σT =

2

iance from two sources, the sine and cosine 2

γcc (0) + γss (0)

coef¬cient time series, into three components

= (11.88)

is not well-de¬ned mathematically. It is even 2

possible to obtain negative variances with this 1

2

= cc (ω) + ss (ω) dω.

partitioning under some conditions. There is

probably no universal method of partitioning 0

space“time variance into standing and propa-

11.5.4 The Variance of Eastward and West-

gating components.

ward Propagating Waves. The next step is

The expression space“time spectral analysis to split the total variance given by (11.87) into

indicates that the method requires a spectral the contributions from eastward and westward

decomposition of the process in space, that is, the travelling waves, so that

calculation of the Fourier coef¬cients ckt and skt ,

σT = σ E + σW ,

2 2 2

and a spectral decomposition of the temporal

co-variability of these coef¬cients. However, the where σ E and σW represent the components

2 2

analysis is far from being symmetric in terms that propagate eastward (E) and westward (W)

of space and time. The spatial decomposition is respectively.

only geometrical in nature; there are no sampling Formally, we write

problems. The temporal decomposition, on the

σ E = 1 σT + R

2 2

other hand, is heavily loaded with sampling 2

(11.89)

problems and non-trivial assumptions, as we will σW = 1 σT ’ R,

2 2

2

see in the following.

where R is currently unknown. What properties

should R have?

11.5.3 The Total Variance of the Waves. We

assume that the space“time stochastic process has 1 The westward and eastward variance should

been been decomposed, by means of (11.85), be non-negative. Thus

into sine and cosine coef¬cient stochastic proces-

ses Ckt and Skt . For convenience we will use 12

|R| ¤ σT .

index k only when necessary for clarity. The 2

bivariate process formed by the sine and cosine

coef¬cients is denoted 2 If the bivariate Xt (11.86) contains no noise

and consists of a single, undamped, eastward

Xt = (Ct , St )T . travelling wave, then all of σT should be

2

(11.86)

attributed to the ˜eastward™ compartment.

We also assume, for convenience, that the means of That is, we would have R = 1 σT .2

2

the sine and cosine coef¬cient processes are zero

(i.e., E(Ct ) = E(St ) = 0). Then, the total variance 3 If Xt is such that the sequences of eastward

of the space“time stochastic process F(x, t) at travelling waves are randomly overlaid by

spatial wavenumber k, say σT 2 , is

noise, then only part of σT should be

2

attributed to the eastward variance. The

L

remaining ˜unaccounted™ for variance should

σT = Var Ct cos( 2πLkx ) + St sin 2πLkx d x

2

be distributed evenly between the eastward

0

and westward component. In this case R <

L

= E C2 cos2 ( 2πLkx ) d x

2 σT .

12

t

0

L

4 If the components of Xt vary in an unrelated

+ E S2 sin2 ( 2πLkx ) d x

t

manner at all time scales, then there is no

0

preference for a direction and R = 0.

L

+ 2E(Ct St ) cos( 2πLkx ) sin( 2πLkx ) d x

21 The total variance in the ¬eld at wavenumber k is half of

0

Var(Ct ) + Var(St ) the sum of the variances of the coef¬cients because of the way

= . (11.87) in which coef¬cients Ct and St are de¬ned.

2

11: Parameters of Univariate and Bivariate Time Series

244

ct’ζ = 0 and st’ζ = ’1). Thus, for small

values of „ , γcs („ ) is positive and γcs (’„ ) is

t+ζ

negative. For a suf¬ciently well-behaved process,

the quadrature spectrum cs (ω) will be positive

for most negative ωs (cf. (11.68)) when there are

t

eastward travelling waves. Therefore R is positive

and σ E is greater than σW (cf. (11.89)).

2 2

Similarly, when there are westward travelling

waves, negative sine functions will tend to replace

t- ζ cosines so that R is negative and the westward

travelling variance is larger than the eastward

E travelling variance.

If the sine and cosine coef¬cient processes are

Figure 11.11: A schematic diagram illustrating one unrelated, then the quadrature spectrum is zero.

wavelength of a wave that travels eastward 1/4 of Thus equation (11.90) satis¬es the requirements

a wavelength every ζ time units. listed above. The concept can be extended so

that the propagating variance can be attributed to

speci¬c frequency ranges. To do that, we de¬ne the

One quantity that satis¬es all of these requirements frequency“wavenumber spectrum as

is the integral of the quadrature spectrum over

cc (ω) + ss (ω)

negative frequencies, fw

cs (ω) = ’ cs (ω). (11.91)

2

0

R= cs (ω) dω. (11.90) This is a two-sided spectrum with different

’1

2

densities for negative and positive frequencies.

To motivate this choice for R, let us consider Since

what an eastward travelling wave is. Suppose that

1

the ¬eld f (x, t) consists of a pure cosine pattern fw

2

cs (ω) dω

at time t and that, a short time „ later, the cosine 0

pattern has been slightly damped and a weak sine 1 1

1 2 cc (ω) + ss (ω) 2

= dω ’ cs (ω) dω

pattern has been added. The effect of adding the

20 2

sine pattern is that the crest of the wave moves 0

from the x = 0 location to some point to the right 0

12

= σT + cs (ω) dω

of x = 0. If our conceptual diagram is oriented 2 ’1 2

with north at the top of the page, the wave will have

12

moved eastward during the interval. Eventually,

= σT + R

after a quarter of a period, the cosine pattern is 2

replaced by a sine pattern, and after half a period