High-resolution wave-equation AVA imaging: Algorithm and tests with a data set from the Western Canadian Sedimentary Basin

Geophysics ◽  
2005 ◽  
Vol 70 (5) ◽  
pp. S91-S99 ◽  
Author(s):  
Juefu Wang ◽  
Henning Kuehl ◽  
Mauricio D. Sacchi
Keyword(s):  
Inverse Problem ◽  
Wave Equation ◽  
Least Squares ◽  
Oil And Gas ◽  
Sedimentary Basin ◽  
Model Space ◽  
Sampled Data ◽  
Inversion Algorithm ◽  
Data Set ◽  
Data Space

This paper presents a 3D least-squares wave-equation migration method that yields regularized common-image gathers (CIGs) for amplitude-versus-angle (AVA) analysis. In least-squares migration, we pose seismic imaging as a linear inverse problem; this provides at least two advantages. First, we are able to incorporate model-space weighting operators that improve the amplitude fidelity of CIGs. Second, the influence of improperly sampled data (footprint noise) can be diminished by incorporating data-space weighting operators. To investigate the viability of this class of methods for oil and gas exploration, we test the algorithm with a real-data example from the Western Canadian Sedimentary Basin. To make our problem computationally feasible, we utilize the 3D common-azimuth approximation in the migration algorithm. The inversion algorithm uses the method of conjugate gradients with the addition of a ray-parameter-dependent smoothing constraint that minimizes sampling and aperture artifacts. We show that more robust AVA attributes can be obtained by properly selecting the model and data-space regularization operators. The algorithm is implemented in conjunction with a preconditioning strategy to accelerate convergence. Posing the migration problem as an inverse problem leads to enhanced event continuity in CIGs and, hence, more reliable AVA estimates. The vertical resolution of the inverted image also improves as a consequence of increased coherence in CIGs and, in addition, by implicitly introducing migration deconvolution in the inversion.

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.

A DETERMINISTIC METHODOLOGY FOR ESTIMATION OF PARAMETERS IN DYNAMIC MARKOV CHAIN MODELS

2011 ◽  
Vol 19 (01) ◽  
pp. 71-100 ◽  
Author(s):  
A. R. ORTIZ ◽  
H. T. BANKS ◽  
C. CASTILLO-CHAVEZ ◽  
G. CHOWELL ◽  
X. WANG
Keyword(s):  
Inverse Problem ◽  
Markov Chain ◽  
Least Squares ◽  
Large Population ◽  
Ordinary Least Squares ◽  
Large Sample Size ◽  
Data Set ◽  
Markov Chain Models ◽  
Sample Paths ◽  
Dynamic Stochastic

A method for estimating parameters in dynamic stochastic (Markov Chain) models based on Kurtz's limit theory coupled with inverse problem methods developed for deterministic dynamical systems is proposed and illustrated in the context of disease dynamics. This methodology relies on finding an approximate large-population behavior of an appropriate scaled stochastic system. The approach leads to a deterministic approximation obtained as solutions of rate equations (ordinary differential equations) in terms of the large sample size average over sample paths or trajectories (limits of pure jump Markov processes). Using the resulting deterministic model, we select parameter subset combinations that can be estimated using an ordinary-least-squares (OLS) or generalized-least-squares (GLS) inverse problem formulation with a given data set. The selection is based on two criteria of the sensitivity matrix: the degree of sensitivity measured in the form of its condition number and the degree of uncertainty measured in the form of its parameter selection score. We illustrate the ideas with a stochastic model for the transmission of vancomycin-resistant enterococcus (VRE) in hospitals and VRE surveillance data from an oncology unit.

scholarly journals Applying compactness constraints to differential traveltime tomography

Geophysics ◽  
2007 ◽  
Vol 72 (4) ◽  
pp. R67-R75 ◽  
Author(s):  
Jonathan B. Ajo-Franklin ◽  
Burke J. Minsley ◽  
Thomas M. Daley
Keyword(s):  
Stable Solution ◽  
Model Space ◽  
Time Lapse ◽  
Process Time ◽  
Spatial Process ◽  
Weighted Norm ◽  
Inversion Algorithm ◽  
Time Data ◽  
Data Set ◽  
Traveltime Tomography

Tomographic imaging problems are typically ill-posed and often require the use of regularization techniques to guarantee a stable solution. Minimization of a weighted norm of model length is one commonly used secondary constraint. Tikhonov methods exploit low-order differential operators to select for solutions that are small, flat, or smooth in one or more dimensions. This class of regularizing functionals may not always be appropriate, particularly in cases where the anomaly being imaged is generated by a nonsmooth spatial process. Time-lapse imaging of flow-induced velocity anomalies is one such case; flow features are often characterized by spatial compactness or connectivity. By performing inversions on differenced arrival time data, the properties of the time-lapse feature can be directly constrained. We develop a differential traveltime tomography algorithm whichselects for compact solutions, i.e., models with a minimum area of support, through application of model-space iteratively reweighted least squares. Our technique is an adaptation of minimum support regularization methods previously explored within the potential theory community. We compare our inversion algorithm to the results obtained by traditional Tikhonov regularization for two simple synthetic models: one including several sharp localized anomalies and a second with smoother features. We use a more complicated synthetic test case based on multiphase flow results to illustrate the efficacy of compactness constraints for contaminant infiltration imaging. We apply the algorithm to a [Formula: see text]-sequestration-monitoring data set acquired at the Frio pilot site. We observe that in cases where the assumption of a localized anomaly is correct, the addition of compactness constraints improves image quality by reducing tomographic artifacts and spatial smearing of target features.

Sparseness‐constrained least‐squares inversion: Application to seismic wave reconstruction

Geophysics ◽  
2003 ◽  
Vol 68 (5) ◽  
pp. 1633-1638 ◽  
Author(s):  
Yanghua Wang
Keyword(s):  
Least Squares ◽  
Seismic Data ◽  
Reconstruction Method ◽  
Sampled Data ◽  
Linear Inversion ◽  
Data Set ◽  
Least Squares Solution ◽  
Working Definition ◽  
The Stability ◽  
Irregularly Sampled Data

The spectrum of a discrete Fourier transform (DFT) is estimated by linear inversion, and used to produce desirable seismic traces with regular spatial sampling from an irregularly sampled data set. The essence of such a wavefield reconstruction method is to solve the DFT inverse problem with a particular constraint which imposes a sparseness criterion on the least‐squares solution. A working definition for the sparseness constraint is presented to improve the stability and efficiency. Then a sparseness measurement is used to measure the relative sparseness of the two DFT spectra obtained from inversion with or without sparseness constraint. It is a pragmatic indicator about the magnitude of sparseness needed for wavefield reconstruction. For seismic trace regularization, an antialiasing condition must be fulfilled for the regularizing trace interval, whereas optimal trace coordinates in the output can be obtained by minimizing the distances between the newly generated traces and the original traces in the input. Application to real seismic data reveals the effectiveness of the technique and the significance of the sparseness constraint in the least‐squares solution.

Nonlinear 3D tomographic least-squares inversion of residual moveout in Kirchhoff prestack-depth-migration common-image gathers

Geophysics ◽  
2008 ◽  
Vol 73 (5) ◽  
pp. VE13-VE23 ◽  
Author(s):  
Frank Adler ◽  
Reda Baina ◽  
Mohamed Amine Soudani ◽  
Pierre Cardon ◽  
Jean-Baptiste Richard
Keyword(s):  
Inverse Problem ◽  
Least Squares ◽  
Nonlinear Least Squares ◽  
Velocity Model ◽  
Migration Velocity ◽  
Prestack Depth Migration ◽  
Data Set ◽  
Depth Migration ◽  
Layer Velocity ◽  
Map Migration

Velocity-model estimation with seismic reflection tomography is a nonlinear inverse problem. We present a new method for solving the nonlinear tomographic inverse problem using 3D prestack-depth-migrated reflections as the input data, i.e., our method requires that prestack depth migration (PSDM) be performed before tomography. The method is applicable to any type of seismic data acquisition that permits seismic imaging with Kirchhoff PSDM. A fundamental concept of the method is that we dissociate the possibly incorrect initial migration velocity model from the tomographic velocity model. We take the initial migration velocity model and the residual moveout in the associated PSDM common-image gathers as the reference data. This allows us to consider the migrated depth of the initial PSDM as the invariant observation for the tomographic inverse problem. We can therefore formulate the inverse problem within the general framework of inverse theory as a nonlinear least-squares data fitting between observed and modeled migrated depth. The modeled migrated depth is calculated by ray tracing in the tomographic model, followed by a finite-offset map migration in the initial migration model. The inverse problem is solved iteratively with a Gauss-Newton algorithm. We applied the method to a North Sea data set to build an anisotropic layer velocity model.

scholarly journals Linear inverse Gaussian theory and geostatistics

Geophysics ◽  
2006 ◽  
Vol 71 (6) ◽  
pp. R101-R111 ◽  
Author(s):  
Thomas Mejer Hansen ◽  
Andre G. Journel ◽  
Albert Tarantola ◽  
Klaus Mosegaard
Keyword(s):  
Inverse Problem ◽  
Least Squares ◽  
Covariance Function ◽  
A Priori ◽  
Model Space ◽  
Sequential Gaussian Simulation ◽  
A Priori Information ◽  
Sequential Approach ◽  
Linear Inverse Problem ◽  
Priori Information

Inverse problems in geophysics require the introduction of complex a priori information and are solved using computationally expensive Monte Carlo techniques (where large portions of the model space are explored). The geostatistical method allows for fast integration of complex a priori information in the form of covariance functions and training images. We combine geostatistical methods and inverse problem theory to generate realizations of the posterior probability density function of any Gaussian linear inverse problem, honoring a priori information in the form of a covariance function describing the spatial connectivity of the model space parameters. This is achieved using sequential Gaussian simulation, a well-known, noniterative geostatisticalmethod for generating samples of a Gaussian random field with a given covariance function. This work is a contribution to both linear inverse problem theory and geostatistics. Our main result is an efficient method to generate realizations, actual solutions rather than the conventional least-squares-based approach, to any Gaussian linear inverse problem using a noniterative method. The sequential approach to solving linear and weakly nonlinear problems is computationally efficient compared with traditional least-squares-based inversion. The sequential approach also allows one to solve the inverse problem in only a small part of the model space while conditioned to all available data. From a geostatistical point of view, the method can be used to condition realizations of Gaussian random fields to the possibly noisy linear average observations of the model space.

3D magnetic data-space inversion with sparseness constraints

Geophysics ◽  
2009 ◽  
Vol 74 (1) ◽  
pp. L7-L15 ◽  
Author(s):  
Mark Pilkington
Keyword(s):  
Least Squares ◽  
Synthetic Data ◽  
Model Space ◽  
Gradient Algorithm ◽  
Magnetic Data ◽  
Model Parameters ◽  
Linear System Of Equations ◽  
Data Space ◽  
System Versus ◽  
Sparse Inversion

I have developed an inversion approach that determines a 3D susceptibility distribution that produces a given magnetic anomaly. The subsurface model consists of a 3D, equally spaced array of dipoles. The inversion incorporates a model norm that enforces sparseness and depth weighting of the solution. Sparseness is imposed by using the Cauchy norm on model parameters. The inverse problem is posed in the data space, leading to a linear system of equations with dimensions based on the number of data, [Formula: see text]. This contrasts with the standard least-squares solution, derived through operations within the [Formula: see text]-dimensional model space ([Formula: see text] being the number of model parameters). Hence, the data-space method combined with a conjugate gradient algorithm leads to computational efficiency by dealing with an [Formula: see text] system versus an [Formula: see text] one, where [Formula: see text]. Tests on synthetic data show that sparse inversion produces a much more focused solution compared with a standard model-space, least-squares inversion. The inversion of aeromagnetic data collected over a Precambrian Shield area again shows that including the sparseness constraint leads to a simpler and better resolved solution. The degree of improvement in model resolution for the sparse case is quantified using the resolution matrix.

Prestack correlative least-squares reverse time migration

Geophysics ◽  
2017 ◽  
Vol 82 (2) ◽  
pp. S159-S172 ◽  
Author(s):  
Xuejian Liu ◽  
Yike Liu ◽  
Huiyi Lu ◽  
Hao Hu ◽  
Majid Khan
Keyword(s):  
Objective Function ◽  
Least Squares ◽  
Step Length ◽  
Inversion Algorithm ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Initial Image ◽  
Data Set ◽  
Time Migration ◽  
Gradient Descent Algorithm

In the correlative least-squares reverse time migration (CLSRTM) scheme, a stacked image is updated using a gradient-based inversion algorithm. However, CLSRTM experiences the incoherent stacking of different shots during each iteration due to the use of an imperfect velocity, which leads to image smearing. To reduce the sensitivity to velocity errors, we have developed prestack correlative least-squares reverse time migration (PCLSRTM), in which a gradient descent algorithm using a newly defined initial image and an efficiently defined analytical step length is developed to separately seek the optimal image for each shot gather before the final stacking. Furthermore, a weighted objective function is also designed for PCLSRTM, so that the data-domain gradient can avoid a strong truncation effect. Numerical experiments on a three-layer model as well as a marine synthetic and a field data set reveal the merits of PCLSRTM. In the presence of velocity errors, PCLSRTM shows better convergence and provides higher quality images as compared with CLSRTM. With the newly defined initial image, PCLSRTM can effectively handle observed data with unbalanced amplitudes. By using a weighted objective function, PCLSRTM can provide an image with enhanced resolution and balanced amplitude while avoiding many imaging artifacts.

Reflection angle/azimuth-dependent least-squares reverse-time migration

Geophysics ◽  
2021 ◽  
pp. 1-68
Author(s):  
Eric Duveneck ◽  
Michael Kiehn ◽  
Anu Chandran ◽  
Thomas Kühnel
Keyword(s):  
Inverse Problem ◽  
Wave Equation ◽  
Least Squares ◽  
Reflection Coefficients ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Linear Inverse Problem ◽  
Full Wave ◽  
Time Migration ◽  
Reflection Angle

Seismic images under complex overburdens like salt are strongly affected by illumination variations due to overburden velocity variations and imperfect acquisition geometries, making it difficult to obtain reliable image amplitudes. Least-squares reverse-time migration (LSRTM) addresses these issues by formulating full wave-equation imaging as a linear inverse problem and solving for a reflectivity model that explains the recorded seismic data. Because subsurface reflection coefficients depend on the incident angle, and possibly on azimuth, quantitative interpretation under complex overburdens requires LSRTM with output in terms of image gathers, e.g., as a function of reflection angle or angle and azimuth. We present a reflection angle- or angle/azimuth-dependent LSRTM method aimed at obtaining physically meaningful image amplitudes interpretable in terms of angle- or angle/azimuth-dependent reflection coefficients. The method is formulated as a linear inverse problem solved iteratively with the conjugate gradient method. It requires an adjoint pair of linear operators for reflection angle/azimuth-dependent migration and demigration based on full wave-equation propagation. We implement these operators in an efficient way by using a mapping approach between migrated shot gathers and subsurface reflection angle/azimuth gathers. To accelerate convergence of the iterative inversion, we apply image-domain preconditioning operators computed from a single de-remigration step. An angle continuity constraint and a structural dip constraint, implemented via shaping regularization, are used to stabilize the solution in the presence of limited illumination and to control the effects of coherent noise. We demonstrate the method on a synthetic data example and on a wide-azimuth streamer dataset from the Gulf of Mexico, where we show that angle/azimuth-dependent LSRTM can achieve significant uplift in subsalt image quality, with overburden- and acquisition-related illumination variation effects on angle/azimuth-dependent image amplitudes largely removed.

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