Multivariate analysis of geomagnetic array data: 1. The response space

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Abstract
In this paper we develop a new approach to the multiple‐station analysis of geomagnetic array data. Our approach is based on the observation that the space of external sources, and hence the space of observable electromagnetic fields at the Earth's surface (the “response space”), can often be well approximated by spaces of very low dimensionp. We demonstrate that the simplifying assumption of a finite dimensional external source space is required for a rigorous justification of transfer function (TF) methods in general and show that whenp= 2, the response space is equivalent to all possible interstation and intercomponent TFs. Heuristically, estimation of the response space by thepdominant eigenvectors of the spectral density matrix (i.e., the frequency domain matrix of averaged cross products for all components measured in the array) can be easily justified. A more rigorous statistical treatment is also discussed. Our statistical model (a complex errors‐in‐variables model) allows for (potentially correlated) errors in all measured field components and treats all stations in a symmetric fashion. The model is thus substantially more reasonable than the usual statistical model used in TF estimation which assumes that the fields at a reference site are noise free and “normal.” We illustrate our approach with a small five‐station magnetotelluric array which was run as part of the EMSLAB experiment. For this small array,p= 2 is a good approximation, and the two dominant eigenvectors provide an estimate of the response of the Earth to (approximately) uniform sources. There is, however, substantial additional structure in the spectral density matrix (SDM), and further analysis of the eigenvalues and eigenvectors of the SDM reveals that the “noise” (i.e., the portion of the data which does not fit a uniform source model) is dominated by source effects. These results illustrate a significant advantage of multiple‐station techniques: they allow us to test model assumptions and to separate noise into coherent and incoherent components, allowing a clearer understanding of the problems, and possibilities, to be found in the interpretation of geomagnetic induction data.