Linear inverse problems in wave motion: nonsymmetric first-kind integral equations

Abstract

We present a general framework to study the solution of first-kind integral equations. The integral operator is assumed to be compact and nonself-adjoint and the integral equation can possess a nonempty null space. An approach is presented for adding contributions from the null space to the minimum-energy solution of the integral equation through the introduction of weighted Hilbert spaces. Stability, accuracy, and nonuniqueness of the solution are discussed through the use of model resolution, data fit, and model covariance operators. The application of this study is to inverse problems that exhibit nonuniqueness.