Interpretation of Inaccurate, Insufficient and Inconsistent Data

Abstract

Summary Many problems in physical science involve the estimation of a number of unknown parameters which bear a linear or quasi‐linear relationship to a set of experimental data. The data may be contaminated by random errors, insufficient to determine the unknowns, redundant, or all of the above. This paper presents a method of optimizing the conclusions from such a data set. The problem is formulated as an ill‐posed matrix equation, and general criteria are established for constructing an ‘inverse’ matrix. The ‘solution’ to the problem is defined in terms of a set of generalized eigenvectors of the matrix, and may be chosen to optimize the resolution provided by the data, the expected error in the solution, the fit to the data, the proximity of the solution to an arbitrary function, or any combination of the above. The classical ‘least‐squares’ solution is discussed as a special case.

Generalized least-squares solutions to quasi-linear inverse problems with a priori information.

“…This solution can be regarded as a natural extension of the classical least-squares solution, and provides a theoretical basis on the "sharp cutoff approach" of WIGGINS (1972) and also JACKSON (1972) in their formalism. On the other hand, the minimum variance solution of FRANKLIN (1970) and also JACKSON (1979), which gives another generalization of the classical theory, provides a theoretical basis on the "tapered cutoff approach" of LEVENBERG (1944) and MARQUARDT (1963MARQUARDT ( , 1970.…”

“…variables, and constructed a general inverse formalism in the context of determining the earth's structure from free oscillation observations. General treatments of the linear inverse problem have been developed by JACKSON (1972) andWIGGINS (1972) in terms of a simple matrix algebra. In their formalism, the inverse oper-ator which optimizes the trade-off between resolution and estimation error is directly constructed from the "natural inverse" of LANCZOS (1961) for a coefficient matrix by truncating the smallest eigenvalues in the inverse so that the estimation error remains less than the certain maximum error determined a priori from requirement for definiteness in the solution.…”

Reconstructing source terms from atmospheric concentration measurements: Optimality analysis of an inversion technique

“…The inverse operator in it, R 1 , is called the Moore-Penrose pseudoinverse. Several criteria can be used to measure the quality of an inverse operator: ''an operator will be a good inverse'' if it satisfies (i) RR 1~I n , (ii) R 1 R~I m , and if (iii) ''the uncertainties in the estimate are not too large, i.e., its variance is small'' [Jackson, 1972]. The first criterion ensures, in case of noiseless data, that the measurements predicted with the source estimate (l5Rr == 5RR 1 l) correspond with the true measurements.…”

“…Moreover, it is shown that this inverse operator satisfies criteria used in geophysics to define a ''good inverse'' [Jackson, 1972] and that it fulfils several optimal properties for the localization of single point sources. This is demonstrated by using the so-called Model Resolution Matrix (MRM) formalism, initially defined for discrete geophysical data analysis by Menke [1984] and currently used to evaluate solutions to the neuroelectromagnetic inverse problem [Grave de Peralta et al, 2009].…”

Bayesian estimation of hypocenter with origin time eliminated.

“…To find a compromise between estimation error and resolution is the central subject in geophysical inverse theory developed by BACKUS and GILBERT (1967, 1968, FRANKLIN (1970), WIGGINS (1972), andJACKSON (1972). Examples of using geophysical inverse theory are found in the papers of CROSSON (1976), Am andLEE (1976), andSPENCER andGUBBINS (1980) for the simultaneous estimation of hypocenter and velocity structure.…”