Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet
Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet
The iteration is carried out either alonq rows or colums. Generally the GauO-Seidel iteration procedure is used with a successive over- relaxation factor to speed up convergence.. 3.3. P.nomalous slab as basic domain In practice it is not necessary to solve the diffusion equation by finite differences in t.he total conductor and the air half-space. Instead it is sufficient to treat the equation only in that slab which contains the anomalous domain. Let the anomalous slab be confined to the depth range zl 5 z 5 z2. Within this domain we have to solve the inho3nogeneous equation (considering for the moment only the TE-case) subject to two homogeneous boundary conditiolls at z = z and z2, 1 ~Thich involve a for z < z and z > z2 respectively and account for n 1 the vanishing anomalous field for z -t 2 m. Yihen (3.25) is solved by finite differences, the discretization invol.ves also the field values one grid point width above and below the anomalous slab. The idea is to express these values in terms of a line integral over Ea at z = z and z2 respectively. I Let V and V2 be the half-planes z - < zl and z -- > z2, respectively. ~ e G t r [r). 2, r, be Green's functions which satisfy " "m (-0 - subject to the boundary condition In V and V2, E is a solution of 1 a Now Green's formula for two-dimensions states that
The RHS i.s a closed line integral bordering the area over which the LIIS integral is performed. I11 the RHS differentiation in direction to the outr,?ard normal is involved. From (3.28) and (3.26) fol.lows Identi.fying U with GI V with Ea, Eq. (3.29) yields in virtue of (3.27) E - a Q around Vm I r - a an G ( ~ (&IL)ds. ) Now G'") and its normal derivative vanish at infinity. Hence, only the part of the line integral along the axis z = z contrihut-.es. a a m Por m=? : - a - - m=2 : - a =-- 2n az' an az' Hence, Because of (3.26) , Eq. (3.30) depends only on the difference y-yo. Defining q . (3.30) reads shorter For a layered structure in Vm, the kernels R (m)' are easily deter- mined:
- Page 1 and 2: Electromagnetic Induction in the Ea
- Page 3 and 4: 6.2. Generalized matrix inversion 6
- Page 5 and 6: A1~Lernativel.y p = 30. m, where T
- Page 7 and 8: The e1ectrica:L effect - of the cha
- Page 9 and 10: The vari.ables x,y, and t which do
- Page 11 and 12: - d IufM12 > 0 . dz - - On the othe
- Page 13 and 14: a) TM-mode From (2.251, (2.26); (2.
- Page 15 and 16: with " the abbreviation + - 1 - ? =
- Page 17 and 18: 2n -i~cr cos (8-$1 -iKrcU J e ~B=J
- Page 19 and 20: In the 1irnl.t a -+ o, I +- m, M =
- Page 21 and 22: -- 2.5. Definition -- of the transf
- Page 23 and 24: we arrive at - v The same appl-ies
- Page 25 and 26: ) Computation of ---- C for a laxez
- Page 27 and 28: The approximate interpretation of C
- Page 29 and 30: I ~ispersi-on relations I Dispersio
- Page 31 and 32: where L is a positively oriented cl
- Page 33 and 34: 1 2 3 4 CPD 1 2 3 4 CPD - g) Depend
- Page 35 and 36: The TE-mode has no vertical electri
- Page 37 and 38: i I Earth Anomalous domain 3.2. Air
- Page 39: Hence, the conductivity is to be av
- Page 43 and 44: 4. Having determined B;, the coeffi
- Page 45 and 46: 3.4. Anomalous region as basic doma
- Page 47 and 48: - 6 and 6= can be so adjusted that
- Page 49 and 50: From the generalized Green's theore
- Page 51 and 52: and y can again be so adjusted that
- Page 53 and 54: 4.2. In3ral - --- equation method L
- Page 55 and 56: The element GZx is needed for all z
- Page 57 and 58: With this knowledge of the behaviou
- Page 59 and 60: After having determined Qzr VJ,; @,
- Page 61 and 62: 4.3. The surface inteyral approach
- Page 63 and 64: F At the vertical boundaries the co
- Page 65 and 66: The four equations A A A A H = i sg
- Page 68 and 69: 6. Approaches to the inverse proble
- Page 70 and 71: to minimize the quantity a s = 12 /
- Page 72 and 73: It remains to show a way to minimiz
- Page 74 and 75: Agai-n, from a finite erroneous dat
- Page 76 and 77: Here lJ - is a N x P matrix contain
- Page 78 and 79: small eigenvalues. The parameter ve
- Page 80 and 81: Then - 77 - A(E2 - E ) = iwu U (E -
- Page 82 and 83: whence 2k d -2k d where a = CA:(A;)
- Page 84 and 85: . 7. Basic concepts of geomagnetic
- Page 86 and 87: orders of magnitude smaller' than t
- Page 88 and 89: Elimination of - E or .,. H yields
The RHS i.s a closed line integral bordering the area over which the<br />
LIIS integral is performed. I11 the RHS differentiation in direction<br />
to the outr,?ard normal is involved.<br />
From (3.28) and (3.26) fol.lows<br />
Identi.fying U with GI V with Ea, Eq. (3.29) yields in virtue of<br />
(3.27)<br />
E -<br />
a Q<br />
around Vm<br />
I<br />
r - a an G ( ~ (&IL)ds.<br />
)<br />
Now G'") and its normal derivative vanish at infinity. Hence, only<br />
the part of the line integral along the axis z = z contrihut-.es.<br />
a a m<br />
Por m=? : - a - - m=2 : - a =--<br />
2n az' an az'<br />
Hence,<br />
Because of (3.26) , Eq. (3.30) depends only on the difference y-yo.<br />
Defining<br />
q . (3.30) reads shorter<br />
For a layered structure in Vm, the kernels R (m)' are easily deter-<br />
mined:
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