Schmucker, 1970 (Scripps) - MTNet

Schmucker: Geomagnetic Variations 69
5.6 Direct Inversion of Surface Observations
The inversion of the previous section can be generalized as follows. Suppose
the impedance or the Z/H ratio above a multilayered substratum has been
observed or calculated. We apply (5.55) to these ratios and obtain for each
resolved frequency component depth and conductivity of a uniform substitute
conductor. This substitution is meaningful only when the arguments of the
ratios lie between i 1T and l 1T, thereby yielding a positive depth h. A multilayered
substratum is in tE-at case indistinguishable from and therefore replaceable
by a uniform conductor at the depth h, as far as its response to a
single frequency component of the incident variation field is concerned. It is
presumed, of course, that (h + Pc) is small against the wave length of the
source field. Cf. p.108 for an extension to spherical conductors.
The depth h" from (5.56), however, is always positive or zero, since the
arguments of E/H and Z/H cannot be smaller than zero. Hence, we can interpret
the out-of-phase component of any given ratio E/H or Z/H in terms
of a perfect substitute conductor at the depth
h* = w -1 Irn(E/H) (S.S7a)
while the in-phase component yields in p the apparent conductivity
c
ITc = '!J {81T[Re(E/H)]2 } -1 (S.S7b)
at that depth. Plots of h* versus IT c for a number of different frequencies
represent good approximations of the true conductivity distribution as demonstrated
in figure 31.
Cagniard's (1953) definition of the "apparent resistivity" which is commonly
used in this context is based on the modulus of the impedance. It would
yield as apparent skin-depth value the modulus of cfl, while the parameters
h* and ITc as defined above give proper regard to the in-phase and out-ofphase
component of the impedance.
5.7 Special Case III: Limitations of Price's Method
Repeated use will be made of Price's method when dealing with the induction
in thin sheets Or shells (cf. sec. 1.2). This method becomes inadequate
(a) when the skin-depth value of the sheet is equal to or even smaller than
its thickness, and (b) when the sheet is underlain at shallow depth by a good
conductor. In both cases the tangential electric field is distinctly attenuated
within the sheet and the integration of equation 1. 8 cannot be carried out with E
as a constant. Since the terrestrial surface layers, to which this method
will be applied, are indeed underlain at some depth by highly conductive
mantle material, such limitations for Price's method deserve careful consideration.