Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet
Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet
The field within the conducting sphere, r - < a, is derived from - P as described in Sec. 7.3. Observing that the radial component of rot rot - P reduces by the use of spherical harmonics to the field components of the spherical harmonic of degree n and order m are: and The impedance of the field at spherical sur'aces, expressed in terms of c:, is then as in the case of plane donductors given by For the field outside of the conducting sphere, r - > a, the solution of the radial function is (in quasi-stationary approximation) Hence, the surface value of the charlac-terisll-ic scale length for r = a is found to be The ratio of internal. -to external. parts of the magnetj.~ surfacc fleld is by definition
yielding The ratio of internal to external parts is therefore derived from cm according to n which demonstrates the r$le of n/a, respec-tive1.y (n+l)/a, as equi- c valent wavenumber of the source fielh in spherical coordinates (cf. Sec. 9.1). The spherical version of the input function for the inverse problem, based on a linearisation according to the $-algorithm, is con- veniently defined as with K~ = itopO/pO. It w ill be shown that the spherical ccrrection 0 is of the order (n[~[/a)', if the conductivity is more or less uniform. If degree and order of the spherical. harmonic representation of the source field are independent of frequency (this is true for;D st but -- not for S ), the inverse problem can be solved by deriving 9 from spherical response functions a prel-iminary plane-earth model h This model. is subsequently transformed into a spherical. Earth model p(r) with WEIDELT's transformation formula: with
- 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
- 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 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
- 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
yielding<br />
The ratio of internal to external parts is therefore derived from<br />
cm according to<br />
n<br />
which demonstrates the r$le of n/a, respec-tive1.y (n+l)/a, as equi- c<br />
valent wavenumber of the source fielh in spherical coordinates<br />
(cf. Sec. 9.1).<br />
The spherical version of the input function for the inverse problem,<br />
based on a linearisation according to the $-algorithm, is con-<br />
veniently defined as<br />
with K~ = itopO/pO. It w ill be shown that the spherical ccrrection<br />
0<br />
is of the order (n[~[/a)', if the conductivity is more or less<br />
uniform.<br />
If degree and order of the spherical. harmonic representation of<br />
the source field are independent of frequency (this is true for;D st<br />
but -- not for S ), the inverse problem can be solved by deriving<br />
9<br />
from spherical response functions a prel-iminary plane-earth model<br />
h<br />
This model. is subsequently transformed into a spherical. Earth<br />
model p(r) with WEIDELT's transformation formula:<br />
with
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