Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet
Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet
Example: n = 12 and @ = 95%: 1 n = 12 and @ = 99%: I/-- G = ~'1003 - 1 = ~43 = 1.22 11- 2 1 ' - 2 5 n = 12 and B = 50%: 4 - - G = 11- 2 42 - 1 = a = 0.39 1 For large n and n; >> lny (y = -1, the following approxiinations are 1-F valid : - 2 - 2 G - - lny n-2 n This approximation. exemplifies the general propaga-ti-on-of-error law, namely that the errors are reduced proportional. to the square root of the number of observations. Ln the more general case that Z depends on X and Y with a non-zero coherence between X and Y confidence limits can be obtained in a similar way: Let z = a x + boyo + 6zo, 0 0 assuming that now X -- and Y are real-ized error free. Then the Fisher- distributed ratio, involving the deviation of A and B from a and 0 b ; turns out to be which has a f (4, n-4) dis.tri.buti.on. The confidence limi-ts for A and F R cannot be calculated individuall.y, unless of course S is taken "jr to be zero. On the other hand, if we assume that I~-a,l and I R-boI are equal, Chen
whi.ch allows the determination of cornbined confidence limits for A and B. The threshold value of F for a given probability fi can be derived from 1-6 = - 1 (1 t -- 26 - n- 4 ) - (- ) with m = - 2 ' 2 m Cl + %G) m+2G 8.4. Data analysis in the time domain, pilot studies Suppose the time func-tions to be related linearly to each o-ther are not oscillatory but more like a one-sided asymtotic return to the undisturbed normal level after a step-wise deflection from it: It may then be preferable to avoid a Fourier transformation and to derive i-nductive response functions in the the domain. If Z depends linearly only on X, i.e. * Z(w) = A(w) X(m) + 6ZCw) in the frequency domain, then Z(t) will be derived from X(t) in the time_domain by- a convolution of X(t) with the Fourier trans- form of A, f" 1 - i.wt A(t) = - J A(w) e dm. 2s -m Since ZCt) cannot depend on X(t) at some future time T > 't, the response function Act) must be zcro for t < 0, yielding -
- 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 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 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
- 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
whi.ch allows the determination of cornbined confidence limits for<br />
A and B. The threshold value of F for a given probability fi can<br />
be derived from<br />
1-6 = - 1<br />
(1 t -- 26 - n- 4<br />
) - (- ) with m = -<br />
2 '<br />
2 m<br />
Cl + %G) m+2G<br />
8.4. Data analysis in the time domain, pilot studies<br />
Suppose the time func-tions to be related linearly to each o-ther are<br />
not oscillatory but more like a one-sided asymtotic return to the<br />
undisturbed normal level after a step-wise deflection from it:<br />
It may then be preferable to avoid a Fourier transformation and to<br />
derive i-nductive response functions in the the domain.<br />
If Z depends linearly only on X, i.e.<br />
*<br />
Z(w) = A(w) X(m) + 6ZCw)<br />
in the frequency domain, then Z(t) will be derived from X(t) in<br />
the time_domain by- a convolution of X(t) with the Fourier trans-<br />
form of A,<br />
f"<br />
1 - i.wt<br />
A(t) = - J A(w) e dm.<br />
2s -m<br />
Since ZCt) cannot depend on X(t) at some future time T > 't, the<br />
response function Act) must be zcro for t < 0, yielding<br />
-
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