Jökull - 01.01.2014, Page 18

W. Menke and H. Menke
1000
Figure 2. (a) Results of numerical simulations demonstrate that the parameter α increases with signal-to-noise
ratio (s.n.r.). Useful variance reduction occurs for values of α below about 1/4, corresponding to s.n.r.’s below
about 10. (b) Log-log graph of error reduction predicted by equation (5), for various combinations of N and
α. – Niðurstöður útreikninga úr tilbúnum skjálftagögnum. Þær sýna að stærðin α eykst með hækkandi merkis–
suðs hlutfalli í gögnunum. Þegar α er minna en 1/4 eða svo, gefur aðferðin úr 1. mynd umtalsverða minnkun
ferviks í metnum tímahliðrunum. (b) Graf úr jöfnu (5), þar sem sést hvernig fervikið og þarafleiðandi óvissa í
hliðruninni breytist með α og með N fjölda tímaraða.
is correlated with all the terms in the sum for τAij , the
two quantities are not statistically independent, but in-
stead has a covariance that can be shown to be 2ασ2.
Since the variance of τAij is approximately equal to its
covariance, averaging τestij and τ
A
ij does not result in
any further variance reduction.
In the high s.n.r. limit, Marquering et al.’s (1999)
formula can be used to show that the error δestij in the
differential delay time τestij obeys the first-order ap-
proximation:
δτij ≈ gT (δu(i) − δu(j)) with g = ∆tu̇/(uT ü)
(6)
Here the dot signifies ∂/∂t. This rule implies that
α = 1/2, exactly the critical value and well above
the level of α ≈ 1/4 where a useful amount of er-
ror reduction occurs. Thus, τAij can only offer an im-
provement over τestij in cases where the approximation
does not apply; that is, at low signal-to-noise ratios.
Fortunately, this noisy data limit is precisely the one
where improving the accuracy of the estimated lag is
the most important.
We have investigated the functional behavior of
α(r) as a function of s.n.r. r using Monte-Carlo sim-
ulation. Our procedure is to create an ensemble of
N = 100 time series, each consisting of a transient
band-limited pulse with superimposed band-limited
noise of specified s.n.r. All the deterministic pulses
are aligned, so that the true τij’s are zero. Any devia-
tion from zero is attributable to the effect of the noise.
We calculate the differential delay times for all pairs
of noisy time series by numerical cross-correlation.
The time of the maximum in the cross-correlation pro-
vides an estimate of the time delay accurate to within
one sample; we then refine it to sub-sample accuracy
using quadratic interpolation. The resulting tables of
τestij ’s and τ
A
ij ’s each contains about 5000 elements –
sufficiently large for statistical analysis.
The parameter α can be estimated by identifying
all pairs of τestij ’s that share a common index and cal-
culating their sample covariance (Figure 3). These
tests clearly demonstrate that α  1/2 at the lower
s.n.r.’s. While the results are somewhat dependent on
the spectral characteristics of signal and noise, we find
18 JÖKULL No. 64, 2014