use of conditional stochastic models to generate climate change scenarios

USE OF CONDITIONAL STOCHASTIC MODELS TO GENERATE

CLIMATE CHANGE SCENARIOS

RICHARD W. KATZ Environmental and Societal Impacts Group, National Center for Atmospheric Research, Boulder,

CO 80307, U.S.A.

Abstract. Stochastic models have been proposed as one technique for generating scenarios of future climate change. One particular daily stochastic weather generator, termed Richardson's Model or WGEN, has received much attention. Because it is expressed in a conditional form convenient for simulation (e.g., temperature is modeled conditional on precipitation occurrence), some of its statistical characteristics are unclear. In the present paper, the theoretical statistical properties of a simplified version of Richardson's model are derived. These results establish that when its parameters are varied, certain unanticipated effects can be produced. For instance, modifying the probability of daily precipitation occurrence not only changes the mean of daily temperature, but its variance and autocorrelation as well. A prescription for how best to adjust these model parameters to obtain the desired climate changes is provided. Such precautions apply to conditional stochastic models more generally.

1. Introduction

The issue of what methods should be employed to generate scenarios of future climate change has received increased attention recently (e.g., Robock et al., 1993). General circulation models (GCMs) have serious limitations, including their inabil- ity to reproduce the climate on a regional scale, that make their direct use in impact studies questionable. One alternative involves the combination of GCM exper- imental output with stochastic models (Wilks, 1992). In this way, the climate change scenarios to be generated could at least be consistent with the statistical characteristics of climate variables.

Numerous stochastic weather generators exist, having been designed primarily for simulating the present climate. The reliance on one particular stochastic model, termed Richardson's model or WGEN (Richardson, 1981; Richardson and Wright, 1984), has been prevalent in climate impact studies. For instance, Riha et al. (1996) and Meatus et al. (1996) adapt this generator to produce climate change scenarios as input to crop-climate models. Richardson's model simulates daily time series of precipitation amount, maximum and minimum temperature, and solar radiation. It is expressed in a conditional form (i.e., the other variables are modeled conditional on precipitation occurrence), which is especially convenient for computer algorithms.

Although some physical scientists have questioned whether the use of stochastic weather generators is appropriate (Robock et al., 1993; Young, 1994), research for the most part has focused on the performance of such models and how they might be improved (e.g., Hanson et aL, 1994; Johnson et al., 1996; Wallis and Griffiths,

Climatic Change 32: 237-255, 1996. (~) 1996 Kluwer Academic Publishers. Printed in the Netherlands.

238 RICHARD W. KATZ

1995). This work has identified some inadequacies, but has not focused on a fundamental understanding of how these models function. In the present paper, the theoretical properties of a simplified version of Richardson's model will be derived. Because of its conditional form, some of its statistical characteristics may not be obvious to the user. In particular, when its parameter values are varied, certain unanticipated effects can be produced. A prescription for how best to adjust these model parameters to obtain the desired climate changes is provided. Such precautions apply to stochastic models more generally, in raising the important distinction between conditional and unconditional statistics. Daily precipitation and temperature observations for Denver, CO, are utilized for illustrative purposes.

2. Description of Richardson's Model

A simplified version of Richardson's model for simulating time series of daily climate variables at a given site is treated (Richardson, 1981). With these simplifi- cations, the most important features of the model become more salient. Rather than simultaneously modeling daily minimum and maximum temperature and solar radiation, all conditional on daily precipitation occurrence, only one of these variables is considered at a time. Specifically, either the minimum, maximum, or mean temperature is modeled conditional on precipitation occurrence. Instead of allowing the model parameters to vary daily with the annual cycle, they are held constant within an individual month. On the other hand, certain constraints imposed by Richardson and Wright (1984) are not adopted. All the model parameters are 'tuned' to the particular site, whereas Richardson and Wright constrain certain parameters to be invariant across the U.S. Moreover, their constraint that certain model parameters be independent of precipitation occurrence is relaxed. In this way, some potential sources of discrepancies when fitting the model to climate data are eliminated.

2.1. DEFINITION OF MODEL

Let {Jr: t = 1 ,2 , . . .} denote the sequence of daily precipitation occurrence (i.e., Jt = 1 indicates that the tth day is 'wet', Jt = 0 a 'dry day') at a given site. To represent the tendency of wet or dry weather to persist, assume that this process is a first-order, two-state Markov chain (e.g., Katz, 1977). This model is completely characterized by the transition probabilities

Pij = Pr{Jt --- j I Jt-~ -- i}, i , j = O, 1. (1)

Note that P0o + P0J = Plo + Pll -- 1, so that only two transition probabilities (e.g., P01 and Pll) need be specified.

An alternative parameterization of the Markov chain model has parameters ~r, the probability of a wet day and d, the first-order autocorrelation coefficient (or 'persistence parameter'); that is,

7r = Pr{Jt = 1}, g = corr(Jt_l, Jt). (2)

USE OF CONDITIONAL STOCHASTIC MODELS 239

Here the parameters 7r and d are related to the transition probabilities by

7r = POl/ (Plo +/901), d = Pll - P01- (3)

For sequences of daily precipitation occurrence, generally the persistence parameter d > 0 and this constraint is imposed in the subsequent discussion. The assumption of a first-order Markov chain may be an oversimplification and could be relaxed to allow for a higher-order chain (Gates and Tong, 1976). Richardson's model also includes provision for the simulation of daily precipitation amount. This aspect does not enter into the manner in which the model links temperature with precipitation. Because the distribution of daily precipitation amount has received much study, it is not treated here.

Let {Xt: t = 1 ,2 , . . . } denote the corresponding time series of daily temperature (either minimum, maximum, or mean). To account for the statistical dependence between daily temperature and precipitation (e.g., the tendency of the maximum temperature to be lower on wet days than on dry days), the Xt-process is defined condit ional on the dr-process. Given the precipitation occurrence state on the tth day, say Jt = i, the conditional distribution of temperature on that day is assumed to be normal with mean #i and variance or/2. This conditional dependence is written a s

( X t I Jt = i) ,.~ N ( # i , a]), i = 0, 1. (4)

To allow for day-to-day dependence of temperature (i.e., the tendency of hot or cold weather to persist), a randomly standardized (termed 'residual' by Richardson, 1981) time series of daily temperature is constructed. That is, given ,It = i, define

z t = (x t - i = 0 , 1. ( 5 )

Because it is unknown a priori which parameters will actually appear in (5) for a particular day [i.e., whether (#0, (70) or (#l, al)], this standardization is termed 'random'. By definition, the randomly standardized variable Zt has a N(0, 1) distribution.

Next the Zt-process is modeled as a first-order autoregressive JAR(l)] process:

Zt = ~ Z t - l + ct, t = 2 , 3 , . . . . (6)

Here the parameter 4) is the first-order autocorrelation coefficient for the Zt-time series, and the error terms et are taken as independent N(0, 1 - ~b2). This parameter ~b represents the 'conditional' autocorrelation of the temperature time series, given the states of the precipitation occurrence process. In the subsequent discussion, the physically reasonable constraint that q~ > 0 is imposed. The assumption of an AR(1) process may be an oversimplification and could be relaxed to allow for a higher-order autoregressive process (Katz, 1982).

For the purposes of simulating time series of daily precipitation occurrence and temperature, this definition is both complete and convenient to implement.

240 RICHARD W. KATZ

But for understanding how best to change the parameters of Richardson's model, the statistical properties of the equivalent unconditional model for daily precipitation occurrence and temperature are required. Expressions for these unconditional statistics will be presented in Section 3.

2.2. FIT TO DENVER DATA

The simplified version of Richardson's model is fitted to 30 years, 1965-1994, of daily observations at Denver during two months, January and July. The three temperature variables should be viewed as different applications of Richardson's model, as they are not simultaneously treated. Because our purpose in utilizing the Denver data is simply to illustrate how the model works, formal significance tests are not applied to determine whether individual temperature parameters actually need to be varied conditional on whether or not precipitation occurs.

The shift in the conditional normal distribution of temperature between dry and wet days is the key to the link between precipitation occurrence and temperature in Richardson's model. Figure 1 shows this effect in the case of maximum temperature. For both months, the parameter estimates satisfy #1 < #0 and or1 > or0. A larger difference in the daily temperature mean, depending on whether or not precipitation occurs, is evident in January, whereas a greater effect on the daily temperature variability is evident in July.

Table I lists the complete set of parameter estimates (denoted by a 'caret' over the corresponding parameter) for the fit of Richardson's model to the Denver observations. These statistics are fitted by maximum likelihood for temperature and by approximate maximum likelihood for precipitation occurrence. It is evident that the daily minimum temperature exhibits a smaller, but still substantial, difference in conditional means in January than that for the maximum. On the other hand, little or no dependence of the conditional distribution of minimum temperature in July on precipitation occurrence is apparent. With the exception of the maximum temperature in July, only a small effect of precipitation occurrence on the conditional standard deviation appears. The conditional standard deviation is greater on wet days than on dry days (except for the possibly insignificant difference for the minimum temperature in July), and this behavior is consistent with the general tendency for time periods (or for locations) with cooler weather on the average to be more variable as well.

The conditional day-to-day temperature autocorrelation is greatest for the minimum temperature in January and least for the maximum temperature in July. Natu- rally, the effects of precipitation occurrence on the conditional parameter estimates for the daily mean temperature fall intermediate between those for the minimum and maximum. Wet days are nearly twice as frequent in July than in January, and have a slightly greater tendency to persist. In Section 3, these parameter estimates will be employed to demonstrate some of the theoretical properties of Richardson's model.


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Figure 1. Conditional normal distributions given a dry day (solid curve) or a wet day (dashed curve) fit to Denver daily maximum temperature in: (a) January; (b) July. Also included are exact (dotted curve) and approximate normal (dot-dashed curve) unconditional distributions.

3. Properties of Richardson's Model

Richardson's model involves the specification of the conditional distribution of daily temperature given the occurrence (or non occurrence) of daily precipitation.

242 RICHARD W. KATZ

Table I

Parameter estimates for fit of Richardson's model to Denver daily observations during January and July, 1965-1994 (30 • 31 = 930 days each)

Daily Conditional Conditional Conditional temperature mean standard deviation autocorrelation variable /20 (~ /21 (~ &0 (~ ~1 (~

January a

Minimum -7.76 -11.34 6.07 6.22 0.683 Maximum 7.99 0.02 6.68 7.00 0.550 Mean 0.12 -5.66 6.03 6.12 0.663

July a Minimum 14.84 14.79 2.31 2.28 0.582 Maximum 32.03 28.97 3.28 4.33 0.426 Mean 23.43 21.88 2.40 2.92 0.557

a Parameter estimates of precipitation occurrence process: ~- = 0.178 and d ---- 0.219 (January); ~ = 0.325 and d = 0.262 (July).

Consistent with these conditional models, an unconditional (or ' induced') model for the time series of daily precipitation occurrence and temperature can be derived. Such a combination of conditional distributions is sometimes referred to as a 'mixture' in the statistics literature (e.g., Everitt and Hand, 1981; Titterington et al., 1985). Due to their technical nature, details of the derivation of these properties are relegated to the Appendix.

3.1. UNCONDITIONAL MODEL

First, the unconditional statistical properties of the Markov chain model for daily precipitation occurrence are briefly reviewed. Employing the alternative parameterization, rc and d, the daily mean # j = E(Jt), daily variance ~r~ = var(Jt) , and day-to-day autocorrelation function pj (1) = corr(J t - t , ,It), l = 1, 2, . . . . are given by

# j = T r , c r ~ = T r ( 1 - T r ) , p j ( 1 ) = d , I = 1 , 2 , . . . . (7)

The first two properties actually hold for any process that assumes only two possible states, '0 ' and '1 ' , whereas the geometric decay of the autocorrelation is a characteristic of the first-order Markovian dependence.

3.1.1. Distribution According to Richardson's model [i.e., from (4)], the unconditional (or 'marginal ') distribution of the daily temperature is a mixture of two normal distributions. That is, the probability density function f ( x ) of Xt can be expressed as

f ( x ) = ( l - 7r)f0(x, /zo, o 0 ) + 7rfl(x; /~1, crl), (8)


where fi(x; #i, cr2i) denotes the probability density function for a X(#i , o-2) distribution [as in (4)], i = 0, 1. A mixture of two normal distributions (8) cannot be the normal (excluding the trivial cases of 7r = 0 or 1, or both #0 = #1 and ~r 0 = ~r~) and may exhibit substantial degrees of skewness or kurtosis. If the difference between the two conditional means, #0 and #1, is sufficiently large (relative to a0 and crl), this unconditional distribution may even be bimodal (e.g., Johnson and Kotz, 1970, pp. 8%92). Nevertheless, for many configurations of the parameters that appear in (8), the density function f (x) is still a close approximation to the normal.

3.1.2. Mean For this model, the unconditional mean of daily temperature, r = E(Xt ) , is just a weighted average of the two conditional means. That is,

# x = (1 - 7r)r + 7r#1. (9)

This property of mixtures can be regarded as 'conservatism of center of mass'.

3.1.3. Variance The unconditional variance of daily temperature, or} = var(Xt) is not simply a weighted average of the conditional variances. It is shown in the Appendix that

@( = [(1 - 7r)cro 2 + ~-cr~] + 7r(1 - ~r)(#l - #0) 2. (10)

The basic idea underlying (10) is that two sources of variation arise in the mixture (8), the conditional variances themselves, [i.e., the term inside the square brackets in (10)] and the variation in the conditional means [i.e., the other term on the right-hand side of (10)]. As such, this property (10) can be viewed as a statistical version of the Pythagorean Theorem.

3.1.4. Autocorrelation Function Despite the randomly standardized temperature time series (5) being modeled as an AR(1) process (6), the autocorrelation function of the unconditional model for daily temperature, px(1) = corr(Xt_l, Xt), l = 1,2, . . . . is much more complex than that for an AR(1) process. As variance is a special case of covariance, the expression for Px (1) constitutes a generalization of (10). Again, the unconditional autocovariance is not simply a weighted average of the conditional covariances, as another source of variation must be taken into account. One way to express this autocorrelation function is

px(1) = { r 2 + ~ ] q-Tr(1 - 7r)[dl(/zl - /z0) 2 - r - dt)(crl - cr0)2J}/~r~( , (11)

l = 1, 2 , . . . , where o-} is given by (10) (see Appendix). It is evident that the unconditional autocorrelation function (11) for daily temper-

ature is a complicated combination of the autocorrelation functions of the randomly standardized temperature time series [i.e., pz(1) = et] and of the precipitation

244 RICHARD W. KATZ

occurrence time series [i.e., pj (1) -= d t as in (7)]. If d < r (as is the case when com- paring the degree of persistence of the daily precipitation occurrence and randomly standardized temperature time series), then it is straightforward to show that

p x ( 1 ) < = p z ( 0 , l = l , 2 , . . . . (12)

In other words, the unconditional autocorrelation is smaller than the conditional autocorrelation at every lag. A heuristic explanation for this behavior is that the random switching between the two conditional normal distributions for temperature (as governed by the states of the Markov chain, see Fig. 1) serves as an additional source of variation, thus effectively reducing the extent of temporal dependence. Nevertheless, if the degree of persistence d of the precipitation occurrence process were great enough, then it is theoretically possible that the unconditional temperature autocorrelations could be larger than the corresponding conditional values.

3.1.5. Cross Correlation Function The last statistic required to essentially characterize the unconditional model for daily precipitation occurrence and temperature is the cross correlation function, pjx(1) = corr(Jt, Xt+l), 1 ----- . . . . -1, 0, 1 . . . . . This function can be expressed in a simpler form than the autocorrelation function for temperature (11), being

jx(z) = - - 1 . . . . , - 1 , 0 , 1 , . . . , (13)

where ~rx can be obtained from (10) (see Appendix). Closely resembling the autocorrelation function for the precipitation occurrence time series [i.e., p j (1) in (7)], it involves an adjustment for the difference between the two conditional temperature means (relative to the unconditional temperature standard deviation), as well as for the standard deviation of the occurrence time series [i.e., ~rj in (7)]. This cross correlation function (13) is independent of the conditional autocorrelation coefficient r for the randomly standardized temperature time series. If #1 < #0 (e.g., as in Table I), then pjx(1) < 0 for all 1.

3.2. APPLICATION TO DENVER DATA

Both the exact unconditional distribution (8) and its normal approximation [i.e., N ( # x , a2x) with /zx and ~r~r given by (9) and (10), respectively] are included in Figure 1 for the case of Denver maximum temperature. The unconditional (or marginal) distribution of the stochastic model for daily temperature (whether minimum, maximum, or mean) is quite close to the normal for both the Denver January and July data. In particular, the sometimes substantial differences in conditional means are still not nearly large enough (relative to the corresponding conditional standard deviations) to produce bimodality. Nevertheless, one advantage of this form of model is its capability of reproducing skewness, such as that typical of the

USE OF CONDITIONAL STOCHASTIC MODELS

Table II Unconditional daily temperature statistics derived from fit of Richardson's model to Denver data, along with corresponding sample statistics (given in parentheses)

Daily Mean Standard Autocorrelation Cross temperature #x (~ deviation px (1) correlation variable a x (~ pJX (0)

January Minimum

Maximum

Mean

July Minimum

Maximum

Mean

-8.40 6.25 0.661 -0.219 (-8.40) (6 .25) (0.739) (-0.221)

6.57 7.40 0.494 -0.412 (6.54) (7 .41) (0.616) (-0.415)

-0.91 6.44 0.611 -0.343 (-0.93) (6 .44) (0.729) (-0.346)

14.82 2.30 0.582 -0.010 (14.82) (2 .30) (0.583) (-0.009)

31.04 3.93 0.399 -0.365 (3!.06) (3 .91) (0.518) (-0.363)

22.93 2.68 0.532 -0.271 (22.94) (2 .68) (0.611) (-0.270)

245

unconditional distribution of daily solar radiation data (Bruhn et al., 1980). Recall thatthis variable is also treated in the complete version of Richardson's model.

Table II contains some unconditional daily temperature statistics derived from the parameter estimates for the fit of Richardson's model to the Denver data (Table I). At this point in the analysis, these parameter estimates are treated as if they constitute the actual parameter values of the model (i.e., any sampling error is ignored). Also included are the corresponding statistics calculated directly from the data. For theoretical reasons, Richardson's model necessarily essentially reproduces the unconditional mean, standard deviation, and contemporaneous cross-correlation coefficient (as is evident in Table II). Nevertheless, some additional insight is provided. For instance, the unconditional standard deviations of all three temperature variables in January are greater than both of the corresponding conditional standard deviations (cf. Tables I and II), with the second term on the right-hand side of (10) providing a theoretical explanation for this phenomenon.

It is evident from Table II that Richardson's model has a marked tendency to underestimate the unconditional first-order autocorrelation coefficient of daily temperature (despite the fact that the conditional autocorrelation ~b is automatically reproduced). Only when the conditional mean and standard deviation effectively

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do not depend on the occurrence of precipitation (i.e., for July minimum temperature) is the discrepancy negligible. How this discrepancy might be removed will be briefly mentioned in the Discussion (Section 5). Finally, Figure 2 illustrates the result (12) that the unconditional autocorrelations are smaller than the corresponding conditional autocorrelations of the randomly standardized time series, using the parameter values of Richardson's model fit to Denver daily maximum temperature (see Table I). These differences are more substantial in January than in July.

4. Climate Change Scenario Generation

4.1. EFFECTS OF CHANGES IN PARAMETERS

The theoretical relationships discussed in Section 3 are now utilized to examine more closely how the unconditional temperature statistics depend on the individual parameters of Richardson's model. Only one or two parameters at a time are adjusted, holding all other model parameters fixed. For clarity, new parameter values are sometimes distinguished by asterisks.

4.1.1. Conditional Means It is simplest to suppose that the conditional means of temperature on dry and wet days are adjusted by the same amount (i.e., #* = #i + A, i ---- 0, 1). Then the unconditional temperature mean (9) naturally also would change by this amount


(i.e., # ~ = # x + A). Because the unconditional temperature variance (7~, autocorrelation function Px (1), and cross correlation function PJX (1) depend on the conditional means only through their difference (i.e., #1 - #0), all of these statistics would remain the same in the new climate [see (10), (11), and (13)]. If the two conditional temperature means were adjusted by differing amounts (i.e, #1 - #0 does not remain constant), then not only the unconditional mean would change, but all the other unconditional statistics as well.

4.1.2. Conditional Standard Deviations Adjustments to the conditional standard deviations of temperature on dry and wet days, (70 and (rl, naturally would change the unconditional standard deviation (Tx. Because the difference in conditional means is not affected, crx would change by less than a proportionate amount [see (10)]. Specifically, if e I = 6(7i, i = 0, 1, then (7~7 < (5(7x, for 1 < (5 < oo, and (7~: > (5(7x, for 0 < (5 < 1. Figure 3 shows this relationship between the unconditional standard deviation and the scaling factor fi, using the parameter values of Richardson's model fit to Denver daily maximum temperature (see Table I). The lack of proportionality is quite evident. For instance, when the two conditional standard deviations are halved (i.e., (5 = 0.5), the unconditional standard deviation would actually only decrease by about 39% in January and 41% in July (not 50%); when (5 = 2, crx would only increase by about 87% in January and 90% in July (not 100%). The unconditional mean # x is, of course, unaffected by any changes in conditional variances [see (9)], but changes in the autocorrelation and cross correlation functions would be unavoidable [see (11) and (13)].

4.1.3. Conditional Autocorrelation A change in the conditional first-order autocorrelation coefficient ~b of the randomly standardized time series of daily temperature, not surprisingly, would result in the same directional change in the unconditional autocorrelation function px~l) at every lag. But this change would not be proportionate [see (11)]. Adjustments in ~b would not have any effect on the other unconditional statistics.

4.1.4. Precipitation Occurrence Figure 4 shows the relationships between certain unconditional temperature statistics and the probability of precipitation occurrence It, using the parameter values of Richardson's model fit to Denver daily maximum temperature (see Table I). As is evident in Figure 4a, the unconditional mean temperature # x is a linear function of 7r, with a slope greater (in absolute value) in January than in July because 1#1 - #01 is larger [see (9)]. The relationship between the unconditional temperature standard deviation cr x and 7r is nonlinear [see (10)], in this case reaching a maximum at about 7r = 0.5 for January and 7r = 0.9 for July and varying less than # x does (Fig. 4b). Both the unconditional temperature first-order autocorrelation Px (1) and the contemporaneous cross correlation PJx (0) are nonlinearly related

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to 7r [see (11) and (13)]. In this case, the3( each reach minima at about 7r = 0.5 for January and 7r = 0.4 for July, with the cross correlation being more sensitive to 7r (Figs. 4c and 4d). The fact that the unconditional autocorrelation Px (1) is smaller than the conditional autocorrelation ~b [recall (12)] is demonstrated in Figure 4c, with this reduction only vanishing in the extreme limiting cases of 7r = 0 or 7r = 1 see (11)]. Both statistics are more sensitive to 7r in January than in July, again primarily because I#l - #01 is larger. Whether any of these re la t io~sbiN make physical sense would be subject to considerable debate.

Changes in the persistence parameter d of the daily precipitation occurrence model would affect both the temperature autocorrelation and cross correlation functions. The unconditional autocorrelation function px(l) is positively linearly related to the autocorrelation function of the Markov chain, p j(1) = d l, which is intuitively reasonable as making precipitation occurrences more persistent ought to result in the associated temperatures being more persistent as well [see (11)]. The cross correlation function pjx(1) is directly proportional to p j(1), but it is difficult to attach a physical interpretation to (13).

USE OF CONDITIONAL STOCHASTICNI~I~ELS 249

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250 RICHARD W. KATZ

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4.2. HYPOTHETICAL EXAMPLE

Having clarified the relationships among the conditional parameters and unconditional statistics, the use of Richardson's model to construct scenarios of climate change is now addressed. As an example, Table III gives unconditional statistics when hypthetical adjustments are imposed on the parameters for Richardson's model fit to Denver daily mean temperature in January (see Table I). Suppose it is desired to produce climate scenarios with a relative frequency of wet days that is double the present value. Table III includes, as Scenario No. 1, this situation in which the probability of precipitation is increased to 7r* = 0.355, but the other parameters of Richardson's model are held fixed at their current values. This adjustment in 7r induces a decrease in the unconditional temperature mean # x of about 1.0 o C, an increase in the unconditional temperature standard deviation ~x of about 3.5%, a decrease in the unconditional first-order temperature autocorrelation px(1) of about 4.0%, and an increase (in absolute value) in the contemporaneous cross correlation PJx (0) of about 20.9%.

Next Table III gives, as Scenario No. 2, the situation in which the two conditional temperature means are both increased by about 1.0 ~ C (i.e.,/z~ = 1.14, #~ = - 4.64 ~ to counteract the effect of increasing 7r on the unconditional mean/~x. This adjustment has no additional effect on the other unconditional statistics. Table III also includes, as Scenario No. 3, the situation in which the two conditional standard


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Figure 4d.

Figure 4. Unconditional statistics as function of probability of occurrence of precipitation 7c using the parameter values of Richardson's model fit to Denver daily maximum temperature in January (solid curves) and July (dashed curves): (a) mean #x; (b) standard deviation crx; (c) first-order autocorrelation px (1); (d) contemporaneous cross correlation pJx (0).

deviations are both decreased by about 4.1% (not 3.5%) (i.e., o-~ = 5.78, cr~ = 5.87 ~ C) to counteract the effect of increasing 7r on the unconditional standard deviation ~rx (still changing /t0 and #1 as in Scenario No. 2). Now the values of both the mean and standard deviat ion, / tx and ~rx, are preserved, but the first-order autocorrelation Px (1) and contemporaneous cross correlation pjx(O) have been driven slightly further away from their values for the present climate. Finally, Scenario No. 4 involves increasing the conditional autocorrelation ~b by about 5.6% (i.e., ~b* = 0.700) to counteract the combined effects of increasing 7r and of decreasing o0 and or1 on the unconditional first-order autocorrelation px(1) (still changing the other parameters as in Scenario No. 3). Now the values of all of the unconditional statistics that appear in Table III, with the exception of the contemporaneous cross correlation PJx (0), have been preserved [it would be impossible to preserve the value of PJX (0) as well].

If the same sort of exercise were applied to July mean temperature instead, then these effects of unconditional statistics would be less substantial because of the smaller shift in conditional distributions (see Fig. 1). Starting with the

252 RICHARD W. KATZ

Table III Unconditional statistics for hypothetical adjustments in parameters of Richardson's model fit to Denver daily mean temperature in January

Climate change scenario Mean Standard Auto- Cross /zx (~ deviation correlation correlation

~x (~ px(1) p~x(o)

Present climate a -0.91 6.44 0.611 -0.343

Scenario No. 1 (same as -1.93 6.66 0.586 -0.415 present, except 7r* = 0.355)

Scenario No. 2 (same as No. 1, -0.91 6.66 0.586 -0.415 except #~ = 1.14,/~ = -4.64 o C)

Scenario No. 3 (same as No. 2, -0.91 6.44 0.581 --0.430 except ~r~ = 5.78, cry' = 5.87 ~

Scenario No. 4 (same as No. 3, -0.91 6.44 0.611 -0.430 except 4" = 0.700)

a Parameter values for present climate are 7r = 0.178, d = 0.219, #0 ----- 0.12 ~ /sl = -5.66 ~ a0 = 6.03 ~ al = 6.12 ~ ~b = 0.663.

adjustments as in Scenario No. 4, it would be straightforward to produce more complex scenarios, such as for a climate that is not only wetter, but warmer as well.

5. Discussion

The unconditional statistical properties of a simplified version of Richardson's model have been described in detail, demonstrating the importance of making the distinction between conditional and unconditional statistics. It would be straight-

forward, but tedious, to extend the approach to more complete versions of this model, such as the simultaneous modeling of daily minimum and maximum tem-

perature conditional on precipitation occurrence. In particular, the same approach outlined in the Appendix could be utilized to obtain an analogous expression for the unconditional cross correlation function between minimum and maximum daily temperature. Among other things, it can be established that if the probability of precipitation occurrence were modified, then changes would be induced in the unconditional cross correlation function between minimum and maximum temperature as well.

Another research issue relates to the observation that Richardson's model fails to reproduce the first-order autocorrelation of daily temperature (see Section 3). Future research will deal with the problem of how to generalize the model to remove this deficiency. One approach would be to condition the distribution of temperature


on the occurrence of precipitation, not only on the present day, but on the previ- ous day as well. Past research by Bruhn et aL (1980) and Feyerherm and Bark (1973) does provide evidence of such a dependence. Of course, generalizations like this one would further complicate the issue of how best to employ these models to generate scenarios of climate change. A related issue concerns the fact that stochastic models for daily weather variables tend to underestimate the observed variance of monthly mean temperature (Hansen and Driscoll, 1977; Young, 1994) or monthly total precipitation (Gregory et al., 1993; Katz and Parlange, 1993). How much of this underestimation is attributable to being an inadequate model for daily weather (e.g., a higher-order Markov chain for precipitation occurrence or a higher-order autoregressive process for randomly standardized temperature being a better model) and how much reflects the fact that low frequency variation is not taken into account by such models is an open question. Finally, a challenging statistical problem is the development of stochastic models that simulate weather variables simultaneously over both space and time (Hutchinson, 1995).

Ackowledgements

The comments of four anonymous reviewers are gratefully acknowledged, and the encouragement of Linda Meatus and Dan Wilks is also appreciated. This research was partially supported by the Geophysical Statistics Project of the National Center for Atmospheric Research (NCAR) through NSF Grant DMS-9312686. NCAR is sponsored by the National Science Foundation.

Appendix

Derivation of Unconditional Statistics

To derive the relationships presented in Section 3, some general properties of conditional moments are employed. Let U, V, and W denote arbitrary random variables. A well known property of the conditional expected value is:

E(u) = E[E(U I V)]. (A1)

The corresponding property for the conditional variance (e.g., Lindgren, 1968, p. 118) is:

var(u) = E[var( ; I v)] + var[E(u I v)]. (A2)

The generalization of (A2) to the conditional covariance is straightforward, but not often explicitly mentioned in the statistics literature:

cov(U, W) : E[cov(U, W ] V)] + cov[E(U [ V), E ( W I V)]. (A3)

25 4 RICHARD W. KATZ

In (A 1)-(A3), the conditioning variable V may actually be a set of random variables (e.g., the past history of a time series).

Expression (9) for the unconditional mean follows directly from (A1), through conditioning on Jr. Expression (10) for the unconditional variance follows from (A2), through conditioning on Jt and then deriving that

E[va r (x , I J,)] = (1 - ~),~o ~ + ~ , var[E(Xt [ Jt)] = 7 r ( 1 - 7 r ) ( t z 1 - l z o ) 2 .

(A4)

Expression (11) for the unconditional autocorrelation function follows from (A3), through conditioning on Jt-t and Jt and then deriving that

E[cov(X,_,, x , [ J,-, , J,)] = ~ ' [ (1 - ~),~o 2 + ~

cov[E(Xt_l I Jr-t, Jr), E(Xt I Jr-t, Jr)] = dZTr( 1 - 7 r ) ( # l - # 0 ) 2.

(A5)

Finally, expression (13) for the cross correlation function could also be derived from (A3), but follows most directly from utilization of (A1) in the form of

E(JtXt+t) = E[E(JtXt+t l Jt, dt+t)], (A6)

and then deriving an expression for the right-hand side of (A6).

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(Received 9 May 1995; in revised form 31 October 1995)

use of conditional stochastic models to generate climate change scenarios

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