discussion).

The particular time series model used is the displayed as a function of the lag-1 day correlation

auto-regressive model of order 1 (10.3). With this when within season variations behave as red noise.

model it is assumed that day-to-day variations

within a season behave as red noise (see Sections

0.2 0.4 0.6 0.8 1.0

10.3 and 17.2). If we let {Wi jt : t = 1, . . . , 90}

Shrinkage Factor

represent the daily weather within season i of

simulation j after removal of the annual cycle,

then the red noise assumption states that

Cor Wi, j,t1 , Wi, j,t2 = ρ |t2 ’t1 | , (9.35)

where ρ is the correlation between, say, 850 hPa

temperature on adjacent days.

The monthly means, which are the object of our

0.0 0.2 0.4 0.6 0.8 1.0

study, are given by

Lag-1 day correlation

l30

Yi jl = Wi jt /30. (9.36)

t=(l’1)30+1

Figure 9.6: Shrinkage factor for the unadjusted

Using (9.35) and (9.36) we obtain, after some F-ratio for interaction effects.

simpli¬cation, that

σW 29

„

2

(1 ’ )ρ „

= 1+2

Var Yi jl (9.37) We substituted the exact expressions (9.37) and

30 30

(9.38) into (9.33) and evaluated n — as a function

„ =1

of ρ (see Figure 9.5). We see that n — > 1.95 for

and

ρ < 0.9. This was expected because n — = 2

σW ρ 30(k’1)

2

in the absence of terms affected by ρ2 , which

Cov Yi, j,l , Yi, j,(l+k) =

becomes important only when ρ is very large.

30

29

„ „ Hence, the degrees of freedom of the test for

ρ 30 ρ „ .

—ρ + + 1’

30

interaction effects need only be adjusted if day-to-

30 30

„ =1

day dependence is very strong.

(9.38)

We also substituted the exact expressions (9.37)

Further simpli¬cation yields that, for ρ < 0.9, and (9.38) into (9.34). The fraction 1/C, used

ρ to shrink the unadjusted F-ratio for interaction

ρ1 ≈ effects, is illustrated in Figure 9.6. The shrinkage

302 (1 ’ ρ)2

factor decreases slowly with increasing ρ when ρ

ρ 31

is small, and drops very quickly as ρ approaches

ρ2 ≈ .

302 (1 ’ ρ)2 1. When ρ = 0.9, it is necessary to shrink the

It is reasonable to assume that ρ2 = 0, except when unadjusted F-ratio for interaction effects by a

ρ is large (ρ > 0.9). factor of approximately 32%.

9.5: Tuning a Basin Scale Ocean Model 191

Figure 9.7: Lag-1 day correlation for 850 hPa Figure 9.8: The natural log of the F-ratios for

DJF temperature in the CCCma six member the interaction effect for 850 hPa temperature in

AMIP ensemble. Correlations greater than 0.4 are the CCCma AMIP experiment using the variance

shaded. component adjustment method. Each contour

indicates a doubling of the F-ratio. The shading

Results for the CCCma AMIP indicates ratios which are signi¬cantly greater

9.4.11

Experiment. Estimates of lag-1 day correlation than 1 at the 10% signi¬cance level.

ρ for DJF 850 hPa temperature computed from the

CCCma AMIP simulations using (9.37) and (9.38)

are shown in Figure 9.7. We see that the simulated the no interaction effect hypothesis is rejected at

lower tropospheric temperature is generally most the 10% signi¬cance level over only 12.4% of

persistent on a day-to-day time scale where there the globe in DJF and there does not appear to

is subsidence, and least persistent in the tropics be a preferred location for the signi¬cantly large

and in the extratropical storm tracks. Estimated F-ratios.

lag-1 day correlations range between ρ = 0.0765

and ρ = 0.891. Corresponding values for C

(9.34) range between 1.005 and 1.409, and those 9.5 Tuning a Basin Scale Ocean

for n — range between n — = 2 and n — = 1.96. Model

The varying amounts of dependence result in

substantial spatial variation in the adjustment to 9.5.1 Tuning an Ocean Model. We now brie¬‚y

the F-ratio but almost no spatial variation in the describe a designed experiment of a different

degrees of freedom of the F test for interaction sort. As discussed previously, geophysical models

effects. use parameterizations to describe sub-grid scale

The adjusted F-ratios (9.31) required to test processes (see [6.6.6]). The sensitivity of such

H0 (9.30) are displayed in Figure 9.8. The null a model to a small number of parameters

hypothesis of the absence of the interaction can be explored systematically with designed

effect is rejected over 17.5% of the globe at experiments provided individual runs of the model

the 10% signi¬cance level. Experience suggests can be made at reasonable computational cost.

that this rate of rejection is ¬eld signi¬cant. Even today, this constraint places fairly tight

The structure of this ¬eld of F-ratios is very bounds on the complexity of models that can

similar to that obtained with the ˜rough-and-ready™ be studied in this way and ingenuity is required

test, but the rate of rejection is higher because to develop experimental designs that adequately

all of the data are used, rather than only two- explore parameter space.

thirds. Gough and Welch [145] describe a study

Figure 9.8 illustrates that the interaction effects of an isopycnal mixing parameterization in an

are con¬ned primarily to locations over land. ocean general circulation model7 (OGCM) that

As noted in [9.4.9], this suggests that land has seven adjustable parameters (diapycnal and

surface properties do not evolve identically in isopycnal diffusivity, vertical and horizontal eddy

each AMIP simulation. The effects of slow

7 Isopycnal parameterizations represent mixing processes

variations in soil moisture and surface albedo are

apparently detectable in the temperature of the on surfaces of constant density (isopycnals) and their

lower troposphere. These effects do not appear to perpendiculars[72] and Cox [93]) representparameterizations

(diapycnals). Conventional

(as in Bryan these processes

be detectable in the mean ¬‚ow of the atmosphere as on surfaces of constant height (horizontal levels) and their

represented by 500 hPa geopotential. In this case, perpendiculars.

9: Analysis of Variance

192

viscosity, horizontal background eddy diffusivity, coverage of the parameter space. One indicator

maximum allowable isopycnal slope, and peak of success in this regard is low correlation

wind stress). Had they used a standard factorial between the selected values of pairs of OGCM

design (see [9.1.1]) with, say, three different parameters. The objective is not always achieved

values of each parameter, it would have been with the randomization procedure because large

necessary to integrate the model 37 = 2187 correlations can occur by chance. Iman and

times. Instead, they used a design called a random Conover [192] describe a method for transforming

Latin hypercube (McKay, Conover, and Beckman a given random Latin hypercube into one with

[271]) that enabled them to adequately explore the better correlation properties. Gough and Welch

model™s parameter space with just 51 runs.8 All used this method iteratively to improve their

runs were 1500 years long and were started with experimental design.

the ocean at rest. A dif¬culty encountered by Gough and Welch

The design employed by Gough and Welch is that the parameter space for which the

exploits the fact that OGCMs are fully determin- OGCM converges to a steady state is not a

istic and converge to a steady state at long times, hypercube (i.e., a seven-dimensional rectangle).

given a particular set of parameter values and no In fact, four of the 26 runs displayed explosive

random forcing. Thus the experimental outcomes behaviour, and one evolved to an ˜unconverged™

do not contain random noise in the conventional oscillatory solution. The regression-like analysis

sense. This means that stochastic variation can be methods alluded to above were applied to the

introduced into the response by means of the pa- 21 successful runs to estimate the relationship