Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

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with N (w,y) being the Fourier transform of C- (w,k 1. Insertion I n1 Y above gives 9.4 Non-uniform layers .above and within a layered structure The source of the surface variation anomaly is assumed to be an anomalous slab between z = zo and z = z + D in which the re- 0 sistivity changes in vertical &horizontal direction: p=p,+pa. The region above and below the slab are taken to be layered, p = pn being here a sole function of depth (cf. Sec.3). The response functions for the normal variatLon field are defined for the normal structure as given by p . If the anomaly lies at n the transj.tion between two extended normal regions, two normal sol.utions havk to be formulated. It should be no-Led that in that case the anomalous field w i l l not disappear bu-t converge outside of the anomaly toward the difference of the two normal soluti.ons. Henceforth., the normal structure , the normal response functions J and the normal fields En and E will be assumed to be known -n . . throughou-t the lower conducting half-space. The source-field will be regarded as quasi-uniform except for source fields in E-polari-
sation in the case of longated anomalies when a non-uniform source can be permitted (cf. Sec. 9.3, calculation of E from -nx H 1). nY The principal problem in basing -the interpretation of actual field data on this type of model consists in a proper choice of the upper and lower bounds of the anomalous slab. The following argumepts may be useful. for a sensible choice: The anomalous region must be reached by the normal variation field) i.e. z should not be made larger than the depth of pene- 0 tration of the normal field as given by I C (w,0)1 at the highest n frequency of an anomalous response. A lower bound for the depth of the slab is not as readly formulated because the ano~nalous response does not disappear necessarily when the depth of penetration is much larger than z + D, i.e. when no significant 0 normal induction takes places within the anomalous slab. Only if the anomaly is elongated and the source field in E-polarisation, can it be said that the depth zo + 1) must be at least com- parable to the normal depth of penetration at the lowest frequency of an anomalous response. Pilot studies: In section 8.2 the general properties of the impedance tensor Z above a non-layered structure have been discussed and the following rule for elongated anomalies was establi.shec1: The impedance for E-polarisation does not diverge markedly from the impedance of a hypothetical. one-dimensional response for the local resistivity-depth profile, while the impedance for H-polarisation will do.so unless the depth of penetration is small in comparison to the depth of the internal resistivi.-ty anomaly and the inductive response nearly normal anyway. Suppose then that an impedance tensor has been obtained at a'lo- cation y on a profile across a quasi-twodimensional anomaly which by rotation of coordinates has zero or allnos-t zero diagonal elements and that a distinction of the offdiagonal elements for E- and H-polarisation can be made (cf.Sec.8.2). Regarding the E-polari- sztion response for a first approximation as quasi-normal., a local inductive scale length
Page 1 and 2:
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
Page 27 and 28:
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
Page 49 and 50:
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
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
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 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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Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

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