Theory for Errors, Resolution, and Separation of Unknown Variables in Inverse Problems, with Application to the Mantle and the Crust in Southern Africa and Scandinavia

Abstract

The geophysical inverse problem, the determination of the properties of the subsurface from a limited set of measurements, is basically non‐unique. Separation of the unknown variables, the depth resolution, and the accuracy of the parameter estimates are three competing objectives which are reciprocally related. The investigation of a set of free oscillation measurements having errors based upon those given by Derr revealed that the addition of overtones with low radial order numbers to the set of fundamental mode observations does not greatly improve the shear velocity depth resolution but does facilitate the separation of shear velocity from density. Two sets of multi‐mode surface wave dispersion measurements were inverted: the surface wave observations for Southern African reported by Bloch, Hales & Landisman, and a set of data for central Scandinavia as reported by Noponen, Porkka, Pirhonen and Luosto, and Crampin. Several models were obtained for each data set by varying the constraints used in the inversion process. The errors of the dispersion measurements were propagated into estimates of the accuracy of the inverted crustal models for the first time. Inclusion of shear and compressional velocity refraction measurements as a constraint on the inversion process leads to models for Scandinavia having some indication of velocity inversions at the base of the sialic portion of the crust at the base of the crust and in the first few tens of kilometres of the uppermost mantle. Copyright © 1972, Wiley Blackwell. All rights reserved