Funnel functions in linear and nonlinear appraisal

This paper presents new methods for appraising the nonuniqueness inherent in linear and nonlinear inverse problems. The usual approach of constructing unique averages of the model is abandoned in favor of computing upper and lower bounds to inner products of the model with exact boxcar functions of variable width. The bounds are found by extremizing a suitable objective function in a linear programing problem. Information about the average value of the model is presented in terms of funnel functions and uncertainty functions. In an alternate approach, information about the model averages is summarized by global bound curves. These functions are upper and lower bounds to the indefinite integral of the model, and they provide, at a glance, insight about the resolving power of the data. Once computed, they can also be used to generate approximate funnel functions. The methods presented here are also applicable to the nonlinear problem, but first a reformulation is necessary so that a global norm of the model can be extremized. This has been done, but the global extremum is achieved only through iteration. Consequently, the potential difficulties arising from nonconvergence of a linearized algorithm and possible convergence to a local, rather than global, extremum are ever present. As an example of the nonlinear problem, I have analyzed the magnetotelluric responses taken near an ocean spreading center in the Pacific. The data are shown to possess very poor resolving power, and hence the large variability seen in acceptable models is now understood.