Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

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- - . the same or from different sites. Let Z(w>, X(w), Y(w) be the Fouri-er transforms of Z(t), XCt) and Y(t). Then a linear relation of the form - - is established in which A and B represent the desired transfer func- - tions between Z on the one side and X and Y on the other side; 6Z is the uncorrelated "noise" in 2, assuming X and Y to be noise-free. As the best fitting transfer - functions will be considered those whicf produce minLmum noise < 16zI2 > in the statistical average. Here the average is to be taken either over a number of records -- or within extended frequency bands of the width which is L times greater than the ultimate spacing IL'T of individual spectral estimates. The noise: signal ratio defines the residual e(w) ,%atio of related to observed signal the . --- coherence R(w): The coherence in conjunction with the degree of freedom of the - - functions A and B. averaging procedure estab1ishe.s confidence limits for -the transfer The averaged products of Fourier transforms are denoted as S ZZ ., - = < Z Z * >: power spectrum of Z. - - S = < Z Y * >: cross spectrum between Z and Y ZY wi-th S = S* . ZY Y = In summary, the data reduction involves the following steps (a) Fourier transformation of tiine records (b) Calculations of power and cross-spectra (c) Calculation of transfer functions (d) )COlculation of confidence limits for the trans- ferfunctions. Steps Ca) and (bf can be substi.cuted by the foll.owing alternatives: (ax) : Calculate auto-correlation functions ) , .. . and cross- correlation f.urictf.ons R , . . . with T being a time lag, ZY
with 0 < T < 'rmax. R (T) = / Z(+-t) Z(t)dt Z z T R (T) = / Z(+-t) YCt)dt ZY T x (b ): Take the Fourier transforms of the correlation functions and obtain as in (b) power- and cross-spectral estimates, if the averaging is done within frequency bands of the width m. There is a formal correspondence between the -- -1. maximum lag and Af . The actual performance of the steps (a) to (d) with one or more setsof records is now described in detail. -- a. Fourier transformation. - The time series and their spectra are given, respectively to be found, at discrete values of t and w, which are equally spaced in time and frequency. Let Z e-tc. be an n .instantenuous value of ZCt) for t = tn, n = 0,1,2, . . . N with At = tntl - t . Outside of the record, extending from t to t n o N Z(t) is assumed to be zero. - Let Z be the Fourier transform of Z(t) for the frequency f = f . n1 IT1 Because of the finite length T of the rec6rd the lowest resolvable frequency and thereby the frequency spacing w i l l be given by T-I = Af = f and f has to be a multiple of Af. Because Z(t) is 1 m gj-ven at discrete instances of -time, A t apart, the hi.ghest resol- -, vable frequency, called the Ny quist frequenz, w i l l be the reci- procal of (At-2), i.e. and - I fH - - = MAf 2At
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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
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 114 and 115: -- Structural soundi~z with station
Page 116 and 117: Since it follows that - E 1 = - T E
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
Page 164 and 165: - 160 - below can neither enter nor
Page 166 and 167: 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
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One of the thin plates represents t
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low conductivity requires the use o
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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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