Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet
Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet
For a dipole - in z-direction this is equal to a e-kR A e -kR G = (k" - - grad-) = - curl2 (z -- 4a~k' since -kR A (e-kR/@~r~) ) = lc2e / (4aR) - -z a~ 4Ta2 - 1 I Comparison of (4.16) and (4.12) shows that for a whole space The absence of a TE-potential is clear from physical reasons, since the magnetic field of a vertical. dipole must be confined to hori- zontal planes (i.e. no vertical. magnetic field, which can only be produced by a TE-field). Using Somrnerfeld's integral, C4.17) is written The f5.el.d of a horizontal electric dipole (in x-direction, say) has both an electric and magnetic component in z-direction. Hence, a TM- and TE-potential are needed. 9 + Since ,- J%i. a2 a2 1 a2 e --kl?. m -a/ z-z ] G =-(- + -7) 4Jx=--y - -=-- I I A2e O J~ (Xr)dX cos$ Xz ax2 ay 41~k axaz R 4ak20 and we have and from then follows xsiqn(z-z ) 0 J, (Xr)dXcos$ * sign (z-zo) (4.19) f
With this knowledge of the behaviour of $,, @,, QX in the uniform whole-space, these functions for a layered medium can be easily obtained. Let in the m-th layer hm < z < hmt1 where where + -1- ~ ~ A i f i z - < z0 + -I- ~ ~ B I ~ f i > z o z - + and f-. = (z-h )Il a2 = h2-tk2. Then starting with n~ _ in n~ m m - t - + - + - + 1, A =O, B =Or BL=l1 C =I, C=O, D =01 D =I. ko 0 L 0 0 JJ 1, The boundary conditions(4.13) lead to the recurrence relations
- 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 and 40: Hence, the conductivity is to be av
- Page 41 and 42: The RHS i.s a closed line integral
- 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: The element GZx is needed for all z
- 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
- Page 90 and 91: Observing that rot pot rot g = - ro
- Page 92 and 93: Two special types of such anomalies
- Page 94 and 95: Model : wo+ Solution for uniform ha
- Page 96 and 97: parameter u and that the pressure d
- Page 98 and 99: (=disturbed)-variations: After magn
- Page 100 and 101: with 4 as geographic latitude. From
- Page 102 and 103: Very rapid oscillations with freque
- Page 104 and 105: ! 8. Data Collection - and Analysis
For a dipole - in z-direction this is equal to<br />
a e-kR A e -kR<br />
G = (k" - - grad-) = - curl2 (z --<br />
4a~k'<br />
since<br />
-kR<br />
A (e-kR/@~r~) ) = lc2e / (4aR) -<br />
-z a~ 4Ta2 - 1 I<br />
Comparison of (4.16) and (4.12) shows that for a whole space<br />
The absence of a TE-potential is clear from physical reasons, since<br />
the magnetic field of a vertical. dipole must be confined to hori-<br />
zontal planes (i.e. no vertical. magnetic field, which can only<br />
be produced by a TE-field).<br />
Using Somrnerfeld's integral, C4.17) is written<br />
The f5.el.d of a horizontal electric dipole (in x-direction, say) has<br />
both an electric and magnetic component in z-direction. Hence, a<br />
TM- and TE-potential are needed. 9 +<br />
Since ,- J%i.<br />
a2 a2 1<br />
a2 e<br />
--kl?. m<br />
-a/ z-z ]<br />
G =-(- + -7) 4Jx=--y - -=-- I I A2e O J~ (Xr)dX cos$<br />
Xz ax2 ay 41~k axaz R 4ak20<br />
and<br />
we have<br />
and from<br />
then follows<br />
xsiqn(z-z )<br />
0<br />
J, (Xr)dXcos$ * sign (z-zo) (4.19)<br />
f
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