Data reconstruction with shot-profile least-squares migration

Kaplan, Sam T.; Naghizadeh, Mostafa; Sacchi, Mauricio D.

doi:10.1190/1.3478375

Cited by 44 publications

(6 citation statements)

References 22 publications

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“…These weight functions often include several components in a chain of operators, some of which play a geophysical preconditioner role. For example, W m may contain an illumination compensation function, as discussed later, a Laplacian filter to decrease the effect of low wavenumbers in the gradient (Rocha, Sava and Guitton ) and smoothness constraints on an extended model (Kühl and Sacchi ; Kaplan, Naghizadeh and Sacchi ; Trad ; Hou and Symes ). When the inversion is performed in the image space instead of the data space (Schuster )

J = false∥ W_{m}^{H} L^{H} boldd - W_{m}^{H} L^{H} boldL W_{m} {m ∥}^{bold2},

where W m can take more complex forms like matching filters to remove spreading functions (Yu et al .…”

Section: Nuts and Bolts Of Least Squares Migrationmentioning

confidence: 99%

Assumptions and goals for least squares migration

Trad

2019

Geophysical Prospecting

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Least squares migration uses the assumption that, if we have an operator that can create data from a reflectivity function, the optimal image will predict the actual recorded data with minimum square error. For this assumption to be true, it is also required that: (a) the prediction operator must be error‐free, (b) model elements not seen by the operator should be constrained by other means and (c) data weakly predicted by the operator should make limited contribution to the solution. Under these conditions, least squares migration has the advantage over simple migration of being able to remove interference between different model components. Least squares migration does that by de‐convolving or inverting the so‐called Hessian operator. The Hessian is the cascade of forward modelling and migration; for each image point, it computes the effects of interference from other image points (point‐spread function) given the actual recording geometry and the subsurface velocity model. Because the Hessian contains illumination information (along its diagonal), and information about the model cross‐correlation produced by non‐orthogonality of basis functions, its inversion produces illumination compensation and increases resolution. In addition, sampling deficiencies in the recording geometry map to the Hessian (both diagonal and non‐diagonal elements), so least squares migration has the potential to remove sampling artefacts as well. These (illumination compensation, resolution and mitigating recording deficiencies) are the three main goals of least squares migration, although the first one can be achieved by cheaper techniques. To invert the Hessian, least squares migration relies on the residual errors during iterations. Iterative algorithms, like conjugate gradient and others, use the residuals to calculate the direction and amplitudes (gradient and step size) of the necessary corrections to the reflectivity function or model. Failure of conditions (a), (b) or (c) leads the inversion to calculate incorrect model updates, which translate to noise in the final image. In this paper, we will discuss these conditions for Kirchhoff migration and reverse time migration.

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J = false∥ W_{m}^{H} L^{H} boldd - W_{m}^{H} L^{H} boldL W_{m} {m ∥}^{bold2},

where W m can take more complex forms like matching filters to remove spreading functions (Yu et al .…”

Section: Nuts and Bolts Of Least Squares Migrationmentioning

confidence: 99%

Assumptions and goals for least squares migration

Trad

2019

Geophysical Prospecting

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“…Imaging can be viewed as the exact or approximate inversion of this relationship. What follows is a brief outline and simplified implementation of the shot-profile algorithm due to Kaplan, Naghizadeh, and Sacchi (2010a); Kaplan, Routh, and Sacchi (2010b).…”

Section: A Multidimensional Model Of Time-lapse Difference Datamentioning

confidence: 99%

Perturbation methods for two special cases of the time‐lapse seismic inverse problem

Innanen

Naghizadeh

Kaplan

2014

Geophysical Prospecting

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Scattering theory, a form of perturbation theory, is a framework from within which time‐lapse seismic reflection methods can be derived and understood. It leads to expressions relating baseline and monitoring data and Earth properties, focusing on differences between these quantities as it does so. The baseline medium is, in the language of scattering theory, the reference medium and the monitoring medium is the perturbed medium. The general scattering relationship between monitoring data, baseline data, and time‐lapse Earth property changes is likely too complex to be tractable. However, there are special cases that can be analysed for physical insight. Two of these cases coincide with recognizable areas of applied reflection seismology: amplitude versus offset modelling/inversion, and imaging. The main result of this paper is a demonstration that time‐lapse difference amplitude versus offset modelling, and time‐lapse difference data imaging, emerge from a single theoretical framework. The time‐lapse amplitude versus offset case is considered first. We constrain the general time‐lapse scattering problem to correspond with a single immobile interface that separates a static overburden from a target medium whose properties undergo time‐lapse changes. The scattering solutions contain difference‐amplitude versus offset expressions that (although presently acoustic) resemble the expressions of Landro (). In addition, however, they contain non‐linear corrective terms whose importance becomes significant as the contrasts across the interface grow. The difference‐amplitude versus offset case is exemplified with two parameter acoustic (bulk modulus and density) and anacoustic (P‐wave velocity and quality factor Q) examples. The time‐lapse difference data imaging case is considered next. Instead of constraining the structure of the Earth volume as in the amplitude versus offset case, we instead make a small‐contrast assumption, namely that the time‐lapse variations are small enough that we may disregard contributions from beyond first order. An initial analysis, in which the case of a single mobile boundary is examined in 1D, justifies the use of a particular imaging algorithm applied directly to difference data shot records. This algorithm, a least‐squares, shot‐profile imaging method, is additionally capable of supporting a range of regularization techniques. Synthetic examples verify the applicability of linearized imaging methods of the difference image formation under ideal conditions.

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“…We can distinguish between methods that rely on a direct inversion of the combination of migration and demigration (Ferguson, 2006;Stolt, 2002) and methods that consecutively apply the migration and modeling operators to reconstruct the data at the new locations. The latter methods can be separated into algorithms based on partial prestack migration (Chemingui & Biondi, 2002;Ronen, 1987) and those based on full prestack migration, either with a Kirchhoff operator (Duquet et al, 2000;Nemeth et al, 1999;Santos et al, 2000) or with a wavefield-continuation operator (Kaplan et al, 2010;Kühl & Sacchi, 2003;Trad, 2003).…”

Section: Pre-processing For Reducing Migration Artifactsmentioning

confidence: 99%