Next: WEMVA sensitivity kernels Up: Wave-equation migration velocity analysis Previous: Cycle skipping in image

# Linearized image perturbations

I address the problem of cycle skipping by employing linearized image perturbations. If we define , we can write a discrete version of the image perturbation using a Taylor series expansion of psrm as
 (69)
where the ' sign denotes differentiation relative to the velocity ratio parameter . For the image perturbations computed with storm.differential, I use the name linearized image perturbations . dip graphically illustrates this procedure.

WEP2.imag
Figure 6
Comparison of image perturbations obtained as a difference between two migrated images (b) and as the result of the forward WEMVA operator applied to the known slowness perturbation (c). Panel (a) depicts the background image corresponding to the background slowness. Since the slowness perturbation is large (), the image perturbations in panels (b) and (c) are different from each-other.

The linearized prestack Stolt residual migration operator can be computed analytically, as described in Appendix C. With this operator, I can compute linearized image perturbations in two steps. First, I run residual migration for a large range of velocity ratios and pick at every image point the ratio which maximizes flatness of the gathers. Then, I apply the operator in storm.differential to the background image and scale the result with the picked .

WEP2.rays
Figure 7
Comparison of slowness backprojections using the WEMVA operator applied to image perturbations computed as a difference between two migrated images (b,d) and as the result of the forward WEMVA operator applied to a known slowness perturbation (c,e). Panel (a) depicts the background image corresponding to the background slowness. Since the slowness perturbation is large (), the image perturbations in panels (b) and (c) and the fat rays in panels (d) and (e) are different from each-other. Panel (d) shows the typical behavior associated with the breakdown of the Born approximation.

The linearized image perturbations approximate the non-linear image perturbations caused by arbitrary velocity model changes. They are based on the gradient of the image change relative to a velocity model change, and are less restrictive than the Born approximation limits.

WEP2.raan shows how the linearized image perturbation methodology applies to the synthetic example used earlier in this section. All panels are similar to the ones in WEP1.rays and WEP2.rays, except that the left panels (b and d) correspond to linearized image perturbations, instead of simple image perturbations. Again, I compare image and slowness perturbations with the ideal perturbations obtained by the forward WEMVA operator (c and e). Both the image and slowness perturbations are identical in shape and magnitude.

 dip Figure 8 A schematic description of the method used for computing linearized image perturbations. The dashed line corresponds to image changes described by residual migration with various values of the velocity ratio parameter (). The straight solid line corresponds to the linearized image perturbation computed with an image gradient operator applied to the reference image scaled at every point by the difference of the velocity ratio parameter .

WEP2.raan
Figure 9
Panels (a) and (c) correspond to Cartesian coordinates, and panels (b) and (d) correspond to ray coordinates. Velocity model with an overlay of the ray coordinate system initiated by a D source at the surface (a); image obtained by downward continuation in Cartesian coordinates with the equation (c); velocity model with an overlay of the ray coordinate system (b); image obtained by wavefield extrapolation in ray coordinates with the equation (d).espite the fact that the slowness perturbation is large (), the image perturbations in panels (b) and (c) and the fat rays in panels (d) and (e) are practically identical, both in shape and in magnitude.

Next: WEMVA sensitivity kernels Up: Wave-equation migration velocity analysis Previous: Cycle skipping in image
Stanford Exploration Project
11/4/2004