Calculating finite-frequency sensitivity kernels using the Gaussian beam method Xiao-Bi Xie* Institute of Geophysics and Planetary Physics, University of California at Santa Cruz Summary

We propose to use the Gaussian beam method to calculate finite-frequency sensitivity kernels for transmitted wave and for prestack migration geometry. The Gaussian beam method has the advantage of high efficiency and does not have angle limitations. Thus it is suitable for generating sensitivity kernels for high-frequency and long-distance propagation and for wide-angle waves including the turning waves. The limitation of this method is that the velocity model should be relatively smoothed. We use numerical examples to demonstrate this technique. The resulted sensitivity kernels can be used for velocity tomography and migration velocity analysis.

Introduction

Traditionally, both velocity tomography and migration velocity analysis (MVA) have been dominated by the ray-based techniques assuming an infinite frequency. When a broadband signal propagates through a complex region, the ray theory often poorly approximates the actual wave propagation. Thus, a robust velocity tomography method based on the wave equation theory is preferred. As a substitution, we can use wave equation based sensitivity kernel to replace the geometrical ray. The sensitivity of finite-frequency signal to velocity perturbation has been extensively investigated and successfully used in velocity tomography (e.g., Woodward, 1992; Vasco et al., 1995; Dahlen et al., 2000; Huang et al., 2000; Zhao et al., 2000; Spetzler and Snieder, 2004; Jocker, et al., 2006; and Liu et al., 2009) and MVA (e.g., Sava and Biondi, 2004; de Hoop, 2006; Xie and Yang, 2007, 2008a, b; Fliedner et al., 2008; Shen and Symes, 2008; and Sava and Vlad, 2008). Sensitivity kernels are usually derived based on the scattering theory. Several methods can be used to calculate finite-frequency sensitivity kernels. The method based on the full-wave FD calculation (e.g., Zhao et al. 2005) can deal with complex velocity models but is very time consuming. Besides, the FD calculation often includes waves from all paths. It is usually difficult to separate different arrivals to generate kernels for specific phases. Alternatively, the one-way wave equation based propagator can be used (e.g., Xie and Yang 2007, 2008a). The one-way method is efficient and can properly handle wave phenomena such as scattering, diffraction and interference. Given the large data size involved, this method is particularly useful in exploration seismology. The disadvantage of the one-way propagator is that it has angle limitations. The wide-angle signals and turning waves cannot be properly handled.

A Gaussian beam (BG) is an asymptotic solution to the wave equation constructed in a curvilinear coordinate system. Compared to the geometrical ray method, the GB method overcomes some critical difficulties such as the two-point boundary value problem and the caustic problem. Compared to the full-wave equation, the GB method is very efficient, particularly at high frequencies and for long propagation distance. Compared to the one-way propagator, the GB method does not have angle limitations, thus it can handle wide-angle waves including turning waves. The disadvantage of the GB method is that it is usually accurate in relatively smoothed models. For sharply changed models, velocity smoothing is often required and it may cause accuracy deterioration. The GB method has been widely used to construct Green’s functions for seismic modeling and imaging (e.g., Wapenaar et al., 1989; Nowack and Aki, 1984, Hill, 1990, 2001; Gray, 2005; and Nowack, 2008). One may think it is odd to use the GB method to generate sensitivity kernels because it is a ray based method. However, as Nolet (2008) pointed out: one could improve on ray theory using ray theory. Dahlen et al. (2000) showed how to use dynamic ray tracing to calculate Frechet kernels efficiently. In this paper, we demonstrate how to use the GB method to calculate finite-frequency sensitivity kernels for velocity tomography and the MVA.

Theory and Methods

The Gaussian beam method. A GB solution of wave equation can be expressed as (e.g., Nowack and Aki, 1984; and Hill 1990)

1 2

2, , exp2

v s P siu s n i s n

Q s Q s

, (1)

where s is the coordinate along the ray path, n is the coordinate perpendicular to the ray, is the travel time along the ray, is the frequency, v is the velocity field, and P and Q are complex scalar functions. The P and Q

can be obtained by solving

dQ sv s P s

ds , (2)

2

2 2

,1dP s v s nQ s

ds v s n

, (3)

with initial values 0 0P i V and 20 0 0rQ w V , where r

is a reference frequency, 0w is the initial beam width and

0V is the velocity at the starting point of the ray. A Green’s

function can be constructed using equation (1) by shooting multiple GBs along required directions.

© 2011 SEGSEG San Antonio 2011 Annual Meeting 38923892

Calculating sensitivity kernels using the GB method

The sensitivity kernel for transmitted waves. Several authors derived equations for calculating sensitivity kernels for transmitted waves (e.g., Woodward, 1992; Spetzler and Snieder, 2004; and Jocker, et al. 2006). The single frequency phase sensitivity kernel FK can be expressed as

20

; , ; ,, , , imag 2

; ,S GF

S GG S

G GK k

G

r r r rr r r

r r, (4)

where 0k v r is the background wavenumber, v r is

the background velocity, G is the Green’s function, r is the space location, Sr and Gr are the source and receiver

locations, and imag denotes taking imaginary part. In

equation (4), the sensitivity kernel is composed by the correlation of two Green’s functions, one radiated from the source and another radiated from the receiver, normalized by the source wavefield at the receiver location. Adopting the previously introduced GB Green’s function into equation (4) can generate the sensitivity kernel.

Figure 1. Monotonic (left column) and broadband (right column) sensitivity kernels. Different rows are for different frequencies. The distance between the source and the receiver is 5 km. Shown in Figure 1 are examples of travel time sensitivity kernels calculated using the GB method. These kernels are for a finite-frequency signal which is radiated from the source and observed at the receiver location. A 2D homogeneous model with a 3 km/s velocity is used in the calculation. Illustrated in the left column are monotonic kernels and in the right column are for broadband waves. Different rows are for different frequencies. The sensitive area is apparently broader than a thin ray. We call these sensitivity maps “travel time sensitivity kernels” because

they tell how a finite-frequency signal radiated from a source can “sense” velocity variations along its way to the receiver. The sensitivity shows both positive and negative signs indicating the same perturbation may cause either positive or negative travel time delays depending on its location. For single frequency kernels, its sensitivity shows oscillations in a broad region. The central non-zero region is the first Fresnel zone. For broadband signals, the sensitivity kernel can be obtained by stacking single frequency kernels weighted by the source spectrum (Woodward, 1992; Spetzler and Snieder, 2004; and Xie and Yang, 2008a). The oscillating parts are mostly canceled out due to interference and its non-zero sensitivity shrink to the first Fresnel zone linking the source and the receiver. Figure 2 checks the sensitivity kernels calculated using the GB method by comparing it with that calculated using the analytical solution. With the increase of frequency, the broadband sensitivity kernel becomes narrower and its behavior approaches a ray. The sensitivity kernel links the observed travel time residual to the velocity perturbation and sets up a basis for velocity tomography.

Figure 2. Comparison between 5 Hz broadband sensitivity kernels calculated using the GB method and the analytic method. Shown here is a profile located in the middle between the source and the receiver, and perpendicular to the geometrical ray.

Figure 3. 20 Hz sensitivity kernels calculated using GB Green’s functions with different radiation angle coverage, where (a) through (d) are single frequency kernels with angle coverage of ±180º, ±90º, ±60º, and ±30º, and (e) and (f) are broadband kernels with angle coverage of ±30º and ±180º.

© 2011 SEGSEG San Antonio 2011 Annual Meeting 38933893

Calculating sensitivity kernels using the GB method

The GB method is particularly useful in calculating the sensitivity kernels for long distance propagation and at high frequencies. Under these circumstances, other methods are usually very time-consuming. In addition, a high frequency, long distance kernel usually involves narrower radiation direction. Thus, less GB rays are required to compose the Green’s functions. Figure 3 compares the sensitivity kernels constructed using GB Green’s functions with different radiation-angle coverage. Figures 3a through 3d are single frequency kernels calculated using radiation angles ranging ±180º, ±90º, ±60º, and ±30º, respectively. The sensitivity kernels calculated using different radiation-angle coverage show differences mostly in the surrounding areas, and particularly around the back-azimuth relative to the receiver direction. These are mostly oscillating areas in a single frequency kernel, while the region in the first Fresnel zone is less affected. For a broadband signal, the sensitive area is mainly in the first Fresnel zone which is described by the broadband kernel. Illustrated in Figures 3e and 3f are broadband sensitivity kernels calculated using GB Green’s functions with radiation-angle coverage of ±30º and ±180º, respectively. Although the single-frequency kernels show apparent differences, the broadband kernels using different radiation-angle coverage are less affected. Therefore, in a relatively smoothed area, only limited numbers of GB rays around the receiver direction are required to compose a high-frequency/long-distance broadband kernel. At other directions, the beams can either be neglected or traced only for a very short distance, which makes this method highly efficient.

Figure 4. (a) the ray diagram and (b) the wave front in a 1D velocity model used to calculate the sensitivity kernels for turning waves. The sensitivity kernels for turning waves. One of the advantages for the GB method is that it can handle turning waves which are very useful in near-surface seismic tomography and in imaging the steep dip structures in seismic migration. To calculate the turning wave sensitivity

kernels, we use a model which has a linear vertical velocity gradient with the velocities of 2.0 km/s at the surface and 5.0 km/s at a depth of 6 km. For a typical GB Green’s function, the ray diagram and the wave front are shown in Figure 4. Illustrated in Figure 5 are turning wave sensitivity kernels, where the left and right columns are for monotonic and broadband kernels, and different rows are for different frequencies. We see that turning wave kernel with frequency up to 40 Hz can be properly handled.

Figure 5. Sensitivity kernels for turning waves. Illustrated in the left column are monotonic kernels and in the right column are broadband sensitivity kernels. Individual rows are for different frequencies. The sensitivity kernel for migration velocity analysis. In seismic migration, the information regarding the velocity model error is extracted from the residual moveout (RMO) in depth image. Based on the Born and Rytov approximations, Xie and Yang (2007, 2008a) formulated the sensitivity kernel for the shot-index common image gather. This sensitivity kernel is composed of two parts, the down-going leg and the up-going leg. The kernel for the down-going leg can be expressed as

20

; , ; ,, , , imag 2

; ,D S IF

D S ID I S

G GK k

G

r r r rr r r

r r, (5)

where DG is the down-going wave Green’s function

radiated from the source, and Ir is the image location.

Equation (5) is similar to equation (4) except the receiver

© 2011 SEGSEG San Antonio 2011 Annual Meeting 38943894

Calculating sensitivity kernels using the GB method

location in equation (4) is replaced by the image location. However, the kernel for the up-going leg is different from the transmitted wave. In prestack depth migration, the up-going wave is obtained from the time-reversed reflection wave. The time reversal in the time domain is equivalent to the complex conjugate in the frequency domain. Thus the equation for up-going leg is similar to equation (5) but with the source wavefield replaced by the conjugate of the target reflected wave (Xie and Yang, 2008a), i.e.,

20

; , ; ,, , , imag 2

; ,U S IF

U S IU I S

G GK k

G

r r r rr r r

r r , (6)

where UG is the Green’s function for up-going reflected

wave, and the asterisk denotes taking complex conjugate.

Figure 6. The GB Green’s functions used to construct the sensitivity kernel for migration velocity analysis. (a) Down-going Green’s function ; ,D SG r r , (b) up-going Green’s

function ; ,U SG r r , and (c) Green’s function ; ,IG r r .

To calculate the sensitivity kernel, we use a model with a constant velocity of 3 km/s and a horizontal reflector buried at a depth of 2.5 km. Illustrated in Figure 6 are GB Green’s functions

DG , UG and G which are used in equations (5)

and (6) for constructing the sensitivity kernel. Shown in Figure 7a is a 10 Hz broadband sensitivity kernel for migration geometry calculated using the GB method. The source side sensitivity kernel is similar to that for transmitted waves. However, the receiver side kernel is quite different. Near the source location and immediately above the image point, there are two sensitive regions where the velocity model affects the RMO the most. As comparisons, shown in 7b is the same sensitivity kernel calculated using the one-way propagator, and shown in 7c

is the actually measured sensitivity map from the migration process (Xie and Yang, 2008a). We see these kernels show excellent consistency.

Figure 7. Comparison between sensitivity kernels obtained using (a) the GB method, (b) the one-way method, and (c) actually measured.

Conclusions

We have demonstrated that the GB method can be used to calculate sensitivity kernels for velocity tomography and MVA. The GB method has the advantages of high efficiency and without angle restriction. It is suitable for calculating sensitivity kernels for high frequency and long propagation distance. Under these circumstances, other methods often encounter difficulties or are too time consuming when used in exploration practice. The disadvantage for the GB method is that it requires the model to be relatively smoothed. In very complex media, smoothing is required. The sensitivity kernels are formed by interference between Green’s functions from the source, receiver or image locations. Accurate phase (or travel time) is crucial to obtain correct sensitivity kernels.

Acknowledgement

The author wish to thank Yingcai Zheng, Yaofeng He and Yu Geng for their helpful discussions. This research is supported by the WTOPI Research Consortium at the University of California, Santa Cruz.

© 2011 SEGSEG San Antonio 2011 Annual Meeting 38953895

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