Jökull - 01.01.2014, Side 19

Improved Precision of Delay Times Determined Through Cross Correlation
10 20 30 40 50 60 70 80 90
dest(t)
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
0
1
2
lag
p
(l
a
g
)
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
0
5
lag
p
(l
a
g
)
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
0
5
lag
c
o
u
n
ts
-5 -4 -3 -2 -1 0 1 2 3 4 5
x 10
-3
0
500
lag
p
(l
a
g
)
-5 -4 -3 -2 -1 0 1 2 3 4 5
x 10
-3
0
500
lag
p
(l
a
g
)
-5 -4 -3 -2 -1 0 1 2 3 4 5
x 10
-3
0
500
lag
c
o
u
n
ts
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
tau(i,k)
ta
u
(j
,k
)
s.n.r = 3.12 alpha = 0.047
-3 -2 -1 0 1 2 3
x 10
-3
-3
-2
-1
0
1
2
3
x 10
-3
tau(i,k)
ta
u
(j
,k
)
s.n.r = 7701.49 alpha = 0.495
s.n.r. = 100 s.n.r. = 2
τ(i,k) τ(i,k)
τ(
j,k
)
τ(
j,k
)
t
u
(t
)
u
(t
)
τ τ
p
(τ
)
p
(τ
)
p
(τ
)
p
(τ
)
(a) (e)
(f)
(g)
(h)
(c)
(b)
τ τ
0 2048Δt 0 2048Δt
0
.1
Δ
t
-0
.1
Δ
t
-0.1Δt 0.1Δt
-0.1Δt 0.1Δt
-1
0
Δ
t
-10Δt 10Δt
10Δt
-10Δt 10Δt
α=0.50 α=0.04
(d) 0. Δt
τ τ
10 20 30 40 50 60 70 80 90
dest(t)
t
Figure 3. (a) Exemplary pair of time series (black and grey curves) with a high s.n.r. of 100. (b) Scatter plot
of differential delays of two pairs of time series that share a common member, u(k). The delays are correlated
with α = 0.50. (c) Empirical probability density function for differential delay τestij , with empirical variance σ
2
(grey). (d) Empirical probability density function for differential delay τAij , with empirical variance σ
2
A (grey)
and that predicted by equation (5) (black). Note that σ2A ≈ σ2 at this high s.n.r. (e)-(h) Same as (a)-(d), except
for a low s.n.r. of 2. Note that σ2A < σ
2 in this case. – (a) Dæmi um tvær tilbúnar tímaraðir, með hátt merk-
is-suðs hlutfall r = 100. (b) Dreifing matsgilda tímahliðrunar milli um 5000 para úr 100 slíkum tímaröðum.
(c) Tíðniþéttleiki matsgildanna og staðalfrávik þeirra (stutt grá strik) þegar aðferð greinarinnar er ekki notuð.
(d) Tíðniþéttleiki og staðalfrávik (stutt svört strik) fundið með aðferð greinarinnar. (e)-(h) eiga á sama hátt við
tímaraðir með r = 2.
that α < 1/4 when r<10 (Figure 2). Estimates of σA
from histograms of τAij ’s are in very close agreement
with the predictions of equation (5). Variance reduc-
tion σ/σA of 2–4 is typical for a s.n.r.’s of r ≈ 4 –
definitely an amount large enough to justify the extra
computational effort of the procedure.
In many practical cases, the deterministic part of
the time series u(i)0 has a shape that evolves system-
atically with index i. For example, the time series
might represent a P wave whose pulse shapes broaden
systematically with source-receiver offset xi, due to
anelasticity. In this case, using the full set of τestij ’s in
JÖKULL No. 64, 2014 19