Linear and non-linear appraisal using extremal models of bounded variation

Abstract

Summary Model features may be appraised by computing upper and lower bounds for the average value of the model over a specified region. The bounds are computed by constructing extremal models which maximize and minimize this average. In order to compute the most meaningful bounds, it is important that the allowed models are geophysically realistic. In this paper, the appraisal analysis of Oldenburg (1983) is extended to incorporate a bound on the total variation of the extremal models. Restricting the variation discriminates against highly oscillatory models and, as a consequence, the difference between upper and lower bounds is often considerably reduced. The original presentation of the funnel function bound curves is extended to include the variation of the model as another dimension. The interpreter may make use of any knowledge or insight regarding the variation of the model to generate realistic extremal models and meaningful bounds. The appraisal analysis is extended to non-linear problems by altering the usual linearized equations so that a global norm of the model can be used in the objective function. The method is general, but is applied here specifically to compute bounds for localized conductivity averages of the Earth by inverting magnetotelluric measurements. The variation bound may be formulated in terms of conductivity or log conductivity. The appraisal is illustrated using synthetic data and field measurements from southeastern British Columbia, Canada. Bounding the total variation may be viewed as constraining the flatness of the model. This suggests a new method of calculating (piecewise-constant) l  1 flattest models by minimizing the norm of the total variation. Unlike l  2 flattest models which vary in a smooth, continuous manner, the l  1 minimum-variation model is a least-structure model that resembles a layered earth with structural variations occurring at distinct depths.