Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

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- 1. Introduction. basic equations -- I.. 1.. General. ideas The fo1.lowing rnodel j.s the basis ~fall e1ectromagne'ti.c lne-thods which try to infer the electrical conductivity stl2ucture in the Earth's subsurface from an anal.ysis of the natural or art.lfici.a~l electromagnetic surface field: On or above the surface of -the Earth is situated a tine-dependent electr'omagnetic source. By Fiiraday's 1.a.w the time-varying inagnetic field - 13 induces an elec.trica1 field - - this cu~,rent F, which drives within the conductor a current 1. By kinp5re's law has a magnetic fi.eld which again is an inducing ageri-t, and so on. Hence, there is the closed chain . induction __I_ i -> .-source l5 // E -. - *mp.res '011.ct:o. law - The mathematical exprfssion of this closed chain is a second orde-r ~art?.al. differen-tia:l equation resulting after the e1iminati.c.n of two field quantities from the two first order equations 1 + 3 and At the surface the ~lectrornagnetic field of the source is dl.sturbcd by the internal fie1.d~. This di.sirurbance depends on the conduc'ti-
A1~Lernativel.y p = 30. m, where T in h. (lb 1 Eqs, (la,b) can also be used as a rule of thumb if p varies with depth and an average resistivity is inserted. The natural magnetic source fields of the ionospheric and magneto- spheric currents offer periods approxiniately between 0.1 see and 20 permitting crustal and upper mantle surveys down to a depth of approximately 800 - 1000 km. On the other side periods between 0.91 and 0.002 sec are most commonly used for Sounding of? -?he first 300 c with artificial fields. Electromagnetic methods are applicab1.e for two objectives: 1) ~nvestigation of the change of conductivity with depth, in par.ti- cular detection and delineation of horizontal interfaces marking a change of stratigraphy or temperature. 2) Investigation of lateral conductivity variations, in particular search for local regions wiCh abnormal conductivities (e.g. meta1lj.c ore deposi-ts, salt domes, sedimeni-dry' basins, zones of eleva-ted temperature). .The results of the first investigation are often used. to construct a normal. conductivi-ty model from which local deviations are measure1 In an interpretation of natural fields the c:-.ange of the electromagnetic field quantities both with frequency and with position are used. Broadly speaking .the dependence on the position provides the lateral reso1.u-tion and the dependence on frequency gives the reso- lu-tion with depth. 1.2. Dasic equations - Let - r be .the position vector and let H, g, 2nd + I he the vectoi-s of the magnetic field, electrical field and curl-2n.t density, ~?especti- vely. Using SI u11.i.-ts and a vacuurrl permeability throughout:, the 0 pertinent equations are cupl - H = - I i- L 11.2) . -e CUI?].' - E = --kt H (1.3) 0- . . '- 1 = 0- C (l.4) div - !i = 0 (1.5)
Page 1 and 2: Electromagnetic Induction in the Ea
Page 3: 6.2. Generalized matrix inversion 6
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 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
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
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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
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 162 and 163:
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
Page 168 and 169:
This law can be used to i-nterpret
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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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