Joint Seismic Deconvolution and Geostatistical Extrapolation in a Bayesian Framework Using a B-spline Discontinuous Galerkin Method

PreviousNext No AccessSEG Technical Program Expanded Abstracts 2012Joint seismic deconvolution and geostatistical extrapolation in a Bayesian framework using a B-spline discontinuous Galerkin methodAuthors: Jonathan KaneWilliam RodiJonathan KaneShell International Exploration and ProductionSearch for more papers by this author and William RodiEarth Resources Laboratory, MITSearch for more papers by this authorhttps://doi.org/10.1190/segam2012-1582.1 SectionsAboutPDF/ePub ToolsAdd to favoritesDownload CitationsTrack CitationsPermissions ShareFacebookTwitterLinked InRedditEmail Abstract We propose a method for simultaneously performing seismic deconvolution and geostatistical well log extrapolation in order to optimally estimate subsurface reflectivity. Traditional zero-offset seismic deconvolution and Kriging (geostatistical extrapolation of point samples) can be seen as special cases of this more general problem. We model the reflectivity field as a continuous function and incorporate prior geological knowledge in the form of local correlation information and discontinuities (faults) obtained from the processed seismic data. We use a variational method to define the optimal solution of the continuous problem. It requires minimization of a Lagrangian functional composed of three terms: an integral of a differential operator acting on the interior of the model - this imposes spatial continuity on the solution; an integral on the boundary of the model - in order to specify boundary conditions; and a norm on the data misfit in order to fit the data. Minimization of this functional via the Euler-Lagrange equations leads to defining our reflectivity field as the solution to the weak form of a partial differential equation. Further representing our solution in a basis of well-defined functions allows us to discretize the problem via the Galerkin method. We choose B-splines as our basis due to optimal approximation and computational properties. Faults are modeled by breaking the B-splines at fault locations and combining broken pieces with neighboring basis functions on the other side of the fault in a principled way, thus leading to a basis of ‘extended B-splines’. This guarantees no degradation in accuracy of the final piecewise-continuous solution. By further allowing the differential operator to be anisotropic and location-dependent enables us to model the local nonstationary correlation structure in the geology. We invoke a Bayesian interpretation of the discrete system of equations and weight each data set according its assumed variance. The model that minimizes the Lagrangian is then equivalent to the Bayesian Maximum A-Posteriori (MAP) model. One particular differential operator used in the Lagrangian hasa well defined covariance function (variogram) as it’s inverse. This makes a connection between classical geostatistics and Tikhonov-style methods of regularizing inverse problems. Permalink: https://doi.org/10.1190/segam2012-1582.1FiguresReferencesRelatedDetails SEG Technical Program Expanded Abstracts 2012ISSN (print):1052-3812 ISSN (online):1949-4645Copyright: 2012 Pages: 4609 Publisher:Society of Exploration Geophysicists HistoryPublished: 25 Oct 2012 CITATION INFORMATION Jonathan Kane and William Rodi, (2012), "Joint seismic deconvolution and geostatistical extrapolation in a Bayesian framework using a B-spline discontinuous Galerkin method," SEG Technical Program Expanded Abstracts : 1-5. https://doi.org/10.1190/segam2012-1582.1 Plain-Language Summary PDF DownloadLoading ...