Jökull - 01.01.2014, Page 16

W. Menke and H. Menke
application of computers to seismology allowed au-
tomated picking, but while this facility sped up the
location process, enabling near real-time monitoring,
it did little to improve the accuracy of locations. The
accuracy of a pick, whether determined by a human
being or a machine algorithm, is limited by the ability
of either to pick the onset of a seismic wave on a noisy
record. Instead, it was the ability to compute differen-
tial arrival times (the time difference between two ar-
rivals) using cross-correlation that provided the leap
in data quality that underpinned the improvement in
accuracy. Cross-correlations could routinely be com-
puted from digital data, something that was not pos-
sible with the older analogue recordings. Early suc-
cesses with the new approach (Rögnvaldsson, 1994;
Got et al., 1994; Slunga et al., 1995; Rögnvaldsson et
al., 1998; Rubin et al., 1999; Waldhauser et al., 1999)
revealed a richness in the pattern of seismicity that
hitherto fore had been missed. It drove further im-
provements in the method (e.g. the Double Difference
Method of Waldhauser (2001)) and its extension to re-
lated problems (e.g. the joint location – tomographic
imaging method of Zhang and Thurber (2003)).
The cross-correlation process exploits the whole
waveform, not just its onset, and so achieves an ac-
curacy in aligning two time series that far exceeds
what can be achieved by differencing two picks (al-
beit without determining the absolute timing of ei-
ther), at least when the two waveforms have very sim-
ilar shapes. Furthermore, the error decreases indefi-
nitely with the length of the waveform and (somewhat
surprisingly) can be smaller than the digital sampling
interval of the time series (Cespedes et al., 1995).
Although the high accuracy of the double difference
method arises from several sources, the most impor-
tant is the precision of the data (Menke and Schaff,
2004). Simply put, differential travel times deter-
mined through cross-correlation have at least an or-
der of magnitude smaller error than traditional arrival
times determined through onset picking.
The lower noise differential times enable more
accurate locations. In this paper, we explore the
statistics of differential travel time measurements and
demonstrate that a factor of about two to four addi-
tional error reduction in timing can be achieved when
the relative timing of a large group of time series (as
contrasted to a single pair of time series) is determined
simultaneously.
OUT-MEMBER AVERAGING TO
REDUCE THE VARIANCE OF
DIFFERENTIAL DELAY TIMES
We consider the commonly-encountered signal corre-
lation problem of aligning time series that differ only
by a delay plus additive noise. The N time series
u(i) are all of length M and sampling interval ∆t and
consist of a deterministic part u(i)0 and additive noise
δu(i):
u(i) = u
(i)
0 + δu
(i) (1)
All the deterministic parts have the same shape u0, up
to a time shift ti. The signal to noise ratio (s.n.r.) r is
given by r−2 = (δuT δu)/(uTu). The noise δu(i) in
one time series is assumed to be uncorrelated with the
noise δu(j) in another, though both may have non-
trivial autocorrelation functions. The objective is to
determine the differential time delay τij = ti − tj be-
tween all pairs of time series. We start with the rule
that a good estimate of τij is the one that maximizes
the cross-correlation cij : and
cestij = maxtu
(i) ? u(j)
and
τestij = argmaxtu
(i) ? u(j) (2)
Here ? signifies cross-correlation. As is customary,
we normalize each u(i) by the square root of its en-
ergy, so that |cestij | ≤ 1. Since it depends upon noisy
time series, the estimated delay τestij differs from the
true delay, τestij = τ
true
ij + δτij by an error δτij with
variance σ2. Two delays, τestik and τ
est
kj that share
one (and only one) index k are correlated, with co-
variance, say, ασ2 (with 0≤ α ≤ 1) since they both
depend upon the time series u(k). Two delays that
do not share a common index are uncorrelated. As
we will discuss below, the quantities σ2 and α depend
upon u0 and the statistics of δu.
We now consider whether we can improve upon
the estimate τestij . We have available a total of N − 2
16 JÖKULL No. 64, 2014