Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

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The Fourier integral - +- -io t T -iw t n z(un) = I Z(t)e dt = J Z(t)e n dt -@2 0 will be evaluated r.umerica1l.y according to th,e trapezoidal formula of approximation. Setting and observing tha-t zo = $[zct 0 ) + 2(tN)-j "n?n = 2a fmtn = -- 2 limn N' the discrete Fourier transform of Z(t) is 2 L * A linear trend of the record within the chosen interval may be wri.tte11 as with d = Z(tN) - Z(to). A cor3rection for this trend i.n the frequency domain implies that the imaginary Fourier transform of Z1(t), - is substracted from Z . m A second correction arises from the fact that Z(t) does not v?.nish necessarily outside of the chosen intervall. In that case the original time function can he multiplied in the rime domain wirh n weight function W(t) which is zero for t < to and t > tN. The Fourier transform of the product W(t) . Z(t) - - - - - is given by7a convolution of Z with the Fourier transform of W: - 1 W(o) 35 ZCo) - Ff C Wm-& Z; . A m A frequent.1~ used weight function is ' 1 2li T - WCt) = -11 + cos --(t-t .- 2 T 0 2
which has 'the Fourier transform T for m = 0 "1% 0 71- t -5 G({l where < 'm > is the discs-te Fourie~ transform of W ( t ) * Z(t). This smoothing procedure of the origi.na1 spectrum is czlled "harming" after Julius von Hann. \ \ \ \ \ / -. L- q-4----& b , I.\/' 3 1 The convolution of W with descrefe values of Z reduces then to - 1 - - m T m-m m - 1- -(Z. + ZIY m r O ( 2 0 = - C bi A* ZA = m = 1, 2, . . . PI-1 -(Z 4-2 m = M 2 M-1 M In certain cases it will. be necessary to apply a numerical fi1.ti.r to the time series to be analysed beforc -. the Fourier transformation. This filtering process consi.sts i.n a convolution of 2c.t) with a filter function \d(t) - in the - time do1naj.n and thus corresponds to a multiplication of Z with W in the frequency domain. But because the filtering procedure is intended to prepare the time series for the Fourier transformation it must be carried out in the time domain If W denotes the filter weight for t = t17, the discrete form of n the convolu-ti.on is Usually even 'filters are employed, W ( t ) = W(-t), to preserve .the corr-ec-t phase of the Fourier coniponents, The transform of is chose: in such a way that it acts as a heigh-, lord- or bandpass fj.1-ter for fr.equency-independent (="whitet') spectra:
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
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In the 1irnl.t a -+ o, I +- m, M =
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-- 2.5. Definition -- of the transf
Page 23 and 24:
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
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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
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
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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
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 118 and 119: - - . the same or from different si
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
Page 168 and 169: This law can be used to i-nterpret
Page 170 and 171:
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
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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