Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

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Let q be the tranfer function between Hnx(H at the surfade and nY ) at the depth z = d: Hnx(H nY Then H (d) = q Hnx(0) where I q 1 < 1. nx Single site geo-letic structural sounding: The source field is regarded as quasi-uniform (k = 0) and the ver:tical magne-tic component therefore as anomalous, arising solely from lateral changes of the resistivity within the depth-distance range of penetration, where p = Pn(z) + P,(X,Y,Z)\ In this special case the resulting anomalous magnetic vector H = (Ha,, H , H = H is linearly dependent on the quasi-uniform -a ay az z normal magnetic vector H = (Ilnx, H 0) iii the frequency-distance -n ny' domain : denotes a matrix of linear transfer functions as functions of frequency and surface location. This i.mplies that also linear relations exist between HZ and the total (=observed) ho17izontal variations: with These alternative transfer functions A and B can be del-ived now from observations at a single site. Their graphical display in the form of Parkinson-Wiese induction arrows indicates the trend of -the subsurface resistivity structure which is responsible for the appearance of anomalous '2-var.i.ations :
The in-phase induction arrow is defined by (in Parkinson's sense of orientation) by h p = - R&(x A + B] and the ou-t' of-phase arrow by q + Imagl; A + Bl h h where x and y are unit vectors in x- and y-direction. General-ly speaking, the in-phase arrows point toidards internal concentrations of induced currents, i.e. to zones of lower than "normal" re- sistivi.ty at one particular depth.TheYmay point also away from high resistiv:ity zones around which the induced currents are diverted. Vertical soundings with station arrays: The resistivity structure is regarded as layered, p = p(z),but the inducing source field as non-uniform. Inducing and induced fields will have ms.tching wave-number spectren with well defined ratios between spectral components in accordance to the subsurface re- s i s t ivi-ty structure . h ea-t-k-+h~++t-~o-~~&e-~~idc~~~ e.ve;i:~-~:gh-i-s-k~ol.a-7,-u~.-f m~rnGty--rnaap-~be--~i-~-~P+f+a Let U and V be field components of the surface field in the frequency distance domain: U = UCw ,R) , V = VCw ,R). They are deco~nposed into h h A the wave-number spectren U(w,k), VCw,k) according to with irnh ikR - - U(w,R) = I I UCw,L,)e dk dk - '0 Y X
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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
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 108 and 109: For a data reducti.on in the fr3equ
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
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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
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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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