Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

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Eqs. (4.35aIb) admit a representation of the field values outside the anomalous layer in terms of the boundary values of the con- tinuous tangential component of E . -a A physical interpretation of Green's vector G subject to (4.33) -i is as follows: Reflect the normal conductivity structure for z < z and z > z2 at the planes z = z and z = z and place a unit 1 1 2 dipole in x -direction at r and an image dipole at i ,. --o " 'm r' = r + 2 (zm - z0)zI the moment being, the same for the vertical -0 -0 dipole and the opposite for the two horizonta~l dipoles. Then the tangential. component of G!") vanishes at z = z and G!~) is a -1 111' -1 solution of (4.32) for - r E Vm. Hence, if Vm is a uniform half-space, G (m) is constructed from the -i whole-space formula (4.15) . Eq. (4.3'5) then reads : m Eaz = (-1) / J? ( ~ {x-x j )E (g)+ (y-yo)Eo (5) ld~, ax CIY 'm where R = [r - - r 1, k2 = iww cr --O 0 Eqs. (4.36a-c) contain as important subcase the condition at the air-earth interface (m = 1, zl = 0, k = 0). 0 Because of the limited range of the kernelstin applications of the surface integral on1y.a small portion of Smm:ust be cocsj.dered. For Eax and E the contribution of the region nearest to r is aY -0 most important. Assuming E and E to be cionstant within's sma1.l ax aY disc of radius p centered perpendicularly over r the weight from -0 (4.36a.b) is where h = lz - zo[ . There is a contributiorr to E onl-y if Ea,, and m az E have a gradient along x and y direction kespectively. aY
F At the vertical boundaries the condition E = 0 --a is relaxed to the impedance boundary condition A A kga.t = n x curl zar - n b E =Of -a whiere gat is the tangential component and - & the outward normal. 5. Conversion - formulae for a two-dbensional TE-field 5.1. Separation formu1.ae In this subsection 5.1. it is assumed that the conductivity does not change in x-direction and that the inducing magnetic field is in the (y,z)-plane, i.e. the conditions for the E-polarization case of Sec. 3.1. Then from a knowledge of the H and Hz component Y along a profile from -a to +a at z = o it is possible to separate the magnetic field into its parts of internal and external origin without having an additional knowledge of the underlying conductivity structure. The pertinent differential equation is C3.5a), i.e. which reducing in z - < 0 to which has the general solution where A: describes the external and A: the internal part of the fie1 The magnetic field components at z = 0 are (cE. (3.2.3) and (3.3a)): Let . . - - - - . . H =W +H HZ = H i.I.Izi, Y ye yi' z e
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 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: 4.3. The surface inteyral approach
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 =
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
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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
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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
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
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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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