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The theoretical foundations of the various methods of magneto-variational sounding (MVS) are
developed from first principles. Because only time variations of the Earth’s magnetic field are
involved, these methods respond exclusively to the tangential-electric (TE) mode of the electromagnetic
field on and above ground, presuming that the inducing source field is also in this mode.
With increasing period MVS results become less and less sensitive to lateral resistivity contrasts
at shallow depth, which is demonstrated. It sets these methods apart from magneto-telluric soundings
(MTS) subject to persistent surface effects due to anomalous electric field variations in the
tangential-magnetic (TM) mode. This work concentrates on the analysis of daily variations and
associated activity-related variations, yielding response estimates for periods between three hours
and two days. The relevant depth range of penetration extends from 250km to 750km and includes
the transition from a resistive upper mantle (# 100Wm) to a conducting deeper mantle ( 1Wm).
The ultimate purpose of this study is to obtain information about the degree of lateral uniformity in
resistivity beneath Europe.
Among MVS methods the gradient method is the most versatile one, relating the vertical component
of geomagnetic variations to the spatial derivatives of their horizontal components. The connecting
transfer function is the C-response. Within certain limits, which are specified, the method can be
applied without concern about the spatial structure of the inducing source field, which is tested with
response estimates for variations from two different sources: Quasi-periodic daily variations and
transient storm-time variations. All calculations are carried out in spherical coordinates, Alternative
MVS methods based on global presentations of the horizontal components by one or more spherical
harmonics are of restricted applicability. Tests show that the gradient method gives the best results.
A new generalised version of this method removes the constraint about one-dimensionality. It combines
gradient sounding with geomagnetic depth sounding (GDS), provided the source field is of
sufficient spatial complexity to rule out representation by a single spherical harmonic. The resulting
multivariate relation involves up to five transfer functions, including a tensor C-response in close
relation to the tensor impedance for the TE mode in the electric field. Input variables are three of
the four spatial derivatives of the horizontal components and these components themselves. They
are derived from polynomials fitted to the horizontal components in a network of observing sites.
An eigen-value analysis ascertains that the performed fit of 2-dimensional second degree polynomials
is a numerically stable process. A second eigen-value analysis concerns the inversion of the
spectral matrix in multivariate regressions, indicating that regularisation is required, when all five
transfer functions are to be found. Eigen-solutions identify the two GDS transfer functions together
with the Berdichevsky-average of the tensor C-response as the best resolvable combination. Two
kinds of errors are derived for (robust) estimates of transfer functions, distribution-dependent errors
and jack-knife errors. For univariate regressions both errors are of comparable size, but for trivariate
regressions the former turn out to be twice as large. The data base for exemplary soundings are
two years of hourly mean values (1964-65) at 35 European observatories.
