Magnetotelluric sampling theorem

We consider a general case of a magnetotelluric (MT) study to reveal three-dimensional (3D) distribution of the electrical conductivity within the Earth based on measurements of electromagnetic (EM) fields by a two-dimensional (2D) array. Such an MT array observation can be regarded as a spatially discrete sampling of the MT responses (impedances), and each observation site can be regarded as a sampling point. This means that MT array measurements must follow the Nyquist-Shannon sampling theorem. This paper discusses how the sampling theorem is applied to MT array studies and what kind of consideration is required in the application on the basis of synthetic model calculations, with special attention to spatial resolutions. With an aid of the EM scattering theory and the sampling theorem, we can show that an observation array resolves some features of the MT impedance but does not others. We call the resolvable and unresolvable features the MT signal and noise, respectively. This study introduces the spatial Fourier transform of array MT data (impedances) which helps us investigating sampling effects of lateral heterogeneity from a different angle (in the wavenumber domain). Shallow heterogeneities cause a sharp spatial change of impedance elements near structural boundaries. High wavenumber Fourier components are required to describe such a feature, which means the site spacing must be sufficiently short to be able to resolve such features. Otherwise, a set of array MT data will suffer from aliasing, which is one of the typical causes of MT distortion (MT geologic noise). Conversely, a signal due to a deep-seated conductivity anomaly will have more reduced amplitude at higher wavenumbers, which means focused imaging of such an anomaly is generally difficult. Finally, it is suggested to properly consider the sampling theorem in an observation array design, so as to have best performance in resolving MT signals.