Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

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For a data reducti.on in the fr3equency domain (cf. Sec. 8.3) the following set of transfer or response functions w i l l be defined: For a frequency-space factor Of the source field the response function C(w) = C n which - characterises the downward depth of penetration is connected to these transferfunctions accor- ding to A Z n x y n ikx iwv0 c =- - The Cagniard apparent resistivity if given hy the phase of the impedance by I$(w) = arg(Z ) = arg(C ) + 5 , xY n 2 the modified apparent resistivity by which can he used as an estima-tor of the true resistivi-ty at the depth z"(~) = R~IC~I. (cf. Sec. 2.5 and 9.1) A cornpairison ,of the various equivalent respcrlse func-l-j.ons for a . , given substructus&ives the fol.lowing displ.ay: L? *
Tests for the assumption of a layered dis.tri.bution and the source In addition there are certain constraints with regard to modu.lus, phase and frequency dependence of the transfer functions. (cf. Sec. 2.6). The compatability of MTS and GDS vertical sounding results can be tested by the requirement that asreadily seen form the second field equation rotE = -imp H . -n o -n If the wave-number structure of the source field is not lcnown, a - Horizontal Gradient GDS can be formed: Observing that - the differentation of E = Cn impo H with respect to y and of nx nY = -iou C H wi.th respect to x yields .Env o n nx Since the electric surface field may be distorted by lateral non- uniformi-ties even when their scale length and dep-th is smal.1 in comparison -to the depth of. penetration, -the E-field observa.tj.oi~s may be replaced by observing the tangen-tj.al- magnetic field also at some dep-th d below -the surface, i.e. by conducting a 'vertical - ----- Gradient GIIS. No knowledge of the wave-number st~uc-ture of the source field w i l l be required, but"the resi;tivity po between the ' surface and the subsurface points of observations will be reqci-red. Assume that the lateral gradients of IJnz are small in comparison to the vertical gradients of H and H in v5.e~ of the condi.tion n x nY I -. I ~ c [
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
Page 9 and 10:
The vari.ables x,y, and t which do
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- 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 - ? =
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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
Page 37 and 38:
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
Page 49 and 50:
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
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 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 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.
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
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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
Page 178 and 179:
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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