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 migration with a blockwise Hessian matrix: A prestack time-migration approach

Geophysics ◽  
2019 ◽  
Vol 84 (4) ◽  
pp. R625-R640 ◽  
Author(s):  
Bowu Jiang ◽  
Jianfeng Zhang
Keyword(s):  
Least Squares ◽  
Field Data ◽  
Hessian Matrix ◽  
Total Variation Regularization ◽  
Data Sets ◽  
Reflection Point ◽  
Inverse Approach ◽  
Time Migration ◽  
Prestack Time Migration ◽  
Least Squares Migration

We have developed an explicit inverse approach with a Hessian matrix for the least-squares (LS) implementation of prestack time migration (PSTM). A full Hessian matrix is divided into a series of computationally tractable small-sized matrices using a localized approach, thus significantly reducing the size of the inversion. The scheme is implemented by dividing the imaging volume into a series of subvolumes related to the blockwise Hessian matrices that govern the mapping relationship between the migrated result and corresponding reflectivity. The proposed blockwise LS-PSTM can be implemented in a target-oriented fashion. The localized approach that we use to modify the Hessian matrix can eliminate the boundary effects originating from the blockwise implementation. We derive the explicit formula of the offset-dependent Hessian matrix using the deconvolution imaging condition with an analytical Green’s function of PSTM. This avoids the challenging task of estimating the source wavelet. Moreover, migrated gathers can be generated with the proposed scheme. The smaller size of the blockwise Hessian matrix makes it possible to incorporate the total-variation regularization into the inversion, thus attenuating noises significantly. We revealed the proposed blockwise LS-PSTM with synthetic and field data sets. Higher quality common-reflection-point gathers of the field data are obtained.

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.

Least‐squares migration of incomplete reflection data

Geophysics ◽  
1999 ◽  
Vol 64 (1) ◽  
pp. 208-221 ◽  
Author(s):  
Tamas Nemeth ◽  
Chengjun Wu ◽  
Gerard T. Schuster
Keyword(s):  
Least Squares ◽  
Frequency Distribution ◽  
Field Data ◽  
Radar Data ◽  
Data Set ◽  
Frequency Distributions ◽  
Vertical Seismic Profiling ◽  
Reflection Data ◽  
Ground Penetrating ◽  
Least Squares Migration

A least‐squares migration algorithm is presented that reduces the migration artifacts (i.e., recording footprint noise) arising from incomplete data. Instead of migrating data with the adjoint of the forward modeling operator, the normal equations are inverted by using a preconditioned linear conjugate gradient scheme that employs regularization. The modeling operator is constructed from an asymptotic acoustic integral equation, and its adjoint is the Kirchhoff migration operator. We tested the performance of the least‐squares migration on synthetic and field data in the cases of limited recording aperture, coarse sampling, and acquisition gaps in the data. Numerical results show that the least‐squares migrated sections are typically more focused than are the corresponding Kirchhoff migrated sections and their reflectivity frequency distributions are closer to those of the true model frequency distribution. Regularization helps attenuate migration artifacts and provides a sharper, better frequency distribution of estimated reflectivity. The least‐squares migrated sections can be used to predict the missing data traces and interpolate and extrapolate them according to the governing modeling equations. Several field data examples are presented. A ground‐penetrating radar data example demonstrates the suppression of the recording footprint noise due to a limited aperture, a large gap, and an undersampled receiver line. In addition, better fault resolution was achieved after applying least‐squares migration to a poststack marine data set. And a reverse vertical seismic profiling example shows that the recording footprint noise due to a coarse receiver interval can be suppressed by least‐squares migration.

scholarly journals Viscoacoustic least-squares migration with a blockwise Hessian matrix: an effective Q approach

2020 ◽  
Author(s):  
Mingpeng Song ◽  
Jianfeng Zhang ◽  
Jiangjie Zhang
Keyword(s):  
Least Squares ◽  
Field Data ◽  
Hessian Matrix ◽  
Data Sets ◽  
Matrix Approach ◽  
Inverse Approach ◽  
Time Migration ◽  
Q Model ◽  
Prestack Time Migration ◽  
Least Squares Migration

Abstract We present an explicit inverse approach using a Hessian matrix for least-squares migration (LSM) with Q compensation. The scheme is developed by incorporating an effective Q-based solution of the viscoacoustic wave equation into a blockwise approximation to the Hessian in LSM, which is implemented after the so-called deabsorption prestack time migration (PSTM). The effective Q model used fully accounts for frequency-dependent traveltime and amplitude at the same imaging location. We can extract the effective Q parameters by scanning during previous deabsorption PSTM. This avoids the challenging task of building the Q model. The blockwise Hessian matrix approach decomposes the full Hessian matrix into a series of computationally tractable small-sized matrices using a localised approach. We derive the explicit formula of the offset-dependent Hessian matrix using an analytical Green's function obtained from deabsorption PSTM. In this way, we can approximate a reflectivity imaging for the targeted zone by a spatial deconvolution of the migrated result with an explicit inverse. The resulting scheme broadens the frequency-band of imaging by deabsorption, and improves the subsurface illumination and spatial resolution through the inverse Hessian. A high-resolution, true-amplitude migrated gather can then be obtained. Synthetic and field data sets demonstrate the proposed blockwise LS-QPSTM.

A compressed data approach for image-domain least-squares migration

Geophysics ◽  
2021 ◽  
pp. 1-37
Author(s):  
Ram Tuvi ◽  
Zeyu Zhao ◽  
Mrinal Kanti Sen
Keyword(s):  
Least Squares ◽  
Hessian Matrix ◽  
Wide Band ◽  
Summation Method ◽  
Basis Functions ◽  
Image Domain ◽  
Spectral Localization ◽  
Imaging Point ◽  
Least Squares Migration ◽  
Compressed Data

We consider the problem of image-domain least-squares migration based on efficiently constructing the Hessian matrix with sparse beam data. Specifically, we use the ultra-wide-band phase space beam summation method, where beams are used as local basis functions to represent scattered data collected at the surface. The beam domain data are sparse. One can identify seismic events with significant contributions so that only beams with non-negligible amplitudes need to be used to image the subsurface. In addition, due to the beams' spectral localization, only beams that pass near an imaging point need to be taken into account. These two properties reduce the computational complexity of computing the Hessian matrix - an essential ingredient for least-squares migration. As a result, we can efficiently construct the Hessian matrix based on analyzing the sparse beam domain data.

Adaptive sampling of potential-field data: A direct approach to compressive inversion

Geophysics ◽  
2014 ◽  
Vol 79 (1) ◽  
pp. IM1-IM9 ◽  
Author(s):  
Nathan Leon Foks ◽  
Richard Krahenbuhl ◽  
Yaoguo Li
Keyword(s):  
Potential Field ◽  
Adaptive Sampling ◽  
Field Data ◽  
Computational Cost ◽  
Large Field ◽  
Magnetic Data ◽  
Data Sampling ◽  
Data Types ◽  
Data Set ◽  
Potential Field Data

Compressive inversion uses computational algorithms that decrease the time and storage needs of a traditional inverse problem. Most compression approaches focus on the model domain, and very few, other than traditional downsampling focus on the data domain for potential-field applications. To further the compression in the data domain, a direct and practical approach to the adaptive downsampling of potential-field data for large inversion problems has been developed. The approach is formulated to significantly reduce the quantity of data in relatively smooth or quiet regions of the data set, while preserving the signal anomalies that contain the relevant target information. Two major benefits arise from this form of compressive inversion. First, because the approach compresses the problem in the data domain, it can be applied immediately without the addition of, or modification to, existing inversion software. Second, as most industry software use some form of model or sensitivity compression, the addition of this adaptive data sampling creates a complete compressive inversion methodology whereby the reduction of computational cost is achieved simultaneously in the model and data domains. We applied the method to a synthetic magnetic data set and two large field magnetic data sets; however, the method is also applicable to other data types. Our results showed that the relevant model information is maintained after inversion despite using 1%–5% of the data.

Migration deconvolution using domain decomposition

Geophysics ◽  
2021 ◽  
pp. 1-61
Author(s):  
Luana Nobre Osorio ◽  
Bruno Pereira-Dias ◽  
André Bulcão ◽  
Luiz Landau
Keyword(s):  
Domain Decomposition ◽  
Least Squares ◽  
Iterative Algorithms ◽  
Image Resolution ◽  
Image Domain ◽  
Hessian Operator ◽  
Incomplete Coverage ◽  
And Migration ◽  
Least Squares Migration ◽  
Data Frequency

Least-squares migration (LSM) is an effective technique for mitigating blurring effects and migration artifacts generated by the limited data frequency bandwidth, incomplete coverage of geometry, source signature, and unbalanced amplitudes caused by complex wavefield propagation in the subsurface. Migration deconvolution (MD) is an image-domain approach for least-squares migration, which approximates the Hessian operator using a set of precomputed point spread functions (PSFs). We introduce a new workflow by integrating the MD and the domain decomposition (DD) methods. The DD techniques aim to solve large and complex linear systems by splitting problems into smaller parts, facilitating parallel computing, and providing a higher convergence in iterative algorithms. The following proposal suggests that instead of solving the problem in a unique domain, as conventionally performed, we split the problem into subdomains that overlap and solve each of them independently. We accelerate the convergence rate of the conjugate gradient solver by applying the DD methods to retrieve a better reflectivity, which is mainly visible in regions with low amplitudes. Moreover, using the pseudo-Hessian operator, the convergence of the algorithm is accelerated, suggesting that the inverse problem becomes better conditioned. Experiments using the synthetic Pluto model demonstrate that the proposed algorithm dramatically reduces the required number of iterations while providing a considerable enhancement in the image resolution and better continuity of poorly illuminated events.

Image-Domain Least-Squares Migration for RTM Surface-Offset Gathers

2019 ◽  
Author(s):  
W. Dai ◽  
Z. Xu ◽  
X. Cheng ◽  
K. Jiao ◽  
D. Vigh
Keyword(s):  
Least Squares ◽  
Image Domain ◽  
Least Squares Migration

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.

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