Common Angle Gathers Kirchof Migration

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CHINESE JOURNAL OF GEOPHYSICS Vol.53, No.3, 2010, pp: 430∼439

COMMON-ANGLE GATHERS BASED ON KIRCHHOFF

PRE-STACK TIME MIGRATION

ZOU Zhen1,2, LIU Hong1, LIU Hong-Wei1,2

1 Key Laboratory of Petroleum Resources Research, Institute of Geology and Geophysics,

Chinese Academy of Sciences, Beijing 100029, China

2 Graduate University of Chinese Academy of Sciences, Beijing 100049, China

Abstract In this paper we present the improved algorithm for common-angle gathers based on Kirchhoff

pre-stack time migration. The algorithm can calculate relatively true incident angle under three-dimensional

circumstance, while the traditional ray-parameter method could not gain the incident angle between incident ray

and dipping reflector normal. Compared with straight-ray Kirchhoff angle domain gathers method proposed in

recent years, our method is able to extract much wider angle for taking interval velocity and ray bending into

account. Moreover, we apply non-hyperbolic moveout equation to calculate the travel time, thus the common-

angle gathers are flatter at large angle than straight ray method. Efficient true-amplitude weighting function

as well as hit-count correction is beneficial for AVA inversion. The results of synthetic model data have verified

that, compared with angle gathers transformed from offset to angle domain with CMP gathers or CRP gathers

after migration, common-angle gathers based on Kirchhoff pre-stack time migration have following advantages:

much more accurate incident angle, the amplitude versus angle (AVA) is robust and clear. The real 3-D data test

shows that angle gathers provided with our method are beneficial for AVA analysis and AVA inversion, and could

improve pre-stack inversion’s resolution.

Key words Kirchhoff PSTM, Common-angle gathers (CAG), Angle-domain common image gathers (ADCIG)

1 INTRODUCTION

The technique of amplitude versus offset (AVO) is widely used in reservoir inversion and petrophysi-

cal prediction, and it efficiently enhances the accuracy of prediction. The theory of AVO technique is based

on amplitude versus angle (AVA), the two are equivalent only when the subsurface reflectors are horizontal [1].

Transformation from offsets to incident/reflection angles is the key step of AVO. Common-angle gathers (CAG),

known to domestic oil-industry, are obtained from mapping Common middle point gathers (CMP) after NMO

correction or Common imaging gathers (CIG)/Common reflection point gathers (CRP) after pre-stack migra-

tion. It seldom uses the ray tracing method to map from offset domain to angle domain, owing to its high

requirement of velocity accuracy. The Walden approximation[2] is usually used for the transformation. The

formula is based on horizontal layered medium assumption, thus it couldn’t gain the incident angle between the

incident ray and normal direction of inclined reflector. Analyzing the AVO effect of CMP gathers after NMO,

DMO and CRP gathers after pre-stack time migration, Resnick et al.[3] proposed to use Kirchhoff migration in

conjunction with AVO analysis. Mosher et al.[4] derived the ray parameter p gathers during plane wave pre-stack time migration, and compared them with CMP gathers. It showed the value of pre-stack time migration

in improving stratigraphy and AVO resolution.

In recent years, angle domain common imaging gathers (ADCIG) for velocity and AVA analysis aroused

academic attention widely[5∼10]. Compared to offset domain common imaging gathers (ODCIG), ADCIG could

reduce artifacts considerably. These studies are most based on pre-stack depth migration (PSDM). Other than

the ones got from surface offset approximation during Kirchhoff PSTM, the incident angles got during PSDM

are relatively true angles of subsurface reflector when the interval velocity is accurate. However, building

velocity model of PSDM is challenging and entails high calculation cost, so the ADCIG based on PSDM,

E-mail: [email protected]



ZOU Z et al.: Common-Angle Gathers Based on Kirchhoff Pre-Stack Time Migration 431

especially wave-equation PSDM, has not been used for 3D AVA inversion. Kirchhoff integration pre-stack

time migration with good adaptability to observing system, costing low calculation, dominates in practical

application. Thus common-angle gathers extracted during Kirchhoff PSTM are applicable to AVA analysis.

Zheng[11] and Perez[12] proposed an angle-domain imaging approach of Kirchhoff PSTM, which is based on

straight ray approximation, then applied it to high resolution imaging and fracture analysis (AVAZ). Cheng [3]

proposed the algorithm of table-driven angle-domain imaging approach of Kirchhoff PSTM, with the travel time

calculated using ray tracing method, and non-hyperbolic moveout method[14]. For taking ray bending effect and

the equivalent horizontal isotropy into account, the relatively accurate value of incident angles and travel time

could be calculated. Wang[15] analyzed the ray parameter p gathers acquired from plane wave decomposition

pre-stack time migration, and applied them to 2D AVO analysis. But extending it to 3D cases is difficult.

In this paper, we firstly analyze the deficiencies of the traditional conversion algorithm from offset domain

to angle domain, and then propose a new method to extract common-angle gathers during 3D Kirchhoff PSTM.

The travel time is calculated with the optimized six-order NMO equation [16] that adapts to far offset (large

incidence angle), or the non-symmetric travel-time algorithm[17,18] which has better focus in linear lateral

velocity variation medium. For the purpose of “true-amplitude”, we adopt the relatively amplitude-preserving

weighting factor and hit-count correction technique. The theoretical model and real data tests show thatcommon-angle gathers during Kirchhoff pre-stack time migration could obtain more accurate mapping between

offset and incident angle, the amplitude is relatively well preserved, and this kind of CAG is suitable for AVA

analysis and inversion.

2 CMP, CRP AND CAG

In contemporary oil industry, Formula (1)[2] is usually used to convert CMP, CRP/CIG gathers to common-

angle gathers

α = sin−1vintx

v2Rmst

, (1)

where α stands for the angle between incident ray and time axis (Fig. 1), vint for interval velocity, vRms is RMS

velocity, t is two-way travel time, x is offset. This formula is valid only in the 2D horizontal layered medium.

In Fig. 1, when the subsurface reflector is horizontal, common depth point M is the reflecting point. While the

formation is tilted, the reflecting point is located at point N . However, we will range the reflecting information

of point N to point M when sorting CMP gathers. The incident angle β at point N is different from the one α

of horizontal layer. With dip angle value increasing, deviation between them enlarges. Moreover, the amplitude

of point M will be contaminated. Muerdter et al.[19] numerically analyzed the relationship between offsets and

reflecting angles when dip angle varies, and showed that the effect of slope strata must be considered in AVA

analysis. In addition to the angle factors, lateral discontinuity of fault and etc bring fault plane reflection and

diffraction wave, which will interfere with the reflected wave of target object in CMP gathers. AVO/AVA

analysis based on the horizontal cases is not credible.

Thus, the AVA analysis based on pre-stack migration

becomes necessary. Pre-stack migration could migrate

the tilted reflectors to their real locations, converge the

diffraction wave, and improve the lateral resolution as

well as signal to noise ratio. In one word, pre-stack mi-

gration provides relatively good preserved-amplitude

data to attribute predication and inversion[20∼23].

Conversion from CRP gathers to angle gathers

has following problems: when the subsurface structures

are complicated, ray shadow area appears, where CRP

gathers exhibit kinematic artifacts[7]; as CRP gathers

Fig. 1 Contrast between CMP, CRP

and CAG gathers



432 Chinese J. Geophys. Vol.53, No.3

record the coordinate of neither shots nor receivers and ignore azimuth factor, the incident angle according to

Walden formula will far deviate from its true value when dip angle of tilted reflector is large enough; the improper

stack times of near, middle and far offsets and sorting interval of CRP gathers will result in amplitude anomalies

and lead AVA to misunderstanding. Angle domain pre-stack migration algorithm recording incident/reflecting

angle information simultaneously during migration procedure, not only retains the advantages of CRP gathers

but also avoids the dependency on horizontal layered medium assumption for converting CRP gathers to CAG

gathers. The appropriate amplitude weighting factor and hit-count correction technique enable angle-domain

imaging gathers suitable for AVA inversion. Besides, because the stretch factor of angle gathers is time-invariant,

it is very convenient to remove the stretch effect of large-offset traces, which affects resolution of imaging and

AVA inversion[24].

3 CAG PRODUCED BY KIRCHHOFF PSTM

Common-angle gathers extracted by pre-stack time migration take incident/reflection angle as sorting

header words instead of offsets. Different offsets of seismic traces are mapped into different angles. Fig. 2a

shows the ray path of straight ray PSTM. Travel time is calculated using DSR hyperbolic moveout equation(Formula (2)):

t = ts + tg =

ti

2

2

+r2si

v2Rms

+

ti

2

2

+r2ig

v2Rms

, (2)

rsi =

(xi − xs)2 + (yi − ys)2

rig =

(xi − xg)2 + (yi − yg)2, (3)

where ti is the two way travel-time of zero offset, rsi, rig is horizontal distance from shot and geophone points to

the imaging points respectively. The coordinate of shot point S is (xs, ys, 0), (xg, yg, 0) stands for the geophone

point R, (xi, yi, ti) is the coordinate of imaging point I .

Fig. 2 Ray path and incident angle of (a) straight-ray and (b) bending-ray

Incident angle β , half of the angle between incident ray SI and reflected ray IR, could be calculated

by cosine law[12]. This algorithm overcomes the dependence of Formula (1) on the horizontal layered medium

assumption and could obtain the incident angle between incident ray and reflector normal. However, the

straight ray angle-domain PSTM method has the following shortcomings: the hyperbolic travel time (Formula

(2)) retains only 2-order term of offset r, as r increases, the deviation between travel time t and the true one

rises, as a result the events of far offset/angle could not be flattened. As the ray bending effect is not taken into

account, the incident angles calculated by straight ray PSTM have poor illumination on target stratum, and




the loss of illumination badly affects the imaging of inclined reflectors with large dip angle and AVA inversion.

Large offset acquisition is widely used in oil-industry. The largest offset of land acquisition system reaches

5000∼6000 meter, or even longer. The straight ray method couldn’t meet the need of travel-time and incident-

angle accuracy. In order to improve the accuracy of AVO inversion, Ross[25] and Bale et al.[26] proposed to

combine with AVO and NMO correction using non-hyperbolic moveout equation. We combine and extend

their ideas to Kirchhoff pre-stack time migration, using non-hyperbolic moveout equation, such as optimized

6-order NMO equation and the asymmetric travel-time algorithm that adapts linear lateral velocity variation,

to calculate travel-time. According to ray parameter method and taking the high order terms of offset into

account, we obtain higher precision at large angle.

We take optimized 6-order NMO equation for example to indicate the principle of extracting CAG during

non-hyperbolic (bending ray) Kirchhoff PSTM, Formula (4) is double square root (DSR) optimized 6-order

NMO travel-time equation

t = ts + tg ≈ t3s + ccc4r

6si

t3s+ t3g + cc

c4r6ig

t3g, (4)

where t3s = c1 + c2r2

si+ 4c3r4

si, t3g = c1 + c2r

2

ig+ 4c3r4

ig, c1, c2, c3 and c4 are equal to the first four coeffi-

cients proposed by Tanner[27,28], cc is a constant, ts is the travel time from shot to imaging point and tr is

the time from geophone to imaging point. According to Snell theorem namely the horizontal ray parameter p

invariance principle, the derivations of ts and tr along r direction is given by following expressions

dtsdrsi

=1

t3s(c2rsi + 8c3r

3si + 6cc ∗ c4r5si) −

cc∗c4r6si(c2rsi + 8c3r

3si)

t33s,

dtgdrig

=1

t3g(c2rig + 8c3r

3ig + 6cc ∗ c4r5gi) −

cc∗c4r6ig(c2rig + 8c3r

3ig)

t33g, (5)

α1 =sin−1vint

dtsdrsi

,

α2 =sin−1

vint

dtg

drig. (6)

When computing α1, α2, we retain the terms up to 5-order, with the aim of saving computation cost. This action

has little effect on the accuracy of incident angle measured in degrees. In this article, both the synthetic model

data and real data adopt this truncation. Alternative superposition between angle intervals is implemented in

order to improve the signal to noise ratio. As shown in Fig. 2b, the angle α1 between ray MI and time axis t0

and angle α2 between ray NI and timeline are calculated with Eq.(6). The incident angle β could be given by

vector space law[29]

β =1

2cos−1

−−→MI · −→NI

|−−→MI ||−→NI |

, (7)

where · stands for vector dot product, the unit vector of the angle between incident/reflection ray and time axis

could be denoted as Formula (8), module of the vector equals to 1:

−−→MI =

sinα1

xs − xi

rsi, sinα1

ys − yi

rsi,

cosα1

,

−→NI =

sinα2

xg − xi

rig, sinα2

yg − yi

rig,

cosα2

. (8)

We adopt following weight factor, that is adaptive to ray bending effect, for amplitude-preserved Kirchhoff

pre-stack time migration[30∼32]

w =

√ cosαs0 cosαr0

v0

√ tstr

ts

tr+

tr

ts

1

tr+

1

ts

. (9)

Where coss0, cosr0 represents the angle between incident/emergent ray and time axis on the surface respectively,

v0 demonstrates the surface velocity. The uneven illumination of subsurface targets caused by the uneven




surface acquisition is the main problem for 3D AVA. Data regularization before migration could only reduce

the uniformity of surface cover, and the hit-count correction in angle domain should be applied to weaken the

AVA illusion[33]. The idea is to record the number of effective hit-count at imaging point meanwhile calculating

incident angle, and then get the corrected angle gathers through regularization division in angle domain[8].

4 MODEL TEST

In order to test the correctness of the algorithm proposed in this article, we design a 2D model, as shown

in Fig. 3, the velocity model consists of three reflectors, the P-wave velocity of the four layers are 3000 m/s,

3500m/s, 4500 m/s and 5000 m/s, the maximum depth is 6000 m, there are 1700 CMP in the horizontal x

axis with spacing 10 m. The synthetic wave field is simulated by acoustic wave equation with high-order finite

difference method. The maximum offset is 6000 m, recording length lasts for 4 s, sampling is 4 ms, shot interval

40 m and group interval 20 m, unilateral spreading, one side shooting.

Fig. 3 Velocity profile of depression model

Common-angle gathers, at the point A (x=10000

m) as shown in Fig. 3, are produced with straight ray

and bending ray Kirchhoff PSTM. We also transform

the CMP and CRP gathers on point A to common-

angle gathers using Hampson-Russell (HRS) software,

which calculates the incident angles with Formula (1).

All results are shown in Fig. 4. As illustrated in Fig. 5,

normalized amplitude peak of the common-angle gath-

ers transformaed from the CMP, CRP gathers and pro-

duced by our method, are compared with the theoret-

ical reflection coefficient calculated using Aki formula,

with the aim of inspecting their respective AVA effect.

Fig. 4 CAG at location A in depression model extracted by different algorithms

(a) CAG transformed with CMP gathers; (b) CAG transformed with Kirchhoff PSTM CRP gathers; (c) CAG extracted

with straight-ray Kirchhoff PSTM; (d) CAG extracted with non-hyperbolic travel time Kirchhoff PSTM.




In Fig. 4a, common-angle gathers transformed from CMP has low signal to noise ratio, and the diffracted

wave produced by the second inclined reflector couldn’t be migrated. Comparing AVA curves with theoretical

reflection coefficient, the AVA curve of first reflector is in good agreement with the theoretical curve (Fig. 5a), as

the diffracted wave produced by second reflector contaminates the reflected wave of second and third reflector,

the amplitude curves of this two reflectors deviate from the theoretical value greatly (Fig. 5(b,c)); Fig. 4b

demonstrates the common-angle gathers transformed from migration CRP gathers. The reflection wave is

migrated to its correct location and the diffraction is convergent. The amplitude curves of the first and third

flat layer match relatively well to the theoretical values, while the second inclined reflector doesn’t match well

with the theoretical curve, as Formula (1) ignores the case of inclined layer. Fig. 4c shows CAG produced by

the straight ray Kirchhoff PSTM. Fig. 4d illustrates CAG extracted by our method, which has the advantages

of CRP gathers and overcomes the defects of CRP gathers. All the curves of the first and third horizontal

reflector, and the second slope reflector are consistent with theoretical curves (Fig. 5). Compared with the

straight ray PSTM, our method enlarges the range of incident angle. The incident angle of middle and deep

reflector increases by about 15 degree. Retaining large angle information is quite important for target imaging

and AVA inversion. According to the numerical results, common-angle gathers produced by our method, with

more accurate incident angle, travel-time and relatively well preserved amplitude, are suitable to pre-stack AVAinversion.

Fig. 5 Contrast between amplitude variation with angle, extracted with CMP, CRP,

our PSTM-CAG method, and theoretical reflection coefficient curve using Aki formula

(a) The first reflector; (b) The second reflector; (c) The third reflector; Aki: the theoretical reflection coefficient curve;

CMP-CAG, CRP-CAG: Common-angle gather transformation based on CMP, CRP gathers; PSTM CAG:

CAG gathers extracted by our PSTM mehod.

5 REAL DATA

We applied pre-stack AVA inversion to one certain work area in western China, where the targets belong

to Ordovician carbonate reservoirs, located at 5000∼5700 m. Owing to strong surface absorption, seismic data

has low S/N and the reflected wave of carbonate reservoirs is messy. And these features bring difficulties to

seismic imaging and inversion. Because of the low S/N ratio, CMP gathers are not suitable for AVO inversion in

this area. As Kirchhoff PSTM is able to get good imaging of fractured-cave reservoir, CRP gathers are usually




transformed to CAG as basic data for AVA inversion. Fig. 6 shows CAG produced by straight ray PSTM,

Fig. 7 shows CAG transformed from CRP with HRS software, while CRP are produced by the bending-ray

PSTM. Fig. 8 shows CAG produced by our method. Comparing these three figures, CAG produced by straight

ray PSTM have small range of incident angle, with the maximum angle of target objects about 30 degree;

the others could reach 42 degree, that means they meet the need of two terms AVA analysis. We compare

AVA curves of these CAGs near the well around 3452 ms. In order to compare, the amplitude peak value

around this time is normalized. Fig. 9 shows the amplitude of CAG converted from CRP fluctuates obviously,

and CAG gained by straight ray PSTM has narrower range; while the amplitude of CAG, obtained with our

method, varies gently with incident angles and could meet the requirement of inversion. Furthermore, there are

amplitude anomalies at the maximum/minimum angle transformed from CRP, therefore in practice terms, we

should limit maximum and minimum angle used for inversion.

Fig. 6 Section of common-angle gather extracted by Straight ray PSTM

Fig. 7 Section of common-angle gather (transformed CRP with HRS Software)

Fig. 8 Section of common-angle gather extracted by bending-ray Kirchhoff PSTM




Combining the common-angle gathers pro-

duced by our method with the well logging data of

this work area, we carry out pre-stack simultaneousinversion. Fig. 10 shows the profile of transverse wave

impedance with the well, horizontal and vertical dis-

tributions of the “string bead” reservoir are well de-

scribed. The results demonstrate CAG produced by

this algorithm are much more suitable for AVA pre-

stack inversion. Fig. 9 AVA curve contrast between CAG transformed

based on CRP, by straight ray PSTM and

bending ray PSTM at 3452 ms

Fig. 10 Section of shear wave impedance inversion with CAG extracted by our method

Tz72 is well number, the color scale stands for the value of shear wave impedance in unit of (m ·s−1)·(g·cm−3).

6 CONCLUSION

In this paper, we propose a new method for extracting common-angle gathers during non-hyperbolic travel-

time Kirchhoff pre-stack time migration. Different from traditional CAG converted from offset gathers, it has

following advantages: more precise angle value, larger angle range and relatively well preserved amplitude. On

the other hand, computation cost increases limitedly. So the kind of CAG is suitable for pre-stack AVA analysis

and inversion. The hit-count correction technique is available only in simple geological situation. Once thegeological structure is complex, there will be error in coverage statistics of reflecting point and it will bring

artificial error to amplitude. Thus it is necessary to take further study on the topic how to reduce the uneven

illumination and coverage of imaging point in angle domain as to improve the accuracy of 3D AVA.

Amplitude preserving includes seismic preserved processing and preserved migration, both are indispens-

able. If amplitude anomaly appears during processing, it will affect the following amplitude-preserved migration.

When amplitude anomaly shows up either in processing or migration, it will make the AVA analysis and inver-

sion inaccurate. So quantification of preserved processing and interpretation is urgent for the lithologic reservoir

description.

PSTM is applicable to the medium with smooth lateral velocity variation, when the geological structure

is complex with intense lateral velocity variation, the common-angle gather produced by Kirchhoff PSTM will




not be fit for AVA analysis anymore. Using common-angle gathers produced by true amplitude pre-stack depth

migration is the development direction of AVA inversion.

ACKNOWLEDGMENTS

This research was supported by the Project of National 863 Plan of China (2006AA09A102-08), and the

Project of National 973 Plan of China (2007CB209603). We thanks Zhang Bing (Geophysical Technology

Research Institute, Sinopec Nanjing) for providing the modeling data, and CGG for its “University donation

scheme” providing the HRS software.

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Common Angle Gathers Kirchof Migration