wilks 1992 - adapting atochastic weather generation algorithms for climate change studies

A D A P T I N G S T O C H A S T I C W E A T H E R G E N E R A T I O N

A L G O R I T H M S F O R C L I M A T E C H A N G E S T U D I E S

D A N I E L S. W I L K S

Department of Soil, Crop and Atmospheric Sciences, Cornell University, Ithaca, N Y 14853, U.S.A.

Abstract. While large-scale climate models (GCMs) are in principle the most appropriate tools for predicting climate changes, at present little confidence can be placed in the details of their projections. Use of tools such as crop simulation models for investigation of potential impacts of climatic change requires daily data pertaining to small spatial scales, not the monthly-averaged and large-scale information typically available from the GCMs. A method is presented to adapt stochastic weather generation models, describing daily weather variations in the present-day climate at particular locations, to generate synthetic daily time series consistent with assumed future climates. These assumed climates are specified in terms of the commonly available monthly means and variances of temperature and precipitation, including time-dependent (so-called 'transient') climate changes. Unlike the usual practice of applying assumed changes in mean values to historically observed data, simulation of meteorological time series also exhibiting changes in variability is possible. Considerable freedom in climate change 'scenario' construction is allowed. The results are suitable for investigating agricultural and other impacts of a variety of hypothetical climate changes specified in terms of monthly-averaged statistics.

1. Introduction

Concern about potential climate changes resulting from anthropogenic emissions of 'greenhouse' gases has lead, among other things, to efforts to estimate consequences of these changes on a variety of natural and human-mediated systems (e.g. LLNL et al., 1990; Riebsame, 1989). Typically the preferred source of climate projections for impact studies is one or more atmospheric general circulation model (GCMs). This choice stems from the fact that, at least in principle, GCMs can provide a complete (including simulation of geographical and seasonal distributions of changes for a variety of variables) and internally consistent view of future climate changes (Gates, 1985). While current GCMs perform reasonably in simulating the present climate with respect to annual or seasonal averages over large (perhaps continental) geographical areas, they are considerably less reliable in portraying the smaller-scale features relevant to most impact questions (Grotch and MacCracken, 1991; Schlesinger and Mitchell, 1987). This is particularly the case for precipitation, to which many sectors of interest are especially sensitive.

Accordingly, little faith can be placed at present in the fine-scale details of GCM projections, and the impact analyst is reduced to the consideration of a spectrum of 'scenarios' of climate change, Typically these are constructed using observed records of temperature and precipitation, adjusted to reflect climate changes

Climatic Change 22: 67-84, 1992. �9 1992 Kluwer Academic Publishers. Printed in the Netherlands.

68 Daniel S. Wilks

derived from monthly averages of GCM results. Usually the adjustments are addi- tive for temperature and multiplicative for precipitation, and may vary from month to month in order to reflect seasonality in the assumed changes (e.g. Cohen, 1990). Two parallel time series of daily meteorological variables, the 'control' (observed instrumental record) and 'changed climate', are then available to drive a model or suite of models of a climate-sensitive process or enterprise of interest. To date the adjustments have been based on the differences between 1 x CO 2 and 2 x CO 2 GCM integrations, although in principle the procedure could be implemented using one or more of the recent time-dependent warming integrations (Hansen et

al., 1988; Manabe et al., 1991; Washington and Meehl, 1989). A variety of practical needs of the impact analyst are satisfied by this straight-

forward procedure. First, many response models (e.g. 'physiological' crop models) require daily, rather than monthly-averaged meteorological information in order to capture adequately processes important to potential impacts. It is possible using the procedure described above to produce such daily time series using the monthly-averaged information typically available from GCM integrations. In addition, the use of observed time series provides at least an accurate baseline climate, from which relative changes portrayed by the GCM(s) employed can be constructed.

This now traditional procedure does suffer some important limitations, however. The first derives from the usual practice of adopting specific relative changes at particular gridpoints from particular pairs of GCM runs. It is commonly agreed that GCM results are unreliable at this, the limit of their horizontal resolution (e.g. Grotch and MacCracken, 1991; Wilson and Mitchell, 1987). Furthermore, since it is unlikely that results from any of the available GCM integrations are correct, there seems little reason to limit attention only to scenarios based literally on the few which do exist.

More troubling, however, is the fact that the traditional procedure is incapable of depicting changes in climatic variability. Changes in various aspects of climatic variability could have important consequences (Katz and Brown, 1992; Mearns et

al., 1984; Neild et al., 1979; Parry and Carter, 1985). Indeed, it is the capacity to respond to daily weather variations that motivates, for example, use of the more detailed crop models in climate impact studies. Simply adjusting observed time series using scalar constants insures that both the nature and degree of variability exhibited by the historical record will be reflected in any scenario constructed, regardless of whether changes in variability are exhibited by the GCM integration(s) from which the adjustment parameters are taken.

A related problem pertains to construction of the time-dependent climate change scenarios which should become the basis for the next generation of impact studies. Time horizons on the order of a century will be appropriate for these, and instrumental records of this length do not exist for most locations. Basing scenarios on observed data severely limits the number of realizations of the modeled impacts which can be simulated for, say, a particular decade in the future. Re-use of

Climatic Change September 1992

Stochastic Weather Generation Algorithms for Climate Change Studies 69

segments of the observed record are likely to lead to impact variability being underestimated.

Presented here is an alternative to the conventional procedure, which addresses the shortcomings just outlined. It employs a time-domain stochastic model representing daily weather variations at a location. Characteristics of the present-day climate are represented in terms of the parameters of the daily stochastic process. Changed climate scenarios are produced by adjusting these parameters in a manner consistent with imposed changes in monthly statistics, using the statistical properties of the sampling distributions of the appropriate sums or averages of the daily values. The model is then used in a Monte-Carlo sense (as a 'synthetic weather generator') to produce streams of daily weather values used to drive response models. Section 2 reviews the structure of an appropriate stochastic weather model, Section 3 presents the procedure for parameter adjustment, and Section 4 illustrates this procedure for a small subset of the possible climate changes which could be assumed.

2. Structure of the Stochast ic Weather Generator

A variety of stochastic weather generation algorithms has been developed (see Hutchinson (1986) for a review), and the general procedures outlined in the following sections could be employed with most of them. The model employed here is that of Richardson (1981). The daily variables simulated are precipitation occurrence and amount, maximum temperature, minimum temperature, and solar radiation. The choice of this set is motivated by their common use in many crop response models.

2.l. Precipitation Component

Precipitation occurrence is described by a two-state, first order Markov chain. That is, precipitation either occurs or it does not (the two states), and the conditional probability of precipitation occurrence depends only on whether precipitation occurred on the previous day. There are thus two parameters describing the precipitation occurrence process: the transition probabilities P01, the probability of a wet day following a dry day, and P l l, the probability of a wet day following a wet day. Representing the occurrence process in terms of the transition probabilities is convenient for Monte-Carlo simulation, although an equivalent alternative is to use the two parameters

P01 , ( 1 )

1 +P01 - P l l

the (unconditional) probability of a wet day, and the 'dependence parameter'

d = p l l -pro, (2)


70 Daniel S. Wilks

which indexes the strength of the persistence (Katz, 1983; 1985). The precipitation occurrence parameters for a given location are allowed to vary through an annual cycle, which is usually accomplished by defining separate parameter pairs for each of the twelve calendar months.

Variation of precipitation amounts on wet days is characterized using the gamma distribution. The probability density function for the gamma distribution is

(r/fl) ~ - I exp(-r//3) fr= (3)

where the distribution parameters are a (the shape parameter) and /3 (the scale parameter), r is the daily precipitation amount, and F( . ) denotes the gamma function. In terms of the two distribution parameters, the mean precipitation amount (considering only wet days) is #r = a/3, and the corresponding variance is a~ = a/32. Again, separate pairs of precipitation amount parameters are defined for each month, so that the precipitation part of the algorithm is characterized by 4 parameters for each month.

2.2. Temperature and Radiation Components

The maximum temperature, minimum temperature, and surface solar radiation components are represented as a first-order trivariate autoregression,

2(t) = [A l2 ( t - 1)+ [Ble(t ). (4)

The vector x consists of three elements, xk; with Xa, x2, and x 3 denoting maximum temperature, minimum temperature, and solar radiation, respectively. The random forcing vector e consists of three independent standard normal (Gaussian) variates. The tildes denote standardization by subtraction of means and division by standard deviations, i.e.,

2k(t) = xk(t) - #kj(t) ; k = 1, 2, 3; j = 0, 1; (5) m:(t)

so that each of the three standardized time series, x~(t) will have zero mean and unit variance. Here the means #kj(t) and standard deviations ~rkj(t ) are defined separately for the three weather variables (index k), and for wet and dry days (index j) as simulated in the precipitation part of the algorithm. The six means and six standard deviations characterizing a particular location will exhibit annual cycles, which will be most conveniently modeled using single Fourier harmonics.

The (3 by 3) matrices [A] and [B] in (4) are obtained using the matrices of lag-0 and lag-1 correlations among the three elements of x (e.g. Matalas, 1967), and are assumed to be equal for wet and dry days. (Note that this convenient assumption may not necessarily be satisfied.) Elements of [A] are the primary means of representing time dependence in (4), while [B] serves mainly to produce appropriate



simultaneous correlations among the three simulated variables. Actually, both matrices depend on both lagged and unlagged correlations among all three variables, and result from the multivariate generalization of the Yule-Walker recur- sion (e.g. Kendall and Ord, 1990). While a common implementation of this algorithm treats [A] and [B] as being constant in time and equal for all locations in the conterminous U.S. (Richardson and Wright, 1984), this assumption is dubious, particularly for those elements depending strongly on the unlagged correlations between solar radiation and the temperature variables.

2.3. Implementation

Generation of synthetic sequences of daily weather variables using this above model is straightforward. Knowing whether precipitation occurred O n the previous simulated day, the appropriate transition probability, P01 or PH is compared to a newly generated uniform [0, 1] random number. A wet day is simulated if the random number is less than the transition probability. If this is the case, a random precipitation amount is generated for the current day using the appropriate gamma distribution. The previous day's temperature and radiation variables, xk(t, 1), are standardized using (5), with means and standard deviations appropriate to the time of year and the precipitation status of the previous day. Three independent standard normal variates are then generated as the elements of e (t), and (4) is used to calculate standardized values 2(t) for the current day's temperature and radiation. Finally, these values are transformed to their dimensional values by inverting (5).

3. Parameter Adjustment Procedure

Information pertaining to climate changes (e.g. from GCM integrations) will often be available in terms of monthly, not daily, values. In particular, monthly means and the variances of those means (reflecting interannual variability) will commonly constitute the basic information. Scenarios for changed climates will be constructed on the basis of changes in these statistics between 'control' and changed (or possibly gradually changing) GCM climates.

Daily time series generated using the procedure described in Section 2 are of course random variables. Their sums (or, equivalently, averages) will therefore exhibit sampling variability, which can be described by probability distributions known as sampling distributions. These sampling distributions in turn exhibit statistical characteristics which will depend on the structure and parameters of the daily generation process. This link between the statistics (means and variances) of monthly quantities, and the parameters of the daily processes, can be used to adjust the daily parameters describing the present-day climate at a location in a manner consistent with imposed changes in monthly statistics.


72 Daniel S. Wilks

3.1. Precipitation Component

Define S N as the sum of N daily precipitation amounts generated as described in Section 2a. In terms of (1)-(3), the mean and variance of this quantity can be written as

jb~(SN) = No, aft,

and, for large N,

[ 1+ 1 o2(s N) = N~a/32 1 + a(1 - 0,) 7 2 ~ j ,

respectively (Katz, 1983; 1985). Clearly these correspond to the average monthly precipitation and its variance, for a month comprised of N days.

Considering the ratios of the above quantities in a changed climate to those characterizing the baseline climate, and denoting those symbols pertaining to the changed climate with primes, yields

F,(s;,) _ ~ '~ '~ '

~ (s~) ~ar '

and

o2(si~)

o~(S~)

(6)

l+d ,J

[/1 + ~(1 - ~) 1 + d Lq 7ga/~ 2 [ 1 - d j

(7)

All the unprimed quantities on the right-hand sides of (6) and (7) correspond to characteristics of the present-day climate, and are known. The ratios pertaining to the monthly statistics, on the left-hand sides, constitute the basic information describing the change in the precipitation climate for the month in question.

The precipitation part of the changed climate scenario will be determined by the four parameters a', fl', ~', and d' (with the latter two implying values for p< and P]I). Two additional constraints are required to determine these from (6) and (7). The nature of these constraints will depend on other information which may be available or the objectives of a particular climate impact study, and may be varied as part of sensitivity analyses. In general all four of the parameters may change, but some constraints may result in one or two of them remaining constant. Both the ratios of the monthly statistics and the nature of the constraints may change through time for impact studies treating time-dependent climate changes. Some examples are considered in Section 4.

The conceptual framework is illustrated schematically in Figure 1. Observed daily data both define the nature of the observed monthly statistics (means and


Stochastic Weather Generation Algorithms for Climate Change Studies 7 3

f Stochastic Model f~or Present Climate J

..~ Stochastic Model for /lChanged/Changing Climate

Synthetic Weather Sequences for Present Climate

Synthetic Weather Sequences for

Changed/Changing Climate

Fig. 1. Schematic conceptual framework of the procedure. The stochastic model for daily weather variations in a changed/changing climate is developed by modifying the corresponding model describing observed daily data, as constrained by the relationship between the daily statistics and the means and variances of the monthly values. The resulting stochastic models can be used to generate sequences of synthetic daily values of arbitrary length.

variances), and are used to fit the parameters of the stochastic model characterizing daily weather variations in the present climate. These observed monthly statistics and 'control' daily stochastic model parameters are used with the projected/ assumed changes in monthly statistics and scenario-specific constraints, to define the parameters (primed quantities in (6) and (7)) of the daily stochastic model representing the changed climate. The two daily stochastic models (control and changed/changing climates) can then be used to simulate sequences of synthetic daily weather values of arbitrary length.

3.2. Temperature Components

An analogous approach can be taken for the variables modeled with the multivariate autoregression (4). Only changes for the temperature variables will be treated here. The procedure could also be applied to the surface radiation component if information regarding its behavior in a changed climate were available. However, given that the primary determinants of surface solar radiation are astro-


74 Daniel S. Wilks

nomical (changing only very slowly) or pertain to clouds (the distribution and properties of which in current GCMs are of questionable accuracy, e.g. Schlesinger and Mitchell, 1987), impact analysts will probably want to assume no changes in the statistics pertaining to this quantity. Note, however, that since the means and standard deviations in (5) are defined separately for wet and dry days, changes in the patterns of precipitation occurrence will produce consistent changes in the synthetic radiation series.

Of primary interest will be treatment of changes in temperature means. Changes in the mean of observations (i.e. daily temperature) making up a time average (i.e. average monthly temperature) will be reflected directly in the expected value of the time average regardless of the presence of dependence in the underlying observations (Katz, 1982; 1985). That is,

1 T(t) ~ ~ [~max(t) -b ~min(t)],

where the conventional representation of the mean temperature as the average of maxima and minima has been adopted. It will in general be desirable to specify annual cycles of temperature change, separately for the maxima and minima. The assumption of equal changes in the two means may be unavoidable in some appli- cations however, for example when working from GCM results where only a single temperature value has been saved. For simplicity, the same changes will be applied here to means for wet and dry days in (5). Note that, because separate mean temperatures are defined for wet and dry days, some changes in average temperature will result if changes in rainfall frequency (i.e. Jr' r Jr) are specified. In practice these additional changes are generally small unless very extreme changes in the precipitation climatology are assumed.

A convenient approach in the present context is to define separate annual Fourier harmonics for the changes in the maximum and minimum temperature components,

1 - = {Al'ma (t) +

2 CX~ + C X 1 cos ~

+ CN~ + C N 1 cos 365 '

where C X o and C N o are the annual average changes in maximum and minimum temperatures, respectively, and CX~ and CNI are the corresponding amplitudes of the annual harmonics. The phase angle ~ is written here as the julian date, t, of the greatest temperature increase, assumed to be the same day for both maximum and minimum temperature changes. Here :~ = 3.14 .... As it is a general consensus that



warming would be greater in winter than summer (Grotch and MacCracken, 1991; Schlesinger and Mitchell, 1987), it will be assumed in the following that r = 21 days (i.e. maximum warming on 21 January) for both maximum and minimum temperatures. The resulting mid-winter maximum warming is consistent with the above- mentioned consensus for the northern hemisphere extratropics. Other choices for will be appropriate for other locations, particularly in the southern hemisphere.

Clearly, specification of an annual average temperature increase at a location provides only one constraint on the remaining four parameters in (8), i.e.

1 AT = ~ (CXo + CUo), (9)

where the lack of time dependence on the left-hand side denotes the annual average change. As before, three other constrants on the average temperature changes are necessary to determine the other parameters in (8). While alternative choices are possible, it will be convenient here to use the change in temperature range between summer and winter,

A [ ]e (JJA) - T(DJF)] = -0.8 9 5 (CX~ + CN, ), (lO)

the change in average diurnal temperature range (DR = tAmax - - / / r a i n ) in winter,

ADR(DJF) = C X o - C N o + 0.895(CX~ - CN~), (11)

and the corresponding change for summer

ADR(JJA) = C X o - C N o - 0 . 8 9 5 ( C X 1 - CN~). (12)

Here the constant 0.895 arises in the average of the cosine functions (8) over each of the three-month seasons.

The variance of a time average (here the interannual variance of average monthly temperature) of correlated variables (daily temperatures) depends on the variance of the underlying daily temperature values, 02, the number of days averaged N, and the autocorrelation function of the daily values, Pk (Katz, 1985):

a? Var(7 ~)-~ ~ l + 2 ~ , p k , (13)

k=I

which again is a large-sample approximation. Thus changes in the interannual variability of monthly temperatures can imply changes in the variance of daily temperatures and/or the structure of the autocorrelation (persistence) exhibited by daily temperatures. The behavior of this latter aspect of daily temperature variability in GCM integrations has been investigated only to a limited extent, but without consistent changes being noted between control and doubled CO 2 (Mearns et al., 1990; Rind et

al., 1989; Wilks, 1986) or quadrupled CO 2 (Wilson and Mitchell, 1987) integrations.


76 Daniel S. Wilks

Accordingly the simplifying assumption will be adopted here that the nature of daily temperature autocorrelations remains the same in the changing climates. This implies that the matrices [A] and [B] in (4) will not be adjusted in the stochastic model. This corresponds physically to changes in temperature variability being controlled by changes in airmass characteristics only, with the nature of the timing of alrmass transitions at a location remaining unchanged. This assumption leads to

[ Var(lP')] 1/2= a~ (14) Var(T) a a '

so that changes in interannual temperature variability are controlled only by changes in the variance of the daily temperature series. Note that (13) takes on a simpler form if a first-order autoregressive process is assumed for the daily temperature (Katz, 1985), which relationship would allow a relatively straightforward adjustment involving autocorrelation changes to be implemented if warranted by future climate model results.

Separate standard deviations for maximum and minimum temperature are required in (5). Again invoking the conventional assumption that daily temperature is the average of the maximum and minimum,

1 2 + 2pmax min0.max0.min), (15) 0"2 ~ 4 (0"2ax "[- 0.min

where Prnax-min is the unlagged correlation between daily maximum and minimum temperature. Again barring evidence to the contrary, the simplest approach regarding changes in maximum and minimum temperature variances consistent with (14) and (15) is to assume the equal proportional changes a'a/a d = 0.max/0.max = 0.min/0"mi,, for the temperature standard deviations in (5), for both wet and dry days. Alternatives, consistent with (15) and explicitly using Pmax-min (knowledge of which is also required for calculation of [A] and [B]), could also be employed. As before, an annual cycle of changed temperature variability can be imposed, for example by evaluating (14) on a month-by-month basis, or by specifying (14) in terms of the smoothly varying average temperature change (8).

4. Some Possible Changed Climate Scenarios

In this section the procedure presented so far will be illustrated by constructing example climate change scenarios. That is, adjustments to parameters of the Richardson (1981) daily stochastic weather generation model will be specified in terms of changes in monthly and annual statistics derived primarily from GCM results. Time-dependent, or 'transient', changes will be specified to emphasize that the procedure is not limited to supporting 'equilibrium doubling' studies. Rather than assume accuracy of particular results at specific gridpoints and times of the year, information from larger space and time scales, and indeed from combined results of different GCMs, will be used. This approach is as reasonable as any, given



the relatively low confidence which can currently be placed in GCMs at subconti- nental resolution. The assumptions assembled here are meant to be illustrative, but could easily be used in impact studies as the base around which to construct sensitivity analyses. It should be emphasized that the suggestions in this section are a very small subset of the possible scenarios which could be constructed.

4.1. Temperature

The most basic, and perhaps the most reliable, statistic to be extracted from increased-CO 2 GCM integrations is the annual- and global-averge temperature change, ATglobal: TWO reasonable choices for the time course of global warming over the next century are given by results from 'Scenario A' and 'Scenario B' of Hansen et al. (1988), obtained with the Goddard Institute for Space Studies (GISS) GCM. The global temperature changes for these two scenarios can be represented approximately, at least through the year 2060, as

I 0.229 + 0.0185y + 0.00039y 2 ('Scenario A') A Tglobal( y) (1 6)

l 0.192 + 0.0076y + 0.00028y 2 ('Scenario B'),

where y = (year-1980). Other functions could also be used, based for example on particular desired trajectories of greenhouse gas emissions (Rotmans et al., 1990).

The consensus of GCM results is that enhanced greenhouse warming would be greater at higher than lower latitudes, and more pronounced in winter than summer. This result has been found both in integrations for equilibrium CO 2 doubling (Grotch and MacCracken, 1991; Schlesinger and Mitchell, 1987) and gradually increasing greenhouse forcing (Hansen et al., 1988; Manabe et al., 1991; Washington and Meehl, 1989). The seasonal pattern has also been noted in the observed warming in the recent instrumental record (Angell, 1986). One attractive approach to incorporating the geographic and intra-annual dependence of the warming is through use of the standardized average temperature changes for winter and summer, Alp *(DJF) and AlP *(JJA), respectively (Santer et al., 1990). These statistics express the average, over 5 GCMs, of the ratio of the temperature increase in the specified season at a location to the 2 x CO 2 sensitivity of the respective climate model. For example, figures in Santer et al. (1990) show the standardized temperature changes for central New York state being approximately 1.35 for winter and 1.10 for summer. That is, in the consensus of 5 GCMs, the equilibrium doubling climate at this location warms 35% more in winter and 10% more in summer than the annual and global average temperature. Much larger values for these parameters are found in high latitudes in winter, and values less than unity occur in the tropics throughout the year. These standardized average temperature changes fix the values for

Alp = A Tglobal(Y ) [Alp* (DJF) + A lp* (JJA)] 2

in (9), and


78 Daniel S. Wilks

A [T(J JA) - T(DJF)] = A Tgloba I (y) IT* (JJA) - A T* (DJF)]

in (10). Results from some doubled CO2 GCM integrations have indicated decreases in

the diurnal range (i.e. minimum temperatures increasing more than maximum temperatures) particularly in summer (Rind et al., 1989). Such changes have also been detected in the instrumental records of stations in the U.S. and Canada (Karl et al.,

1984). Here it will be assumed that the diurnal range decreases primarily in summer, with little decrease in winter. The change is varied linearly through time such that the summer diurnal range decreases by 0.7 ~ when the global temperature change (16) equals the equilibrium 2 x CO 2 sensitivity, AT0oba ~ (2 x CO2) , in this case equal to 4.2 ~ for the GISS GCM. This yields

A D R ( D J V ) = 0

for (11) and

A Tgl~ 1 ADR(JJA) = -0 .7 ~ ATg~oba,(2 x CO2) '

for (12). Thus the linear system defined by (9)-(12) can be solved simultaneously, as a function of the year during the warming (16), for the four temperature average parameters in (8). For the case of 'Scenario A', the year 2060, and the winter and summer standardized temperature changes cited above for central New York, this yields C X o = 4.97, C N o = 5.32, CX1 = 0.78, and CN~ = 0.39 ~

Decreases in dally temperature variability have been noted in association with greenhouse-gas induced temperature increases in most GCM results where this has been investigated (Rind et aL, 1989; Wilks, 1986; Wilson and Mitchell, 1987). Again, it seems simplest to linearly tie decreases in temperature variability to overall temperature changes. Failing contrary information concerning separate adjustments to the daily variances for maximum and minimum temperature, it is reasonable to assume that these are equal, i.e.

L [ g,oua,(zxc ) ATg,oU~,(y) ] Od-- O'max-- O'min 1 -0 .1 A T- - - ;~ -- =O 2. Q

(7 d O'ma x O'mi n

Here the 10% decrease, at the time of global warming equal to the equilibrium doubling sensitivity, is a plausible but arbitrary choice.

4.2. Prec ip i ta t ion

Although the consensus of GCM studies is that the global hydrologic cycle would intensify in a greenhouse warming (e.g. Grotch and MacCracken, 1991), modeled local precipitation changes are highly variable, and indeed include both increases and decreases. Considering the relatively poor representation of small-scale



features of the observed precipitation climatology in control GCM integrations, it would be unwise to put much faith in any of the available local projections. In the face of this uncertainty, it should be the precipitation component of impact studies which receives the most attention in the construction of alternative scenarios. That is, exploration of response sensitivities to different aspects of precipitation changes should be explored more exhaustively than those for temperature, as the latter element is better portrayed in the climate models.

Here the method will be illustrated assuming a 10% increase in mean monthly precipitation, again linearly coupled to the global temperature increase (16), so that

/x (S;v) 1.1 (17) #(aN ) AZglobal(2 X CO2)

would be substituted into (6). Interannual variability of precipitation has been reported to change in the same direction as the mean in GCM studies (Rind et al., 1989), and consistent results (across different locations, not through time) are evident in observed data (Waggoner, 1989). Here the result from Waggoner (1989) will be used to impose the condition

L (18)

in (7). Recall that two additional constraints are required to determine the four param-

eters of the precipitation process. One of the simplest courses is to assume no changes in the nature of the precipitation occurrence process (i.e. d ' /d = zr'/~ = 1) so that the only changes are in the gamma distribution parameters. Again for the year 2060 in 'Scenario A', these assumptions together with (17) and (18) lead to

f i ' - l . 0 3 - 0 . 0 7 a ( 1 - s r ) ~

and

a ' _ 1.1__fi " a fl'

The resulting parameter changes for Ithaca, New York, are given in the second column of Table I.

Alternatives involving changes in the precipitation occurrence parameters result in relatively complicated general expressions, but solutions for particular cases are straightforward to evaluate. For example, increasing (on average) the lengths of runs of wet and dry days by imposing d' = 1.5d while preserving the climatological probability for the number of wet days (~' = ~), again phased in as a linear function of the warming, leads to results for Ithaca given in the third column of Table I. The means (aft) of the gamma distributions for the two changed climate scenarios are the same except for rounding error, but the variance (aft 2) in the d' = d case is


8 0 Daniel S. Wilks

TABLE I: July daily precipitation parameters for Ithaca, New York, characterizing observed data, and reflecting two climate change adjustments. Both adjustments are based on a 10% increase in the monthly mean, and an increase in interannual variability specified by (18)

Parameter Observed Adjusted

d'/d = 1.0 d'M = 1.5

a 0.715 0.80 0.92 fl (ram) 10.6 10.4 9.0

PIll 0.284 0.284 0.257 P l I 0.445 0.445 0.498

larger, since no contribution to the increased interannual variability is made by the parameters of the occurrence process in that case.

Figure 2 illustrates the combined results for all of the variables simulated. Shown are three realizations of July weather for Ithaca, generated using the example parameters just derived. Open bars show synthetic data generated using parameters fit to the observed record. The synthetic time series shown in black result from modifications consistent with 'Scenario A' in 2060, with d' = d, while hatched symbols show the same case except for increased persistence of precipitation occurrence given by d' = 1.5d. The three realizations have been constructed using the same underlying random number streams, so that the day-by-day results may be compared.

The most striking differences are of course in the temperatures. The changed climates are distinctly warmer although, because of the reduced temperature variability, the individual daily temperature differences are not equal. While the daily temperature variability decreases, the frequency of extreme warm events increases, as would be expected (Mearns et aL, 1984), unless the mean is increasing slowly in relation to the decrease in variability. For example, the number of days with temperatures of at least 35 ~ increases from one to four, and three of these occur con- secutively.

The three precipitation series are similar, except for the 21st and 29th of the month. For this realization, extending average lengths of wet and dry spells with d' = 1.5 d has resulted in two fewer wet days which, coincidently, were simulated to be major rainfalls. Increased numbers of rain days would of course be simulated in other realizations for this case, since Jr' = vr has been used. Temperature differences between the two changed climate scenarios occur only for those two days when there are differences in the precipitation occurrence. As might be expected, maximum temperatures are higher and minimum temperatures are lower in the d' = 1.5d case, for which these two days are dry. Similarly, since the radiation



40 "o

30

g

}--

= 20

y -

lO

3O

c-J

E -3 y-

20

o =

10 rv

o U3

0

30

1980s I 2060s, d'= d

2060s, d'= I .S d _I

,jllllllll j!llJllljjjllll!lilll 1 3 5 7 9 II 13 15 17 19 21 23 25 27 29 31

I 3 5 7 9 II 13 15 17 19 21 23 25 27 29 31

E v 20

U I 0 P El-

1 3 5 7 9 II 13 15 17 19 21 23 25 27

Day of the Month

2 9 ' 3'I

Fig. 2. Realizations of simulated July weather for Ithaca, New York. Open symbols, current climate; solid symbols, changed climate with unchanged persistence of precipitation occurrence; hatched symbols, changed climate with increased persistence of precipitation occurrence. The realizations result from identical underlying random number streams, so that day-to-day comparisons are appropriate.

parameters do not change, the three solar radiation series are identical except on these same two days, for which the radiation increases in the (dry) d ' = 1.5d case.

5. Summary and Conclusions

While GCMs are the best available tools for estimating the nature of possible future climate changes, they are far from ideal for investigating impacts of those changes.


82 Daniel S. Wilks

Their deficiencies for this purpose include inaccuracies at the smaller time and space scales of importance to many potential impacts, as well as the limited availability of GCM information. This limited availability derives both from the compu- tational expense involved, constraining numbers of climate change experiments and the lengths of their output time series; and also from the abbreviated nature of the relevant surface information saved, typically monthly means of temperature and precipitation.

This paper has advocated the construction of climate change scenarios for use in impact studies using stochastic weather generation algorithms modified in light of the information available from GCM integrations. The procedure is entirely analogous to the conventional practice of simply applying changes in the first- moment quantities to observed data. The fundamental difference is that changes to elements of a richer parameter set are allowed. A wide variety of adjustments are possible, which allows a wide range of sensitivity studies to be easily accessible.

Monthly values typically saved from GCMs are used as the basis of the adjustment procedure. This is accomplished by exploiting the link between daily and monthly-averaged data, provided by statistical results pertaining to the sampling distributions of the sums or averages of daily values. The chosen monthly statistics can be drawn from a particular gridpoint of a particular GCM, reflect wider geographic features of a changed climate integration, or reflect 'consensus' changes among GCMs, at the discretion of the impact analyst. The issue of making the transition between the GCM grid and local scales (Karl et aL, 1990; Kim et aL, 1984; Wilks, 1989) has been sidestepped, but the procedure could easily be extended to accommodate links between these scales.

Of considerable importance is the fact that changes in variability (including different aspects of variability), to whicla many potential impacts may be sensitive, can be represented. A major portion of climate impact work focuses on agricultural and other plant-dominated systems, and the procedure described here represents weather variables in a form needed by many dynamic crop growth simulation models. Thus the method allows simulation of the effects of changing means and variability on these and possibly other systems.

Finally, the approach is capable of producing output time series of arbitrary length, which is necessary for conducting Monte-Carlo impact studies. It is important to consider interannual variability of the responses of a system of interest, and multiple realizations of a given climate are necessary to evaluate this. It is hoped that the method presented here will contribute to investigation of impacts of gradual changes in means and variability, and that the procedure can be used to begin addressing the important and interesting issues relating to the nature and rates of possible adaptations of a variety of enterprises.


Stochastic Weather Generation Algorithms for Climate Change Studies 8 3

Acknowledgements

This work was supported by the USDA under projects 91-34244-5917, 58- 37~EM-0-80053, and NYC125430.

References

Angell, J. K.: 1986, 'Annual and Seasonal Global Temperature Changes in the Troposphere and Low Stratosphere, 1960-1985', Mon. Wea. Rev. 114, 1922-1930.

Cohen, S. J.: 1990, 'Bringing the Global Warming Issue Closer to Home: The Challenge of Regional Impact Studies', Bull. Amer. Meteorol. Soc. 71, 520-526.

Gates, W. L.: 1985, 'The Use of General Circulation Models in the Analysis of the Ecosystem Impacts of Climate Change; Clim. Change 7, 267-284.

Grotch, S.L. and MacCracken, M.C.: 1991, 'The Use of General Circulation Models to Predict Regional Climatic Change' J. Clim. 4, 286-303.

Hansen, J., Fung, I., Lacis, A., Rind, D., Lebedeff, S., Ruedy, R., and Russell, G.: 1988, 'Global Climate Changes as Forecast by Goddard Institute for Space Studies Three-Dimensional Model', J. Geo- phys. Res. D93, 9341-9364.

Hutchinson, M. E: 1986, 'Methods of Generation of Weather Sequences; in Bunting, A. H. (ed.), Agri- cultural Environments, C.A.B. International, Walling:fiord, 149-157.

Karl, T. R., Kukla, G., and Gavin, J.: 1984, 'Decreasing Diurnal Temperature Range in the United States and Canada from 1941 through 1980; J. Clim. Applied Meteorol. 23, 1489-1504.

Karl, T. R., Wang, W.-C., Schlesinger, M. E., Knight, R. W., and Portman, D.: 1990, 'A Method of Relating General Circulation Model Simulated Climate to the Observed Local Climate. Part I: Seasonal Statistics; J. Clim. 3, 1053-1079.

Katz, R. W.: 1982, 'Statistical Evaluation of Climate Experiments with General Circulation Models: A Parametric Time Series Modeling Approach', J. Atrnos. Sci. 39, 1446-1455.

Katz, R. W.: 1983, 'Statistical Procedures for Making Inferences about Precipitation Changes Simu- lated by an Atmospheric General Circulation Model; J. Atmos. Sci. 40, 2193-2201.

Katz, R.W.: 1985, 'Probabilistic Models', in Murphy, A.H., and Katz, R.W. (eds.), Probability, Statistics, and Decision Making in the Atmospheric Sciences, Westview, 261-288.

Katz, R. W. and Brown, B.G.: 1992, 'Extreme Events in a Changing Climate: Variability is More Important than Averages', Clim. Change, in press.

Kendall, M. and Ord, J. K.: 1990, Time Series, third edition, Edward Arnold, 296 pp. Kim, J.-W., Chang, J.-T., Baker, N. L., Wilks, D. S., and Gates, W. L., 1984, 'The Statistical Problem of

Climate Inversion: Determination of the Relationship between Local and Large-Scale Climate', Mon. Wea. Rev. 112, 2069-2077.

LLNL (Lawrence Livermore National Laboratory) et aL, 1990, Energy and Climate Change, Report of the Multi-Laboratory Climate Change Committee, Lewis, 161 pp.

Manabe, S., Stouffer, R. J., Spelman, M. J., and Bryan, K.: 1991, 'Transient Response of a Coupled Ocean-Atmosphere Model to Gradual Changes of Atmospheric CO2. Part I: Annual Mean Response; J. Clim. 4, 785-818.

Matalas, N. C.: 1967, 'Mathematical Assessment of Synthetic Hydrology; Water Resourc. Res. 3, 937- 945.

Mearns, L. O., Katz, R. W., and Schneider, S. H.: 1984, 'Extreme High-Temperature Events: Changes in Their Probabilities with Changes in Mean Temperature', J. Clim. Appl. Meteorol. 23, 1601-1613.

Mearns, L. O., Schneider, S. H., Thompson, S. L., and McDaniel, L.R.: 1990, 'Analysis of Climate Variability in General Circulation Models: Comparison with Observations and Changes in Varia- bility in 2 x COz Experiments', J. Geophys. Res. D9g, 20469-20490.

Neild, R. E., Richman, H. N., and Seeley, M. W.: 1979, 'Impacts of Different Types of Temperature Change on the Growing Season for Maize', Agricult. Meteorol. 20, 367-374.

Parry, M. L. and Carter, T. R.: 1985, 'The Effect of Climatic Variations on Agricultural Risk; Clim. Change 7, 95-110.


84 Daniel S. Wilks

Reed, D. N.: 1986, 'Simulation of Time Series of Temperature and Precipitation over Eastern England by an Atmospheric General Circulation Model', J. Climatol. 6, 233-257.

Richardson, C. W.: 1981, 'Stochastic Simulation of Daily Precipitation, Temperature, and Solar Radia- tion', Water Resourc. Res. 17, 182-190.

Richardson, C.W. and Wright, D.A.: 1984, 'WGEN: A Model for Generating Daily Weather Variables', U.S. Dept. of Agriculture, ARS-8, 83 pp.

Riebsame, W.: 1989, Assessing the Social Implications of Climate Fluctuations: A Guide to Climate Impact Studies, World Climate Impacts Programme, UNEP, Nairobi, 83 pp.

Rind, D., Goldberg, R., and Ruedy, R.: 1989, 'Change in Climate Variability in the 21st Century', Clim. Change 14, 5-37.

Rotmans, J., de Boois, H., and Swart, R.J.: 1990, ~ n Integrated Model for the Assessment of the Greenhouse Effect: The Dutch Approach', Clim. Change 16, 331-356.

Santer, B. D., Wigley, T. M. L., Schlesinger, M. E., and Mitchell, J. E B.: 1990, 'Developing Scenarios from Equilibrium GCM Results', Report No. 47, Max-Planck-Institut flit Meteorologic, Hamburg, 29 pp.

Schlesinger, M.E., and Mitchell, J. E B.: 1987, 'Climate Model Simulations of the Equilibrium Climatic Response to Increased Carbon Dioxide', Rev. Geophys. 25, 760-798.

Waggoner, P. E.: 1989, 'Anticipating the Frequency Distributions of Precipitation if Climate Change Alters its Mean', Agricult. Forest Meteorol. 47, 321-337.

Washington, W. M. and Meehl, G. A.: 1989, 'Climate Sensitivity due to Increased CO2: Experiments with a Coupled Atmosphere and Ocean General Circulation Model', Clim. Dynam. 4, 1-38.

Wilks, D. S.: 1986, 'Specification of Local Surface Weather Elements from Large-Scale General Circu- lation Model Information, with Application to Agricultural Impact Assessment', SCIL Report 86- 1, Department of Atmospheric Sciences, Oregon State University, Corvallis 97331, 233 pp.

Wilks, D.S.: 1989, 'Statistical Specification of Local Surface Weather Elements from Large-Scale Information', Theor. Appl. Clim. 40, 119-134.

Wilson, C. A. and Mitchell, J. E B.: 1987, 'Simulated Climate and CO2-Induced Climate Change over Western Europe', Clim. Change 10, 11-42.

(Received 11 February, 1991; in revised form 13 January, 1992)


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