.5
4.0
time [s]
CMP location [km]
2 4 6 8 10
(a) CRS stack section (with noise and artifical
static time shifts)
0.8
1.0
2
CMP location [km]
2 4 6 8 10
(c) zoom of first event (0.8 - 1.05 s)
CMP location [km]
2 4 6 8 10
(e) zoom of second event (1.7 - 2.6 s)
time [s]
time [s]
1.0
1.5
2.0
2.5
3.0
3.5
4.0
time [s]
CMP location [km]
2 4 6 8 10
(b) CRS stack section after first iteration
0.8
1.0
2
CMP location [km]
2 4 6 8 10
(d) zoom of first event (0.8 - 1.05 s)
CMP location [km]
2 4 6 8 10
(f) zoom of second event (1.7 - 2.6 s)
Figure 6.10: Comparison of the CRS stack section with noise and artifical static
time shifts added and after applying the residual static correction of the first iteration.
(a), (c), (e) with artifical static time shifts applied and (b) , (d), (f) after the
first iteration.
52

time [s]
time [s]
1.0
1.5
2.0
2.5
3.0
3.5
4.0
time [s]
CMP location [km]
2 4 6 8 10
(a) CRS stack section after second iteration
0.8
1.0
2
CMP location [km]
2 4 6 8 10
(c) zoom of first event (0.8 - 1.05 s)
CMP location [km]
2 4 6 8 10
(e) zoom of second event (1.7 - 2.6 s)
time [s]
time [s]
1.0
1.5
2.0
2.5
3.0
3.5
4.0
time [s]
6.2 Residual static correction
CMP location [km]
2 4 6 8 10
(b) CRS stack section with noise but without
artifical static time shifts
0.8
1.0
2
CMP location [#]
200 400 600
(d) zoom of first event (0.8 - 1.05 s)
CMP location [#]
200 400 600
(f) zoom of second event (1.7 - 2.6 s)
Figure 6.11: Comparison of the CRS stack section after applying the residual statics
of the second iterations and the stack without artifical static time shifts. (a),
(c), (e) after second iteration and (b), (d), (f) without artifical static time shifts.
53

Chapter 6. Synthetic data example
statics [ms]
statics [ms]
15
10
5
0
−5
−10
−15
4000 4200 4400 4600 4800 5000
source position [m]
5200 5400 5600 5800 6000
(a) red line: original static time shifts; green line: estimated statics with Method 1 after 2
iterations; yellow line: estimated statics with Method 1 after 2 iterations but by means of the
initial CRS stack
15
10
5
0
−5
−10
−15
4000 4200 4400 4600 4800 5000
receiver position [m]
5200 5400 5600 5800 6000
(b) red line: original static time shifts; green line: estimated statics with Method 1 after 2
iterations; yellow line: estimated statics with Method 1 after 2 iterations but by means of the
initial CRS stack
Figure 6.12: subsets of source statics time shifts with a maximum allowable time
shift of 24 ms and with a time window of 600 ms - 4400 ms. Estimated (a) source
and (b) receiver statics by means of the initial CRS stack and optimized CRS stack,
respectively.
reason for this is that the moveout correction, which is performed on the basis of
the CRS attributes, is very important for the following process. Thus, the results
of the residual static approach strongly depend on the accuracy of the CRS attributes.
With this results, it is advisable that an optimization process should be
performed before starting the residual static correction.
As mentioned in Chapter 5, it appears reasonable to apply a combination of the
three methods. However, the fact that the statics converge to zero after the second
iteration prevented such a test on this data example. In the next chapter such a
54

6.2 Residual static correction
possible combination is presented for a real data example.
This data example is also not suited to show that cycle skips occur if the maximum
allowable time shift is to large. The residual static program was tested with
different maximum time shifts up to 2 s but no changes in the estimated static
corrections are noticeable. The distance between the events is about 1.5 s to 2 s.
This would allow cycle skips for a chosen maximum time shift of 2 s. They often
occur due to the presence of short-period multiples or reverberations, or with
data with narrow bandwidth and/or high noise level. The main reason that cycle
skips do not occur in this synthetic example is that the three events are well
separated, and that the S/N ratio is relatively high in comparison to many real
datasets. The other reason is that, in the new approach, the cross correlation is
performed between the pilot trace and the CRS supergather and not only between
the pilot trace and the traces of the corresponding CMP gather. Therefore,
much more traces contribute to the search of the static corrections for each source
and receiver location and the single result of the cross correlation between two
traces has only little influence. To choose the right maximum allowable time shift
is more difficult. On the one hand, the maximum allowable time shift should,
of course, be larger than all possible total static time shifts at any given location
along the profile. On the other hand, cycle skips, especially under poor S/N ratio
conditions, are more likely to occur if the maximum allowable shift exceeds
the dominant period of the data. In the example presented here, the static time
shifts are added artifically and, thus, it was known from the beginning that the
maximum sum of receiver and source static time shifts are ± 20 ms. However, to
find an appropriate value of the maximum allowable time shift for real data, it
is recommended to make some tests with different settings, e. g., on a subset of
the dataset. For this data example, a maximum allowable time shift of 24 ms has
shown to be most suitable and all results presented are calculated with this value.
The other parameter which can be varied is the length of the cross correlation
time window. This new approach is a trace-based method, there also exist other
methods which are based on cross correlation over single events. At the analysis
of this window length it turned out that the best value is a window from 600 to
4200 ms (see Table 6.2 and Figures 6.13 and 6.14). The first part of the traces, here
the first 600 ms, should always be discarded. In this part, the assumption of surface
consistency is not always satisfied. The wavefront propagates not vertically
in the LVL if the offset from source to receiver is much larger than the depth in
which the ray is reflected. The reason why the last 151 ms are cut off is that this
part of the traces contains only noise, and a smaller time window in the cross
correlation process saves CPU time.
Furthermore, if the cross correlation is performed with a time window only over
the third event (2600 ms - 4200 ms) the result is inferior compared to the result
of the cross correlation with a time window over the second event (1700 ms -
2600 ms). The noise is added to the data using a program of the Seismic Unix
55

Chapter 6. Synthetic data example
statics [ms]
statics [ms]
statics [ms]
56
15
10
5
0
−5
−10
−15
4000 4200 4400 4600 4800 5000
source position [m]
5200 5400 5600 5800 6000
(a) red line: original static time shifts; green line: estimated statics with Method 1 after 1
iteration; black line: estimated statics with Method 1 with a time window which covers nearly
the whole trace (0 s - 4.4 s)
15
10
5
0
−5
−10
−15
4000 4200 4400 4600 4800 5000
source position [m]
5200 5400 5600 5800 6000
(b) red line: original static time shifts; green line: estimated statics with Method 1 after 1
iteration; black line: estimated statics with Method 1 with a time window which covers the
second event (1.7 s -2.6 s)
15
10
5
0
−5
−10
−15
4000 4200 4400 4600 4800 5000
source position [m]
5200 5400 5600 5800 6000
(c) red line: original static time shifts; green line: estimated statics with Method 1 after 1
iteration; black line: estimated statics with Method 1 with a time window which covers the
third event (2.6 s - 4.2 s)
Figure 6.13: Subsets of source statics - maximum time shift of 24 ms (III).

statics [ms]
statics [ms]
statics [ms]
15
10
5
0
−5
−10
6.2 Residual static correction
−15
4000 4200 4400 4600 4800 5000
receiver position [m]
5200 5400 5600 5800 6000
(a) red line: original static time shifts; green line: estimated statics with Method 1 after 1
iteration; black line: estimated statics with Method 1 with a time window which covers nearly
the whole trace (0 s - 4.4 s)
15
10
5
0
−5
−10
−15
4000 4200 4400 4600 4800 5000
receiver position [m]
5200 5400 5600 5800 6000
(b) red line: original static time shifts; green line: estimated statics with method 1 after 1
iteration; black line: estimated statics with Method 1 with a time window which covers the
second event (1.7 s -2.6 s)
15
10
5
0
−5
−10
−15
4000 4200 4400 4600 4800 5000
receiver position [m]
5200 5400 5600 5800 6000
(c) red line: original static time shifts; green line: estimated statics with Method 1 after 1
iteration; black line: estimated statics with Method 1 with a time window which covers the
third event (2.6 s - 4.2 s)
Figure 6.14: subset of receiver statics - maximum time shift of 24 ms (III).
57

Chapter 6. Synthetic data example
Frequency [Hz]
10
20
30
40
50
CMP location [km]
2 4 6 8 10
(a)
CMP location [km]
2 4 6 8 10
Amplitude [arb. unit]
0 0.2 0.4 0.6 0.8 1.0
(b)
CMP location [km]
2 4 6 8 10
Figure 6.15: A comparison of the amplitude spectra (a) without static time shifts,
(b) with artifical residual static time shifts applied, and (c) after residual static
correction (Method 1, second iteration). The synthetic dataset was generated with
a Ricker wavelet with a dominant frequency of 30 Hz.
package (see Appendix C). This program uses the maximum amplitude of the
prestack dataset as reference for the S/N ratio. Thus, the local S/N ratio at the
second event is much higher than at the third event. This fact and the fact that
the third event has a triplication leads to the bad result. If the window covers the
triplication, there are at some traces two or even three branches of the triplication
within the time window.
The ZO sections which result from the CRS stack of the data set with added artifical
static time shifts after the first and second iteration, and without artifical static
time shifts are shown in Figures 6.10 and 6.11. The used correlation window is
0.6 s to 4.2 s, the maximum allowable time shift is 24 ms. The used parameters
for the CRS stack are displayed in Table 6.1. The parts (c)-(f) of Figures 6.10 and
6.11 show magnifications of the first and second event, respectively. After the
first iteration, it is possible to recognize the wavelet and the artifacts around the
wavelet are reduced. The events after the first iteration clearly deviate from the
original linear shapes. The maximum deviation is around 8 ms. After the second
iteration (Figure 6.11 (a), (c), (e)), these deviations are smaller (around 4 ms) and
the shapes of the events approximate to the original ones (Figure 6.11 (b), (d),
(f)). Furthermore, the amplitude spectrum of the CRS stack section shows the expected
behavior. After applying the artifical time shifts, the amplitudes reduce in
an area around the dominant frequency (see Figure 6.15). It is not possible to relate
this behavior directly to Figure 2.2 because stacking is a summation of many
traces with different static time shifts and Figure 2.2 is related to two traces, only.
58
(c)

time [s]
1.8
2.0
2.2
2.4
2.6
CMP location [km]
2 4 6 8 10
time [s]
1.8
2.0
2.2
2.4
2.6
a)
CMP location [km]
2 4 6 8 10
Coherence
0 0.1 0.2 0.3 0.4 0.5 0.6
6.2 Residual static correction
CMP location [km]
2 4 6 8 10
Figure 6.16: Comparison of subsets (second event, 1.7 - 2.6 s) of the coherence
section associated with the optimized CRS stack result from (a) original data, (b)
data with static time shifts, and (c) data with residual static correction.
time [s]
1.8
2.0
2.2
2.4
2.6
CMP location [km]
2 4 6 8 10
time [s]
1.8
2.0
2.2
2.4
2.6
a)
CMP location [km]
2 4 6 8 10
Emergence angle [°]
c)
CMP location [km]
2 4 6 8 10
-30 -20 -10 0 10 20 30
Figure 6.17: Comparison of subsets (second event, 1.7 - 2.6 s) of the emergence
angle section associated with the optimized CRS stack result from (a) original
data, (b) data with static time shifts, and (c) data with residual static correction.
c)
b)
b)
59

Chapter 6. Synthetic data example
time [s]
1.8
2.0
2.2
2.4
2.6
CMP location [km]
2 4 6 8 10
time [s]
1.8
2.0
2.2
2.4
2.6
a)
CMP location [km]
2 4 6 8 10
Radius of NIP wavefront [km]
1 2 3 4 5
CMP location [km]
2 4 6 8 10
Figure 6.18: Comparison of subsets (second event, 1.7 - 2.6 s) of the RNIP section
for the dominant events associated with the optimized CRS stack result from (a)
original data, (b) data with static time shifts, and (c) data with residual static
correction.
time [s]
1.8
2.0
2.2
2.4
2.6
CMP location [km]
2 4 6 8 10
time [s]
1.8
2.0
2.2
2.4
a)
CMP location [km]
2 4 6 8 10
2.6
c)
Curvature of normal wavefront [1/km]
c)
CMP location [km]
2 4 6 8 10
-0.2 -0.1 0 0.1 0.2
Figure 6.19: Comparison of subsets (second event, 1.7 - 2.6 s) of the curvature of
the normal wavefront, i. e., KN, section associated with the optimized CRS stack
result from (a) original data, (b) data with static time shifts, and (c) data with
residual static correction.
60
b)
b)

6.2 Residual static correction
However, the amplitude spectrum of the stack section is almost recovered after
the residual static correction (Figure 6.15(c)).
Furthermore, the coherence and the CRS attributes are almost recovered (Figures
6.16 - 6.19 part (c)), only at the CMPs between 0 m and 40 m the coherence
sections differ. Most traces corresponding to these CMPs have receivers with bad
static corrections because of the low coverage and the effect of the CRS aperture.
Consequently, the correct CRS attributes in this parts of the events can hardly be
found, even after the residual static correction.
Compromising, the new residual static correction approach by means of the CRS
attributes was able to compensate the previously added static time shifts almost
perfectly, after only two iterations. However, even with more iterations the approach
was not able to fully recover the static time shift in this example. The
reason for this is that the CRS operator is only an approximation of the true reflection
response and the solution of the residual static correction is not necessarily
unique (see Chapter 3.1). However, after the residual static correction the
ZO stack sections are very similar to those ones that were obtained before the artifical
static time shifts have been added. Furthermore, Method 2 and Method 3
yield also acceptable results with less CPU time.
61

Chapter 7
Real data examples
After presenting the results of the new approach on synthetic data in Chapter 6,
here the approach is applied to real and modified real data. The term modified
stands for a real data set with added artifical, random but surface consistent
residual static time shifts. Method 1 and Method 3 as well as a combination of
these methods are tested on these dataset and the advantages and drawbacks are
analyzed.
7.1 The real dataset and the real dataset with artifical
static time shifts
The real dataset consists of 100 CMP gathers with a fold of up to 31 and contains
3028 traces. The dataset was recorded in Northern Germany and underwent routine
processing, including the application of some iterations of a standard residual
static program. The shot points range from 4.2 kft to 10.8 kft and the receiver
points from 4.2 kft to 10.74 kft. Furthermore, the minimum offset is 60 ft and the
maximum offset is 3.6 kft. The data were acquired in a so called split-spread geometry:
receivers are positioned on both sides of the source on the seismic line.
Moreover, the traces are NMO corrected and must be inverse NMO corrected
before it is possible to apply the new approach. As can be seen from Figure 7.3(a),
the optimized CRS stack obtained from these input traces is of fairly good quality
with only slightly dipping reflectors. Furthermore, the stack does not exhibit any
visible static problems. The parameters which are used to perform the CRS stack
are listed in Table 7.1.
This dataset is also discussed in the work of Kirchheimer (1990). He applied a
CMP-based residual static correction method to the dataset, perturbed by synthetic
receivers static time shifts. Here in this work, the new approach is firstly
tested on the original dataset to see whether there are some residual static time
shifts in the original dataset (Section 7.2.1).
63

Chapter 7. Real data examples
Context Processing parameter Setting
Dominant frequency 40 Hz
General Coherence measure Semblance
parameters Data used for coherence analysis Original traces
Temporal width of coherence band 22 ms
Velocity and Near surface velocity 4000 ft/s
constraints Tested stacking velocities 2500 . . .20 000 ft/s
Simulated ZO traveltimes 0 . . .2.2 s
Target Simulated temporal sampling interval 2 ms
zone Number of simulated ZO traces 101
Spacing of simulated ZO traces 30 ft
Minimum ZO aperture 200 ft @ 0.15 s
Aperture Maximum ZO aperture 595 ft @ 1.75 s
and Minimum CMP aperture 400 ft @ 0.15 s
taper Maximum CMP aperture 3636 ft @ 1.75 s
Relative taper size 30 %
Automatic Initial moveout increment for largest offset 2 ms
CMP stack Number of refinement iterations 3
Linear Tested emergence angles −60 . . .60◦ ZO Initial emergence angle increment 1◦ stack Number of refinement iterations 3
Hyperbolic Initial moveout increment for largest ZO distance 2 ms
ZO stack Number of refinement iterations 3
Hyperbolic Initial moveout increment for largest offset 2 ms
CS/CR stack Number of refinement iterations 3
Conflicting Maximum number of dips 2
dip Absolute coherence threshold for global maximum 0.5
handling Relative coherence threshold for local maxima 0.35
Coherence threshold for smallest traveltime 0.025
Coherence threshold for largest traveltime 0.0125
Maximum number of iterations 100
Local Maximum relative deviation to stop 10−4 optimization Initial variation of emergence angles 6◦ Initial variation of RNIP
5 %
Initial variation of transformed RN
6◦ Transformation radius for RN
100 m
Table 7.1: Real data example: processing parameters used for the ZO simulation
by means of the CRS stack.
In the next step, random but surface consistent residual static time shifts between
-10 ms and +10 ms are added to every source or receiver, respectively. This artifical
static time shifts are shown in Figure 7.4 (b) and (c). The resulting static time
shifts for every trace of the prestack dataset are between -20 ms and +20 ms with
a mean of zero. Figure 7.4 (a) shows the optimized CRS stack result for the ar-
64

correlation stacks [#]
5000
4000
3000
2000
1000
7.2 Results of the new residual static correction approach
0
4 5 6 7 8 9 10 11
source/receiver position [kft]
Figure 7.1: Number of traces contributing to the cross correlation stack for receivers
(blue line) and sources (red line), respectively.
tifically distorted prestack traces. The synthetic residual static time shifts almost
completely destroyed the stacking result.
7.2 Results of the new residual static correction approach
7.2.1 Original data
As mentioned before, the new approach is first tested on the original dataset.
Method 1 (see Section 6.2) was applied with a maximum allowable time shift of
30 ms and a cross correlation window from 0 s to 2.2 s , i.e., the whole trace is
used. In this example the whole trace is used, as reference result for the dataset
with artifical static time shifts added (Section 7.2.2). The number of contributing
cross correlation results to the cross correlation stack is shown in Figure 7.1. The
number of contributions for the receiver stack (blue line) is only the half of the
number for the source stack (red line) due to the split-spread acquisition geometry.
For this acquisition geometry, there exist for each source location two receiver
locations with the same offset and, consequently, two times more contributions to
the source correlation stack. The bell-shapes of the two curves are caused by the
chosen aperture and the coverage of the source and receiver locations within the
available small subset of the whole dataset. This subset contains 100 CMP gathers
of almost constant fold. Therefore the number of traces which contain the same
source or receiver locations decrease to the borders of the seismic line.
The estimated static corrections after the first (blue lines) as well as the second
iteration (green lines) are displayed in Figure 7.2. In (a), the section with source
static time shifts, the estimated static time shifts for the most points are equal to
zero. Only at 11 points the values are ±1 ms and at the receiver locations 10.68
65

Chapter 7. Real data examples
statics [ms]
statics [ms]
6
4
2
0
−2
−4
−6
6
4
2
0
−2
−4
−6
5 6 7 8
source position [kft]
9 10
(a) source statics
5 6 7 8
receiver position [kft]
9 10
(b) receiver statics
Figure 7.2: Source and receiver statics for the original dataset. Blue lines: estimated
static time shifts with Method 1 after 1 iteration; green lines: estimated
statics with Method 1 after 2 iteration.
and 10.8 kft they are -4 ms. After the second iteration every estimated value is
equal to zero. Consequently, the curve after the second iteration does not differ
from the curve after the first iteration.
The receiver static section looks a little bit different. The values range from -5 ms
to +6 ms. Nevertheless, for the receiver statics, a lot of values are 0 ms, especially
between 8.34 kft and 9.6 kft. After the second iteration, the estimated values are
0 ms or ±1 ms and, thus, the curve after the second iteration (green line in Figure
7.2 (b)) is very close to the curve after the first iteration.
The difference of the curves for receiver and source statics shows the advantage
of the new approach. In this data example, as mentioned above, the number of
contributions for the receiver correlation stack is only the half of the contributions
to the corresponding source correlation stack. This could be the reason that
the conventional approach was not able to do provide more accurate values for
66

time [s]
time [s]
7.2 Results of the new residual static correction approach
CMP [#]
200
0
220 240 260 280 300
0.5
1.0
1.5
2.0
(a) CRS stack section of original
dataset
0.19
0.20
0.21
0.22
0.23
0.24
260 265
CMP [#]
270 275 280
(c) subset of CRS stack section in (a)
time [s]
time [s]
CMP [#]
200
0
220 240 260 280 300
0.5
1.0
1.5
2.0
(b) CRS stack section of original
dataset after second iteration
0.19
0.20
0.21
0.22
0.23
0.24
260 265
CMP [#]
270 275 280
(d) subset of the CRS stack section
in (b)
Figure 7.3: Optimized CRS stack section of (a) the original dataset and of (b) the
dataset after 2 iterations and (c), (d) subsets of these two stacks.
67

Chapter 7. Real data examples
the receiver static time shifts. The new approach is now able to perform better
because it uses the entire CRS supergathers for the cross correlation process.
The optimized CRS stack after applying two iterations of the new residual static
correction approach (Figure 7.3(b)) shows more distinctive reflection events and
also some gaps are closed, e. g., around CMP 250 at the event at around 0.25 s and
around CMP 270 at the event at around 0.7 s. In the subsets in Figures 7.3(c) and
(d), the already discussed behavior can be observed: the shape of the wavelets
in the stacked ZO section are improved and the amplitudes are higher after the
residual static correction. This is especially evident at CMP gathers 268 and 277.
7.2.2 Original data with artifical static time shifts
As mentioned before, artifical residual static time shifts were added to the original
dataset. The static time shifts for sources and receivers as well as the result
of the optimized CRS stack can be seen in Figure 7.4. Method 1 of the new approach,
with a maximum allowable time shift of ± 30 ms and a cross correlation
window from 0 s to 2.2 s, is applied to this data. In this example the whole trace is
used because the artifical static time shifts are surface consistent for all times and
offset (see Section 3.2). In the Figures 7.5 - 7.9, the result of the estimated static
time shifts as well as the resulting CRS stack after different iterations (up to five)
are presented.
Compared to the synthetic dataset, one difference for this modified real dataset
is that more than two iterations are needed to obtain a satisfactory result. After
the first iteration, the estimated statics for the source locations (Figure 7.5 (b)) are
almost equal to zero. Only a few of the estimated statics match to the artifical
static time shifts, e. g., at the source location 5.04 kft. For unknown reasons, the
statics at the source points with a low coverage are approximated more accurate.
However, the receiver statics (Figure 7.5 (c)) are also not satisfactory after the first
iteration. The CRS stack after applying the corrections of the first iteration reveals
no significant improvements compared to the stack without static corrections.
One can probably suspect the second main event (around 0.25 s).
Consequently, a second iteration is necessary. The estimated statics from this
iteration together with the statics of the first iteration approximate the artifical
static time shifts already well. The results for the receiver locations (Figure 7.6 c)
match better to their artifical counterparts than the results for the source locations
(Figure 7.6 b). This can be also seen in Table 7.2, where the mean deviation of the
estimated statics from the artifical static time shifts is given. Furthermore, it is
clearly visible that the second iteration yields smaller deviations for the source
as well as for the receiver statics. In the CRS stacked section after application of
the static corrections from the second iteration it is possible to recognize the main
events, e. g., the slightly dipping event at a time of 1.5 s is now visible across the
entire section. Looking at the mean deviation, this second iteration yields the
68

statics [ms]
statics [ms]
15
10
5
0
−5
−10
−15
15
10
5
0
−5
−10
−15
time [s]
7.2 Results of the new residual static correction approach
CMP [#]
200
0
220 240 260 280 300
0.5
1.0
1.5
2.0
5 6 7 8
source position [kft]
9 10
5 6 7 8
receiver position [kft]
9 10
Figure 7.4: (a) CRS stack section with artifical static time shifts added; (b) artifical
source static time shifts and (c) artifical receiver static time shifts.
(a)
(b)
(c)
69

Chapter 7. Real data examples
statics [ms]
statics [ms]
15
10
5
0
−5
−10
−15
15
10
5
0
−5
−10
−15
time [s]
CMP [#]
200
0
220 240 260 280 300
0.5
1.0
1.5
2.0
5 6 7 8
source position [kft]
9 10
5 6 7 8
receiver position [kft]
9 10
Figure 7.5: (a) CRS stack section after the first iteration, artifical static time shifts
(red lines) and estimated statics (black lines) (b) source and (c) receiver static corrections.
70
(a)
(b)
(c)

statics [ms]
statics [ms]
15
10
5
0
−5
−10
−15
15
10
5
0
−5
−10
−15
time [s]
7.2 Results of the new residual static correction approach
CMP [#]
200
0
220 240 260 280 300
0.5
1.0
1.5
2.0
5 6 7 8
source position [kft]
9 10
5 6 7 8
receiver position [kft]
9 10
Figure 7.6: (a) CRS stack section after the second iteration and (b) estimated
source and (c) estimated receiver static corrections. Red lines: original; black
lines: first iteration of Method 1; green lines: second iteration of Method 1.
(a)
(b)
(c)
71

Chapter 7. Real data examples
statics [ms]
statics [ms]
15
10
5
0
−5
−10
−15
15
10
5
0
−5
−10
−15
time [s]
CMP [#]
200
0
220 240 260 280 300
0.5
1.0
1.5
2.0
5 6 7 8
source position [kft]
9 10
5 6 7 8
receiver position [kft]
9 10
Figure 7.7: (a) CRS stack section after the third iteration and (b) estimated source
and (c) estimated receiver static corrections. Red lines: original; green lines: second
iteration of Method 1; blue lines: third iteration of Method 1.
72
(a)
(b)
(c)

statics [ms]
statics [ms]
15
10
5
0
−5
−10
−15
15
10
5
0
−5
−10
−15
time [s]
7.2 Results of the new residual static correction approach
CMP [#]
200
0
220 240 260 280 300
0.5
1.0
1.5
2.0
5 6 7 8
source position [kft]
9 10
5 6 7 8
receiver position [kft]
9 10
Figure 7.8: (a) CRS stack section after the fourth iteration and (b) estimated source
and (c) estimated receiver static corrections. Red lines: original; blue lines: third
iteration of Method 1; yellow lines: fourth iteration of Method 1.
(a)
(b)
(c)
73

Chapter 7. Real data examples
statics [ms]
statics [ms]
15
10
5
0
−5
−10
−15
15
10
5
0
−5
−10
−15
time [s]
CMP [#]
200
0
220 240 260 280 300
0.5
1.0
1.5
2.0
5 6 7 8
source position [kft]
9 10
5 6 7 8
receiver position [kft]
9 10
Figure 7.9: (a) CRS stack after the fifth iteration and (b) estimated source and
(c) estimated receiver static corrections. Red lines: original; yellow lines: fourth
iteration of Method 1; black lines: fifth iteration of Method 1.
74
(a)
(b)
(c)

7.2 Results of the new residual static correction approach
Method 1
number of mean deviation mean deviation
iteration receiver [ms] source [ms]
1 3.97 3.42
2 3.08 2.30
3 2.32 1.77
4 2.27 1.55
5 2.34 1.45
Table 7.2: Mean deviation of the estimated statics (Method 1) from the artifical
statics. The calculation is performed with a time window from 0 s to 2.2 s and a
maximum time shifts of ± 30 ms.
Method 1 - initial statics
number of mean deviation mean deviation
iteration receiver [ms] source [ms]
1 3.98 3.56
2 2.97 2.64
3 1.94 1.54
4 1.89 1.25
5 1.88 1.11
Table 7.3: Mean deviation of the estimated statics (Method 1) from the artifical
statics, considering the initial static time shifts of the original dataset. The calculation
is performed with a time window from 0 s to 2.2 s and a maximum time
shifts of ± 30 ms.
most significant improvement.
The third iteration yields the strongest improvements in the image quality of the
CRS stacked section (see Figure 7.7 (a)): all main events can be recognized in the
ZO section. Most of the estimated statics after the third iteration approximate
the artificial static time shifts well. However, e. g., at Figure 7.7 (b) the receiver
locations 6.66 kft to 6.78 kft the estimated statics are far from their artifical counterparts.
The fourth iteration (Figure 7.8 (b)) yields further, although small, improvements.
Outstanding in the CRS stacked section is the gap in the first main event around
CMP 240, whereas the stacked ZO section of the original data shows a gap around
CMP 255. This ZO section is closer to the ZO section which is obtained after applying
two iterations of the new residual static approach to the original data than
to the stacked ZO section of the original data. As shown in Section 7.2.1, the new
approach was able to determine statics in the original data. As the artifical static
75

Chapter 7. Real data examples
statics [ms]
statics [ms]
15
10
5
0
−5
−10
−15
15
10
5
0
−5
−10
−15
5 6 7 8
source position [kft]
9 10
(a) source statics
5 6 7 8
receiver position [kft]
9 10
(b) receiver statics
Figure 7.10: Estimated (a) source and (b) receiver static time shifts for the dataset
with and without consideration of the initial static time shifts of the original
dataset. Red lines: original static time shifts; black lines: estimated statics with
Method 1 after the fifth iteration; green lines: estimated statics with Method 1
after the fifth iteration under consideration of the static time shifts of the initial
dataset.
time shifts are added to the original data, the initial static time shifts are still contained
in the prestack data. Thus, the mean deviation of the estimated statics and
the artifical static time shifts can also be calculated considering the initial static
time shifts of the original dataset (see Table 7.3). The mean deviations show that
the estimated statics are closer to the artifical static time shifts if the initial statics
are considered, especially for the receiver statics. Apart from few exceptions, this
can be also seen in Figure 7.10. The estimated statics, which consider the initial
statics (green lines), are closer to the artifical static time shifts (red lines) than the
estimated statics which do not consider the initial statics (black lines). A difference
of this two approaches is that, if the initial statics are considered, the best
76

7.2 Results of the new residual static correction approach
Method 3
number of mean deviation mean deviation
iteration receiver [ms] source [ms]
1 n/a n/a
2 3.34 2.70
3 2.90 2.28
4 2.85 2.00
5 3.34 2.17
6 4.15 2.81
7 4.76 3.32
Table 7.4: Mean deviation of the estimated statics (Method 3) from the artifical
statics. The calculation is performed with a time window from 0 s to 2.2 s and a
maximum time shifts of ±30 ms.
Method 3 - initial statics
number of mean deviation mean deviation
iteration receiver [ms] source [ms]
1 n/a n/a
2 3.22 2.66
3 2.65 2.17
4 2.48 2.85
5 2.89 2.02
6 3.72 2.66
7 4.36 3.17
Table 7.5: Mean deviation of the estimated statics (Method 3) from the artifical
statics, considering the initial static time shifts of the original dataset. The calculation
is performed with a time window from 0 s to 2.2 s and a maximum time
shifts of ±30 ms.
estimation of the static correction is reached at the fifth iteration, whereas at the
approach which does not consider the initial values reaches the best estimation
after the fourth iteration. However, the most of the estimated static time shifts
at the fifth iteration are equal to zero. This indicates that the iterative process
converges. That the approach which considers the initial values yields the best
estimation of the static correction at the fifth iteration is consequently the desired
result.
As implied in the last paragraph, the new residual static correction approach
is also be performed with Method 3. On the one hand, the mean deviation of
the estimated residual statics with Method 3 from the artifical static time shifts is
77

Chapter 7. Real data examples
Method combined
number of mean deviation mean deviation
iteration receiver [ms] source [ms]
1 3.97 3.42
2 3.34 2.70
3 2.90 2.28
4 2.44 1.30
5 2.73 1.60
Table 7.6: Mean deviation of the estimated statics (combined Method) from the
artifical statics. The calculation is performed with a time window from 0 s to 2.2 s
and a maximum time shifts of ±30 ms.
Method combined - initial statics
number of mean deviation mean deviation
iteration receiver [ms] source [ms]
1 3.98 3.56
2 3.22 2.66
3 2.65 2.17
4 1.94 1.19
5 2.23 1.53
Table 7.7: Mean deviation of the estimated statics (combined Method) from the
artifical statics, considering the initial static time shifts of the original dataset. The
calculation is performed with a time window from 0 s to 2.2 s and a maximum
time shifts of ±30 ms.
higher than the mean deviation of Method 1 (see Table 7.2). On the other hand,
Method 3 yields the smallest deviation after the fourth iteration. The estimated
statics which consider the initial static time shifts yields with this method better
results, too (see Tables 7.4 and 7.5), but the smallest deviation is, this time,
reached after the fourth iteration. The statics obtained with Method 3 converge
after seven iterations. At that time, the deviations of the estimated residual statics
are higher than after the first iteration of Method 1. The violation of the surface
consistency (see Section 5.3) might be the reason for this behavior. Nevertheless,
if the process is stopped after the fourth iteration, the image quality of the
stacked ZO section is nearly as good as after the fourth iteration of Method 1
(compare Figures 7.11(b) and 7.11(c)). Both, Method 1 and Method 3, are applied
to this dataset, as Method 3 is the fastest of these three methods and Method 1 is
the most accurate.
With this dataset, it was now possible to test a combination of this two methods.
78

7.2 Results of the new residual static correction approach
The methods are combined as follows: first iteration with Method 1, second and
third iteration with Method 3. The fourth iteration with Method 1 and the last
and fifth iteration again with Method 3. The mean deviations from the estimated
residual statics from the artifical static time shifts for all iterations are shown in
Table 7.6. The best value is obtained after the fourth iteration and a comparison
with Tables 7.2 and 7.4 shows that this method yields the best result. However,
as with the other methods the estimated statics improve if the initial static time
shifts are taken into account and the best result is also obtained after the fourth
iteration (see Table 7.7). The fifth iteration with Method 3 deteriorates the results.
In general, it is, of course, not possible to determine the mean deviation of the
estimated statics and the true static time shifts. Therefore, as mentioned in Chapter
6, an automatic criterion is necessary to decide whether more iterations are
needed or not. One possibility is the deviation of the mean of the estimated statics
of the current iteration from zero. This would work very well with Method 1
but it would not work with Method 3 or with the combined method. In future
work, it is necessary to investigate the behavior of the static correction values at
more iterations to find a suitable stop criterion for these methods. However, the
CRS stack in each single iteration should be kept in view according to Method 1.
Figure 7.11 shows a comparison of the optimized CRS stack for different methods
after four iterations (Figures 7.11(b) - (d)) to the original stack (Figure 7.11(a)). The
stacked ZO sections of the different methods only slightly differ. A combination
of those methods appears to be a reasonable method. On the one hand, the error,
made by the violation of the assumption of surface consistency by Method 3 is
kept small if after a small number of iterations a complete CRS stack is performed.
On the other hand, the combination of the methods requires far less CPU time as
Method 1 on its own. This must also be confirmed by future research work.
79

Chapter 7. Real data examples
time [s]
time [s]
CMP [#]
200
0
220 240 260 280 300
0.5
1.0
1.5
2.0
(a) CRS stack of the original dataset
CMP [#]
200
0
220 240 260 280 300
0.5
1.0
1.5
2.0
(c) CRS stack section of the dataset
with artifical static time shifts after 4
iterations of Method 3
time [s]
time [s]
CMP [#]
200
0
220 240 260 280 300
0.5
1.0
1.5
2.0
(b) CRS stack section of the dataset
with artifical static time shifts after 4
iterations of Method 1
CMP [#]
200
0
220 240 260 280 300
0.5
1.0
1.5
2.0
(d) CRS stack section of the dataset
with artifical static time shifts after 4
iterations of the combined Method
Figure 7.11: Comparison of the optimized CRS stack for the different methods.
80

Chapter 8
Summary
In this thesis, a new residual static correction approach by means of CRS attributes
is presented. This method aims to eliminate the influence of the weathering
layer on the traveltimes which remains after the datum static correction. This
new approach is based on the stack power maximization method from Ronen and
Claerbout (1985). In contrast to the conventional residual static methods, the new
approach uses the CRS attributes for the moveout correction and the CRS stacked
ZO traces as pilot traces. The result of the cross correlation of each pre-stack trace
with the pilot trace is assigned to the corresponding source and receiver locations.
As several traces share the same source or receiver, respectively, a cross correlation
stack is generated for every source or receiver. In general, the maximum of
this cross correlation stack corresponds to the surface consistent residual static
time shift of the respective source or receiver.
Furthermore, the new approach makes use of comparatively large subsets of the
full multi-coverage dataset for the residual static correction procedure. Consequently,
many more traces compared to the method of Ronen and Claerbout
(1985) contribute to the cross correlation stacks. The residual static correction
is, in general, an iterative process. Basically, three different methods exist how to
perform in the new approach. These three methods differ in their required CPU
time and in the accuracy of their results.
These three methods and some utilities, e. g., adding synthetic statics to seismic
data, were implemented in a C++ code. Investigations on synthetic data demonstrated
that the new approach is able to estimate residual statics very well with,
in this example, only two iterations. First tests on a real dataset which is already
residual static corrected by means of a conventional method, showed that the new
approach yields more accurate results than conventional methods. Considering
a real data example with artifical statics, the new approach demonstrated that it
is possible to find the residual statics even if no contiguous events are visible in
the initial ZO section.
The development of the new residual static program by means of CRS attributes
81

Chapter 8. Summary
is still going on. New ideas or extensions, like different search strategies for the
search of the correct maximum in the cross correlation stack or that the moveout
corrected supergathers can be produced ’on the fly’ wait to be implemented.
Furthermore, in the context of this work and in cooperation with Dr. Franz
Kirchheimer (WesternGeco), a program was developed which allows to use the
moveout-corrected CRS supergathers in the residual static correction of a conventional
processing system. Unfortunately, no results of this approach have been
available yet.
82

Appendix A
Refraction seismics in relation to the
near-surface
A seismic ray which crosses a boundary between two formations of different velocities
is refracted according to Snell’s law (see Figure A.1). This law states that
the ratio of the sines of the incident angle θ1 and refracted angle θ2 is equal to the
ratio of the velocities of the two formations v1 and v2:
sin θ1
sin θ2
= v1
. (A.1)
v2
As long as the velocity increases with depth, the ray is refracted away from the
interface normal. For the so called critical angle θ1 = θc, the refracted angle is
θ2 = 90 ◦ . The critical angle θc follows from Equation A.1 as
sin θc = v1
. (A.2)
If a wavefront reaches the interface under the critical angle, it propagates along
the boundary with the velocity of the lower medium. At every point of the raypath
along the the boundary, there exists a ray from the boundary to the surface.
The angle between all this rays and the normal to the boundary is the incident
angle θc. This can be explained by considering the corresponding wavefront to
the refracted ray (see, e. g., Telford et al., 1976).
The most convenient way to represent refraction data is to plot the first-arrival
time tx vs. the source-receiver distance x (see Figure A.2 (a)). In the following,
the time-distance relations for the case of two layers with velocities v1 and v2,
separated by a horizontal discontinuity at depth z0 is derived (illustrated in Figure
A.2 (b)). The total time along the refraction path ABCD in Figure (b) is
83
v2

Chapter A. Refraction seismics in relation to the near-surface
V 1
V 2 V 1
θ
1
θ
> 2
Figure A.1: A seismic ray which crosses a boundary. The ratio between the sines
of the incident angle θ1 and refracted angle θ2 is equal to the ratio of the velocities
of the two formations v1 and v2 (Snell’s law)
tx = tAB + tBC + tCD = 2tAB + tBC (A.3)
= 2
= 2
z0
v1 cos θc
z0
v1 cos θc
v2
+ x − 2z0 tan θc
v2
− 2z0 sin θc
v2 cos θc
(A.4)
+ x
, (A.5)
v2
or expressed in terms of velocities, only,
tx = x
+ 2z0
�
v2 2 − v2 1 . (A.6)
On a tx vs. x plot, this is the equation of a straight line which has a slope of 1/v2
and which intersects the tx axis (x = 0) at the so-called intercept time
ti = 2z0
�
v2 2 − v2 1 . (A.7)
v1v2
The direct arrival is simply given by a straight line with a slope of 1/v1 that, in
a tx vs. x plot, intersects the tx axis (x = 0) at t = 0. In the time-distance plot,
the traveltime curves of the direct and refracted wave intersects each other at the
crossover distance
v1v2
� v2 + v1
xcross = 2z0 . (A.8)
v2 − v1
At offsets smaller than xcross, the direct wave along the top of the upper layer
reaches the receiver first. At larger offsets the refracted wave along the interface
arrives earlier than the direct wave.
84

a)
b)
t i
z
A
slope=1/v 1
x c
second arrivals
x cross
θ θ θ θ θ
C C C C C
first arrivals
D
slope=1/v2
B C v
v 1
2
x
location along the
seismic line
x
Figure A.2: (a) Traveltime curves of the refracted and the direct wave. (b) Refracted
and direct rays in the corresponding model with two layers separated by
a horizontal interface.
The depth z0 of the interface can be calculated by means of Equation A.7. In terms
of ti and the velocities v1 and v2, Equation A.7 can be solved for z0 to obtain
z0 = ti v1v2
�
2 v2 2 − v2 . (A.9)
1
One can see in Figure A.2 that the first refracted ray intersects the surface at
the critical distance xc. This corresponds to the source-receiver offset where the
length of the ray along the refractor is zero, i.e., the case of critical reflection. The
critical distance can be expressed as
xc = 2z0v1
�
v2 2 − v2 . (A.10)
1
In real data, it is often difficult to use first breaks to estimate the intercept time and
velocities for the weathering layer vw (v1 in Equation A.1) and the bedrock ve (the
layer below the weathering layer, v2 in Equation A.1). Some reasons responsible
for this are, e. g., the undulation of the base of the weathering, or a low number of
receivers in between the source and crossover distance xcross. Different methods
z0
85

Chapter A. Refraction seismics in relation to the near-surface
exists to solve this problem. A descriptive method was introduced by Hagedoorn
(1959), the so-called Plus-Minus method.
It is important to note that a problem occurs if the principle of the refraction seismics
is explained with zero-order ray theory. The energy of the incident wavefront
dispenses along refracted wavefront. Consequently, the energy of one refracted
wavefront is very small and could usually not be observed in seismic data.
Therefore, other wavefronts have to produce the seismic signal at the same traveltime
as the refracted wavefronts. The theory explained above has illustrated
the basic idea behind refraction seismics for planar reflectors, only.
86

Appendix B
Cross Correlation
Cross correlation is an operation that provides a measure of the similarity and
time displacement of two traces. The cross correlation function φ between two
traces f (t) and g(t) is given by
φ f g(t) =
� ∞
−∞
f (τ )g(τ + t)dτ ≡ f (−t) ∗ g(t) (B.1)
The cross correlation is the convolution of the two traces f (−t) and g(t).
The cross correlation is often only performed in a defined window, and in many
cases it is advantageous to divide Equation B.1 by the cross signal energy to obtain
a normalized cross correlation
� t2
f (τ )g(τ + t)dτ
t1 φ f g(t) = �� t2
f t1 2 (τ )dτ � t2 g t1 2 , (B.2)
(τ + t)dτ
where t1 and t2 are the data (window) start and end times, respectively. For discrete
timeseries, this becomes
φ f g(t) =
� �t2
� t2
t 1 f (τ )g(τ + t)Δτ
t 1 f 2 (τ )Δτ � t 2
t 1 g 2 (τ + t)Δτ
, (B.3)
with the sampling interval Δτ . The cross correlation is widely used in various
stages of data processing, e. g., vibroseis correlation or Wiener filter. When normalized,
an amplitude of the cross correlation (correlation coefficient) of 1 indicates
a perfect match, values close to zero indicate very little similarity. Furthermore,
values of -1 indicate that traces are identical but differ in sign. If both traces,
f (t) and g(t), are identical the term auto correlation is used instead of cross correlation.
The correlation coefficient can also be used as a weighting criterion for the
picked times.
87

Chapter B. Cross Correlation
time [s]
1.0
1.2
Amplitude
0
(a) trace # 1
time [s]
1.0
1.2
Amplitude
0 1
(b) trace # 2
Amplitude
1
0
-0.4 -0.2 0 0.2 0.4
time [s]
(c) cross correlation result
Figure B.1: Example of a cross correlation result: trace # 2 in (b) is the same trace
as in (a) but with a static time shift of 100 ms. Figure (c) shows the normalized
cross correlation of the traces of Figure (a) and (b).
time [s]
1.0
1.2
Amplitude
0
(a) trace 1
time [s]
1.0
1.2
Amplitude
0
(b) trace 2
Amplitude
1
0
-0.4 -0.2 0 0.2 0.4
time [s]
(c) cross correlation result
Figure B.2: Same traces as in Figure B.1 but now with a S/N ratio of 2.
In Figure B.1, an example of a cross correlation result is shown. The trace # 1
from Figure B.1 is a subset of a synthetic seismic trace with a sampling interval of
4 ms. Trace # 2 is the same trace but now shifted by 100 ms. This shift can be seen
in the cross correlation result in Figure B.1(c), namely the time coordinate of the
global maximum which is exactly at the time shift τ = 100 ms. Furthermore, the
amplitude of 1 at the maximum shows that both traces are identical but shifted
against each other. Figure B.2 shows again the two traces from Figure B.1, but
now with synthetic random white noise added with a S/N ratio of 2. In the cross
correlation result in Figure B.2(c), a global maximum can hardly be identified.
Also, all correlation coefficients are lower than 0.3, which indicates that the two
traces have little similarities.
88

Appendix C
Used hard- and software
The computations were performed on a SILICON GRAPHICS ORIGIN 3200
computeserver (IRIX 6.5) with six processors and on dual-processor PCs (with
S.u.S.E. Linux 8.1).
The residual static correction algorithm is implemented in C++.
The synthetic data set was produced with the software package NORSAR.
For numerical calculations and for the visualization of the static correction
values, Matlab 6.5 Release 13 was used.
The 2-D figures of the seismic data sets and the CRS attributes were generated
with the Seismic Unix package (Center of Wave Phenomena, Colorado School
of Mines). Furthermore, some utilities of this package were used to generate
synthetic noise and to apply artifical static time shifts to the prestack dataset.
This thesis was written on a PC (S.u.S.E. Linux 8.1) using the freely available
word processing package TEX, the macro package L ATEX 2ε, and several extensions.
The bibliography was generated with BIBTEX. Schematic figures were
mainly constructed with Xfig 3.2.
89

List of Figures
Chapter 1 – Introduction 1
1.1 3D data volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 CMP array with plane and dipping reflector . . . . . . . . . . . . . 3
1.3 Example for a raypath in a model with a weathering layer. . . . . . 5
1.4 Example of residual static correction enhancement . . . . . . . . . . 6
Chapter 2 – Static correction 7
2.1 Some possible near-surface conditions . . . . . . . . . . . . . . . . . 8
2.2 Static time shifts act as a high-cut filter in the amplitude spectrum. 9
2.3 Rays in a surface consistent and in a not surface consistent model. . 10
2.4 Difference between weathering correction and elevation correction. 11
2.5 CMP stacked data example with static correction. . . . . . . . . . . 14
Chapter 3 – Conventional methods for residual static correction 17
3.1 Example for different cross correlation results. . . . . . . . . . . . . 21
3.2 Sketch of the residual static correction method by Ronen and Claerbout
(1985) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Simulated annealing technique; normalized stack power plotted
against the number of iterations. . . . . . . . . . . . . . . . . . . . . 23
Chapter 4 – Common Reflection Surface stack 25
4.1 Illustration of eigenwaves experiments . . . . . . . . . . . . . . . . . 26
4.2 CRS stack operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3 CRS stack operator and the first interface Fresnel zone . . . . . . . . 30
4.4 ZO, CMP, and spatial CRS apertures. . . . . . . . . . . . . . . . . . . 31
Chapter 5 – Residual static correction by means of CRS-attributes 33
5.1 Flowchart of the residual static correction method by means of CRS
attributes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
91

List of Figures
5.2 CRS operator, CRS aperture, and the moveout corrected reflection
event. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Chapter 6 – Synthetic data examples 39
6.1 The model and the resulting stacked ZO section. . . . . . . . . . . . 40
6.2 Stacked ZO section of the dataset with noise and with noise and
artifical static time shifts. . . . . . . . . . . . . . . . . . . . . . . . . . 42
6.3 Example for the CRS aperture and a single CMP gather from the
CRS supergather. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.4 Number of traces contributing to the cross correlation stack. . . . . 44
6.5 Comparison of subsets of the source static time shift. Method 1 -
iteration 1 and 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.6 Comparison of subsets of the receiver statics. Method 1 - iteration
1 and 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.7 Comparison of subsets of the source statics. Method 1, Method 2
and Method 3 - iteration 2. . . . . . . . . . . . . . . . . . . . . . . . . 48
6.8 Comparison of subsets of receivers statics. Method 1, Method 2
and Method 3 - iteration 2. . . . . . . . . . . . . . . . . . . . . . . . . 49
6.9 Subset of the total statics after two iterations, estimated with
Method 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.10 Comparison of the CRS stack section with noise and artifical static
time shifts added and the stack after applying the residual static
correction of the first iteration. . . . . . . . . . . . . . . . . . . . . . . 52
6.11 Comparison of the CRS stack section after applying the residual
statics of the second iteration and the stack without any artifical
static time shifts added. . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.12 Comparison of source and receivers statics: Method 1 by means of
initial CRS attributes and by means of optimized CRS attributes. . . 54
6.13 Comparison of source statics. Method 1 with different cross correlation
windows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.14 Comparison of receiver statics. Method 1 with different cross correlation
windows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.15 Comparison of the amplitude spectra. . . . . . . . . . . . . . . . . . 58
6.16 Comparison of subsets of the coherence section . . . . . . . . . . . . 59
6.17 Comparison of subsets of the emergence angle section . . . . . . . . 59
6.18 Comparison of subsets of the section of radius of curvature of the
NIP wavefront . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.19 Comparison of subsets of the section of curvature of the normal
wavefront . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
92

List of Figures
Chapter 7 – Real data examples 63
7.1 Number of traces contributing to the cross correlation stack. . . . . 65
7.2 Source and receiver statics for the original dataset after first and
second iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
7.3 Optimized CRS stack section of the original dataset and after two
iterations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
7.4 CRS stack section with artifical static time shifts added. . . . . . . . 69
7.5 CRS stack section after the first iteration. . . . . . . . . . . . . . . . . 70
7.6 CRS stack section after the second iteration. . . . . . . . . . . . . . . 71
7.7 CRS stack section after the third iteration. . . . . . . . . . . . . . . . 72
7.8 CRS stack section after the fourth iteration. . . . . . . . . . . . . . . 73
7.9 CRS stack after the fifth iteration. . . . . . . . . . . . . . . . . . . . . 74
7.10 Estimated source and receiver static time shifts for the dataset with
and without consideration of the initial static time shifts of the original
dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
7.11 Comparison of the optimized CRS stack section for the three different
methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Chapter A – Refraction seismic in relation to the near-surface 83
A.1 Illustration of Snell’s law . . . . . . . . . . . . . . . . . . . . . . . . . 84
A.2 Traveltime curves of the refracted wave and the direct wave and
the corresponding rays in a 2-D model. . . . . . . . . . . . . . . . . 85
Chapter B – Cross Correlation 87
B.1 Example of a cross correlation result without noise. . . . . . . . . . 88
B.2 Example of a cross correlation result with noise. . . . . . . . . . . . 88
93

List of Tables
Chapter 6 – Synthetic data examples 39
6.1 Processing parameters used for the ZO simulation by means of the
CRS stack. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.2 Mean deviation of the estimated static corrections (different methods
and different settings) from the artifical static time shifts. . . . . 45
Chapter 7 – Real data examples 63
7.1 Processing parameters used for the ZO simulation by means of the
CRS stack. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
7.2 Mean deviation of the estimated statics (Method 1) from the artifical
statics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
7.3 Mean deviation of the estimated statics (Method 1) from the artifical
static time shifts, in consideration of the initial static time shifts
of the original dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . 75
7.4 Mean deviation of the estimated statics (Method 3) from the artifical
static time shifts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
7.5 Mean deviation of the estimated statics (Method 3) from the artifical
static time shifts, in consideration of the initial static time shifts
of the original dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . 77
7.6 Mean deviation of the estimated statics (combined Method) from
the artifical static time shifts. . . . . . . . . . . . . . . . . . . . . . . . 78
7.7 Mean deviation of the estimated statics (combined Method) from
the artifical static time shifts, in consideration of the initial static
time shifts of the original dataset. . . . . . . . . . . . . . . . . . . . . 78
95

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Danksagung
An erster Stelle danken ich meinen Eltern, für die gute Unterstützung während
meines Studiums.
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.
Ingo Koglin danke ich für die Betreuung bei der Erarbeitung dieser Diplomarbeit.
Er beantwortete immer alle Fragen und stand mir bei allen Problemen
hilfreich zur Seite.
Dr. Franz Kirchheimer (Western Geco) danke ich für den Austausch von Ideen
und der Diskussion über theoretische und praktische Probleme.
Für die Hilfe bei der Lösung programmiertechnischer Probleme danke ich vor
allem Jürgen Mann.
Für das Korrekturlesen danke ich: Jürgen Mann, Thomas Hertweck, Zeno
Heilmann und Sonja Greve.
Ich danke aber auch allen Leuten außerhalb der Geophysik für eine schöne
Studienzeit.
99