the cross-correlation function between Xt and

Zt is positive everywhere with a maximum

where σx = E Xt X— and σ y is de¬ned

2 2

t

near lag zero.

similarly. The correlation ρ is then written in

polar coordinates as

Thus the cross-covariance functions in Figure 11.5

suggest that, in the extratropics, the heat ¬‚ux

R

ρ = eiξ ,

(Z) drives the SST (X), which in turn exerts a

σx σ y

(small) negative feedback on the heat ¬‚ux.15 In the

tropics the shape of the cross-correlation function

where eiξ R is the complex covariance

suggests that there is weak positive feedback.

E Xt Yt— . The angle ξ is used as an estimate

of the veering angle; R, σx , σ y , and ξ

11.3.12 Example: Ekman Veering at the

are estimated from the ¬nite sample in the

Bottom of the Ocean. Kundu [233] describes

usual manner by forming sums. Note that

an interesting application of a complex cross-

estimate of the complex cross-covariance can

correlation (at lag zero). Theoretical arguments

be written in the form

based on the Ekman theory for boundary layers

predict that the currents near the bottom of the

eiξ R = eiξ j R j , (11.62)

ocean will veer counter-clockwise in the Northern

j

Hemisphere (i.e., a current close to the bottom of

where R j is the product X j Yj— expressed

the ocean will be directed somewhat more to the

left than a current above). in polar coordinate form. Thus, the veering

Kundu [233] used a two-month long time estimate obtained from (11.62) can be

series of current data collected off the Oregon interpreted as the mean of all observed angles

(USA) coast to search for observational evidence weighted by the strength of the instantaneous

supporting the theory. Data from two current ¬‚ow.

meters moored 5 m and 20 m above the bottom

was ¬rst ¬ltered to eliminate the effects of tidal Kundu [233] obtained veering estimates of 3—¦

and inertial modes. The ˜veering angle™ was then from the angle spanned by the mean currents, 7—¦

estimated from these data using three approaches. from the average angle, and 6—¦ from the complex

correlation.

1 The currents were averaged and the angle

spanned by the mean currents 5 m and

20 m above the bottom was computed. The 11.4 The Cross-spectrum

problem with this approach is that the mean

can be strongly in¬‚uenced by a few large 11.4.0 General. The purpose of cross-spectral

analysis is to learn how the variability of two

events in the time series.

time series is interrelated in the spectral domain”

2 The angle between the currents was computed that is, to determine the time scales on which

at every observing time. These angles were variability is related as well as the characteristics

of that covariation. Conceptually, we could split a

15 Note, however, that the similarity of cross-correlation

16 Kundu did not really calculate the correlation; he did not

functions is not proof that the proposed statistical model, say

(11.60, 11.61), is correct. subtract the mean values.

11.4: The Cross-spectrum 235

pair of time series into slowly and quickly varying 2 The cross-spectrum can be written in polar

parts, say coordinates as

x y (ω)

f

x y (ω) = A x y (ω) ei .

Xt = Xt + Xs

t

f

Yt = Yt + Ys ,

t Then A x y and x y are called the amplitude

spectrum and phase spectrum respectively.

where f denotes the fast components, and s the

The amplitude spectrum is given by

slow components. We want to know, for example,

whether the slow components of Xt and Yt vary 2 1/2

A x y (ω) = x y (ω) + x y (ω) .

2

together in some way. If at a certain time t there

is a ˜slow positive (negative) bump,™ is there a

The phase spectrum is given in three parts:

characteristic time lag „ , such that, on average,

= tan’1

there will also be a ˜slow positive (negative) bump™

x y (ω) x y (ω)/ x y (ω) (11.64)

in Yt+„ ? If so, the two slow components vary

˜coherently™ with a ˜phase lag™ of „/„s , where „s x y (ω) = 0 and x y (ω) = 0,

when

is the time scale of the slow variability.

x y (ω) >0

Just as with spectral analysis [11.2.1], our 0 if

x y (ω) = (11.65)

±π x y (ω) < 0

purpose in the next several subsections is to re¬ne if

these concepts in such a way that the nature of the

x y (ω) = 0, and

when

covariability of a process can be examined over a

continuum of time scales.

π/2 x y (ω) >0

if

x y (ω) = (11.66)

’π/2 x y (ω) < 0

if

11.4.1 De¬nition: The Cross-spectrum. Let

x y (ω) = 0.

Xt and Yt be two weakly stationary stochastic when

processes with covariance functions γx x and γ yy ,

3 The (squared) coherency spectrum

and a cross-covariance function γx y . Then the

cross-spectrum x y is de¬ned as the Fourier

A2 y (ω)

transform of γx y : x

κx y (ω) = (11.67)

x x (ω) yy (ω)

x y (ω) = F γx y (ω)

expresses the amplitude spectrum in dimen-

∞

’2πi„ ω

= γx y („ )e sionless units. It is formally similar to a con-

(11.63)

„ =’∞ ventional (squared) correlation coef¬cient.

for all ω ∈ [’1/2, 1/2].

The cross-spectrum is generally a complex- 11.4.2 Some Properties of the Cross-spectrum.

valued function since the cross-covariance func- 1 The cross-spectrum is bilinear. That is, for

tion is, in general, neither strictly symmetric nor jointly weakly stationary processes Xt , Yt ,

and Zt , and arbitrary constants ± and β,

anti-symmetric.

The cross-spectrum can be represented in a

—

±x,βy+z (ω) = ±β x y (ω) + ± x z (ω).

number of ways.

1 The cross-spectrum can be decomposed into This follows from the linearity of the Fourier

its real and imaginary parts as transformation and the bilinearity of the

cross-covariance function (11.44).

x y (ω) = x y (ω) + i x y (ω).

2 The cross-covariance function can be recov-

ered from the cross-spectrum by inverting the

The real and imaginary parts x y and x y

Fourier transform (11.63)

are called the co-spectrum and quadrature

spectrum17 respectively. 1

2

2iπ „ ω

γx y („ ) = x y (ω)e dω.

17 Note that we de¬ne the quadrature spectrum as the