Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet
Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet
Field equations and boundary condition together constitute a set of four linear equations. for an equal number of unlcnown field components. They are uniquely determined in this way. A solution toward the anomalous tangential electric field in 8-polarisation, for instance, gives A similar solution toward the anomalous current density in H- polarisation gives For a given model-distribution these equations are solved numeri- cally, by setting up a system of linear equations for the unknown field components at a finite number of grid points along y. The convolution integrals, involving derivatives of the unknown field componentswith respect to y, are preferably treated by partial integration which in effect leads to a convolution of the unknown field components themselves with the derivatives of the kernels. It should be noted that the kernels L approach for y -t finite -1 limiting values, given by' { 2 ~ i ( w ,011 . Inverse problem for two-dimensional strnc_j;Lms -. 1.t -is. also possib1.e to consider the anomalous conductance -ra a.s the unknown quantity to be determined from an observed elongated variation anomaly, in the actual calculations to be represented by a set of respective transfer functions. The normal. sheet conductance T and the resistivity of the substructure, for which the kernels n L have to be determined, enter into the calculations as free model parameters. They can be varied to get the best agreement in ~ ~ ( y ) when uslng more than one frequency of the variation anomaly.
Another point of concern is the reality of the resulting numeri- cal values of ra. Usually empirical data wi1.l give complex values and the free parameters should be adjusted al.so to minimize the imaginary part of the calculated conductance. In E-polari.sation the elimination of H- gives a Y tie can then eliminate either HC and Haz, if -ra is to be determined aY from anomalous electric field,or we can derive ra from the anoma- lous magnetic field by observing tha-t by integration . of the second field equation. The norrnal electric field is derived from the normal magnetic field by setting when the source field is quasi-uniform and when the source field is non-uniform; N (w,y) is the Fourier I1 transform of * '%. (w,k). Cf. Sec. 8.2, "Vertical soundings with Cln11 station arrays". In H-polarisation only the anolnalous electric Field is observable at the surface. For the eli.mination of -the anolnalous magneti.~ field at the lower face of the sheet from the field equations we use the generalized inlpedance boundary condition for the ano- malous TM-field at the surface of the substructure (cf. Sec.7.3 und 8.2) :
- 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
- 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 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
Another point of concern is the reality of the resulting numeri-<br />
cal values of ra. Usually empirical data wi1.l give complex values<br />
and the free parameters should be adjusted al.so to minimize the<br />
imaginary part of the calculated conductance.<br />
In E-polari.sation the elimination of H- gives<br />
a Y<br />
tie can then eliminate either HC and Haz, if -ra is to be determined<br />
aY<br />
from anomalous electric field,or we can derive ra from the anoma-<br />
lous magnetic field by observing tha-t<br />
by integration<br />
. of the second field equation. The norrnal electric field is<br />
derived from the normal magnetic field by setting<br />
when the source field is quasi-uniform and<br />
when the source field is non-uniform; N (w,y) is the Fourier<br />
I1<br />
transform of * '%. (w,k). Cf. Sec. 8.2, "Vertical soundings with<br />
Cln11<br />
station arrays".<br />
In H-polarisation only the anolnalous electric Field is observable<br />
at the surface. For the eli.mination of -the anolnalous magneti.~<br />
field at the lower face of the sheet from the field equations<br />
we use the generalized inlpedance boundary condition for the ano-<br />
malous TM-field at the surface of the substructure (cf. Sec.7.3<br />
und 8.2) :
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