Degeneracy, singularity and multiple solutions in

Geophys. J . Int. (1993) 113,434-448

Degeneracy, singularity and multiple solutions in geophysical inversion D. W. Vasco Seismographic Station and Center for Computational Seismology, University of California, Berkeley, CA 94720, USA

Accepted 1992 October 5. Received 1992 August 18; in original form 1991 July 13

Key words: inverse problem.

INTRODUCTION Many geophysical inverse problems have no unique solution. How do such situations arise? In linear problems non-uniqueness manifests itself as a singular matrix and a linear system of equations with an infinite number of solutions. In non-linear problems two, three, up to an infinite number of solutions may occur. The number of solutions to an inverse problem depends on the structure of the geophysical experiment. By this I mean the physics used in modelling the phenomena under study as well as the particulars of the experiment. Such factors may often be described by a finite of parameters. An important factor which should be considered before an experiment is conducted is the role of these experimental or control parameters, such as station locations or an assumed medium property, on the number of solutions. For example, symmetries or other degeneracies in the experimental construction can lead to multiple solutions. Certain combinations of control parameters result in degenerate inverse problems where small perturbations will change the number of minima. To date, I know of no systematic exploration of the relation between experimental parameters and the stability of solutions to general geophysical inverse problems. Here, techniques are presented to examine such a relationship. The approach is based on the theory of singularities of smooth mappings as developed by

434

Morse (1931), Whitney (1955), Arnold (1974, 1975), Thom (1975), Mather (1968a, .1968b, 1969a, 1969b, 1970) and others. The basic dilemma is that all mathematical descriptions of geophysical experiments are approximate. Therefore, we as geophysicists are never solving the correct inverse problem. We may conceive of the unmodelled aspects of the experiment as perturbations of our description of the inverse problem. Therefore, one would like to know if the solutions to a given inverse problem are stable with respect to these small changes in the construction and description of the geophysical experiment. If not, when the description of a degenerate geophysical experiment is perturbed slightly the number and types of solutions to the inverse problem will change. In such a case it is desirable to know how the solutions change. For example, how many additional minima appear in the perturbed problem? In general, the solution to a geophysical inverse problem may be phrased as the minimization of a functional which depends on the Earth model parameters as well as on the experimental control parameters. The approach taken here is to reduce this functional to the lowest order polynomial which has the same number and type of minima. The first portion of this work is concerned with techniques to accomplish this reduction. It is then possible to derive a relationship between the resulting low-order terms of the functional and the number of solutions which appear when

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SUMMARY The relationship between the parameters describing a geophysical experiment and the number of solutions to the inverse problem is explored. It is found that degenerate situations, such as an inherent symmetry in an experimental set up, can give rise to multiple or even an infinite number of solutions. Techniques are developed to determine the simplest multivariable polynomial which represents the essential properties of the misfit functional. From the polynomial, the stability of a given solution to an inverse problem may be evaluated. It is shown how to calculate additional terms necessary to represent all the possible perturbations of a given minima due to changes in the description of the geophysical experiment. These terms indicate the number and types of solutions to the inverse problem which result from such changes. An example, the earthquake location problem, is considered. For differering station configurations the nature of the solutions to the location problem behave dramatically different with respect to changes in station position.

Geophysical inversion

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the functional is perturbed. In the later part of this paper a method is developed to characterize the changes produced by the perturbation of a functional. Geophysical experiments and inversion A geophysical experiment involves gathering a set of data d, the response of the Earth to some phenomena measured at a number of stations. The experiment depends on k experimental or control parameters, c, such as station locations. The parameters may include inherent assumptions in the formulation of a geophysical model such as a fixed density or a symmetric velocity model used in the analysis. For a finite dimensional inverse problem the Earth properties of interest, the model parameters x, may be considered to be a vector in an n dimensional space. If a set of measurements is independent, the probability of I measurements is given by the product of the probabilities of each datum,

df c ) the zeros of ___. (x3

dXi

For a general objective functional f(x, c ) there may be one, two, to an infinity of minima. For a given set of experimental parameters c the objective functional defines a mapping 9"-+ 9 from the space of models to the real line. When the experimental or control parameters are explicitly considered f(x,c ) defines an W k dimensional family of mappings from 9"+9,or the mapping 9 .X W k + 9. Each member of this family, each objective functional for a particular set of experimental control parameters c = co has a specific set of minima. As c is vaned the number of minima may change.

P(d;).

i=l

For data with Gaussian errors this is given by,

where g,(x, c ) is the functional relationship between the geophysical model and the predicted observations based on that model, it has a parametric dependence on the k experimental parameters c. The confidence interval on the data is 6d and the variance of the observations is u2.There is only one model, the Earth, not a statistical distribution of models and just a statistical distribution of data sets drawn from it. Therefore, one seeks a model which makes the data the most likely to have occurred (maximum likelihood model) (Menke 1984). The probability of the data is maximized when either the maximum of P ( d ) is computed, or the minimum of its negative logarithm,

is found. Data errors are not always Gaussian and other distributions, such as the two-sided exponential distribution

Singularity and degeneracy As mentioned, an objective functional f(x,c ) defines a mapping from 9"X gk to W. The set of extrema of the objective functional maximum, minimum, saddle points and the like, points where the gradient (Of],) vanishes, are all termed critical points. Generally, a critical point, x,, may be isolated, no other such points lie within some small neighbourhood. An important categorization of critical points is based on the Hessian of the objective functional:

D'fl, =

~

2f .

A critical point is non-degenerate if the

d X i dXi

Hessian is non-singular. That is, a critical point is degenerate if it cannot be locally described by a quadratic form and higher order terms are required. Non-degenerate critical points are isolated while degenerate critical points may be either isolated or non-isolated. As will be shown, the perturbation of a functional at a degenerate critical point will change the number .and type of critical points. Depending on the Hessian, an objective functional may be transformed into a number of normal or canonical forms. The Hessian matrix is a function of experimental control parameters and as these parameters vary the matrix may become non-singular. A related concept is that of a singular point of a mapping f, say from W" to 9". A point xoE R" is a singular point off if the rank, r, of the Jacobian matrix

ah

or the Cauchy distribution

may be more appropriate. A general form for the probability of a set of independent data may be written,

n exp I

P(d)=

[-p(d,, x, c)16d

i=l

where p(di, x, c ) is the negative logarithm af the probability density (Menke 1984). Maximizing the likelihood leads to

- is not maximal i.e. r