| Abstract: | We consider how to treat a finite-dimensional linear inverse problem when the form of the forward problem is known exactly, but is dependent upon some parameters whose exact value is uncertain and which enter the forward problem multiplicatively. We show one way to proceed when the uncertainty is treatable in a statistical manner. Predicting the secular variation ∂tB(t) produced by a particular fluid flow V at the core-mantle boundary (when magnetic diffusion is ignored) is one such example, because the results depend on the main magnetic field B(t) originating in the core which is improperly known because of contamination by the crustal magnetic field. This infinite-dimensional inverse problem is often solved by projection on to a finite-dimensional basis, and the resulting parameters found by a maximum likelihood technique. If the main field is contaminated with errors from a Gaussian distribution, this paper describes how the maximum likelihood solution can take this into account, and we show the probability density function that must be maximised in this case. We give an example of the effects for a simple model system, and suggest possible areas of application. |
