Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

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with ~ = - T E + as sheet current density; Htg is the tangential magnetic field above the sheet at z = 0, H- the tangential magnetic field ;;tg below the sheet at z = d; z is the unit vector downward. 1 ... , . . G' 3 for example a sedimentary cover with I exceeding 4 km i n thickness. Then condition (i) will be satis- fied for periods up to 2 minutes and condition (ii) for deep resi.stivities in excess of, say, 50 Slm. The response functions for the normal field above the sheet at z = 0 are readily found from where T is the constant normal part of T, while C; has to be n derived from the given substructure resistivi-ty profile. The field equations.to be considered for the evaluation of the va- . . . riation anomaly are with rot E = impo EIa, Z -a
The second field equation implies that the vertical magnetic field is also to be regarded as a constant between z = 0 and z = d. In addition there are boundary conditions for the anolnalous field, arising from the fact that its primary sourced lie within the sheet. Pilot interpretation: In many cases the anomalous field has no inductive coupling with the substructure because its half-width is small in comparison to (c,[. In that case no currents are in-- duced by Ha in the subs-tructure and the anomalous magnetic field is solely due to the anomalous sheet current ja within the sheet. For reasons of symmetry Observe that the inductive coupling of -the normal field has not been neglected, i.e. its internal sources lie within the sheet as well as within the substructure. Assuming that T~ is known and that it is possib1.e to identify the normal part of the total electric field - E = En + Ea either by calculation 01, by observations outside of the anomaly, the anomalous part of the conductance T., a t is readily found from observations of Hatg and -a E . In this way, using the example from above,the variable dep-ti1 of the basement bel-ow the sedimen.ks, can be estimated. If an el-ongated structure is in H-polarisation with respect to the normal field, the magnetic variation anomaly will disappear (cf. Sec. 7.3). The pilot investigation of the conductance is done 11~ even inore simply because the sheet current densj.ty must be a constant in th-e direction normal to the trend:
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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
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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
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orders of magnitude smaller' than t
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Elimination of - E or .,. H yields
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Observing that rot pot rot g = - ro
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Two special types of such anomalies
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Model : wo+ Solution for uniform ha
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parameter u and that the pressure d
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(=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 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 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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Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

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