seismic depth migration using non-uniform gabor frames · 2020-01-17 · seismic migration by phase...

EAGE 69th Conference & Exhibition — London, UK, 11 - 14 June 2007

Seismic Depth Migration using Non-uniform Gabor Frames

Yongwang Ma and Gary F. Margrave CREWES, University of Calgary

Summary We present a seismic depth migration algorithm which uses a discrete Gabor transform over the lateral spatial coordinates and a discrete Fourier transform over time. These transforms enable a wavefield depth extrapolation by laterally variable, frequency and wavenumber dependent, phase shift. The discrete Gabor transform is implemented as a windowed discrete Fourier transform where the windows are constrained to form a partition of unity (POU), meaning that they sum to one. For efficiency, an adaptive partitioning scheme that relates window width to the lateral velocity variation is developed, and defines a non-uniform (or adaptive) Gabor frame. Within each window, the Gabor method uses the familiar split-step Fourier technique. The construction of the adaptive partition of unity is guided by an accuracy threshold that constrains the spatial positioning error, for each depth step. The spatial positioning error is estimated by comparing the Gabor method to a nonstationary phase shift that changes according to the local velocity at each position. We present the details of construction of the adaptive POU for both 2D and 3D imaging. The performance of Gabor depth imaging using this partitioning algorithm is illustrated with imaging results from prestack depth migration of the Marmousi dataset.


INTRODUCTION Seismic migration by phase shift (Gazdag, 1978) (also referred to as wavefield extrapolation with phase shift) was proposed as an accurate and efficient method, that is unconditionally stable, provided that there are no lateral velocity variations. An approximate extension to lateral velocity variations, called PSPI (phase shift plus interpolation) was presented by Gazdag and Sguazzero (1984), who proposed spatial interpolation between a set of constant velocity reference extrapolated wavefields. This method has proven popular and, although it is no longer unconditionally stable (e.g. Etgen 1994), it is more stable than a typical explicit space-frequency method (Margrave et al., 2006). Other ways of extending the concept of a phase shift include the split step Fourier method (Stoffa et al., 1990), the various phase screen methods (e.g. Wu and Huang, 1992; Roberts et al., 1997; Rousseau and de Hoop, 2001; Jin et al., 2002), the generalized phase shift plus interpolation (GPSPI) and nonstationary phase shift methods (NSPS) (Ferguson and Margrave, 1999), and the Gabor method (Grossman et al., 2002; Ma and Margrave, 2005). Ferguson and Margrave (1999) showed that the GPSPI Fourier integral is the limit of PSPI in the extreme case of using a distinct reference velocity for each output location. Writing outside the typical seismic literature, Fishman and McCoy (1985) derived the GPSPI formula as a high frequency approximation to the exact wavefield extrapolator for a laterally variable medium. They refer to the GPSPI formula as the locally homogeneous approximation (LHA) and we adopt that nomenclature. While the LHA is called a high frequency approximation it is still much more accurate than raytracing because the approximation is done at a different place in the theoretical development. (See Fishman and McCoy for more discussion.) In fact, it can be shown that virtually all explicit depth stepping methods in practice today, including those mentioned above, are approximations to the LHA formula. Our Gabor method also approximates the LHA formula and is derived from Grossman et al. (2002) and Ma and Margrave (2005) but is also closely related to the windowed screen method described in Jin and Wu (1998). Our main idea is to develop an optimal set of spatial windows that localize the wavefield more strongly (i.e. narrow windows) as the velocity variation increases. Conversely, when the lateral velocity gradient is weak very little localization is required and much wider windows are used. Since computation costs are directly proportional to the number of windows, this then controls cost; however, with some sacrifice of accuracy. We analyze the tradeoff between localization and accuracy by estimating the phase error between our Gabor method and its logical limit the LHA approximation. We present an algorithm that constructs a complete set of spatial windows, called a partition of unity (POU) because it sums to one, such that the lateral position error is bounded below any desired threshold. The windows in this set have spatially variable widths linked directly to the spatial variation of velocity through an analysis of the wavefield positioning error. Such a window set allows the construction of a non-uniform, or adaptive, Gabor frame. Locally, within each window, we apply the split-step Fourier method. Our algorithm applies easily in either 2D or 3D. For simplicity, our theoretical development is in 2D.

METHOD The LHA Wavefield Extrapolator The LHA wavefield extrapolator, also known as the GPSPI extrapolator, is formulated as (Margrave and Ferguson, 1999)

( ) ( ) ( ) ( )1 ˆ ˆ, , , , ( ), , exp d ,

2 x x x xx z z k z W k x k z ik x k+∞

−∞

+∆ = ∆ −∫ψ ω ψ ωπ

(1)

where ( ), ,x zψ ω is the seismic wavefield in the space-frequency domain at depth z , ω is temporal frequency, xk and zk (to be defined) compose the total wavenumber vector with a magnitude of ( )k x in the Fourier domain, x denotes transverse coordinates, ( )ˆ , ,xk zψ ω is the Fourier transform of

( ), ,x zψ ω , z∆ is the step size of extrapolation in z (vertical)

direction, and W is the LHA wavefield extrapolator, which is a spatial phase shift term and defined as


( ) ( )( )ˆ ( ), , exp ( ),x z xW k x k z ik k x k z∆ = ∆ (2)

2 2 2 2

2 2 2 2

( ) , ( )

( ), ( )

x xz

x x

k x k k x kk

i k k x k x k

⎧⎪ − >⎪⎪=⎨⎪ − <⎪⎪⎩, ( ) ,

( )k x

v x= ω

(3)

where ( )v x denotes the lateral velocity at x along a slab with thickness z∆ .

Approximation of the LHA Extrapolator We approximate LHA extrapolator using

( ) ( )ˆ ˆ( ), , ( ) , , ,x j j j xj

W k x k z S x W k k z∆ ≈ Ω ∆∑ (4)

where jΩ is a set of windows forming a partition of unity (POU), meaning

( ) 1jj

xΩ =∑ , the split-step Fourier operator (Stoffa et al., 1990) is

( )( )1 1( ) exp ( )j jS x i z v x v− −= ∆ −ω and ˆ ( , , )j xW k k z∆ is defined by equation (2) using

( ) /j jk x k v= = ω with the reference velocities jv defined as the average velocity across

window jΩ . Equation (4) asserts that the LHA extrapolator can be approximated by a split-

step Fourier extrapolator within each window. Using approximation (4) in equation (1), the LHA formula becomes

( ) ( ) ( ) ( )1 ˆ ˆ, , ( ) ( ) , , , , exp d ,2 j j x j x x x

j

x z z x S x k z W k k z ik x k+∞

−∞

+∆ ≈ Ω ∆ −∑ ∫ψ ω ψ ωπ

(5)

where jΩ and ( )jS x are xk independent and taken out of the integrand. Equation (5) is the

formula used in the Gabor wavefield extrapolation method in this paper. Aside from this novelty, our migration algorithm is a standard shot-record migration with a deconvolution imaging condition. Building Non-uniform Gabor Frames (Adaptive Partitions)

The simplest implementation of equation (5) would be to have ( )j xΩ be a set of constant-

width windows, one of which we call the atomic window, uniformly distributed along the x coordinate. However, this is generally inefficient as the atomic window width must be chosen to accommodate the most rapid velocity change and this may be completely unnecessary elsewhere. For efficiency, the number of windows should be reduced to as few as possible while maintaining acceptable accuracy. This can be done by summing adjacent atomic windows to form molecules in regions where there is little velocity variation. For this purpose, a criterion is required which specifies when atomic windows can be consolidated. There are several methods (Grossman et al., 2002; Ma and Margrave, 2005) available but their consolidation criteria are not very physical. In this paper, a new scheme is developed using “lateral position error” as a criterion.

Considering a straight ray emanating at angle θ from a point ( ),x z and traversing to

( ),x z∆ we have the relationships tanx z=∆ θ and sin

xkk

v= = ω

θ. Differentiating these

results and rearranging gives 2secx z vv

∂=∆∂θδ θ δ and

secxk

v

∂ =∂θ θ

ω, from which we

conclude

3cos.

sin

xv v

z=

∆θ δδθ

(6)


Equation (6) relates the position error expected in a single depth step, xδ , to a velocity error of vδ . This formula can be used to build the desired adaptive partition, given choices for the allowable positioning error and the maximum scattering angle of interest. To specify reference velocities, 1v is chosen as the most frequently occurring velocity in a

given depth step (easily done from a histogram of the discretely sampled ( )v x ) and then 1vδ

is obtained from (6) as 1 1v av=δ , where 3 1cos sin /a x z−= ∆θ θδ . Letting 2v denote the

next higher reference velocity and 3v the next lower, we set the conditions

( ) ( )1 1 3 3 3 1 1 2 2 2/ 2 / 2 1 / 2 , / 2 / 2 1 / 2 .v v v v v a v v v v v a− = + = + + = − = −δ δ δ δ (7)

Using (7), calculation of 2v and 3v gives

( ) ( ) ( ) ( )2 1 1 3 1 12 / 2 , 2 / 2 .v v v a v v v a= + − = − +δ δ (8)

Figure 1 illustrates this process.

(a) (b)

Figure 1: (a) The reference velocities chosen according to equation (6) are shown on top of a

velocity histogram. (b) Velocity profile ( )v x is shown in the solid curve and the reference

velocities created in (a) are shown in horizontally dashed lines.

Proceeding in a similar way until a reference velocity exceeds the range of ( )v x , which

defines a complete set of reference velocities. Once the reference velocities are selected, they

can be used as measures to build indicator functions ( )jI x , using

1, ( ) min( )

0, otherwise,

jj

v x vI x

⎧⎪ − =⎪=⎨⎪⎪⎩ (9)

where ( 1, 2,..., )jv j n= are reference velocities created by equation (8). We convolve the

indicator function ( )jI x with the atomic window to create the adaptive POU by

( )( ) ( ),j jx I xΩ = ∗Θ

(10)

where Θ is the atomic window (e.g. Gaussian), ∗ means normalized convolution. Thus there will be one window for each reference velocity even if the regions where that velocity is required are disjoint. Although we do not discuss it here, this method extends easily to 3D.

0 200 400 600 800 1000 12000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Distance (m)

velocity (km/s)window 1window 2

0 200 400 600 800 1000 1200

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Distance (m)

velocity (km/s)window 1window 2......

(a) (b)

Figure 2: Non-uniform partitions of unity as created by the algorithm described here.


In Figure 2 (a), a bump velocity profile (shown in cyan colour) is used; two reference velocities are selected. Accordingly, two windows (green-dashed and blue-solid lines) have been chosen given a lateral position error criterion. Figure 2 (b) shows the partitioning algorithm can deal with such complicated lateral velocity variations as those in the Marmousi velocity model; a specific colour shows a single partition (frame) created by the algorithm.

GABOR IMAGING EXAMPLE The Marmousi velocity section used in Gabor depth imaging in this paper is shown in Figure 3 (a). In the Marmousi dataset, there are 240 shot records. All depth images have been created using prestack shot migration, with step size of 12.5z∆ = meters.

(a) (b)

Figure 3: (a) Marmousi velocity model. (b) Gabor imaging result using a lateral position error of 5 m. Figure 3 (b) shows the prestack, shot-record, depth migration of the Marmousi dataset using a position-error criterion of 5 m. Taking a closer look at the imaging target (a reservoir extending from distance 6000 m to 7500 m in depth about 2500 m, see also Figure 3 (a)), we know that the target reservoir has been accurately imaged. Observing the rest of the areas in the image, we see that they are all well imaged.

CONCLUSIONS The approximation of LHA extrapolator using the adaptive Gabor wavefield extrapolation algorithm gives good results. The adaptive partitioning algorithm helps to achieve efficient depth imaging and the imaging accuracy and speed can be controlled using the adaptive partitioning scheme.

ACKNOWLEDGEMENTS We thank the Consortium of Research in Elastic Wave Exploration Seismology (CREWES) and the sponsors for financial support of this research.

REFERENCES Etgen, J. T., [1994] Stability analysis of explicit depth extrapolation through laterally-varying

media. Expanded abstracts 64th SEG International Convention, 1266-1269. Gazdag, J. and Sguazzero, P. [1984] Migration of seismic data by phase shift plus interpolation.

Geophysics, 49, 124-131. Grossman, J. P., Margrave, G. F., and Lamoureux, M. P. [2002] Fast wavefield extrapolation by

phase-shift in the nonuniform Gabor domain. CREWES Research Report, 14. Jin, S., Xu, S., and Mosher, C. C. [2002] Migration with a local phase screen propagator. SEG

Expanded Abstracts, 21, 1164. Margrave, G. F., and Ferguson, R. J. [1999] Wavefield extrapolation by nonstationary phase

shift. Geophysics, 64, 1067-1078 Margrave, G. F., Geiger, H. D., Al-Saleh, S. M., and Lamoureux, M. P. [2006] Improving

explicit seismic depth migration with a stabilizing Wiener filter and spatial resampling. Geophysics, 71, S111-S120.

Stoffa, P. L., Fokkema, J. T., de Luna Freire, R. M., and Kessinger, W. P. [1990] Split-step Fourier migration. Geophysics, 55, 410-421.

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Wu, R., and Huang, L. [1992] Scattered calculation in heterogeneous media using a phase-screen propagator. SEG Expanded Abstracts, 11, 1289.

seismic depth migration using non-uniform gabor frames · 2020-01-17 · seismic migration by phase...

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