Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

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-- Structural soundi~z with station arrays: - If not only the source field but also the resistivity structure are laterally non-uniform, no generally valid linear relations between normal and anomalous field components exist. In the f~llowing it will be assumed that the source field is either quasi-uniform or for a given frequency well represented by a single set of wave numbers. Then the relations between the various field components can be expressed by linear transfer functions in the frequency-distance (w,R) domain. They can be formulated either for the "normal" horizontal field of the whole array or, if necessary; also for the local horizontal field only. In either case it is necessary to remove from the ob- served vertical magnetic component its normal part in accordance to the normal resistivity dis.tributi.on and the source field structure. It is assumed that the normal depth of penetration is smal.1 in comparison to the scale length of non-uniformity of the source ~wI,:~I. ;%pi i
. . These sets of tPansfe$fUnCtions renresek the input ior the model calculations to explain the anomalous fields H -a a resistivity anomaly or E -a in terms of pn(z) being known. P,(x,Y,~) = P - Pn(z), There exist various constraints about acceptable sets of transfer functions, for examples the functi.ons in either one column of W must describe a magnetic field of solely internal origin as dis- cussed below. The basic complication in the presence of 3-dimensional structures is due to the fact that the TE and TM mode of the anomalous field cannot be separated. This separation is possible, however, if the resistivity structure is 2-dimensional, say Pa = Pa(yl>z). If then the normal E-field is linearly .! , , .. .. .l. ,,' ' :. ;, ;/ ... Y polarised in xl-direction, the anomaly ,/' ,{ ,> ,, contains only TE modes with E parallel 1:; % -a ,,; 71 Y' A to x1 and H in planes XI-constant: -a E-polarisatio~, If the normal. magnetic field is paral.lsl to x', the anomalous field is in the TM mode with zero ~nagnetic fields above the ground. Hence, Ha = 0 for z = 0 and E lies in planes x1- -a const. : H-polarisation Ccf. Sect. 7. 3 ). The new sets of transfer functions in (xl,yl,z) coordinates are given hy >?here the subscript (\I refers to E-polarisation and -the subscrip-? (l.) to ~-~olarisation. The resul-ting symmet~y relations for W and Z in general (x,y,z) coordinates can be derived as follows: Let be rotation matrices, transforming .- E and -- H fron: (x,y ,z) -to (x' ,yl ,z) coordinates, c = cosa-and s = sina: , , , ! ..,
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
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The e1ectrica:L effect - of the cha
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The vari.ables x,y, and t which do
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- d IufM12 > 0 . dz - - On the othe
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a) TM-mode From (2.251, (2.26); (2.
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with " the abbreviation + - 1 - ? =
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2n -i~cr cos (8-$1 -iKrcU J e ~B=J
Page 19 and 20:
In the 1irnl.t a -+ o, I +- m, M =
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-- 2.5. Definition -- of the transf
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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
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where L is a positively oriented cl
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1 2 3 4 CPD 1 2 3 4 CPD - g) Depend
Page 35 and 36:
The TE-mode has no vertical electri
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i I Earth Anomalous domain 3.2. Air
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Hence, the conductivity is to be av
Page 41 and 42:
The RHS i.s a closed line integral
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4. Having determined B;, the coeffi
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3.4. Anomalous region as basic doma
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- 6 and 6= can be so adjusted that
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From the generalized Green's theore
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and y can again be so adjusted that
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4.2. In3ral - --- equation method L
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The element GZx is needed for all z
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With this knowledge of the behaviou
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After having determined Qzr VJ,; @,
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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
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 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.
Page 142 and 143: 9.2 Layered Sphere - The sphericity
Page 144 and 145: The field within the conducting sph
Page 146 and 147: and An algorithm for the direct pro
Page 148 and 149: with I - and- a = gn g-n I 1 6-n-1
Page 150 and 151: with ~ = - T E + as sheet current d
Page 152 and 153: E~~ T r: j = const. or E T + E a r
Page 154 and 155: Field equations and boundary condit
Page 156 and 157: with N (w,y) being the Fourier tran
Page 158 and 159: is calculated as function of freque
Page 160 and 161: Both types of anomaly can be explai
Page 162 and 163: A field line segment with the horiz
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- 160 - below can neither enter nor
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I '. - L.. . . - I . --.> . ~ 4 The
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This law can be used to i-nterpret
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Only in this special case will be j
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anomalous conductivity oat OP the a
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the product WUL' constant with L de
Page 176 and 177:
One of the thin plates represents t
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low conductivity requires the use o
Page 180 and 181:
Other derivable properties of mantl
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11. References for general reading
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Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

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