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Descripción
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| In this paper, we present a methodology to perform geophysical inversion of large- scale linear systems via a covariance-free orthogonal transformation: the discrete cosine transform. The methodology consists of compressing the matrix of the linear system as a digital image and using the interesting properties of orthogonal trans- formations to define an approximation of the Moore?Penrose pseudo-inverse. This methodology is also highly scalable since the model reduction achieved by these tech- niques increases with the number of parameters of the linear system involved due to the high correlation needed for these parameters to accomplish very detailed forward predictions and allows for a very fast computation of the inverse problem solution. We show the application of this methodology to a simple synthetic two-dimensional gravimetric problem for different dimensionalities and different levels of white Gaus- sian noise and to a synthetic linear system whose system matrix has been generated via geostatistical simulation to produce a random field with a given spatial correla- tion. The numerical results show that the discrete cosine transform pseudo-inverse outperforms the classical least-squares techniques, mainly in the presence of noise, since the solutions that are obtained are more stable and fit the observed data with the lowest root-mean-square error. Besides, we show that model reduction is a very effective way of parameter regularisation when the conditioning of the reduced dis- crete cosine transform matrix is taken into account. We finally show its application to the inversion of a real gravity profile in the Atacama Desert (north Chile) ob- taining very successful results in this non-linear inverse problem. The methodology presented here has a general character and can be applied to solve any linear and non-linear inverse problems (through linearisation) arising in technology and, par- ticularly, in geophysics, independently of the geophysical model discretisation and dimensionality. Nevertheless, the results shown in this paper are better in the case of ill-conditioned inverse problems for which the matrix compression is more efficient. In that sense, a natural extension of this methodology would be its application to the set of normal equations. | |
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Internacional
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Si |
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JCR del ISI
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Si |
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Título de la revista
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Geophysical Prospecting |
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ISSN
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1365-2478 |
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Factor de impacto JCR
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1,846 |
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Información de impacto
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Volumen
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65 |
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DOI
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10.1111/1365-2478.12548 |
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Número de revista
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S1 |
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Desde la página
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94 |
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Hasta la página
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111 |
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Mes
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DICIEMBRE |
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Ranking
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