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- Volume 59, Issue 5, 2011
Geophysical Prospecting - Modelling Methods for Geophysical Imaging: Trends and Perspectives, 2011
Modelling Methods for Geophysical Imaging: Trends and Perspectives, 2011
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A review of the spectral, pseudo‐spectral, finite‐difference and finite‐element modelling techniques for geophysical imaging
Authors Jean Virieux, Henri Calandra and René‐Édouard PlessixABSTRACTModelling methods are nowadays at the heart of any geophysical interpretation approach. These are heavily relied upon by imaging techniques in elastodynamics and electromagnetism, where they are crucial for the extraction of subsurface characteristics from ever larger and denser datasets. While high‐frequency or one‐way approximations are very powerful and efficient, they reach their limits when complex geological settings and solutions of full equations are required at finite frequencies. A review of three important formulations is carried out here: the spectral method, which is very efficient and accurate but generally restricted to simple earth structures and often layered earth structures; the pseudo‐spectral, finite‐difference and finite‐volume methods based on strong formulation of the partial differential equations, which are easy to implement and currently represent a good compromise between accuracy, efficiency and flexibility and the continuous or discontinuous Galerkin finite‐element methods that are based on the weak formulation, which lead to more accurate earth representations and therefore to more accurate solutions, although with higher computational costs and more complex use. The choice between these different approaches is still difficult and depends on the applications. Guidelines are given here through discussion of the requirements for imaging/inversion.
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From modelling to inversion: designing a well‐adapted simulator
Authors William W. Symes, Dong Sun and Marco EnriquezABSTRACTThis paper describes a few mild design constraints that permit rapid adaptation of the modelling code for linear wave propagation to imaging/inversion or design optimization applications, retaining parallelism and other performance enhancements of the underlying simulator. It also describes an abstract software framework preserving the modularity of both optimization and modelling software in building inversion applications and illustrates this possibility via an example framework implemented in C++. Wave inverse problems tend to be afflicted by a variety of features, including extreme ill‐conditioning and nonlinearity, which degrade the performance of optimization formulations. Extended modelling variants of least‐squares inversion, motivated by migration velocity analysis, may relieve some of these difficulties. The framework described also accommodates these extensions to standard inversion.
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Three‐dimensional parallel frequency‐domain visco‐acoustic wave modelling based on a hybrid direct/iterative solver
Authors Florent Sourbier, Azzam Haidar, Luc Giraud, Hafedh Ben‐Hadj‐Ali, Stéphane Operto and Jean VirieuxABSTRACTWe present a parallel domain decomposition method based on a hybrid direct‐iterative solver for 3D frequency‐domain modelling of visco‐acoustic waves. The method is developed as a modelling engine for frequency‐domain full waveform inversion. Frequency‐domain seismic modelling reduces to the solution of a large and sparse system of linear equations, resulting from the discretization of the heterogeneous Helmholtz equation. Our approach to the high‐performance, scalable solution of large sparse linear systems is to combine direct and iterative methods. Such a hybrid approach exploits the advantages of both direct and iterative methods. The iterative component uses a small amount of memory and provides a natural way for parallelization. The direct part has favourable numerical properties for multiple right‐hand side modelling. The domain decomposition is based upon the algebraic Schur complement method, which allows for the iterative solution of a reduced system, the solution of which is the wavefield at the interfaces between the subdomains. Once the interface unknowns have been computed, the wavefield at the interior of each subdomain is efficiently computed by local substitutions. The reduced Schur complement system is solved with the generalized minimum residual method and is preconditioned by an algebraic additive Schwarz preconditioner. A direct solver is used to factorize local impedance matrices defined on each subdomain. Theoretical analysis shows that the time complexity of the hybrid solver is the same as that of iterative solver and time‐domain approaches for single frequency modelling. Simulations are performed in the SEG/EAGE overthrust and the salt models for frequencies up to 12.5 Hz. The number of iterations increases linearly with the number of subdomains for a given computational domain but the elapsed time of the iterative resolution remains almost constant. The number of iterations also increases linearly with frequencies, when the grid interval is adapted to the frequencies and the size of the subdomains is kept constant over frequency. These results make the cost of the hybrid solver of the same order as that of finite‐difference time‐domain modeling for one‐frequency modelling. Although the hybrid approach allows one to tackle larger problems than the direct‐solver approach, further improvements are needed to mitigate the computational burden of the iterative component in the context of multisource modelling. On the numerical side, the use of block iterative solvers and of incremental two‐level deflating preconditioners and on the parallel implementation side, the use of two levels of parallelism in the domain decomposition method should mitigate this computational burden.
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On 3D modeling of seismic wave propagation via a structured parallel multifrontal direct Helmholtz solver
Authors Shen Wang, Maarten V. de Hoop and Jianlin XiaABSTRACTWe consider the modeling of (polarized) seismic wave propagation on a rectangular domain via the discretization and solution of the inhomogeneous Helmholtz equation in 3D, by exploiting a parallel multifrontal sparse direct solver equipped with Hierarchically Semi‐Separable (HSS) structure to reduce the computational complexity and storage. In particular, we are concerned with solving this equation on a large domain, for a large number of different forcing terms in the context of seismic problems in general, and modeling in particular. We resort to a parsimonious mixed grid finite differences scheme for discretizing the Helmholtz operator and Perfect Matched Layer boundaries, resulting in a non‐Hermitian matrix. We make use of a nested dissection based domain decomposition, and introduce an approximate direct solver by developing a parallel HSS matrix compression, factorization, and solution approach. We cast our massive parallelization in the framework of the multifrontal method. The assembly tree is partitioned into local trees and a global tree. The local trees are eliminated independently in each processor, while the global tree is eliminated through massive communication. The solver for the inhomogeneous equation is a parallel hybrid between multifrontal and HSS structure. The computational complexity associated with the factorization is almost linear in the size, n say, of the matrix, viz. between O(n log n) and O(n4/3 log n), while the storage is almost linear as well, between O(n) and O(n log n). We exploit the use of a regular (Cartesian) mesh common in many seismic applications.
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Three‐dimensional seismic full‐waveform inversion using the finite‐difference contrast source inversion method
Authors A. Abubakar, G. Pan, M. Li, L. Zhang, T.M. Habashy and P.M. van den BergABSTRACTWe present the extension of the so‐called finite‐difference contrast‐source inversion method for the three‐dimensional interpretation of full‐waveform seismic data using the acoustic approximation. The finite‐difference contrast‐source inversion method does not simulate a full forward problem in its inversion process, hence it is computationally more efficient than the standard non‐linear inversion methods. Furthermore it allows the use of an inhomogeneous background medium, hence this method has a great potential for carrying out time‐lapse seismic inversion. However, the price that we have to pay is the storing of the LU decomposition arrays of the impedance matrix of the background medium. We show some numerical examples to illustrate the (dis)advantages of this method.
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New curvilinear scheme for elastic wave propagation in presence of curved topography
Authors Issam Tarrass, Luc Giraud and Pierre ThoreABSTRACTWe study a new curvilinear scheme for wave propagation modelling in presence of topography. The discrete scheme takes advantage of recent developments in areoacoustics. Our new scheme relies on the conventional grid coupled with optimized filters to remove numerical noise in case of strong material heterogeneity. We used non‐centred stencils for free surface implementation and optimized the explicit Runge‐Kutta scheme for the time differencing. We performed a complete theoretical stability and dispersion analysis of the discrete scheme. Finally, we illustrate the numerical accuracy of the new scheme by intensive experiments.
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Numerical performances of a hybrid local‐time stepping strategy applied to the reverse time migration
Authors Caroline Baldassari, Hélène Barucq, Henri Calandra and Julien DiazABSTRACTThe numerical simulation of wave propagation in heterogeneous media involves very significant computational costs that can be considerably reduced by using finite elements whose order is adapted to the characteristics of the medium. In this paper, we show that we can further reduce these costs by using a mixed time scheme, combining two discrete formulas and using local time steps. We illustrate the effectiveness of the new scheme considering an application to seismic imaging treated by reverse time migration. We also show that the new scheme greatly reduces the numerical dispersion problems that appear frequently when adapting the orders of approximation in space only.
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Discontinuous Galerkin frequency domain forward modelling for the inversion of electric permittivity in the 2D case
Authors Mohamed El Bouajaji, Stéphane Lanteri and Matthew YedlinABSTRACTWe have recently developed a discontinuous Galerkin frequency domain modelling algorithm for the solution of the 2D transverse magnetic Maxwell equations. This method is formulated on an unstructured triangular discretization of the computational domain and makes use of a high order polynomial interpolation of the electromagnetic field components within each triangular element. The discontinuous nature of the approximation naturally allows for a local definition of the interpolation order that is, in combination with a possibly non‐conforming local refinement of the mesh, a key ingredient for obtaining a flexible and accurate discretization method. Moreover, heterogeneity of the propagation media is easily dealt with by assuming element‐wise values of the electromagnetic parameters. In this paper, we propose the use of this discontinuous Galerkin frequency domain method as the forward modelling algorithm for solving the inverse problem for the electric permittivity in the 2D case. The inversion process is based on a gradient minimization technique developed by Pratt for seismological applications. Preliminary numerical results are presented for the imaging of a simplified subsurface model with the aim of assessing the performances of the proposed inversion methodology with regards to the number of frequencies, the number of recorded data and the number of sources.
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Seismic prediction of geological structures ahead of the tunnel using tunnel surface waves
Authors Stefan Jetschny, Thomas Bohlen and André KurzmannABSTRACTIn order to keep up with the economic and safety demands of modern tunnel construction projects, especially in urban areas, there is a need to detect safety threats in real time while the construction is advancing. Tunnel prediction methods accompanying the drilling process can help to correlate and update a priori information on expected geological structures with their actual spatial location or even existence ahead of the tunnel face. We recently presented a seismic prediction approach using tunnel surface waves, which has already proven its potential during field surveys. However, common tunnel seismic data interpretation, regardless of the prediction method, requires human interaction. Either a specially trained field technician have to be present at the construction site or the data has to be uploaded to an office for further interpretation. In this work we present a simple but stable approach to automatically detect major geological structures ahead of the tunnel face. We focus on the accurate determination of the distance of fault zones or lithological boundaries from the tunnel face without any a priori information. By 3D seismic finite difference modelling we simulated a synthetic tunnel seismic survey that includes typical features encountered in tunnel construction. The developed prediction sequence was tested on these data and later successfully applied to two different tunnel data sets acquired at the Gotthard Base Tunnel (Switzerland) and during the construction of the ‘Neuer Schlüchterner Tunnel’ close to Fulda (Germany).
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Marine electromagnetic inverse solution appraisal and uncertainty using model‐derived basis functions and sparse geometric sampling
More LessABSTRACTWe summarize, for marine electromagnetic inverse problems, a newly developed inverse solution appraisal and non‐linear uncertainty estimation method based on parameter reduction techniques and efficient posterior model space sampling. This method uses model compression methods to decorrelate parameters in an inverse solution and represent all feasible posterior models as linear combinations of a small number of model‐derived basis vectors and corresponding coefficients. This allows us to reduce the posterior sampling problem by orders of magnitude. We further contract this reduced‐dimensional posterior space by confining all acceptable models to a set of bounds mapped from our original parameter space. As a final step to increase efficiency, we implement a geometric sampling scheme that we use to approximate our restricted posterior by generating feasible models on adaptive, optimally‐sparse grids. The sampled equi‐feasible models are accepted according to a data misfit threshold and constitute an optimally‐sparse representation of the restricted posterior model space. Although very efficient, our method imposes a bias in the posterior space by truncating the basis expansion during the model reduction step. To investigate this, we compare two types of fast and scalable bases, the discrete cosine transform and singular value decomposition. We demonstrate that while the choice of base does influence the type of models sampled and the model rejection rates, the posterior statistics are generally compatible between the methods providing confidence in the uncertainty estimations. For the marine electromagnetic problem, we show that a representative ensemble of equivalent inverse solutions can be generated for realistically‐sized inverse problems and that solution appraisal and uncertainty inference follow directly from ensemble statistics.
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Three‐dimensional gravity modelling and focusing inversion using rectangular meshes
More LessABSTRACTRectangular grid cells are commonly used for the geophysical modeling of gravity anomalies, owing to their flexibility in constructing complex models. The straightforward handling of cubic cells in gravity inversion algorithms allows for a flexible imposition of model regularization constraints, which are generally essential in the inversion of static potential field data. The first part of this paper provides a review of commonly used expressions for calculating the gravity of a right polygonal prism, both for gravity and gradiometry, where the formulas of Plouff and Forsberg are adapted. The formulas can be cast into general forms practical for implementation. In the second part, a weighting scheme for resolution enhancement at depth is presented. Modelling the earth using highly digitized meshes, depth weighting schemes are typically applied to the model objective functional, subject to minimizing the data misfit. The scheme proposed here involves a non‐linear conjugate gradient inversion scheme with a weighting function applied to the non‐linear conjugate gradient scheme's gradient vector of the objective functional. The low depth resolution due to the quick decay of the gravity kernel functions is counteracted by suppressing the search directions in the parameter space that would lead to near‐surface concentrations of gravity anomalies. Further, a density parameter transformation function enabling the imposition of lower and upper bounding constraints is employed. Using synthetic data from models of varying complexity and a field data set, it is demonstrated that, given an adequate depth weighting function, the gravity inversion in the transform space can recover geologically meaningful models requiring a minimum of prior information and user interaction.
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Volumes & issues
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Volume 72 (2023 - 2024)
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Volume 71 (2022 - 2023)
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Volume 70 (2021 - 2022)
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Volume 69 (2021)
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Volume 68 (2020)
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Volume 67 (2019)
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Volume 66 (2018)
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Volume 65 (2017)
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Volume 64 (2015 - 2016)
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Volume 63 (2015)
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Volume 62 (2014)
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Volume 61 (2013)
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Volume 60 (2012)
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Volume 59 (2011)
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Volume 58 (2010)
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Volume 55 (2007)
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Volume 53 (2005)
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Volume 52 (2004)
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Volume 51 (2003)
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Volume 50 (2002)
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Volume 49 (2001)
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Volume 48 (2000)
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Volume 47 (1999)
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Volume 46 (1998)
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Volume 45 (1997)
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Volume 44 (1996)
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Volume 43 (1995)
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Volume 41 (1993)
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Volume 40 (1992)
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Volume 39 (1991)
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Volume 38 (1990)
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Volume 37 (1989)
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Volume 36 (1988)
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Volume 35 (1987)
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Volume 34 (1986)
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Volume 33 (1985)
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Volume 32 (1984)
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Volume 31 (1983)
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Volume 30 (1982)
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Volume 29 (1981)
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Volume 28 (1980)
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Volume 27 (1979)
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Volume 26 (1978)
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Volume 25 (1977)
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Volume 24 (1976)
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Volume 23 (1975)
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Volume 22 (1974)
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Volume 21 (1973)
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Volume 20 (1972)
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Volume 19 (1971)
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Volume 18 (1970)
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Volume 17 (1969)
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Volume 16 (1968)
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Volume 15 (1967)
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Volume 14 (1966)
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Volume 13 (1965)
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Volume 12 (1964)
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Volume 11 (1963)
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Volume 10 (1962)
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Volume 9 (1961)
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Volume 8 (1960)
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Volume 7 (1959)
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Volume 6 (1958)
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Volume 5 (1957)
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Volume 4 (1956)
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Volume 3 (1955)
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Volume 2 (1954)
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Volume 1 (1953)