Jökull - 01.01.2014, Page 17

Improved Precision of Delay Times Determined Through Cross Correlation
pairs of differential delays τik and τkj (with k 6=i
and k 6=j) and which can also be used to estimate τij ,
since they obey the rule:
τkij = τ
est
ik + τ
est
kj (3)
Their arithmetic mean τAij is another estimate for the
differential delay:
τAij =
1
N − 2
N−2∑
k=1
τkij (4)
We term this new estimate an out-member average,
for it uses time series outside a given pair to esti-
mate the differential delay of that pair. Equations (3)
and (4) define a linear relationship between τAij and
(2N − 4) distinct τestij ’s. The usual rule of error prop-
agation for linear systems (Figure 1) then gives:
σ2A =
(2N − 4)
[
1 + (N − 2)α
]
N − 2)2
σ2 (5)
In the N →∞ limit, (6) reduces to σ2A = 2ασ2 (Fig-
ure 2). Thus, σ2A < σ
2 when α < 1/2.
Note that computation of τAij does not depend on
the value of the corresponding τestij ; the formula for
the former (equation 4) is not a function of the latter.
A comparison of the two estimates is useful because a
significant mismatch can flag gross errors in the data
processing, such as cycle-slips (two oscillatory wave-
forms misaligned by exactly one cycle). Since τestij
Figure 1. Analysis used to derive equation (5), based on examining the differential delay matrix τij for the
case N = 6. We focus here on τ23 (circle). Any of N − 2 = 4 pairs of delays of the form τ2k, τk3 can be
used to estimate τ23 (highlighted boxes connected in pairs). The mean delay τA23 is constructed by averaging
these 2N − 4 = 8 highlighted delays, which are grouped together in the vector v. The covariance matrix Cv
has diagonal elements σ2 and off-diagonal elements that are either ασ2 or zero. The non-zero elements of Cv
correspond to pairs of vi’s corresponding to pairs of τij’s that share a common index. Every vk corresponds to
a τij taken from either a row or column of τ , and so shares an index with the N − 3 = 3 elements in that row or
column (triangles). Furthermore, it shares an index with the other member of its pair (rectangles). Thus, a row
of Cv contains one instance of σ2 and N −2 = 4 instances of ασ2, the other elements being zero. Averaging is
a linear operation of the form τA23 = Av with A = (N − 2)−1[1, 1, 1...1]. The usual rule of error propagation
(e.g. Menke and Menke, 2012, their equation 3.36) gives the variance of τA23 as σA = ACvA
T which when
multiplied out yields equation (5). – Aðferð til að leiða út jöfnu (5) fyrir N tímaraðir skjálftamæligagna. Þær
innihalda allar samskonar merki ásamt suði. Hliðrun er hinsvegar í merkinu milli hvers tímaraða-pars, og þarf
að meta hana úr gögnunum. Aðferðin er hér sett upp í fylki fyrir tilfellið N = 6. Vinstra megin er athygli beint
að mati á hliðruninni milli tímaraða 2 og 3 (hringur). Fervik dreifinga á því mati má minnka með því að nýta á
tiltekinn hátt mæligögnin úr hinum tímaröðunum.
JÖKULL No. 64, 2014 17