Formulation of a statistical approach for estimating external source excitation characteristics toward consistent magnetotelluric modeling

The conventional magnetotelluric (MT) framework, formulated in Cartesian coordinates using the 4-element impedance and the tipper, assumes a spatially uniform external source for which these MT responses are source independent. Recent theoretical studies have shown that, in spherical geometry under the dipole source assumption, the same responses can exhibit source-amplitude dependence, implying that internally consistent modeling requires knowledge of the source statistics that characterize the ensemble of magnetic field variations from which the responses are inferred. However, no method has existed to infer these statistics directly from single-site MT observations. This study introduces a statistical formulation that reconstructs the source covariance matrix from observed magnetic spectra and derives an eigen-source-scaling matrix that characterizes the second-order spectral excitation statistics. When the observable rank of the magnetic covariance matrix remains three, this matrix enables source-consistent modeling in spherical geometry, providing a physically grounded alternative to the conventional MT formulation. When the observable excitation space becomes effectively two-dimensional, the data cannot resolve all three excitation modes. Although this observational degeneracy simplifies the apparent MT response, spherical forward modeling may still exhibit source-amplitude dependence, indicating that a reduced representation is not always physically adequate. In practical applications, measurement noise and non-dipolar contributions affect the estimated source statistics. Remote-reference (RR) processing helps recover the underlying excitation characteristics by suppressing uncorrelated noise and improving the effective rank of the magnetic covariance matrix. The proposed framework provides a diagnostic for assessing the validity of the conventional MT formulation and a practical means to incorporate source-consistent excitation statistics when necessary.