Least‐squares wave‐equation migration for AVP/AVA inversion

Geophysics ◽  
2003 ◽  
Vol 68 (1) ◽  
pp. 262-273 ◽  
Author(s):  
Henning Kühl ◽  
Mauricio D. Sacchi
Keyword(s):  
Wave Equation ◽  
Least Squares ◽  
Inversion Algorithm ◽  
Constrained Least Squares ◽  
Data Set ◽  
Parameter Domain ◽  
First Case ◽  
Least Squares Migration ◽  
Ray Parameter ◽  
Seismic Wavefield

We present an acoustic migration/inversion algorithm that uses extended double‐square‐root wave‐equation migration and modeling operators to minimize a constrained least‐squares data misfit function (least‐squares migration). We employ an imaging principle that allows for the extraction of ray‐parameter‐domain common image gathers (CIGs) from the propagated seismic wavefield. The CIGs exhibit amplitude variations as a function of half‐offset ray parameter (AVP) closely related to the amplitude variation with reflection angle (AVA). Our least‐squares wave‐equation migration/inversion is constrained by a smoothing regularization along the ray parameter. This approach is based on the idea that rapid amplitude changes or discontinuities along the ray parameter axis result from noise, incomplete wavefield sampling, and numerical operator artifacts. These discontinuities should therefore be penalized in the inversion. The performance of the proposed algorithm is examined with two synthetic examples. In the first case, we generated acoustic finite difference data for a horizontally layered model. The AVP functions based on the migrated/inverted ray parameter CIGs were converted to AVA plots. The AVA plots were then compared to the true acoustic AVA of the reflectors. The constrained least‐squares inversion compares favorably with the conventional migration, especially when incompleteness compromises the data. In the second example, we use the Marmousi data set to test the algorithm in complex media. The result shows that least‐squares migration can mitigate kinematic artifacts in the ray‐parameter domain CIGs effectively.

High-resolution wave-equation AVA imaging: Algorithm and tests with a data set from the Western Canadian Sedimentary Basin

Geophysics ◽  
2005 ◽  
Vol 70 (5) ◽  
pp. S91-S99 ◽  
Author(s):  
Juefu Wang ◽  
Henning Kuehl ◽  
Mauricio D. Sacchi
Keyword(s):  
Inverse Problem ◽  
Wave Equation ◽  
Least Squares ◽  
Oil And Gas ◽  
Sedimentary Basin ◽  
Model Space ◽  
Sampled Data ◽  
Inversion Algorithm ◽  
Data Set ◽  
Data Space

This paper presents a 3D least-squares wave-equation migration method that yields regularized common-image gathers (CIGs) for amplitude-versus-angle (AVA) analysis. In least-squares migration, we pose seismic imaging as a linear inverse problem; this provides at least two advantages. First, we are able to incorporate model-space weighting operators that improve the amplitude fidelity of CIGs. Second, the influence of improperly sampled data (footprint noise) can be diminished by incorporating data-space weighting operators. To investigate the viability of this class of methods for oil and gas exploration, we test the algorithm with a real-data example from the Western Canadian Sedimentary Basin. To make our problem computationally feasible, we utilize the 3D common-azimuth approximation in the migration algorithm. The inversion algorithm uses the method of conjugate gradients with the addition of a ray-parameter-dependent smoothing constraint that minimizes sampling and aperture artifacts. We show that more robust AVA attributes can be obtained by properly selecting the model and data-space regularization operators. The algorithm is implemented in conjunction with a preconditioning strategy to accelerate convergence. Posing the migration problem as an inverse problem leads to enhanced event continuity in CIGs and, hence, more reliable AVA estimates. The vertical resolution of the inverted image also improves as a consequence of increased coherence in CIGs and, in addition, by implicitly introducing migration deconvolution in the inversion.

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

Geophysics ◽  
2009 ◽  
Vol 74 (6) ◽  
pp. WCA95-WCA107 ◽  
Author(s):  
Yaxun Tang
Keyword(s):  
Wave Equation ◽  
Least Squares ◽  
Green's Functions ◽  
Green’S Functions ◽  
Random Phase ◽  
Inverse Filtering ◽  
Data Set ◽  
Phase Encoding ◽  
Hessian Operator ◽  
Least Squares Migration

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 Green’s 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 Green’s 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.

Wave‐equation migration velocity analysis based on common imaging gathers in the ray parameter domain

2003 ◽  
Author(s):  
Shuqing Yang ◽  
Huazhong Wang ◽  
Jiubing Cheng ◽  
Zaitian Ma ◽  
Jianping Chen ◽  
...  
Keyword(s):  
Wave Equation ◽  
Migration Velocity ◽  
Velocity Analysis ◽  
Parameter Domain ◽  
Migration Velocity Analysis ◽  
Ray Parameter

Kirchhoff modeling, inversion for reflectivity, and subsurface illumination

Geophysics ◽  
2000 ◽  
Vol 65 (4) ◽  
pp. 1195-1209 ◽  
Author(s):  
Bertrand Duquet ◽  
Kurt J. Marfurt ◽  
Joe A. Dellinger
Keyword(s):  
Least Squares ◽  
A Priori ◽  
Computational Cost ◽  
Image Point ◽  
Surface Sampling ◽  
Constrained Least Squares ◽  
Amplitude Variation With Offset ◽  
Depth Migration ◽  
Kirchhoff Migration ◽  
Least Squares Migration

Because of its computational efficiency, prestack Kirchhoff depth migration is currently one of the most popular algorithms used in 2-D and 3-D subsurface depth imaging. Nevertheless, Kirchhoff algorithms in their typical implementation produce less than ideal results in complex terranes where multipathing from the surface to a given image point may occur, and beneath fast carbonates, salt, or volcanics through which ray‐theoretical energy cannot penetrate to illuminate underlying slower‐velocity sediments. To evaluate the likely effectiveness of a proposed seismic‐acquisition program, we could perform a forward‐modeling study, but this can be expensive. We show how Kirchhoff modeling can be defined as the mathematical transpose of Kirchhoff migration. The resulting Kirchhoff modeling algorithm has the same low computational cost as Kirchhoff migration and, unlike expensive full acoustic or elastic wave‐equation methods, only models the events that Kirchhoff migration can image. Kirchhoff modeling is also a necessary element of constrained least‐squares Kirchhoff migration. We show how including a simple a priori constraint during the inversion (that adjacent common‐offset images should be similar) can greatly improve the resulting image by partially compensating for irregularities in surface sampling (including missing data), as well as for irregularities in ray coverage due to strong lateral variations in velocity and our failure to account for multipathing. By allowing unstacked common‐offset gathers to become interpretable, the additional cost of constrained least‐squares migration may be justifiable for velocity analysis and amplitude‐variation‐with‐offset studies. One useful by‐product of least‐squares migration is an image of the subsurface illumination for each offset. If the data are sufficiently well sampled (so that including the constraint term is not necessary), the illumination can instead be calculated directly and used to balance the result of conventional migration, obtaining most of the advantages of least‐squares migration for only about twice the cost of conventional migration.

Modified imaging principle: An alternative to preserved-amplitude least-squares wave-equation migration

Geophysics ◽  
2007 ◽  
Vol 72 (6) ◽  
pp. S221-S230 ◽  
Author(s):  
Denis Kiyashchenko ◽  
René-Edouard Plessix ◽  
Boris Kashtan
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Acoustic Wave ◽  
Least Squares ◽  
Forward Modeling ◽  
Acoustic Wave Equation ◽  
Impedance Contrast ◽  
Least Squares Migration

Impedance contrast images can result from a least-squares migration or from a modified imaging principle. Theoretically, the two approaches should give similar results, but in practice they lead to different estimates of the impedance contrasts because of limited acquisition geometry, difficulty in computing exact weights for least-squares migration, and small contrast approximation. To analyze those differences, we compare the two approaches based on 2D synthetics. Forward modeling is either a finite-difference solver of the full acoustic wave equation or a one-way wave-equation solver that correctly models the amplitudes. The modified imaging principle provides better amplitude estimates of the impedance contrasts and does not suffer from the artifacts at-tributable to diving waves, which can be seen in two-way, least-squares migrated sections. However, because of the shot-based formulation, artifacts appear in the modified imaging principle results in shadow zones where energy is defocused. Those artifacts do not exist with the least-squares migration algorithm because all shots are processed simultane-ously.

Least-squares Gaussian beam migration

Geophysics ◽  
2016 ◽  
Vol 81 (3) ◽  
pp. S87-S100 ◽  
Author(s):  
Hao Hu ◽  
Yike Liu ◽  
Yingcai Zheng ◽  
Xuejian Liu ◽  
Huiyi Lu
Keyword(s):  
Least Squares ◽  
Gaussian Beam ◽  
Spatial Resolution ◽  
Field Data ◽  
Synthetic Data ◽  
Data Sets ◽  
Data Set ◽  
Seismic Acquisition ◽  
Gaussian Beam Migration ◽  
Least Squares Migration

Least-squares migration (LSM) can be effective to mitigate the limitation of finite-seismic acquisition, balance the subsurface illumination, and improve the spatial resolution of the image, but it requires iterations of migration and demigration to obtain the desired subsurface reflectivity model. The computational efficiency and accuracy of migration and demigration operators are crucial for applying the algorithm. We have developed a test of the feasibility of using the Gaussian beam as the wavefield extrapolating operator for the LSM, denoted as least-squares Gaussian beam migration. Our method combines the advantages of the LSM and the efficiency of the Gaussian beam propagator. Our numerical evaluations, including two synthetic data sets and one marine field data set, illustrate that the proposed approach could be used to obtain amplitude-balanced images and to broaden the bandwidth of the migrated images in particular for the low-wavenumber components.

scholarly journals Image domain least-squares migration with a Hessian matrix estimated by non-stationary matching filters

2019 ◽  
Vol 17 (1) ◽  
pp. 148-159 ◽  
Author(s):  
Song Guo ◽  
Huazhong Wang
Keyword(s):  
Least Squares ◽  
Numerical Experiments ◽  
Field Data ◽  
Hessian Matrix ◽  
Computational Cost ◽  
Image Deblurring ◽  
Data Set ◽  
Image Domain ◽  
Slow Convergence ◽  
Least Squares Migration

Abstract Assuming that an accurate background velocity is obtained, least-squares migration (LSM) can be used to estimate underground reflectivity. LSM can be implemented in either the data domain or image domain. The data domain LSM (DDLSM) is not very practical because of its huge computational cost and slow convergence rate. The image domain LSM (IDLSM) might be a flexible alternative if estimating the Hessian matrix using a cheap and accurate approach. It has practical potential to analyse convenient Hessian approximation methods because the Hessian matrix is too huge to compute and save. In this paper, the Hessian matrix is approximated with non-stationary matching filters. The filters are calculated to match the conventional migration image to the demigration/remigration image. The two images are linked by the Hessian matrix. An image deblurring problem is solved with the estimated filters for the IDLSM result. The combined sparse and total variation regularisations are used to produce accurate and reasonable inversion results. The numerical experiments based on part of Sigsbee model, Marmousi model and a 2D field data set illustrate that the non-stationary matching filters can give a good approximation for the Hessian matrix, and the results of the image deblurring problem with combined regularisations can provide high-resolution and true-amplitude reflectivity estimations.

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