Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-29T12:37:46.575Z Has data issue: false hasContentIssue false
  • English
  • Français

Orthogonal representations of random fields and an application to geophysics data

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

M. D. Ruiz-Medina  and
M. J. Valderrama
M. D. Ruiz-Medina*
Affiliation:
University of Granada
M. J. Valderrama*
Affiliation:
University of Granada
*
∗Postal address: Department of Statistics and Operations Research, University of Granada, 18071 Granada, Spain.
∗Postal address: Department of Statistics and Operations Research, University of Granada, 18071 Granada, Spain.
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a brief summary of some results related to deriving orthogonal representations of second-order random fields and its application in solving linear prediction problems. In the homogeneous and/or isotropic case, the spectral theory provides an orthogonal expansion in terms of spherical harmonics, called spectral decomposition (Yadrenko 1983). A prediction formula based on this orthogonal representation is shown. Finally, an application of this formula in solving a real-data problem related to prospective geophysics techniques is presented.

Keywords

ORTHOGONAL KARHUNEN-LOÈVE EXPANSIONLAPLACE RANDOM FIELDMINIMUM-SQUARE LINEAR PREDICTIONSPECTRAL TECHNIQUES

MSC classification

Primary: 60G60: Random fields 60G12: General second-order processes
Type
Research Article
Information
Journal of Applied Probability , Volume 34 , Issue 2 , June 1997 , pp. 458 - 476
Copyright
Copyright © Applied Probability Trust 1997 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Gikhman, I. I. and Skorokhod, A. V. (1974-1979) The Theory of Stochastic Processes. Vol. 1. Springer, Berlin.Google Scholar
Gutierrez, R., Ruiz-Medina, M. D. and Valderrama, M. J. (1993) Análisis Estocástico de Campo de Kiefer. In Homenaje a Pablo Bobillo Guerrero. Servicio de Publicaciones de la Universidad de Granada, Granada. pp. 213–227.Google Scholar
Gutierrez, R, Ruiz-Medina, M. D. and Valderrama, M. J. (1994) Problemas de representación y predicción sobre el proceso de Ornstein-Uhlenbeck multiparamétrico. Rev. Real Acad. Ciencias Exactas, Fis. Nat. 2, 285–299.Google Scholar
Hochstadt, H. (1973) Integral Equations. Wiley, New York.Google Scholar
Hutson, V. and Pym, J. S. (1980) Applications of Functional Analysis and Operator Theory. Academic Press, London.Google Scholar
Karhunen, K. (1947) Uber lineare Methoden in der Wahrscheinlichkeitsrechung. Ann. Acad.: Sci. Fennicae. Al, 37.Google Scholar
Orellana, E. (1992) Prospección Geoeléctrica en Corriente Continua. Paraninfo, Madrid.Google Scholar
Pietch, A. (1987) Eigenvalues and s-Numbers. Cambridge University Press, Cambridge.Google Scholar
Piterbarg, L. (1981) Investigation of a class of integral equations. Diff. Uravnenija 17, 2278–2279.Google Scholar
Ramm, A. G. (1978) Investigation of some classes of integral equations and their applications. In Abel Inversion and its Generalizations. ed. Preobrazhensky, N. Siberian Dept. Acad. Sci. USSR, Novosibirsk. pp. 120–179.Google Scholar
Ramm, A. G. (1990) Random Fields Estimation Theory. (Pitman Monograph Surveys in Pure and Applied Mathematics 48.) Longman, New York.CrossRefGoogle Scholar
Riesz, F. and Sz-Nagy, B. (1990) Leçons d'Analyse Fonctionnelle. ed. Gabay, J., 3rd edn. Gauthier-Villars.Google Scholar
Ripley, B. D. (1991) Statistical Inference for Spatial Processes. Cambridge University Press, Cambridge.Google Scholar
Walsh, J. B. (1987) An Introduction to Stochastic Partial Differential Equations. (Lecture Notes in Mathematics 1180.) Springer, Berlin.Google Scholar
Yadrenko, M. I. (1983) Spectral Theory of Random Fields. Publication Division, New York.Google Scholar