Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet
Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet
Two special types of such anomalies can be distinguished. If the electric vector of .the primary fields is linearly polarized in x-direction, i.e. and consequently E = (E , 0, 0) -n nx H = (0, H , Hnz), -n nY the anomaly has an electric vector likewise only in x-direction: because the flow of eddy currents will not be changed in direction. Hence, the anomalous field is a TE-field - This polarisation of the primary field vector is termed 3ol.arisati If the electric vector of the primary field is 1inearl.y polarised in y-direction, the normal field is and consequently E = (0, Eny' 0) -n H = (H 0, 0) -n nx' provided that its depth of penetration is small in comparison to its reciprocal wave number, yielding H = 0. Only with this con- the n z straint is flow of eddy currents to vertical planes x = const. and the resulting anomalous field w i l l be a TM-fie1.d - with zero magnetic field above the ground: E = (0, E -a ayy Eaz) H = (Hax, 0, 0). -a This pola.risation .of the primary field is termed "H-polarisation". - -. For three-dimensional structures O = Ci(x,y,z) = u Cz) + Cia(x,y,z) n the anomaly of the induced field will be composed of TE-and TM- fields which cannot be separated by a special choice of coord:. 7 nates. There is, however, the following possibility to suppress in model calculation the TM-mode of the anomalous field:
Suppose the lateral variations u are confined -to a "thin sheet". a This sheet may bz imbedded into a layered conductor from which it must be separated by thin non-conducting layers. Then no currents can leave or enter the non-uniform sheet and the TM-mode of the induced field is suppressed. Such models are used to describe the induction in oceans, assumed to be separated from zones of high mantle conductivity by a non-conducting crust. Schemat:ic summary: Source field Induced field ,,,\~. x,'~ - . . normal anomalous TE { TEtTM(genera1) TE (thin sheet) xAppGn?3iX to'?, 3 : Recurrence formula for the calculation of the depth of penetration C for a layered substratum (cf. chapter 2) 7, r Definition: C = - 0 aPolaz a2p Differentjal equation to be solved: -L? = (iw~~~o + lkI2)po which satisfies az2 V - ~ = P iwpo~ . Continuity condictions: 1. TE-field: - H and E must be continuous which implies that C-is continuous - 2. TM-field: H and (Ex,E ,oEZ) are continuous which imp]-ies v that oC is continuous.
- 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 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
- Page 90 and 91: Observing that rot pot rot g = - ro
- 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
- Page 106 and 107: A horizontal electric -- field comp
- Page 108 and 109: For a data reducti.on in the fr3equ
- Page 110 and 111: Let q be the tranfer function betwe
- Page 112 and 113: . A as transfer function between A
- Page 114 and 115: -- Structural soundi~z with station
- Page 116 and 117: Since it follows that - E 1 = - T E
- Page 118 and 119: - - . the same or from different si
- Page 120 and 121: The Fourier integral - +- -io t T -
- Page 122 and 123: The weigh-t . function W is then fo
- Page 124 and 125: Two convenient filters are 3 sinx I
- Page 126 and 127: (e.g. X), their realizations by obs
- Page 128 and 129: Observe that the residual, of which
- Page 130 and 131: Example: n = 12 and @ = 95%: 1 n =
- Page 132 and 133: - As a consequence, the real and im
- Page 134 and 135: This relati-on implies .that .the l
- Page 136 and 137: 9. --- Data 5.nterpretatj.on on the
- Page 138 and 139: The "modified apparent - - resistiv
- Page 140 and 141: Exercise Geomagne-tic varj.ations.
Suppose the lateral variations u are confined -to a "thin sheet".<br />
a<br />
This sheet may bz imbedded into a layered conductor from which it<br />
must be separated by thin non-conducting layers. Then no currents<br />
can leave or enter the non-uniform sheet and the TM-mode of the<br />
induced field is suppressed. Such models are used to describe the<br />
induction in oceans, assumed to be separated from zones of high<br />
mantle conductivity by a non-conducting crust.<br />
Schemat:ic summary:<br />
Source field Induced field<br />
,,,\~. x,'~ - . . normal anomalous<br />
TE {<br />
TEtTM(genera1)<br />
TE (thin sheet)<br />
xAppGn?3iX to'?, 3 : Recurrence formula for the calculation of the<br />
depth of penetration C for a layered substratum (cf. chapter 2)<br />
7,<br />
r<br />
Definition: C = - 0<br />
aPolaz<br />
a2p<br />
Differentjal equation to be solved: -L? = (iw~~~o + lkI2)po<br />
which satisfies<br />
az2<br />
V - ~ = P iwpo~ .<br />
Continuity condictions:<br />
1. TE-field: - H and E must be continuous which implies<br />
that C-is continuous<br />
-<br />
2. TM-field: H and (Ex,E ,oEZ) are continuous which imp]-ies<br />
v<br />
that oC is continuous.
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