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Jacobian matrix

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Journal Article
Journal: Geophysics
Published: 18 June 2021
Geophysics (2021) 86 (4): R447–R461.
...-Richards approximations), which are not suitable for formations with strong contrast interfaces and seismic data with large incidence angles. Based on the previous derivation of the accurate Jacobian matrix, we have found that the sign of each partial derivative of reflection coefficient with respect...
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Illustration of the <b>Jacobian</b> <b>matrix</b> in equation  17  in logarithmic scale. ...
Published: 21 December 2018
Figure 3. Illustration of the Jacobian matrix in equation  17 in logarithmic scale. Each column of the matrix represents an unknown parameter. Twenty-seven parameters including 2 electrical conductivity values, the thickness of the first soil layer, 12 multiplicative factors ( MF 1
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Components of the <b>Jacobian</b> <b>matrix</b> for      S  11  TI     . Analytically det...
Published: 27 July 2016
Figure 4. Components of the Jacobian matrix for S 11 TI . Analytically determined values (equation  48 ): lines, values obtained directly from the fitting algorithm (equation  47 ): symbols. All model parameters are given in units used for and obtained from fitting (see Table  1
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Depth-resolution plot for the <b>Jacobian</b> <b>matrix</b> calculated using the preferre...
Published: 03 March 2014
Figure 14. Depth-resolution plot for the Jacobian matrix calculated using the preferred model (see Figure  8b ) and the field acquisition geometry. Each column represents the L 2 -norms for depth slices k = 1 … N z , calculated for i = 1 … 2 N d singular
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Generalized flow diagram demonstrating the <b>Jacobian</b> <b>matrix</b> computation for ...
Published: 30 September 2010
Figure 2. Generalized flow diagram demonstrating the Jacobian matrix computation for the inversion problem. The Jacobian matrix is divided evenly among slave nodes in terms of storage and computation, where each node stores and computes approximately the same number of rows. As the Jacobian
Journal Article
Journal: Geophysics
Published: 28 April 2011
Geophysics (2011) 76 (3): F173–F183.
...Maokun Li; Aria Abubakar; Jianguo Liu; Guangdong Pan; Tarek M. Habashy Abstract We developed a compressed implicit Jacobian scheme for the regularized Gauss-Newton inversion algorithm for reconstructing 3D conductivity distributions from electromagnetic data. In this algorithm, the Jacobian matrix...
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Journal Article
Journal: Geophysics
Published: 03 June 2013
Geophysics (2013) 78 (4): E161–E171.
... is mainly comprised of the multiple solutions of the forward problem and linear algebraic operations using the Jacobian matrix for calculating the GN step. For large-scale 2.5D and 3D problems with multiple sources and receivers, the corresponding cost grows enormously for inversion algorithms using...
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Journal Article
Journal: Geophysics
Published: 04 August 2021
Geophysics (2021) 86 (5): E333–E342.
... by extracting from part of the global model mesh, the relationship between the local and global meshes is straightforward, so we can easily create a projection of the Jacobian matrix between global and local meshes and rapidly construct the global Jacobian matrix for 3D EM inversions. After formulating...
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Journal Article
Journal: Geophysics
Published: 08 June 2001
Geophysics (2001) 66 (6): 1856–1863.
... in the earth's subsurface. This sensitivity, or Jacobian, matrix is usually quite expensive to estimate for all but the simplest model parameterizations. We exploit the numerical structure of the finite-element method, modern sparse matrix technology, and source–receiver reciprocity to develop an algorithm...
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Journal Article
Published: 01 December 2010
Journal of Environmental and Engineering Geophysics (2010) 15 (4): 203–218.
... (Method 1). In this scheme, ray tracing was not a part of the inversion algorithm and the Jacobian matrix was calculated by numerical differentiation. Travel-time calculations were performed by a finite-difference eikonal equation solver. Model velocity fields were updated by matrix inversion techniques...
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Journal Article
Journal: Geophysics
Published: 28 September 2009
Geophysics (2009) 74 (5): WB153–WB163.
... the resolution capability of the data and possibly reduce the dimensions of the Jacobian matrix. Based on the general expressions, relevant formulas for isotropic and 2.5D and 3D tilted transversely isotropic (TTI) media are derived. Two computational schemes, constant-point and constant-block parameterization...
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Journal Article
Journal: Geophysics
Published: 01 September 1986
Geophysics (1986) 51 (9): 1789–1800.
... the difficult distinction between multiples and primaries. Equations are provided for adding the surface to the inversion procedure. This involves adding the surface effects to the Jacobian matrix as well as to the forward modeling procedure. To speed computation, the addition of the surface effects...
Journal Article
Journal: Geophysics
Published: 01 January 2006
Geophysics (2006) 71 (1): G1–G9.
... iteratively with a constrained regularized Gauss-Newton minimization scheme. As documented in the literature, the main computational bottleneck when solving this full 3D inverse problem is the CPU time associated with constructing the Jacobian matrix. In this study, to achieve the inversion results within...
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Journal Article
Journal: Geophysics
Published: 01 December 1994
Geophysics (1994) 59 (12): 1868–1881.
... on the reflectivity method where the solution for each frequency-wavenumber component is found by computing the generalized reflection and transmission matrices recursively. The gradient of the objective function is computed from analytical expressions of the Jacobian matrix derived directly from the recursive...
Journal Article
Published: 01 April 2003
Bulletin of the Seismological Society of America (2003) 93 (2): 904–921.
... in seismic imaging and inversion because we can economically calculate a Jacobian matrix using the reciprocity. Successful simulation of a large-size model shows that our method can be used for the simulation of wave propagation in the geological model needed in the reverse-time migration or seismic...
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Journal Article
Journal: Geophysics
Published: 10 July 2012
Geophysics (2012) 77 (4): WB171–WB177.
... calculation. Equally important, the data noise and the number of data points are integrated into the calculation. Our methodology is based on a recalculated sensitivity (Jacobian) matrix of the final model and thus it can be used on any model type for which a sensitivity matrix can be calculated. Unlike other...
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Journal Article
Journal: Geophysics
Published: 12 March 2018
Geophysics (2018) 83 (3): E133–E150.
... standard methods: (1) It avoids the storage of the usually large covariance matrix, which contains the geostatistical information, by factorizing it in a product of low-rank matrices. (2) It does not assemble the Jacobian matrix per se. We determine the robustness of this approach with three examples. We...
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Journal Article
Journal: Geophysics
Published: 01 January 2003
Geophysics (2003) 68 (6): 2026–2039.
... data that addresses several of these issues. The salient features of our algorithm include an efficient approach to sensitivity computation, a strategy for band-limiting the Jacobian matrix, and a novel approach to computing regularization weight that is iteration adaptive. In this paper, we first...
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Journal Article
Journal: Geophysics
Published: 01 January 2000
Geophysics (2000) 65 (2): 452–464.
... of our inversion algorithm is that the partial derivatives of the response functions, with respect to model parameters, are given in analytical forms, which results in an efficient computation of the Jacobian matrix. Theoretical studies show good convergence and good resolution of the model parameters...
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Journal Article
Journal: Geophysics
Published: 24 April 2018
Geophysics (2018) 83 (3): R245–R255.
... of Levenberg’s algorithm and the split Bregman iterative scheme, in which we have to refine the data and model parameters at each iteration. We refine the data via the original nonlinear equations, but we use the traditional cost-effective linearized AVO inversion to construct the Jacobian matrix and update...
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