# Fast and Accurate Maximum-Likelihood Estimation of Parameterized Spectral Densities that Jointly Characterize Bivariate Two-Dimensional Random Fields

Monday, October 22, 2018 - 11:10am - 12:00pm

Keller 3-180

Frederik Simons (Princeton University)

The problem posed in this contribution is as follows. Two unobservable random fields of a certain unknown spectral density structure are the input to a pair of coupled differential equations with a common set of unknown parameters, which generates a pair of observable output random fields. What are the parameters of the differential equations, and what are the spectral densities of the input fields? Without (much) loss of generality, we assume a Matern covariance for the input fields, and attempt to derive their parameters as part of the set to be estimated. The statistically interesting problems arise when the fields are finite-dimensional, stationary over subdomains of arbitrary geographical description, and potentially haphazardly observed within those. The geophysically interesting problem is to derive process from their values, and to formulate geological hypotheses as to their origin. The science questions motivating the mathematical and computational approach put a real premium on estimating those parameters in an unbiased, robust, fast and flexible manner, and on formulating sensitive hypothesis tests for the applicability of the modeling framework. Our spectral-domain approach relies on the Whittle likelihood with important new, theoretically justified modifications for the debiasing necessary to treat incompletely observed finite-field summed-output problems. Our =main examples are from problems in planetary science that seek to elucidate the joint structure of topography and gravitational anomalies in order to decipher the mechanical structure of a planetary interior.