Two Classic Models of the Distortion Theory
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Two Classic Models of the Distortion Theory
In the subsequent chapters we will consider how the near-surface and deep inhomogeneities distort the magnetotelluric and magnetovariational response functions observed on the Earth’s surface. We are going to examine a number of models with a view to find the characteristic evidences of two-dimensional and three-dimensional galvanic and induction distortions and look for ways of their recognition. All models to be examined are excited by a plane monochromatic wave vertically incident on the Earth’s surface. We have to analise the informativeness of MT- and MVsoundings and provide the background necessary for qualitative interpretation of experimental data and identification of objective geoelectric structures. As a preliminary we would like to review two, now classic, models put forward by Obukhov (1962), d’Erceville and Kunetz (1962) and Rankin (1962) at dawn of magnetotellurics. A special feature of these two-dimensional models is that they offer analytical solutions for the TM-mode.
6.1 The Vertical-Interface Model We start with a vertical-interface model shown in Fig. 6.1. Meaningful analysis of this model can be found in works by Weaver (1963, 1994), Berdichevsky (1968), Jones and Price (1970) and Fischer et al. (1992). The model consists of the nonconductive air and the conductive Earth that includes two quarter-spaces of different resistivities, and , divided by the infinite vertical interface y = 0, 0 ≤ z ≤ ∞. The problem for the TM-mode has been solved independently by Obukhov (1962) and d’Erceville and Kunetz (1962). Following these pioneering works, we write ⎧ N A ⎪ y ≤ 0, z ≥ 0 ⎨ H˙ x (z) + H˙ x (y, z) (6.1) Hx (y, z) = ⎪ ⎩ H¨ N (z) + H¨ A (y, z) y ≥ 0, z ≥ 0, x x N A N A where H˙ x , H˙ x and H¨ x , H¨ x are the normal and anomalous magnetic fields within the left and right quarter-spaces.
M. Berdichevsky, V.I. Dmitriev, Models and Methods of Magnetotellurics, C Springer-Verlag Berlin Heidelberg 2008 DOI 10.1007/978-3-540-77814-1 6,
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6 Two Classic Models of the Distortion Theory
Fig. 6.1 The vertical-interface model
The normal fields are defined as N H˙ x = Hxo eik z ,
N H¨ x = Hxo eik z .
(6.2)
√ √ N N Here k = io / , k = io / , Im k > 0 and Hxo = H˙ x (0) = H¨ x (0) = p p 2 H (0), where H (0) is the primary magnetic field on the Earth’s surface z = 0. The anomalous fields meet the equations A
A
A A ⭸2H¨ x ⭸2H¨ x A + + (k )2H¨ x = 0 ⭸ z2 ⭸ y2
⭸2H˙ x ⭸2H˙ x A + + (k )2H˙ x = 0, ⭸ z2 ⭸ y2
(6.3)
A A with the boundary conditions H˙ x (0) = H¨ x (0) = 0 on the Earth’s surface and the A A conditions H˙ x (z > 0) → 0, H¨ x (z > 0) → 0 at infinity. y→ −∞
y→ ∞
Solving these equations by the method of separation of variables, we get A H˙ x =
∞
am e y
sin m z dm,
A H¨ x =
0
∞
am e− y sin m z dm,
(6.4)
0
where =
m 2 − (k )2 ,
=
m 2 − (k )2 ,
Re > 0.
The constants am and am can be defined from the boundary conditions at the vertical interface y = 0. It follows from
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