Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet
Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet
Both types of anomaly can be explained at a given frequency by the undulating surface S of a perfect conductor below non-con- ducting matter, I-ts variable depth below the surface point (x,y) X x will be denoted as h (x,y) . Outside of the anomaly h shall be constant and equal to the real part of Cn at the considered fre- quenc y . This kind of interpretation is intended to demonstrate the effect of lateral changes of in-ternal resistivity on the depth of penetra- tion as a function of frequency and location, it does not provide, however, quantitative information about -the resistivities involved nor does it allow a distinction of the two types of anomalies mentioned above. Clearly, the magnetic fie1.d below S must be zero and the magnetic field vector on S tangential with respect to S because the con- nor a1 tinui-ty condition for the field conponent toif'?? requires that this component vanishes just above S. ~irect model problem: For a given shape of S the anoinalous surface field can be found with the methods of potential field theory, since Ma will be irrotational and of internal origin above S. If ~ ~ in -particular S has a simple shape independen-t of x, the field % ' lines for E-polarisation in the (y,z)-plane for z - < h (y) can be found by conformal mapping as follows: Let w(y,z) = y(yl,zl) + i.z(yl,zl) be an analytic function whi.ch maps the line zl=O of rectangular (yl ,zl) coordinates into the line * z=l> fy) of rectangular (y,z) coordinates. Lines z1 = const, are in- terpreted as magnetic field lines of a uniform field above a perfect conduc-tor at constan-t depth, their image in the (y,z)-plane as field lines of adistorted field above a perfect conductor at the variable
x: depth h (y), the image of the ultiina.'ce field line zf=O being ta.ngen- tial to the surface of the conductor as required. If gn= (Hlly '0) denotes the uniform horizontal field vector at a point in the original (yl,z') coordinates, the components of the field vector - H= (H ,Hz) at the image point in the (y,z) coordinates Y can be shown to be given by with The difference - H - H represents -n the anomalous field to be ' Inverse problem: The shape of the surface S can be found inversely from a given surface anomaly by constructing the internal field lines of -the field gn+ga. Field lines which have at some distance of the anpmaly the required normal depth R ~ ( C ) define the surface S, n provided of course that the surface thus found does no-t inS:ersec3< the Earth's surface anywhere. The ac-tual calculation of internal field lines requires 2 downward ex-tension of the anomaly through the non-conducting matter above the perfec-t conductor, using the well developed methods of potential field continuation towards its sources, In order to obtain sufficien: stability of thenumericaz process, the anomaly has to be low-pass filtered prior to the downward con'tinuation with a cut-off at a reciprocal spa-tial wave number comparable to the maximum depth of intended downward extrapolation.
- 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 156 and 157: with N (w,y) being the Fourier tran
- Page 158 and 159: is calculated as function of freque
- 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
x:<br />
depth h (y), the image of the ultiina.'ce field line zf=O being ta.ngen-<br />
tial to the surface of the conductor as required.<br />
If gn= (Hlly '0) denotes the uniform horizontal field vector at a<br />
point in the original (yl,z') coordinates, the components of the<br />
field vector - H= (H ,Hz) at the image point in the (y,z) coordinates<br />
Y<br />
can be shown to be given by<br />
with<br />
The difference - H - H represents<br />
-n<br />
the anomalous field to be<br />
' Inverse problem: The shape of the surface S can be found inversely<br />
from a given surface anomaly by constructing the internal field<br />
lines of -the field gn+ga. Field lines which have at some distance of<br />
the anpmaly the required normal depth R ~ ( C ) define the surface S,<br />
n<br />
provided of course that the surface thus found does no-t inS:ersec3<<br />
the Earth's surface anywhere.<br />
The ac-tual calculation of internal field lines requires 2 downward<br />
ex-tension of the anomaly through the non-conducting matter above<br />
the perfec-t conductor, using the well developed methods of potential<br />
field continuation towards its sources, In order to obtain sufficien:<br />
stability of thenumericaz process, the anomaly has to be low-pass<br />
filtered prior to the downward con'tinuation with a cut-off at a<br />
reciprocal spa-tial wave number comparable to the maximum depth of<br />
intended downward extrapolation.
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