Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

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. The electric field of the E-type solution is tangential to the planes 6 = const. IIence, 2-t is called a TE-mode. (tangei~tial -- e:!ectric). Conversely the M-type solutj.on is name3 a TM-mode.!!~ required from (1.5) -the 111agnetic field. of both modes is sol.enoidal (i.e. divl-I - = 0). In addition the E-field of the TE-mode is also solenoidal.. In this mode there is no accumulation of charge, all current flow is parallel to the (x,y)-plane. ldow we have to derive the differen.ti.al equatior~s for the scala~s GE and $M. The TE-field (2.8a,b) satisfies already (1.. 3) - (1.5). It remains to satisfy ~u1~l.H~ = 0 g*. Using (2.5), (2.2), and (2.4) it results from which Eollows (fop if afiax = 0 and af/ay : 0 and fi.0 for x,y -> -, then f=O). From -this equation fol.lows with the same argurnc1-i-ts lil urliform domains, (2.9) and (2.10) agree. Using (2.9), (2.10), (2.5), and (2.4) the TE- and TM-fields in the vacuun oiltside the sources are 11 = 0, 11: = grad - --M -PI a L wL 0. -E H = grad - az' -.E E = y z x grad $E O- Fields on-tside the conductor -- II . .. (2.12
The vari.ables x,y, and t which do not explici.tly occu~ i.n (2.9) and (2.10) can be separa-ted from Q E and OM by exponential factors. Let for OE or OM - x - Y A where K = K x + K and K~ = - ,-. Then (2.3) and (2.10) mad K = - K . (2.14a,l The general solution of (1.2) - (1.5) for a. one-dimensional conductivity structure is the superposition of a TE- and TM-field. The total field then reads in components: --- .--a a20E H = -Co$ ) + --x ay M ax;)~ a ae QE H = - -(uQ ) + Y ax M ay a z H = -(- a z ax2 ay2 WE - 1 a "~l$~) - Ex-o axaz a~ ., 7 1 E = - y o ayaz 3 - "8, a I - (-- + a E~ :)OM ax2 ay + "0 ax General expression of fie1.d components - - At a l>.orizontal di.scon-l-inuity the tangential. compone~~fs of - E and - H and ille ndriaal. con~ponenl: of I1 arc coiitinuous. Let [ ] deno-te Z the j~unp of a p;lrticulaT quanti-ty, Theil frorr~ (2.15~) ]?OF )I z (2.15a) (2.15b) (2.15~ (2.16a) (2.16b) (2.16~
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: The e1ectrica:L effect - of the cha
Page 11 and 12: - d IufM12 > 0 . dz - - On the othe
Page 13 and 14: a) TM-mode From (2.251, (2.26); (2.
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
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 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
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- - . 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
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(e.g. X), their realizations by obs
Page 128 and 129:
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
Page 134 and 135:
This relati-on implies .that .the l
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9. --- Data 5.nterpretatj.on on the
Page 138 and 139:
The "modified apparent - - resistiv
Page 140 and 141:
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
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
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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