Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

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E~~ T r: j = const. or E T + E a r = j = 0. n~ a a This approach has been used by HAACK to obtain a fairly reliable conductance cross-section through the Rhinegraben. In the case of E-pol-arisation a different kind of simplification may be in order: Suppose tHe half-width of the anomaly is sufficiently small in comparison to (W~~T)-' everwhere. Then no significant local self-induction clue to HaZ which produces takes places, i.e. the electric field driving the anomalous current will be the large-scale induced normal field only: Assuming E again to be known, the conductance anomaly is now -n ll derived from the observation of the tangential magnetic var,iation anomaly normal to the trend, In the actual performance with real data all field components may be expressed in terms of their transfer functions W and Z and thus be norrnalised with regard to h+ r1tg' Let T be variable only in one horizontal direction, say, i.n y- dlrec-tion'which. implies that the anomal.ous fj-eld is also variable only in that direction, obeying the field equa-tions CE-polarisation) (H-polarisation) - H+ - 1-1- = H - H.k - ay ay ?ax ax ax - jay with and

Here a denotes the conducti.vity at the top of the substratum 0 and ~a~ the anomalous vertical field at z = d which is responsible for driving currents upwards from the substructure to the sheet and vice versa. The boundary conditions, reflecting the "thin-sheet origin" of the variation anomaly, are (E-polarisation) (H-polarisation) for z = 0: H' = -K x H H+ = 0, a Y az ax - for z = d: H- = LII x Haz E = x Eaz. ay a Y c The kernel function of the convolution integrals - are -n 5 H " ' ~ - K * I ~ ' ~ ~ * 1 r "7 ip H KCy) = - =-> "' = Q ~ Y ) - : ~ Y ~ . ~ - "Y hy? q* c(i) U- +m sin(y k ) -7 LB% HCi; "I 1 LII(w,y) = - I - dk + 't H"b x " oik Y Ha): ! Y Cn~=(W'!C Y 7 ) I L E, Y-,.-r-,--* . ./-: sin(y k ) m' 1 Y LICw,y) = - I - d k . ' LI" Ea; . 2 - . UCLX = -go. "-0 ik ~-(w,k) Y n1 Y {hcifl< s2x.x:,~.c.r := ---- He-e C- n I1 is the response function of the substructure for the anomalous TE-field in the case of E-polarisation and C- the nI response function of the substructure for the anomalous TM-field in the case of H-polarisation. The kernel K is the separation kernel, introduced in Sec.5.1. It i connects K and HaZ in such a manner that the source of the aY anomaly is "internal" wh.cn seen from above the sheet. The kernels L are the Fourier transforms of the response functions, intro- .duced in Sec.7.3, which connect the tangential and horizontal field components above a layered half-space. Their application to the anomalous field at z = d implies that H and E diffuse -a -a downward into tlie substructure antidisappear for z -p a. --- They represent the.inductive coupling of the anomalous field with the substructure .'

E~~ T r: j = const. or E T + E a r = j = 0.<br />

n~ a a<br />

This approach has been used by HAACK to obtain a fairly reliable<br />

conductance cross-section through the Rhinegraben.<br />

In the case of E-pol-arisation a different kind of simplification<br />

may be in order: Suppose tHe half-width of the anomaly is<br />

sufficiently small in comparison to (W~~T)-' everwhere. Then no<br />

significant local self-induction clue to HaZ which produces<br />

takes places, i.e. the electric field driving the anomalous<br />

current will be the large-scale induced normal field only:<br />

Assuming E again to be known, the conductance anomaly is now<br />

-n ll<br />

derived from the observation of the tangential magnetic var,iation<br />

anomaly normal to the trend,<br />

In the actual performance with real data all field components<br />

may be expressed in terms of their transfer functions W and Z<br />

and thus be norrnalised with regard to h+<br />

r1tg'<br />

Let T be variable only in one horizontal direction, say, i.n y-<br />

dlrec-tion'which. implies that the anomal.ous fj-eld is also variable<br />

only in that direction, obeying the field equa-tions<br />

CE-polarisation) (H-polarisation)<br />

-<br />

H+ - 1-1- = H - H.k -<br />

ay ay ?ax ax ax - jay<br />

with<br />

and

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