Double diffraction stack as an
alternative strategy for CRS-based
limited aperture true-amplitude
Kirchhoff depth migration
Diplomarbeit
an der Fakultät für Physik der
Universität Karlsruhe (TH)
eingereicht von
Ines Veile
Referent:
Korreferent:
Abgabe:
Prof. Dr. Peter Hubral
Prof. Dr. Friedemann Wenzel
23. April 2009
Ehrenwörtliche Erklärung:
Hiermit versichere ich, dass ich die vorliegende Arbeit selbstständig und nur mit den
angegebenen Hilfsmitteln verfasst habe.
Karlsruhe, 23. April 2009
Ines Veile
Zusammenfassung
Die vorliegende Diplomarbeit wurde mit Ausnahme dieses Kapitels in Englisch verfasst. Die grundlegenden Aspekte der Arbeit sind im Folgenden zusammengefasst. Da
für die meisten der vorkommenden englischen Fachbegriffe keine gebräuchlichen deutschen Übersetzungen existieren, wurden diese in ihrer Originalform belassen und sind
zur leichteren Erkennbarkeit kursiv gedruckt. Zu Gunsten der Lesbarkeit wurde in dieser Zusammenfassung weitestgehend auf die Angaben von Referenzen verzichtet. Diese
Informationen finden sich im englischen Hauptteil der Arbeit.
Einleitung
Das generelle Ziel der Geophysik ist die Erkundung des Erdinneren mittels physikalischer Methoden. Die Explorationsgeophysik befasst sich dabei im Speziellen mit der Erkundung von Kohlenwasserstoff- und Minerallagerstätten. Als geeignetes Verfahren dient
hierbei die Reflexionsseismik, bei welcher künstlich erzeugte seismische Wellen in den
Erduntergrund eingebracht werden. Diese Wellen werden dort an Grenzschichten transmittiert, gestreut und reflektiert. Geophone registrieren an der Erdoberfläche das elastische Wellenfeld in Form von Bodenbewegung (an Land) bzw. Druckänderung (auf See)
als Funktion der Zeit. Anhand der im aufgezeichneten Wellenfeld enthaltenen Reflexionsereignisse werden anschließend Lage und Krümmung von Reflektoren im Erduntergrund bestimmt.
Naheliegender Weise genügt jedoch ein einzelnes Experiment dieser Art nicht, um daraus
ein strukturelles Abbild des Untergrundes bestimmen zu können. Daher werden viele dieser Experimente mit verschiedenen Quell-/Empfänger-Positionen durchgeführt und kombiniert. Dies liefert einen mehrfach überdeckten Datensatz aus dem letztlich mittels eines
bildgebenden Verfahrens (Migration) ein strukturelles Abbild des Untergrundes generiert
wird. Um diesen mehrfach überdeckten Datensatz leichter handhaben zu können, wird
in der Reflexionsseismik zunächst oft eine zero-offset Sektion simuliert, indem mit Hilfe so genannter Stapelung entlang analytischer Laufzeitapproximationen der Einfluss der
i
Zusammenfassung
Messgeometrie entfernt wird. Werden bei der Migration der reflektierten Wellen auch die
dynamischen Effekte der Wellenausbreitung auf die Amplituden explizit berücksichtigt,
so handelt es sich um eine so genannte true-amplitude Migration. Hierbei dienen die Amplituden als direktes Maß für die Winkel-abhängige Reflektivität an der entsprechenden
Grenzschicht im Untergrund. Neben der Struktur können in diesem Fall auch Aussagen
über die physikalischen Eigenschaften des vorliegenden Reflektors getroffen werden. Der
Schritt der Migration nimmt dabei verständlicherweise eine zentrale Rolle ein und wird
daher in einem separaten Abschnitt diskutiert.
Generell unterscheidet man zwischen zwei Arten der Migration: der Zeitmigration, welche als Ergebnis ein strukturelles Abbild im Zeitbereich liefert und der Tiefenmigration,
die als Ergebnis ein Tiefenabbild erzeugt. Im Rahmen dieser Arbeit habe ich mich ausschließlich mit der letztgenannten Art befasst. Allgemein gilt, dass während der Migration
eine Summation aller Amplituden, die Informationen über denselben Tiefenpunkt tragen
könnten, längs eines Migrationsoperators durchgeführt wird. Das Summationsergebnis
wird dem entsprechenden Bildpunkt im migrierten Zeit- oder Tiefenbereich zugewiesen.
Aus der Gesamtheit dieser so bewerteten Bildpunkte entsteht das Reflektorabbild.
Die Common-Reflection-Surface-Stapelung
Die vor der Migration verwendete Stapelmethode hat einen erheblichen Einfluss auf
das resultierende strukturelle Abbild des Untergrundes und stellt somit einen wichtigen
Schritt in der Datenverarbeitung dar. Durch die Redundanz der mehrfach überdeckten
Daten ist es möglich, verschiedene kinematische Einflüsse auf das Wellenfeld zu berücksichtigen.
Im Rahmen dieser Arbeit kommt die am Geophysikalischen Institut in Karlsruhe entwickelte common-reflection-surface (CRS) Stapelung zum Einsatz. Gegenüber konventionellen Methoden, wie zum Beispiel einer Abfolge von normal-moveout-Korrektur, welche den kinematischen Einfluss des offsets entfernt, dip-moveout-Korrektur, welche den
Einfluss einer eventuellen Neigung korrigiert, und konventioneller Stapelung, birgt sie
vorallem den Vorteil eines sehr hohen Rauschabstandes, das durch die intensive Nutzung
der Redundanz des mehrfach überdeckten Datensatzes mittels eines flächenhaften Operators garantiert wird. Der offset stellt dabei den euklidischen Abstand zwischen Quelle und
Empfänger dar.
Ein Ergebnis der CRS-Stapelung neben der simulierten ZO Sektion sind die so genannten kinematischen Wellenfeld-Attribute, welche eine analytische Laufzeitapproximation
zweiter Ordnung parametrisieren. Wie die Nomenklatur vermuten lässt, beschreiben diese
Attribute generell die kinematischen Eigenschaften des entsprechenden Reflexionsereignisses mit Hilfe des Auftauchwinkels und der Krümmungen von zwei hypothetischen
ii
Kirchhoff-Migration
Wellenfronten. Diese Attribute kommen in vielen Bereichen zur Anwendung. Eine Möglichkeit besteht zum Beispiel in der Bestimmung eines sinnvollen Stapelbereichs des (konventionellen) Migrationsoperators, das heißt der Apertur. Eine weitere wichtige Anwendung der Attribute liegt in der Bestimmung eines Makrogeschwindigkeitsmodells, wie es
für die Migration benötigt wird.
Kirchhoff-Migration
Kirchhoff-Migration ist eine effiziente Migrations-Methode, um ein Abbild des Untergrundes aus reflexionsseismischen Daten zu erzeugen. Diese auf der integralen Lösung
der Wellengleichung basierende Methode stützt sich auf das Tangentialitätskriterium nach
Hagedoorn, welches besagt, dass für einen Reflektorpunkt MR die Diffraktionslaufzeitfläche (Huygens-Fläche) im Zeitbereich tangential zur Reflexionslaufzeitfläche ist. Unter der
Annahme, dass jeder Punkt M in der Tiefe als potentieller Diffraktionspunkt aufgefasst
werden kann, kann unter Verwendung eines geeigneten Geschwindigkeitsmodells die mit
diesen Punkten assoziierte Huygensfläche berechnet werden. Durch die Summation aller
Amplituden entlang der zu M gehörigen Diffraktionslaufzeitfläche (d. h., entlang des Migrationsoperators) ergibt sich das Migrationsergebnis. Handelt es sich bei M um einen
Reflektorpunkt MR , so führt konstruktive Interferenz im Tangentialbereich zwischen Reflexionsereignis und Stapeloperator zu einem Ergebnis, welches sich deutlich von Null
unterscheidet. Handelt es sich bei M hingegen um einen Tiefenpunkt, der nicht in der
Nähe des Reflektors liegt, so führt destruktive Interferenz zu einem vernachlässigbaren
Ergebnis. Im Folgenden werde ich zwei besondere Arten der Kirchhoff-Migration erläutern, welche im Rahmen dieser Arbeit angewendet wurden.
True-amplitude Kirchhoff-Migration
In der Regel werden bei der Migration die dynamischen Einflüsse der Wellenausbreitung
auf die Amplituden nicht explizit berücksichtigt und man erhält dadurch ein ausschließlich strukturelles Abbild des Untergrundes. Berücksichtigt man jedoch diese Einflüsse,
hauptsächlich handelt es sich hierbei um den Einfluss der geometrischen Ausbreitung, so
können nach der Migration auch Aussagen über die physikalischen Eigenschaften des Untergrundes getroffen werden. In diesem Fall stellen die Amplituden ein direktes Maß für
die Winkel-abhängige Reflektivität am entsprechenden Reflektor dar. Mathematisch wird
die Korrektur des Einflusses der geometrischen Ausbreitung durch eine Gewichtsfunktion im Migrationsintegral realisiert, welche mittels der Methode der Stationären Phase
hergeleitet werden kann.
iii
Zusammenfassung
Kirchhoff-Migration mit limitierter Apertur
Wie bereits erwähnt, wird während der Kirchhoff-Migration längs eines Operators alle
Energie aufsummiert, die von einem gemeinsamen Tiefenpunkt resultieren könnte. Theoretisch geschieht dies über eine unbegrenzte Apertur, praktisch muss die Apertur jedoch
aufgrund der räumlich und zeitlich begrenzten Akquisition, sowie der endlichen Rechenkapazitäten eingeschränkt werden. Wählt man für die erforderliche Einschränkung gerade
den Bereich um den stationären Punkt, das heißt um den Punkt, an dem der Operator tangential ist zum Reflexionsereignis, so werden nur Anteile aufsummiert, die auch tatsächlich einen Informationsgehalt über diesen Reflektor tragen. Diese Apertur wird als limitierte Apertur bezeichnet und bringt verschiedene Vorteile mit sich: Durch den Ausschluss
irrelevanter Beiträge wird die Anzahl der durchzuführenden Summationen reduziert und
dadurch die Effizienz erhöht. Darüber hinaus verbessert dies auch die Qualität des Migrationsergebnisses, da Störsignale, die außerhalb des Tangentialbereiches liegen, bei der
Summation nicht berücksichtigt werden. Nicht zuletzt wird dadurch allgemein das Auftreten von Artefakten, im Speziellen von operator aliasing, verhindert, die ausschließlich
durch Summation außerhalb des Tangentialbereichs entstehen.
Die limitierte Apertur ist jedoch leider im Vorfeld der Migration nicht bekannt. Daher
muss der Benutzer beim konventionellen Vorgehen selbst eine angemessene Apertur wählen. Diese ist sowohl durch die maximale Operator-Steigung begrenzt als auch symmetrisch zum stationären Punkt (laterale Position des Bildpunktes M) aufgehängt. Da dieser
jedoch in der Regel ebenfalls unbekannt ist, muss die konventionelle Apertur wesentlich
größer gewählt werden als die unbekannte limitierte Apertur, so dass sie die letztere in
jedem Fall voll miteinschließt. Andernfalls werden stark geneigte Ereignisse nicht abgebildet.
Für die Kirchhoff-Tiefenmigration mit limitierter Apertur entwickelte Jäger (2005a) eine CRS-basierte Methode, mittels der zunächst der stationäre Punkt bestimmt und anschließend die (um diesen Punkt zentrierte) limitierte Apertur berechnet werden kann.
Entsprechend dem bereits erwähnten Tangentialitätskriterium haben der Stapeloperator
und das Reflexionsereignis am stationären Punkt die gleiche Neigung haben, welche proportional zur horizontalen slowness ist. Anhand eines Vergleichs der Neigung entlang des
Operators mit der Neigung des entsprechenden Reflexionsereignisses für jeden Tiefenpunkt M können so die entsprechenden stationären Punkte bestimmt werden. Die hierzu
benötigte horizontale slowness ist für die zero-offset Konfiguration durch einen einfachen
Zusammenhang mit einem der CRS-Attribute, dem Auftauchwinkel des Normalstrahls,
gegeben. Eine Extrapolation des stationären Punktes für finite offsets ist über eine weitere CRS-basierte Methode möglich. Zur Bestimmung der Größe der dazugehörigen optimalen Apertur wird der von Schleicher vorgeschlagene Zusammenhang verwendet, der
besagt, dass die Größe der limitierten Apertur gerade der Größe der projizierten ersten
Fresnel Zone entspricht. Für zero-offset ist letztere, wie die horizontale slowness auch,
iv
Die doppelte Diffraktions-Stapelung
durch die CRS-Attribute gegeben. Wird kein stationärer Punkt gefunden, so findet an dieser Stelle eine Migration mit konventioneller Apertur statt, um Lücken in den Ergebnissen
zu vermeiden.
Die doppelte Diffraktions-Stapelung
Die doppelte Diffraktions-Stapelung beruht auf der zweimaligen Anwendung der
Kirchhoff-Migration auf denselben Datensatz mit zwei verschiedenen Gewichtsfunktionen. Diese Gewichte werden gewissermaßen auf dem Rücken der eigentlichen Daten migriert und können generell beliebig gewählt werden, solange sie innerhalb der Apertur
hinreichend langsam variieren. Berechnet man anschließend das Verhältnis der beiden
Migrationsergebnisse, erscheint das aufmultiplizierte Gewicht direkt an der migrierten
Position.
Im Rahmen dieser Arbeit entspricht das erste Gewicht einem Einheitsgewicht und führt
praktisch zur Durchführung einer konventionellen Kirchhoff Migration. Da ein Ziel die
Bestimmung der stationären Punkte darstellt und diese durch die Spurlokation charakterisiert werden, wird letztere als zweites Migrationsgewicht verwendet. Entsprechend der
Methode der Stationären Phase und mit entsprechendem tapering werden so ausschließlich Beiträge aus der unmittelbaren Umgebung des stationären Punktes erfasst. Berechnet
man anschließend das Verhältnis der beiden Migrationsergebnisse, erhält man die Spurlokation direkt an der entsprechenden Bildposition.
Ursprünglich wurde die doppelte Diffraktions-Stapelung zur wirtschaftlichen Durchführung der true-amplitude Migration entwickelt. Das Ziel dieser Diplomarbeit ist es, zu
zeigen, dass diese Methode auch zum Ansatz der CRS-basierten Kirchhoff-Migration mit
limitierter Apertur beitragen kann.
Praktische Aspekte und Anwendung
Im Rahmen dieser Arbeit wurde aus den zuvor eingeführten Konzepten und unter Verwendung von den am Geophysikalischen Institut der Universität Karlsruhe entwickelten Programmen zur CRS-Stapelung, zur amplitudenbewahrenden Kirchhoff-Migration
mit limitierten Aperturen im Tiefenbereich und zur Ereignis-bewahrenden Glättung ein
Arbeitsablauf für eine vollständige poststack Tiefenmigration mit limitierter Apertur erstellt. Dieser Ablauf wurde sowohl auf ein einfaches synthetisches Datenbeispiel als auch
auf Realdaten angewendet. Die doppelte Diffraktions-Stapelung mit der Spurlokation als
zweitem Migrationsgewicht wird dabei zur Bestimmung der stationären Punkte verwendet. Elementare Probleme des Verfahrens konnten erfolgreich gelöst werden: Durch die
v
Zusammenfassung
Berechnung der Einhüllenden des analytischen Signals beider Ergebnisse der doppelten
Diffraktions-Stapelung vor der Berechnung des Verhältnisses der beiden, konnten die Probleme an den Nulldurchgängen des wavelets umgangen werden. Ein weiterer Schritt stellt
sicher, dass das durch die Migration selbst generierte Hintergrundrauschen unterdrückt
wird: Durch das erstmalige Anwenden von CRS-Methoden auf Ergebnisse der poststackMigration (und die damit notwendig gewordene Transformation der CRS-Stapelung vom
Zeitbereich in die Tiefe) ist es möglich, anhand eines Kohärenz-Schwellwertes zwischen
zuverlässigen und unzuverlässigen stationären Punkten zu unterscheiden. Letztlich vervollständigt ein Ereignis-bewahrender Glättungsalgorithmus den Arbeitsablauf: unphysikalische Schwankungen und Ausreißer in den Lokationen der stationären Punkte werden
beseitigt. Die Kohärenzwerte erlauben ferner auch eine verlässliche Identifikation von
Reflektorabbildern.
Darüber hinaus wurden relevante Aspekte im Zusammenhang mit der Durchführung der
doppelten Diffraktions-Stapelung, wie zum Beispiel der Einfluss der relativen Variation
der Gewichtsfunktion, das Auftreten von operator aliasing und das Problem der Mehrdeutigkeit, angesprochen.
Schlussfolgerungen
Im Rahmen dieser Diplomarbeit habe ich mich mit der Kirchhoff-Tiefenmigration beschäftigt. Der Schwerpunkt lag dabei in der zuverlässigen und stabilen Bestimmung
des stationären Punktes, der zur Abschätzung der Lage der Apertur für die KirchhoffMigration mit limitierter Apertur benötigt wird. Zu diesem Zweck habe ich die Methode
der doppelten Diffraktions-Stapelung aufgegriffen und untersucht, ob dieser Ansatz eine
Alternative zum bislang verwendeten Neigungs-basierten Verfahren nach Jäger (2005a)
darstellt.
Der Kern der vorliegenden Arbeit ist die Erstellung eines mehrstufigen Arbeitsablaufes,
der auf der geschickten Zusammenstellung von verschiedenen bereits existierenden Verfahren beruht und lautet wie folgt:
• Lineare Transformation der Spurlokationen
• Gewichtung der Eingangsspuren mit den transformierten Spurlokationen
• Berechnung der Einhüllenden des analytischen Signals beider Ergebnisse
• Berechnung des Verhältnisses der beiden Ergebnisse
• Lineare Rücktransformation der Spurlokationen
vi
Schlussfolgerungen
• Anwendung der CRS-Stapelung auf das migrierte Abbild
• Kohärenz-basierte Auswahl der zuverlässigen Ergebnisse
• Ereignis-bewahrende Glättung der Ergebnisse
Die dargestellten Ergebnisse zeigen, dass die mit der doppelten Diffraktions-Stapelung
bestimmten stationären Punkte eine zuverlässige Alternative zu den mit Hilfe des
Neigungs-basierten Verfahrens ermittelten darstellen.
vii
Abstract
In Kirchhoff migration, the proper choice of the aperture is essential: the optimum aperture is the limited aperture which itself is linked to the projected Fresnel zone. This is
the smallest aperture providing interpretable amplitudes along with the highest possible
S/N ratio and the minimum number of required summations. In addition, limited-aperture
migration naturally prevents operator aliasing. The common-reflection-surface stack provides kinematic wavefield attributes which allow to estimate the optimum aperture size
for zero-offset and the dislocation of the stationary point with varying offset. The aperture is centered around the stationary point, but this point has to be associated with the
corresponding point in the migrated domain in an additional process. Kirchhoff migration
itself implicitly connects the stationary point and the image point in depth by collecting
the energy in the vicinity of the former and assigning it to the latter. Due to the properties
of Kirchhoff migration, not only the seismic reflection data itself, but also any smoothly
varying property or weight can be migrated ‘on top’ of the seismic data. By a multiple
application of migration with different weight functions, the latter can be retrieved at the
migrated image locations. In this so-called vector diffraction stack, the most generic property to be migrated is the source/receiver midpoint which represents the lateral location of
the stationary point. In this way, we can establish a relation between the image point in the
depth domain and the stationary point in time domain. In this thesis the validity and accuracy of this approach for simple synthetic data is investigated. In addition, the method is
applied to a real land data set. A straightforward extension is introduced to solve some of
the numerical problems linked to this approach and various common-reflection-surfacebased strategies are transferred from the time domain to the depth domain to identify the
reflector images and to attenuate migration noise. Finally, the approach is compared to
another common-reflection-surface-based approach which directly evaluates the tangency
criterion.
ix
Contents
Zusammenfassung
i
1
Introduction
1
1.1
Topic of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2
3
Common-Reflection-Surface Stack
5
2.1
The CRS stacking operator . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2
The Common-Reflection-Point trajectory . . . . . . . . . . . . . . . . .
9
2.3
The projected Fresnel Zone . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.4
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
Kirchhoff migration
13
3.1
True-amplitude Kirchhoff migration . . . . . . . . . . . . . . . . . . . .
13
3.1.1
Basic principles . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
3.1.2
True-amplitude Kirchhoff migration in the depth domain . . . . .
15
3.1.3
2.5 D Kirchhoff migration . . . . . . . . . . . . . . . . . . . . .
21
3.1.4
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
Limited-aperture Kirchhoff migration . . . . . . . . . . . . . . . . . . .
23
3.2.1
CRS-based limited-aperture migration . . . . . . . . . . . . . . .
24
3.2.2
Operator aliasing . . . . . . . . . . . . . . . . . . . . . . . . . .
26
3.2.3
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
3.2
xi
Contents
4
Vector diffraction stack
29
4.1
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
4.1.1
Double diffraction stack . . . . . . . . . . . . . . . . . . . . . .
29
4.1.2
Double diffraction stack with trace location . . . . . . . . . . . .
31
4.1.3
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
Practical aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
4.2.1
Description of the data set . . . . . . . . . . . . . . . . . . . . .
32
4.2.2
Poststack migration . . . . . . . . . . . . . . . . . . . . . . . . .
32
4.2.3
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
4.2
5
6
Real land data example
39
5.1
Description of data set . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
5.2
Poststack double diffraction stack . . . . . . . . . . . . . . . . . . . . .
40
5.3
Workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
5.4
Comparison with the dip-based strategy . . . . . . . . . . . . . . . . . .
47
5.5
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
Conclusions
51
A Variation of weight function
53
B Ambiguities in the imaging methods
57
B.1 Double diffraction stack . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
B.2 Dip-based strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
B.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
C Used Software
59
D Danksagung
61
List of Figures
63
References
65
xii
Chapter 1
Introduction
The primary aim of seismic reflection imaging is to gain a structural image of the subsurface, a task that is usually addressed in several successive processing steps. The last step
in this sequence is the application of depth migration which transforms the time domain
data into interpretable depth images1 . In this thesis, I will consider a particular type of
migration, Kirchhoff migration, and how it can be applied in an optimum and efficient
manner by incorporating the results of the preceding processing steps.
In Kirchhoff migration, summation along the forward-calculated diffraction traveltime
curves (or Huygens surfaces) in the unmigrated time domain constructs the migrated time
or depth image of the searched-for reflector. The basic idea is that a reflector can be
seen as an ensemble of diffractors, each of them representing a secondary source or Huygens source in the subsurface. From a theoretical point of view, the summation has to
be performed within an infinite aperture. Practically, the aperture is limited by various
aspects, such as the acquisition geometry, the recording time, and computational efforts.
However, the appropriate limitation of this migration aperture is a complex issue. Ideally,
the migration operator should be restricted to the region within the reflection signal strip
that contains the information which is of interest. In this region, the migration operator and the reflection event are tangent to each other, the center of this region is called
stationary point. Schleicher et al. (1997) showed that this tangency region is defined by
the so-called limited aperture, the minimum aperture defined by the first projected Fresnel zone (PFZ), the time-domain counterpart of the interface Fresnel zone from which
we expect the constructive contributions to the seismic amplitudes. Combined with appropriate tapering within the adjacent second PFZ, this yields interpretable amplitudes
along the reflector images and the minimum number of required summations. In a fur1
For not too complex models, imaging and interpretation can also be applied in the time domain. However, the actual subsurface is in the depth domain such that some kind of time-to-depth conversion is mandatory in any case.
1
Chapter 1. Introduction
ther extension of Kirchhoff migration with an additional weight function in the migration
integral, the geometrical spreading effect can be removed such that the amplitudes are
approximately proportional to the angle-dependent reflection coefficient. Therefore, this
extension is termed true-amplitude migration. Irrespective of such additional weights, in
limited-aperture migration, summation is only carried out in the tangency region. Thus, it
automatically prevents operator aliasing and minimizes the unwanted contributions from
other events and background noise usually gathered along the remaining, non-tangent part
of the migration operator. In principle, limited-aperture migration requires the size of the
first (and second) PFZ as well as the lateral location of the stationary point, both for all
available offsets.
The common-reflection-surface (CRS) stack method (see, e. g., Mann et al., 1999b; Jäger
et al., 2001) provides so-called kinematic wavefield attributes which allow to estimate
several properties required for the estimation of the limited aperture in a second-order
approximation: the size of the first projected Fresnel zone for offset zero and the commonreflection-point (CRP) trajectory describing the dislocation of the stationary point with
varying offset. However, the absolute location of the aperture, the stationary point, has
to be associated with the corresponding depth point in the migrated domain to actually
perform limited-aperture Kirchhoff migration. This information is not directly available
from the CRS attributes. However, it can be partially retrieved from these attributes in an
additional process.
In this context, one strategy to determine the stationary point associated with a given image location in the migrated domain is to directly evaluate the tangency criterion between
the migration operator and the reflection event. For the latter, the dip for offset zero is
available from the CRS wavefield attributes. Therefore, we only have to determine the
migration operator dip for zero offset (ZO) to evaluate this criterion. In the next step, the
stationary point found for offset zero is extrapolated along the CRP trajectory to finite
offsets.
Spinner and Mann (2006) implemented this concept in true-amplitude limited-aperture
time migration. In this case, the required operator dip can be directly calculated from the
analytic migration operator. Because this operator is based on a straight ray approximation the implementation is straightforward. In the scope of depth migration the calculation
of the migration operator is far more complicated. Due to its complex shape an analytic
approximation is no longer appropriate. To follow the tangency-based strategy here, Jäger
(2005a,b) proposed to numerically calculate the migration operator dip by means of a
finite-difference scheme from the Green’s function tables (GFTs) on the fly during migration. A comparison of the results of these different variants of limited-aperture migration
can be found in Kienast (2007). Unfortunately, the dip-based approach in the depth domain is quite sensitive to the smoothness of the GFTs and, thus, to the smoothness of the
underlying macro-velocity model. Therefore, numerical stability is not ensured such that
the correct determination of the stationary point cannot be guaranteed. Furthermore, the
2
1.1 Topic of this thesis
approach suffers from seconder-order approximations in the CRP trajectory and the ZO
PFZ and lacking information on the variation of the PFZ with offset.
1.1
Topic of this thesis
The main issue of this thesis is the more stable and more reliable determination of the
stationary points. Because Kirchhoff migration itself is far less sensitive to the smoothness
of the GFTs than an operator dip calculated with a finite-difference scheme, it stands to
reason to investigate whether this method itself is suited to determine stationary points
in this context. Other features of Kirchhoff migration include that this method generates
only a significant output amplitude if a stationary point exists in the unmigrated domain.
In addition, due to the linearity of the process it is, in general, possible to migrate any
local slowly-varying superimposed information ‘on top’ of the actual input data. Since
the stationary point is directly related to the trace location it stands to reason to use the
latter quantity as a migration weight and thereby receive this location migrated directly to
the corresponding image point in depth.
In this thesis the validity and accuracy of this so-called double diffraction stack approach
for the determination of the stationary points is investigated. In contrast to the dip-based
approach by Jäger (2005a), there is no second order approximation involved here. The
double diffraction stack method is applied to a simple synthetic data set as well as to a real
land data set. To avoid most of the numerical problems typical for the double diffraction
stack approach, I use the envelopes of the analytic signal after the individual diffraction
stacks and prior to the calculation of the stationary points. Other strategies adopted from
the CRS stack method help to identify the reflector images associated with reliable stationary points and to remove unphysical outliers and fluctuations in these stationary point
results.
1.2
Outline of this thesis
After the initial remarks in this introduction, Chapter 2 gives a brief overview of the CRS
stack method which yields the so-called kinematic wavefield attributes. In addition, this
method is briefly compared to the established concepts of stacking velocity analysis and
the processing sequence normal-moveout (NMO) correction/dip-moveout (DMO) correction/stack.
In Chapter 3 the principle of Kirchhoff migration is introduced. The two special aspects of
the latter employed in the scope of this thesis are addressed as well: true-amplitude Kirchhoff migration which reverses the dominant effect of wave propagation on the amplitudes
3
Chapter 1. Introduction
with the goal to obtain amplitudes directly proportional to the reflection coefficient on
the reflector in depth. Secondly, limited-aperture Kirchhoff migration is discussed which
restricts the migration operator to the size of the PFZ to obtain an improved image of the
reflector in depth in a more efficient manner.
In Chapter 4 I will explain the double diffraction stack, i. e., the double accomplishment
of a Kirchhoff migration with two different weights to obtain the lateral position of the
stationary point. In addition, I will show that in the context of this work it makes sense
to use the trace location as the second migration weight for the double diffraction stack.
Furthermore, I apply this approach to a simple synthetic data example. The corresponding
results are discussed to identify the numerical problems of the double diffraction stack.
In Chapter 5 the double diffraction stack is applied to a real land data set. The expected
problems already encountered in the synthetic data example are confirmed and some effective methods to overcome these problems are implemented and discussed. In addition,
an overview of the applied workflow is given and discussed. Finally, the results of the
proposed approach are compared to the results of its dip-based counterpart.
Finally, in Chapter 6 a summary and conclusions are presented.
4
Chapter 2
Common-Reflection-Surface Stack
For many years the sequence of NMO correction/DMO correction/stack has been commonly used as initial process in seismic imaging. In a first step, a stacking velocity section
is obtained by stacking velocity analysis in common-midpoint (CMP) gathers characterized by a symmetric distribution of source and receivers with respect to the CMP. The
stacking velocity section is used twice: first to perform the NMO correction which allows
to simulate a ZO section from the prestack data, i. e., a section with coincident sources and
receivers. Later on, stacking velocity serves to obtain a macro-velocity model suitable for
depth migration.
Although the practical application of this workflow is more complicated and usually requires two velocity analyses and an intermediate inverse NMO correction, the principle
idea can be reduced to two aspects:
• The redundancy of the seismic prestack data: each (unknown) reflection point is illuminated by various different shot/receiver configurations and, thus, with different
reflection angles.
• Model assumptions to reduce the inherent ambiguity of the imaging problem. These
allow to approximate reflection traveltimes in a parametric form.
Mayne (1962) discussed this concept for 1D models, where the processing sequence reduces to NMO/stack (also called CMP stack). Up to second order, the traveltime can
be parameterized in terms of stacking velocities which, in turn, approximate the rootmean-square velocity of a reflector’s overburden. The traveltime approximation allows to
remove the influence of the acquisition geometry to simulate a ZO section. Furthermore,
the inversion scheme by Dix (1955) allows to invert for the interval velocities of such
5
Chapter 2. Common-Reflection-Surface Stack
1D models from the stacking velocities. CMP stack as well as Dix inversion consider
common-reflection points (CRPs) in the subsurface rather than contiguous reflectors1 .
The DMO correction (Hale, 1991) extends this approach to models with lateral velocity
variations. It approximately transforms the prestack data such that all information belonging to a certain CRP is still collected in a CMP gather. Although the DMO correction
already implicitly relies on the existence of contiguous reflectors (see, e. g., Hertweck
et al., 2007), it still sticks to the concept of (independent) CRPs in the subsurface. As a
consequence, the implied contiguity of reflectors is not exploited to full extent. Therefore, it stands to reason to consider entire common-reflection surfaces (CRSs) rather than
CRPs, only.
This is the basic concept of the CRS stack method which represents a generalization of
the conventional stacking velocity analysis and stacking approach. It is based on paraxial ray theory which, in turn, is an approximation to zero-order ray theory. On the one
hand, CRS stack and the conventional method have in common that they employ analytic
second-order approximations of the reflection traveltime. They also share that a coherence
measure is computed along a variety of different stacking surfaces: the stacking operator
fitting best the actual reflection response is associated with the highest coherence. On
the other hand, the methods differ with respect to the kind of stacking operator and the
number of stacking parameters: the CRS stacking operator spatially extends over several
adjacent CMP gathers as it approximates the reflection response of a whole reflector segment. Thus, more traces contribute to the stack and thereby the S/N ratio is enhanced.
Furthermore, the more general model assumptions naturally require a larger number of
stacking parameters to describe the stacking operator, the so-called kinematic wavefield
attributes or CRS attributes.
Initially, the CRS stack was designed for the simulation of 2D zero-offset sections
(Müller, 1998; Müller et al., 1998). Only little time later the CRS stack was extended
to 2D finite offset (Zhang et al., 2001) and to the simulation of 3D zero-offset volumes
(Bergler et al., 2002). Further extensions of the CRS approach allow to handle data with
rough top-surface topography including redatuming purposes (Zhang et al., 2002). Furthermore, this method can be used for multi-component data and arbitrary acquisition
geometries, e. g. vertical seismic profiling data and ocean-bottom-seismics (see, e. g.,
Boelsen and Mann, 2005a,b; von Steht and Goertz, 2007; von Steht, 2008).
The entire set of wavefield attributes contains more information on the prestack data than
the conventional stacking velocity, only. This information can be used in various applications: e. g., Mann et al. (2000) used these attributes to obtain a time-migrated section in
an automated way without an explicit time migration velocity model. Duveneck (2004)
estimated a velocity model for depth migration. Finally, Jäger (2005a) estimated the size
1
Both methods assume a layered 1D model, thus silently implying the contiguity of reflectors and the
coincidence of CMP gathers and CRP gathers.
6
2.1 The CRS stacking operator
of the projected Fresnel zone and the stationary point for ZO as well as the CRP trajectory
in the context of limited-aperture prestack Kirchhoff depth migration.
2.1
The CRS stacking operator
The CRS stacking operator can be described by a second-order approximation of traveltimes along rays in the vicinity of a known central ray. For the zero-offset configuration
the central ray is considered to be normal to the reflector in the normal-incidence point
(NIP). Assuming continuous reflection events over several neighboring midpoint gathers,
the use of an entire stacking surface leads to multiple advantages. Concerning the determination of the stacking parameters a high stability is achieved. Furthermore, stacking
over a whole surface accounts for the problem that an event in a CMP gather may contain
information from more than only one reflection point in the depth which collides with the
assumption of a CRP in the conventional approach. Figure 2.1 shows the CRS stacking
operator for a simple 2D model.
A descriptive representation of the CRS stacking operator can be given in terms of kinematic wavefield attributes. Schleicher et al. (1993) introduced a hyperbolic approximation
for two-point traveltimes. For normal rays, 2D acquisition, and expressed in terms of CRS
attributes, the traveltime approximation reads (Müller, 1998):
"
#
¸2
·
2
2
2
2t
cos
α
(x
−
x
)
h
2
sin
α
0
m
0
(xm − x0 ) +
+
. (2.1)
t2R (xm , h) = t0 +
v0
v0
RN
RNIP
This equation approximates the traveltime tR along a paraxial ray characterized by its midpoint xm and its half-offset h in terms of the traveltime along the normal ray t0 (x0 , h = 0)
and the CRS wavefield attributes α, RNIP , and RN . Furthermore, v0 denotes the nearsurface velocity assumed to be constant within the stacking aperture.
In the following, the practical application of the CRS stack will be discussed. Without the
knowledge of a velocity model, the CRS attributes define the CRS stacking operator. In
order to simulate a zero-offset section, various CRS operators for different sets of CRS
attributes for each zero-offset sample are calculated. By analyzing the coherence of the
prestack data along these different CRS operators the one fitting best an actual reflection
event in the data is selected. The set of optimum attributes parameterizes the stacking
operator yielding the highest value of coherence. As final output, the CRS stack yields
entire sections of kinematic wavefield attributes (α, RN , RNIP ), a coherence section and,
of course, the stacked section. For details on the determination of the wavefield attributes
in the CRS stack implementation used in this thesis see, e. g., Mann et al. (1999a).
Because the kinematic wavefield attributes play an important role in the context of this
thesis, I will give a short geometrical interpretation of them in the following. According to
7
Chapter 2. Common-Reflection-Surface Stack
CRS stacking surface
t
Depth [m]
Time [s]
0.6
0.4
h
0.2
P0
0
X0
x
−200
400
NIP
−400
300
200
−600
−1000
Half−offset [m]
100
−500
0
Midpoint [m]
500
1000
0
Figure 2.1: The green surface indicates the stacking operator for the zero offset sample
P0 . This operator approximates the reflection response of the red reflector segment that
has locally the same curvature as the reflector at the reflection point R of the normal
ray belonging to sample P0 . The blue curves represent reflection traveltime curves for
different fixed offsets. (Figure modified after Jäger, 1999)
8
2.2 The Common-Reflection-Point trajectory
x0
0
α
v0 = 1.5 km/s
Depth [km]
NIP
RNIP
2
v1 = 2.5 km/s
RN
v2 = 3.0 km/s
4
2
4
Distance [km]
6
Figure 2.2: Sketch of a 2D three-layer model that gives a geometrical interpretation of the
kinematic wavefield attributes. Assuming a point source at the NIP, the NIP wave (red)
with the local radius RNIP propagates towards the surface and arises at the emergence
location x0 of the zero offset central ray (blue) with the angle α. The normal wave (green)
with a local radius RN is associated with an exploding reflector experiment around the NIP
and also emerges with angle α at x0 . (Figure modified after Mann, 2002).
Hubral (1983) α, RN , and RNIP can be related to wavefronts of so-called eigenwaves. As
sketched in Figure 2.2 these eigenwaves result from two hypothetical experiments. The
sketch shows a model consisting of three homogeneous layers with the normal wavefront
in green and the NIP wavefront in red. Assuming a point source at the NIP the corresponding NIP wave is obtained whereas the normal wave is obtained by a hyperthetical
exploding reflector experiment. In the vicinity of x0 , both wavefronts are approximated
by circles with radii of curvatures RNIP and RN , respectively. The common emergence
angle of these two wavefronts is denoted with α. The kinematic reflection response of an
arbitrary reflector in depth can be expressed as a superposition of these two experiments.
2.2
The Common-Reflection-Point trajectory
The CRP trajectory describes all points in the prestack time domain that are associated
with the same reflection point in the depth domain. For curved reflectors and a inhomogeneous overburden, Höcht et al. (1999) present a second-order approximation for the 2D
9
Chapter 2. Common-Reflection-Surface Stack
CRP trajectory in terms of CRS parameters. Its projection onto the midpoint-offset plane
reads
Ãs
!
RNIP
h2
xm (h) = x0 + rT
+1−1
with rT =
.
(2.2)
2
rT
2 sin α
which reduces to xm (h) = x0 for α → 0. The corresponding variation of the traveltime
along the CRP trajectory is not shown here because it is not needed in the context of this
thesis.
2.3
The projected Fresnel Zone
Zero-order ray theory is a high-frequency approximation to the elasto-dynamic wave
equation. In practice, we have to deal with band-limited data such that the concept of elastic energy propagating along a raypath of infinitesimal width is not appropriate. Instead,
recorded amplitudes are influenced by an entire volume, the so-called Fresnel volume,
surrounding the considered ray. The first Fresnel volume is defined by all points which
can be connected with a (non-Snell) ray to source and receiver of the considered ray in a
way that the traveltime along this non-Snell ray does not differ by more than the wavelet
length compared to the traveltime along the considered ray. All constructive contributions
can be expected to originate from this first Fresnel volume.
The intersection of the Fresnel volume with a reflector is called interface Fresnel zone.
Two reflecting points located within the first interface Fresnel zone are not distinguishable. The time domain counterpart of the interface Fresnel zone in depth is called projected Fresnel zone. The latter was introduced in 1983 by Hubral for normal rays and is
descriptively obtained by the projection of the borders of the interface Fresnel zone onto
the measurement surface with the considered acquisition geometry. A general approach
for the estimation of the PFZ from traveltimes is given by Schleicher et al. (1997). In
terms of wavefield attributes an approximation of the size of the PFZ for zero-offset is
given by (Vieth, 2001)
v
WF
1 u
u ¯ v0 T
¯.
= |xm − x0 | =
t
2
cos α 2 ¯¯ 1 − 1 ¯¯
RN
RNIP
(2.3)
The three kinematic wavefield attributes α, RN , and RNIP are known from the CRS stack.
v0 is again the near-surface velocity and T is a measure of the wavelet length.
10
2.4 Summary
2.4
Summary
The CRS stack presents a highly automated data-driven imaging process for the simulation of ZO sections. The two most convenient advantages of this method are the achievement of a highly improved signal-to-noise ratio in the stacked section and the yield of the
useful kinematic wavefield attributes that allow a variety of subsequent processing steps,
such as velocity model building, the estimation of the projected ZO Fresnel zone, or the
estimation of the CRP trajectory.
11
Chapter 3
Kirchhoff migration
3.1
True-amplitude Kirchhoff migration
Migration methods are applied to generate interpretable structural images in the time or
depth domain from seismic reflection prestack or poststack data. If the dynamic aspects of
wave propagation are not considered during the migration process, the amplitudes along
the reflector images lack a well-defined physical meaning. Such schemes are termed
kinematic migration. In contrast, the general idea of true-amplitude migration is to explicitely account for these dynamic aspects such that the amplitudes are proportional to
the angle-dependent reflection coefficient. In the following, I will consider a particular
migration scheme, Kirchhoff depth migration, which is based on an integral solution of
the elasto-dynamic wave equation. The by far most dominant effect of wave propagation
on the recorded amplitudes is the geometrical spreading effect. In true-amplitude Kirchhoff migration, this geometrical spreading effect is estimated and removed from the data
such that the amplitudes become the aimed-for measure of the reflection coefficient. This
method finds its application in the determination of physical properties of the reflector.
For example, the amplitude variation with offset or reflection angle can be used to characterize hydrocarbon reservoirs due to the different physical properties of gas, oil, and
water in the pore space.
3.1.1
Basic principles
The following considerations refer to a three-dimensional model where ray theory is applicable. The specific situation of a 2.5D model and the corresponding simplifications are
explained at the end of the chapter.
13
Chapter 3. Kirchhoff migration
I assume a plane measurement surface (z = 0) that is densely covered with sourcereceiver pairs (S, R). The latter can be described by the 2D configuration vector ξ~ =
(ξ1 , ξ2 ) and decisive configuration matrices ΓS and ΓR :
³ ´
³ ´
~
~
~
S ξ = S0 + ΓS ξ~ ,
(3.1a)
³ ´
³ ´
~ ξ~ = R
~ 0 + ΓR ξ~ ,
R
(3.1b)
~0 and R
~ 0 depending on the origin of the ξ~ plane. For commonly
with the constant vectors S
used configurations these matrices are defined as follows: for common-shot configuration
they are given by ΓS = 0 and ΓR = 1. For the common-offset configuration they are
given by ΓS = 1 and ΓR = 1. In both cases 1 and 0 are 2 × 2 unit and zero matrices,
respectively.
For any considered source-receiver configuration, the stationary point ξ~∗ lies within
³ ´ the
~ t =
aperture A in which the position of the seismic trace is characterized by U ξ,
h ³ ´
³ ´ i
³ ´
~ t
U S ξ~ , R ξ~ , t . Furthermore, a point in the time domain with coordinates ξ,
³ ´
~ t and a point in the depth domain with coordinates
can be described by N = N ξ,
(r1 , r2 , z) can be described by M = M (~r, z). If M lies on an actual reflector the latter
equation reshapes to MR = MR (~r, z). In this context reflectors are parameterized in the
form z = z (~r).
Applying zero-order ray theory the analytic particle displacement of a primary reflection
event can be expressed as follows (Červený, 2001):
³ ´
³ ´
~ t = U0 ξ~ F (t − τR ) = RC A F (t − τR ) ,
(3.2)
U ξ,
L
where F (t) = f (t) + iH [f (t)] is the analytic counterpart of the point source wavelet f (t)
and the operator H denotes the Hilbert transformation. τR denotes the reflection traveltime
along the ray path SMR R. RC denotes the complex plane wave reflection coefficient on
the reflector in the depth and L is the geometrical spreading factor. All other influences
on the amplitude of the wave (as, for example, transmission loss, amplitude loss due to
scattering and absorption, or source and receiver coupling effects) are combined in the
factor A.
Following the goal to consider true amplitudes, the entire factor A/L should be removed.
In practice, only the most dominant effect on the amplitudes, the geometrical spreading
factor L, is considered. The factor A is difficult to assess as it contains various different
contributions. Usually, the factor A is assumed to vary slowly within the migration aperture. Accordingly, approximating A by an (unknown) constant within the aperture the
obtained amplitudes are at least proportional to the angle-dependent reflection coefficient
in depth.
14
3.1 True-amplitude Kirchhoff migration
3.1.2
True-amplitude Kirchhoff migration in the depth domain
Hagedoorn (1954) introduced amongst others the so-called Huygens surface, that is the
surface of maximum convexity and better known as the diffraction traveltime surface.
Geometrically one can say that the Huygens surface presents the kinematic image in time
domain
of a point
in depth. For a fixed point M (~r, z) this surface is the set of all points
³
´
~ t = τD for which the traveltime t equals the sum of the traveltime from the source
N ξ,
S³
to ´
the diffraction point M and the traveltime from the diffraction point to the receiver
R ξ~ ,
³
´
h ³ ´
i
h
³ ´i
~ M = τ S ξ~ , M + τ M, R ξ~ .
τD ξ,
(3.3)
The process of true-amplitude Kirchhoff migration can be split into two parts: the dynamic one that ensures correct amplitudes and the kinematic one that ensures correct
placement of reflector images in depth. In the following paragraph I will address only
the kinematic part of the principles of Kirchhoff migration. The dynamic aspects will be
addressed in the next section.
In Figure 3.1 the principle of Kirchhoff migration is illustrated: we assume that the target
zone, the part of the subsurface for which a migrated image is to be constructed, is built
up by a regular grid of diffraction points M . If a macro velocity model is given the
Huygens surface for every point M can then be calculated and the actual migration is
realized by a stack of all amplitudes along each Huygens surface. The resulting value is
assigned to the corresponding depth point M . If a point M is a point of the reflector in
depth the Huygens surface is tangent to the actual reflection traveltime surface. Therefore,
constructive interference leads to a non-zero stacking result. The point of tangency, that is
the point where the Huygens surface is tangent to the actual reflection traveltime surface,
~ If a point M is not located on the reflector in depth
is called the stationary point ξ.
destructive interference ideally leads to a zero stacking result.
According to our aim to express the amplitudes of the migrated reflector image as a measure of the reflection coefficient we have to remove the geometrical spreading from the
time domain data. Therefore, we apply a specific varying true-amplitude weight factor
during the stacking process.
Mathematically, the true-amplitude Kirchhoff migration can be expressed as the integration of the recorded wavefield and reads in 3D (Schleicher et al., 1993):
³ ´¯
ZZ
~ t ¯¯
´
³
∂U
ξ,
1
~
¯
dξ1 dξ2 WDS ξ, M
.
(3.4)
V (M ) = −
¯
2π
∂t
¯
~ )
A
t=τD (ξ,M
The path of integration is given by the Huygens surface τD for the considered target zone
grid point M . V (M ) is the migration output. To correctly account for the phase shift
15
Chapter 3. Kirchhoff migration
t
Huygens curve
Σ
CMP
x
Migration
Target
Zone
z
Figure 3.1: 2D sketch of a Huygens curve for an arbitrary target point in depth. The actual
Kirchhoff migration is realized by a stack of all amplitudes along the Huygens curve. The
output of this computation is assigned to the corresponding point in depth. Only in case
a point M is located on the reflector the corresponding Huygens surface is tangent to
the actual reflection traveltime surface and therefore constructive interference leads to a
non-negligible result. (Figure taken from Kienast, 2007)
arising from supercritical reflections or caustics this value is kept complex. The region of
integration A in the ξ~ plane is called the aperture and should ideally match the whole ξ~
plane. However, that is impossible due to limited acquisition area and time. In addition,
the aperture has to be restricted to avoid operator aliasing, to reduce the summation of
noise, and to save computational costs. Because the determination of the optimum aperture is a difficult and important task,³the ´
proper choice of this value will be addressed in
~ t consists only of one single primary reflection
the next chapter. We assume that U ξ,
event. Due to the linearity
³ ´of the process this does not imply any loss of generality. The
~ t is needed to properly recover the source pulse (Tygel et al.,
time derivative of U ξ,
1996). Finally, WDS indicates the true-amplitude weight function that will be described in
16
3.1 True-amplitude Kirchhoff migration
the following section.
True-amplitude weight function
In this paragraph the dynamic part of Kirchhoff migration is considered and the trueamplitude weight function for the elimination of the influence of geometrical spreading is
discussed.
To find a suitable expression for the true amplitude weight function a Fourier transformation of equation (3.4) has to be accomplished. For that purpose, an artificial time variable
t is introduced:
ZZ
³
´ ³
´
1
~ M U 0 ξ,
~ t + τD .
V (M, t) = −
dξ1 dξ2 WDS ξ,
(3.5)
2π
A
Using the imaging condition t = 0 the migration output for a point M is later given
by V (M, t = 0). Inserting equation (3.2) as ansatz for a reflection event U the time
dependent diffraction stack reads
ZZ
³
´i
ARC 0 h
1
~
dξ1 dξ2 WDS
F t + τdif ξ, M
,
(3.6)
V (M, t) = −
2π
L
A
³
´
~
~
~
where τdif ξ, M = τD ξ, M − τR ξ, M denotes the difference between the diffraction traveltime at M and the reflection traveltime. Again, RC is the reflection coefficient at
the reflector in the depth and L is the geometrical spreading factor. F 0 (t) is the derivative
with respect to time of the analytic source wavelet. As indicated above, the factor A is
considered as constant and will be set to 1 in the following.
³
´
³
´
Performing a Fourier transformation, equation (3.6) reads in the frequency domain
ZZ
³
´
−iω
~ )
~ M RC eiωτdif (ξ,M
V̂ (M, ω) = F̂ (ω)
dξ1 dξ2 WDS ξ,
,
(3.7)
2π
L
A
where V̂ (M, ω) and F̂ (ω) denote the Fourier transformations of V (M, t) and F (t), respectively. Provided that ω only takes high frequencies this equation can be solved by the
Method of Stationary Phase (Bleistein, 1984). Since all considerations so far are based
on the applicability of zero-order ray theory and, thereby, are based on the assumption of
sufficiently high frequencies the use of this approach is justified.
According to the Method of Stationary Phase there are two types of contributions to the
value V̂ (M, ω): one from the stationary point ξ~∗ within the aperture A and another one
17
Chapter 3. Kirchhoff migration
from the aperture boundaries.
Providing that a stationary point exists, the gradient of the
³
´
~
phase function τdif ξ, M with respect to ξ~ at ξ~∗ vanishes and the first type of contribution
constitutes the migrated image. Otherwise, the results will be dominated by the unwanted
contributions from the aperture boundary which will cause artifacts. A suppression of
the latter is achieved by tapering the input data in the vicinity of the migration aperture
boundaries. Subsequently, in the last case the migration output for a point M is negligible.
The expansion of the phase function into a Taylor series reads
τdif
³
´
³
´ 1³
´
³
´T
∗
∗
∗
~
~
~
~
~
~
ξ, M = τdif ξ , M +
ξ − ξ Hdif ξ − ξ
,
2
(3.8)
with the Hessian matrix

Hdif = 
2
∂ τdif
³
~ M
ξ,
∂ξj ∂ξk
´ ¯
¯
¯
¯
¯
¯~
with j, k = 1, 2 .
(3.9)
ξ=ξ~∗
Due to the assumption that Hdif is non-singular, the determinant of the Hessian matrix
does not vanish. Thus, the Method of Stationary phase is applicable and yields
´
³
1
iπ
R
~∗
p C
V̂ (M, ω) ' F̂ (ω)WDS ξ~∗ , M
eiωτdif (ξ ,M )− 2 [1− 2 Sgn(Hdif )] ,
L |det(Hdif )|
(3.10)
where Sgn (Hdif ) is the signature of Hdif , i. e., the number of positive eigenvalues minus
the number of negative eigenvalues. Up to now the true-amplitude weight function WDS is
undefined. Therefore, we can now choose WDS such that the migration output for a depth
point M is proportional to the reflection coefficient RC . With
³
´
p
iπ
1
(3.11)
WDS ξ~∗ , M = L |det(Hdif )| e 2 [1− 2 Sgn(Hdif )]
as true-amplitude weight function the migration output reads
(
~∗
RC F̂ (ω)eiωτdif (ξ ,M ) ∀ ξ~∗ ∈ A ,
V̂ (M, ω) '
0
∀ ξ~∗ ∈
/ A.
(3.12)
Here, V̂ (M, ω) represents the spectrum of the true-amplitude output and τdif accounts
for the traveltime difference between the reflected and the diffracted wave at ξ~∗ . Finally,
an inverse Fourier transformation and the imaging condition t = 0 are applied. The
migration result for points M that have a stationary point ξ~∗ in the region of integration A
then reads
h ³
´i
∗
~
V (M ) ' RC F τdif ξ , M
.
(3.13)
18
3.1 True-amplitude Kirchhoff migration
t
t
τD ( ξ ,M)
τD ( ξ,MR)
τR ( ξ )
τ R ( ξ)
τdif ( ξ
ξ
ξ*
a)
,M)
ξ*
ξ
b)
Figure 3.2: Depicted are two possible stationary situations in which the gradient of τdif
equals zero at a stationary point ξ~∗ . Therefore both traveltime curves have the same slope
at ξ~∗ . a) The diffraction traveltime curve τD results from a point MR on the reflector.
b) The diffraction traveltime curve τD results from a point M which is located below MR .
(Figure taken from Jäger, 2005a)
As shown in Figure 3.2 this output even holds for points M that are dislocated not too
far vertically away from the specular reflection point MR on the reflector. Depending on
the actual size of the vertical distance τdif we obtain the desired migration result V (M ).
However,
M is located too far away from the actual reflector, that means that
´
³ if a point
∗
~
t = τdif ξ , M does not lie in the interval 0 ≤ t ≤ τ² the migration result V (M ) yields
zero due to finite length τ² of the source-wavelet F (t). On the contrary, if a point M is
only located at a small distance from MR the migration output reforms to
V (M ) ' RC F (t) ,
³
(3.14)
´
where t = τdif ξ~∗ , M . In order to rewrite the weight function (3.11) such that it is
independent of the reflector properties, we start with the factorization of the geometrical
spreading factor L along a reflected ray (Tygel et al., 1994):
L=
LSMR LMR R
,
LF
(3.15)
where LSMR and LMR R denote the point-source geometrical spreading factors of the ray
segments SMR and MR R, respectively. LF denotes the Fresnel geometrical spreading
factor that takes the influence of the reflector on the total geometrical spreading into account. It can be expressed as follows:
LF =
iπ
1
cos αMR
1
p
e 2 [1− 2 Sgn(HF )] .
νR
|det (HF )|
(3.16)
Here, αMR is the angle of incidence with respect to the reflector normal at MR and νR is
the medium macro-velocity at MR . The Fresnel matrix HF describes the Fresnel zone in
19
Chapter 3. Kirchhoff migration
the plane tangent to the reflector and is an approximation of the true Fresnel zone at MR
(Hubral et al., 1992). The signature and the determinant of the latter can be expressed
with the help of the Hessian matrix in the following way:
Sgn [det (HF )] = Sgn [det (Hdif )] ,
p
|det (HF )| =
νR hB
1
p
,
2 cos αMR |det (HF )|
³
´
~ M is the Beylkin determinant (Beylkin, 1985a,b) given by
where hB = hB ξ,
¯
³
´ ¯
¯ ~T
¯
~
¯ ∇ τD ξ, M
¯
³
´ ¯
³
´ ¯
~ M = ¯¯ ∂ ∇
~ M ¯¯ ,
~ T τD ξ,
hB ξ,
¯ ∂ξ1
³
´ ¯
¯ ∂ ~T
¯
~
¯ ∂ξ2 ∇ τD ξ, M ¯
(3.17)
(3.18)
(3.19)
~ T symbolizes a row vector composed of the first derivatives with respect to the
where ∇
depth coordinates at MR . The true-amplitude weight function for depth points MR on the
reflector can now be written as follows:
³
´
2
~ MR = hB νR LSM LM R .
(3.20)
WDS ξ,
R
R
2 cos2 αMR
Since this concept is independent of any reflector properties this equation can be extended
for arbitrary depth points. With αM now being the half-angle between the two ray segments SM and M R and νM being the macro-velocity at a point M the final result for the
true-amplitude weight function reads
´
³
2
~ M = hB νM LSM LM R .
(3.21)
WDS ξ,
2 cos2 αM
There are also other expressions for the true-amplitude weight function for Kirchhoff
depth migration available. A comparison of these expressions can be found in Hanitzsch
(1997).
For zero-offset configuration the true-amplitude weight function simplifies due to the coincidence of source and receiver positions to
³
´
cos αS
cos αR
ZO ~
WDS ξ, M = 4
(−1)κS = 4
(−1)κR ,
(3.22)
νS
νR
where αR = αS represents the emergence angle at the surface. κR and κS denote the
KMAH indices along the ray segments SM and M R assuming a point source in the
source or receiver location. In general, the KMAH index counts the quantity of caustics
along the corresponding ray path. The index increases by one when the ray tube shrinks
to a line and increases by two at focusing points.
20
3.1 True-amplitude Kirchhoff migration
3.1.3
2.5 D Kirchhoff migration
The 2.5D Kirchhoff migration considers a depth model with 3D wave propagation and
the exclusive variation of all parameters in the direction of the acquisition line (denoted
as in-line or in-plane). The notation for the direction perpendicular to the acquisition line
in which all parameters remain constant is out-of-plane or cross-line.
Considering an in-line acquisition geometry the ray paths remain within the in-plane direction. Therefore, the following simplifications can be made (Martins et al., 1997):
S (ξ1 ) = [xS (ξ1 ) , yS = 0] ,
R (ξ1 ) = [xR (ξ1 ) , yR = 0] .
(3.23a)
(3.23b)
As equation (3.23) shows, shots and receivers can be described by only one component
of the 2D configuration vector. Furthermore, the 3D point-source geometrical spreading
factor can be simplified by splitting it into an in-plane component described by L̄SR and
an out-of-plane component described by σSR :
√
(3.24)
LSR = L̄SR σSR ,
RR
with σSR = S ν(s)ds where ν represents the velocity and s the arc length along the ray,
respectively. In 2.5D the Beylkin determinant transforms to
µ
¶
1
1
hB (ξ1 , 0, M ) = h¯B (ξ1 , 0, M )
+
,
(3.25)
σS σR
where σS is calculated along the ray branch SM and σR is calculated along the ray branch
M R and h¯B is
à ∂τD
∂τD !¯¯
∂x
∂z
¯
h¯B = det
.
(3.26)
¯
2
2
∂ τD
∂ τD
¯
∂x∂ξ1
∂z∂ξ1
~
ξ=(ξ
1 ,0) ; M =(x,0,z)
Besides, the first and second spatial derivatives of τD with respect to ξ2 yield
¯
¯
∂ 2 τD ¯¯
1
1
∂τD ¯¯
= 0 and
=
+
.
2 ¯
¯
∂ξ2 ξ=(ξ
∂ξ
σ
σ
~
~
S
R
2
,0)
;
M
=(x,0,z)
ξ=(ξ
,0)
;
M
=(x,0,z)
1
1
(3.27)
Due to the model symmetry in 2.5D data there is no variation along the ξ2 -direction.
Therefore, the diffraction stack integral (3.4) reduces to an in-plane stack where the limits
define the aperture. These limits can be chosen in the way that the integral becomes
an infinite strip over the out-of-plane coordinate ξ2 and results in the simplification of
equation (3.4)
Za2
1
V (M ) =
dξ1 IDS (ξ1 , ξ2 , M ) ,
(3.28)
2π
a1
21
Chapter 3. Kirchhoff migration
with the condition a1 ≤ ξ1 ≤ a2 . IDS is the integral over ξ2 and can be expressed in the
following way
Z∞
IDS (ξ1 , ξ2 , M ) = −
−∞
¯
∂U (ξ1 , ξ2 , t) ¯¯
dξ2 WDS (ξ1 , ξ2 , M )
,
¯
∂t
t=τD (ξ1 ,ξ2 ,M )
(3.29)
Analyzing this integral with the Method of Stationary Phase the following expression is
obtained
!− 12
¯
¯
1
∂ 2 τD ¯¯
¯
2
IDS (ξ1 , ξ2 , M ) ' 2π WDS (ξ1 , 0, M )
∂
U
(ξ
,
0,
t)
.
¯
1
t
2 ¯
∂ξ2 (ξ1 ,0,M )
t=τD (ξ1 ,0,M )
(3.30)
1
2
(Bleistein et al., 2001). This
In this case ∂t represents the anti-causal time half-derivative √
filter corresponds to a multiplication of the input data with −iω in the frequency domain. The 2.5D diffraction stack integral reads
Ã
√
1
V (M ) = √
2π
Za2
¯
1
¯
2.5D
dξ1 WDS
(ξ1 , M ) ∂t2 U (ξ1 , 0, t)¯
.
(3.31)
t=τD (ξ1 ,0,M )
a1
The final 2.5D true-amplitude weight function can be obtained from the corresponding
3D weight function using equation (3.27)
W
3.1.4
2.5D
=W
3D
µ
1
1
+
σS σR
¶− 12
.
(3.32)
Summary
In this section I have given a descriptive characterization of Kirchhoff migration that explicitly considers the effect of wave propagation on amplitudes. This process is based on
the concept of the Huygens surface by Hagedoorn (1954) and the tangency of the latter
to a reflection event in time domain. Furthermore, the derivation of the true-amplitude
weight function from the Kirchhoff migration integral has been shown using the Method
of Stationary Phase. Due to the application of the true-amplitude weight function, amplitudes are freed from geometrical spreading. Therefore amplitudes become a direct
measure of the angle-dependent reflection coefficient. In the following section Kirchhoff
migration with a limited migration operator to optimize the migration process is presented
and discussed.
22
3.2 Limited-aperture Kirchhoff migration
3.2
Limited-aperture Kirchhoff migration
The migration aperture has a significant influence on the kinematic as well as on the
dynamic properties of the migration result. Therefore, the proper choice of this migration
aperture is quite important. If the aperture is chosen too small, one has to accept the
loss of relevant frequency content of the reflector and steeply dipping events might not
be depicted in the correct way. If the size of the aperture falls below the size of the
radius of the projected Fresnel zone and/or the aperture does not fully cover the latter the
amplitudes of the migration result become totally meaningless. If the aperture is chosen
too large, all events will be covered. On the other hand, the computational costs highly
increase and furthermore operator aliasing might be introduced. The latter effect will be
discussed separately at the end of this chapter.
As we have just mentioned the migration result explicitly depends on the applied migration aperture. On this account the general idea of the limited aperture migration is the
restriction of the migration aperture to the constructively contributing part of the migration operator.
With the target of enhancing the efficiency of the migration process as well as reducing
migration artifacts caused by the wrong choice of the aperture Jäger (2005a) proposed the
following workflow for an efficient implementation of limited-aperture Kirchhoff depth
migration. The migration output assigned to the depth point M results from the summation of amplitudes along the diffraction traveltime curve τD . To ensure that the actual
stacking only adds up contributions in the area that lies within the part of the reflection signal strip containing the information which is of interest, the optimum summation along
the migration operator should be restricted to the minimum aperture (Schleicher et al.,
1997) with the stationary point in its center, as Figure 3.3 clearly depicts.
This method retrieves various advantages: because of the exclusion of unwanted contributions from the summation the S/N ratio is enhanced and, thereby, the image quality is
strongly improved. The second advantage is traced back to the fact that less summations
are required. Thus, the performance perceptibly increases. A further advantage is that migration artifacts that are caused by the summation of the data outside of the first Fresnel
zone are reduced.
Figure 3.4 shows the difference between the conventional aperture and the limited aperture. The left part of the figure illustrates the conventional aperture. Since the position of
the stationary point, marked with a black cross, is normally unknown before the migration
the user is forced to define an aperture radius without knowing all required quantities for a
proper choice of the latter. Without the knowledge of the stationary point this aperture has
then to be placed around the apex of the migration operator. In contrast, the right part of
Figure 3.4 illustrates the minimum aperture which is centered around the stationary point.
It is clearly visible that in the context of conventional migration a stringent restriction of
23
Chapter 3. Kirchhoff migration
migration operator
t
reflection event
ξ*
v(x,z)
minimum aperture
ξ
x
M
z
(unknown) reflector
Figure 3.3: Sketch of the limited-aperture Kirchhoff migration: the minimum aperture
is centered around the stationary point ξ ∗ . It contains the part of the migration operator
(bold red line) within the reflection signal strip. (Figure taken from Jäger, 2005a)
the aperture as in the scope of limited-aperture migration is not possible, because steeply
dipping reflectors would not be correctly imaged.
3.2.1
CRS-based limited-aperture migration
For the determination of the minimum migration aperture Jäger introduced a new approach that utilizes the kinematic wavefield attributes α, RN , and RNIP already discussed
in Chapter 2. Amongst others, this approach makes use of a mathematical relation (Schleicher et al., 1997) which says that the radius of the projected Fresnel zone matches the
size of the minimum aperture.
Determination of the stationary point
We remember that ξ~∗ is the point where the migration operator is tangent to the reflection
event. Thus, ξ~∗ can be determined by comparing the predetermined slope of a reflection
event in time to the local slope of the migration operator at the same location within the
aperture. In a strict sense, at an actual stationary point these slopes should coincide. In
24
3.2 Limited-aperture Kirchhoff migration
1.5
2
3
4
5
Distance [km]
6
7
8
9
10 2
3
4
5
Distance [km]
6
7
8
9
10
2.0
Time [s]
2.5
3.0
3.5
4.0
Figure 3.4: Sketch of the poststack migration aperture. The stationary point is presented
by the black cross. In the left part of the sketch the conventional aperture centered around
the apex is displayed. The right part of the sketch shows the minimum aperture centered
around the stationary point. The corresponding parts of the operator are highlighted in
white. (Figure taken from Spinner and Mann, 2006)
practice, the location with the smallest misfit of the slopes is considered as stationary
point.
For the determination of the stationary
´ for offset zero, Jäger (2005b) proposed the
³ point
~ t0 of the zero-offset normal ray. This attribute
usage of the emergence angle α = α ξ,
is directly linked to the midpoint coordinate ξ~ of the input trace and to the corresponding
two-way traveltime t0 . Furthermore, α is associated with the local slope 2p of an event
in the stacked zero-offset section. p is well known as the horizontal slowness. Mathematically, the emergence angle α is related to the slowness p as follows: p = 2 sin α/v0 ,
where v0 is the near-surface velocity which has previously been applied during the CRS
stack. The slope of the migration operator has to be calculated from the available traveltime tables. However, the stationary points determined in this way are only meaningful
for actual reflection points M = MR . The coherence section gained during the CRS stack
can be utilized to determine reliable dips associated with actual reflection events, at all
other locations, the conventional aperture should be applied to avoid gaps in the migrated
image.
For finite offset, the CRS stack does not provide a sufficiently accurate slope of the reflection events such that the tangency criterion cannot be applied. However, a second-order
approximation of the CRP trajectory can be expressed in terms of CRS attributes. Its
25
Chapter 3. Kirchhoff migration
projection onto the midpoint-offset plane (equation 2.2) thus relates the already known
stationary point for offset zero to the stationary points for all other offsets.
Determination of the aperture size
As I have already shown in Chapter 2 the radius of the projected first Fresnel zone for
offset zero can be estimated with CRS wavefield attributes according to equation (2.3).
Unfortunately, the CRS attributes do not allow to directly estimate the PFZ size for finite
offset. For models with moderate complexity, the relative variation of the PFZ size with
offset is usually quite small (Spinner, 2007) such that it can be neglected in comparison
to the uncertainties in the ZO PFZ and the estimated stationary point. For more complex
models as usually considered in depth migration, this variation might be significantly
larger such that a sufficient tolerance in this estimate has to be ensured. The current implementation of limited-aperture migration supports a simple user-defined linear widening
of the PFZ size with offset.
Limitations of this approach
Although the just discussed approach yields acceptable results, some significant problems remain. The tangency criterion is only applied for offset zero. In case the above
mentioned criteria for the acceptance of a stationary point are not met for offset zero, the
limited aperture will not apply irrespective of the event’s behavior for other offsets. In
addition, the numerical calculation of the migration operator slopes is very sensitive to
the smoothness of the GFTs as well as of the macro-velocity model (in contrast to the migrated image itself). Finally, the ZO reflection event dip used to determine the stationary
point is not available and/or reliable at all locations.
Concluding, we have to declare that the determination of the stationary point is not sufficiently solved in this approach. Thus, one aim of this work is the improvement of the
determination of the stationary point in a more reliable way via the implementation of the
vector diffraction stack introduced by Tygel et al. (1993). The theoretical background of
this alternative approach will be discussed in the next chapter. However, as I will follow
the strategy of extrapolating the stationary points along the CRP trajectories, the simplistic extrapolation of the PFZ size to finite offset remains which might be too inaccurate for
complex models.
3.2.2
Operator aliasing
Unlike the typical aliasing effects that have their origin in the incorrect acquisition of the
seismic data, operator aliasing is caused during the imaging process itself. Due to the dis26
3.2 Limited-aperture Kirchhoff migration
crete data recording, traces are recorded with a certain discrete trace spacing. So, operator
aliasing occurs if the temporal difference (concerning the actual reflection traveltime in
the data and the time given by the migration operator) for two neighboring traces exceeds
half of the wavelet length. Concrete, that means that the migration operator under-samples
the wavelet and such that artificial frequencies are aliased. The unfavorable consequence
is the appearance of unwanted migration artifacts. The reason for the latter lies in the fact
that even if there is no contribution to the stack from the vicinity of the stationary point
the migration result yields a value different from zero. For a more detailed description I
refer to Baina et al. (2003).
As already mentioned at the beginning of this chapter, one nice advantage of limited
aperture migration is the avoidance of operator aliasing, due to the exclusive summation
of amplitudes in regions where the migration operator is tangent to the actual reflection
event. Thus, techniques to suppress operator aliasing, as for example the implementation
of anti-aliasing filters are avoided.
3.2.3
Summary
In this section I gave a descriptive overview of the limited-aperture Kirchhoff depth migration approach which is based on the use of the CRS wavefield attributes. We have seen
that without the knowledge of stationary points and the corresponding size of the migration apertures limited-aperture migration might not be carried out. Fortunately, these
requirements are manageable: the location of the stationary point has to be known as well
as its variation with offset. In addition, the size of the limited aperture has to be determined which is defined by the radius of the projected first Fresnel zone. All the required
properties were determined by means of the CRS attributes except for the variation of the
aperture size with offset.
By the restriction of the migration operator to the limited aperture (or minimum aperture)
the summation of amplitudes along the migration operator is restricted to the area of
interest. Thus, several advantages increasing the performance of the migration and raising
the quality of its result arise. However, in Section 3.2.1 I have also mentioned that this
method has several limitations. Some of these problems will be addressed in the next
chapter and an alternative approach will be presented.
27
Chapter 4
Vector diffraction stack
The so-called vector diffraction stack was established by Tygel et al. (1993). The general
idea of this method is the multiple application of Kirchhoff migration upon the same data
set using identical stacking surfaces but different weights wj . In principle, one can use
any kind of weight function w during the migration as long as it is smoothly varying with
~ According to the Method of Stationary Phase (Bleistein, 1984) and in case appropriate
ξ.
tapering is applied, exclusively contributions from the vicinity of the stationary point
are received. The notation ‘vector’ stems from the possible simultaneous application of
different weights. Originally, the vector diffraction stack was intended to economize trueamplitude migration. The goal of this work is to show that this method is also able to
support the CRS-based limited-aperture Kirchhoff migration.
4.1
4.1.1
Theory
Double diffraction stack
As the name already reveals the vector diffraction stack method is based on the concept
of the diffraction stack method that has already been discussed in Chapter 3. In the scope
of this thesis I confine myself to the double diffraction stack, i. e., the double accomplishment of a Kirchhoff migration with two different weights. Accordingly, that means
that a diffraction stack is performed twice with two different weights w1 (ξ1 , ξ2 , M ) with
the migration result V1 (M, t) and w2 (ξ1 , ξ2 , M ) with the migration result V2 (M, t), respectively. Substituting wj for the weight function WDS , equation (3.10) shows that the
diffraction stack output is proportional to the implemented weight function at the stationary point:
Vj (MR , t) ∝ wj (ξ1∗ , ξ2∗ , MR ) .
(4.1)
29
Chapter 4. Vector diffraction stack
Here, ξ~∗ denotes the stationary point associated with the ray connecting the source S
at the surface via point MR in depth located on the actual reflector to the receiver R at
the surface. In the following, I assume that there is only one stationary point ξ~∗ in the
time domain associated with a reflection point MR in the depth domain. The situation of
multiple stationary points is discussed in Appendix B.
Calculating the ratio of the two migration results V1 and V2 provides the ratio of the two
applied weights w1 and w2 (Geoltrain and Chovet, 1991). In case of a non-zero diffraction
stack result, the ratio yields:
w2 (ξ1∗ , ξ2∗ , MR )
V2 (MR , t)
,
=
V1 (MR , t)
w1 (ξ1∗ , ξ2∗ , MR )
(4.2)
for w1 (ξ1∗ , ξ2∗ , MR ) 6= 0. However, in case the denominator becomes zero, this ratio is not
defined anymore. This happens if the diffraction stack or the chosen weight-function w1
become zero.
Assuming that the chosen weight functions can be expressed in terms of the stationary
point ξ~∗ which, in turn is expressible as a function of the actual (searched-for) reflection ray SMR R the accomplishment of two diffraction stacks with two different weight
functions leads to the determination of any seismic quantity c (ξ1∗ , ξ2∗ , MR ). The only requirement is the according definition of the weight functions w1 and w2 :
c (ξ1∗ , ξ2∗ , MR ) =
w2 (ξ1∗ , ξ2∗ , MR )
.
w1 (ξ1∗ , ξ2∗ , MR )
(4.3)
In this context one might not forget that for the correct application of the double diffraction
stack two conditions have to be fulfilled. Firstly, as mentioned above, the implemented
weights have to be functions of the stationary point ξ~∗ , encountered along each diffraction
ray SM R, and the depth point M , irrespective whether M is located on an actual reflector
or not. Secondly, as a condition for the application of the Method of Stationary Phase to
the diffraction stack integral the weights may only vary smoothly within the aperture
centered around the stationary point ξ~∗ .
Inserting the imaging condition t = 0 in equation (4.2) and performing the divison delivers a depth section of the seismic parameter
( V (M ,t=0)
2
R
for M = MR ,
V1 (MR ,t=0)
(4.4)
c(M ) =
0
for M 6= MR .
In case a depth point M is located on the reflector but not illuminated within the migration aperture or not located at all on the reflector the value for c(M ) is set to zero. In case
30
4.1 Theory
a point M in depth matches the corresponding reflection point MR the value for c(M )
is assigned to this point M . Hence, c(M ) is only reliable and meaningful along actual
events. The remaining problem now is to find a reliable criterion for the distinction between points M that correspond to actual reflection points MR and points M that do not
correspond to reflection points. In this context it is not sufficient to avoid the division
by zero in equation (4.4). The reason therefore is obvious: the double diffraction stack
would deliver a quite unstable image of the depth section because the ratio of very small
values will remain significant for many depth points M not located in the vicinity of the
actual reflector.
Note that the double diffraction stack method can be applied for any arbitrary
source/receiver offset. However, in the framework of this thesis, it is only applied for
offset zero for two reasons: Firstly, I will use poststack data as input for the double diffraction stack. Due to the high S/N ratio of poststack data compared to the prestack data, the
results can be expected to be much more reliable and more stable. Secondly, the proposed
poststack strategy can be more easily compared to the alternative dip-based approach used
so far.
4.1.2
Double diffraction stack with trace location
As already mentioned above, the double diffraction stack is accomplished by performing
a diffraction stack twice. Practically, the first migration weight is chosen to be a unit
weight and thereby one diffraction stack is performed in its usual, unweighted way. The
main goal of this work is to find a method for the reliable determination of the stationary
point ξ~∗ . Since the latter is characterized by the trace location it stands to reason that the
trace location is used as the second migration weight. According to equation (4.2), the
ratio of these two migration results directly represent the lateral locations of the stationary
points.
In order to meet the mentioned prerequisits, namely the requirement of a ‘smooth’ variation of the chosen seismic parameter, in Appendix A the choice of trace location as weight
function is further investigated. This investigation has shown that the relative variation of
the weight function causes only negligible effects along the reflector images whereas the
impact on the background noise is significant.
4.1.3
Summary
Besides providing a structural image of the subsurface, Kirchhoff migration also allows
the estimation of useful kinematic and dynamic information about the specular ray that
connects source and receivers via the unknown reflecting surface. In order to gain this
31
Chapter 4. Vector diffraction stack
information of interest a diffraction stack has to be performed more than once upon the
same seismic data set with identical stacking surfaces but with different weights. In this
section I have shown that the method provides the possibility to easily and efficiently
determine the lateral location of the stationary point ξ~∗ . Although the process was originally intended to economize the true-amplitude migration, this work will show that it also
presents a CRS-supporting alternative to the so-far used dip-based approach.
4.2
Practical aspects
In this chapter a very simple synthetic data set is investigated with the goal to get an idea
of the area in which the stationary point is valid and of its sensitivity with respect to the
overall noise level.
4.2.1
Description of the data set
The very simple model consists of two horizontal reflectors at depths of 1000 m and
2500 m with the same reflectivity embedded in a homogeneous background model with
a subsurface-velocity of 2000 m/s. The choice of a simple model provides convenient
properties that allow an unbiased examination of the method. Firstly, because of the possible use of analytic operators no error-prone GFTs are required. Therefore, errors in the
latter as well as in the corresponding macro-velocity model cannot occur and numerical
stability is ensured. Secondly, the flat reflectors allow to use small migration apertures
and thus, to exclude the risk of operator aliasing. Thirdly, picking in the depth domain is
not required because amplitudes can be extracted from the migration results along welldefined constant depth levels. Last but not least, the stationary point should just represent
the image location where it is defined due to the 1D nature of the model. In other words,
the relative displacement of these locations is a direct measure of the obtained error.
For this model, I simulated a zero-offset data set with the primary reflection responses
consisting of 351 traces with a sampling interval of 4 ms and a shot spacing of 20 m. As
signal a Ricker wavelet with a peak frequency of 40 Hz was used. Transmission loss has
not been simulated. Furthermore, colored noise of various different levels has been added
to the data. Despite of the identical reflectivity of the two reflectors, the second reflection
event appears far weaker in the data set due to the larger geometrical spreading.
4.2.2
Poststack migration
The double diffraction stack procedure was processed as shown in Section 4.1.1. For
that purpose, each seismic trace has been multiplied with its shot position to obtain the
32
4.2 Practical aspects
weighted input for the stack. Then, a poststack Kirchhoff migration was applied to the
original as well as to the weighted data with a target zone of 5000 m width and 3000 m
depth at a sampling rate of 20 m in x-direction and 5 m in z-direction, respectively. The
aperture linearly varies between 100 m and 500 m with increasing target depth. Finally,
the ratio of the two migration results was computed to recover the stationary point.
Figure 4.1 (top) shows the section with the ratio of the two migration result for a S/N
ratio1 of 40. At a first glance, the location of the stationary point seems to closely follow
the lateral image location and thereby produces a linear trend in the section. This effect is
even clearly visible in the noisy areas in between the two reflector images. The explanation for this phenomenon is quite easy: the used aperture is symmetric with respect to the
considered lateral image location. In the noisy areas, the migration effectively averages
all trace locations within the aperture with random weighting factors. This average scatters around the unweighted average, i. e., the center of the aperture which coincides with
the lateral image location.
Obviously, this way of displaying the results is of little use as the large variation of the
weight function along the section completely obscures any local variations. For this 1D
model, we know that the actual stationary point along the reflector images should coincide with the lateral image location. Thus, a section of the relative lateral displacement
between the calculated stationary points and the corresponding image points in the depth
domain is far better suited to analyze the validity and accuracy of the results: Figure 4.1
(bottom) immediately reveals that the result in the noisy areas is as random as expected,
whereas the result along the reflector images is close to the expected zero displacement.
The difference in the noise levels along the two reflector images is due to the different
geometrical spreading: despite of the identical reflectivity, the S/N ratio of the second
event in the prestack data is lower than for the first event. Although the true-amplitude
migration restores the original reflectivity, it also boosts the noise level. This effect is
weaker for the shallow reflector image.
To be able to evaluate this displacement as a function of the S/N ratio and the position
within the wavelet in a quantitative manner, the average absolute displacement error (over
all traces) along constant depth levels was observed. For the center of the Ricker wavelet,
the average absolute displacement error is semilogarithmically displayed in Figure 4.2
(top) for various noise levels. The black curve corresponds to the first reflector at a depth
of 1000 m whereas the gray curve corresponds to the second reflector at a depth of 2500 m.
The big difference between the error along the first reflector and the error along the second
reflector can be attributed to the above-mentioned difference in the noise levels but also to
the migration aperture which increases with depth. Obviously, the diffraction stack will
gather more background noise within the larger aperture.
For reasonably high S/N ratios, the average error is far smaller than the size of the pro1
In this work S/N ratio refers to the corresponding parameter of the Seismic Un*x utility suaddnoise.
33
Chapter 4. Vector diffraction stack
0
5
0
1
Distance [km]
2
3
4
5
4
3
2
Depth [km]
1
2
1
0
3
Location of stationary point [km]
0
150
0
1
Distance [km]
2
3
4
5
100
1
0
-50
Depth [km]
50
2
-100
-150
3
Displacement of stationary point [m]
Figure 4.1: Synthetic data: top: location of the stationary point given by the ratio of the
double diffraction stack results. Bottom: displacement of stationary point with respect to
the lateral image position.
34
Average absolute displacement error [m]
4.2 Practical aspects
316.23
100.00
31.62
10.00
3.16
20
40
60
80
100
S/N ratio
Time [s]
0.9
1
5
S/N ratio
10
50
100
1.0
1.1
Traces for different noise levels - first reflector
Time [s]
2.4
1
5
S/N ratio
10
50
100
2.5
2.6
Traces for different noise levels - second reflector
Figure 4.2: Synthetic data: top: semilogarithmic display of the average absolute displacement error for first (black) and second (gray) reflector. Representative traces for different
noise levels for the first (middle) and second (lower) reflector.
35
Chapter 4. Vector diffraction stack
Sample # relative to wavelet center
-5
-5
10
Error [m]
100 1000
-5
Error [m]
10 100
-5
Error [m]
10 100
-4
-3
-2
-1
0
0
0
0
1
2
3
4
5
5
5
S/N = 10
5
S/N = 25
S/N = 100
Figure 4.3: Synthetic data: left: migrated Ricker wavelet (without noise). Right: semilogarithmic representation of the average absolute displacement error along the wavelet for
first (black) and second (gray) reflector for three different noise levels. Note the different
horizontal scales. Depth sampling interval is 5 m.
jected Fresnel2 zone which is ≈ 320 m for the first and ≈ 500 m for the second reflector.
Considering the usually quite unrealistic properties of synthetic noise, we can expect an
even higher accuracy dealing with real data. However, even for quite high S/N ratios a
certain background migration noise remains caused by the imaging method itself. This
aspect will be addressed in the real data example presented in Chapter 5.
The lower part of Figure 4.2 shows some representative traces of the unmigrated data for
different noise levels for the first (middle) and the second (bottom) reflector, respectively.
In these plots mainly two aspects can be observed that are triggered by the influence of
the S/N ratio on the signal itself: as expected, the second reflector appears far weaker
than the first one. Furthermore, the signal is clearly visible for S/N ratios exceeding
approximately 5.
So far we have only observed the occurring error at the center of the used wavelet. In the
next step, the average absolute displacement error is analyzed in a similar manner along
the seismic wavelet. Figure 4.3 displays this error as a function of the distance to the
center of the wavelet for three different noise levels together with the original, noise-free
2
36
Calculated for a monochromatic signal of 40 Hz.
4.2 Practical aspects
Ricker wavelet. On the one hand, it is obvious that the wavelet is reasonably sampled.
On the other hand, it is clearly visible that the average absolute displacement error varies
significantly along the wavelet. Especially close to zero-crossings, the error strongly
increases. This confirms the expected numerical problems occurring for small amplitudes
in both migration results used to compute the stationary points. At some samples, the
displacement error reaches up to 1000 m which is unacceptable compared to the size of
the PFZ. Therefore, this problem will be addressed in the real data example below as well.
4.2.3
Summary
In this chapter I have shown that the double diffraction stack is in principle applicable for
the determination of stationary points. However, analyzing the migration results showed
that only the display of well-chosen quantities is useful for the interpretation of the validity and accuracy of the results: the observation of the relative lateral displacement between
the location of each stationary point and the location of the corresponding image point in
depth turned out to be a helpful quantity. For the 1D model considered in this chapter,
this displacement can be treated as a direct indicator for the occurring error during the migration process. From these results, I observed that the ratio of the two diffraction stack
results is unstable and unreliable close to the zero-crossings of the wavelet. In addition,
a certain background noise level is caused by the migration itself. In the next chapter,
dealing with real land data, some revealing methods to overcome these problems will be
shown and discussed.
37
Chapter 5
Real land data example
In this chapter the double diffraction stack is applied to a real land data set. Some revealing methods to overcome the problems already encountered with the synthetic data set are
presented and applied in this chapter.
5.1
Description of data set
The 2D seismic land data set was acquired by an energy resource company in a fixed
split-spread geometry. The seismic line had a total length of about 12 km with a shot and
receiver spacing of 50 m. The utilized source signal was a linear upsweep from 12 Hz to
100 Hz of 10 s duration whereas the temporal sampling rate is 2 ms.
The data set was processed with a standard preprocessing sequence that can for example
be found in Hertweck (2004). Hence, the setup of the data geometry, trace editing, deconvolution, geometrical spreading correction, field static correction, and bandpass filtering
are applied to the data set.
The underlying structure consists of nearly horizontal layers and some dipping layers due
to dip faulting in some parts. This yields a nearly 1D structure which is easy to handle. A
migrated section of the data set is shown in Figure 5.1 (top). Jäger (2005a) revisited these
data in the context of limited-aperture migration.
An entire imaging sequence consisting of CRS stack, NIP wave tomography, and limited
aperture poststack and prestack Kirchhoff migration for data acquired along an adjacent
line can be found in Hertweck et al. (2004).
39
Chapter 5. Real land data example
0
2
4
Distance [km]
6
8
10
10
1
6
4
2
Depth [km]
8
2
3
0
Location of stationary point [km]
Figure 5.1: Real land data: location of the stationary points given by the ratio of the
double diffraction stack results.
5.2
Poststack double diffraction stack
By the application of the double diffraction stack the CRS-stacked section and the GFTs
referred to Hertweck et al. (2004) are used. All migration parameters have been pertained
to make the results comparable: the target grid is sampled with 20 m in lateral direction
and with 4 m in vertical direction. The symmetric aperture has a half-width varying from
60 m to 450 m from top to bottom.
According to the workflow of the double diffraction stack the seismic input data, i. e.,
the CRS stack result, is initially weighted with the shot location of each seismic trace.
Then, two independent Kirchhoff poststack migration processes are performed, one of
the weighted and one of the unweighted input data, respectively. Finally, the ratio of
these two migration results is calculated.
Figure 5.1 shows the ratio of the two migration results which reveals the locations of the
stationary points. As already expected due to the synthetic data results, the direct display
of the locations of the stationary points is quite pointless as it does not provide the required
resolution. The same applies to similar double diffraction stack results as, e. g., presented
by Chen (2004).
With the aim to obtain an interpretable result, again the relative lateral displacement be40
5.2 Poststack double diffraction stack
tween the stationary points and the corresponding image locations in depth is computed.
In contrast to the simple synthetic example where the lateral displacement between the
two locations is exactly zero due to the 1D nature of the model, the displacement will, in
general, not vanish for the real data. Figure 5.2 (bottom) shows this displacement with
an obvious correlation with the dip of the reflector images: the lateral component of the
well-known up-dip movement during migration can directly be observed. In general, this
figure clearly reveals further weaknesses of the double diffraction stack:
• The stationary point gets unacceptably inaccurate at zero-crossings of the wavelet.
• Migration background noise generated by the imaging method itself impairs the
quality of the migration result.
• There are several regions which are obviously meaningless.
In Figure 5.2 (bottom) all of these problems can clearly be observed. The accuracy of the
stationary point determined via the original form of the double diffraction stack, i. e. the
computation of the ratio of the two diffraction stack results, is below reasonable limits.
As a consequence, the double diffraction stack is of no practical use in its original form.
Unfortunately, other publications on this subject like Chen (2004) do not comment on this
problem and its solution. The aim of this thesis is mainly the removal of these instabilities at the zero-crossings of the wavelet and the attenuation of the background migration
noise. Furthermore, a method for the identification of depth locations with meaningful
results has to be implemented. In the following various steps to overcome the mentioned
problems will be discussed.
To overcome the numerical problems occurring at zero-crossings of the wavelet the phase
behavior of the latter has to be outwitted. This can be achieved in a strikingly straightforward way by the calculation of the envelope of the analytic signal for both diffraction
stack results before computing their ratio. Naturally, the envelope is non-zero along the
entire wavelet. The local amplitude properties of the signal are fully preserved, but the
zero-crossings vanish. Therefore, the numerical problems concerning the zero-crossings
of the wavelet should completely disappear. Indeed, this can be readily seen in the upper part of Figure 5.3: the displacement of the stationary point is now stable along the
entire wavelet. This approach is as simple as efficient and finally provides useful input
for limited-aperture migration. The remaining variations are most likely due to migration
noise and are well below the size of the projected first Fresnel zone.
Since the double diffraction stack result is only well-defined along actual reflector images,
the locations associated with meaningful results have to be identified. For that purpose,
the CRS stacking strategy is partly transferred from the time domain to the depth domain.
Although the coherence-based strategy first was designed to detect and parameterize reflection events in the unmigrated prestack time domain up to second order, in the scope
41
Chapter 5. Real land data example
0
2
4
Distance [km]
6
8
10
8
10
Depth [km]
1
2
3
Migrated section
0
400
2
4
Distance [km]
6
300
1
100
0
-100
Depth [km]
200
2
-200
-300
3
-400
Displacement of stationary point [m]
Figure 5.2: Real land data: top: unweighted conventional poststack depth migration. Bottom: relative lateral displacement of stationary point calculated by the double diffraction
stack with respect to the lateral image position.
42
5.2 Poststack double diffraction stack
0
400
2
4
Distance [km]
6
8
10
300
1
100
0
-100
Depth [km]
200
2
-200
3
-300
-400
Displacement of stationary point [m]
0
400
2
4
Distance [km]
6
8
10
300
1
100
0
-100
Depth [km]
200
2
-200
-300
3
-400
Displacement of stationary point [m]
Figure 5.3: Real land data: relative lateral displacement of stationary points with respect
to the lateral image position, now calculated from the envelopes of the migration results
(top) and after event-consistent smoothing (bottom).
43
Chapter 5. Real land data example
of this thesis the same strategy was applied in the migrated poststack domain: the hyperbolic traveltime expansion in the depth is determined at each depth location along which
a coherence analysis in the migration result, the optimum trajectory yields the highest
coherence value. In this way the reflector images are parameterized in terms of their local dip and curvature. For points which are not located on an actual reflector image the
coherence value will be low and allows to distinguish between reflector images and noise.
To properly attenuate the background migration noise it has first to be clarified where it
originates from. Since the high-frequency variation of the double diffraction stack result
is not consistent with ray theory which predicts a smooth variation of the stationary point
locations along the reflector images, I assume the origin of the noise in the migration
itself. Thus, this effect has to be considered as unwanted and the data should be filtered
along the reflector images. One possible strategy to overcome this problem was again
developed in the framework of the CRS stack: Mann and Duveneck (2004) proposed
an event-consistent smoothing strategy to remove similar unphysical fluctuations from
CRS wavefield attributes in the time domain. The approach uses coherence values and
locals dips of the events to select the samples belonging to the same event, i. e., to ensure
event consistency. Furthermore, this strategy applies a combined median and averaging
filter to the selected samples. Fortunately, the required dips and coherence values are
already available from the identification of the reflector images such that the smoothing
algorithm can be directly applied. The result shown in Figure 5.3 (bottom) reveals a strong
attenuation of the migration noise and thereby a clearer migration result is gained.
In Figure 5.4 the same detail of all plots in Figures 5.2 and 5.3 is displayed. For a closer
view on the observed aspects a striking event at an approximate depth of 1.6 km was
chosen (Figure 5.4a). In all plots illustrating the displacement of the stationary point a
coherence measure semblance and a threshold of 0.6 has been used to mask out samples
not located on actual reflector images. Looking at these sections, the above mentioned
problems as well as the effects of the proposed solutions are clearly visible. As Figure 5.4c
shows, by the calculation of the envelope of the two diffraction stack results prior to the
computation of their ratio, the zero-crossing problem is overcome. Furthermore, by the
application of an event-consistent smoothing approach events are recognized and thereby
the migration background noise is attenuated as can be seen in Figure 5.4d.
5.3
Workflow
Since the established workflow is quite complex this section will give an overview of the
implemented steps and their sequence.
Figure 5.5 shows the utilized sequence for the determination of the stationary points and
their reliability. Before the actual migration process starts a linear transformation is ap44
5.3 Workflow
-200
Depth [km]
0
Depth [km]
200
Depth [km]
Depth [km]
4
Distance [km]
5
6
7
1.5
1.7
a)
1.5
1.7
b)
1.5
1.7
c)
1.5
1.7
d)
Figure 5.4: Real land data: a) Detail of the migrated section shown in Figure 5.2. b) Corresponding detail of displacement shown in Figure 5.2. c) Detail of displacement calculated
from the envelopes of the migration results shown in Figure 5.3. d) Detail of displacement calculated from the envelopes of the migration results and after event-consistent
smoothing (Figure 5.3). Values not located on reflector images have been masked out.
plied to the trace locations used as the migration weight (see Appendix A). Then, the
input data is weighted by multiplying each input trace with this transformed trace location. After the double diffraction stack the envelope of the analytic signal is calculated
for both results. This straightforward step is as easy as efficient and prevents potential
numerical problems occurring at zero-crossings of the wavelet. Subsequently, the ratio of
the two results is calculated and, thereby, the stationary points are determined. In addition, in this step the linear transformation of the trace location is reversed. However, it is
not apparent so far which stationary points are reliable and useful. Accordingly, the next
step towards the actual application of these results is the automatic extraction of the latter: the CRS stack is adopted to the depth domain and applied to the poststack migration
result: a hyperbolic stacking operator is determined at each depth location by maximizing
the coherence along this operator in the migrated poststack data. In this way, the reflector images are parameterized in terms of their local dip and curvature. For points which
are not located on an actual reflector image the coherence value will be low and allows
to distinguish between reflector images and noise. Finally, an event-consistent smoothing approach using local dip and curvature proposed by Mann and Duveneck (2004) is
applied to remove the migration noise. Fortunately, the required properties are already
45
Chapter 5. Real land data example
Perform linear transformation of trace location
Weight each input trace with transformed trace location
Calculate envelopes of analytic signal in both results
Calculate ratio of both results
Linear back transformation of the trace location
Apply CRS stack in poststack depth domain
Coherence based selection of meaningful data
Perform event-consistent smoothing
Figure 5.5: Overview of the applied workflow. The steps marked in red have been introduced in the scope of this thesis to stabilize the results and to identify the locations where
the former are reliable.
46
5.4 Comparison with the dip-based strategy
4.5
Distance [km]
5.0
5.5
PFZ width/displacement [m]
-450 -300 -150 0
150 300 450
Depth [km]
0.5
1.0
1.5
2.0
Figure 5.6: Real land data: left: subset of the migration result. The trace marked in green
is used for the comparison of the two approaches. Right: lateral displacement between
the stationary point and the image point in depth obtained with the dip-based approach
(red crosses) and the double diffraction stack (blue line). The size of the projected first
Fresnel zone for zero offset in the dip-based approach (black) and the double diffraction
stack approach (brown).
available from the identification of the reflector images such that the smoothing algorithm
can be directly applied.
5.4
Comparison with the dip-based strategy
Finally, the stationary points obtained through the workflow presented in this thesis are
compared with the corresponding results based on the direct evaluation of the tangency
criterion proposed by Jäger (2005a,b). To obtain easily comparable results the same conventional aperture size (see Section 5.2) was used in both approaches at locations lacking
identified stationary points. For the remaining image points the size of the projected first
Fresnel zone has been calculated according to equation (2.3). If the both methods yield
different stationary points the CRS attributes will evidently be extracted at different locations. Therefore, the estimated sizes of the projected first Fresnel zone will differ as
well.
47
Chapter 5. Real land data example
For that purpose, Figure 5.6 (left) shows a subset of the conventional migration result.
In the right part of the figure the lateral displacement between stationary points and the
image points is displayed along with the size of the projected first Fresnel zone for the
trace in the center of the migrated section shown in Figure 5.6 (left) for both approaches.
This figure allows to relate the fluctuations and potential errors in the displacement, i. e.,
in the relative position of the migration aperture, to the size of the aperture. Apart from
the discrete nature of the dip-based approach the double diffraction stack delivers more
feasible results. The fluctuations received by the application of the latter are well below
the size of the projected first Fresnel zone and follow a consistent trend. In contrast, the
dip-based approach shows stronger fluctuations, especially at smaller depths. In addition,
at some depths as for example at ≈ 1.6 km the fluctuations of the lateral displacement
are totally unreasonable since they account for more than 2/3 of the according size of the
projected Fresnel zone. In this case study, the almost 1D nature of the considered data
set largely tolerates the mentioned errors. Therefore, the limited-aperture results of both
approaches will be almost indistinguishable for these data. However, this might not hold
for more complex situations.
With the chosen apertures in the double diffraction stack not all events are properly imaged: a comparison of Figure 5.2 (top) and Figure 7.9 in Jäger (2005a) clearly reveals
the lack of such steeply dipping events in my results. Accordingly, no meaningful weight
function has been migrated ‘on top’ of these lacking events. Using a sufficiently large
aperture instead, the double diffraction stack also succeeds in capturing these steep events
and in locating the positions of the actual stationary points (not displayed). However, the
application of such large apertures introduces severe operator aliasing at shallow depths
in the double diffraction stack result. This also leads to artifacts in the locations of the
stationary points associated with spurious reflector images caused by operator aliasing.
Although the limited-aperture migration itself is not subject to operator aliasing, it is affected by the stationary points of these spurious events. Thus, it also tends to generate
artifacts due to systematically misplaced apertures at such locations. This problem has
not been further addressed in the framework of this thesis. Of course, one might apply an
anti-aliasing filter during the double diffraction stack as often done in conventional Kirchhoff migration. However, even far simpler approaches might already suffice: reducing the
frequency content of the data prior to the double diffraction stack will reduce the aliasing
effect. The lower frequency content of the result does not hamper the further processing,
as no structural details are contained in the locations of the stationary points. Note that
the limited-aperture migration is still applied to the original, unfiltered data.
48
5.5 Summary
5.5
Summary
In this chapter the double diffraction stack approach proposed by Tygel et al. (1993) was
revisited with the intention to use it as a simple and efficient method to calculate the relation between the stationary point in the unmigrated time domain and the corresponding
image point in the depth-migrated domain. In principle, this allows the correct placement of the the migration aperture in limited-aperture migration and to extract the CRS
wavefield attributes defined at the stationary point.
The direct calculation of the stationary point as ratio of the two diffraction stack results
turned out to be unstable and unreliable especially at zero-crossings of the wavelet. By
the calculation of the envelopes of the analytic signals of the two double diffraction stack
results prior to the computation of their ratio, the occurring numerical problems at the
zero-crossings can be completely avoided. All kinds of applications of the double diffraction stack can be expected to benefit from this extension in a similar way.
Furthermore, I proposed to transform some CRS strategies from the time domain to the
depth domain. This includes the automatic determination of dip and curvature of the reflector images using the linear and hyperbolic zero-offset stacks (see, e. g. Mann, 2002).
As it is well known from its application in the time domain, the CRS method also provides a coherence section. Using the dip and the coherence values, an event-consistent
smoothing approach proposed by Mann and Duveneck (2004) can be transferred to the
depth domain as well to attenuate the migration background noise. Furthermore, the coherence section combined with a suitable coherence threshold can be used to select the
locations with reliable results.
The stationary points determined in this manner have been compared to their counterparts
computed on-the-fly based on GFTs as proposed by Jäger (2005a). The former turned out
to be more stable and plausible.
49
Chapter 6
Conclusions
In this thesis I have studied the process of Kirchhoff depth migration and, in particular,
investigated the determination of the stationary point required for limited-aperture migration. For the purpose of a reliable and stable determination of the stationary point, I have
revisited the double diffraction stack method (Tygel et al., 1993) and analyzed if this approach presents an alternative to the so far used dip-based approach presented by Jäger
(2005a).
In the scope of this thesis an alternative workflow for Kirchhoff depth migration was established, consisting of various different steps that already existed in other contexts. The
double diffraction stack based on the double accomplishment of a Kirchhoff migration to
the same data set but with different weights is used for the actual determination of the
stationary points. Since the latter are characterized by the trace locations this quantity
is used as the second migration weight. The ratio of these two migration results then
presents the stationary points directly at the migrated locations. However, the investigation revealed that numerical problems, especially at the zero-crossings of the wavelet and
in terms of background migration noise, occur via the application of the double diffraction
stack. These problems can be successfully handled applying a multiple-step strategy. By
the calculation of the envelope of the analytic signal prior to the calculation of the ratio
of the two migration results the problems at the zero-crossings vanish. To distinguish
between reliable and unreliable stationary points the CRS stack is adopted to the depth
domain and applied in the poststack migration. A coherence threshold now allows to sort
out unreliable stationary points. Finally, based on dips and coherence values, an eventconsistent smoothing algorithm is applied to the data set to attenuate the background migration noise: unphysical fluctuations and outliers in the locations of the stationary points
are removed. The coherence values additionally allow to identify reliably parameterized
reflector images.
Furthermore, important aspects relevant for the application of this workflow are discussed
51
Chapter 6. Conclusions
in this thesis: the relative variation of the weight function, problems with operator aliasing, the application of this method to prestack data, and the proper choice of the aperture.
The presented results indicate that the stationary points determined in this way can be
successfully used as an alternative to their counterparts computed according to the dipbased strategy proposed by Jäger (2005a).
52
Appendix A
Variation of weight function
One prerequisite for the application of the Method of Stationary Phase to the diffraction
stack integral is the slow variation of the implemented weight. To better quantify ‘slow’
in this context, some possible variations of the weight are analyzed and the effect on the
data will be discussed in this section.
In section 4.1.1, I mentioned that in the scope of this work the trace location is used as
the second migration weight because it characterizes the stationary point. Hence, the
variation of the trace location is analyzed. The stacked section of the considered real land
data has a line length of about 10.7 km. Therefore, using the units given in the data the
trace location varies from values 0 m to 10672 m in increments of 25 m.
Initially, the Seismic Un*x tool suweight has been used to directly weight the stacked
traces with the trace locations. For the closer observation of the effect of the variation
of the weight function, an additional linear transformation of the latter was applied using the same tool: we added an arbitrary non-zero constant to avoid zero weights and
then rescaled the weight by the multiplication with an arbitrary factor f , 0 < |f | < 1.
After performing the actual double diffraction stack the implemented linear transformation was reversed using the Seismic Un*x tool sugain with accordingly chosen linear
coefficients.
The linear transformations were processed in a way that the trace locations, used as the
second weight for the double diffraction stack, were compressed by factors reaching from
1/10 to 1/1000. The attentive observation showed that the relative variation of the used
weight causes only negligible effects along the reflector images whereas the impact on
the background noise is significant. The section displayed at the top of Figure A.1 shows
the conventional unweighted migration result. This plot only serves as a reference for the
localization of the reflector images. In the lower part of Figure A.1 the difference between
the relative lateral displacement of a double diffraction stack with a weight function ranging from 0 m to ≈ 10000 m and a double diffraction stack with a weight function ranging
53
Appendix A. Variation of weight function
from 20000 m to ≈ 20010 m is displayed. This section is displayed twice within different ranges of values to reveal changes in different orders of magnitude. Here, the above
already mentioned cognition is clearly visible: the relative variation of the weight function mainly and quite systematically affects the background noise rather than the reflector
images.
From a theoretical point of view, we expect stronger artifacts for a larger relative variation
of the weight function. These artifacts mainly appear off the reflector images. Therefore,
we attribute the significant differences in the noisy areas to stronger artifacts in the result
with the larger weight variation. The systematic behavior of the difference there might
be caused by some features of the used migration code, probably the order of summation
or a slight asymmetry in the migration aperture: if we change the sign of the weight, the
polarity of the difference changes as well. Concluding, it is advantageous to apply such a
linear transformation to reduce artifacts. The required pair of linear transformations does
not impair the efficiency of the overall process.
54
0
2
4
Distance [km]
6
8
10
Depth [km]
1
2
3
Migrated section
400
300
1
100
0
-100
Depth [km]
200
2
-200
3
-300
-400
Displacement of stationary point [m]
20
15
1
5
0
-5
Depth [km]
10
2
-10
-15
3
-20
Displacement of stationary point [m]
Figure A.1: All sections display the real land data. Top: The migrated section shows
where the reflector images are located and serves as reference. Bottom: The sections
show the difference between the relative lateral displacement of a double diffraction stack
with a weight function ranging from 0 m to ≈10000 m and a double diffraction stack with
a weight function ranging from 20000 m to ≈20010 m. This section is shown with two
different display ranges.
55
Appendix B
Ambiguities in the imaging methods
In seismic reflection imaging methods we usually encounter ambiguities which are not
fully resolved. Let us first consider the unmigrated time domain: several events might
intersect each other at a certain offset and location, so-called conflicting dip situations.
Stacking methods like the CRS stack locally parameterize the reflection events. Sophisticated strategies are required to find a separate stacking operator for each contributing
event (see, e. g. Mann, 2001). Usually, not all contributions can actually be captured such
that the stack results and the attribute sections will be incomplete: in the worst case, only
the most dominant event is parameterized and stacked for the simulation of a sample in the
stack section. The counterpart of these ambiguities in the migrated time or depth domain
corresponds to situations where several stationary points contribute to the same image
location. In the context of minimum-aperture migration, this implies that we theoretically
should consider several, separate apertures, each of them centered around its respective
stationary point. In the following, I will discuss the behavior of the double diffraction
stack method and the dip-based strategy in such situations.
B.1
Double diffraction stack
As described in Section 4.1.1, the basic idea of the double diffraction stack is that we
can migrate the weight function ‘on top’ of the seismic data. This works because we
can rely on the constructive interference in the vicinity of the stationary point. If there
are several stationary points, we will obviously receive contributions from all stationary
points assigned to the image point. There is no way to separate these contributions at the
image point. Instead, we will receive a weighted average of the individual contributions.
For the determination of the stationary point considered in this thesis, this implies that
the result will, in general, be meaningless at such image points with several contributions.
57
Appendix B. Ambiguities in the imaging methods
In the worst case, the apparent stationary point obtained in this way might not be located
within the projected first Fresnel zone centered around any of the actual stationary points.
B.2
Dip-based strategy
In the dip-based strategy discussed in Section 5.4, the migration operator dip is directly
compared to the dip of the reflection events encountered along this operator. In case of
several stationary points, these dips will (virtually) coincide at several locations along the
operator, each of them indicating a different stationary point. Jäger (2005a) presented an
example of such a situation for the real data set also discussed in this thesis. In principle,
each of the contributing stationary points can be determined separately in this approach
by applying similar strategies as for the above-mentioned conflicting dip situations in the
time domain (Mann, 2001, 2002). However, this is a non-trivial task which would require
a set of thresholds and of other, more or less fuzzy criteria to identify such situations.
According to the related experiences in the unmigrated time domain, this might not be
applicable in practice at all. Therefore, the current implementation of the dip-based strategy only evaluates one stationary point per image point, in analogy to the CRS stack
which usually only captures the most dominant event.
In addition to the difficulties to handle the ambiguity due to multiple stationary points, the
dip-based approach might in parallel suffer from unresolved conflicting dip situations in
the CRS stack results: if the dip of a reflection event corresponding to a certain stationary
point has not been detected due to a stronger interfering event, the dip-based strategy
inevitably fails to detect this stationary point.
B.3
Summary
Both compared methods are not able to properly handle ambiguities due to conflicting dip
situations and/or multiple stationary points: whereas the double diffraction stack is not
capable to handle the latter situation due to its basic principles, the dip-based approach is
instead hampered by numerical and practical problems in both situations.
The special case of diffraction events is, however, properly handled by both strategies—
here, the size of the projected first Fresnel zone tends to infinity such that the determined
position of the stationary point does not matter at all.
58
Appendix C
Used Software
This thesis was written using the freely available word processing package TEX (Knuth,
1991), the macro package LATEX (Lamport, 1986), and several extensions. The bibliography was generated with BIBTEX.
The synthetic data set presented in Section 4.2 was produced using Seismic Un*x (Stockwell, 1997, 1999).
The real land data set presented in Chapter 5 was processed using several programs. Most
of them were developed in the seismics working group of the Geophysical Institute of the
University of Karlsruhe. The main programs are:
• CRS stack program: crsstack (Mann, 2003)
• Conventional depth migration program: UNI3D (Jäger and Hertweck, 2003)
• Limited-aperture Kirchhoff depth migration program: MinApMig
• Event-consistent smoothing program: SMOOTH2D
• The remaining processing was accomplished with programs of the Seismic Un*x
package (Stockwell, 1997, 1999).
For the visualization of the data, again Seismic Un*x was used as well as several small
shell scripts and public domain programs.
59
Appendix D
Danksagung
Neben den im Folgenden aufgezählten Personen möchte ich mich vor allem für die Unterstützung des Wave Inversion Technology Consortium bedanken, welches es mir ermöglicht hat, meinen geophysikalischen und kulturellen Horizont auf zwei internationalen
Tagungen der EAGE und SEG zu erweitern.
Prof. Dr. Peter Hubral danke ich für die Übernahme des Referats, obwohl er sich bereits
zu Beginn meiner Diplomarbeit in seinem wohlverdienten Ruhestand befand.
Prof. Dr. Friedemann Wenzel danke ich für die Übernahme des Korreferats.
Besonderer Dank gilt meinem Betreuer Dr. Jürgen Mann. Seine Anregungen, Hilfestellungen und seine Bereitschaft, sich Zeit für meine Fragen zu nehmen, haben diese Arbeit
überhaupt erst ermöglicht.
Desweiteren gilt mein Dank der ehemaligen Arbeitsgruppe von Prof. Hubral für die Unterstützung während meiner Arbeit, obwohl sich zu diesem Zeitpunkt keiner der folgenden Personen mehr am Geophysikalischen Institut in Karlsruhe befand. Im Besonderen
möchte ich mich bei Dr. Miriam Spinner für die Anregung meines Diplomarbeitsthemas
und das gründliche Korrekturlesen meiner Arbeit bedanken. Ihre entscheidenden Hinweise haben vorallem in der Endphase wesentlich zum Gelingen der Arbeit beigetragen.
Ebenfalls gilt mein Dank Dr. Thomas Hertweck, Dr. Tilman Klüver und Dr. Markus
von Steht für die hilfreichen und interessanten Gespräche.
Ein ganz besonderer Dank gilt an dieser Stelle auch meinen engsten Freunden, die immer
ein offenes Ohr, sowohl für studienrelevante als auch für private Probleme hatten und so
zu einer kleinen Familie für mich geworden sind. Im Speziellen möchte ich mich hier
bei meiner wunderbar verrückten Freundin Christina Stawiarski bedanken, ohne die
vorallem die beiden letzten Studienjahre wesentlich ärmer gewesen wären.
Nicht zuletzt möchte ich ganz herzlich meinen Eltern und Großeltern für die permanente
Unterstützung während meines Studiums danken.
61
List of Figures
Chapter 2 – Common-Reflection-Surface Stack
5
2.1
Description of the CRS stacking operator . . . . . . . . . . . . . . . . .
8
2.2
Geometrical interpretation in 2D of the kinematic wavefield attributes α,
RN and RNIP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
Chapter 3 – Kirchhoff migration
13
3.1
An illustration of a Huygens surface, along which amplitudes are stacked.
16
3.2
Stationary point of the phase function τdif . . . . . . . . . . . . . . . . .
19
3.3
Descriptive sketch to depict the stationary point and the minimum aperture 24
3.4
Descriptive difference between the conventional and the limited aperture .
Chapter 4 – Vector diffraction stack
4.1
4.2
4.3
29
Synthetic data: Location and relative lateral displacement of the stationary point, calculated by the double diffraction stack. . . . . . . . . . . . .
34
Synthetic data: observation of the average absolute displacement error on
the middle of the wavelet. . . . . . . . . . . . . . . . . . . . . . . . . . .
35
Synthetic data: observation of the absolute average displacement error
along the wavelet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
Chapter 5 – Real land data example
5.1
25
39
Real land data: location of stationary points given by the double diffraction stack results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
40
List of Figures
5.2
Real land data: unweighted poststack migration and displacement of stationary point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
Real land data: displacement of stationary points calculated by double
diffraction stack with envelopes (top) and after final processing (bottom).
43
5.4
Real land data: details of Figure 5.2 and Figure 5.3 . . . . . . . . . . . .
45
5.5
Overview of the applied workflow . . . . . . . . . . . . . . . . . . . . .
46
5.6
Real land data: comparison of double diffraction stack and dip-based approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
5.3
Chapter A – Appendix
53
A.1 Observation of the linear transformation of the implemented weight function 55
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