The Common-Reflection-Surface Stack under Consideration ... - Sites

GEOPHYSIKALISCHES INSTITUT
UNIVERSITÄT KARLSRUHE
The Common-Reflection-Surface Stack
under Consideration of the Acquisition Surface Topography
and of the Near-Surface Velocity Gradient
Die Common-Reflection-Surface Stapelung
unter Berücksichtigung der Topographie der Messoberfläche
und des oberflächennahen Geschwindigkeitsgradienten
Diplomarbeit
von
Zeno Heilmann
Referent: Prof. Dr. Peter Hubral
Korreferent: Prof. Dr. Friedemann Wenzel
Karlsruhe, 31. 5. 2002

EHRENWÖRTLICHE ERKLÄRUNG:
Hiermit versichere ich, dass ich die vorliegende Arbeit selbständig und nur mit den angegebenen
Hilfsmitteln verfasst habe.
Karlsruhe, den 31.5.2002.

Zusammenfassung
Vorbemerkung
Diese Diplomarbeit wurde bis auf die Zusammenfassung in Englisch verfasst. 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 geschriebenen
Abkürzungen, kursiv dargestellt. Für die deutschsprachige Zusammenfassung wurde der gleiche
Aufbau gewählt, wie er auch im Hauptteil der Arbeit verwendet wird.
Einleitung
Die Common-Reflection-Surface (CRS) Stapelung, hat sich in den vergangenen fünf Jahren
als vielversprechende Alternative zu den bisher verwandten seismischen Abbildungsverfahren
etabliert. Ein prinzipieller Vorteil gegenüber anderen Methoden, liegt in der rein
datenorientierten Funktionsweise, die anders als z.B. die Pre-Stack-Depth-Migration keine
Vorkenntnis über das zu ermittelnde Geschwindigkeitsmodell voraussetzt. Die Parametrisierung
der CRS Summationsfläche basiert auf einem isotropen, lateral inhomogenen Modell mit
gekrümmten Schichtgrenzen. Im Gegensatz dazu liegen beispielsweise der Common-Midpoint
(CMP) Stapelung oder dem Normal-Moveout/Dip-Moveout (NMO/DMO) Prozess simple
1D-Geschwindigkeitsmodellannahmen zugrunde. Folglich passt sich die CRS Stapelfläche
der Reflexionsantwort im dreidimensionalen Datenvolumen weit besser an als herkömmliche
Stapeloperatoren. Zudem tragen durch die flächenhafte Stapelung wesentlich mehr Signale zum
Stapelergebnis bei, als beim CMP Stack, bei dem entlang von Linien summiert wird. Dies führt
zu einer deutlichen Verbesserung des Signal-to-Noise Verhältnisses der erzeugten Stapelsektion.
Statt einem einzigen Laufzeitparameter wie der NMO Geschwindigkeit, werden im Verlauf der
CRS Stapelung drei bzw. fünf charakteristische Eigenschaften des Wellenfeldes, so genannte
Wellenfeldattribute ermittelt. Diese können anschließend zur Lösung kinematischer und dynamischer
Inversionsprobleme, wie der Amplitude-versus-Offset (AVO) oder Amplitude-versus-Angle
(AVA) Analyse verwendet werden. Ursprünglich zur Erzeugung gestapelter 2D Zero-Offset
(ZO) Sektionen konzipiert (Mann, 2002; Jäger, 1999; Müller, 1999; Höcht, 1998) wurde der
CRS Stack in den letzten Jahren erfolgreich auf 3D erweitert (Höcht, 2002) und dahingehend
ausgebaut, dass auch gestapelte Sektionen mit variablem Offset erzeugt werden können (Bergler,
2001; Zhang et al., 2001a). Diese Art der Stapelung wird als Finite-Offset (FO) CRS Stack
bezeichnet. Der Common-Offset (CO) CRS Stack stellt einen wichtigen Spezialfall der FO CRS
Stapelung dar und dient zur Erzeugung gestapelter CO Sektionen.
Ziel dieser Diplomarbeit ist es den bestehenden 2D CRS Stack Formalismus, der, auf eine
ebene Messoberfläche bezogene, seismische Daten voraussetzt, dahingehend zu erweitern, dass
auch Daten die auf einer gekrümmten Messoberfläche gemessen wurden, ohne vorhergehende
Bearbeitung, verwendet werden können. Zusätzlich soll der Einfluss eines oberflächennahen
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Gradienten der Wellenausbreitungsgeschwindigkeit, wie er in verwitterten Deckschichten
häufig vorzufinden ist, berücksichtigt werden. Bisher war es notwendig diese Faktoren vor
dem Stapeln mit Hilfe statischer Korrekturen zu beseitigt, was jedoch im Allgemeinen nur
näherungsweise möglich ist und oft zu systematischen Fehlern führt. Diese haben meist
nur geringen Einfluss auf das Stapelergebnis, können jedoch, wie in dieser Arbeit gezeigt
wird, zu einer Verfälschung der beim Stapeln ermittelten Wellenfeldattribute führen. Obwohl
die vorgestellte Herleitung der CRS Laufzeitformeln für gekrümmte Messoberflächen mit
oberflächennahem Geschwindigkeitsgradienten die Finite-Offset CRS Stapelung einbezieht,
konzentrieren sich die anschließenden Betrachtungen auf den spezielleren Fall der Zero-Offset
CRS Stapelung. Einige der erzielten Ergebnisse lassen sich auf den FO CRS Stack übertragen,
allerdings besteht für Finite-Offset momentan das Problem noch darin, die einzelnen Parameter
der Laufzeitformel, unter Berücksichtigung der Messoberflächeneigenschaften, durch Wellenfeldattribute
auszudrücken.
Als theoretische Grundlage dieser Arbeit dienten, neben den oben genannten Veröffentlichungen,
die Arbeiten Chira et al. (2001) und Chira and Hubral (2001), sowie die darin enthaltenen Referenzen.
Die dort dargestellten Ergebnisse wurden mit Hinblick auf ihre praktische Anwendung
weiterentwickelt und anhand eines synthetischen Datenbeispiels erprobt. Leider konnte die
Erweiterung der bestehenden 2D ZO CRS Stack Software, auf gekrümmte Messoberflächen
mit oberflächennahem Geschwindigkeitsgradienten, im Rahmen dieser Diplomarbeit nicht
vollständig abgeschlossen werden. Jedoch wurden alle hierbei zu beachtenden Aspekte
gründlich untersucht, sowie ein schneller und allgemein anwendbarer Code zur Bestimmung der
Messoberflächeneigenschaften entwickelt und in die bestehende Software integriert.
Theorie
Herleitung einer CRS Laufzeitformel für gekrümmte Messoberflächen mit oberflächennahem
Geschwindigkeitsgradienten
Nach einer kurzen Einführung in die Paraxiale Strahlentheorie, die sich aus der Zero-Order Ray-
Theory ableitet, wird die so genannte Surface-to-Surface Propagator Matrix für 2D hergeleitet.
Diese 2 × 2 Matrix, die nur von Größen, die sich auf einen so genannten Zentralstrahl beziehen,
abhängt, ermöglicht es näherungsweise jeden beliebigen Strahl zu beschreiben, der in der Umgebung
des Zentralstrahls verläuft. Hierfür wird angenommen, dass das selbe Raytracing-System
welches den Zentralstrahl beschreibt auch näherungsweise für alle benachbarten (paraxialen)
Strahlen gilt (paraxial assumption). Mit Hilfe der Surface-to-Surface Propagator Matrix wird
dann, über die so genannte Hamilton Gleichung, eine Laufzeitformel abgeleitet, die die Laufzeit
paraxialer Strahlen bis zur zweiten Ordnung exakt repräsentiert. Als Variablen, dienen die
Lokation von Quelle und Empfänger entlang der Seismischen Linie, wobei später zu Offset und
Midpoint Koordinaten übergegangen wird. Aus der ursprünglich parabolischen Darstellung der
Laufzeitformel, lasst sich eine hyperbolische Repräsentation ableiten, welche in den meisten

Fällen eine bessere Näherung der wahren Laufzeit darstellt (Höcht, 1998; Müller, 1999; Jäger,
1999; Bergler, 2001).
Spezielle seismische Konfigurationen
Die fünf Parameter, die im Falle eines Zentralstrahls mit finitem Offset die parabolische und die
hyperbolische Laufzeitgleichung bestimmen, lassen sich durch dessen Ab- bzw. Auftauchwinkel
an Quelle und Empfänger und die drei unabhängigen Elemente der, sich auf diesen Zentralstrahl
beziehenden, Surface-to-Surface Propagator Matrix ausdrücken. Um diese Größen für
eine spätere Stapelung zu bestimmen, eignen sich besonders solche Sektionen (oder Gather) des
drei dimensionalen Datenraums, in denen die Laufzeit nicht von allen fünf Parametern gleichzeitig
abhängt. Diese sind, im Falle einer gekrümmten Messoberfläche, das Common-Shot (CS)
Gather, das Common-Receiver (CR) Gather sowie das Odd-Dislocation (OD) Gather und das
Even-Dislocation (ED) Gather. Die letzteren beiden Sektionen sind Verallgemeinerungen, der
für eine ebene Messoberfläche definierten, Common-Midpoint (CMP) und Common-Offset (CO)
Messkonfigurationen.
Einführung der Wellenfeldattribute
Die drei unabhängigen Elemente der Surface-to-Surface Propagator Matrix lassen sich im Fall
einer ebenen Messoberfläche mit drei Wellenfrontkrümmungen verknüpfen, und werden dadurch
geometrisch interpretierbar (Bergler et al., 2001; Zhang et al., 2001a). Wie diese Verknüpfung für
eine gekrümmte Messoberfläche und unter Einbeziehung des Gradienten der oberflächennahen
Geschwindigkeit aussieht, lässt sich anhand der vielfach bestätigten Ergebnisse von Červen´y
(2001) bestimmen, indem man den dort hergestellten Zusammenhang zwischen der Surfaceto-Surface
Propagator Matrix und der strahlzentrierten Propagations Matrize Π nutzt. Die im
Anschluss benötigte Relation zwischen den Elementen der Π-Matrix und geeigneten Wellenfeldattributen
ist allerdings nur für den Fall eines Zentralstrahls mit koincidentem Quell- und
Empfängerpunkt bekannt (Hubral, 1983). Folglich lässt sich auch nur für diesen Fall eine Laufzeitformel
aufstellen, mit der sich alle Wellenfrontattribute, unter Berücksichtigung der Messoberflächenkrümmung
und des oberflächennahen Geschwindigkeitsgradienten, bestimmen lassen.
Auf diese Zero-Offset CRS Stack Laufzeitformel (in parabolischer und hyperbolischer Darstellung)
konzentrieren sich die weiteren Betrachtungen. Durch die vereinfachte Geometrie des ZO
Zentralstrahls verringert sich die Zahl der Parameter von fünf auf drei. Die mit diesen drei Parametern
verknüpften Wellenfeldattribute sind der Abtauchwinkel β 0 des Zentralstrahls am koinzidenten
Quell- und Empfängerpunkt, und die dort theoretisch messbaren Krümmungen K N und
K NIP der so genannten NIP- und N-Welle (Hubral, 1983).
Verhältnis zwischen Messoberflächeneigenschaften und Wellenfeldattributen
Natürlich stellt auch die erweiterte CRS Laufzeitformel, für gekrümmte Messoberflächen mit
oberflächennahem Geschwindigkeitsgradienten, nur eine Näherung zweiter Ordnung der wahren
Laufzeit dar. Aus diesem Grunde ist es nahe liegend zu untersuchen in welcher Beziehung
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die wahren Wellenfeldattribute zu, unter Vernachlässigung der Acquisitionstopographie und des
oberflächennahen Geschwindigkeitsgradienten, mit konventioneller ZO CRS Stack Software bestimmten
scheinbaren Wellenfeldattributen stehen. Es zeigt sich, dass durchaus die Möglichkeit
besteht solche “pseudo” Attribute im Nachhinein zu korrigieren - allerdings nur wenn sie richtig
bestimmt wurden. Hierin liegt das Problem dieses, ansonsten sehr pragmatischen, Ansatzes.
Bestimmung geeigneter Suchparametergrenzen
Durch den Einfluss des Geschwindigkeitsgradienten und vor allem der Topographie gelten für die
mit konventioneller ZO CRS Stack Software bestimmten pseudo Wellenfeldattribute vollkommen
andere Wertebereiche, als für die wahren Wellenfeldattribute gelten würden. Nur wenn dies bei
der Wahl geeigneter Suchgrenzen berücksichtigt wird können diese Parameter korrekt bestimmt
werden. Hinzukommt, dass im Falle des konventionellen CRS Stacks die gleichen Suchgrenzen
für die gesamte Messoberfläche gelten, was bei einer ebenen Oberfläche natürlich sinnvoll ist.
Hier führt dies jedoch dazu, dass die Suchbereiche der einzelnen Parameter sehr groß gewählt
werden müssen, um den verschiedenen Oberflächeneigenschaften aller Stapelmittelpunkte entlang
der Messoberfläche Rechnung zu tragen. Die Folge ist eine starke Erhöhung des für die
Suche benötigten Rechenaufwands und der damit verbundenen Kosten.
Mit der hier abgeleiteten Laufzeitformel werden im Gegensatz dazu direkt die wahren Wellenfeldattribute
bestimmt, die, mit Ausnahme von β 0 unabhängig von den Eigenschaften der
Messoberflächen sind. Somit gelten für K N und K NIP die selben Grenzen wie im Falle des konventionellen
2D CRS Stacks (siehe Mann, 2002). Beim Abtauchwinkel β 0 muss beachtet werden,
dass dieser sich auf die Tangente zur Oberfläche im Abtauchpunkt bezieht, und daher vom
Dip der Messoberfläche abhängt. Dies spielt bei einer Ebenen Oberfläche keine Rolle, muss
hier aber berücksichtigt werden. Leider hängt die Normal-Moveout Geschwindigkeit, die bei
der heute größtenteils verwendeten pragmatischen Suchstrategie (Mann, 2002) als erster Parameter
bestimmt wird, neben β 0 und K NIP , auch von der Oberflächenkrümmung und dem oberflächennahen
Geschwindigkeitsgradienten ab, was die Wahl geeigneter Suchgrenzen erschwert.
Zur Lösung dieses Problems wird ein Verfahren hergeleitet, mit dem es möglich ist aus den v NMO
Grenzen, die für eine fiktive ebene Messoberfläche ohne oberflächennahen Geschwindigkeitsgradienten
gelten würden, neue, den tatsächlichen Akquisitionsbedingungen angepasste Grenzen,
zu berechnen. Diese variieren dann je nach Auftauchpunkt des betrachteten Zentralstrahls. Auf
ähnliche Weise lassen sich die, bei der Verwendung konventioneller Software, zusätzlich notwendigen
Grenzen für die dort bestimmten “pseudo” Attribute, aus den Grenzen, die für die wahren
Wellenfeldattribute gelten würden, berechnen.
Redatuming
Stapelt man nun einen Datensatz, der auf einer gekrümmten Messoberfläche mit oberflächennahem
Geschwindigkeitsgradienten, bestimmt wurde, so beziehen sich die erhaltenen
Stapel- und Attributsektionen natürlich auf diese Oberfläche. Dies hat zur Folge, dass z.B. ebene

Reflektoren in der so simulierten ZO-Sektion, der Topographie der Messoberfläche entsprechend,
gekrümmt abgebildet werden. Dasselbe gilt auch für die Attributsektionen. Um eine spätere Interpretation
und Weiterverarbeitung dieser Sektionen zu erleichtern ist es nahe liegend, den Einfluss
der Oberflächentopographie zu beseitigen, indem man die ermittelten Wellenfeldattribute
und ZO Laufzeiten auf eine fiktive horizontale Messoberfläche bezieht. Dieser Vorgang, wird
dadurch stark erleichtert, dass beim CRS Stack die Abtauchwinkel der Zentralstrahlen ermittelt
werden. Es bietet sich an, die fiktive Messoberfläche oberhalb der wahren Akquisitionsfläche
zu platzieren, da dann die ebenfalls benötigte Geschwindigkeit, innerhalb der so entstandenen
fiktiven Schicht zwischen wahrer und fiktiver Messoberfläche, frei gewählt werden kann. Für
den Fall einer gekrümmten Messoberfläche ohne oberflächennahen Geschwindigkeitsgradienten
kann die Geschwindigkeit im fiktiven Layer gleich der oberflächennahen Geschwindigkeit
v 0 gewählt werden, die beim CRS Stack als bekannt vorausgesetzt wird. Somit muss keine
Refraktion der Zentralstrahlen und N- und NIP-Wellenfronten an der wahren Messoberfläche
berücksichtigt werden. Liegt ein oberflächennaher Geschwindigkeitsgradient vor, so müssen neben
dem Transmissions Gesetz, noch das Refraktionsgesetz (Hubral and Krey, 1980) und das
Snellius’sche Gesetz berücksichtigt werden, da dann die fiktive Geschwindigkeit natürlich nicht
gleich der (variablen) oberflächennahen Geschwindigkeit gewählt werden kann.
Einführung eines globalen Koordinatensystems
Für die schon erwähnte Herleitung der CRS Laufzeitformeln, wird ein lokales kartesisches Koordinatensystem
verwendet, dessen x-Achse tangential zur Messoberfläche im Auftauchpunkt des
Zentralstrahls verläuft und dessen Ursprung im Auftauchpunkt liegt. Im allgemeinen gilt für jeden
Auftauchpunkt ein anderes lokales Koordinatensystem, in welches die jeweiligen 2D Quellund
Empfängerkoordinaten transformiert werden müssen um die in den Laufzeitformeln benutzten
1D Offset und Midpoint Koordinaten zu berechnen. Dies kann vermieden werden, indem
man in den Laufzeitformeln die lokalen Offset und Midpoint Koordinaten durch globale ersetzt,
die in einem für alle Auftauchpunkte geltenden Koordinatensystem gemessen werden. Hierdurch
erscheint der Dipwinkel der Messoberfläche und die x-Koordinate des Auftauchpunktes explizit
in den Laufzeitformeln.
Mit Hilfe der auf diese Weise modifizierten Laufzeitformeln lassen sich die Relationen zwischen
den wahren Wellenfeldattributen und den mit konventioneller ZO CRS Stack Software,
ohne Berücksichtigung der lokalen Koordinaten (d.h. des Dipwinkels) bestimmten, “pseudo”
Attributen ableiten. Dadurch kann diese Software, mit den genannten Einschränkungen, fast
ohne Abänderung des Codes auch für Daten, die auf einer gekrümmten Oberfläche mit oberflächennahem
Geschwindigkeitsgradienten gemessen wurden, verwendet werden. Die einzig
notwendige Änderung ist die, dass für den pseudo Abtauchwinkel, bestimmt unter Verwendung
globaler Koordinaten, auch komplexe Werte berücksichtigt werden müssen, was bei der konventionellen
CRS Stack Software natürlich nicht vorgesehen ist.
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Diskussion der Ergebnisse anhand eines synthetischen Datenbeispiels
Datensatz
Der hier verwendete synthetische Datensatz wurde von ENI/Agip erzeugt und an Pedro Chira
und mich weitergegeben, um daran die Anwendung der in (Chira and Hubral, 2001) vorgestellten
CRS Laufzeitformel für gekrümmte Messoberflächen zu untersuchen. Das zur Erzeugung dieses
Datensatzes verwendete Modell besteht aus vier homogenen, durch ebene und horizontale Reflektoren
voneinander getrennte, Schichten. Die Messoberfläche besitzt eine starke ausgeprägte
Topographie. Ein oberflächennaher Geschwindigkeitsgradient wurde bei der Generierung des
Datensatzes nicht berücksichtigt, da dieser erst später in die Laufzeitformeln integriert worden
ist (Chira et al., 2001).
Standard Processing mit Hilfe statischer Korrekturen
Zusätzlich zum Datensatz bekamen wir Stapel- und Attributsektionen, welche ENI/Agip erzeugt
hatte, indem zuerst mit Hilfe statischer Korrekturen näherungsweise eine ebene Messoberfläche
simuliert, und anschließend der herkömmliche CRS Stack durchgeführt worden war. Diese Ergebnisse
erwiesen sich allerdings nicht als optimal, da die Wirkung der Korrekturen auf die
zu bestimmenden Laufzeitparameter bei der Wahl geeigneter Suchgrenzen nicht berücksichtigt
worden war. Dies wird am Beispiel der NMO Geschwindigkeit untersucht und durch erneutes
Stapeln mit besser geeigneten NMO Geschwindigkeitsgrenzen demonstriert.
Bestimmung der Oberflächenattribute
Die in den hergeleiteten Laufzeitgleichungen verwendeten Eigenschaften der Messoberfläche
im Auftauchpunkt des Zentralstrahls sind die Krümmung, der Dip (explizit oder implizit), die
oberflächennahe Geschwindigkeit und deren Gradient. Die letzteren beiden Größen müssen im
Feld bestimmt werden, beim Dip und der Krümmung ist dies im Allgemeinen nicht sinnvoll.
Diese werden im Nachhinein aus den gemessenen Quell- und Empfängerkoordinaten berechnet.
Hierbei muss beachtet werden, wofür diese Werte später verwendet werden sollen, nämlich zur
Beschreibung der Messoberfläche im Bereich der betrachteten Stapelapertur durch eine Parabel.
Schwanken Oberflächenkrümmung und Dip innerhalb der Stapelapertur, wie es in der Natur
meistens der Fall ist, muss ein gemittelter Wert verwendet werden. Gemittelt bedeutet hier, dass
Krümmung und Steigung der Parabel bestimmt werden müssen, die alle zur Stapelung beitragenden
Quell- und Empfängerpunkte am besten repräsentiert. Die z-Koordinate des Auftauchpunkte,
die genauso wie Krümmung und Dip nicht lokal bestimmt werden sollte, und zu diesen
Werten konsistent sein muss, wird ebenfalls auf diese Weise ermittelt. Sind die Abweichungen
der Quell- und Empfängerpunkte zu dieser Parabel zu groß, um noch als vernachlässigbar zu gelten,
können statische Korrekturen, wie sie in Sektion 3.2 beschrieben werden, angewandt werden
um Quell- und Empfängerpunkte zu simulieren, die auf der Parabel liegen. Zur Erweiterung der
bestehenden 2D ZO CRS Stack Implementation, wurde nach ausführlichen Tests, die mit Hilfe
des Computer Algebra Systems MAPLE durchgeführt worden waren, ein C++ Code geschrie-

en, der die benötigten Oberflächeneigenschaften, für beliebige Messoberflächen bestimmt. Mit
diesem Programm wurden dann aus den Quell- und Empfängerkoordinaten des oben genannten
Datensatzes die gesuchten Messoberflächeneigenschaften berechnet.
Vorwärts modellierte Ergebnisse
Aufgrund der Einfachheit des betrachteten Untergrundmodells, können die Wellenfeldattribute
K N , K NIP und β 0 für beliebige Zentralstrahlen berechnet werden, ohne dass eine spezielle
Raytracing-Software verwendet werden muss. Zusammen mit den zuvor ermittelten Messoberflächenattributen,
können dann sowohl die Normal Moveout Geschwindigkeit, als auch die
“pseudo” Wellenfeldattribute berechnet werden, die hier aus der Anwendung der herkömmliche
2D ZO CRS Stack Implementation resultieren würden. Zudem lassen sich mit Hilfe der Oberflächenattribute
Suchparametergrenzen, die für eine ebene Messoberfläche sinnvoll gewesen
wären, in Grenzen für die aktuelle Messoberfläche umrechnen. Leider war es zeitlich nicht
möglich die herkömmlichen 2D ZO CRS Stack Implementation vollständig auf den Datensatz
anzuwenden und die so erhaltenen Ergebnisse für die pseudo Attribute, mit den vorwärts berechneten
zu vergleichen.
Ausblick
Da es der zeitliche Rahmen dieser Diplomarbeit nicht erlaubt hat, neben den notwendigen theoretischen
Betrachtungen, auch die praktische Implementation der gewonnenen Ergebnisse abzuschließen,
kann ich am Ende dieser Arbeit diesbezüglich nur einige Vorergebnisse präsentieren.
Wie schon erwähnt, werden bei der Durchführung der ZO CRS Stapelung nicht alle drei Laufzeitparameter
gleichzeitig bestimmt. Als ersten Schritt, wird mit Hilfe einer datengesteuerten
CMP Stapelung eine vorläufige ZO Sektion und die zugehörige NMO Geschwindigkeitssektion
erzeugt. Die Ergebnisse dieses so genannten automatic CMP Stacks, angewandt auf den hier
verwendeten Datensatz, werden in diesem Abschnitt diskutiert. Aufgrund der Einfachheit des,
dem Datensatz zugrunde liegenden Untergrundmodells, dürfte sich jedoch die so erzeugte CMP
Stapelsektion von der zu erwartende CRS Stapelsektion nicht grundlegend unterscheiden. Aus
der so gewonnene NMO Geschwindigkeitssektion wurden die v NMO -Werte der einzelnen Reflektoren
extrahiert. Diese stimmen im Rahmen der zu erwartenden Abweichungen mit den in der
letzten Sektion vorwärts berechneten Ergebnissen überein.
Bei beiden Sektionen wurde zudem ein Redatuming durchgeführt, um eine ebene Messoberfläche
zu simulieren. Dies ermöglicht den direkten Vergleich zu den in Sektion 3.2 gezeigten Ergebnissen,
die mit Hilfe, vor dem Stapeln durchgeführter, statischer Korrekturen erzeugt wurden.
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Contents
1 Preface 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Theory 5
2.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Zero-order ray theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Paraxial ray theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.3 Surface-to-surface propagator matrix . . . . . . . . . . . . . . . . . . . 7
2.2 Fundamental traveltime expressions . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 Traveltime of a paraxial ray . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.2 Determination of the coefficients . . . . . . . . . . . . . . . . . . . . . . 15
2.2.3 Acquisition topography and near-surface velocity gradient . . . . . . . . 17
2.3 Zero-offset situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.1 Relationship between (near)surface and wavefield attributes . . . . . . . 26
2.3.2 Normal-moveout (NMO) and root-mean-square (RMS) velocities . . . . 28
2.4 Search-range estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4.1 Search range of K N and K ∗ N
. . . . . . . . . . . . . . . . . . . . . . . . . 33

XIV CONTENTS
2.4.2 Search range of β 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4.3 Search range of the NMO velocity . . . . . . . . . . . . . . . . . . . . . 34
2.5 Redatuming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5.1 Mapping of X 0 and t 0 to the common-datum surface . . . . . . . . . . . 40
2.5.2 Mapping of K N and K NIP to the common-datum surface . . . . . . . . . 41
2.5.3 Mapping of v NMO to the common-datum surface . . . . . . . . . . . . . 42
2.6 Global coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.6.1 The inhomogeneity factor E 0 in global coordinates . . . . . . . . . . . . 46
2.6.2 The NMO and RMS velocities in global coordinates . . . . . . . . . . . 46
2.6.3 The search range of K ∗ N , β ∗ 0 , and v NMO
in global coordinates . . . . . . . 47
2.6.4 Redatuming in global coordinates . . . . . . . . . . . . . . . . . . . . . 48
3 Synthetic Data Example 51
3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Standard processing using elevation-statics . . . . . . . . . . . . . . . . . . . . . 53
3.2.1 Results of the standard processing . . . . . . . . . . . . . . . . . . . . . 56
3.3 Determination of the (near)surface attributes and elevation in X 0 . . . . . . . . . 63
3.3.1 Determination of α 0 , K 0 , and xz by fitting parabolas . . . . . . . . . . . . 64
3.3.2 Determination of α 0 , K 0 , and xz by fitting circles . . . . . . . . . . . . . 65
3.4 Forward calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.4.1 Forward calculated take-off angle β 0 and its search limits . . . . . . . . . 70
3.4.2 Forward calculated RMS and NMO velocities . . . . . . . . . . . . . . . 72
3.4.3 Forward calculated slowness p NMO and its search limits. . . . . . . . . . 74
3.4.4 Forward calculated values of K N and K NIP . . . . . . . . . . . . . . . . . 76
3.4.5 Pseudo attributes β ∗ 0 , K∗ N , and K∗ NIP
and their search limits. . . . . . . . . 77

CONTENTS XV
3.5 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.5.1 Automatic CMP stack plus redatuming . . . . . . . . . . . . . . . . . . 80
3.5.2 Comparison with the predicted NMO velocities . . . . . . . . . . . . . . 88
4 Summary 93
A The scalar Hamilton’s equation 95
B Used hard- and software 97
C Acknowledgment/Danksagung 99
List of figures 100
Bibliography 105

XVI CONTENTS

Chapter 1
Preface
1.1 Introduction
Seismic methods have a wide range of application. Among important applications in civil engineering
and groundwater search, the oil exploration is of particular importance. Today, it is
standard for oil companies to rely on seismic interpretation for selecting the sites for exploratory
oil wells because thereby the likelihood of a successful venture can be highly improved. The
basic technique of reflection seismics consists of generating seismic waves and measuring the
time required for the waves to travel from the source located on the surface downwards to the
reflector in the subsurface and back to the surface where a series of receivers are positioned. The
receivers are usually disposed along a straight line directed toward the source. The traveltime
of reflected waves depends on the elastic properties of the subsurface as well as on the position,
orientation, and curvature of the reflector. Thus, it is possible to deduce information about the
subsurface from the observed arrival times. In general, before an acquired seismic dataset can
be interpreted it is subject to many processing steps. According to Yilmaz (1987), there is a
well-established sequence for standard seismic processing. The three principal processes — deconvolution,
stacking, and migration — make up the foundation of routine processing. In this
work, I address the second step of the processing chain, namely the stacking procedure. The
so-called stacked section gives the interpreter a first image of the investigated area and serves
as input to the subsequent post-stack migration. By moving source and receiver arrays along
a straight line, called the seismic line, a three-dimensional multi-coverage reflection dataset is
acquired. This dataset depends on shot and receiver location along the seismic line as well as
on the recording time. The subsequent processing of the three-dimensional dataset is applied to
get a two-dimensional image of the subsurface. For the processing, the data are conventionally
sorted with respect to the midpoint position xm of shot and receiver and half-offset h, i.e., the

2 Preface
half distance between shot and receiver. Thus, the multi-coverage reflection dataset is given in
the xm − h −t space, where t corresponds to the recording time.
Unfortunately, the dataset contains not only signals, i.e., any events on the seismic record from
which we wish to obtain information about the subsurface, but also noise. Noise is often divided
into coherent and incoherent noise. As most processing schemes are using primary reflections
only, multiple reflections belong to the class of coherent noise. Incoherent noise or random noise
is not predictable, i.e., one cannot say what a trace will be from the knowledge of nearby traces.
Random noise in real data may be, for instance, due to traffic, industry, or wind shaking trees.
The goal of stacking is to enhance the signal and to suppress the noise by summing up correlated
events in the multi-coverage data. Zero-offset (ZO) stacking operators approximate the actual
reflections in the xm − h − t space in the vicinity of a ZO point. This point is associated with a
hypothetical experiment where source and receiver are coincident. The summation result along
the ZO stacking operator is assigned to the respective ZO point. Doing the same for all points of
the ZO gather yields the ZO stack section. Well-known conventional ZO stacking methods are
the common-midpoint (CMP) stack and the normal-moveout/dip-moveout (NMO/DMO) process.
Within the last five years, the Common-Reflection-Surface (CRS) stack has established as a
promising alternative to the seismic reflection imaging methods, used so far. Originally designed
to generate a 2D ZO stack-section (Höcht, 1998; Müller, 1999; Mann, 2002), the CRS stack was
successfully extended to 3D (Höcht, 2002) and to finite-offset (FO) (Zhang et al., 2001a; Bergler,
2001). The FO CRS stack operator approximates the actual reflections in the vicinity of an arbitrary
point in the xm − h − t space. A special case of the FO CRS stack is the common-offset
(CO) CRS stack, which is performed analogously to ZO stacking but for a point in a CO gather.
Therefore, the CO stacking operator approximates the reflection event in the vicinity of a point
with a fixed offset. Summing up correlated events along the summation operator and assigning
the result to the respective CO point for all points in a chosen CO section yields the CO stack
section. Of course, also any other stacked section (consisting of arbitrary points in the xm − h −t
space) can be generated this way, using the FO CRS stack operator.
Conventional stacking methods, like, e.g., NMO/DMO stack, are based on simple 1D velocitymodel
assumptions and use one-parametric traveltime-moveout formulas that are applied to
common-midpoint data only. The CRS stack makes full use of the multi-coverage seismicreflection
data and provides additional traveltime parameters. These parameters are very useful
for the extraction of further attributes of the seismic medium or for an inversion of a meaningful
subsurface velocity model. Another important feature of the CRS stack method is that an a priori
macro-velocity model is not required. For this reason this method is referred to the macro-model
independent methods, which also include the Polystack method (de Bazelaire, 1988; de Bazelaire
and Viallix, 1994) and the Multifocusing method (Gelchinsky et al., 1997). Various aspects of
macro-model-independent reflection imaging methods are discussed in Hubral (1999). As practical
experience has shown, these new methods are particularly successful for seismic land-data.

1.2 Structure of the thesis 3
However, land data suffer in many cases from complex near-surface conditions like laterally
changing near-surface velocities and undulating topography. For this reason, the existing CRS
method was generalized to handle such situations (Chira et al., 2001). Until then all discussions
and derivations in this regard had involved a planar measurement surface. However, the
assumption of a planar measurement surface is, by no means, a requirement for the validity of
the surface-to-surface propagator matrix formalism (Bortfeld, 1989), which is the basis of the
derivation of the CRS traveltime-moveout formulas. It is to be mentioned that an extension of the
Multifocusing method, designed to include the topographic features of the measurement surface,
has been also recently proposed in Gurevich et al. (2001).
1.2 Structure of the thesis
The thesis is divided into two parts:
Theory (Chapter 2):
After giving a short review of the wave-theoretical foundations, on which the theory presented
within this thesis is based, a generalized finite-offset CRS traveltime formula that considers the
topography of the measurement surface as well as the near-surface velocity gradient, is derived.
This formula is reduced to the zero-offset case to which the further discussions are devoted.
The main purpose of these discussions will be to provide the theoretical background that is
necessary to extent the current implementation of the 2D ZO CRS stack, designed for a planar
measurement surface, to the more general case of considering a curved measurement surface and
its near-surface velocity gradient.
Synthetic Data Example (Chapter 3):
In the second part of this thesis, the application of the 2D ZO CRS stack to data measured on
a curved measurement surface is discussed by means of a synthetic dataset. The near-surface
velocity gradient, was not yet included into the theory, when the dataset was made. Thus this
point is not considered. However, this is no severe restriction, as it is reasonable to study the
effect of the topography separately, at first. For comparison, standard processing using static
corrections is applied to the dataset and briefly analyzed. After a brief general discussion on, how
the considered characteristic properties of the measurement surface and its underlying top-layer
are determined, these considerations are applied to the synthetic dataset. The results, obtained
this way, serve together with forward modeled attributes of the seismic medium to point out
important aspects, related to the application of the CRS stack to data measured on a curved
surface. Unfortunately it was not possible to finish the extension of existing 2D CRS stack
software within the narrow time-frame of a diploma thesis. However, some provisional results
are presented in the outlook.

4 Preface

Chapter 2
Theory
2.1 Basics
The purpose of this section is to give a short review of the ray-theoretical foundations, underlying
this thesis. This is done with the intention, to point out the main results as well as the made
assumptions, as far as it is necessary for the development and understanding of the formulas,
used in the latter. For a detailed treatment I refer, e.g., to Červen´y (2001).
2.1.1 Zero-order ray theory
“The propagation of seismic body waves in complex, laterally varying 3D layered structures is
a considerably complicated process” (Vlastislav Červen´y).
Getting an exact description would require to solve the elastodynamic equations for this very
general case with its multitude of degrees of freedom. However, analytical solutions of these
equations are not known and - even if there would be a solution - not applicable to practical
problems. Thus the most common approaches to investigate the seismic wave-field in complex
media are based either on the direct numerical solution of the elastodynamic equations or on
approximate asymptotic solutions of these equation valid only for high frequencies. One of
the latter is the so-called ray method which is today highly developed and widely used. In
the ray method the asymptotic high-frequency solution of the elastodynamic equations for each
elementary body wave can be sought in the form of a so called ray series (Babich, 1956; Karal and
Keller, 1959). In the frequency domain this is a series in inverse powers of the circular frequency
ω. That is the reason why the ray method is often called the ray series method, or the asymptotic
ray theory . In most practical applications in seismology and seismics, only the leading term of

6 Theory
the ray series which is of the order ω 0 is considered. This leads to the zero order ray theory which
is the underlying concept of all the formulas derived and used within the scope of this thesis.
The main results of this method are the eikonal- and the transport equation. Solving the eikonal
equation results in the so-called ray tracing system which determines all kinematic aspects of
ray propagation. The solution of the transport equation describes all dynamic properties of the
wave-field. The latter are not considered within this theses.
The validity conditions of the zero-order ray theory are an extensively discussed topic, e.g., by
Ben-Menachem and Beydoun (1985), Kravtsov and Orlov (1990) or Červen´y (2001). Many
investigations on this subject were made in the past but nevertheless there are only heuristic
criteria to determine if zero-order ray theory is applicable for a particular earth model or not. One
of the most commonly used conditions is that the Fourier spectrum ˇf[ω] of the source wavelet
f[t] is required to effectively vanish for frequencies
ω < ω 0 = v(ˆr)/l 0 ,
where l 0 is the length scale of the inhomogeneities in the medium and v(ˆr) is the wave velocity of
the medium. In the following a 2D earth model is assumed that consists of isotropic, laterally and
vertically inhomogeneous layers separated by continuous and smooth first-order discontinuities
of almost arbitrary shape which fulfills the above criteria and should be close enough to real
conditions to provide practical useful results.
2.1.2 Paraxial ray theory
The term “paraxial” has his seeds in optics were it represents the vicinity of the axis of the optical
system. In our case it denotes the vicinity of the so-called central ray. Paraxial ray theory is an
extension of the standard ray method with the purpose to describe approximatively the behavior
of paraxial rays in the near vicinity of a central ray, which is assumed to be known. This is
done by using the paraxial assumption saying that the ray tracing system of a particular ray is
approximatively valid also in the close vicinity of this ray. The resulting paraxial ray tracing
system is also called dynamic ray tracing system because it provides even dynamic information,
which is very useful, e.g., in true amplitude imaging (Schleicher et al., 2002). Solving the latter
ray-tracing system for arbitrary initial conditions leads to the ray-centered propagator matrix
Π and to the surface-to-surface propagator matrix T (Bortfeld, 1989; Červen´y, 2001). This
matrices describe the traveltime moveout of any arbitrary ray in the vicinity of a central ray
in terms of quantities that refer to the central ray only. The traveltime along the paraxial ray,
obtained in this way, is correct up to the second-order in q,which is the location of the paraxial
ray, expressed in the ray-centered coordinate system of the central ray (see, e.g., Červen´y, 2001).

2.1 Basics 7
2.1.3 Surface-to-surface propagator matrix
In seismics we generally have measurement configurations were rays emanate from sources that
are located on a surface and impinge at receivers that are located on another surface. According
to Bortfeld these surfaces are called the anterior surface and the posterior surface. In the 2D
case, on which the further discussions are focused, these surfaces reduce to lines.
According to Figure 2.1, we introduce two local Cartesian coordinate systems. The first with
x-axis tangent to the anterior surface at the source location S and the second analog with x-axis
tangent to the posterior surface at the receiver location G. All quantities measured in the first
coordinate system are denoted by the subscript S and all quantities measured in the second coordinate
system by the subscript G. The central ray SG is at S (anterior surface) and G (posterior
surface) completely described by its respective location and ray slowness vectors. An analog
description holds for the paraxial ray SG at S and G. The location and slowness vectors of the
central ray at S and G are denoted by x S and p S , and x G and p G , respectively. Analogously, the
paraxial ray SG at S is described by x S and p S , and at G by x G and p G .
Knowing the two surfaces and their near-surface velocities v S and v G it is possible to reduce
the 2D location and slowness vectors to scalar values. The 2D vectors can later be uniquely
reconstructed from their scalar description. In case of source and receiver locations x S ,x S ,x G ,x G
and the slowness vectors p S ,p G of the central ray this is done by a simple projection in z-direction
onto the x-axis. In case of the slowness vectors of the paraxial ray p S and p G we have to perform
a projection cascade consisting of two steps (see Figure 2.2). First the slowness vector, e.g., p S
is projected onto the line tangent to the measurement curve at S. Later the resulting projection
vector has to be projected onto the coordinate axis tangent in S. An analog description holds for
p G , the slowness vector of the paraxial ray at G.
If we assume the central ray SG to be known, we can approximately calculate any paraxial ray
SG using the paraxial ray theory. The parameters describing the paraxial ray with respect to the
known central ray are its distance to the central ray and the deviation of its slowness vector from
the slowness vector of the central ray. Here, paraxial ray theory implies that the values of these
parameters at the anterior surface are linearly dependent on those at the posterior surface. This
can be written as (see, e.g., Bortfeld, 1989; Hubral et al., 1992; Schleicher et al., 1993)
where
Δx G = AΔx S + BΔp S , (2.1a)
Δp G = C Δx S + DΔp S , (2.1b)
Δx S = x S − x S and Δx G = x G − x G (2.2a)

8 Theory
z
O
z
x
p S
£¡£¡£ ¤¡¤
¤¡¤ £¡£¡£
x S
¥¡¥¡¥¡¥¡¥¡¥¡¥¡¥¡¥ ¦¡¦¡¦¡¦¡¦¡¦¡¦¡¦¡¦ �¡��
¦¡¦¡¦¡¦¡¦¡¦¡¦¡¦¡¦ ¥¡¥¡¥¡¥¡¥¡¥¡¥¡¥¡¥
¥¡¥¡¥¡¥¡¥¡¥¡¥¡¥¡¥ ¦¡¦¡¦¡¦¡¦¡¦¡¦¡¦¡¦
¦¡¦¡¦¡¦¡¦¡¦¡¦¡¦¡¦ ¥¡¥¡¥¡¥¡¥¡¥¡¥¡¥¡¥
¥¡¥¡¥¡¥¡¥¡¥¡¥¡¥¡¥ ¦¡¦¡¦¡¦¡¦¡¦¡¦¡¦¡¦
¦¡¦¡¦¡¦¡¦¡¦¡¦¡¦¡¦ ¥¡¥¡¥¡¥¡¥¡¥¡¥¡¥¡¥
G
�¡�¡�¡�¡�¡�¡� �¡�¡�¡�¡�¡�
G ��
�¡�¡�¡�¡�¡� �¡�¡�¡�¡�¡�¡�
�¡�¡�¡�¡�¡�¡� �¡�¡�¡�¡�¡�
�¡�¡�¡�¡�¡� �¡�¡�¡�¡�¡�¡�
�¡�¡�¡�¡�¡�¡� �¡�¡�¡�¡�¡�
�¡�¡�¡�¡�¡�¡� �¡�¡�¡�¡�¡�
p
�¡�¡�¡�¡�¡�¡� �¡�¡�¡�¡�¡�
�¡�¡�¡�¡�¡� �¡�¡�¡�¡�¡�¡�
S
central ray
x S
§¡§¡§¡§¡§¡§¡§¡§ ¨¡¨¡¨¡¨¡¨¡¨¡¨
�¡��
¨¡¨¡¨¡¨¡¨¡¨¡¨ §¡§¡§¡§¡§¡§¡§¡§
§¡§¡§¡§¡§¡§¡§¡§ ¨¡¨¡¨¡¨¡¨¡¨¡¨
¨¡¨¡¨¡¨¡¨¡¨¡¨ §¡§¡§¡§¡§¡§¡§¡§
p S
§¡§¡§¡§¡§¡§¡§¡§ ¨¡¨¡¨¡¨¡¨¡¨¡¨
¨¡¨¡¨¡¨¡¨¡¨¡¨ §¡§¡§¡§¡§¡§¡§¡§
§¡§¡§¡§¡§¡§¡§¡§ ¨¡¨¡¨¡¨¡¨¡¨¡¨
§¡§¡§¡§¡§¡§¡§¡§ ¨¡¨¡¨¡¨¡¨¡¨¡¨
paraxial ray
¢¡¢ ¡ ¡
©¡© �
� ©¡©
©¡© �
©¡© �
p � ©¡©
©¡© �
G ��
� ©¡©
O x
� �¡�
� �¡�
G
x
G
G
©¡© �
� ©¡©
S
x
anterior
surface
posterior
surface
Figure 2.1: Sketch of a two-dimensional inhomogeneous and isotropic medium. The central ray
(depicted in green) passes through this medium starting on the anterior surface at S and ending
on the posterior surface at G. The paraxial ray (depicted in red) is in close vicinity of the central
ray. The quantities describing the central ray at the anterior surface are shown in green, those
describing the paraxial ray in red.

2.1 Basics 9
����
O
z
����
S
p S
��������������������� ��������������������� ��
��������������������� p S
���������������������
S
��������������������� ���������������������
��������������������� ���������������������
paraxial ray
� ���
Figure 2.2: Construction of the ray slowness vector projection: The ray slowness vector is firstly
projected onto the tangent to the anterior surface at S. The resulting vector p S,T is then projected
onto the tangent to the anterior surface at S,which coincides with the x-axis of the coordinate
system [x,z].
are the differences of the projected coordinates of the central and paraxial ray at the anterior and
posterior surface, and
p
S,T
Δp S = p S − p S and Δp G = p G − p G (2.2b)
are the differences of the respective projected ray slowness values.
Equations (2.1) can also be written in vector and matrix form by
� � � �
ΔxG ΔxS
= T . (2.3)
ΔpG ΔpS T is the so-called surface-to-surface ray propagator matrix (Hubral et al., 1992; Schleicher et al.,
1993) and reads
� �
A B
T = . (2.4)
C D
Matrix T has the following important properties:
1. The Symplecticity
x

10 Theory
This property states that the inverse of the propagator matrix T can be written as
Consequently holds
T −1 =
� A B
C D
�−1
=
� D −B
−C A
This implies that the T has only three independent values.
�
. (2.5)
AD − BC = 1 . (2.6)
2. The Chain Rule
The chain rule states that we can introduce an arbitrary point M along the central ray SG
to decompose the propagator matrix T (G,S) in the following way.
T (G,S) = T (G,M)T(M,S). (2.7)
T (M,S) denotes the propagator matrix of the first ray segment and T (G,M) the propagator
matrix of the second one.
3. The Reverse Ray Property
A hypothetical ray, traveling from the geophone location G to the source at S on the same
path as the ordinary ray SG but in the opposite direction is called reverse ray. The propagator
matrix of this reverse ray T ∗ is related to the propagator matrix T of the ordinary ray
by the expression
T ∗ �
A∗ B∗ =
C ∗ D ∗
�−1
=
� D B
C A
�
. (2.8)
By comparing equations (2.8) and (2.5) we can see that the inverse propagator matrix T −1
and the propagator matrix T ∗ of the reverse ray are not identical.
2.2 Fundamental traveltime expressions
As the surface-to-surface propagator matrix T is valid for arbitrary rays, not only the situation
as shown in Figure 2.1 but also a measurement geometry as depicted in Figure 2.3 can be
considered, which is the usual measurement geometry in reflection seismics. In the following
the surface-to-surface propagator matrix T is used to derive in a very convenient way the
traveltime moveout of any paraxial ray. This requires only quantities that refer to the central

2.2 Fundamental traveltime expressions 11
measurement surface
S
Δx S
z S
S
x
S
central ray
β S β G
R
R
G
Δ x G
G
paraxial ray
reflector
Figure 2.3: Sketch of a 2D model with a curved measurement surface. The central ray SRG and
a paraxial ray S R G in its vicinity are shown. The initial point S is the origin of the local axis x S
and the end point G is the origin of the local axis x G . Both axes are tangent to the measurement
surface at the source S and receiver G, respectively.
z G
x
G

12 Theory
S
1
� ���
S
S
paraxial ray
��� �����
central ray
��� ���
Figure 2.4: The paraxial ray from S to G in the vicinity of the central ray from S to G.
ray. For a finite-offset central ray in 2D as shown in Figure 2.3, these quantities are the take-off
angles of the central ray at S and G and the three independent elements of the propagator
matrix T . From now on, we choose the origin of our local coordinate systems, without loss of
generality, to be located in the source and receiver points, respectively, according to Figure 2.3.
As mentioned before Zhang et al. (2001b) and Bergler (2001) have extended the standard
zero-offset (ZO) CRS stack traveltime formula to finite-offset (FO). However, they continued
to restrict the discussion to a planar measurement surface. This is not necessary, because the
surface-to-surface propagator matrix formalism does not require a planar measurement surface
to be valid. We will now following the lines of Chira et al. (2001) and Schleicher et al. (2002),
re-derive and extend their results to the more general case of considering a curved measurement
surface.
2.2.1 Traveltime of a paraxial ray
In this section we discuss how to derive the traveltime moveout of an arbitrary paraxial ray with
respect to the known central ray. We are not looking for a representation of the exact traveltime
moveout but for an asymptotic one that is valid up to the second order of the dislocation between
the source and receiver points of the paraxial and the central ray. In the following the word “exact”
is used synonymous to “exact within the scope of the zero-order ray theory”.
G
1
�
G
G

2.2 Fundamental traveltime expressions 13
We assume that we can always find a wavefront that pertains to both rays, i.e., paraxial and
central ray belong to the same ray family and fulfill the eikonal equation with the same initial
conditions. Only in this situation, the description of the paraxial ray in terms that refer to the
central ray makes sense. This situation is depicted in Figure 2.4. In contrast to the illustration
these traveltime differences shall be infinitesimal small. Considering infinitesimal small traveltime
differences is fully sufficient, as we are only looking for a second-order approximation of
the traveltime moveout. According to Figure 2.4 the difference between the traveltime from S to
G, t(S,G), and the traveltime from S to G, t(S,G), can be expressed as
dt = t(S,G) −t(S,G) = dt G − dt S , (2.9)
where dt S is the traveltime from S 1 to S and dt G is the traveltime from G 1 to G.
The change of the traveltime due to a infinitesimal perturbation of the paraxial, e.g., source point,
d(x S − x S1 ) is given by
dt S = p S · d(x S − x S1 ) , (2.10a)
if p S = p S1 is assumed. Please note the analogy to the well known scalar equation v = ds/dt.
This is equal to
dt S = p S · d(x S − x S ) . (2.10b)
because p S is parallel to d(x S − x S1 ).
Certainly relations (2.10) hold also for the paraxial receiver point G. Inserting this into (2.9)
leads to Hamilton’s equation for two point ray-tracing, which reads
dt = p G · d(x G − x G ) − p S · d(x S − x S ), (2.11)
and which is basically nothing else than an alternative mathematical formulation of Fermat’s
principle, saying that the first derivative of the traveltime in the direction vertical to the ray
vanishes.
Our aim is to obtain a second-order approximation of the traveltime of the paraxial ray, therefore
it is sufficient to consider only linear terms of Δx S and Δx G within the Hamilton’s equation. This
enables us, according to Appendix A, to reduce the vectors in equation (2.11) to scalar values,
because the z-components of the dot products are already of second or higher order in Δx S and
Δx G , respectively.
dt = p G d(Δx G ) − p S d(Δx S ) . (2.12)
Now we can benefit from the surface-to-surface propagator matrix T which we introduced in the
last section. If we insert equations (2.2) into equations (2.1), we get,
x G − x G = Ax S − x S + B p S − p S , (2.13a)
p G − p G = C x S − x S + D p S − p S . (2.13b)

14 Theory
Solving the first of the two equations above for p S and the second for p G , whereas we substitute
p S instantly, leads to
pS = pS + B −1 ΔxG − B −1 AΔxS (2.14a)
pG = pG +C ΔxS + DB −1 ΔxG − DB −1 AΔxS . (2.14b)
Finally we have to insert equations (2.14a) and (2.14b) into the scalar representation of Hamilton’s
equation (2.12) and integrate. Considering the symplecticity of the T matrix, we obtain the
traveltime formula
tpar(ΔxS ,ΔxG ) =tSG + pG ΔxG − pS ΔxS − ΔxS B −1 ΔxG + 1
2 ΔxS B−1 AΔxS + 1
2 Δx′ G DB−1 Δx ′ G ,
(2.15)
where t SG is the two-way traveltime of the central ray SG.
This traveltime representation is known as the parabolic traveltime. By squaring this equation
and retaining only the terms up to the second order in the dislocations Δx S and Δx G we obtain
the hyperbolic traveltime equation (Ursin, 1982).
t 2 hyp (ΔxS ,ΔxG ) = � �2 tSG + pG ΔxG − pS ΔxS �
+ 2tSG −ΔxS B −1 ΔxG + 1
2 ΔxS B−1AΔxS + 1
2 ΔxGDB−1 �
ΔxG .
(2.16)
Ursin suggested, after systematic investigations that the hyperbolic traveltime formula is a better
approximation to the real traveltime response than the parabolic one. This was later verified by
the work of Höcht (1998), Müller (1999), Jäger (1999), and Bergler (2001).
The general representation of the second-order Taylor expansion of the traveltime t(Δx S ,Δx G )
reads
tpar(Δx S ,Δx G ) = t SG + a Δx S + b Δx G + c Δx S Δx G + d (Δx S ) 2 + e (Δx G ) 2 , (2.17)
with the five coefficients {a,b,c,d,e}.
We can also derive a hyperbolic representation of this traveltime formula by squaring and retaining
only terms up to the second order in Δx S and Δx G . We get
t 2 hyp (ΔxS ,ΔxG ) = � �2 �
tSG + a ΔxS + b ΔxG + 2 tSG c ΔxSΔxG + d (ΔxS ) 2 + e (ΔxG ) 2� . (2.18)
Of course, these two traveltime equations do not require any assumptions concerning the subsurface
media or the shape of the acquisition surface. Only in order to relate the general parameters
{a,b,c,d,e} to physical quantities, assumptions respective subsurface structure and measurement
surface shape are necessary (see Section 2.2.3).

2.2 Fundamental traveltime expressions 15
If we compare these two equations with equations (2.15) and (2.16) and introduce the two angles
β S and β G , we can relate {a,b,c,d,e} to the three independent values of the propagator matrix T
and β S and β G . It results
a = − sinβ S /v S , (2.19a)
b =sinβ G /v G , (2.19b)
c = − 1/B , (2.19c)
d =A/2B , (2.19d)
e =D/2B , (2.19e)
where β S is the take-off angle of the central ray in S and β G the respective emergence angle in G.
Both are measured with regard to the surface normal (see Figure 2.3). They are defined as
β S = sin −1 (v S p S · t S ) and β G = sin −1 (v G p G · t G ) , (2.20)
where p S , t S and p G , t G are the slowness and surface-tangent vectors at S and G, respectively.
Equations (2.19 c,e,d) show that it is mandatory to require B not to be equal zero, i.e., the endpoint
of the central ray must not be a caustic point.
2.2.2 Determination of the coefficients
In order to use one of the two traveltime formulas stated above for the purpose of a FO CRS
stack, it is necessary to determine those values for the five parameters {a,b,c,d,e} that parameterize
that traveltime surface that approximates the real reflection response, best. This is done by
means of a CRS stack-based coherency analysis which is directly applied to the multi-coverage
dataset (Höcht, 1998; Müller, 1999; Mann, 2002). There, those values for {a,b,c,d,e} are determined
which yield the highest coherency value. The traveltime t SG of the considered central
ray and the two near-surface velocities v S and v G are assumed to be known. Due to the fact
that a simultaneous search for five parameters would demand an immense computational effort,
Bergler (2001) and Mann (2002) presented strategies for the FO and the ZO CRS stack which
allow to split the search into separate one and two parametric searches, respectively. These
searches are performed within gathers that are related to special seismic configurations, i.e., the
common-midpoint (CMP) gather, common-offset (CO) gather, the common-shot (CS) gather and
the common-receiver (CR) gather. Within these special gathers the traveltime does not depend on
all parameters simultaneously. Consequently it is possible to obtain all parameters by performing
a convenient sequence of one and two parametric searches. However, these parameters are only
an approximation of the ideal parameter set that would result from a simultaneous search.
To apply these strategies to the more general case of a curved measurement surface, we have

16 Theory
to modify the conventional CMP and CO measurement configurations, as described for a planar
measurement surface, slightly. This is necessary, because for a curved measurement surface it
is not reasonable to consider a CMP configuration, where the physical midpoint between source
and receiver (not to confuse with the 1D midpoint coordinate introduced in Section 2.3) is fixed.
The same holds for the CO configuration, where the 2D dislocation (offset) between source and
receiver is constant. (However, it is a matter of interpretation. If midpoint and offset are understood
as 1D midpoint and offset coordinates, according to their definition in Section 2.3 (see
equations (2.39)), then the CMP and CO configurations coincide with their extensions, defined
in the following.)
According to Chira et al. (2001) we introduce two measurement configurations which are natural
extensions of the CMP and CO configurations and are just as useful for the estimation of the
searched parameters. These are :
The Odd-dislocation (OD) gather: This configuration is related to the CMP gather. Paraxial
source point S and receiver point G are dislocated by the same amount in opposite directions
with respect to the source and receiver points S and G of the central ray. The OD condition and
the corresponding OD traveltime moveout are
(OD) Δx = Δx G = −Δx S , t 2 OD = [t SG + (b − a) Δx]2 + 2t SG (d + e − c) (Δx) 2 . (2.21)
The Even-dislocation (ED) gather: This configuration is related to the CO situation. Paraxial
source point S and receiver point G are dislocated by equal amounts in the same direction with
respect to the source and receiver points S and G of the central ray. The ED condition and the
corresponding ED traveltime are
(ED) Δx = Δx G = Δx S , t 2 ED = [t SG + (a + b) Δx]2 + 2t SG (c + d + e) (Δx) 2 . (2.22)
In addition to the OD and the ED gather we have to consider the well known common-shot (CS)
and the common-receiver (CR) gather. Here the conditions and traveltime formulas are
(CS) Δx = Δx G , Δx S = 0 , t 2 CS = [t SG + b Δx]2 + 2t SG e (Δx) 2 . (2.23)
(CR) Δx = Δx S , Δx G = 0 , t 2 CR = [t SG + a Δx]2 + 2t SG d (Δx) 2 . (2.24)
Comparing the traveltime formulas in the different gathers reveals that they are all of the same
structure, i.e., one-variable quadratic polynomials with two independent parameters, which read
t 2 (Δx) = (t SG + m Δx) 2 + 2t SG n (Δx) 2 . (2.25)
The estimation of these two parameters in at least three specific gather via coherency analysis
is sufficient to determine all five traveltime parameters {a,b,c,d,e}. This can be done either
using two one-parametric searches, where first the quadratic coefficient n is assumed to be zero,

2.2 Fundamental traveltime expressions 17
or performing a two-parametric search which is more accurate but also more time consuming
(Bergler, 2001). The relationship between the parameters m and n of the different gathers and
the attributes {a,b,c,d,e} can be derived simply from the comparison of equation (2.25) with
equations (2.21), (2.22), (2.23), and (2.24). This leads to
a = m CR , b = m CS = m OD + m CR , c = n ED − n CS − n CR ,
d = n CR , e = n CS .
where, e.g., m CS is the linear coefficient and n CS is the quadratic coefficient in the CS gather.
(2.26)
We can express the elements of the propagator matrix T in a similar way when we insert equations
(2.26) into (2.19) and make use of the symplecticity relation.
2 nCR 1
A =
, B =
,
nCS + nCR − nED nCS + nCR − nED 4 n CR n CS
2 nCS C =
+ nED − n
nCS + nCR − n CS − nCR , D =
.
ED
nCS + nCR − nED 2.2.3 Acquisition topography and near-surface velocity gradient
(2.27)
So far we made no assumptions concerning the acquisition topography. With respect to the subsurface
structure the only assumption was that the near-surface velocity at S and G is known. Of
course, the described paraxial vicinity is respectively small if the subsurface structure is unfavorable.
Traveltime formulas (2.15) and (2.16) are fully sufficient to describe the second-order
traveltime moveout of any paraxial ray by means of the three independent elements of matrix
T and the two take-off angles β S and β G . The next step is to relate the parameters {a,b,c,d,e}
and thus also the elements of T to so-called wavefield attributes, physical properties referred
to the central ray that are theoretically measurable at the measurement surface. These are, as
mentioned before, very useful, e.g., for subsequent inversion or Fresnel-zone and geometrical
spreading estimation. To do this, previously known information about acquisition surface and
underlying top-layer has to be considered explicitly within the traveltime formulas, in order to
separate this information from the searched-for wavefield attributes. To give an example: At
the moment we are already able to determine two physical properties of the central ray from
the parameters {a,b,c,d,e}, using equations (2.19). These are the take-off angles β S and β G ,
which still include the influence of the topography, as they are measured against the normal of
the surface tangent in S and G, respectively. In the case of β S and β G the topography dependency
does not conceal the physical meaning, thus it is not absolutely necessary to remove it. But it
can be easily removed by introducing the dip angles α S and α G of the surface tangents in S and

18 Theory
G and defining alternative take-off angles, now measured against the horizontal. We do this later
in the zero-offset case (see Section 2.3), where we introduce the global ZO take-off angle β g
0
according to Figure 2.6 and equation (2.52) in order to split the take-off angle measured against
the horizontal, which is an attribute of the central ray, from the surface dip, which is an attribute
of the measurement surface.
In oil exploration, it is usual to drill shallow holes along the seismic line to get information
which can be used for static corrections. Consequently the near-surface velocity gradient is often
known. In the following we do not only assume to know the acquisition topography and
near-surface velocity, but also the near-surface velocity gradient. The propagator matrix T obviously
incorporates, besides the overall properties of the propagating medium and the acquisition
geometry, also these characteristics. More specifically, it will be seen to depend on the near
surface velocity and its gradient, as well as on the curvature of the measurement surface at the
source and receiver points of the central ray. Following Červen´y (2001), Chapter 4, we assume
the measurement surface, in the vicinity of the source and receiver points of the central ray to be
representable by parabolas. These read
zS = − 1
2 KS x2S and zG = − 1
2 KG x2G , (2.28)
whereas the respective local coordinate systems have their origins in S and G (see 2.3) and
the curvature of the surface in this points is denoted by K S and K G . To be in conformance
with Červen´y we choose the negative sign in the equations above. This means that the surface
curvature is defined in analogy to the curvature of an upcoming wavefront. Due to the fact that
Červen´y treats the general 3D case and we are only looking for 2D solutions we assume the
y-coordinate axis always to point into the drawing plane, according to the right hand rule. In
contrast to Červen´y who chooses the surface normals and consequently the z-axes always in a
way that keeps the angles β S and β G acute and positive, we choose both surface normals to point
upwards which leads to acute angles that can be also negative (see (2.20)).
Up to this point the T matrix can be seen as a black box formulation. To determine the dependency
of its elements {A,B,C,D} on the properties of the measurement surface and on the
near-surface velocities at S and G, it is necessary to relate this matrix to a corresponding “intrinsic”
propagator matrix. Such an intrinsic matrix can be readily selected as the 2D, in-plane,
ray-centered propagator matrix Π which is extensively discussed in Červen´y (2001). It reads
Π(G,S) =
� Q1 Q 2
P 1 P 2
�
, with Q1P2 − Q2P1 = 1 . (2.29)
If the acquisition surface in the vicinity of S and G can be represented according to equations
(2.28) and if there are only first order velocity variations in the vicinity of S and G, then the matrix
Π is related to the surface-to-surface propagator matrix T in the following way (see Červen´y

2.2 Fundamental traveltime expressions 19
(2001) equation (4.4.90) with a different notation):
T (G,S) = Y(G) Π(G,S) Y −1 (S) . (2.30)
The two matrices Y(G) and Y −1 (S) that constitute the relationship depend only on properties
of the measurement surface and of the top-layer in the vicinity of S and G (see Červen´y (2001),
equation (4.13.21) with a different notation).
with
� �
1/cosβG 0
Y(G) =
, (2.31)
(EG − cosβG KG /vG )/cosβG cosβG Y −1 � �
cosβS 0
(S) = −
, (2.32)
− (ES + cosβS KS /vS )/cosβS 1/cosβS det(Y(G)) = det(Y −1 (S)) = 1 . (2.33)
E S and E G are the so-called inhomogeneity factors and account for the first-order velocity variations
in the vicinity of the source and receiver points of the central ray, respectively. They are
given by (see Červen´y, 2001, equation (4.13.20) with a different notation) and read
E S = − sinβ S
v 2 S
E G = − sinβ G
v 2 G
[(1 + cos 2 β S )(∂x S v) S + cosβ S sinβ S (∂z S v) S ] , (2.34a)
[(1 + cos 2 β G )(∂x G v) G − cosβ G sinβ G (∂z G v) G ] . (2.34b)
Here, (∂x S v) S and (∂z S v) S denote the 2D in-plane components of the medium velocity gradient
(∇v) at the source point S. An analog meaning holds for (∂x G v) G and (∂z G v) G , thus
(∇v) S =
� ∂x S v
∂z S v
�
| S and (∇v) G =
� ∂x G v
∂z G v
�
| G . (2.35)
Note that the two inhomogeneity factors E S and E G are no pure attributes of the top-layer in
the vicinity of S and G, because they depend not only on the near-surface velocity gradients
(∇v) S and (∇v) G , but also on the take-off angles β S and β G . In the following v S ,v G ,K S ,K G and
(∇v) S ,(∇v) G are called (near)surface attributes. Equation (2.30) enables us to express the components
{A,B,C,D} of the surface-to-surface propagator matrix T , in terms of their counterpart
components {P 1 ,Q 1 ,P 2 ,Q 2 } of the ray-centered propagator matrix Π using characteristic quantities
of the measurement surface and the top-layer in the vicinity of the points S and G. These
quantities are the near-surface velocities v S and v G the near-surface velocity gradients (∇v) S
and (∇v) G and the surface curvatures K S and K G . To do this we have only to insert equations
(2.31), (2.32), (2.34a) and (2.34b) into equation (2.29). Subsequently we can relate the quadratic

20 Theory
coefficients c,d and e of the traveltime equations (2.15) and (2.16) to the elements of the surfaceto-surface
propagator matrix, {A,B,C,D}, using equations (2.19). We remember at this point
that the linear coefficients a and b depend only on the take-off and emergence angles (of the
central ray) β S and β G and are implicitly related to the dip angle of the measurement surface at S
and G.
a = − sinβ S /v S , (2.36a)
b =sinβ G /v G , (2.36b)
c = − 1/B = cosβ S cosβ G /Q 2 , (2.36c)
d =1/2 A/B = 1/2 ((Q 1 /Q 2 ) cos 2 β S − (cosβ S /v S )K S − E S ), (2.36d)
e =1/2 D/B = 1/2 ((P 2 /Q 2 ) cos 2 β G − (cosβ G /v G )K G + E G ). (2.36e)
Under the assumption that the (near)surface attributes {v S ,v G ,,K S ,K G ,(∇v) S ,(∇v) G } are known,
this relation between the different sets of traveltime parameters gives us the opportunity to determine
the values of the T or Π matrix. These can be computed from the values {a,b,c,d,e},
which previously have to be obtained via coherency analysis or directly by searching these values
in a coherency analysis which uses a traveltime formula, where {a,b,c,d,e} are substituted
according to equations (2.36). Most probably, these elements of the propagator matrices T or Π
will become of utmost importance for the inversion of velocity models in the near future. First
results in the zero-offset case, as reported in Biloti et al. (2001), are very encouraging. Despite
all the benefit we can obtain from the five traveltime parameters for inversion problems, the
main purpose of the Common-Reflection-Surface stack method remains the stacking. But also
for the stacking it is important to relate the searched-for parameters {a,b,c,d,e} to physically
meaningful quantities in order to estimate suitable limits in which a parameter is searched.
2.3 Zero-offset situation
In the last section, we have derived the very general expression (2.15) for the traveltime moveout
of a ray that is paraxial to an arbitrary central ray. This expression is called the parabolic
traveltime according to the mathematical form of this equation. From this equation we deduced
the hyperbolic traveltime expression (2.16), which was shown to be more successful in the practical
application of the CRS stack, by Höcht (1998), Müller (1999), Jäger (1999) for zero-offset
and by Bergler (2001) for common-offset. By means of this formulas we are able to create a
stacked section with an arbitrary offset (FO CRS stack), using central rays with arbitrary shot
and receiver dislocation. In the following we want to reduce our formulas to the simpler case of
a central ray that impinges perpendicularly at the reflector. This particular central ray has a coincident
source and receiver point S = G and consequently an offset that is equal to zero (ZO CRS

2.3 Zero-offset situation 21
Figure 2.5: ZO situation: The central ray shown in green hits the reflector perpendicularly at
the normal incidence point (NIP). Consequently, the down-going and up-coming ray paths are
identical. The wavefronts of the NIP-wave depicted in light-blue and the Normal-wave depicted
in blue travel along the central ray emerging at the coincident source and receiver point X 0
with the wavefront curvatures K NIP and K N , respectively. For the illustration, the wavefronts
are approximated by circular segments with the corresponding curvature of the wavefronts at
the central ray. The measurement surface depicted in brown is a smooth curve with negative
curvature K 0 in X 0 .

22 Theory
stack). This situation is shown in Figure 2.5. For the sake of simplicity and for consistency with
former publications we denote the coincident source and receiver point of a central ray by X 0 . In
order to specify the location of the coincident source and receiver point pairs of different central
rays, a global coordinate system is used. In this global coordinate system, the x-coordinate of X 0
is denoted by x 0 and the z-coordinate by z 0 . The origin of the global coordinate system can be
chosen arbitrarily, but the origin of the local coordinate system is always X 0 . In the following X 0
is called the central point, because it is the emergence point of the central ray to which the CRS
traveltime surface t(m,h)| (x0 ,t0 ) is referred, whereas t0 is traveltime of the (zero-offset) central
ray. Please note that we can simulate the zero-offset traveltime for any point in the vicinity of X 0 ,
but these points must not be confused with the central point X 0 even if they are also coincident
source and receiver points. All attributes that refer to the central point X 0 are indicated in the
following with the index 0, to signify the ZO case.
Considering this particular geometry leads to a strong simplification of the problem, because
the up-going and the down-going ray paths coincide. Only the directions of propagation are
opposite. This has two important consequences:
1. As we have now only one point which is both, source and receiver, the take-off angle β S
and the emergence angle β G cannot have different absolute values. But due to the fact that,
per definition (equation (2.20)), the sign of the angle between a ray and a surface depends
on the direction of ray propagation, we have the following relation.
β S = − β G = β 0 . (2.37)
This means that for a ZO central ray the take-off angle β S and the emergence angle β G
reduce to one take-off angle which we will call β 0 .
2. As the two ray paths as well as the source and receiver points coincide, also the surfaceto-surface
propagator matrices of the ray and its respective reverse ray have to coincide.
Using the property of the reverse ray propagator matrix T ∗ (2.8) stated in Section 2.1.3 we
can write
T ZO =
Consequently holds in the ZO case
� A B
C D
�
=
� D B
C A
�
∗
= T ZO . (2.38a)
A = D . (2.38b)
Thus, remembering the symplecticity relation (2.5), the ZO propagator matrix T ZO has
only two independent values.
The FO CRS stack traveltime formulas were dependent on five attributes: two angles and three
independent values of the propagator matrix T or Π, respectively. In the ZO case this dependency

2.3 Zero-offset situation 23
is reduced to one angle and two matrix elements. If we introduce midpoint and half-offset coordinates,
instead of the source and receiver dislocations Δx S and Δx G , according to the relations
m = Δx S + Δx G
2
, and h = Δx G − Δx S
2
, (2.39)
we can express the parabolic (2.15) and hyperbolic (2.16) FO traveltime equations, now reduced
to ZO, using only three parameters. This reads
t par
ZO = t 0 + σ 1 m + σ 2 m2 + σ 3 h 2 ,
t hyp
ZO = (t 0 + σ 1 m)2 + 2t 0 (σ 2 m 2 + σ 3 h 2 ) .
(2.40)
In view of our results for the FO case, equations (2.36), we readily find the relations that hold
for the three coefficients {σ 1 ,σ 2 ,σ 3 } in the ZO case. We have only to replace β S and β G with
β 0 , according to relation (2.37), and v S and v G with v 0 and substitute the matrix element D by A
according to (2.38), within equations (2.19) and subsequently insert these into the parabolic and
hyperbolic traveltime equations (2.17) and (2.18). Doing this we find the relations
σ 1 = 2sinβ 0 /v 0 , σ 2 = (A − 1)/B and σ 3 = (A + 1)/B . (2.41)
To find now the relation between the traveltime coefficients {σ1 ,σ2 ,σ3 }, and the (near)surface
attributes {v0 ,K0 ,(∇v) 0 } at the coincident source and receiver point X0 we can follow the same
strategy as in the FO case upon the use of the “intrinsic” ray-centered propagator matrix ΠZO � �
Q1 Q
ΠZO =
2 , (2.42)
P 1 P 2
and reduce the expressions (2.36), which we derived by means of results from Červen´y (2001),
to the ZO case like we did above. This, together with equations (2.41), leads after some straightforward
algebraic calculations to the searched-for relationship between the traveltime parameters
{σ 1 ,σ 2 ,σ 3 } and the surface characteristics {v 0 ,K 0 ,(∇v) 0 } in S 0 = G 0 ,
where
σ 1 = 2sinβ 0 /v 0 ,
σ 2 = ((Q 1 + 1)/Q 2 ) cos 2 β 0 − (cosβ 0 /v 0 )K 0 − E 0 ,
σ 3 = ((Q 1 − 1)/Q 2 ) cos 2 β 0 − (cosβ 0 /v 0 )K 0 − E 0 ,
E0 = − sinβ0 v2 �
(1 + cos
0
2 �
β0 )(∂xv) 0 + cosβ0 sinβ0 (∂zv) 0
= E S (S = S 0 )
= −E G (G = G 0 ) .
(2.43)
(2.44)

24 Theory
For the zero-offset situation it is possible to express the elements of the ray-centered propagator
matrix Π by means of two wavefield attributes. These are the wavefront curvatures of two
conceptual eigenwaves (see Figure 2.5) which were firstly introduced in Hubral (1983). For the
wavefront curvatures the sign convention according to Hubral and Krey (1980) is used. When a
wavefront lags behind its tangent plane then the wavefront curvature is defined as positive. If the
wavefront is ahead of its tangent plane the wavefront curvature is negative.
1. The Normal-Incidence-Point-wave (NIP-wave) is related to a ray that originates from a
point source at the coincident source and receiver point X 0 , hits the reflector perpendicular
and travels back to its origin, using the same path, but in opposite direction. If we assume
now a wave traveling along this ray with half the true medium velocity and starting at the
reflector, its wavefront would reach X 0 with the curvature K NIP . Consequently a downward
propagating wavefront with K(X 0 ) = −K NIP would shrink to a point reaching the reflector
and would be reflected “into itself” and reach X 0 again with the curvature K NIP .
2. The Normal-wave (N-wave) originates as a wavefront that coincides with the reflector in
the vicinity of the reflection point NIP and propagates upwards reaching the surface at X 0
with the curvature K N . If we propagated this wavefront back to the reflector, it would also
be reflected “into itself” and reach X 0 again with the curvature K N .
According to Hubral (1983) the elements of the ray-centered propagator matrix Π are related to
these eigenwave curvatures as
Π ZO =
−v 0
K NIP − K N
� (KN + K NIP )/v 0
2K N K NIP /v 2 0
If we insert this into equation (2.43) we obtain
2
(K N + K NIP )/v 0
σ 2 = (cos 2 β 0 /v 0 )K N − (cosβ 0 /v 0 )K 0 − E 0 ,
σ 3 = (cos 2 β 0 /v 0 )K NIP − (cosβ 0 /v 0 )K 0 − E 0 .
�
. (2.45)
(2.46)
Note, in analogy to the FO case, the measurement surface in the vicinity of X 0 is assumed to be
representable, in the local coordinate system of X 0 , by the parabola
where K 0 is the measurement surface curvature at X 0 .
z = − 1
2 K 0 x2 , (2.47)

2.3 Zero-offset situation 25
Finally we can rewrite our traveltime expressions for zero-offset, using only parameters that are
related to physical properties that refer to the central ray and can be measured in the vicinity of
the coincident source and receiver point X 0 . We get
tpar(m,h) = t0 + 2 sinβ0 m +
v0 1 �
KN cos
v0 2 �
2
β0 − K0 cosβ0 − v0 E0 m (2.48a)
+ 1 �
KNIP cos
v0 2 �
2
β0 − K0 cosβ0 − vo E0 h , and
t 2 �
hyp (m,h) = t0 + 2 sinβ �2 0
m +
v0 2 t �
0
KN cos
v0 2 �
2
β0 − K0 cosβ0 − v0 E0 m (2.48b)
+ 2 t �
0
KNIP cos
v0 2 �
2
β0 − K0 cosβ0 − v0 E0 h .
Under the assumption of a slowly varying or even constant near-surface velocity, the inhomogeneity
factor E 0 that accounts for the gradient of the medium velocity at the source and receiver
points virtually vanishes. This leads to a simplified version of equation (2.48b), which was previously
presented by Chira and Hubral (2001). For E 0 = 0 this reads
tpar(m,h) = t0 + 2 sinβ0 m +
v0 1 �
KN cos
v0 2 � 2
β0 − K0 cosβ0 m
+ 1 �
KNIP cos
v0 2 � 2
β0 − K0 cosβ0 h , and
t 2 hyp (m,h) =
�
t0 + 2 sinβ �2 0
m +
v0 2 t �
0
KN cos
v0 2 � 2
β0 − K0 cosβ0 m
+ 2 t �
0
KNIP cos
v0 2 � 2
β0 − K0 cosβ0 h .
(2.49a)
(2.49b)
For comparison we can reduce this equations to the conventional 2D ZO CRS stack traveltime
formulas, valid for a planar measurement surface (Höcht, 1998; Müller, 1999; Jäger, 1999).
These can be obtained by setting K 0 = 0 in the formulas above and reads
tpar(m,h) = t0 + 2 sinβ0 m +
v0 1 �
KN cos
v0 2 � 2
β0 m
+ 1 �
KNIP cos
v0 2 �
2
β0 h , and
t 2 �
hyp (m,h) = t0 + 2 sinβ �2 0
m +
v0 2 t �
0
KN cos
v0 2 �
2
β0 m
+ 2 t �
0
KNIP cos
v0 2 �
2
β0 h .
(2.50a)
(2.50b)
Of course it is also possible to consider the case of a planar measurement surface having a near
velocity gradient. This case is committed to the reader.

26 Theory
Please note that, in practice, the traveltime formulas stated above are very inconvenient to use,
because for different locations of X 0 different local coordinate systems have to be used. In Section
(2.6) we introduce slightly modified traveltime formulas referring to the global coordinate system
in which the source and receiver coordinates are usually measured. In general, these traveltime
formulas are much more convenient to implement, because the source and receiver coordinates
do not have to be transferred to different local coordinate systems.
2.3.1 Relationship between (near)surface and wavefield attributes
If we assume a curved acquisition surface, where the near-surface velocity gradient (∇v) 0 and the
curvature K 0 is known in every central point X 0 , then we can approximate the traveltime response
of an arbitrary reflector segment up to the second order, by using either the parabolic (2.48a) or
the hyperbolic (2.48b) traveltime equation. This means that we have to find either the parabolic
or the hyperbolic traveltime surface that fits best to the reflection response of the considered (but
unknown) reflector segment. Doing this, e.g., by means of a coherency analysis, we obtain values
of the three wavefield attributes {β 0 ,K N ,K NIP } that parameterize our traveltime formula optimal,
and have in the context of the made approximations their defined physical meaning. However it is
not mandatory to use these set of equations to obtain a stacked ZO section from data measured on
a curved measurement surface with near surface velocity gradient. Equations (2.49) or (2.50) are
also valid to represent a second-order approximation of the considered traveltime response. They
have the same mathematical form as equations (2.48) - they are of second order (parabolic and
hyperbolic, respectively) and have three free parameters. In other words, equations (2.49) and
(2.50) describe the same surfaces in the [t,m,h]-space as equations (2.48), but with other values
for the parameters {β 0 ,K N ,K NIP }. Of course these values are not the real values of the physical
wavefield attributes {β 0 ,K N ,K NIP }. They implicitly contain the influence of the topography and
of the near-surface velocity gradient, which is not explicitly considered within the traveltime
formulas. We will denote this apparent wavefield attributes with a star in the following and call
them pseudo attributes. If we know the (near)surface attributes {K 0 ,v 0 ,(∇v) 0 } and the dip angle
of the surface in X 0 , i.e., α 0 (see Figure 2.6), we can convert these pseudo attributes to their
true counterparts. The searched-for relationship can be found by comparing the coefficients of
formulas (2.48) and (2.50). It makes no difference if we compare the parabolic or the hyperbolic
equations.
Doing this we find:
β 0 = β ∗ 0 , (2.51a)
K N = K ∗ N + K 0 cos(β ∗ 0 ) + E 0 v 0
cos 2 (β ∗ 0 )
, (2.51b)

2.3 Zero-offset situation 27
central ray
Figure 2.6: The relationship between the take-off angles of the normal ray, β0 and β g and the
0
dip angle α0 for a curved measurement surface. Note that β0 is measured in the local and β g
0 in
the global coordinate system. The angles are defined in the mathematical positive direction of
rotation (counterclockwise). Consequently β0 has a negative value in the figure above.
Please note: For this figure the origin of the global coordinate system is chosen to coincide with
X0 , which is also the origin of the local coordinate system. Of course, in general this is not the
case.

28 Theory
and
K NIP = K ∗ NIP + K 0 cos(β ∗ 0 ) + E 0 v 0
cos 2 (β ∗ 0 )
. (2.51c)
We observe that the obtained take-off angle β 0 remains the same, irrespective of whether we use
equations (2.49), (2.50) or equations (2.48). The reason for this is that the linear term in our
traveltime formulas depends only on the angle between the tangent to the measurement surface
in X 0 and the tangent to the considered reflector in the normal incidence point (NIP). This angle
is, within the range of the underlying assumptions, equal to β 0 . Thus, it makes no difference to
the linear term if the surface is planar or not.
There is one point that should be emphasized. If we use the plain topography formula (2.50) and
do not know the real topography, we cannot interpret the obtained take-off angle β 0 , because it
is measured against the unknown surface normal in X 0 . Only if we know the dip angle α 0 we are
able to interpret β 0 geometrically. In some cases it can be useful to transfer β 0 from the local to
a surface independent global coordinate system (see Figure 2.6), according to the relation
β g
0 = β0 + α0 . (2.52)
Note that here and in the following the global coordinate system is defined as:
x-axis parallel to the horizontal and z-axis parallel to the depth direction.
2.3.2 Normal-moveout (NMO) and root-mean-square (RMS) velocities
In the following discussion we will no longer consider the parabolic traveltime formulas, because
they are not used in the current implementations of the CRS stack (see Höcht, 1998; Müller,
1999; Bergler, 2001). Nevertheless, similar considerations would also hold in this case.
NMO velocity
To look at the odd-dislocation (OD) gather 2.2.2, which is closely related to the CMP gather, we
have to insert the condition m = 0 into the traveltime formula (2.48b). We get
t 2 (hyp,OD) = t2 0 + 2 t 0
v 0
�
KNIP cos 2 �
2
β0 − K0 cosβ0 − v0 E0 h . (2.53)
In analogy to the NMO velocity defined for a planar measurement surface (Shah, 1973), we can
introduce the NMO velocity for a curved measurement surface, having a near-surface velocity

2.3 Zero-offset situation 29
gradient, by rewriting equation (2.53) in the following way:
t 2 (hyp,OD) (h) = t2 0
v 2 NMO =
4h2
+
v2 ,with
NMO
(2.54a)
2v
� 0
t0 KNIP cos2 � .
β0 − K0 cosβ0 − v0 E0 (2.54b)
According to this definition the terms NMO velocity and stacking velocity can be used synonymously.
This definition is common within the context of the CRS stack. It defines the NMO
velocity as that velocity that specifies those hyperbolas in the CMP gather which yield the highest
coherence and thus the best stacking result. Please note that this definition of the NMO
velocity is a little bit different from the classical definition as that velocity which is best to reduce
the quasi-hyperbolic taveltimes in the CMP gather to a horizontal line. We would get a
slightly different expression for the NMO velocity if we would consider the parabolic instead
of the hyperbolic traveltime formula. E.g., the parabolic NMO velocity does not depend on t 0 .
Please note that even for a planar acquisition topography with constant near-surface velocity,
v NMO can have imaginary values if K NIP < 0. However, this case is much more probable for a
curved measurement surface with near-surface velocity gradient, as it can be seen by looking at
equation (2.54b).
It is evident that to obtain the NMO velocity for a curved measurement surface without nearsurface
velocity gradient one has only to set E0 = 0 in the equation above. Similar if the measurement
surface is planar, one has to set K0 = 0. An important question is: what NMO velocity
do we find if we use, e.g., the traveltime formula valid for a planar measurement surface without
near surface velocity gradient to stack data that was measured on a curved surface with
(∇v) 0 �=�0? The answer can easily be found by looking at equation (2.54a). (4/v2 NMO ) is always
the coefficient of the h 2 term of the considered traveltime formula. Irrespective of the used traveltime
formula ((2.48b), (2.49c) or (2.50c)), this coefficient has to have the same value, if the
resulting traveltime surface shall fit to the respective reflection response in the data. Thus the
found NMO velocity has also the same value. However, the subsequently obtained wavefield
attribute K NIP depends on, whether the real curvature and velocity gradient of the measurement
surface is considered within equation (2.54b) or not. This can also be demonstrated by inserting
equations (2.51a) and (2.51c) into equation (2.54b).
Finally we want to derive the relation between the actually measured NMO velocity, which
strongly depends on the surface curvature and the near-surface velocity gradient, and the NMO
velocity which we would obtain if the measurement surface would be horizontal and without a
near surface velocity gradient. Please note that the term “horizontal” always includes flatness. If
we imagine to have a planar dipping measurement surface, tangent to the real curved measure-

30 Theory
ment surface in X 0 and with E 0 = 0, we would measure the NMO velocity
v 2 NMO,P =
2v 0
t 0 K NIP cos 2 β 0
, (2.55)
according to equation (2.54b). Similarly on a fictitious planar and horizontal measurement surface
through X 0 with E 0 = 0 the obtained NMO velocity reads
v 2 NMO,H =
2v0 t0KNIP cos2 β g . (2.56)
0
K NIP would have the same value as K NIP found for the planar but dipping measurement surface
because the dip of the measurement surface is considered within the moveout formula by spec-
ifying the respective take off angle β0 , according to its definition (2.20), relative to the local
coordinate system. For the relation between β0 and β g see Figure 2.6 and equation (2.52). The
0
NMO velocities that would be measured in this two hypothetical experiments differ, because
they are no pure subsurface attributes like KNIP and KN but depend on the surface on which they
are measured. These two hypothetical experiments lead to the same KNIP that we would obtain
by applying equation (2.48b) to the data measured on the real curved measurement surface with
near velocity gradient. Thus, we can solve equation (2.56) for KNIP and insert into the equation
for vNMO (2.54b) to obtain the relationship between the measured NMO velocity and its corresponding
value vNMO,H that would have been measured on a fictitious horizontal surface through
X0 without near-surface velocity gradient. We find
and vice versa,
v 2 NMO =
t0 v 2 NMO,H =
2v 0
�
2v0 cos2 β0 t0v2 NMO,H cos2 β g − K0 cosβ0 − vo E0 0
2v 2 NMO v 0 cos2 β 0
� , (2.57a)
cos2 β g
0 (v2 NMOK0 cosβ0t0 + v2 NMOv0E0t 0 + 2v . (2.57b)
0 )
v NMO,H can be seen as the NMO velocity that would be obtained after applying a perfect static
correction to pre-stack data that was measured on a surface which is in the vicinity of X 0 , representable
according to equation (2.47), and which has only first order velocity variations near
X 0 .
Zero-dip NMO velocity
For a planar measurement surface, without near-surface velocity gradient, it is possible to separate
the influence of the overburden from the influence of the take off angle β 0 , within equation

2.3 Zero-offset situation 31
(2.54) by introducing the zero-dip NMO velocity,
and rewriting equation (2.54a) in the following way:
v 2 NMO,ZD = 2v0 = v
t0KNIP 2 NMO cos2 β0 , (2.58)
t 2 (hyp,CMP) (h) = t2 0 + 4h2 cos 2 β 0
v 2 NMO,ZD
= t 2 0
(2.59a)
4h2
+
v2 −
NMO,ZD
4h2 sin2 β0 v2 . (2.59b)
NMO,ZD
The second h 2 -term, which depends on the reflector dip, can be removed in a velocity independent
way by applying the so-called Gardener Dip Moveout (DMO) procedure described in
(Gardener et al., 1990). Afterwards the zero-dip NMO velocity can be determined on the conventional
way. In the early days of reflection seismics all reflecting layers were assumed to
be horizontal and thus it was the zero-dip NMO velocity, which appeared in the first moveout
formulas.
Solving the left side of equation (2.58) for K NIP results
K NIP =
2v 0
t 0 v 2 NMO,ZD
. (2.60)
Inserting this in equation (2.54b) leads to the useful relation between the zero-dip NMO velocity
which would have been found on a fictitious planar measurement surface through X 0 without
near-surface velocity gradient and the actual NMO velocity found on the real measurement surface
described by K 0 , v 0 and (∇v) 0 . It reads
Vice versa
v 2 NMO =
t0 2v 0
�
2v0 cos2 β0 t0v2 − K0 cosβ0 − v0E0 NMO,ZD
2v 2 NMO v 0 cos2 β 0
� . (2.61a)
v 2 NMO,ZD =
v2 NMOK0 cosβ0t0 + v2 NMOv0E0t 0 + 2v . (2.61b)
0
Note that we would have, in general, a different fictitious planar measurement surface for every
single coincident source and receiver position, whereas only the elevation, not the dip, has an
influence on the zero-dip NMO velocity. The same holds for the RMS velocity.

32 Theory
RMS velocity
The RMS velocity constitutes the average wave velocity, above a certain level, within a medium
composed of parallel and planar homogeneous layers and can be represented as (Dix, 1955)
v 2 RMS,i = v2 1 Δt 1 + v2 2 Δt 2 + ... + v2 i Δt i
t 0,i
i = 1,...,N , (2.62)
where vi is the velocity in layer i, t0,i is the two way traveltime of the central ray reflected at layer
i and Δti = t0,i − t0,i−1 is the two way traveltime within layer i. The interval velocity of layer i
can be computed using the formula
�
v2 RMS,i t0,i − v
vi =
2 RMS,i−1 t �1/2 0,i−1
. (2.63)
t 0,i −t 0,i−1
Provided that we have a medium with parallel and planar homogeneous layers, and offsets which
are small compared to depth, we can express approximatively the traveltime within a CMP gather
for a planar measurement surface without near-surface velocity gradient, using the RMS velocity.
This reads
t(h) 2 = t 2 0 + 4cos2 β0 v2 h
RMS
2 . (2.64)
The comparison with equations (2.54), using K0 and E0 equal zero, leads to the relation between
vRMS and KNIP , i.e.,
KNIP = 2v0 t0v2 . (2.65)
RMS
It is easy to see that the zero-dip NMO velocity coincides with the (approximative) RMS velocity
determined according to equation (2.64), if the conditions under which the latter is defined are
met. In the context of this approximation, the RMS velocity can be seen as a special case of the
more general zero-dip NMO velocity.
Inserting equation (2.65) into equation (2.54b) yields the relation between the actual measured
NMO velocity and the RMS that would be obtained according to equation (2.64) after removing
the inhomogeneity of the top layer and the curvature of the measurement surface. We find
and vice versa,
v 2 NMO =
t0 2v 0
�
2v0 cos2 β0 t0v2 − K0 cosβ0 − v0E0 RMS
2v 2 NMO v 0 cos2 β 0
� , (2.66a)
v 2 RMS =
v2 NMOK0 cosβ0t0 + v2 NMOv0E0t 0 + 2v . (2.66b)
0
Note that these equations can only be used, if the subsurface can be assumed, to consist of planar,
homogeneous and parallel layers. Otherwise it makes no sense to use the RMS velocity. For more
general media one has to replace the RMS velocity with the zero-dip NMO velocity.

2.4 Search-range estimation 33
2.4 Search-range estimation
In the implementation of the CRS stack, it is very important for the efficiency of the coherency
based search that the search ranges of the different search parameters are defined as narrow as
possible. A variable measurement surface curvature, dip, and near-surface velocity gradient lead
here to a considerable complication, because some of the found parameters strongly depend on
these values, namely the NMO velocity, the take-off angle β0 , and the pseudo attributes K∗ N
and K∗ NIP . In order to solve this problem, it is useful to relate the actual search limits to those
limits that would hold if the real measurement surface would have been replaced by a horizontal
measurement surface without near-surface velocity gradient. How to chose the latter search limits
is well known from the conventional 2D CRS stack (see Mann, 2002). To derive these relations,
between the actual search limits and those reference limits is the purpose of this section.
, respectively, are no search
In the current implementations of the CRS stack KNIP and and K∗ NIP
parameters. They are computed from the obtained values of vNMO and β0 , according to equation
(2.54b), whereas K∗ NIP results if K0 and E0 are neglected. Thus, we will not consider KNIP and
K∗ NIP as search parameters in the following, but similar derivations as for KN and K∗ N would hold
in this case, too.
2.4.1 Search range of K N and K ∗ N
If traveltime formula (2.48b) or (2.48a) is used and the (near)surface attributes are considered,
than we search for the real wavefield attribute K N , which has of course the same search range
as in case of a conventional 2D CRS stack with horizontal measurement surface and without
near-surface velocity gradient. But if we use a traveltime formula that does not consider the
(near)surface attributes (or not all of them), we have to transfer the search range that holds for
K N in order to get a appropriate search range for K ∗ N . If we solve equation (2.51b) for K∗ N and
insert at the righthand-side the search limits of K N , we get
K ∗max
N = Kmax N cos2 (β0 ) − K0cos(β0 ) − E0v0 cos2 , (2.67a)
(β0 )
K ∗min
N = Kmin N cos2 (β0 ) − K0cos(β0 ) − E0v0 cos2 . (2.67b)
(β0 )
Please note that the use of these relations demands that β 0 is already determined. In all current
implementations of the 2D ZO CRS stack K N is the last parameter that is searched-for. Otherwise
a procedure similar to the v NMO search-range determination, presented in 2.4.3, would be
necessary.

34 Theory
Figure 2.7: This figure shows the search range [−60◦ ,60◦ ] for the take-off angle β0 (black) and
its global counterpart β g (red). In order to search within the same physical limits, the two search
0
ranges have to coincide physically. This can be achieved by subtracting α0 from the search limits
that were used for β0 . Note that |β0 | must not get bigger than 90◦ . To keep the figure simple, the
z-axes are not displayed.
2.4.2 Search range of β 0
For the conventional CRS stack, a planar measurement surface is generally assumed to be perpendicular
to the depth direction and it is not distinguished between β0 and β g.
According to
0
Mann (2002) the search limits of β0 = β g can be defined on the basis of the maximal CMP event
0
dip that should be considered and can depend on t0 .
How this limits for β0 = β g
0 can be related to those limits for β0 that hold in case of a curved measurement
surface with near surface velocity gradient, is shown in Figure 2.7. There the search
limits for β 0 are obtained by subtracting the dip-angle α 0 from the search limits that hold for β g
0 .
Of course, it has to be considered that it makes no sense to allow values for β 0 that are bigger
than 90◦ or smaller than −90◦ . Accordingly, we obtain for the search limits β max
0
2.4.3 Search range of the NMO velocity
and β min
0
β max
�
0 = min β g,max
− α 0 0 , π
�
, (2.68a)
2
β min
�
0 = max β g,min
− α 0 0 , − π
�
. (2.68b)
2
Usually the coefficient of the h 2 -term of the traveltime formula, and thus the NMO velocity, is
the first parameter which is determined during the application of the CRS stack. This search is
performed in the CMP and OD gather, respectively. For the conventional CRS stack a multitude
of experience exists how the search limits of the NMO velocity can be chosen with respect to an
assumed subsurface model. Consequently zero-dip NMO velocity limits (or limits for K NIP ) can

2.4 Search-range estimation 35
easily be derived from the v NMO limits and the β 0 limits by means of equation (2.58). These zerodip
NMO velocity limits that hold for the conventional CRS stack, shall serve in the following
as a general, acquisition independent, basis, from which the v NMO search limits that hold for the
real measurement surface, can be derived. Of course the zero-dip NMO velocity limits can be
replaced by RMS velocity limits, if the conditions under which the RMS velocity is defined are
met.
To find a relation between the limits of the actually measured NMO velocity and the zero-dip
NMO velocity range mentioned above, we have to analyze equation (2.61a). The first observation
is that one has to know the take off angle β 0 to relate the measured NMO velocities to the
respective zero-dip NMO velocity values. This leads to a severe problem, because, as stated
above, the NMO velocity is the first parameter that is determined and thus the angle β 0 is still
unknown at this stage. Unfortunately it is hardly possible to permute the order in which the
parameters are determined. To find a solution, we have to look at the situation more closely.
Actually we are not searching the limiting values of vNMO , but the limits of the coefficient of the
h2-term of the traveltime, which depends, according to equation (2.54a), on v−2 . If we call the
NMO
inverse of vNMO NMO slowness pNMO , then we have to find the global extrema of the function
p 2 NMO = 2v 0 cos2 β 0 − K 0 t 0 cosβ 0 v 2 NMO,ZD − v 0 t 0 E 0 v2 NMO,ZD
2v 0 v 2 NMO,ZD
, (2.69)
which is the inverse of equation (2.61a). At this point, we have to remember that the inhomogeneity
factor E 0 is no pure (near)surface attribute, but depends on the take-off angle β 0 . If we
substitute E 0 in equation (2.69), according to equation (2.44), we get
p 2 NMO = 2v 0 cos2 β 0 − K 0 t 0 cosβ 0 v 2 NMO,ZD
+
This function of v 2 NMO,ZD and β 0
extrema reads
t 0 sinβ 0
2v 0 v 2 NMO,ZO
(2.70)
�
(1 + cos 2 �
β0 )(∂xv0 ) + cosβ0 sinβ0 (∂zv0 )
. (2.71)
2v 2 0
describes a surface in the 3D space, and the condition for the
∂ p 2 NMO
∂(v 2 NMO,ZD
) = 0 and
∂ p 2 NMO
∂(β 0 )
= 0 . (2.72)
The first of the two conditions above, can be evaluated without problems. If we take the derivative
of pNMO with respect to v2 NMO,ZD , we obtain
∂ p2 NMO
∂(v2 NMO,ZD ) = − cos2 β0 v2 . (2.73)
NMO,ZO

36 Theory
values and
Thus p2 NMO is a monotone function of v2 NMO,ZD . It decreases for positive v2 NMO,ZD
increases for negative v2 NMO,ZD values. Consequently the extrema lie at the borders of the p2 NMO
surface, at
p 2 NMO = p2 NMO
�
v2 �min NMO,ZD
��v �min
�
2
NMO,ZD ,β0
�
v2 �max NMO,ZD
and p 2 NMO = p2 NMO
��v �max
�
2
NMO,ZD ,β0 , (2.74)
where
and
are the pre-estimated limiting values of the squared
�
zero-dip NMO velocity. Depending on the assumed subsurface model v2 �min NMO,ZD can be
both positive and negative.
The first derivative of p 2 NMO with respect to β 0 reads
∂ p 2 NMO
∂β 0
= sinβ 0 (K 0 t 0 v2 NMO,ZD − 4v 0 cosβ 0 )
2v 2 0 v2 NMO,ZD
+ 3t 0 v2 NMO,ZD cos3 β 0 (∂xv) 0 −t 0 v 2 NMO,ZD cosβ 0 (∂xv) 0
2v 2 0 v2 NMO,ZD
+ 3t 0 v2 NMO,ZD cos2 β 0 sinβ 0 (∂zv) 0 −t 0 v 2 NMO,ZD sinβ 0 (∂zv) 0
2v 2 0 v2 NMO,ZD
.
(2.75)
Now the next step would be to set the righthand side of this equation equal zero, and to solve the
resulting equation for β 0 , but this is analytically hardly possible. The computer algebra system
MAPLE is not able to find a solution and it is most probably that even if a solution would be
found, it would be difficult to apply, due to a multitude of different cases which would be to consider,
depending on the signs and possibly also on the relative values of K 0 ,v 2 NMO,ZD ,(∂xv) 0 and
(∂zv) 0 . If we assume the near surface velocity to be constant and accordingly E 0 = 0, the extremal
values of p NMO are much easier to find. Therefore, we will discuss two different strategies, one
for E 0 = 0 and one for E 0 �= 0.
Strategy 1 (E0 = 0): If we set E0 = 0 within equation (2.69), the first derivative of p2 NMO (E0 = 0)
with respect to β0 reads
∂ p 2 NMO
∂β 0
= sinβ 0 (K 0 t 0 v2 NMO,ZD − 4v 0 cosβ 0 )
2v 0 v 2 NMO,ZD
. (2.76)
If we set this equation to be equal zero, solve for β0 , and consider that the extrema have to lie at
surface we find,
the v min,max
NMO,ZD borders of the p2 NMO
β (1)
0
= 0 and cosβ (2)
0 =
� K0 t 0 (v min,max
NMO,ZD )2
4v 0
�
=
K0 2Kmin,max , (2.77)
NIP

2.4 Search-range estimation 37
�
v2 �min,max NMO,ZD
were we have used equation (2.60) to express
It holds
K max 2v0 NIP =
t0 (vmin NMO,ZD )2 and K min 2v0 NIP =
t0 (vmax .
)2
NMO,ZD
(2.78)
in terms of Kmin,max. NIP
At this point, we have to consider the limits of the search range of the take-off angle β 0 , derived
in the last section. These are
−π/2 ≤ β min
0
≤ β 0
≤ β max
0
Due to this limitation, the second solution for β0 , i.e., β (2) , is only valid for,
0
K0 2Kmin,max ≥ min
NIP
� cosβ min
≤ π/2 . (2.79)
0 ,cosβ max
0
� . (2.80)
If this condition is met, we get possible candidates for the global extrema, which read
C (21) = p2 NMO ((v2 NMO,ZD )min ,β (2)
0 ) , C(22) = p2 NMO ((v2 NMO,ZD )max ,β (2)
0
) . (2.81)
Otherwise, if β (2) lies outside of the considered range, possible extrema which correspond to this
0
solution lie at the respective edges of the p2 NMO (v2 NMO,ZD ,β0 ) surface. These are
C (a) = p2 NMO ((v2 NMO,ZD )min ,β min
0 ) , C (b) = p2 NMO ((v2 NMO,ZD )min ,β max
0 ) , and
C (c) = p2 NMO ((v2 NMO,ZD )max ,β min
0 ) , C (d) = p2 NMO ((v2 NMO,ZD )max ,β max
0 ) ,
respectively.
The first solution β (1) is always valid and yields
0
C (11) = p2 NMO ((v2 NMO,ZD )min ,β (1)
0 ) , C(12) = p2 NMO ((v2 NMO,ZD )max ,β (1)
0
(2.82)
) . (2.83)
To make things clearer, will discuss an example of how, according to our derivations, the maximum
and minimum values of the NMO velocity are obtained.
If, e.g.,
K0 2Kmin ≥ min
NIP
� cosβ min
0 ,cosβ max
�
0 , and
K0 2Kmax < min
NIP
� cosβ min
0 ,cosβ max
�
0 , (2.84)
then we have the possible extrema {C (11) ,C (12) ,C (a) ,C (b) ,C (22) }.
The extremum C (21) is not valid, because the value of β (2)
0
outside the considered range. Consequently C (21) has to be replaced by C (a) and C (b) .
, which results for KNIP = Kmax NIP is

38 Theory
In our example the limiting values for v 2 NMO
� �max 2
vNMO =
� �
2 min
vNMO =
1
� �
p2 min =
NMO
1
� �
p2 max =
NMO
would be
Of course an analogous procedure is valid in any other case, too.
1
min{C (1) ,C (2) ,C (a) ,C (b) ,C (4) and (2.85)
}
1
max{C (1) ,C (2) ,C (a) ,C (b) ,C (4) . (2.86)
}
For the sake of simplicity and brevity I abstained from completely analyzing the second derivatives
of p2 NMO in order to determine in which case an extremum is a maximum or minimum.
Doing this, one has to consider many different cases depending on the sign of K0 , Kmin,max and
NIP
(K0 − 2Kmin,max), and in view of the practical application it is faster and much more convenient
NIP
to perform the strategy stated above, which is to compute first all possible candidates for the
extrema and to decide afterwards which their maximum and minimum value is.
Note, as mentioned before, all above derivations hold also for the RMS velocity, which is nothing
more than the zero-dip NMO velocity for a special subsurface structure. Thus, if the subsurface
can be assumed to be constituted of homogeneous layers, divided by planar and parallel reflectors,
RMS velocity limits can be used instead of zero-dip NMO velocity limits, simply by
substituting vNMO,ZD by vRMS in the equations above.
Strategy 2 (E 0 �= 0): If the near-surface velocity gradient can not be assumed to be zero, it is
hardly possible to derive analytically the extremal values of the NMO velocity. In this case it is
proposed to determine the v NMO search limits, assuming E 0 = 0 and to enlarge the search range
subsequently by a certain value that accounts for the unknown influence of E 0 . In general, the
influence of E 0 should be small, compared to the influence of K 0 , so that the search range has not
to be extended considerably. To study this point more closely lies unfortunately outside the time
frame of this thesis.
2.5 Redatuming
The final goal of the CRS stack is to provide a time domain image of the subsurface structure,
i.e., the ZO section, and the also very important attribute sections, which are the β 0 section, the
K N section, the K NIP section, and the v NMO section. It is evident that due to the fact that we
are only interested in the subsurface structure, we do not want these sections to depend on the
characteristics of the measurement surface and the potentially inhomogeneous top layer. If the
acquisition topography meets the required conditions, then the traveltime equations (2.48) enable
us to determine the (near)surface attribute independent wavefield attributes K N and K NIP and the
only surface dip dependent take-off angle β 0 . According to Figure 2.6 and equation (2.52), β 0 can

2.5 Redatuming 39
Figure 2.8: To remove the influence of the acquisition surface topography from the obtained ZO
and attribute sections, we simulate a situation where all central rays end on the same horizontal
measurement surface, e.g., the common-datum surface. This is shown exemplarily, for one central
ray. In order to keep the figure simple, only the situation (∇v) 0 = 0 is displayed, were we
have no refraction at the measurement surface, because we can choose v f = v 0 .

40 Theory
easily be transferred to the surface- dip independent take-off angle β g,
which is measured with
0
respect to the horizontal. Also, the NMO velocity values should be corrected to those values that
would be obtained on a horizontal measurement surface without near-surface velocity gradient.
This can be done by means of equation (2.57b). Alternatively, it is possible to compute the
NMO velocity measured on a fictitious surface through X0 with arbitrary (near)surface attributes,
from the value of KNIP . This is done by inserting KNIP into equation (2.54b), together with the
(near)surface attributes of the fictitious measurement surface.
The application of these corrections to the determined wavefield attributes, leads to attribute
sections, which represent those values of the respective attribute which refer to a horizontal
measurement surface, without near-surface velocity gradient. However, it has to be taken into
account that, in general, the elevation of these fictitious horizontal measurement surfaces is different
for different central points X0 . This means for the attribute sections Ai (x0 ,t0 ) and also
for the ZO section AZO (x0 ,t0 ) that in general every x0 is related to a different elevation z0 . The
consequence is that, e.g., for the case of a measurement surface with a sinusoidal shape and a
subsurface constituted of homogeneous layers separated by horizontal reflectors, we would find
sinusoidal images of the reflectors in the ZO section. The same observation would hold for the
attribute sections. This can be seen very well in the synthetic data example, presented in Section
3.5. To remove this undesired topography effect from the ZO and attribute sections, we introduce
a fictitious horizontal measurement surface to which all attributes and traveltimes are related. In
other words, we simulate a situation in which all central rays start and end at the same horizonal
measurement surface. Thus, we have to transfer the zero-offset traveltimes and also the attributes
to those values, which would be measured on this common-datum surface. Such a procedure is
called redatuming. In the following, all values that pertain to this virtual measurement surface
are denoted with a prime. The key information for this procedure is the knowledge of the takeoff
angle β0 , which is provided by the CRS stack. If the common-datum surface is assumed to
be above the actual topography, it is possible to choose an arbitrary velocity v f for the fictitious
layer between topography and new datum. If the near-surface velocity gradient is zero, then the
most convenient choice is to set v f equal v0 , because this avoids that the topography has to be
considered as a additional reflector. For (∇v) 0 �= 0 Snell’s Law has to be considered to derive
β ′ 0 , the fictitious take-off angle at the fictitious coincident source and receiver point X ′ 0 . Knowing
the take-off angles and the wave velocity within the fictitious layer, it is not difficult to forward
propagate the Normal- and NIP-wave fronts up to the common-datum surface.
2.5.1 Mapping of X 0 and t 0 to the common-datum surface
To map the coincident source and receiver point X0 of a central ray from the original measurement
surface to its corresponding location X ′ 0 at the common-datum surface, one has to transfer its

2.5 Redatuming 41
coordinates x0 and z0 to their new values x ′ 0 and z′ 0 . Of course, z′ 0 is given by the elevation of the
common-datum surface. To transfer x0 we have to know the emergence angle of the central ray
after being refracted at the measurement surface. We will denote this angle as β f . According to
0
Snell’s Law, we have the relation,
v0 =
v f
sinβ0 , (2.87)
sinβ f
which leads to
β f
0
� �
v f
= arcsin sinβ
v 0 . (2.88)
0
In analogy to equation (2.52), the take-off angle measured at the common-datum surface β ′ 0 is
given by
β ′ f
0 = β0 + α0 . (2.89)
If we denote the vertical distance between X0 and the common-datum surface as Δz, we can
transfer now x0 to x ′ 0 . This reads
x ′ 0 = x0 + Δztanβ ′ 0 . (2.90)
For the new two-way traveltime of the central ray t ′ 0 , we get, after simple trigonometric considerations,
t ′ 0 = t0 + 2Δz
v f cosβ ′ . (2.91)
0
2.5.2 Mapping of K N and K NIP to the common-datum surface
In order to transfer the values of the wavefield attributes K N ,K NIP to those values which would
be measured at the common-datum surface we have to use the refraction law (Hubral and Krey,
1980) that gives us the curvature of the N- and NIP-wave, respectively, after passing the mea-
surement surface. It results
K f
N,NIP = KN,NIPv f cos2 β0 v0 cos2 β f +
0
K0 cos2 β f
0
� v f
vo
cosβ0 − cosβ f
�
0
, (2.92)
where K f are the refracted wavefront curvatures on the upper side of the measurement sur-
N,NIP
face. Subsequently, we use the transmission law to propagate the wavefronts of the N- and NIPwave
through the fictitious layer above the real measurement surface, up to the common-datum
surface.
1
K ′ �
=
N,NIP
1
K f
+
N,NIP
1
2 v f t �
f , (2.93)
with the two-way traveltime within the fictitious layer
t f = t ′ 0 −t 0
2Δz
=
v f cosβ ′ . (2.94)
0

42 Theory
Inserting equations (2.92) and (2.94) into (2.93) leads to the final equations for the wavefront
curvatures K ′ N and K′ NIP measured at the common-datum surface
1
K ′ N (x′ 0 ,t′ ⎛
= ⎝
o )
cos2 β f
�0 v f
vo cosβ � +
0 − cosβ f
0
Δz
cosβ ′ ⎞
⎠ ,
0
(2.95a)
1
K ′ NIP (x′ 0 ,t′ ⎛
= ⎝
o )
KN (x0 ,t0 ) v f
v cos
0
2 β0 + K0 cos2 β f
0
KNIP (x0 ,t0 ) v f
v cos
0
2 β0 + K0 � v f
vo cosβ 0
� +
− cosβ f
0
Δz
2.5.3 Mapping of v NMO to the common-datum surface
cosβ ′ 0
⎞
⎠ . (2.95b)
In Section 2.3.2 we have derived equation (2.57b) to transfer the NMO velocity measured at the
real curved measurement surface, with near-surface velocity gradient to that NMO velocity which
would be measured at a corresponding horizontal measurement surface without near surface
velocity gradient, i.e., vNMO,H . Now we want to transfer vNMO,H to v ′ NMO , which is the NMO
velocity which would be found at the common-datum surface. To do this, we assume an auxiliary
subsurface model, for which the RMS velocity is defined. It holds
v RMS = v NMO,H cosβ g
0
, (2.96)
according to Section 2.3.2. This is allowed because it does not change anything on the actual
measured NMO velocity, whether the real model or the auxiliary model is considered. According
to equation (2.62) we would obtain at the common-datum surface
v ′ RMS =
�
�
�
� t0v2 RMS +t f v2 f
. (2.97)
t0 +t f
The corresponding NMO velocity, v ′ NMO , at the common-datum surface is
v ′ NMO = v′ RMS
cosβ ′ . (2.98)
0
By inserting the equations (2.88), (2.97), (2.96) and (2.94) into the equation above we get as
final result the relation between the actually measured NMO velocity and the NMO velocity that
would be measured at the common-datum surface,
v ′ NMO =
cos(β0 + α0 )
�
cos arcsin
�
v2 t
NMO,H 0v f cosβ ′ 0 +2Δz v f
t0v f cosβ ′ 0 +2Δz
� �
v f
v 0
sinβ 0
+ α 0
� , (2.99a)

2.6 Global coordinate system 43
with
v 2 NMO,H =
according to equation (2.57b).
2v 2 NMO v 0 cos2 β 0
cos2 (β0 + α0 )(v2 NMOK0 cosβ0t0 + v2 NMOv0E0t 0 + 2v , (2.99b)
0 )
Of course it is also possible to derive v ′ NMO from K′ NIP and β ′ 0
in this case
v ′ NMO =
�
2.6 Global coordinate system
2v f
t ′ 0 K′ NIP cos2 β ′ 0
using equation (2.56), which reads
. (2.100)
Figure 2.9: The transformation of the local 1D midpoint and half-offset coordinates h and m to
the respective global 1D coordinates hg and mg.
The traveltime formulas that we have derived up to this point assume a Cartesian coordinate system
with x-axis tangent to the measurement surface and origin in X 0 (see Figure 2.6). This is the
coordinate system, where, according to (2.39), half-offset h and midpoint m are defined as 1D
coordinates measured along the x-axis. But for practical application, it is very inconvenient to
transfer the measured source and receiver coordinates into the specific local coordinate system
of every central point, in order to determine the required offset and midpoint coordinates. Particularly,
if one wants to use a conventional CRS stack software and correct the found pseudo
attributes afterwards, this is not possible. To solve this problem caused by the curvature of the
measurement surface is the aim of this section.

44 Theory
Let us first look at the much simpler case of having a planar measurement surface (see, with a
different notation, Höcht, 1998). Here it is easy to extend our traveltime formulas, which were
derived for a single central point, to hold for all considered central points at the same time. To do
this we define a global coordinate system, equal oriented as the local ones, whereas the location
of the origin is arbitrary. Then we substitute the local midpoint coordinate m by mg − x 0 ; mg
is the location of the midpoint in the global coordinate system and x 0 is the x-coordinate of the
central point X 0 that is considered. The local half-offset coordinate h is substituted by hg. It
results
tpar(mg,hg) = t0 + 2 sinβ0 (mg − x
v 0 ) +
0
1 �
KN cos
v0 2 �
β0 (mg − x0 ) 2
(2.101a)
+ 1 �
KNIP cos
v0 2 �
2
β0 hg , and
t 2 hyp (mg,hg)
�
= t0 + 2 sinβ �2 0
(mg − x
v 0 ) +
0
2 t �
0
KN cos
v0 2 �
β0 (mg − x0 ) 2 (2.101b)
+ 2 t �
0
KNIP cos
v0 2 �
2
β0 hg .
It is evident that this approach can not be directly applied to the case of a curved measurement
surface, because here the local coordinate systems are in general different oriented. This case
requires an additional coordinate rotation to transfer m and h to their global counterparts hg and
mg. According to Figure 2.9 we find the transformation
h = 1
hg and m =
cosα0 1
(mg − x
cosα 0 ) . (2.102)
0
If we apply this coordinate transformation to the general set of traveltime equations (2.48), we
get a new one which does now explicitly depend on the dip angle α 0 of the measurement surface
and the global x-coordinate x 0 of the central point X 0 . This reads
tpar(mg,hg) = t 0 + 2 sinβ 0
t 2 hyp (mg,hg) =
+
+
v 0 cosα 0
1
v 0 cos 2 α 0
1
v 0 cos 2 α 0
�
t 0 + 2 sinβ 0
v 0 cosα 0
+ 2 t 0
v 0 cos 2 α 0
+ 2 t 0
v 0 cos 2 α 0
(mg − x 0 ) (2.103a)
�
KN cos 2 �
β0 − K0 cosβ0 − v0 E0 (mg − x0 ) 2
�
KNIP cos 2 �
2
β0 − K0 cosβ0 − vo E0 hg , and
�2 (mg − x0 )
�
KN cos 2 �
β0 − K0 cosβ0 − v0 E0 (mg − x0 ) 2
�
KNIP cos 2 �
2
β0 − K0 cosβ0 − v0 E0 hg .
(2.103b)

2.6 Global coordinate system 45
From these two equations one can deduce eight different subsets, by setting one, two, or all
of the (near)surface attributes {K 0 ,α 0 ,(∇v) 0 } equal zero. Note that in the equations above,
β0 is still related to the local coordinate system, but it can be easily substituted by its global
according to equation (2.52), if this is desired.
counterpart β g
0
Equal to Section 2.3.1 we can relate the pseudo wavefield attributes {β ∗ 0 ,K∗ NIP ,K∗ N }, to their real
values {β0 ,KNIP ,KN } by comparing the coefficients of equations (2.103) and equations (2.50).
We can use either the hyperbolic or the parabolic equations. This results the following relations
∗
sinβ0 = cosα0 sinβ0 , (2.104a)
KN = K∗ N cos2 α0 cos2 β ∗ 0 + K0 �
1 − cos2 α0 sin2 β ∗ 0 + v0E0 , and (2.104b)
K NIP = K∗ NIP cos2 α 0 cos 2 β ∗ 0 + K 0
1 − cos 2 α 0 sin 2 β ∗ 0
�
1 − cos 2 α 0 sin 2 β ∗ 0 + v 0 E 0
1 − cos 2 α 0 sin 2 β ∗ 0
. (2.104c)
Please note that β ∗ 0 becomes complex if | sinβ0 | > 1. Of course, the current implementations of
cosα0 the conventional 2D ZO CRS stack assume K0 , E0 and α0 to be zero, because they were designed
for a planar measurement surface and 1D source and receiver coordinates, measured along these
surface. Consequently, complex values of β ∗ 0 have to be taken into account, if a conventional
implementation is applied to data measured on a curved surface. In other words, they have to be
considered within the search range of β ∗ 0 . Unfortunately, this is generally not possible without
changing the code, since this case is not intended, as the software was designed for a planar
measurement surface only.
To avoid confusion we will discuss a little bit closer, how to use the relations above. E.g., if one
wants to compute the real wavefield attributes {β 0 ,K NIP ,K N } from the pseudo attributes obtained
by using a traveltime formula that considers K 0 , but not E 0 and α 0 , one has to set K 0 = 0 in the
equations above and insert the pseudo wavefield attributes together with the actual values of E 0
and α 0 into the righthand sides. At this point, it must be stressed again that α 0 is implicitly
considered within the traveltime formulas that use local midpoint and offset coordinates. Not
considering α 0 means, e.g., using a conventional CRS stack software that does not transfer the
source and receiver coordinates into the respective local coordinate systems, but uses global
midpoint and offset coordinates.
The validity of the relations above can be demonstrated by setting K 0 = 0, E 0 = 0 and α 0 = 0.
Doing this pseudo and real attributes are equal as it should be the case for a traveltime formula
which considers all (near)surface attributes.

46 Theory
2.6.1 The inhomogeneity factor E 0 in global coordinates
According to the definition of the inhomogeneity factors E S , E G (2.34), and E 0 (2.44), the velocity
gradient has to be measured within the respective local coordinate system. One can imagine
that in practical application, it may often be more convenient to determine the velocity gradient
within the global coordinate system. To transpose the velocity gradient from local to global coordinates
one has to perform a rotation of the coordinate system by the dip-angle α 0 , which leads
to the relation,
� �
cosα0 sinα
(∇v) 0 =
0 (∇v) g
. (2.105)
0
−sinα 0 cosα 0
If we insert this relation into (2.44), we obtain the inhomogeneity factor E 0 in global coordinates,
E 0 = − sinβ 0
v 2 0
+ cosβ 0 sinβ 0
�
�1 �
2
+ cos β0
� �
∂v
cosα0 � �
∂v
−sinα0 ∂xg
�
∂xg
� � � �
∂v
+ sinα0 0
∂zg 0
� � ��
∂v
.
+ cosα0 0
2.6.2 The NMO and RMS velocities in global coordinates
∂zg
0
(2.106)
In analogy to Section 2.3.2, the following definition holds for the (hyperbolic) NMO velocity in
global coordinates.
t 2 (hyp,OD) (hg) = t 2 0 + 4h2g v2 ,with (2.107a)
NMO
(v g
NMO )2 =
2v 0 cos 2 α 0
�
t0 KNIP cos2 � . (2.107b)
β0 − K0 cosβ0 − v0 E0 Comparing equations (2.107) and (2.54) one can easily see, that the NMO velocity measured in
global coordinates differs from its counterpart, obtained in local coordinates. We find the relation
(v g
NMO )2 = cos 2 α0v 2 NMO . (2.108)
Inserting now equation (2.60), which does not depend on the choice of the coordinate system,
leads to the useful relation between the found NMO velocity and the zero-dip NMO velocity
which would be found on a fictitious planar measurement surface without near-surface velocity
gradient.
(v g
NMO )2 =
t0 2v0 cos2 α
� 0
2v0 cos2 β0 t0v2 − K0 cosβ0 − v0E0 NMO,ZD
� , (2.109a)

2.6 Global coordinate system 47
and vice versa:
v 2 NMO,ZD =
2(v g
NMO )2 v 0 cos 2 β 0
((v g
NMO )2 K 0 cosβ 0 t 0 + (v g
NMO )2 v 0 E 0 t 0 + 2v 0 cos 2 α 0 )
. (2.109b)
To obtain the respective relations for a case in which the RMS velocity is defined one has only to
set vRMS = vNMO,ZD in the equations above. Equation (2.109b) provides the possibility to derive
the zero-dip NMO or the RMS velocities from the vg values that are obtained by fitting the
NMO
respective traveltime surface to the pre-stack data. The opposite direction, equation (2.109a) is
very useful to estimate the search limits of v g
NMO by means of the estimated range of v RMS and
v NMO,ZD , respectively, according to Section 2.4.3.
A similar derivation to that in Section 2.3.2, now for global coordinates, leads to the relationship
between the measured NMO velocity and its corresponding value measured on a fictitious planar
and horizontal surface without near velocity gradient. It reads
Vice versa, we get
v 2 NMO,H =
(v g
NMO )2 =
t0 2v0 cos2 α
� 0
2v0 cos2 β0 t0v2 NMO,H cos2 β g − K0 cosβ0 − vo E0 0
2(v g
NMO )2 v 0 cos 2 β 0
cos 2 β g
0 ((vg
NMO )2 K 0 cosβ 0 t 0 + (v g
NMO )2 v 0 E 0 t 0 + 2v 0 cos 2 α 0 )
� . (2.110a)
. (2.110b)
Note that v NMO,H has the same value whether obtained in global or in local coordinates because
the two systems coincide in case of a horizontal measurement surface. The same holds for the
values of v NMO,ZD and v RMS , which do not depend on the dip of the local coordinate system and
thus can be seen as measured in a horizontal 1D coordinate system that is equal to the global one.
2.6.3 The search range of K∗ N , β ∗ 0 , and vNMO in global coordinates
Of course the search range of the Normal-wave curvature KN remains the same, but in case of
the limits of the pseudo attribute K∗ N one has to use slightly modified formulas that account for
the additional option not to consider α0 . These are given by
= Kmax
�
N 1 − cos2 α0 sin2 β ∗ � �
0 − K0 1 − cos2 α0 sin2 β ∗ 0 − v0E0 , (2.111a)
and
K ∗max
N
K ∗min
N
cos 2 α 0 cos 2 β ∗ 0
�
Kmin N 1 − cos2 α0 sin
= 2 β ∗ � �
0 − K0 1 − cos2 α0 sin 2 β ∗ 0 − v0E0 cos2 α0 cos2 β ∗ , (2.111b)
0

48 Theory
which result from solving equation (2.104b) for K ∗ N
and inserting the respective limiting values.
The found take-off angle β 0 has the same value in both coordinate systems. Consequently the
strategy to obtain the search range for β 0 does not change. But the pseudo take off angle β ∗ 0
that one obtains using global coordinates and neglecting the surface dip α 0 in equation (2.103c)
differs from β 0 . In this case we have to choose other search limits. Combining equations (2.68)
and equation (2.104a) yields
β ∗max
0 =
⎛
arcsin⎝
sin
� �
min β g,max − α
0 0 , π β
��⎞
2
⎠ ,
cosα0 (2.112a)
∗min
0 =
⎛
arcsin⎝
sin
� �
max β g,min − α
0 0 , − π ��⎞
2
⎠ .
cosα0 (2.112b)
As mentioned before, β ∗ 0 has, under certain circumstances, complex values. Consequently, in
any case were complex β ∗ 0 are possible, the respective limiting values have to be complex, too.
Such a situation is always given if the argument of the arcsin in the equations above is larger than
one.
In global coordinates the squared NMO slowness reads
(p g
NMO )2 = 2v0 cos2 β0 − K0t0 cosβ0 v2 NMO,ZD − v0t0E0v2 NMO,ZD
2v0 cos2 α0v2 , (2.113)
NMO,ZO
according to the inverse of equation (2.109a).
According to equation (2.108) holds for the relation between (p g
NMO )2 and the squared NMO
slowness in local coordinates p 2 NMO
From this equation follows that (p g
(p g
NMO )2 = p2 NMO
cos2 , (2.114)
α0 have their extrema at the same locations,
NMO )2 and p2 NMO
and their extremal values differ only by the factor cos2 α0 . Thus the whole strategy to determine
the NMO velocity search range remains valid even for global coordinates. Only the values at the
extremal points have to be computed using equation (2.113) instead of equation (2.69).
2.6.4 Redatuming in global coordinates
The ZO traveltimes and the wavefield attributes {K N ,K NIP ,β 0 } are certainly the same, whether
obtained in global or local coordinates. Only the NMO velocities which are measured at the

2.6 Global coordinate system 49
is equal to vg , be-
NMO,H
cause for a horizontal measurement surface, the global and the local coordinate systems coincide.
Thus all derivations in Section 2.5 can be used for global coordinates, too, with one exception.
In equation (2.99a), equation (2.110b) has to be used to relate vNMO,H to the actually measured
NMO velocity.
curved measurement surface differ (vNMO �= vg NMO ). However, vNMO,H

50 Theory

Chapter 3
Synthetic Data Example
The aim of this chapter is to study the application of the 2D ZO CRS stack to data measured on
a curved measurement surface by means of a synthetic data example. To do this I have used a
synthetic dataset which was created by ENI (Agip) and that was relinquished to Pedro Chira and
me to test his extended CRS traveltime formula (2.49c), valid for a smooth curved measurement
surface (Chira and Hubral, 2001; Chira et al., 2001). In addition to the dataset, we received results
which ENI obtained via standard data processing, using static corrections and the conventional
CRS stack software. This approach and its results are discussed at the beginning of this chapter.
3.1 Model
The synthetic seismic dataset used in this chapter is based on the depth model shown in Figure
3.1. This model consists of four homogeneous layers separated by three horizontal reflectors, and
lying underneath a curved acquisition surface. Each layer has a different P-wave velocity. The
values are specified in the figure. The model does not include a near surface velocity gradient,
because it was designed to study the influence of the acquisition surface topography, only. Due
to the different scale of the axes, the topography looks steeper than it is, but anyway the changes
in height can be compared to the changes that can be found, e.g., at the border of the Apennine
Mountains. At the surface between x=10.0 km and x=15.0 km a small-scale undulation of the
topography can be observed. The term small-scale refers to an approximate stacking aperture
of 2 km, measured in the global coordinate system. This part of the measurement surface does
not meet the assumptions, required for the validity of the CRS traveltime formulas, derived in
Section 2.3. For central points that lie in this area one can hardly approximate the surface within
the stacking aperture by a parabola. Of course the CRS traveltime formula is still the best second-

52 Synthetic Data Example
Figure 3.1: Model, consisting of four homogeneous layers with different P-wave velocities. These
are separated by three horizontal reflectors and lie underneath a curved measurement surface.

3.2 Standard processing using elevation-statics 53
order approximation of the actually measured traveltime response, but the obtained wavefield
attributes K N ,K NIP and β 0 have lost their expected physical meaning to a certain extent. This
area provides a good example to study the validity limits of the CRS stack procedure presented
within this thesis (for results see Section 3.5).
3.2 Standard processing using elevation-statics
A standard method to apply the conventional CRS stack, which uses traveltime formula (2.101),
also to seismic data that was recorded on a curved acquisition surface is to subtract so-called
static corrections from the measured traveltimes with the aim to simulate data that corresponds
to a planar measurement surface. In this case ENI approximatively related all traveltimes to
sources and receivers located at the sea level (see Figure 3.1). Basically, this downward continuation
of the measured traveltimes requires at least the knowledge of two parameters: The take-off
angle β 0 of every central ray and the velocity of the medium between the surface and the desired
virtual measurement level. The upward continuation to a common level is also possible and requires
besides β 0 only the knowledge of the near-surface velocity v 0 , because the velocity within
the fictitious layer above the real measurement surface can be chosen arbitrarily. Nevertheless,
the downward continuation is very common in oil exploration because in addition to the pure
topography influence also the effect of a slow laterally inhomogeneous top-layer can be partly
compensated.
To determine the necessary near-surface velocity values it is usual practice to drill a sufficient
number of shallow holes along the seismic line or to use informations from the shot holes. The
emergence angles of the upcoming rays are normally not measured and thus unknown in real
pre-stack data. However, the rays can be assumed to emerge vertical, if the velocity altogether
increases strongly with depth, the considered reflectors lie deep compared to the stacking aperture,
and are not too steep. Another point which justifies this assumption is that in many cases
the velocity difference between the top-layer and the layer below is particularly high and they are
divided by a more or less horizontal interface which causes the upcoming rays to change their
direction towards the vertical. An approximatively vertical emergence of the rays at the surface
justifies the so-called surface-consistency assumption which assumes the static correction time
to depend only on the source and receiver location and not on the angle of emergence. Using
this assumption, the traveltimes t(m,h) can be related to their corresponding values t ∗ (m,h),
“measured” at the sea level, by the equation
t ∗ (m,h) ≈ t(m,h) − Δt stat with Δt stat = (e S + e G )/v 0 , (3.1)
where Δt stat is the static correction which has to be subtracted from the considered trace and e S
and e G are the elevations of the source and the receiver above the new datum. This kind of static

54 Synthetic Data Example
Z g
S
measurement surface
S* S’ G’ G*
sea level = new datum
reflector
Figure 3.2:
Visualization of the applied static corrections. The figure shows two homogeneous layers separated
by a planar dipping reflector. Please note that distance and traveltime are displayed
simultaneously assuming units in which the velocity has the value one. To simulate a planar
measurement surface at sea level, ENI subtracted from every trace a correction time, according
to equation (3.1), which is displayed here as the distance (SS ∗ + GG ∗ ). The figure shows that
this correction is, due to the non-vertical ray-path too small (blue line segment) to obtain the
real traveltime t ′ (m ′ ,h ′ ) from S ′ to G ′ , whereas S ′ and G ′ are the fictitious source and receiver
locations at the new datum that pertain to the ray that joins S and G. But on the other hand this
correction is too big (red line segment) to yield the searched-for traveltime t ∗ (m,h) from S ∗ to
G ∗ .
G
Xg

3.2 Standard processing using elevation-statics 55
corrections are usually called surface-consistent elevation-statics.
On the first sight, it seems that this correction produces traveltimes which are to large for nonzero
offset, because the wave propagates obliquely through the top-layer and not vertical like the
correction assumes. However, Figure 3.2 shows that it is also necessary to consider the consequence
of keeping offset and midpoint fixed, which is implicitly included within the assumption
of vertical emergence and equation (3.1). Accordingly, subtracting the static correction Δt stat
results for non-zero offset too small instead of too large traveltimes. Thus, this kind of static correction
tends to pretend NMO velocities that are higher than those NMO velocities which would
really be measured at the new datum.
To avoid negative traveltimes, caused by the static correction, a bulk-shift of 1400 m/s was applied
to the traces before they were corrected. Here it has to be remarked that doing this is not
mandatory, because negative traveltimes caused by the static correction correspond to reflection
events located above the simulated measurement surface. These can be neglected therefore.
Applying a bulk-shift changes the later found NMO velocities, too. This is shown by the following
equation, which holds for the NMO velocity according to equation (2.54a).
v 2 NMO
= 4h2
t 2 −t 2 0
. (3.2)
Due to the squares, the denominator on the righthand side gets bigger if the same positive value is
added to t and t 0 . Consequently a positive bulk-shift causes the measured NMO velocities to have
smaller absolute values than those NMO velocities that would be measured without bulk-shift.
For the used synthetic example one finds, after the elevation-statics and the bulk-shift were applied,
NMO velocities that are smaller than those values that are provided by the known depth
model. Thus in this example the v NMO lowering effect of the bulk-shift overcompensates the
v NMO raising effect of the static correction. As expected, both effects decrease, with increasing
distance between the reflector and X 0 . Evidently similar effects also occur for the found wavefield
attributes β 0 ,K NIP , and K N .
Finally, it can be said that the application of static corrections, bulk-shift and a subsequent stack
with the conventional traveltime formula (2.50c) leads, under certain conditions, to a good stack
result, but the found attributes {β 0 ,K N ,K NIP } loose their expected geometrical meaning, up to a
certain degree. In addition it is more difficult to define appropriate search limits for the traveltime
parameters. However, applying this strategy without a bulk-shift in case of a subsurface structure,
which justifies the surface-consistency assumption better than the model used here, should
yield satisfying results, even for the wavefield attributes.
In this case, one finds for the first reflector, after bulk-shift and elevation-statics were applied,
a NMO velocity of approximately 1900 m/s. According to the model it should be 2500 m/s.
Consequently, if the v NMO search limits for this layer are chosen too close around 2500 m/s,

56 Synthetic Data Example
assuming to know the velocity roughly, one obtains a poor or wrong stack result, as the search
algorithm finds stacking surfaces which are the optimal within the chosen limits, only. Exactly
this happened when ENI stacked the dataset, as it is shown in the following section.
3.2.1 Results of the standard processing
In this section results, obtained by the aforementioned standard processing, are displayed. For
a better comparability to the results presented in Section 3.5 only the results of the so-called
Automatic CMP stack are shown (see Mann, 2002). This data-driven CMP stack has the purpose
to provide the NMO velocity values and a first provisional ZO section, both needed in the subsequent
steps of the split traveltime-parameter search. However, the obtained CRS stack section
looks very similar, due to the simpleness of the used subsurface model.
The first three sections, Figures 3.3, 3.4, and 3.5, were obtained using wrong NMO velocity
constraints, ignoring the influence of the elevation-statics and the bulk-shift. This is clearly
visible in the coherency section, which is displayed in Figure 3.4. The overall v NMO lowering
effect of the applied corrections leads to NMO velocities that are below the search range for
nearly the whole first reflector and half of the second reflector. Consequently, the coherency
values in this area are very low. At the right side, where the topography is higher we find a smaller
v NMO lowering effect, because the static corrections are bigger and the original traveltimes larger.
This can also be verified by looking at the v NMO section (Figure 3.5).
For all that, the obtained ZO section (Figure 3.3), looks very satisfying, particularly because
images of the reflectors are absolutely planar. However, this is caused by the simpleness of the
used model, where all central rays emerge vertical at the surface. This has the consequence that
the applied corrections adulterate the curvature of the hyperbolas found in the OD gather, but
do not cause an error respective the new location of their apex, given by x0 ,t ∗ 0 . For an arbitrary
layered model, comparable results can not be expected, because in that case the error in t ∗ 0 caused
by the static correction, depends on the respective global take-off angle β g and thus changes for
0
every central ray.
The second three sections, Figures 3.6, 3.7, and 3.8, were generated using NMO velocity limits
that take the effect of the applied corrections into account. Unfortunately, the lower limit is still
higher than the actual NMO velocity within those areas of the first reflector that are nearest to
the surface, and where the effect of the bulk shift is highest. Nevertheless, it is evident that the
results are enhanced considerably. Proper search limits should provide coherency values, close
to one for all three reflectors.

3.2 Standard processing using elevation-statics 57
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
-1.0
Time [s]
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
3.8
4.0
4.2
Global x-coordinate [km]
0 2 4 6 8 10 12 14
Figure 3.3: Automatic CMP stack section, created from the original dataset after applying static
corrections and a bulk-shift. Ignoring the influence of the applied corrections to the found attributes,
wrong search limits for the NMO velocity were used.

58 Synthetic Data Example
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.60
0.55
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
Time [s]
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
3.8
4.0
4.2
Global x-coordinate [km]
0 2 4 6 8 10 12 14
Figure 3.4: Coherency section of the Automatic CMP stack, created from the original dataset
after applying static corrections and a bulk-shift. Due to the wrong search limits for the NMO
velocity, the coherency values for the first and most of the second reflector are very low.

3.2 Standard processing using elevation-statics 59
2900
2880
2860
2840
2820
2800
2780
2760
2740
2720
2700
2680
2660
2640
2620
2600
2580
2560
2540
2520
2500
2480
2460
2440
2420
2400
2380
2360
2340
2320
2300
Time [s]
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
3.8
4.0
4.2
Global x-coordinate [km]
0 2 4 6 8 10 12 14
Figure 3.5: NMO velocity section, created from the original dataset after applying static corrections
and a bulk-shift. Due to the wrong search limits of the NMO velocity, the found NMO
velocity values for the first reflector and for the left side of the second reflector are not optimal.
Please note that the found NMO velocities are generally lower than those velocities that would
be expected according to the used subsurface model (Figure 3.1).

60 Synthetic Data Example
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
Time [s]
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
3.8
4.0
4.2
Global x-coordinate [km]
0 2 4 6 8 10 12 14
Figure 3.6: Automatic CMP stack section, created from the original dataset after applying static
corrections and a bulk-shift. Considering the influence of the applied corrections to the found
attributes, more appropriate v NMO search limits were used.

3.2 Standard processing using elevation-statics 61
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.60
0.55
0.50
0.45
0.40
0.35
0.30
0.25
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0.15
0.10
0.05
Time [s]
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
3.8
4.0
4.2
Global x-coordinate [km]
0 2 4 6 8 10 12 14
Figure 3.7: Coherency section of the Automatic CMP stack, created from the original dataset
after applying static corrections and a bulk-shift. Due to the better suited search limits for the
NMO velocity, the coherency values are close to one for the major parts of the three reflectors.

62 Synthetic Data Example
3100
3000
2900
2800
2700
2600
2500
2400
2300
2200
2100
2000
1900
1800
1700
1600
1500
Time [s]
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
3.8
4.0
4.2
Global x-coordinate [km]
0 2 4 6 8 10 12 14
Figure 3.8: NMO velocity section, created from the original dataset after applying static corrections
and a bulk-shift. Due to the better suited search limits of the NMO velocity, the found NMO
velocity values are correct except of the two spots beneath the lowest points of the measurement
surface. Please note that found NMO velocities are generally lower than those velocities that
would be expected according to the used subsurface model (Figure 3.1).

3.3 Determination of the (near)surface attributes and elevation in X 0
3.3 Determination of the (near)surface attributes and elevation
in X 0
In order to stack with traveltime formulas (2.48) and (2.103), respectively, or to compute the real
wavefield attributes from pseudo attributes according to equations (2.51) and (2.104), it is necessary
to know the values of the four (near)surface attributes {K 0 ,α 0 ,v 0 ,(∇v) 0 } in all central points
X 0 . The near-surface velocity and its gradient have to be obtained directly in the field, whereas
the latter can be measured either to global coordinates or to local coordinates (see Section 2.6.1).
Theoretically, also the curvature K 0 and the dip-angle α 0 could be measured directly, or be approximatively
computed from the measured source and receiver positions, vicinal to X 0 . But this
makes only sense if the measurement surface is actually smooth and representable in the vicinity
of each central point according to equation (2.47). If there are little deviations from this ideal
shape, then the local values of the dip and curvature can fluctuate, within a large range. Such
a situation is displayed in Figure 3.11, where the dip-angle and curvature of the measurement
surface (depicted in blue) show considerable changes within the considered stacking aperture.
Similar situations are very common in the land-data acquisition. Thus, in practice it is rarely
reasonable to work with the local values of K 0 and α 0 . Here, it makes more sense to determine
those values of K 0 and α 0 that represent the locations of all sources and receivers that contribute
to the stack, best (according to equation (2.47)).
There are different options how to get this averaged surface attributes. For example, one could
use spline functions to smooth the measurement surface and determine the surface attributes afterwards.
The problem of this approach is how to define the degree of smoothness. According
to the conditions demanded in the derivation, the smoothed measurement surface has to be representable
by a parabola within the stacking aperture of every point X 0 . But on the other hand,
it is evident that the measurement surface must not be smoothed more than necessary. This is
very difficult to achieve, especially if the respective smoothing algorithm shall work automatically,
i.e., without further human interaction, for different datasets. The approach I used to get
averaged surface attributes requires some computational effort, but seems to me to be the most
natural way to solve this problem. For every central point X 0 a parabola (or a circle) is fitted
to the source and receiver points that lie within the respective stacking aperture. Accordingly,
this parabola is the best parabolic representation of the measurement surface within this area and
constitutes the new averaged measurement surface in the vicinity of X 0 . If the stacking aperture
depends on t 0 , it would be possible to determine values of K 0 and α 0 that depend on X 0 and t 0 ,
but this would demand a very high computational effort. I propose in this case to use a average
stacking aperture for every point X 0 , independent of t 0 .
In addition to the values of the surface attributes K 0 and α 0 , the elevation z 0 of the central points
X 0 has to be determined. Here holds the same as for K 0 or α 0 . This value could be measured
directly, or be approximatively computed from the neighboring source and receiver positions.
63

64 Synthetic Data Example
Figure 3.9: Example for the determination of the averaged measurement surface dip α 0 by fitting
a straight line (red) to the source and receiver locations within the stacking aperture (blue).
The knowledge of α 0 enables to transfer the global source and receiver coordinates to the local
coordinate system.
However, if the measurement surface is not really smooth and representable by a parabola in the
vicinity of every central point, one has to choose an averaged value, consistent with the averaged
surface attributes. This value is given by the z-coordinate of the fitted parabola or circle, respectively,
at xg = x 0 (see Figures 3.10, 3.11 and 3.14). If the distance between the considered source
and receiver points and the best fitting parabola is to big to be assumed as negligible than it is
proposed to apply static corrections, as described in Section 3.2 , to simulate source and receiver
points lying on these parabola.
3.3.1 Determination of α 0 , K 0 , and xz by fitting parabolas
For the first tests I used the computer-algebra system MAPLE to determine K 0 , α 0 , and z 0 by
fitting parabolas to the respective source and receiver locations. MAPLE provides an algorithm
to fit a parabola of the form z = ax 2 + b to a set of points. This equation describes a parabola
in the local coordinate system, so that I had firstly to transfer the respective source and receiver
coordinates from the global to the local coordinate system. To determine the orientation of the
local coordinate system I fitted a straight line to the considered surface points. To fit the parabola,
MAPLE uses an iterative algorithm that minimizes the so called object function, i.e., in this case
the sum of the squared distances of the single points to the parabola. The straight line can be

3.3 Determination of the (near)surface attributes and elevation in X 0
Figure 3.10: Example for the determination of the averaged measurement surface curvature and
elevation in X 0 by fitting a parabola (red) to the the source and receiver locations within the
stacking aperture (blue). This fit is performed in the local coordinate system. Note that in the
local coordinate system, the stacking aperture is not symmetrical with respect to X 0 anymore.
fitted analytically in one step by solving a system of linear equation. Finally α 0 is given by the
angle between this line and the x-axis of the global coordinate system, K 0 is provided by the
coefficient a of the fitted parabola (see equation (2.28)) and z 0 can be computed by means of α 0
and the coefficient b. These two steps of fitting have to be performed for every central point X 0 .
An example for this procedure is displayed in Figures 3.9 and 3.10.
Please note that applying this strategy means to split the primary three parametric search for the
best fitting arbitrarily oriented parabola, to a one parametric search for α 0 and a subsequent two
parametric search for K 0 and z 0 . Doing this saves much computation time in comparison to the
three parametric search. However, searching the minimum of the object function with respect
to α 0 does not inevitably provide those value of α 0 that also minimizes the object function with
respect to all three parameters. In this case, it works very well, but this does not hold generally
and has always to be checked.
The primary tests with MAPLE brought results, very close to the results that I achieved with the
C++ implementation presented in the following. Thus, I have abstained from displaying them
here.
3.3.2 Determination of α 0 , K 0 , and xz by fitting circles
As my final aim was to extend the existing 2D CRS stack implementation, I wrote a C++ program
to determine K 0 , α 0 , and z 0 that could be integrated in the existing C++ code. To achieve
a maximum at accuracy I decided to use the alternative strategy of fitting three parameters simultaneously.
An arbitrarily oriented parabola has no unique explicit description in the global
coordinate system and thus is very inconvenient to handle. For this reason, I used circles instead
of parabolas. An example is shown in Figure 3.11. As a circle is a very good approximation of a
parabola, in the vicinity of the apex, the error produced by fitting circles instead of parabolas is
negligible. The three parameters that define a circle are the radius and the x- and z-coordinates
of the midpoint. The curvature in X 0 is given by the inverse of the radius of the circle, the dip-
65

66 Synthetic Data Example
Figure 3.11: Example for the determination the averaged measurement surface dip, curvature,
and elevation in X 0 by fitting a circle (red) to the the source and receiver locations within the
stacking aperture (blue).
angle is provided by its midpoint location, and the elevation z 0 is defined by its zg-coordinate at
xg = x 0 .
First implementation
In my first C++ implementation, I performed an independent search for every central point X 0 .
I did not use a minimization algorithm, like the Gradient or Newton method but tried every
possible combination of parameters within a certain three dimensional search grid to find that
combination that minimizes my object function. Unfortunately, this strategy required an immense
computational effort. To demonstrate this I will give an example. For the initial search,
it was necessary to sample the search range for every parameter in at least 40 steps, this means
40 3 = 64000 possible circles. For every circle, the squared distances to, e.g., 200 source and
receiver points have to be computed and summed up. This means 1.28 · 10 7 computations for
one central point. My synthetic dataset had 1539 central points, thus at the end approximatively
2 · 10 10 computations were necessary to determine α 0 , K 0 , and z 0 in all central points.
Subsequently, every found parameter triple was optimized by an additional iterative search process
that was performed within the close vicinity of the obtained values. For every iteration the
search grid was refined. This was done until the achieved value of the object function changed
only by a amount, smaller than a certain limit. The optimization does not demand much computation
time, because for this step it is sufficient to divide the search space in, e.g., five samples
per dimension. The usual number of iterations lies between one and three.
The current implementation of the 2D CRS stack uses a taper function that gives the signals measured
at the borders of the stacking aperture a lower weight. I have used the same taper function
in my object function by weighting the squared distance of source and receiver points that lie at
the border of the stacking aperture less than the distance of points from the center. The results,
obtained with this implementation were very similar to the results that I obtained with MAPLE.
Maybe they were slightly improved, but the run-time was five times longer and thus far beyond
any acceptable limit.
By the way, in principal this search process is very similar to that one applied within the 2D

3.3 Determination of the (near)surface attributes and elevation in X 0
ZO CRS stack itself, where for every central point X 0 and every Z0-traveltime t 0 the hyperbolic
traveltime surface has to be found which yields the highest coherency value within the pre-stack
data. Usually this search for β 0 , K N , and K NIP is splited to save computation time (see, e.g.,
Mann, 2002).
Final implementation
The crucial point of any optimization process is the previously known information that can be
used to confine the search and to tailor the search algorithm to the specific problem. In this case
we have two informations that are very useful to restrict the search process. These are:
1. The source and receiver points within the stacking aperture do not change considerable
from one central point to the next.
2. The source and receiver points are not arbitrarily distributed in space but lie on a real
measurement surface which can only be rough within certain “natural” limits.
According to this considerations, my final implementation determines only the first parameter
triple using large search ranges. From this triple suited search ranges for the next triple are derived
and so on. This works very well, because most of the source an receiver points that lie in
the stacking aperture of one central point, lie also within the stacking aperture of its neighboring
central points. Of course, it is very important for the stability of this approach that the search
ranges are not too narrow. If the search fails in one central point because the optimal parameter
triple lays outside of the search space, then it fails in general in all following points, too. To handle
this problem I modified the optimization step. Due to the fact that the search ranges can be
chosen much smaller, since the information from the previous search is used, it is not necessary
anymore to refine the search grid considerable within the optimization. Now it is possible to use
the iterations that were needed before solely to refine the search grid, also to move the search
grid within the parameter space towards the searched minimum, if this lays outside. The modified
optimization procedure serves both, to refine the initial result and to make the search more
flexible and stable. I will give an 1D example to explain the functionality of this procedure. If the
minimum of the object function is given for the parameter value 10, but the initial search range
of this parameter is from 1 to 7, then the search will yield the value 7. For the first optimization
step the new search range is, e.g., 5 to 9 with a slightly refined grid. Accordingly the new result
would be 9. For the second optimization step the grid is refined again and the respective search
range is 8 to 10. This leads to the result 10. The third search step with a range from 9.5 to
10.5 achieves no considerable improvement, thus the search is finished. Note that this procedure
demands that the object function is well behaved, i.e., has no further local minima in the vicinity
of the global minima.
This modified search algorithm leads to a drastic reduction of the run-time. Since this imple-
67

68 Synthetic Data Example
Figure 3.12: The measurement surface curvature K 0 determined by fitting circles to all source
and receiver locations within the respective stacking aperture. This figure shows the periodic
small scale undulation of the measurement surface very clearly.
mentation provides much more options to adjust the algorithm to the used topography data it is
possible to achieves the same results as with the former implementation in 1% of the time.
3.4 Forward calculation
In this section, forward calculated properties of the subsurface model, displayed in Figure 3.1,
were used to study by means of a practical example the validity of the search-range determination,
derived in Chapter 2. Due to the simplicity of the model it is not necessary to use a
ray-tracing software to obtain the wavefield attributes K N , K NIP , and β 0 . In Chapter 2 all equations
were derived, needed to compute these values from the known properties of the subsurface
model and the measurement surface. Knowing the wavefield attributes {K N ,K NIP ,β 0 } and the
surface attributes {K 0 ,α 0 ,z 0 }, determined in the last section, it is possible to estimate the measured
NMO velocity and every particular set of pseudo attributes, both in global and in local
coordinates. In addition, one can transfer the search limits that would hold for a fictitious horizontal
measurement surface to appropriate limits that hold for the actual surface. For the sake
of brevity, the discussion will be focused upon the case of applying a conventional 2D ZO CRS

3.4 Forward calculation 69
Figure 3.13: The measurement surface dip α 0 determined by fitting circles to all source and
receiver locations within the respective stacking aperture.
Figure 3.14: Comparison between the real measurement surface, provided by the source and
receiver locations (red) and that surface that is given by the determined locations of the central
points X 0 (x 0 ,z 0 ) (black).

70 Synthetic Data Example
Figure 3.15: The forward calculated take-off angle β 0 (red) and its search limits (black). Due to
the simplicity of the model the same take-off angle holds for all central rays that impinge at a
certain central point X 0 .
stack to the synthetic dataset, used in this chapter. As mentioned before, the conventional CRS
stack, designed for a planar measurement surface, uses global coordinates, and does not consider
α 0 , K 0 , and E 0 .
3.4.1 Forward calculated take-off angle β 0 and its search limits
The three reflectors of the subsurface model are horizontal, thus all central rays propagate in
vertical direction and their global take-off angles β g are zero. For this very simple situation, the
local take-off angles β 0 can be easily computed according to equation (2.52). Please note, in
general every central ray has its specific take-off angle, but in this particular situation, the same
β 0 holds for all central rays that impinge at the same central point. Here it is possible to plot β 0
over x 0 without considering a specific reflector.
In the following the search range of β g
0 is assumed to be [−80◦ ,80 ◦ ]. Based on this, the search
limits of β0 can be computed according to equation (2.68). In practice it can be reasonable to
choose the β g
0 search range in dependence of t0 . This case is not considered here. The resulting
values of β0 and the respective search limits are displayed in Figure 3.15.
0

3.4 Forward calculation 71
Figure 3.16: The sine of the forward calculated pseudo take-off angle β ∗ 0 (red) and its search
limits (black). Here the sine of β ∗ 0 is used for the plot, because under certain circumstances
sinβ ∗ 0 becomes larger than one, and consequently β ∗ 0 complex. Due to the simplicity of the
model the same pseudo take-off angle holds for all central rays that impinge at a certain central
point X0 .
Figure 3.17: The RMS velocities calculated for the model shown in Figure 3.1. The RMS velocities
for the first layer are depicted in red those that corresponds to the bottom of the second layer
in blue and those that correspond to the bottom of the third layer in green.

72 Synthetic Data Example
Figure 3.18: The forward calculated NMO velocity for the first reflector. Negative values correspond
to imaginary velocities, according to convention (3.4).
3.4.2 Forward calculated RMS and NMO velocities
The RMS velocities that hold for the three reflectors of the subsurface model (Figure 3.1), are
computed according to equation (2.62). The results are shown in Figure 3.17, where the RMS
velocities that correspond to the first reflector and the whole first layer are shown in red, those
that correspond to the second reflector in blue and those that correspond to the third reflector in
green. These colors are also used in the following figures, to signify the considered reflector. It
is evident that the RMS velocity of the first layer is constant due to the fact that it is the average
velocity within a homogeneous layer. However, the RMS velocity at all points below the first
reflector depends on the vertical extension and of the overlaying layers.
After the RMS velocities are determined, the NMO velocities that are expected to be obtained
applying the conventional CRS stack can be computed by means of equation (2.109a). Please
note that it is necessary to use the ”averaged”X 0 elevation, i.e., z 0 , for the computation of the
values of t 0 . The results are shown in the Figures 3.18, 3.19, and 3.20. The NMO velocities that
are expected to be obtained applying the conventional CRS stack using local coordinates are in
principle very similar, and can be derived according to equation (2.108). As mentioned before,
the NMO velocity is imaginary if
K NIP cosβ 0 − K 0 < 0, (3.3)

3.4 Forward calculation 73
Figure 3.19: The forward calculated NMO velocity for the second reflector. Negative values
correspond to imaginary velocities, according to convention (3.4).
Figure 3.20: The forward calculated NMO velocity for the third reflector. Negative values correspond
to imaginary velocities, according to convention (3.4).

74 Synthetic Data Example
Figure 3.21: The forward calculated NMO slowness for the first reflector (red) and its search
limits (black).
according to equation (2.109a) with E0 equal zero. Therefore, the so-called signed square root
was used to visualize imaginary NMO velocities in the plots. This reads
vNMO = sign � v 2 �
NMO
�
|v2 | . (3.4)
NMO
Consequently imaginary values are displayed by negative values in the Figures 3.18, 3.19, and
3.20. Please note that the analytical function given by equation (2.109a) provides of course infinite
values for the squared NMO velocity, at any location where the denominator changes it
sign. However, the displayed values of the NMO velocity correspond to discrete points at the
measurement surface, where in general the denominator of equation (2.109a) is not exactly zero.
K NIP is always positive for the assumed subsurface model. Thus the NMO velocity only becomes
imaginary, if the surface curvatures K 0 is positive and larger than the respective value of
K NIP cosβ 0 .
3.4.3 Forward calculated slowness p NMO and its search limits.
As mentioned before, the first parameter, searched within the current implementations of the 2D
ZO CRS stack is the squared NMO slowness p2 NMO . In Section 2.4.3 a strategy was derived, to
transfer the search limits that would hold for a planar measurement surface to limits that account

3.4 Forward calculation 75
Figure 3.22: The forward calculated NMO slowness for the second reflector (blue) and its search
limits (black).
Figure 3.23: The forward calculated NMO slowness for the third reflector (green) and its search
limits (black).

76 Synthetic Data Example
Figure 3.24: The forward calculated values of the NIP-wave curvature K NIP . The values that
correspond to the first reflector are depicted in red, those for the second reflector in blue and
those for the third reflector in green
for the actual measurement surface, if E0 = 0. In order to treat the case of applying a conventional
CRS stack, global coordinates have to be considered according to Section 2.6.3. To study this
are assumed:
strategy by means of the used dataset, the following search limits for v RMS and β g
0
vmin RMS (x0 ,t0 ) = vRMS (x0 ,t0 ) − 500m/s ,
g,min
β = −80
0
◦ , (3.5a)
vmax RMS (x0 ,t0 ) = vRMS (x0 ,t0 ) + 500m/s ,
g,max
β = +80 ◦ , (3.5b)
wheres, without loss of generality, v RMS limits instead of v NMO,ZD limits are used in this case.
In the plots, p NMO instead of p 2 NMO is displayed, in order to simplify the comparison to v NMO .
Therefore the signed square root was used, according to its definition in equation (3.4). The results
are displayed in the Figures 3.21, 3.22, and 3.23. One can see that a remarkable search-range
reduction can be achieved, if the influence of the measurement surface is considered, particularly
in surface areas, with large and positive K 0 values.
3.4.4 Forward calculated values of K N and K NIP .
It is evident that the Normal-wave curvature K N is zero for the first reflector. Looking at the
refraction and transmission laws, equations (2.92) and (2.93), reveals that K N is zero for the
0

3.4 Forward calculation 77
Figure 3.25: The forward calculated values of K∗ N
fact that the reflectors are planar, both, the limiting values and the obtained values of K ∗ N
same for all three layers.
(red) and its search limits (black). Due to the
are the
second and third reflector, too. In other words, a planar wavefront does not change its curvature,
passing through planar reflectors. Consequently it can be abstained from displaying the values
of K N here.
For the calculation of K NIP one could apply the refraction and transmission laws, too. However,
this is not necessary for this simple case, were the obtained values of the RMS velocity can be
used to calculate K NIP according to equation (2.65). The results are displayed in Figure 3.24.
3.4.5 Pseudo attributes β ∗ 0 , K∗ N , and K∗ NIP and their search limits.
In case of the pseudo take-off angle β ∗ 0 it was mentioned before that a conventional CRS stack
software, has generally to consider complex values of β ∗ 0 within the search range, since β ∗ 0 is
complex, if sinβ ∗ 0 is larger than one (see equation (2.104a)). Usually, this requires a slightly
modification within the code of the used software.
To avoid complex values in the plot, the sine of β ∗ 0 and its search limits are displayed in Figure
3.16. The latter correspond to the search range of β g
0 , i.e., [−80◦ ,80◦ ]. It can be seen that the
pseudo take-off angles that would be obtained for this dataset by a conventional CRS stack have
no complex values. This is due to the small take-off angles β0 (Figure 3.15). However the black

78 Synthetic Data Example
Figure 3.26: The forward calculated values of K ∗ NIP
sponding search limits (black).
Figure 3.27: The forward calculated values of K ∗ NIP
responding search limits (black).
for the first reflector (red) and the corre-
for the second reflector (blue) and the cor

3.4 Forward calculation 79
Figure 3.28: The forward calculated values of K ∗ NIP
sponding search limits (black).
for the third reflector (green) and the corre-
lines show that generally it would have been possible to obtain complex values for the pseudo
take-off angles β ∗ 0 on this measurement surface.
were chosen:
To display the respective search range of K ∗ N the following search limits for K N
K min
N = − 1
1000m
and Kmax N = 1
. (3.6)
1000m
On the basis of this values, the corresponding limits of K∗ N can be determined, according to equations
(2.111). The values of K∗ N that would be obtained with the conventional CRS stack are
computed by means of equation (2.104b). Due to the fact that the reflectors are planar, the values
of K∗ N , are the same for each of the three layers. Of course, the same holds for the search limits.
The results are depicted in Figure (3.25).
Starting from the forward modeled values of KNIP , those values of K∗ NIP can be derived which
are expected to be obtained with the conventional CRS stack. This is done by means of equation
(2.104c). KNIP is with exception of confliction dip situations (see Mann, 2002) no search parameter
in the current implementations of the 2D ZO CRS stack. Nevertheless, those limits for
K∗ NIP that correspond to the used vRMS limits are displayed, too. These are computed according
to equation (2.111), where the subscript N has to be substituted by the subscript NIP. The results
for the different layers are displayed in Figures 3.26, 3.27, and 3.28.

80 Synthetic Data Example
3.5 Outlook
As mentioned before, the final objective of my work presented in this thesis is to extent the existing
2D ZO CRS stack software for planar measurement surfaces, such as to handle data measured
on a curved surface and to consider the near surface velocity gradient. In addition, the obtained
results shall be referred to a fictitious planar measurement surface to simplify interpretation and
further processing. Unfortunately, it was not possible to finish theses extensions within the temporal
range of this thesis. The 2D ZO CRS stack software for planar measurement surfaces is
highly developed and accordingly very complex. Every single change or extension of the code
has to be considered carefully and thoroughly tested.
In this section I show results that were obtained, performing the first step of the CRS stack procedure
(see Mann, 2002), i.e., the Automatic CMP stack, using global coordinates for half-offset
and midpoint. As mentioned before, these data-driven CMP stack has the purpose to provide the
NMO velocity values (here vg ) and a first provisional ZO section, both needed in the subse-
NMO
quent steps of the split traveltime-parameter search. The strategy to determine appropriate search
limits for the NMO velocity, depending on the dip and curvature of the measurement surface of
every central point (see Section 2.4.3 and 2.6.3) was not jet implemented. Thus a sufficiently
large global search range for the NMO velocity and a coarsely sampled search grid were used.
3.5.1 Automatic CMP stack plus redatuming
As expected, the time domain images of the horizontal reflectors are, due to the influence of
the topography, not planar, but curved (see Figure 3.29). The same holds for the v g
NMO section,
represented in Figure 3.31. To remove these curvature, a planar measurement surface at sea-level
was simulated by mapping the central points and the respective traveltimes t 0 to this surface,
according to the derivations made in in Section 2.5.1. Please note, the fictitious measurement
surface was chosen at sea level to simplify the comparison to the results obtained via static corrections
and a conventional CRS stack (see Section 3.2). In general, the velocity of the first layer
is unknown and the fictitious measurement surface must be chosen above the real measurement
surface. The resulting ZO section is shown in Figure 3.34, the vg section in Figure 3.31. It
NMO
is clearly observable that also the obtained values of the NMO velocity are strongly influenced
by the acquisition topography. Below the highest points of the topography, the denominator of
equation (2.54b) is negative and one finds imaginary values for the NMO velocity. These are displayed
using convention (3.4). Consequently, the simulation of a planar measurement surface is
not complete, as it would also be necessary to transfer the values of vg to those values which
NMO
would be obtained at the fictitious measurement surface at sea level. An eye-catching property
of these results, are the vertical strips, most clearly visible in the coherency section depicted in

3.5 Outlook 81
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
Time [s]
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
3.8
Global x-coordinate [km]
0 2 4 6 8 10 12 14
Figure 3.29: Automatic CMP stack section. Due to the influence of the acquisition topography
to the measured traveltimes, the time domain images of the reflectors are curved. The effect of
a, from left to right increasing, periodic small-scale topography undulation that does not satisfy
the smoothness requirements, can be observed.

82 Synthetic Data Example
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.60
0.55
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
Time [s]
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
3.8
Global x-coordinate [km]
0 2 4 6 8 10 12 14
Figure 3.30: Coherency section of the automatic CMP stack. Due to the influence of the acquisition
topography to the measured traveltimes, the time domain images of the reflectors are
curved. The effect of a, from left to right increasing, periodic small-scale topography undulation
that does not satisfy the smoothness requirements, can be observed most clearly in this section.

3.5 Outlook 83
4000
3500
3000
2500
2000
1500
1000
500
0
-500
-1000
-1500
-2000
-2500
-3000
Time [s]
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
3.8
Global x-coordinate [km]
0 2 4 6 8 10 12 14
Figure 3.31: NMO velocity section. Negative values correspond to imaginary velocities, according
to convention (3.4). Due to the influence of the acquisition topography to the measured
traveltimes, the time domain images of the reflectors are curved. The found NMO velocities are
strongly influenced by the shape of the measurement surface. The effect of a, from left to right
increasing, periodic small-scale topography undulation that does not satisfy the smoothness requirements
can be observed.

84 Synthetic Data Example
Time [s]
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
-1500 -1000 -500
Offset [m]
0 500 1000 1500
Figure 3.32: CMP gather (x 0 = 10.8 km). The periodic small-scale undulation of the measurement
surface is clearly visible in the data. Please note the curvature of the traveltime functions
that corresponds to imaginary NMO velocity values.

3.5 Outlook 85
Time [s]
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
-1500 -1000 -500
Offset [m]
0 500 1000 1500
Figure 3.33: CMP gather (x 0 = 10.7 km). The periodic small-scale undulation of the measurement
surface is hardly visible, because this CMP is located at a zero-crossing of the periodic
undulation. Here, due to the horizontal reflectors, the traveltime variation caused by the undulation
has the same absolute value on the right and on the left branch of the ray path, but the
opposite sign, and thus the effect of the undulation to the whole traveltime is equal zero for all
CMP experiments in this gather.

86 Synthetic Data Example
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
Time [s]
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
Global x-coordinate [km]
0 2 4 6 8 10 12 14
Figure 3.34: Automatic CMP stack section after redatuming. After transferring the ZO traveltimes
to those values which would be measured on a horizontal measurement surface at sea
level, the reflectors are planar and horizontal. However, the effect of the small-scale topography
undulation cannot be removed by this procedure.

3.5 Outlook 87
4000
3500
3000
2500
2000
1500
1000
500
0
-500
-1000
-1500
-2000
-2500
-3000
Time [s]
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
Global x-coordinate [km]
0 2 4 6 8 10 12 14
Figure 3.35: NMO velocity section after redatuming. After transferring the ZO traveltimes to
those values which would be measured on a horizontal measurement surface at sea level, the
reflectors are planar and horizontal. However, the values of the NMO velocity are not transferred
to those values that would be measured at this fictitious measurement surface. The black lines
denote the picked ZO traveltimes that refer to the reflectors. Of course, the effect of the smallscale
topography undulation remains visible.

88 Synthetic Data Example
Figure 3.30. They are caused by a, from left to right increasing, periodic small-scale topography
undulation (see Figure 3.1) that does not satisfy the smoothness requirements of the used traveltime
formula. Of course, it is still the best second order approximation of the real traveltime, but
the approximation is much worse than it would be for a measurement surface representable by a
second order function. In addition the obtained wave-field attributes lose their physical meaning
to a certain extent. High coherency values are obtained only at those central points, where the
small-scale undulation crosses zero. This is caused by the periodicity of the undulation, and can
be verified by, looking at the single CMP gathers at x 0 = 10.7 km and x 0 = 10.8 km, pictured in
Figures 3.33 and 3.32. This strips should be hardly visible in the final CRS stack sections, because
there, the data is stacked along surfaces and the effect of the undulation should be smeared.
Nevertheless, I propose to remove such small-scale undulations as far as possible by static corrections
before the CRS stack is performed. For very complex top-surface topography I would
like to refer to an alternative approach, presented in Zhang et al. (2002), which enables to handle
an arbitrary surface topography by considering the elevation of every single source and receiver
point explicitly within a modified CRS traveltime formula. The implementation of this approach
is still in work, but first results are very promising.
3.5.2 Comparison with the predicted NMO velocities
In order to compare the obtained values for the NMO velocity with those values, predicted in
Section 3.4.2, the NMO velocities for the three reflectors were extracted from the v g
NMO section.
This was done on the basis of picked ZO traveltimes, represented as black lines in Figure 3.35.
The extracted NMO velocities of the three reflectors are depicted in Figures 3.36, 3.37, and
3.38. Due to the influence of the small-scale topography undulation, the picked NMO velocities
fluctuate considerably in the affected region. In addition, one has no usable values for the NMO
velocity in the vicinity of the left and right boarders of the acquisition surface, as there are not
enough traces within the CMP gathers. Nevertheless, an overall consistency to the predicted
values is observable. However, the results for the first reflector match only qualitatively, because
it was not considered that the used CRS stack implementation determines the stacking aperture
depending on t 0 . Contrary to this, the largest possible stacking aperture, containing all existing
traces within the CMP gather, was used for the determination of the surface attributes α 0 and
K 0 and for the central-point elevation z 0 . Thus the apertures actually used for the first reflector
were smaller than the aperture used for the prediction. Therefore, the absolute values of the
determined curvature and dip were too small with respect to the averaged values within the
real stacking aperture. For the second and the third reflector the maximum stacking aperture
was used, as expected in the prediction. Here the predicted and the actually determined values
match also quantitatively. This shows how important it is to determine the values of the dip and
particularly of the curvature consistently with the later used stacking procedure.

3.5 Outlook 89
VNMOg [m/s]
x10
7
4
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Xg [km]
Figure 3.36: Extracted NMO velocity for the first layer. Negative values correspond to imaginary
velocities, according to convention (3.4). To extract these velocity values, picked ZO traveltimes
were used, which are represented in Figure 3.35 by black lines.

90 Synthetic Data Example
VNMOg [m/s]
x10
3
4
2
1
0
-1
-2
-3
-4
-5
-6
-7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Xg [km]
Figure 3.37: Extracted NMO velocity for the second layer. Negative values correspond to imaginary
velocities, according to convention (3.4). To extract these velocity values, picked ZO traveltimes
were used, which are represented in Figure 3.35 by black lines.

3.5 Outlook 91
VNMOg [m/s]
x10
5
4
4
3
2
1
0
-1
-2
-3
-4
-5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Xg [km]
Figure 3.38: Extracted NMO velocity for the third layer. Negative values correspond to imaginary
velocities, according to convention (3.4). To extract these velocity values, picked ZO traveltimes
were used, which are represented in Figure 3.35 by black lines.

92 Synthetic Data Example

Chapter 4
Summary
The presented extension of the 2D ZO CRS stack enables to stack seismic data, considering
the average dip and curvature of the measurement surface within the stacking aperture and the
near-surface velocity gradient at the emergence point of the central ray. As in case of a planar
measurement surface, three wavefield attributes are determined that provide important informations
about the investigated subsurface structure. These can find applications in a number of
kinematic and dynamic modeling, inversion and stacking problems. The stack remains purely
data-driven, i.e., independent of the 2D laterally inhomogeneous velocity model. Only the nearsurface
velocity and its gradient in the vicinity of the coincident source and receiver points are
required.
To simplify subsequent interpretation and further processing, it is reasonable to perform a redatuming
process that relates the obtained results to a planar measurement surface, with constant
near-surface velocity. All relations needed for that purpose were derived and discussed within
this thesis.
Theoretically, it is possible to use the conventional CRS stacking software, assuming a planar
measurement surface with constant near-surface velocity, even though the data is measured on
a curved surface having a near-surface velocity gradient. It was shown that the found pseudo
wavefield attributes can be corrected afterwards. All necessary relations were derived, and the
advantages and disadvantages of this approach were extensively discussed.
A modified formulation of the CRS traveltime formula, for data measured on a curved surface
having a near-surface velocity gradient, presented in Chira et al. (2001), was derived. This
formulation is better suited for the practical application, as the used midpoint and half-offset
coordinates are referred to a global coordinate system, instead of the local coordinate system of
the considered central point.

94 Summary
Very important for an efficient implementation of the CRS stack formalism are the search limits
of the traveltime parameters. It was shown, that the found NMO velocity and also the pseudo
wavefield attributes, determined by conventional software, are strongly affected by near-surface
velocity variations and particularly by the surface topography. For this reason, strategies were
derived to calculate appropriate search limits for the actual measurement surface from those
values that would be valid for a planar measurement surface with constant near-surface velocity.
The validity of the theoretical considerations mentioned above was studied by means of a synthetic
dataset, whereby the near-surface velocity gradient was not considered. In addition, the
main steps on the way to a satisfying implementation of the presented theory, on the basis of the
existing 2D CRS stack software, were made. Unfortunately, is was not possible to finish this
part completely within the temporal range of this diploma thesis. Therefore final results of the
presented CRS stack extension could not be shown.
For comparison, results obtained by ENI/Agip via standard processing using static corrections
and the conventional CRS stack software were discussed. Afterwards, these results were enhanced
by re-processing the dataset using better suited NMO velocity limits that take the effect
of the applied corrections into account. Advantages and disadvantages of this approach in comparison
to the presented CRS stack extension were pointed out.
A fast and stable C++ code to determine the surface attributes α 0 , K 0 , and z 0 of an arbitrarily
shaped measurement surface was written. This code was tested by means of the synthetic dataset,
and included into the existing 2D CRS stack software. The used algorithm was described in detail
and the obtained results were displayed.
On the basis of the determined surface attributes and the forward modeled wavefield attributes
β 0 , K N , and K NIP the validity of the proposed search range estimation was demonstrated and
other important aspects related to the application of the CRS stack to data measured on a curved
surface were pointed out.
Finally, the first step of the CRS stack procedure, the Automatic CMP stack, was performed.
In addition, a redatuming process was applied to the resulting NMO velocity and CMP stack
sections. The obtained NMO velocities were compared to those values that were predicted on
the basis of the forward modeled wavefield attributes and the determined surface attributes. It
was shown that they were consistent within the expected range.

Appendix A
The scalar Hamilton’s equation
According to Section (2.2.1), equation (2.11), the Hamilton’s equation for two-point ray tracing
reads
dt = p G · d(x G − x G ) − p S · d(x S − x S ). (A.1)
Let us look at the infinitesimal source dislocation d(x S − x S ) and the slowness vector of the
paraxial ray at the anterior surface p S , keeping in mind that the following derivations are also
valid for the corresponding values d(x G − x G ) and p G at the posterior surface. If we assume the
anterior surface in the vicinity of S to be representable by the smooth analytical function f (x),
we can express d(x S − x S ), in linear approximation, as
d(x S − x S ) =
�
Δx S ,
∂ f
∂x (S) Δx S
� T
, (A.2)
where Δx S , the x-coordinate of d(x S − x S ), results from the projection of d(x S − x S ) onto the
tangent of the surface in S, i.e., the x-axis of the local coordinate system at the anterior surface.
This means that we assume the infinitesimal source dislocation d(x S − x S ) to be tangent to the
measurement surface in S, as we only consider the first derivative of f (x) at S. Consequently we
can express the dot product at the righthand side of equation (A.1) as
p S · d(x S − x S ) = p S,T · d(x S − x S ) , (A.3)
with p S,T being the projection of p S onto the tangent to the surface in S (see Figure (2.2)).
In the same way as above, we can express the slowness of the paraxial ray at the anterior surface,
i.e., pS , in linear approximation, as
�
∂ f
pS,T = pS ,
∂x (S) p �T S , (A.4)

96 The scalar Hamilton’s equation
where p S , the x-coordinate of p S,T , results from the projection of p S,T onto the tangent to the
surface in S, according to Figure 2.2.
Due to our choice of the local coordinate system, f (x) is tangent to the x-axis at origin S and
consequently, the first derivative of f (x) has only first or higher order terms. This has the consequence
that the z-components of the dot products on the righthand side of equation (A.1) have
only terms that are of second order or higher in Δx S and Δx G , respectively (see equations (A.2),
(A.3), and (A.4)). These terms can be neglected within the derivation of a traveltime formula
which shall only be valid up to the second order of Δx S and Δx G , respectively, because they get
third and higher order when we finally integrate the Hamilton’s equation to obtain an expression
for the traveltime. Accordingly, we can reduce in this case the vector representation of the
Hamilton’s equation (A.1) to a scalar expression. This reads
dt = p G d(Δx G ) − p S d(Δx S ). (A.5)

Appendix B
Used hard- and software
The computations were done on a dual-processor Linux PC with (S.u.S.E. Linux 6.1), on
HEWLEDTT PACKARD workstations 9000 with HP-UX 10.20 and on a SILICON GRAPH-
ICS ORIGIN 3200 (6 processors) with IRIX 6.5.
For various analytical calculations and for visualization of data, I used Maple V Release 5.1
(Waterloo Maple) and MATLAB 6.1 Release 12.1.
To visualize the various stack and attribute sections I used the Seismic Unix package (Center of
Wave Phenomena at Colorado School of Mines).
This thesis was written on a PC (S.u.S.E. Linux 6.1) using the freely available word processing
package TEX, the macro package LATEX 2 ε , and several extensions.
The bibliography was generated with BIBTEX.
Figures were mainly constructed with Xfig 3.2 (rev2) and Corel Draw 8.0.

98 Used hard- and software

Appendix C
Acknowledgment/Danksagung
An erster Stelle möchte ich mich bei meiner Familie für die Unterstützung während meines
Studiums bedanken.
Herrn Prof. Dr. Peter Hubral danke ich für die Übernahme des Hauptreferats und sein stets
freundliches Interesse an meiner Arbeit. Er hat mir sehr viel Förderung zuteil werden lassen
und zugleich alle Freiheiten gegeben.
Herrn Prof. Dr. Friedemann Wenzel danke ich für die Übernahme des Korreferats.
Steffen Bergler, Dr. German Höcht und besonders Jürgen Mann möchte ich für ihre vielfältige
Hilfe in allen den CRS-Stack betreffenden aber auch allgemein computerspezifischen Fragen,
sowie für viele nützliche und interessante Diskussionen danken.
Großen Dank möchte ich Prof. Dr. Jörg Schleicher aussprechen, der sich während seines Aufenthaltes
in Karlsruhe viel Zeit genommen hat um mir bei der Lösung einiger wichtiger theoretischer
Fragen zu helfen.
Danken möchte ich Prof. Dr. Martin Tygel, der in der Zeit, in der er hier in Karlsruhe zu Gast war,
meiner Arbeit viel Interresse entgegen gebracht hat und mir einige gute Anregungen gegeben hat.
Meinem Zimmerkollegen Pedro Chira möchte ich für seine Unterstützung und freundschaftliche
Zusammenarbeit danken.
Yonghai Zhang danke ich für seine freundliche und kompetente Hilfe, vor allem in Fragen der
Wellentheorie.
Ingo Koglin möchte ich besonders für die Unterstützung beim Picken und Extrahieren der NMO-
Geschwindigkeiten danken.

100 Acknowledgment/Danksagung
Für das Korrekturlesen meiner Arbeit möchte ich mich bedanken bei: Jürgen Mann, Steffen
Bergler, Ingo Koglin, Eric Duveneck und Prof. Dr. Jörg Schleicher.
Zu guter Letzt möchte ich auch noch allen anderen Mitarbeiterinnen und Mitarbeitern der Universität
Karlsruhe danken, deren Hilfe ich im Laufe meines Studiums in Anspruch nehmen durfte.

Abbildungsverzeichnis
2.1 Sketch of a two-dimensional inhomogeneous and isotropic medium. . . . . . . . 8
2.2 Construction of the ray slowness vector projection. . . . . . . . . . . . . . . . . 9
2.3 Sketch of a 2D model with a curved measurement surface. . . . . . . . . . . . . 11
2.4 The paraxial ray from S to G in the vicinity of the central ray from S to G. . . . . 12
2.5 ZO situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.6 The relationship between the take-off angles of the normal ray, β 0 and β g
0 and
the dip angle α 0 for a curved measurement surface. . . . . . . . . . . . . . . . . 27
2.7 The search range of β 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.8 Redatuming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.9 The transformation of the local 1D coordinates h and m to the global 1D coordinates
hg and mg. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1 Model, consisting of four homogeneous layers with different P-wave velocities.
These are separated by three horizontal reflectors and lie underneath a curved
measurement surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2 Visualization of the applied static corrections . . . . . . . . . . . . . . . . . . . 54
3.3 Standard processing, using wrong v NMO search limits: Automatic CMP stack
section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4 Standard processing, using wrong v NMO search limits: Coherency section of the
Automatic CMP stack. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

102 ABBILDUNGSVERZEICHNIS
3.5 Standard processing, using wrong v NMO search limits: NMO velocity section. . . 59
3.6 Standard processing, using better suited v NMO search limits: Automatic CMP
stack section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.7 Standard processing, using better suited v NMO search limits: Coherency section
of the Automatic CMP stack. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.8 Standard processing, using better suited v NMO search limits: NMO velocity section. 62
3.9 Example for the determination of the averaged measurement surface dip α 0 by
fitting a straight line to the source and receiver locations within the stacking
aperture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.10 Example for the determination of the averaged measurement surface curvature
and elevation in X 0 by fitting a parabola to the the source and receiver locations
within the stacking aperture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.11 Example for the determination the averaged measurement surface dip, curvature,
and elevation in X 0 by fitting a circle to the the source and receiver locations
within the stacking aperture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.12 The determined measurement surface curvature K 0 . . . . . . . . . . . . . . . . . 68
3.13 The determined measurement surface dip α 0 . . . . . . . . . . . . . . . . . . . . 69
3.14 Comparison between the real measurement surface, provided by the source and
receiver locations and that surface that is given by the determined locations of
the central points X 0 (x 0 ,z 0 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.15 The forward calculated take-off angle β 0 and its search limits. . . . . . . . . . . 70
3.16 The sine of the forward calculated pseudo take-off angle β ∗ 0
and its search range. 71
3.17 The forward calculated RMS velocities. . . . . . . . . . . . . . . . . . . . . . . 71
3.18 The forward calculated NMO velocity for the first reflector. . . . . . . . . . . . . 72
3.19 The forward calculated NMO velocity for the second reflector. . . . . . . . . . . 73
3.20 The forward calculated NMO velocity for the third reflector. . . . . . . . . . . . 73
3.21 The forward calculated NMO slowness for the first reflector and its search limits. 74

ABBILDUNGSVERZEICHNIS 103
3.22 The forward calculated NMO slowness for the second reflector and its search
limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.23 The forward calculated NMO slowness for the third reflector and its search limits. 75
3.24 The forward calculated values of K NIP . . . . . . . . . . . . . . . . . . . . . . . . 76
3.25 The forward calculated values of K ∗ N
3.26 The forward calculated values of K ∗ NIP
and its search limits. . . . . . . . . . . . . . 77
for the first reflector and the corresponding
search limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
for the second reflector and the corresponding
search limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.27 The forward calculated values of K ∗ NIP
for the third reflector and the corresponding
search limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.28 The forward calculated values of K ∗ NIP
3.29 Automatic CMP stack section. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.30 Coherency section of the automatic CMP stack. . . . . . . . . . . . . . . . . . . 82
3.31 NMO velocity section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.32 CMP gather (x 0 = 10.8 km). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.33 CMP gather (x 0 = 10.7 km). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.34 Automatic CMP stack section after redatuming. . . . . . . . . . . . . . . . . . . 86
3.35 NMO velocity section after redatuming. . . . . . . . . . . . . . . . . . . . . . . 87
3.36 Extracted NMO velocity for the first layer. . . . . . . . . . . . . . . . . . . . . . 89
3.37 Extracted NMO velocity for the second layer. . . . . . . . . . . . . . . . . . . . 90
3.38 Extracted NMO velocity for the third layer. . . . . . . . . . . . . . . . . . . . . 91

104 ABBILDUNGSVERZEICHNIS

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