z t+„ = ξ „ e’i

2π„

F

z tF„

zt (15.21)

T

Var Var z t

where T = 2π/· is the period of the POP

E |z tF„ ’ z t | ,

S„ = (15.23)

and ξ = |»|. Equation (15.21) describes a

damped persistence forecast in the complex plane

where, as above, z tF„ is the (complex) forecast of

(Figure 15.1) which corresponds to a damped

z t made at time t ’ „ , and z t is the estimated state

propagating mode in physical space. Forecasts

at time t that is used to verify the forecast. Note

are made by identifying the current state of the

that the diagnosed forecast skill depends upon the

POP coef¬cient process and then applying (15.21).

skill of both z F„ and z t as estimators of z t .

Depending upon whether the practitioner thinks

The correlation skill score ρ„ is an indicator

a mature or growing oscillatory mode has been

only of phase errors since it is insensitive to

detected, the forecast will either be a damped

amplitude errors. This makes ρ„ a suitable skill

persistence forecast (i.e., ξ = |»| < 1 in (15.21))

score for POP forecasts since we anticipate that

or a persistence forecast in terms of amplitude (i.e.,

most of their utility lies in the phase component.

ξ = 1 in (15.21)). These forecasts will have some

The mean squared error S„ , which is sensitive to

skill at short leads, but at longer lead times the

both phase and amplitude errors, tends to be less

built-in linearity of the POP analysis, as well as the

¬‚attering of POP forecasts.

unpredictable noise, will result in a deterioration of

The skill of the POP forecast is put in

forecast skill.

perspective by comparing with the skill of

A basic limitation of POP forecasts is that,

the persistence forecast z tP„ = z t’„ , which

although they can predict the regularly changing

freezes patterns in time and space. As shown by

phase of the oscillation, they cannot predict an

(15.21), persistence and POP forecasts are close

intensi¬cation of amplitude. However, a phase

neighbours in the hierarchy of forecast schemes.

forecast is valuable even if the amplitude is not

Thus comparison of their skills is well justi¬ed.

well predicted.

Forecasting is complicated by the substantial

15.3.3 Example: The Madden-and-Julian

amount of noise in the analysed ¬eld, resulting

Oscillation. The skill of the POP forecasts of

in estimates of the POP coef¬cient that may not

the MJO (see [15.2.4]) was examined in [388,

be very reliable on a given day. Thus some sort

401]. The forecasts were initialized with the ˜time

of ˜initialization™ is necessary. ˜Time ¬ltering™

averaging™ technique using information from days

initialization [432] uses a one-sided digital ¬lter

0 through ’4 (i.e., l = 4; see [15.3.1]). The POP

to suppress variance on short time scales before

amplitude |z t | was predicted by persistence (i.e.,

estimating the POP coef¬cient in the usual way.

ξ = 1 in (15.21)).

˜Time averaging™ initialization begins with direct

Individual forecasts are presented as harmonic

estimates of the POP coef¬cients realized at the

dials that display the evolution of the POP

last few time steps, say z t , z t’1 , . . . , z t’„ . Then

coef¬cients before and after the forecast date, and

(15.21) is used to produce a one-lag ahead forecast

the forecast itself. Two cases are considered: 30

z tF1 of z t from z t’1 , a two-lag forecast z tF2 of z t

January 1985 and 1 December 1988. Dynamical

from z t’2 , and so on. Finally, an improved esti-

forecasts, produced with the NCAR CCM, were

mate of z t is obtained by computing a weighted av-

also made for a number of cases.11

erage of z t , z tF1 , . . . , z tF„ . More weight is given to

10 See [18.2.3] for details about these measures of forecast

the recent information than the older information.

skill.

Small POP coef¬cients that move irregularly in 11 The dynamical model was used to forecast 15 cases.

the two-dimensional phase space indicate that the According to the correlation skill score, the POP forecasts

process represented by the POP is not active, in outperformed the dynamical forecasts in these cases (see

which case it is reasonable not to rely on the formal Figure 18.8 and [18.4.4]).

15: POP Analysis

346

b)

a)

Figure 15.10: MJO: Forecasts of the POP coef¬cient z t . The forecasts are presented in the

two-dimensional POP-coef¬cient plane with the x-axis representing the z r -coef¬cient, and the y-axis

the z i -coef¬cient. The POP forecast model (15.21) implies a trajectory that rotates clockwise.

The dashed line that connects the open circles represents the observed evolution, the solid line that

connects the solid circles represents a dynamical forecast, and the POP forecast is given by the crosses.

From von Storch and Baumhefner [388].

a) Initialized 30 January 1985.

b) Initialized 1 December 1988.

squared error, S„ , reaches its saturation level at

Figure 15.10a shows the predicted and analysed

evolution for 30 days beginning on 30 January about the same time. The skill of the POP forecast

1985. The MJO evolved smoothly, with a decreases more slowly with time, reaching a value

clockwise rotation in the POP coef¬cient plane, of 0.5 at a lead of 9 days. Also note that the

until about 25 February. It reversed direction after mean squared error of the POP forecast has not yet

that day. Both the POP forecast and the NCAR reached saturation at a 24-day lead.

CCM forecast are skilful in predicting the regular

evolution in the ¬rst 25 days, but they fail to predict 15.4 Cyclo-stationary POP Analysis

the phase reversal on 25 February.

Figure 15.10b shows the less successful forecast The POP analysis described in Section 15.1

of 1 December 1988. The MJO POP coef¬cient assumes temporal stationarity while observed

was small at the time of initialization and remained processes are often cyclo-stationary, that is, the

so. The velocity potential ¬eld did not contain a ¬rst and second moments depend on an external

well-de¬ned wavenumber 1 pattern, and thus the cycle, such as the annual cycle. In this section

failure of both forecasts is not unexpected. we present a generalization of the conventional

The correlation skill score, ρ„ , and the root POP analysis that explicitly accounts for this

non-stationarity.12

mean square error, S„ , derived from a large

(n ≈ 1500) ensemble of forecast experiments 12 Cyclo-stationary POP analysis was ¬rst suggested by

are shown in Figure 15.11 for the POP scheme Klaus Hasselmann in an unpublished manuscript in 1985.

and for persistence. Persistence is more skilful Two groups, namely Maria Ortiz and her colleagues at the

University of Alcala in Spain and Benno Blumenthal from

than the POP forecast during the ¬rst 2 days, but

the Lamont Doherty Geological Observatory in Palisades,

rapidly loses skill at longer leads. Persistence has New York, showed how to implement the cyclo-stationary

a minimum in ρ„ at about 20 days, consistent POP analysis independently in 1989/1990. Only Blumenthal

with the 30“60 day period of the MJO. The mean published his results [50].

15.4: Cyclo-stationary POP Analysis 347

we ¬nd

Xt+1,„ = B „ Xt,„ + noise (15.25)

where

n

B„ = A„ +s’1 (15.26)

s=1

and where the noise in (15.25) is a moving average