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.

Nonlinear inversion of seismic reflection data in a laterally invariant medium

Geophysics ◽  
1990 ◽  
Vol 55 (3) ◽  
pp. 284-292 ◽  
Author(s):  
A. Pica ◽  
J. P. Diet ◽  
A. Tarantola
Keyword(s):  
Conjugate Gradient ◽  
Velocity Model ◽  
Stratified Medium ◽  
Single Shot ◽  
Nonlinear Inversion ◽  
Reverse Time ◽  
Data Set ◽  
Seismic Reflection Data ◽  
Seismic Waveforms ◽  
Reflection Data

Interpretation of seismic waveforms can be expressed as an optimization problem based on a non‐linear least‐squares criterion to find the model which best explains the data. An initial model is corrected iteratively using a gradient method (conjugate gradient). At each iteration, computation of the direction of the model perturbation requires the forward propagation of the actual sources and the reverse‐time propagation of the residuals (misfit between the data and the synthetics); the two wave fields thus obtained are then correlated. An extra forward propagation is required to compute the amplitude of the perturbation along the conjugate‐gradient direction. The number of propagations to be simulated numerically in each iteration equals three times the number of shots. Since a 2-D finite‐difference code is employed to solve forward‐ and backward‐propagation problems, the method is general and can handle arbitrary 2-D source‐receiver configurations and lateral heterogeneities. Using conventional velocity analysis to derive an initial velocity model, the method is implemented on a real marine data set. The data set which has been selected corresponds approximately to a horizontally stratified medium. Consequently, a single‐shot gather has been used for inversion. In spite of some simplifying assumptions used for wave‐field propagation (acoustic approximation, point source), and using synthetics generated by a nearby sonic log to calibrate amplitudes, our final synthetics match the input data very well and the inversion result has clear similarities to the log.

Migration‐based traveltime waveform inversion of 2-D simple structures: A synthetic example

Geophysics ◽  
2001 ◽  
Vol 66 (3) ◽  
pp. 845-860 ◽  
Author(s):  
François Clément ◽  
Guy Chavent ◽  
Susana Gómez
Keyword(s):  
Least Squares ◽  
Waveform Inversion ◽  
Domain Of Attraction ◽  
Optimization Methods ◽  
Data Sets ◽  
Seismic Reflection Data ◽  
Prestack Migration ◽  
Reflection Data ◽  
Full Wave ◽  
The Time Domain

Migration‐based traveltime (MBTT) formulation provides algorithms for automatically determining background velocities from full‐waveform surface seismic reflection data using local optimization methods. In particular, it addresses the difficulty of the nonconvexity of the least‐squares data misfit function. The method consists of parameterizing the reflectivity in the time domain through a migration step and providing a multiscale representation for the smooth background velocity. We present an implementation of the MBTT approach for a 2-D finite‐difference (FD) full‐wave acoustic model. Numerical analysis on a 2-D synthetic example shows the ability of the method to find much more reliable estimates of both long and short wavelengths of the velocity than the classical least‐squares approach, even when starting from very poor initial guesses. This enlargement of the domain of attraction for the global minima of the least‐squares misfit has a price: each evaluation of the new objective function requires, besides the usual FD full‐wave forward modeling, an additional full‐wave prestack migration. Hence, the FD implementation of the MBTT approach presented in this paper is expected to provide a useful tool for the inversion of data sets of moderate size.

scholarly journals Study on the limitations of travel-time inversion applied to sub-basalt imaging

Solid Earth ◽  
2013 ◽  
Vol 4 (2) ◽  
pp. 543-554 ◽  
Author(s):  
I. Flecha ◽  
R. Carbonell ◽  
R. W. Hobbs
Keyword(s):  
Oil Industry ◽  
Velocity Model ◽  
Time Inversion ◽  
Wide Angle ◽  
Seismic Reflection Data ◽  
Additional Information ◽  
Velocity Models ◽  
Reflection Data ◽  
Travel Time Inversion ◽  
Tomographic Inversions

Abstract. The difficulties of seismic imaging beneath high velocity structures are widely recognised. In this setting, theoretical analysis of synthetic wide-angle seismic reflection data indicates that velocity models are not well constrained. A two-dimensional velocity model was built to simulate a simplified structural geometry given by a basaltic wedge placed within a sedimentary sequence. This model reproduces the geological setting in areas of special interest for the oil industry as the Faroe-Shetland Basin. A wide-angle synthetic dataset was calculated on this model using an elastic finite difference scheme. This dataset provided travel times for tomographic inversions. Results show that the original model can not be completely resolved without considering additional information. The resolution of nonlinear inversions lacks a functional mathematical relationship, therefore, statistical approaches are required. Stochastic tests based on Metropolis techniques support the need of additional information to properly resolve sub-basalt structures.

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.

Precision analysis of first‐break times in grid models

Geophysics ◽  
1998 ◽  
Vol 63 (3) ◽  
pp. 1062-1065 ◽  
Author(s):  
Thomas Gruber ◽  
Stewart A. Greenhalgh
Keyword(s):  
Numerical Models ◽  
Grid Point ◽  
Tomographic Reconstruction ◽  
Velocity Model ◽  
Model Parameters ◽  
Seismic Reflection Data ◽  
Arrival Times ◽  
Velocity Models ◽  
Reflection Data ◽  
Inversion Techniques

Rectangular grid velocity models and their derivatives are widely used in geophysical inversion techniques. Specifically, seismic tomographic reconstruction techniques, whether they be based on raypath methods (Bregman et al., 1989; Moser, 1991; Schneider et al., 1992; Cao and Greenhalgh, 1993; Zhou, 1993) or full wave equation methods (Vidale, 1990; Qin and Schuster, 1993; Cao and Greenhalgh, 1994) for calculating synthetic arrival times, involve propagation through a grid model. Likewise, migration of seismic reflection data, using asymptotic ray theory or finite difference/pseudospectral methods (Stolt and Benson, 1986; Zhe and Greenhalgh, 1997) involve assigning traveltimes to upward and downward propagating waves at every grid point in the model. The traveltimes in both cases depend on the grid specification. However, the precision level of such numerical models and their dependence on the model parameters is often unknown. In this paper, we describe a two‐dimensional velocity model and derive an error bound for first‐break times calculated with such a model. The analysis provides clear guidelines for grid specifications.

Shot-domain deblending using least-squares inversion

Geophysics ◽  
2017 ◽  
Vol 82 (4) ◽  
pp. V241-V256 ◽  
Author(s):  
Shaohuan Zu ◽  
Hui Zhou ◽  
Qingqing Li ◽  
Hanming Chen ◽  
Qingchen Zhang ◽  
...  
Keyword(s):  
Inverse Problem ◽  
Least Squares ◽  
Conjugate Gradient ◽  
Acquisition Cost ◽  
Field Data ◽  
Gradient Algorithm ◽  
Convolution Operators ◽  
Excellent Performance ◽  
Simultaneous Source ◽  
Temporal Overlap

Simultaneous source acquisition, which allows a temporal overlap between shot records, has significant advantages to improve data quality (e.g., denser shooting) and reduce acquisition cost (e.g., efficient wide-azimuth shooting). We have developed a novel shot-domain deblending approach for a wide-azimuth simultaneous shooting survey based on an inversion scheme. The inverse problem is formulated by introducing two convolution operators that can respectively destruct the signal and interference and is solved by a conjugate-gradient algorithm in the least-squares sense. Our approach breaks down the limit that source separation can only be implemented in some domains other than the shot domain based on the incoherency principle. Our shot-domain approach does not require the random dithering time; thus, it is very flexible to use in a wide-azimuth simultaneous shooting survey. Three numerically blended synthetic examples were developed to demonstrate the excellent performance of our method. The feasibility has been further validated via a field-data example that is numerically blended from two realistic marine towed streamers.

scholarly journals Reconstructing the primary reflections in seismic data by Marchenko redatuming and convolutional interferometry

Geophysics ◽  
2016 ◽  
Vol 81 (2) ◽  
pp. Q15-Q26 ◽  
Author(s):  
Giovanni Angelo Meles ◽  
Kees Wapenaar ◽  
Andrew Curtis
Keyword(s):  
Velocity Model ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Data Set ◽  
Seismic Reflection Data ◽  
Surface Reflection ◽  
Reflection Data ◽  
Reference Velocity ◽  
Time Migration ◽  
Recorded Data

State-of-the-art methods to image the earth’s subsurface using active-source seismic reflection data involve reverse time migration. This and other standard seismic processing methods such as velocity analysis provide best results only when all waves in the data set are primaries (waves reflected only once). A variety of methods are therefore deployed as processing to predict and remove multiples (waves reflected several times); however, accurate removal of those predicted multiples from the recorded data using adaptive subtraction techniques proves challenging, even in cases in which they can be predicted with reasonable accuracy. We present a new, alternative strategy to construct a parallel data set consisting only of primaries, which is calculated directly from recorded data. This obviates the need for multiple prediction and removal methods. Primaries are constructed by using convolutional interferometry to combine the first-arriving events of upgoing and direct-wave downgoing Green’s functions to virtual receivers in the subsurface. The required upgoing wavefields to virtual receivers are constructed by Marchenko redatuming. Crucially, this is possible without detailed models of the earth’s subsurface reflectivity structure: Similar to the most migration techniques, the method only requires surface reflection data and estimates of direct (nonreflected) arrivals between the virtual subsurface sources and the acquisition surface. We evaluate the method on a stratified synclinal model. It is shown to be particularly robust against errors in the reference velocity model used and to improve the migrated images substantially.

scholarly journals Study on the limitations of traveltime inversion in the presence of extreme velocity anomalies

2013 ◽  
Vol 5 (1) ◽  
pp. 189-226
Author(s):  
I. Flecha ◽  
R. Carbonell ◽  
R. W. Hobbs
Keyword(s):  
Oil Industry ◽  
Velocity Model ◽  
Wide Angle ◽  
Seismic Reflection Data ◽  
Traveltime Inversion ◽  
Additional Information ◽  
Velocity Models ◽  
Reflection Data ◽  
Velocity Anomalies ◽  
Tomographic Inversions

Abstract. The difficulties of seismic imaging beneath high velocity structures are widely recognised. In this setting, theoretical analysis of synthetic wide-angle seismic reflection data indicates that velocity models are not well constrained. A two-dimensional velocity model was built to simulate a simplified structural geometry given by a basaltic wedge placed within a sedimentary sequence. This model reproduces the geological setting in areas of special interest for the oil industry as the Faroe-Shetland Basin. A wide-angle synthetic dataset was calculated on this model using an elastic finite difference scheme. This dataset provided travel times for tomographic inversions. Results show that the original model can not be completely resolved without considering additional information. The resolution of nonlinear inversions lacks a functional mathematical relationship, therefore, statistical approaches are required. Stochastical tests based on Metropolis techniques support the need of additional information to properly resolve subbasalt structures.

Sign in / Sign up

Export Citation Format

Share Document