Yaxun Tang

Target-oriented wave-equation least-squares migration/inversion with phase-encoded Hessian

Prestack depth migration produces blurred images resulting from limited acquisition apertures, complexities in the velocity model, and band-limited characteristics of seismic waves. This distortion can be partially corrected using the model-space least-squares migration/inversion approach, where a target-oriented wave-equation Hessian operator is computed explicitly and then inverse filtering is applied iteratively to deblur or invert for the reflectivity. However, one difficulty is the cost of computing the explicit Hessian operator, which requires storing a large number of Greens functions, making it challenging for large-scale applications. A new method to compute the Hessian operator for the wave-equation-based least-squares migration/inversion problem modifies the original explicit Hessian formula, enabling efficient computation of this operator. An advantage is that the method eliminates disk storage of Greens functions. The modifications, however, also introduce undesired crosstalk artifacts. Two different phase-encoding schemes, plane-wave-phase encoding and random-phase encoding, suppress the crosstalk. When the randomly phase-encoded Hessian operator is applied to the Sigsbee2A synthetic data set, an improved subsalt image with more balanced amplitudes is obtained.

Least‐squares migration/inversion of blended data

We present a method based on wave-equation least-squares migration/inversion to directly image data collected from recently developed wide-azimuth acquisition geometry, such as simultaneous shooting and continuous shooting, where two or more shot records are often blended together. We show that by using least-squares migration/inversion, we not only enhance the resolution of the image, but more importantly, we also suppress the crosstalk or acquisition footprint, without any preseparation of the blended data. We demonstrate the concept and methodology in 2-D and apply the data-space inversion scheme to the Marmousi model, where an optimally reconstructed image, free from crosstalk artifacts, is obtained.

Wave‐equation Hessian by phase encoding

I demonstrate a method for computing wave-equation Hessian operators, also known as resolution functions or point-spread functions, under the Born approximation. The proposed method modifies the original explicit Hessian formula, enabling efficient computation of the operator. A particular advantage of this method is that it reduces or eliminates storage of Green’s functions on the hard disk. The modifications, however, also introduce undesired crosstalk artifacts. I introduce two different phase-encoding schemes, namely, plane-wave phase encoding and random phase encoding, to suppress the cross-talk. I applied the Hessian operator obtained by using random phase encoding to the Sigsbee2A synthetic data set, where a better subsalt image with higher resolution is obtained.

Gradients For Wave-equation Migration Velocity Analysis

Wave-equation migration velocity analysis (WEMVA) is an image-domain tomography based on a wave-equation propagator (one-way or two-way). Different algorithms for residual calculation, propagation and optimization add varying flavors to WEMVA. The two most popular ways to compute the residual are either using differential semblance optimization (DSO) or using differential residual migration (DRM). In this paper, we present relevant theory to understand the difference these two algorithms make in the final gradient calculation. We compare and contrast the two methods with the help of numerical examples. Both methods have some advantages and disadvantages over the other and they should be kept in mind while making the choice. The theory and results presented are based on one-way wave-equation migration.

Joint preconditioned least‐squares inversion of simultaneous source time‐lapse seismic data sets

We present a joint least-squares inversion method for imaging simultaneous source (or blended) time-lapse seismic data sets. Non-repeatable shot and receiver positions introduce undesirable artifacts into time-lapse seismic images. We conjecture that more artifacts will result from relative shot-timing nonrepeatability when data sets are acquired with several simultaneously shooting sources. We show that these artifacts can be attenuated by joint inversion of such data sets without need for initial separation. Preconditioning with non-stationary dip filters and with temporal smoothness constraints ensures stability and geologically consistent time-lapse images. Results from a modified Marmousi 2D model show that this method yields reliable time-lapse images.

Wave-equation tomography using image-space phase encoded data

Wave-equation tomography in the image-space is a powerful technique that promises to yield more reliable velocity models than ray-based migration velocity analysis in areas of complex overburden. Its practical use, however, has been limited because of the high computational cost. Applying a target-oriented approach and using data reduction can make wave-equation tomography in the image space of practical use. Here, we present results of applying image-space wave-equation tomography in the generalized source domain, where a small number of synthesized shot gathers are generated. Specifically, we generate synthesized shot gathers by image-space phase encoding. This technique can also be used in a target-oriented way. The comparison of the gradients of the tomography objective functional obtained using image-space encoded gathers with those obtained using the original shot gathers shows that those encoded shot gathers can be used in wave-equation tomography problems. Velocity inversion using image-space phase-encoded gathers converges to reasonable results when compared to the correct velocity model. We illustrate our method by applying it to the Marmousi model.

Selective stacking in the reflection‐angle and azimuth domain

I analytically demonstrate the artifacts in angle-domain commonimage gathers caused by sparsely sampled wavefields, from a perspective of shot-profile migration. The local-offset gather is linearly related to the angle gather in locally constant-velocity media when wavefields are sufficiently well sampled, but not when wavefields are poorly sampled. Hence, linear slant-stack or radial-trace transform in local-offset gathers will produce angle gathers with artifacts, which may hinder further interpretation or analysis and reduce the quality of the final stacking image in the angle and azimuth domain. Instead of simply stacking along reflection angle and azimuth axes, I present a method to compute the stacking weights as functions of angle and azimuth and make the stacking process selective. My method is tested on the synthetic wide-azimuth version of the SEG/EAGE salt data set, where a cleaner image with higher signal-to-noise ratio is obtained.

Target‐oriented wavefield tomography using demigrated Born data

We present a method to reduce the computational cost of image-domain wavefield tomography. Instead of using the originally recorded data for velocity estimation, the proposed method simulates a new data set obtained using Born modeling or demigration based on the initial image and gathers. The modeling can be performed in a target-oriented fashion, and it can use arbitrary types of source functions and acquisition geometries. Hence the size of the new data set can be substantially smaller than the original one. We demonstrate with numerical examples that the new data set correctly preserves velocity information useful for velocity estimation, and that it generates wavefield-tomography gradient similar to that obtained using the original data set. We apply the proposed method to a modified version of the Sigsbee2A model, where two square anomalies below the salt have been successfully recovered in a target-oriented fashion at much lower computational cost.

Subsalt velocity analysis by target‐oriented wavefield tomography: A 3‐D field‐data example

We apply target-oriented wavefield tomography to a 3-D field data set acquired from the Gulf of Mexico. Instead of using the original surface-recorded data set, we use a new data set synthesized specifically for velocity analysis to update subsalt velocities. The new data set is generated based on an initial unfocused target image and by a novel application of 3-D generalized Born wavefield modeling, which correctly preserves velocity kinematics by modeling non-zero subsurface-offsetdomain images. We show that the target-oriented inversion strategy drastically reduces the data size and the computation domain for 3-D wavefield tomography, greatly improving its efficiency and flexibility. We apply differential semblance optimization (DSO) using the synthesized new data set to optimize subsalt velocities. The updated velocity model significantly improves the continuity of subsalt reflectors and yields flattened angle-domain common-image gathers.

Selection of reference‐anisotropy parameters for wavefield extrapolation by Lloyd's algorithm

We propose a method for selecting reference anisotropic parameters in laterally varying anisotropic media for mixed Fourier-space domain waveeld extrapolation. We treat the selection problem as a quantization procedure, and use a modied version of the 3D Lloyd’s algoritm for reference-parameter selections. We demonstrate that our method yields a more accurate discription of the anisotropy model with fewer reference parameters than the uniform sampling approach. Real data examples illustrate the performance of our method.