Open AccessInferring the Subsurface Structural Covariance Model Using Cross‐Borehole Ground Penetrating Radar Tomography
Author(s) -
Hansen Thomas M.,
Looms Majken C.,
Nielsen Lars
Publication year - 2008
Publication title -
vadose zone journal
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.036
H-Index - 81
ISSN - 1539-1663
DOI - 10.2136/vzj2006.0144
Subject(s) - covariance , covariance function , ground penetrating radar , rational quadratic covariance function , tomography , inverse problem , a priori and a posteriori , covariance intersection , general covariance , covariance matrix , estimation of covariance matrices , radar , algorithm , matérn covariance function , computer science , mathematics , statistics , physics , mathematical analysis , optics , telecommunications , philosophy , epistemology
We address a fundamental problem inherent in least squares based ground penetrating radar tomography problems, and linear inverse Gaussian problems in general: how should the a priori covariance model be chosen? The choice of such a prior covariance model is most often a very subjective task that has major implications on the result of the inversion. We present a method that allows quantification of the likelihood that a given choice of prior covariance model is consistent with the observed tomography data. This is done by comparing statistical properties of samples of the prior and posterior probability density function of the tomographic inverse problem. In essence, if samples of the posterior are unlikely samples of the prior, then such a choice of a priori covariance model is deemed unlikely. This enables one to quantify the consistency of a number of equally probable prior covariance models to data observations. A synthetic data set was used to describe and validate the approach. We determined how a known covariance model could be inferred from a synthetic tomography problem. The methodology was then applied to a nonlinear ground penetrating radar tomography case study. The covariance model deemed most likely was consistent with nearby ground penetrating radar reflection profiles. The method provides useful results even if just a subset as small as 10% of the available data is considered.