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.

Least‐squares DMO and migration

Geophysics ◽  
2000 ◽  
Vol 65 (5) ◽  
pp. 1364-1371 ◽  
Author(s):  
Shuki Ronen ◽  
Christopher L. Liner
Keyword(s):  
Least Squares ◽  
Survey Design ◽  
A Priori ◽  
Computational Cost ◽  
Earth Model ◽  
Data Sampling ◽  
Sampled Data ◽  
Amplitude Variation With Offset ◽  
And Migration ◽  
Irregularly Sampled Data

Conventional processing, such as Kirchhoff dip moveout (DMO) and prestack full migration, are based on independent imaging of subsets of the data before stacking or amplitude variation with offset (AVO) analysis. Least‐squares DMO (LSDMO) and least‐squares migration (LSMig) are a family of developing processing methods which are based on inversion of reverse DMO and demigration operators. LSDMO and LSMig find the earth model that best fits the data and a priori assumptions which can be imposed as constraints. Such inversions are more computer intensive, but have significant advantages compared to conventional processing when applied to irregularly sampled data. Various conventional processes are approximations of the inversions in LSDMO and LSMig. Often, processing is equivalent to using the transpose of a matrix which LSDMO/LSMig inverts. Such transpose processing is accurate when the data sampling is adequate. In practice, costly survey design, real‐time coverage quality control, in‐fill acquisition, redundancy editing, and prestack interpolation, are used to create a survey geometry such that the transpose is a good approximation of the inverse. Normalized DMO and migration are approximately equivalent to following the application of the above transpose processing by a diagonal correction. However, in most cases, the required correction is not actually diagonal. In such cases LSDMO and LSMig can produce earth models with higher resolution and higher fidelity than normalized DMO and migration. The promise of LSMig and LSDMO is reduced acquisition cost, improved resolution, and reduced acquisition footprint. The computational cost, and more importantly turn‐around time, is a major factor in the commercialization of these methods. With parallel computing, these methods are now becoming practical.

scholarly journals Multilevel weighted least squares polynomial approximation

2020 ◽  
Vol 54 (2) ◽  
pp. 649-677 ◽  
Author(s):  
Abdul-Lateef Haji-Ali ◽  
Fabio Nobile ◽  
Raúl Tempone ◽  
Sören Wolfers
Keyword(s):  
Least Squares ◽  
Polynomial Approximation ◽  
Weighted Least Squares ◽  
A Priori ◽  
Computational Cost ◽  
Discretization Error ◽  
Logarithmic Factor ◽  
Complexity Bounds ◽  
Single Level ◽  
Random Samples

Weighted least squares polynomial approximation uses random samples to determine projections of functions onto spaces of polynomials. It has been shown that, using an optimal distribution of sample locations, the number of samples required to achieve quasi-optimal approximation in a given polynomial subspace scales, up to a logarithmic factor, linearly in the dimension of this space. However, in many applications, the computation of samples includes a numerical discretization error. Thus, obtaining polynomial approximations with a single level method can become prohibitively expensive, as it requires a sufficiently large number of samples, each computed with a sufficiently small discretization error. As a solution to this problem, we propose a multilevel method that utilizes samples computed with different accuracies and is able to match the accuracy of single-level approximations with reduced computational cost. We derive complexity bounds under certain assumptions about polynomial approximability and sample work. Furthermore, we propose an adaptive algorithm for situations where such assumptions cannot be verified a priori. Finally, we provide an efficient algorithm for the sampling from optimal distributions and an analysis of computationally favorable alternative distributions. Numerical experiments underscore the practical applicability of our method.

True-amplitude Gaussian-beam migration

Geophysics ◽  
2009 ◽  
Vol 74 (2) ◽  
pp. S11-S23 ◽  
Author(s):  
Samuel H. Gray ◽  
Norman Bleistein
Keyword(s):  
Wave Equation ◽  
Gaussian Beam ◽  
Opening Angle ◽  
Amplitude Wave ◽  
Amplitude Variation With Offset ◽  
Imaging Condition ◽  
Depth Migration ◽  
Kirchhoff Migration ◽  
Beam Depth ◽  
Gaussian Beam Migration

Gaussian-beam depth migration and related beam migration methods can image multiple arrivals, so they provide an accurate, flexible alternative to conventional single-arrival Kirchhoff migration. Also, they are not subject to the steep-dip limitations of many (so-called wave-equation) methods that use a one-way wave equation in depth to downward-continue wavefields. Previous presentations of Gaussian-beam migration have emphasized its kinematic imaging capabilities without addressing its amplitude fidelity. We offer two true-amplitude versions of Gaussian-beam migration. The first version combines aspects of the classic derivation of prestack Gaussian-beam migration with recent results on true-amplitude wave-equation migration, yields an expression involving a crosscorrelation imaging condition. To provide amplitude-versus-angle (AVA) information, true-amplitude wave-equation migration requires postmigration mapping from lateral distance (between image location and source location) to subsurface opening angle. However, Gaussian-beam migration does not require postmigration mapping to provide AVA data. Instead, the amplitudes and directions of the Gaussian beams provide information that the migration can use to produce AVA gathers as part of the migration process. The second version of true-amplitude Gaussian-beam migration is an expression involving a deconvolution imaging condition, yielding amplitude-variation-with-offset (AVO) information on migrated shot-domain common-image gathers.

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.

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.

Reflector imaging using trial reflector and crosscorrelation: Application to fracture imaging for sonic data

Geophysics ◽  
2016 ◽  
Vol 81 (6) ◽  
pp. S433-S446 ◽  
Author(s):  
Nobuyasu Hirabayashi
Keyword(s):  
Fresnel Zone ◽  
A Priori ◽  
Image Point ◽  
Complex Structures ◽  
A Priori Information ◽  
Kirchhoff Migration ◽  
Common Depth Point ◽  
Resolution Imaging ◽  
The Common ◽  
Priori Information

I have developed two high-resolution imaging methods that do not require a priori information on the structural dip. The dip information is often necessary to image complex structures using Kirchhoff migration by selecting only the constructively interfering parts of waveforms, especially for data with limited acquisition geometry. However, such dip information is not generally available. The methods that I evaluated use a trial reflector, which is defined for each image point and source-receiver pair, to search for the true geologic reflector. The coincidence of these reflectors is judged by a coherency analysis of event signals for the trial reflector using the crosscorrelations, and the coherency is converted to a weight. The weight is combined with the stacking methods of waveform samples in migration. In the first method, a waveform sample summed at an image point for a source-receiver pair is obtained by the common-depth-point stack of array data for the trial reflector. In the second method, a waveform sample of a source-receiver pair at the traveltime of the reflected ray for the trial reflector is smeared in the Fresnel zone computed for the trial reflector. My methods were applied to image fractures for sonic data, whose frequency range is centered approximately 8 kHz, and they provide higher resolution images than those given by conventional Kirchhoff migration.

Imaging salt bodies using explicit migration operators offshore Norway

Geophysics ◽  
2009 ◽  
Vol 74 (2) ◽  
pp. S25-S32 ◽  
Author(s):  
Børge Arntsen ◽  
Constantin Gerea ◽  
Tage Røsten
Keyword(s):  
Image Quality ◽  
Computational Cost ◽  
Fourier Method ◽  
Frequency Content ◽  
Data Set ◽  
Depth Migration ◽  
Kirchhoff Migration ◽  
Velocity Changes ◽  
The Cost ◽  
Better Than

We have tested the performance of 3D shot-profile depth migration using explicit migration operators on a real 3D marine data set. The data were acquired offshore Norway in an area with a complex subsurface containing large salt bodies. We compared shot-profile migration using explicit migration operators with conventional Kirchhoff migration, split-step Fourier migration, and common-azimuth by generalized screen propagator (GSP) migration in terms of quality and computational cost. Image quality produced by the explicit migration operator approach is slightly better than with split-step Fourier migration and clearly better than in common-azimuth by GSP and Kirchhoff migrations. The main differences are fewer artifacts and better-suppressed noise within the salt bodies. Kirchhoff migration shows considerable artifacts (migration smiles) within and close to the salt bodies, which are not present in images produced by the other three wave-equation methods. Expressions for computational cost were developed for all four migration algorithms in terms of frequency content and acquisition parameters. For comparable frequency content, migration cost using explicit operators is four times the cost of the split-step Fourier method, up to 260 times the cost of common-azimuth by GSP migration, and 25 times the cost of Kirchhoff migration. Our results show that in terms of image quality, shot-profile migration using explicit migration operators is well suited for imaging in areas with complex geology and significant velocity changes. However, computational cost of the method is high and makes it less attractive in terms of efficiency.

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