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A series expansion approach to the inverse problem

Part of: Stochastic analysis Stochastic processes

Published online by Cambridge University Press:  14 July 2016

J. M. Angulo  and
M. D. Ruiz-Medina
J. M. Angulo*
Affiliation:
Universidad de Granada
M. D. Ruiz-Medina*
Affiliation:
Universidad de Granada
*
∗Postal address: Departamento de Estadística e I.O. Universidad de Granada, Campus de Fuentenueva s/n, E-18071 Granada, Spain.
∗Postal address: Departamento de Estadística e I.O. Universidad de Granada, Campus de Fuentenueva s/n, E-18071 Granada, Spain.
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Abstract

We consider the inverse problem of estimating the input random field in a stochastic integral equation relating two random fields. The purpose of this paper is to present an approach to this problem using a Riesz-based or orthonormal-based series expansion of the input random field with uncorrelated random coefficients. We establish conditions under which the input series expansion induces (via the integral equation) a Riesz-based or orthonormal-based series expansion for the output random field. The estimation problem is studied considering two cases, depending on whether data are available from either the output random field alone, or from both the input and output random fields. Finally, we discuss this approach in the case of transmissivity estimation from piezometric head data, which was the original motivation of this work.

Keywords

Linear predictionmean-square convergenceorthonormal-based series expansionRiesz-based series expansionthe inverse problem

MSC classification

Primary: 60G60: Random fields 60G12: General second-order processes
Secondary: 60G25: Prediction theory 60H15: Stochastic partial differential equations
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 2 , June 1998 , pp. 371 - 382
Copyright
Copyright © Applied Probability Trust 1998 

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