Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

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1 2.2 Free modes of decay- Whereas external fields can exci-te only the TE-mode within the conductor, there exist free modes of current decay for both TE- and TM-fields, Consider the typical potential where @ and f stand for OE, $M and f,, fM, respectively. First it L wlll be shown that the decay constants B (ei.genvalues) must be positive quantities. Inserting (2.23) in (2.9) and (2.10) we obtain fECz) =. { K ~ - f3~~0(z)l 'fE(z), (2.24) where K' = K' -1- K'. The ej-genfunctions have -to satisfy the boundary X Y conditions fE = (km) = 0, f (to) = 0, fr(m) = 0 M M If there is a perfect conductor at finite depth d then (2.26) fE(d) = 0, fi(d) = 0 (2.2Ea) Multiplying (2.24) and (2.25) by f" aid of" and integrating over z' E M from -m to +m and 0 to -, respectively, we obtain on integrating by parts and using (2.26) Hence, all decay constants f3 are real and positive. The currcnt flow for TE-decays is in horizontal planes, whereas the TM-decay currcnts flow predominantly in vertical planes. Example : u = O 0 + 0,- - z - 0
a) TM-mode From (2.251, (2.26); (2.26a) foll.ous f (0) = 0 yields fM[z) = sinyz, whereas f' (d) = 0 requires cosyd=O. M M Hence, The unnormalized eigenfunctions are Theye exists also an electrical (po-ten.tj.al) field outside the conductor. Find it! b) TE-mode - From (2.24), (2.251, (2.26~1) follows "CZ) = K" fd), Z < 0 fE E 2.. . f;[z)+Y.'f (z)=O, 0 < z < d, y -f3000 - K2 . E with f ( -m) = 0, fE, fc continuous across z = 0, f (dl = 0. E E E Then f (d) = 0 leads to the eigenvalue condition E casyd -I- (a/y)sinyd = 0. For I: = 0 the ei.genval.u.es zgree with those of 'i'JI, For 1:c1
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
Page 7 and 8: The e1ectrica:L effect - of the cha
Page 9 and 10: The vari.ables x,y, and t which do
Page 11: - d IufM12 > 0 . dz - - On the othe
Page 15 and 16: with " the abbreviation + - 1 - ? =
Page 17 and 18: 2n -i~cr cos (8-$1 -iKrcU J e ~B=J
Page 19 and 20: In the 1irnl.t a -+ o, I +- m, M =
Page 21 and 22: -- 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
Page 31 and 32: where L is a positively oriented cl
Page 33 and 34: 1 2 3 4 CPD 1 2 3 4 CPD - g) Depend
Page 35 and 36: The TE-mode has no vertical electri
Page 37 and 38: i I Earth Anomalous domain 3.2. Air
Page 39 and 40: Hence, the conductivity is to be av
Page 41 and 42: The RHS i.s a closed line integral
Page 43 and 44: 4. Having determined B;, the coeffi
Page 45 and 46: 3.4. Anomalous region as basic doma
Page 47 and 48: - 6 and 6= can be so adjusted that
Page 49 and 50: From the generalized Green's theore
Page 51 and 52: and y can again be so adjusted that
Page 53 and 54: 4.2. In3ral - --- equation method L
Page 55 and 56: The element GZx is needed for all z
Page 57 and 58: 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
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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
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;)
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. 7. Basic concepts of geomagnetic
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orders of magnitude smaller' than t
Page 88 and 89:
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
Page 108 and 109:
For a data reducti.on in the fr3equ
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Let q be the tranfer function betwe
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. A as transfer function between A
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-- Structural soundi~z with station
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Since it follows that - E 1 = - T E
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- - . the same or from different si
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The Fourier integral - +- -io t T -
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The weigh-t . function W is then fo
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Two convenient filters are 3 sinx I
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(e.g. X), their realizations by obs
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Observe that the residual, of which
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Example: n = 12 and @ = 95%: 1 n =
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- As a consequence, the real and im
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This relati-on implies .that .the l
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9. --- Data 5.nterpretatj.on on the
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The "modified apparent - - resistiv
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Exercise Geomagne-tic varj.ations.
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9.2 Layered Sphere - The sphericity
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The field within the conducting sph
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and An algorithm for the direct pro
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with I - and- a = gn g-n I 1 6-n-1
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with ~ = - T E + as sheet current d
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E~~ T r: j = const. or E T + E a r
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Field equations and boundary condit
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with N (w,y) being the Fourier tran
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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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