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Regularized inversion of a two‐dimensional integral equation with applications in borehole induction measurements
Author(s) -
Arikan Orhan
Publication year - 1994
Publication title -
radio science
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.371
H-Index - 84
eISSN - 1944-799X
pISSN - 0048-6604
DOI - 10.1029/93rs02006
Subject(s) - integral equation , deconvolution , mathematical analysis , mathematics , singular value decomposition , projection (relational algebra) , inversion (geology) , regularization (linguistics) , convolution (computer science) , radon transform , algorithm , computer science , geology , paleontology , structural basin , artificial intelligence , machine learning , artificial neural network
Well bore measurements of conductivity, gravity, and surface measurements of magnetotelluric fields can be modeled as a two‐dimensional integral equation with additive measurement noise. The governing integral equation has the form of convolution in the first dimension and projection in the second dimension. However, these two operations are not in separable form. In these applications, given a set of measurements, efficient and robust estimation of the underlying physical property is required. For this purpose, a regularized inversion algorithm for the governing integral equation is presented in this paper. Singular value decomposition of the measurement kernels is used to exploit convolution‐projection structure of the integral equation, leading to a form where measurements are related to the physical property by a two‐stage operation: projection followed by convolution. On the other hand, estimation of the physical property can be carried out by a two‐stage inversion algorithm: deconvolution followed by back projection. A regularization method for the required multichannel deconvolution is given. Some important details of the algorithm are addressed in an application to wellbore induction measurements of conductivity.