Report

Sensitivity kernels for finite-frequency signals: Applications in migration velocity updating and tomography Xiao-Bi Xie University of California at Santa Cruz Sanya, China July 24-28, 2011 Outline A brief introduction Data domain vs. depth domain Sensitivity Kernel for Migration Velocity Analysis The Inversion System Velocity model partitioning and sensitivity kernel storage Numerical Result Conclusions Outline A brief introduction Data domain vs. depth domain Sensitivity Kernel for Migration Velocity Analysis The Inversion System Velocity model partitioning and sensitivity kernel storage Numerical Result Conclusions Imag df =arg(u/u0) arg(u/u0) = imag(U/u0) U u U / u0 u0 Real u / u0 u0/ u0 u U imag u0 u0 d arg G r; rs , G r; rg , 2 K r, rs , rg , imag 2k0 G rg ; rs , In applied seismology Huge data size. Efficiency is crucial. Suggested methods could be one-way propagator or Gaussian beam method. Complex background models. The velocity perturbations overlapped on the initial model are large (some times are more than 100%). Including not only transmitted observations, where the information is from the surface data, but also reflection type observation, where the information is collected in image domain. Outline A brief introduction Data domain vs. depth domain Sensitivity Kernel for Migration Velocity Analysis The Inversion System Velocity model partitioning and sensitivity kernel storage Numerical Result Conclusions Complexity in data domain ▼ Simplicity in depth domain A synthetic shot record. The shot is located above relatively complicated structures. There are many complicated features in this synthetic section. Outline A brief introduction Data domain vs. depth domain Sensitivity Kernel for Migration Velocity Analysis The Inversion System Velocity model partitioning and sensitivity kernel storage Numerical Result Conclusions Migration velocity updating Data Source Target Back project to modify the velocity Measuring incoherence in image The incoherence information are RMOs from different common image gathers. Offset index CIG Shot index CIG Angle index CIG The methods that converting the RMO into velocity corrections. Parameterized semblance Ray-based tomography Wave-equation based inversion How the RMO sense the velocity perturbation: --- Direct measurements The actual sensitivity map for a shot image (how the depth image senses the V-model error). To generate this map, we use an velocity error patch to scan the model. At each location, we conduct a migration and measure the RMO from the depth image. The RMOs are then presented in the model to show the sensitivity of the depth image to the velocity error. The sensitivity map is very complex. A positive error can generate either positive or negative RMOs; the sensitivity area is much broader than the ray based theory predicted. Our goal is to derive theoretical equations to express this sensitivity map and use it for velocity updating. Direct measured sensitivity maps for shots at different locations in 2D SEG/EAGE salt model Source side kernel K DF 2 GD r; rS G r; rI r , r , r imag S I 2k0 GD rI ; rS Receiver side kernel KUF 2 GU r; rS G r; rI r, rS , rI imag 2k0 G r ; r U I S Source Image point Imaginary source The GB Green’s functions used to construct the sensitivity kernel for migration velocity analysis. (a) Down-going Green’s function , (b) up-going Green’s function , and (c) Green’s function . Comparison of different kernels for a shot gather The sketch of a ray-based kernel The sensitivity kernel calculated using the finitefrequency theory The actual sensitivity map directly measured from migration Outline A brief introduction Data domain vs. depth domain Sensitivity Kernel for Migration Velocity Analysis The Inversion System Velocity model partitioning and sensitivity kernel storage Numerical Result Conclusions R rI , rS v0 rI m r K D r, rS , rI KU r, rS , rI dv V 2 cos rI , rS = RMO ∫ B δv/v Velocity model error B dV Sensitivity kernel Comparison between inversions using a finite-frequency sensitivity kernel and a ray kernel Sketch illustrating the relative residual moveout measurement from a pair of shots The differential sensitivity kernel for a pair of shots. Note the complexity and volumetric distribution of a finitefrequency kernel The ray based kernel for a pair of shots. Note the sensitivity distribution is unrealistic and the uneven ray distribution can cause singularities in inversions. Differential RMO Differential kernel d R rS1, rS 2 , rI V m r K B r, rS1, rS 2 , rI dv K B r, rS1 , rS 2 , rI v0 rI K DB r, rS 2 , rI KUB r, rS 2 , rI 2 cos rS 2 , rI v0 rI K DB r, rS1 , rI KUB r, rS1 , rI 2 cos rS1 , rI Outline A brief introduction Data domain vs. depth domain Migration Velocity Analysis The Inversion System Velocity model partitioning and sensitivity kernel storage Numerical Result Conclusions A 5-layer velocity model used to demonstrate the migration velocity analysis. How to partition the model? B d R rS1, rS 2 , rI V m r K r, rS1, rS 2 , rI dv How to store huge amount of kernels? d R rS1 , rS 2 , rI V r m r K B r, rS1 , rS 2 , rI dv k k B m r K r, rS1 , rS 2 , rI dv V r i V r a1 a2 i a3 a4 f1 r f 2 r B K r dv f3 r f4 r a1 1 0 0 0 m r1 m r a 1 1 0 0 2 2 a3 1 0 1 0 m r3 a4 1 1 1 1 m r4 m r V r Pij m r j fi r K B r , rS1 , rS 2 , rI dv k Pij m r j FKi k Pij m r j V r fi r K B r , rS1 , rS 2 , rI dv k FK1 FK 2 FK3 Unknown perturbation at cell corners 1 0 0 0 m r1 1 1 0 0 m r 2 FK 4 1 0 1 0 m r3 m r 1 1 1 1 4 Actual output and stored kernels FK1 V r f1 r K B r dv i FK 2 V r f 2 r K B r dv i FK3 V r f3 r K B r dv i FK 4 V r f 4 r K B r dv i Parameter matrix Stored parameter a1. The 4 groups of kernels are for 4 reflectors; the horizontal coordinate is for different image points and the vertical coordinate is for different sources. Model grid size 10m x 10m Cell size 500m x 500m 31shot x 31 imaging point x 4 reflectors, 32x10 cells spend 286Mb. Outline A brief introduction Data domain vs. depth domain Sensitivity Kernel for Migration Velocity Analysis The Inversion System Velocity model partitioning and sensitivity kernel storage Numerical Result Conclusions migration velocity updating process (1) Conduct the migration using an initial model. (2) Calculate the RMOs from the shot-index CIGs. (3) Pick the reflector position from the initial depth image. (4) Use the initial model and reflector locations to calculate sensitivity kernels. (5) Input the RMOs and the sensitivity kernels to the inversion system to do the tomography. (6) Use the inverted errors to update the initial model and use it for the next iteration. A 5-layer velocity model used to demonstrate the migration velocity analysis. Comparison between the theoretically calculated kernels (left column) and actually measured sensitivity maps (right column). From top to bottom are for different reflectors. Coverage of sensitivity kernels in the model. Panels (a) to (d) are kernel coverage for image points on the 4 reflectors. Panel (e) is the coverage from all kernels. Shown here is the summed positive parameter FK1. Velocity models in updating process, with (a) initial model and (b) model after two iterations. Depth image improved in the velocity updating process. (a) Image calculated using the initial model and (b) image calculated using the updated velocity model. CIGs before and after the velocity updating, with (a) CIGs in the initial model and (b) CIGs in the updated velocity model. Outline A brief introduction Data domain vs. depth domain Sensitivity Kernel for Migration Velocity Analysis The Inversion System Velocity model partitioning and sensitivity kernel storage Numerical Result Conclusions Summary (1) Based on the finite-frequency sensitivity theory, we present a migration velocity analysis method. The new approach is a wave-equation based method which naturally incorporates the wave phenomena and is best teamed with the wave-equation based migration for velocity analysis. (2) The finite-frequency sensitivity kernel is used to link the observed shot gather RMO with the errors in the migration velocity model. Angle domain decomposition is not required. (3) We developed method to calculate the broadband sensitivity kernel in complex velocity models and for irregular reflectors. Summary (continues) (4) A new velocity model partitioning approach is tested. This method partitions the model into small cells and uses interpolation function to represent the velocity model within cells. (5) To store the sensitivity kernels, we use interpolation functions as basis and expanded kernels to these basis. Thus we only need to store the expansion coefficients. The accuracy of the kernel is adaptive to the required accuracy of the velocity model. In this way, we significantly reduce the storage space of sensitivity kernels while without losing the required accuracy. Summary (continues) (6) Using this approach, we demonstrate the velocity model updating. The updated velocity model improves the depth image by both flattened the common image gather and bring the image to the original location of reflectors. EOF 44