ses with positive ±1 (i.e., „ D > 1). Tren-

berth [369] calls it a ˜persistence time scale™

and maps it for the Southern Hemisphere

geopotential height ¬eld. Processes with neg-

ative ±1 are the ultimate weakly stationary

˜oscillatory™ processes because they tend to

change sign at every time step. Thus „ D is

less than 1 even though the auto-correlation

˜envelope™ |ρ(k)| decays at the same rate as

that of an AR(1) process with coef¬cient |±1 |.

Thus a reasonable indicator of memory or

persistence that applies to all AR(1) processes Figure 17.2: The dependency of the dimensional

decorrelation time „ D,k on the time increment k

˜

is

and on the coef¬cient ±.

1 + |±1 |

„D =

1 ’ |±1 |

„D if ±1 > 0 is an AR(1) process with a unit time increment,

= ’1

„D if ±1 < 0. then we can construct other AR(1) processes with

k unit time increments by noting that

• Similar dif¬culties occur with oscillatory X = ± k X

1 t’k + Zt (17.13)

t

AR(2) processes since the decorrelation time

k’1 l

tends to be smaller than that indicated where Zt = l=0 ±1 Zt’l . The corresponding

by the decay of the correlation envelope dimensional decorrelation times (17.6) are

1 + 4 Im(a)2r ’k . In this case, a better

1 + ±1

„ D,1 =

˜

indicator of physical memory is

1 ’ ±1

∞ 1 + ±1

k

’k „ D,k = k

˜ .

„ D = 1 + 2 1 + 4 Im(a) 2 r

1 ’ ±1

k

k=1

Thus „ D,k ≥ k for all k when ±1 ≥ 0.

˜

2r 1 + 4 Im(a)2 + r ’ 1

=

r ’1 That means that the decorrelation time is at least

as long as the time increment. In the case of

white noise, with ±1 = 0, the decorrelation

where a and r are de¬ned as before. The

AR(2) process with (±1 , ±2 ) = (0.9, ’0.8) time is always equal to the time increment. Some

has „ D = 0.33 and „ D ≈ 20. dimensional decorrelation times are plotted in

Figure 17.2. The longer the time increment, the

• We showed in [11.1.9] that the auto-

larger the decorrelation time. Note that „ D,k = k

˜

correlation function of an AR( p) process

for suf¬ciently large time increments. For small

can be decomposed into a sum of decaying

±1 -values, such as ±1 = 0.5, „ D,k = k for k ≥ 5.

˜

persistent and oscillatory terms. As above, a

If ±1 = 0.8 then „ D,1 = 9, „ D,11 = 13.1 and

˜ ˜

meaningful indicator of physical memory can

„ D,21 = 21.4. Thus the decorrelation time of an

˜

be obtained by summing the envelope that

±1 = 0.8 process is 9 days or 21 days depending

contains all of these terms.

on whether we sample the process once a day or

In summary, „ D must be interpreted carefully. once every 21 days.

We conclude that the speci¬c value of the

It represents a physical time scale only when

decorrelation time may not be very informative.

the auto-correlation function coincides with the

However, comparison between time series with the

˜auto-correlation envelope™, as it does in white

noise processes.6 same sampling interval helps us identify which

processed have larger memory.

17.1.6 The Dependence of the Decorrelation

Time on the Time Increment. If 17.2 Potential Predictability

Xt = ±1 Xt’1 + Zt (17.12)

17.2.1 Concept of ˜Potential Predictability.™ It is

6 But see the caveat discussed in the next subsection. generally accepted that the skill of the short-term

17.2: Potential Predictability 375

˜climate™ forecasts7 derives primarily from the atmosphere™s lower boundary and other sources

persistence of the atmosphere™s lower boundary not related to the weather variability.

Since T S and T F are assumed to be inde-

conditions. Thus the potential for short-term

pendent, the variance of the „ -mean of Tt may

climate predictability, or potential predictability,

is often estimated using a time-domain analysis be separated into a part re¬‚ecting the integrated

of variance technique. This technique assumes weather noise and another part stemming from the

that variations in, say, seasonal mean sea-level low-frequency process(es):

pressure arise from two sources: one source

„

„

= Var TtS + Var TtF .

represents the effect of the daily weather variations Var Tt (17.16)

and the other re¬‚ects the effect of presumably

unrelated processes, such as tropical sea-surface Since the weather ¬‚uctuations are ˜unpredictable™

on time scales of the order of „ , only that

temperature or the presence of volcanic aerosols

part of the „ -mean of Tt accounted for by the

in the atmosphere. Variation from the ¬rst source

is known to be unpredictable for lead times slow process is potentially predictable. Thus a

of more than, say, 10 days, but the second reasonable measure of the relative importance of

source is thought to be predictable, at least the potentially predictable component in (17.15) is

in principle. Madden [263] ¬rst described a the variance ratio

time domain ANOVA technique for diagnosing „

Var Tt

potential predictability in 1976. His technique

S„ = . (17.17)

„

tries to infer from time series the strength of TtF

Var

the predictable contribution without identifying

A variance ratio S„ = 1 indicates that all

its dynamical source. The statistical aspects were

further elaborated by Zwiers [440]. See also low-frequency variability originates from weather

„

noise whereas S„ > 1 indicates that Tt contains

Zwiers et al. [449] and [9.4.7“11].

more variability than can be explained by weather

17.2.2 Formal De¬nition of Potential Pre- noise alone. Hence there is„the potential to forecast

dictability.8 The following statistical model is some of the variance of Tt .

used in the analysis of potential predictability. The

variable, say the temperature, Tt , is assumed to be 17.2.3 Estimating the Variance Ratio S .

„

the sum of two independent processes T S and T F : The numerator and denominator of variance

ratio S„ (17.17) are estimated separately. We

Tt = TtS + TtF . (17.14)

assume that we have a sample {t jk : j =

1, . . . , n; k = 1, . . . , „ } that consists of n

Furthermore, T S is assumed to vary slowly, and

chunks of „ consecutive observations. In typical

T F quickly. The latter is sometimes assumed to be

applications, each chunk is a daily time series

a red noise process (e.g., see [9.4.10] and [449]).

We also assume that the averaging time „ is short observed over a season, say DJF, and different

chunks represent different years. Ordinarily, the

relative to the characteristic time for the slow

annual cycle is removed so that, to ¬rst order,

process so that

the chunks can be assumed to be independent