Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet
Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet
Two convenient filters are 3 sinx I, .the Parzen filter Q(f) = - T = Min! '626~ m m Hence., the derivatives of S6z6z with respeci - to ,. the real and imaginary parts' of the transfer functions A and B have to be zero: 1% nl aS~z6~ +i---- aS&z6z = = 0 etc. aRe(An) a1rn(An),
using the notations as - introduced - above. The subscripts m are ommittec! Elimination of either A or B yields the basic formula& for the de- termination of transfer functions in geomagnetic and magnetotelluric The coherence can be derived from -. * R' = (AS x z + BSyZ)/SZz as it is readily seen from If only a relation between z and X 01: Y. is sought or if X and Y are linearly independent (S = O), then the above derived relations XY reduce to ~2 = a--- S S S xx yy zz d. Cal.culation of confi-dence liniits -a,- for the transfer functions -- - " In order to establish confidence limits for the transfer functions A, B of the previous section it is necessary .to fj.nd -the probabili-ty density func-tion ("pd f") of their devia-tioils fr>om their "true" " - - - values Ao, Do. W; assulne that esti.mates of A and B accordi.ng to the least-square method of the previous sec-tion are wj:thout bias, i. t?. bl?th~~t sys-tematic errors (E = expected value): I-lencefor-th, random variables will be wri.tten with capj:l-a1 letters
- 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 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 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
using the notations as - introduced - above. The subscripts m are ommittec!<br />
Elimination of either A or B yields the basic formula& for the de-<br />
termination of transfer functions in geomagnetic and magnetotelluric<br />
The coherence can be derived from<br />
-. *<br />
R' = (AS x z + BSyZ)/SZz<br />
as it is readily seen from<br />
If only a relation between z and X 01: Y. is sought or if X and Y<br />
are linearly independent (S = O), then the above derived relations<br />
XY<br />
reduce to<br />
~2 = a---<br />
S S S<br />
xx yy zz<br />
d. Cal.culation of confi-dence liniits -a,- for the transfer functions<br />
--<br />
- "<br />
In order to establish confidence limits for the transfer functions<br />
A, B of the previous section it is necessary .to fj.nd -the probabili-ty<br />
density func-tion ("pd f") of their devia-tioils fr>om their "true"<br />
" - - -<br />
values Ao, Do. W; assulne that esti.mates of A and B accordi.ng to the<br />
least-square method of the previous sec-tion are wj:thout bias, i. t?.<br />
bl?th~~t sys-tematic errors (E = expected value):<br />
I-lencefor-th, random variables will be wri.tten with capj:l-a1 letters
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