Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

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A field line segment with the horizontal increments dx and dy has then -the vertical increment H H a z a z az dz =-dx +-dy = ax 3~ +H H +H Q2 nx ax nY aY and the entire field line is the,,constructed by numerical integra: tion; An analy-tically solvable inverse problem: Suppose the observed anomaly' is 2-dimensional and sinusoi.da1, H (y,z=O) = c sin(k.y) H . aY nY. Because of the internal origin of the anomaly the internal field increasing exponential1.y with depth, Hence, the slope of field lines at the depth z i s by integration z(y) = z - c/k.sin(k-y1.e k.z(y) 0 as the implicit solution with zo as t5e constant of integration. Here it w i l l be chosen in such a way that the highest point of the field line is still below the Earth's surface z=0. We infer from the fi.eLd line equation z(y) that field lines above sinusoida!. anomalies oscillate non-symmetrically around the depth -1. z=z with Lj=27i/k as spatial wave length. Their deepest points z=z 0 are at y=L/4,-3L/4,+. ., . .. y= -L/4,3L/4 ,T.. ... . . . . . . , their highest points 2.2- at u,z 'jar 7, , :f-, , . / ,- - 0 -3 x - L ....-a' \ 1 f .- , / , -r/ ..- ,/' 7 -k> '../%,./' \.- A-- s u, j., , 0.- , rr .L 1 Z Y - i . . '. \ 1 Y 1 F: ,.C'C p, -3
Suppose the spatial wave length of the anomaly is large in'com- -1. parison to the deepest point of penetration z . Then the approximate field line equation can be used wi-th the requirement tha-t zo is equal to or larger than c/k. The mean amplitude of the field line oscillations is readily expressed now in terrns of. the amplitude of the observed surface anomaly as given by the factor c, namely If, for example, a moderate anomaly has the ampli.tudd cr0.25 and a wave length of 628 km, the mean amplitude of the field line' oscil.lations are given by (zo+'100)/4km, zo? 25 km. Using z =30 km 0 gives a mean ainplitude of 33 km and'the field line oscill.ates between 4 km and 70 km, This numerical example demonstrates that even minor ano~nalies require rather large undula-tions in the depth of a perfect substitute conductor. 'Multi-frequency - - interpretation buocal. ir~duc$ion in isolated bodies Most geomagnetic induction anomalies arise from a local rearrange- ment of large-scale induced currents, In some cases, however, it may be justified to assume that the source of the anomaly is a conducting body which is isolated by non-conducting mat-ter from the large-scale curcent systems. Such a body can be, fop instance, a crustal lense of high conductivity within the normal1.y highly resistive crust. The direc-t - problem: Consider a body of (variable) conductivity between z=zl and z= 2. ' The sui7roundi.ng matter. at the same depth is effectively non-conducting at the considered frequency, i,e, large- scale induced curren-ts which flow in the normal structure above and
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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
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with 4 as geographic latitude. From
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Very rapid oscillations with freque
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! 8. Data Collection - and Analysis
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A horizontal electric -- field comp
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For a data reducti.on in the fr3equ
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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 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 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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