Flexibly preconditioned extended least-squares migration in shot-record domain

Geophysics ◽  
2016 ◽  
Vol 81 (5) ◽  
pp. S299-S315 ◽  
Author(s):  
Yin Huang ◽  
Rami Nammour ◽  
William Symes
Keyword(s):  
Least Squares ◽  
Conjugate Gradient ◽  
Numerical Experiments ◽  
Computational Effort ◽  
Amplitude Variation ◽  
Image Stack ◽  
Amplitude Variation With Offset ◽  
Image Volume ◽  
Spatially Varying ◽  
Least Squares Migration

We have developed a method for accelerating the convergence of iterative least-squares migration. The algorithm uses a pseudodifferential scaling (dip and spatially varying filter) preconditioner together with a variant of conjugate gradient (CG) iteration with iterate-dependent (flexible) preconditioning. The migration is formulated without the image stack, thus producing a shot-dependent image volume that retains offset information useful for velocity updating and amplitude variation with offset analysis. Numerical experiments indicate that flexible preconditioning with pseudodifferential scaling not only attains considerably smaller data misfit and gradient error for a given computational effort, but also produces higher resolution image volumes with more balanced amplitude and fewer artifacts than is achieved with a nonpreconditioned CG method.

Model misspecification and bias in the least-squares algorithm: Implications for linearized isotropic AVO

2021 ◽  
Vol 40 (9) ◽  
pp. 646-654
Author(s):  
Henning Hoeber
Keyword(s):  
Least Squares ◽  
Model Misspecification ◽  
Forward Model ◽  
Model Parameters ◽  
Omitted Variables ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Omitted Variable Bias ◽  
Least Squares Fit ◽  
Variable Bias

When inversions use incorrectly specified models, the estimated least-squares model parameters are biased. Their expected values are not the true underlying quantitative parameters being estimated. This means the least-squares model parameters cannot be compared to the equivalent values from forward modeling. In addition, the bias propagates into other quantities, such as elastic reflectivities in amplitude variation with offset (AVO) analysis. I give an outline of the framework to analyze bias, provided by the theory of omitted variable bias (OVB). I use OVB to calculate exactly the bias due to model misspecification in linearized isotropic two-term AVO. The resulting equations can be used to forward model unbiased AVO quantities, using the least-squares fit results, the weights given by OVB analysis, and the omitted variables. I show how uncertainty due to bias propagates into derived quantities, such as the χ-angle and elastic reflectivity expressions. The result can be used to build tables of unique relative rock property relationships for any AVO model, which replace the unbiased, forward-model results.

scholarly journals The conjugate gradient method

2018 ◽  
Vol 37 (4) ◽  
pp. 296-298 ◽  
Author(s):  
Karl Schleicher
Keyword(s):  
Least Squares ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Waveform Inversion ◽  
Linear Inversion ◽  
Large Computer ◽  
Traveltime Tomography ◽  
Full Waveform ◽  
Least Squares Migration

The conjugate gradient method can be used to solve many large linear geophysical problems — for example, least-squares parabolic and hyperbolic Radon transform, traveltime tomography, least-squares migration, and full-waveform inversion (FWI) (e.g., Witte et al., 2018 ). This tutorial revisits the “Linear inversion tutorial” ( Hall, 2016 ) that estimated reflectivity by deconvolving a known wavelet from a seismic trace using least squares. This tutorial solves the same problem using the conjugate gradient method. This problem is easy to understand, and the concepts apply to other applications. The conjugate gradient method is often used to solve large problems because the least-squares algorithm is much more expensive — that is, even a large computer may not be able to find a useful solution in a reasonable amount of time.

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.

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.

Accelerating extended least-squares migration with weighted conjugate gradient iteration

Geophysics ◽  
2016 ◽  
Vol 81 (4) ◽  
pp. S165-S179 ◽  
Author(s):  
Jie Hou ◽  
William W. Symes
Keyword(s):  
Least Squares ◽  
Conjugate Gradient ◽  
Degrees Of Freedom ◽  
Velocity Model ◽  
Domain Model ◽  
Gradient Algorithm ◽  
Range Data ◽  
Seismic Reflection Data ◽  
Reflection Data ◽  
Least Squares Migration

Least-squares migration (LSM) iteratively achieves a mean-square best fit to seismic reflection data, provided that a kinematically accurate velocity model is available. The subsurface offset extension adds extra degrees of freedom to the model, thereby allowing LSM to fit the data even in the event of significant velocity error. This type of extension also implies additional computational expense per iteration from crosscorrelating source and receiver wavefields over the subsurface offset, and therefore places a premium on rapid convergence. We have accelerated the convergence of extended least-squares migration by combining the conjugate gradient algorithm with weighted norms in range (data) and domain (model) spaces that render the extended Born modeling operator approximately unitary. We have developed numerical examples that demonstrate that the proposed algorithm dramatically reduces the number of iterations required to achieve a given level of fit or gradient reduction compared with conjugate gradient iteration with Euclidean (unweighted) norms.

scholarly journals Experimental verification of spatially varying fracture-compliance estimates obtained from amplitude variation with offset inversion coupled with linear slip theory

Geophysics ◽  
2018 ◽  
Vol 83 (1) ◽  
pp. WA1-WA8 ◽  
Author(s):  
Shohei Minato ◽  
Ranajit Ghose ◽  
Godfred Osukuku
Keyword(s):  
Bulk Modulus ◽  
Frequency Dependence ◽  
Hydraulic Properties ◽  
Slip Model ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Avo Inversion ◽  
Fractured Medium ◽  
Spatially Varying ◽  
Linear Slip Model

The elastic compliance of a fracture can be spatially varying, reflecting the variation of microscale properties of the fracture, e.g., aperture, contact asperities, and fracture infill. Characterizing the spatial heterogeneity of a fracture is crucial in explaining the apparent frequency dependence of fracture compliance and in addressing the spatially varying mechanical and hydraulic properties of the fractured medium. Apparent frequency dependence of the estimated fracture compliance is caused when the used seismic wavelength is very large compared to the scale of heterogeneity. We perform ultrasonic laboratory experiments, and characterize the spatially varying compliance along a fluid-filled fracture. We simulate a horizontal fracture, and introduce heterogeneous fluid distribution along the fracture. We perform amplitude variation with offset (AVO) inversion of the P-P reflections, in which we obtain the theoretical angle-dependent reflection responses by considering the linear-slip model. The estimated compliance distribution clearly separates the dry region from the wet region of the fracture. The effective bulk modulus of the fluid is estimated using the derived values of the compliance. We find that the obtained bulk modulus is well-explained by the presence of minute quantity of air bubbles in the water. We also find new evidence of the existence of scattered waves generated at the boundary representing a sharp change in fracture compliance. The estimated boundary between the dry and the wet regions of the fracture, which is detected by AVO inversion, is slightly shifted compared with the actual location. This is possibly due to the interference of the scattered waves that are generated at the boundary. The linear-slip model can represent thin structures in rocks in a wide range of scale. Therefore, our methodology, results, and discussion will be useful in developing new applications for assessing laterally varying mechanical and hydraulic properties of thin nonwelded discontinuities, e.g., fractures, joints, and faults.

Least Squares Migration - Improvements in Quantitative Analysis

2019 ◽  
Author(s):  
Bruno Dias ◽  
Cláudio Guerra ◽  
André Bulcão ◽  
Roberto Dias
Keyword(s):  
Quantitative Analysis ◽  
Least Squares ◽  
Least Squares Migration

Iterative least-squares migration for high-resolution angle gathers

2020 ◽  
Author(s):  
Lian Duan ◽  
Alejandro Valenciano ◽  
Nizar Chemingui
Keyword(s):  
High Resolution ◽  
Least Squares ◽  
Least Squares Migration
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