The Backus-Gilbert method and their minimum-norm solution

Abstract

This paper has two objectives. One is to review the Backus-Gilbert (BG) method. Backus and Gilbert investigate continuous inverse problems in a radially symmetric earth using an approach completely different from that of standard inversion methods. If [Formula: see text] is an earth property of interest, the BG method provides average values of [Formula: see text] over some radial intervals. If an interval [Formula: see text] is relatively short and centered near a particular radius [Formula: see text], the BG average will be a local average of [Formula: see text] near [Formula: see text] with a resolving length of [Formula: see text]. Backus and Gilbert also introduce the concept of trade-off between resolution and error, which is essential in inverse theory regardless of the inversion method. In spite of its importance, inverse theory texts neither describe the BG method in full nor show how to apply it in all its generality. Because the three papers that introduce the BG method are mathematically difficult, this paper provides background material and a detailed derivation of the method, as well as two completely worked-out synthetic 1D examples. In addition, the application of the method to disciplines other than solid-earth geophysics is discussed. The second objective is to derive the minimum-norm solution for the continuous case, also introduced by Backus and Gilbert. Their derivation is little known, and it is completely different from a better known, simpler derivation that has some logical difficulties. The minimum-norm solution lacks a constraint that the BG averages have, which may lead to results affected by significant error. To address this problem, a constrained minimum-norm solution is derived here. The two examples for the BG method were used to compute minimum-norm solutions. Interestingly, these examples show that the BG averages can be closer to the true values than the corresponding minimum-norm solutions, although these results should not be considered general.