Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

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9. --- Data 5.nterpretatj.on on the basis of selected -- iriodels 9.1 - Layered H a l f - s ~ The interpretation of geomagnetic sounding data by a layered resistivity distribution p = p (z) is appropiate at those single n sites or for those arrays which do not show anomalous magnetic Z-Variati.ons and which have a polarisation-independent magneto- telluric inwedance: The transfer functions to be used for the interpretation are the inductive scale length C (w,O) n for zero wavenumber the impedance the ratio'of internal to external parts in the wavenuinber domain The transfer function of the Y-algorithm to be uscd for the lineari- sation of the inverse problem is where p is arbitrary reference resistivity Ccf. chap. 6.4) 0 Considering these transfer functions a.t one particular frequency > the parameters of the following models can be derived directly from tl-tem. These parameters may then be used to resen~ble as functions of frequency certain characteristics of the true resistivity distribution. All formulas are readily derived from the formulas for TE-fields at zero wavenumber in chap. 7.3, annex. (a) Single frequency interpretation of the modulus of transfer- functions (Cagniard-Tilchonov): the model c0nsi.s-ks. of a uniform half-space. Its resistivity is
if the period T is inensured in seconds, E in mV/Icm and H in y. X Y is the "Cagniardapparent resistivi'ty" of the substrat~uil pa at the considered frequency. (b) Single 1 frequency interpretation of the real part of Cn: The model consists of a perfect conductor at W X A f=m :* layer. Im(Z .. . 3: ~ , , ~. n , .. ;\ z = R~IC I = -- \ \ \ g ,. = 0 ., , n \. .,, \\... u"c is the depth of the "perfect substitute conductor" at the con- \ .'.., \ . X the depth z = z beneath a non-conducting -top sidered frequency, indicating the depth of penetration into the substratum at that frequency. (c) Single frequency interpretation of the imaginary part of C : n -7- %* The model consists of a thin conductive top - ;' f - '- layer of the conductance I-~ = 9 cr(z)dz, \ \. , . ., 0 covering a non-conducting half-space. This , \ \ . \ \ apparen-t condoctance is given by (d) Sing1.e frequency interpretation of amplitude - and phase of resp3nse functions; 4 = arg{Zn}. 1 0 < @ < ~ / 4 : The model consists of X a thin top layer of the conductance T above a uniform substratum OF resistivi-ty p"": -ER TX = p"" = PaS -Im(C n ) - Real(C n ) , -- Re(Zn) ,- Im(Zn) -- ullo ! cn I * 1znI2 - 1 + cotZ@ 2 2 n/]; < @ < ~ / 2 : the model consists of a non-conducting 'top -- layer of thickness h above a uniform substratum of the re- X sistivity p ; Im(Zn) - Re(Zn) h = Real(C ) + Im(C,) = n u"o -- p" = 2 wl,;{l:rn(~~)~~ = 2 : Pa 5 , . 1i.tan2@ '
Page 1 and 2:
Electromagnetic Induction in the Ea
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6.2. Generalized matrix inversion 6
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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
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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
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) Computation of ---- C for a laxez
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The approximate interpretation of C
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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
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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
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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
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F At the vertical boundaries the co
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The four equations A A A A H = i sg
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6. Approaches to the inverse proble
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to minimize the quantity a s = 12 /
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It remains to show a way to minimiz
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Agai-n, from a finite erroneous dat
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Here lJ - is a N x P matrix contain
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small eigenvalues. The parameter ve
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Then - 77 - A(E2 - E ) = iwu U (E -
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whence 2k d -2k d where a = CA:(A;)
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. 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 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 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
Page 164 and 165: - 160 - below can neither enter nor
Page 166 and 167: I '. - L.. . . - I . --.> . ~ 4 The
Page 168 and 169: This law can be used to i-nterpret
Page 170 and 171: Only in this special case will be j
Page 172 and 173: anomalous conductivity oat OP the a
Page 174 and 175: 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
Page 182: 11. References for general reading
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