Trimming and procrastination as inversion techniques

By examining the processes of truncating and approximating the model space (trimming it), and by committing to neither the objectivist nor the subjectivist interpretation of probability (procrastinating), we construct a formal scheme for solving linear and non-linear geophysical inverse problems. The necessary prior information about the correct model x(E) can be either a collection of inequalities or a probability measure describing where x(E) was likely to be in the model space X before the data vector y(0) was measured. The results of the inversion are (1) a vector z(0) that estimates some numerical properties z(E) of x(E); (2) an estimate of the error delta z = z(0) - z(E). As y(0) is finite dimensional, so is z(0), and hence in principle inversion cannot describe all of x(E). The error delta z is studied under successively more specialized assumptions about the inverse problem, culminating in a complete analysis of the linear inverse problem with a prior quadratic bound on x(E). Our formalism appears to encompass and provide error estimates for many of the inversion schemes current in geomagnetism, and would be equally applicable in geodesy and seismology if adequate prior information were available there. As an idealized example we study the magnetic field at the core-mantle boundary, using satellite measurements of field elements at sites assumed to be almost uniformly distributed on a single spherical surface. Magnetospheric currents are neglected and the crustal field is idealized as a random process with rotationally invariant statistics. We find that an appropriate data compression diagonalizes the variance matrix of the crustal signal and permits an analytic trimming of the idealized problem.