00 0

calculated by replacing (11.53) with its in¬nite

moving average representation (see [10.5.2]) or so that

with the mechanics developed above. σx 0

2

Σx y (0) = ,

0 σz2

The ¬rst step in the latter approach is to

represent (11.53) as bivariate AR(1) process

where σx = σz2 /(1 ’ ±1 ). Next, we compute,

2

2

(11.45)

±1 ’1

„

„

±1 ±1

X 1 X N „

= + , A=

Yt 0 0 Y Mt 0 0

t’1

11.3: The Cross-covariance Function 233

and ¬nally use (11.50) to ¬nd that for positive „ 0.6

CORRELATION

„2 „ 0.4

γx x („ ) = ±1 σx = σz2 ±1 /(1 ’ ±1 )

2

(11.54)

0.2

γxm („ ) = ±1 ’1 σz2 ,

„

CROSS

(11.55) 0

γmx („ ) = 0. -0.2

(11.56) 12

-12 -6 0 6

LAG (month)

Equation (11.54) is the auto-covariance function

of AR(1) process (11.53) and it holds for all

„ . However, note that, since Mt was de¬ned as

Figure 11.8: Estimated cross-correlation functions

Zt+1 , equations (11.55, 11.56) describe the cross-

ρ SST,S L P between two monthly indices of the

covariance function between the AR(1) process

dominant variability of SST and SLP over the

and its driving noise one time step in the future.

North Paci¬c. One estimate (closed dots connected

Thus the cross-covariance function of Xt and Zt

by a thin line) is estimated from data. A second

are given by

estimate (open dots connected by a heavy line)

„2

γx z („ ) = ±1 σz for „ ≥ 0, (11.57) is obtained from a stochastic climate model. The

SST leads for negative lags. From Frankignoul and

γx z („ ) = γzx (’„ ) = 0 for „ < 0. (11.58) Hasselmann [133].

Note that γx z („ ) is highly non-symmetric. It

is nonzero for all non-negative lags „ , that is, 11.3.11 The Effect of Feedbacks. The contin-

the current Xt value ˜remembers™ the preceding uous version of an AR(1) process is a ¬rst-order

and present noise with a memory that dims differential equation of the form14

exponentially. On the other hand, γx z („ ) is zero for

‚Xt

all negative lags. Hence Xt ˜knows™ nothing about

= ’»Xt + Zt . (11.59)

‚t

future noise.12

Unfortunately, equation (11.59) is of limited

11.3.10 Paci¬c SST and SLP. The following physical interest because the ˜forcing™ Zt acts on

example, which is taken from Frankignoul and Xt without feedback. Frankignoul [128] added

Hasselmann [133], illustrates that (11.57, 11.58) such feedbacks by replacing (11.59) with a system

can be of some practical use. of two equations

Frankignoul and Hasselmann considered two

‚Xt

indices which are representative of the large-scale = ’»o Xt + Zt + Ntx (11.60)

‚t

monthly variability of sea-surface temperature

Zt = »a Xt + Ntz (11.61)

(SST) and sea-level air pressure (SLP) in the

North Paci¬c. The cross-correlation function

with two white noise forcing terms Ntx and Ntz .

ρ SST,S L P estimated from monthly mean data is

For example, we could think of variable Xt as

non-symmetrical with values that are essentially

SST and variable Zt as the turbulent heat ¬‚ux

zero for negative lags. Correlations for lags

into the ocean (as in [11.3.4]). Then the change

between zero and about six months are positive

in SST is in¬‚uenced by its current state (i.e.,

(Figure 11.8; closed dots connected by thin line

memory, represented by the parameter »o >

segments). A stochastic climate model,13 which

0), by the instantaneous heat ¬‚ux forcing, and

is slightly more complex than the simple AR(1)

by some random variations. The heat ¬‚ux, on

process (11.53) with ±1 = 5/6, was also used to

the other hand, depends on the current SST and

estimate ρ SST,S L P . The resulting cross-correlation

random noise induced by the turbulent ¬‚ow of the

function (Figure 11.8; open dots connected by

atmosphere.

heavy line segments) is similar to that computed

The cross-correlation function between Xt and

from the observations, and has structure similar

Zt depends on the value of the ˜feedback™

to that predicted by (11.57, 11.58). To a ¬rst-order

parameter »a . The following can be shown (cf.

approximation, the North Paci¬c SST may be seen

[128]).

as an integrated response to atmospheric forcing

14 We will avoid mathematical questions such as the

which is independent of the SST variability.

de¬nition of continuous white noise. We use the continuous

12 Some authors also use the term innovations to describe the

representation for reasons of convenience, and to clarify the

noise processes that force AR processes. underlying physics. In practice, the derivatives are replaced by

13 This model is derived from a one-dimensional mixed layer ¬nite differences, and the problem of how to de¬ne continuous

ocean model. See [10.4.3] for more discussion on stochastic noise, for example, disappears. A good introduction can be

climate models. found in Koopmans [229].

11: Parameters of Univariate and Bivariate Time Series

234

• There is no feedback when »a = 0. In this subsequently averaged. The disadvantage of

case we get the result developed in [11.3.9] this approach is that weak currents are

and discussed in [11.3.10]. Cross-correlations associated with highly variable angles. Thus

between SST and heat ¬‚ux are zero for the weak current events substantially increase

negative lags and positive for lags „ ≥ 0. the uncertainty of the veering angle estimate.

3 The complex ˜correlation™16 between the two

• There is a negative feedback when »a < 0.

complex random variables Xt = U5m (t) +

When the SST anomaly is positive (Xt > 0),

i V5m (t) and Yt = U20m (t) + i V20m (t) was

the anomalous heat ¬‚ux is negative so that the

SST-tendency becomes negative (i.e., ‚Xt < estimated. For simplicity we assume E(Xt ) =

‚t

E(Yt ) = 0. Then, the complex correlation is

0) on average. The cross-correlation function

is anti-symmetric in this case.

E Xt Yt—

ρ= ,