Multivariate analysis of geomagnetic array data: 2. Random source models

Abstract

In this paper I develop a spatial stochastic process model for randomly varying electromagnetic source fields observed over a one‐dimensional Earth. With this model the complex covariance between any two electromagnetic field components, measured at the same or different locations on the Earth's surface, can be computed. These calculations yield the expected values for the elements of the spectral density matrix (SDM; the matrix of averaged cross products of all components measured in the array) for synthetic arrays with arbitrary station locations. The model can thus be used to study the properties of the eigenvalues and eigenvectors of these synthetic array SDMs. For the case where both the array size and the skin depth of the electromagnetic fields in the conductive Earth are small compared to typical source length scales, the synthetic SDMs exhibit two larger eigenvalues, which correspond to nearly uniform source fields, followed by three smaller eigenvalues, which correspond to a set of canonical gradients. Qualitative consideration of more general models with three‐dimensional conductivity and arbitrary random sources shows that the eigenvalues of a geomagnetic array SDM represents a sort of discrete spatial power spectrum. The corresponding eigenvectors form a natural basis which any realizable field can be expanded in. These results provide strong justification for a generalized transfer function/response space model for the analysis of geomagnetic array data.