Geology Reference
In-Depth Information
comparisons, with clearly defined statistics and hypothesis testing capabilities.
Using the Thomson multitaper approach, the cross-correlation of two time
series x(n) and y(n) is assessed with the coherence and cross-phase functions:
)
K
(S
K
S
SS
(
f)
(f)
imag
x,y,D
x,y,D
C(f)
=
andf(f)
= arctan
)
K
(S
K
K
(f)
(f)
(f)
real
x,D
y,D
x,y,D
where
1
K
−
∑
1
]
K
K
[S
K
S
S
(f)
=
(f)
*
(f)
x,y,D
(
K
+
1
)
y,D
x,D
k
=
0
The coherency is commonly reported as “magnitude-squared coherence
(MSC)” |C(f )|
2
; the basic hypothesis test considers the null model for zero
coherency (i.e., no correlation) between the two time series x and y versus
the alternative model for nonzero coherency. Thus, when |C(f )|
2
is close to
1, the two series are correlated; specific frequencies may show strong corre-
lation and others may show weak to no correlation (noise) with |C(f )|
2
→ 0.
As discussed below, the properties of |C(f )|
2
indicate that there is strong
uncertainty about what constitutes zero coherence and what does not and
related to the dof of the estimation.
Basic application of these estimators is demonstrated in Figure 4.26 on
two insolation models calculated for a ~0.8 Myr interval from 36.2 Ma to
37.0 Ma, one lagging the other by 5 kyr. The differences in noise content of
the two models are quite evident in the power spectra: in Series 1, there is
a hint of orbital eccentricity variation at frequencies 1/405, 1/128, and
1/95 kyr, which is not at all evident in the Series 2 spectrum. The effective
dof are lower for the uniform noise background of Series 1. Otherwise,
both models have high spectral peaks and maximum dof at the obliquity
(1/41 kyr) and precession index (1/24, 1/22, 1/19, and 1/17 kyr) frequencies,
which are strongly correlated, as borne out by the coherency estimates
near 1.0. The cross-phase ϕ(f ) shows a flattening of values across the fre-
quency bands with high |C(f )|
2
that are consistent with a constant lag of
5-kyr of Series 2 with respect to Series 1. For example, at f = 22 kyr, the
phase is +90º, which is one-fourth of a cycle, i.e., 22 kyr/4 = 5 kyr.
|C(f )|
2
has a probability distribution that was originally described for the
general cross-correlation coefficient by R. A. Fisher and is given in Table 1
of Carter et al. (1973). The estimated value of |C(f )|
2
has a strong bias and
variance as a function of dof K, shown in Figure 4.27, which must be
accounted for in coherency estimation. Consequently, the level of “zero
coherence” is not zero, and can be drawn at specific significance levels
(Figure 4.27a), progressively lower for higher K (representative list in
Table 4.1). Finally, these uncertainties propagate into the cross-phase
