Schmucker, 1970 (Scripps) - MTNet

Schmucker: Geomagnetic Variations'
SH = (C C':' )/T
H H 0
SZH :: SHZ = (CZ CH)/T 0
zH = SZH/SH .
17
(3.5)
Notice that To drops out of the last equation which expresses the transfer
function in terms of power and cross spectra.
Suppose Z(t) contains some unrelated "noise" a Z(t). Its power is given
evident! y by
since IZH . cHI2 is the power of the related portion of Z(t). The square root
of the normalized related power (SZ - S a Z )/S Z is commonly referred to as
"coherence" Co(f) between Z(t) and H(t), while the square root of the normalized
unrelated power Saz /SZ shall be denoted as "residual" E(f):
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Co:: 1 - E = ISZHI /(SZ· SH) . (3.6)
The coherence varies between zero in the case of no correlation for a'
certain frequency component and unity in the case of a one-to-one relationship.
Spectra of empirical time series are determined necessarily with
some inaccuracy, since they have to be derived from digitized or analog
records of finite length. This may simulate a nonexisting coherence which
scatters .around a mean value of 12/7,1 for unrelated records, v denoting the
degrees of freedom of the analysis. As a rule, Co(f) should exceed v 4/7,1 to
imply a significant dependence of Z(t) from H(t) for a given frequency.
There are two possibilities to derive spectral values with more than one
degree of freedom: (a) by averaging the spectra of N short intervals, or (b)
by smoothing the spectrum of one long interval To within frequency bands
of the width i\f. In the first case we have 2(N - 1) degrees of freedom, in
the second case 2 i\f To, since Tol would be the ultimate frequency spacing
of standard harmonic coefficients. The factor 2 reflects in either case the
use of sine and cosine terms for the calculation of spectral values (cf. eq.
3.10).
3.6 Spectral Analysis of Single Events
Most conspicuous are bay-shaped deflections during the night hours, which
occur a few times each month and last for about two hours (PI. I). Similar
. events of shorter duration appear from time to time as "sudden impetus, "
"sudden storm commencement," or fast daytime fluctuations in general.
Even though these events are nonperiodic time functions, we can use a standard
harmonic analysis for their representation in the frequency domain.
At the outset a time interval of the length To is chosen equal to or a bit
longer than the duration of the disturbance and the traces are scaled within