Realizations of Daily Weather in Forecast Seasonal Climate

APRIL 00 WILKS 195 Realizations of Daily Weather in Forecast Seasonal Climate D. S. WILKS Department of Earth and Atmospheric Sciences, Cornell University, Ithaca, New York (Manuscript received 6 March 001, in final form 10 September 001) ABSTRACT Stochastic daily weather time series models ( weather generators ) are parameterized consistent with both local climate and probabilistic seasonal forecasts. Both single-station weather generators, and spatial networks of coherently operating weather generators, are considered. Only a subset of parameters for individual station models (proportion of wet days, precipitation mean parameters on wet days, and daily temperature means and standard deviations) are found to depend appreciably on the seasonal temperature and precipitation outcomes, so that extension of the single-station models to coherent multisite weather generators is straightforward. The result allows stochastic simulation of multiple daily weather series, conditional on seasonal forecasts. Example applications of spatially integrated extreme daily precipitation and snowpack water content are used to illustrate the method. 1. Introduction Recent advances in understanding the climate system have allowed successful forecasts of seasonal temperature and precipitation at lead times up to a year in advance. At least two groups currently produce operational seasonal forecasts: the Climate Prediction Center (CPC) of the U.S. National Centers for Environmental Prediction (Barnston et al. 1999), and the International Research Institute (IRI) for Climate Prediction (Mason et al. 1999). While much of their predictive ability is thought to derive from the effects of ENSO on other parts of the climate system (Barnston et al. 1994), the operational forecasts have demonstrated predictive skill during non-enso periods as well (Wilks 000a; Wilks and Godfrey 000, 00). Because the dynamics of the climate system are chaotic, seasonal forecasts are necessarily less specific than weather forecasts. In particular, the evolution of individual weather events cannot be explicitly forecast at these timescales with any credibility. Rather, the predictands in existing operational seasonal forecasts are seasonal average temperature and seasonal total precipitation. Furthermore, since the chaotic dynamics of the system preclude even approaching exact specifications of these seasonally aggregated variables, the forecasts are expressed as probability distributions rather than deterministic point values. Comparison of past seasonal forecasts with corre- Corresponding author address: Dr. D. S. Wilks, Dept. of Earth and Atmospheric Sciences, 113 Bradfield Hall, Cornell University, Ithaca, NY 14853-1901. E-mail: dsw5@cornell.edu sponding observed seasonal outcomes has demonstrated real and potentially useful information content (Wilks 000a; Wilks and Godfrey 000, 00) but the temporally aggregated nature of the forecast quantities may be difficult for some decision makers to incorporate into their operations. In particular, many models of agricultural, hydrological, and other weather- and climate-sensitive managed systems operate on a daily time step. (In part this is due to the historical availability of climatological data at this time scale.) Since the daily weather events for upcoming seasons cannot be specifically anticipated, they are best regarded as random phenomena. However, seasonally aggregated statistics are comprised of individual daily values, and the statistics of a season cannot change without a corresponding change in the statistics of the daily events comprising it. For example, the daily weather within a particularly wet subset of, say, summer seasons, must necessarily exhibit a higher probability of rain days, or greater average precipitation on rain days, or both. A simple and successful way of representing the statistics of daily weather variations is the class of time series models for surface weather data known as weather generators (Richardson 1981; Wilks and Wilby 1999). Weather generators are straightforward to fit to observed data, and the fitted parameters can be regarded as means of summarizing the surface climate of a location. These models are also easily linked to random number generation algorithms to yield stochastic realizations of daily weather series that resemble real weather data with respect to a variety of relevant statistics. Furthermore, their mathematical structure is sufficiently simple that the implied seasonal statistics (to 00 American Meteorological Society

196 JOURNAL OF HYDROMETEOROLOGY VOLUME 3 which seasonal forecasts pertain) can be computed from the parameters governing the daily stochastic weather processes (e.g., Katz 1985; Wilks 199). Briggs and Wilks (1996) proposed that subseasonal statistics consistent with a given probabilistic seasonal forecast could be estimated by resampling the observed climate record for a location according to the probabilities in that forecast. Essentially, the procedure produces climatological statistics by weighting the contributions of data from a particular year according to the probabilities in a forecast and the seasonal mean in that year, rather than weighting all years equally. One class of subseasonal statistics that can be estimated in this way (i.e., conditionally, on a seasonal forecast) is the parameter set of a stochastic weather generator. This paper describes the specification of weather generator parameters, both for single sites and spatially coherent networks, according to seasonal forecasts in the format that is currently used operationally (Barnston et al. 1999; Mason et al. 1999). Consistency of the resulting daily series with the seasonal statistics of temperature and precipitation specified by the forecasts is examined, and two simple examples of the use of the resulting synthetic daily time series are offered.. Climate forecasts Both the CPC and IRI seasonal forecasts are issued in a discrete, tercile format. That is, each forecast consists of a triplet of probabilities {p B, p N, p A } pertaining to the three events below-normal, near-normal, and above normal. The three categories are defined by the two terciles of the relevant climatological probability distribution, q 1/3 and q /3, which divide the climatological distribution into three equal parts: CL CL q 1/3 f (x) dx 1/3 and (1a) q /3 f (x) dx /3, (1b) where f CL (x) is the climatological probability density function for the climate variable x for a location and season of interest. The outcome in a given year is below normal if x q 1/3, near normal if q 1/3 x q /3, and above normal if x q /3. For seasonally averaged temperature, the Gaussian distribution is an excellent probability model on the strength of the central limit theorem (Wilks 1995); in which case q 1/3 CL 0.4307 CL and q /3 CL 0.4307 CL, where CL and CL are the mean and standard deviation, respectively, of the climatological distribution (Briggs and Wilks 1996; Wilks 000b). The choice of a parametric distribution to represent interannual variations in seasonal precipitation totals is less clear theoretically, but in practice gamma distributions appear to work well for this purpose (Briggs and Wilks 1996; Wilks 000b). Because the three categories are mutually exclusive and collectively exhaustive, the three forecast probabilities are subject to the constraint p B p N p A 1, so that each forecast is fully specified by any two of the three probabilities. Operationally the three probabilities are also subject to additional constraints. In most situations (when either p B or p A is the largest of the three probabilities) the CPC forecasts specify that p B /3 p A, so that p N 1/3. While the IRI forecasts do not explicitly adopt this constraint, usually p N 0.35 when one of the two extreme categories is most likely. Both CPC and IRI also occasionally issue forecasts in which p N is the largest probability, in which cases usually (IRI) or always (CPC) p B p A (1 p N )/. However, for both the CPC and IRI forecasts these near-normal forecasts have been found to identify differences in future seasonal outcomes that are generally no better than the unconditional climatological probabilities (Wilks 000a; Wilks and Godfrey 000, 00) and therefore should be regarded cautiously by users. Note that the development in section 4 below depends critically on the underlying seasonal forecasts being well calibrated, or reliable. That is, it is assumed that the forecasts mean what they say in the sense that the conditional relative frequencies of each of the three outcome categories corresponds well (within reasonable sampling variations) to the respective forecast probabilities. To the extent that the forecasts may be miscalibrated (Wilks 000a; Wilks and Godfrey 000, 00), adjustments need to be made before the development below can be validly applied. 3. Weather generator a. Single-site generator A simple but robust weather generator based on the ideas of Richardson (1981) will be used in the following. It consists of separate parts for daily precipitation occurrence and amount, and daily maximum and minimum temperatures, with the statistics of simulated temperature being conditional on the precipitation occurrence on a given day. The structure of the model for a single location is described in this subsection. Precipitation occurrences are modeled using a twostate (dry or wet), first-order (probabilities depend only on the previous day s precipitation occurrence) Markov chain. The two transition probabilities, p01 Pr{today is wet yesterday was dry} and (a) p11 Pr{today is wet yesterday was wet}, (b) are sufficient to define this process because the complementary probabilities p 00 1 p 01 (probability that today is dry given that yesterday was dry) and p 10 1 p 11 (probability that today is dry given that yesterday was wet). Parameterizing the Markov chain in terms of the transition probabilities is convenient for stochastic simulation, but mathematical analysis and manipulation

APRIL 00 WILKS 197 can be easier in terms of the unconditional (that is, climatological) probability of precipitation: p 01 1 p p 01 11 (3a) and the lag 1 autocorrelation of precipitation occurrence d p11 p 01. (3b) The parameter pairs in Eqs. () and (3) contain identical information, and indeed each pair is fully defined by the other. Precipitation amounts on wet days are assumed to be temporally independent and drawn from the mixed exponential distribution x 1 x f ME(x) exp exp. (4) 1 1 This distribution is a simple probability mixture of two one-parameter exponential distributions, with the mixing parameter controlling the use of the larger ( 1 ) or smaller ( ) exponential means. In addition to providing better fits (Foufoula-Georgiou and Lettenmaier 1987; Wilks 1998; Woolhiser and Roldan 198) and a better representation of precipitation extremes (Wilks 1999a) than more conventional choices such as the gamma distribution, use of this distribution also improves the spatial coherency of precipitation simulated at a network of locations (Wilks 1998). Daily temperature simulations are based on a simple Gaussian bivariate autoregression, [ ] [ ] [ ] z max(t) z max(t 1) e 1(t) [ ] [B], (5) z (t) z (t 1) e (t) min min where z max (t) and z min (t) are standard (zero mean, unit variance) Gaussian variates for maximum and minimum temperatures, respectively, on day t; e 1 (t) and e (t) are independent standard Gaussian forcing noise; and the ( ) parameter matrices [ ] and [B] are computed from the simultaneous and lagged correlations between z max and z min (for example, Richardson 1981; Wilks 1995). Dimensional temperatures are then constructed as T max(t) max,i(t)z max(t) max,i(t) and (6a) T (t) (t)z (t) (t); i 0, 1, (6b) min min,i min min,i where the additional subscript i on the standard deviations and means indicates that different parameter sets are used for dry (i 0) and wet (i 1) days. All of the daily weather generator parameters exhibit annual cycles, although because the present application relates to seasonal forecasts, only parameter variations within 3-month periods need be considered here. Within each 3-month season the eight temperature parameters max,0, max,1, min,0, min,1, max,0, max,1, min,0, and min,1 are modeled as separate quadratic functions of the date t. The autoregressive parameter matrices [ ] and [B], the precipitation amount parameters, 1, and, and the precipitation occurrence parameters and d, are all estimated separately for each calendar month and regarded as being constant within the month. b. Spatially coherent generator Single-site weather generators of the kind described in section 3a have been designed to reproduce means, variances, and time and cross correlations between weather variables at one location. A collection of such single-site models, each fit to weather data from one in a network of stations, can be run in parallel to produce spatially coherent synthetic time series that capture the substantial and practically important spatial correlations in daily weather data (Wilks 1998, 1999b). Extension of the single-site precipitation models to spatially coherent weather generation is the more difficult part of the process, but can be accomplished by forcing Eqs. () and (4) with random numbers having the proper spatial correlation structure (Wilks 1998). The tendency for precipitation amounts to be less, on average, near the edges of wet areas on a given day is then easily accommodated by choosing the smaller precipitation mean,, at stations for which the random number forcing the precipitation occurrence is relatively near the wet/dry threshold [Eq. ()], and choosing the larger precipitation mean 1 otherwise (this dividing point is chosen in a way that preserves the unconditional mean). Because of the spatial correlation of the random forcing for precipitation occurrence, there is a strong tendency for precipitation amounts at the edges of wet areas to be generated using the smaller mean.itis useful and compact to parameterize these correlations, separately for the precipitation occurrence and precipitation amounts processes, according to station separation distances, using functions that produce positive definite (physically realizable) correlation matrices (e.g., Cressie 1993). Extension of the temperature generation [Eq. (5)] to multiple stations is straightforward. Instead of a bivariate autoregression for a single station, it is only necessary to construct an autoregression of dimension K, where K is the number of stations in the network (Wilks 1999b). The (K K) parameter matrices [ ] and [B] then depend on the simultaneous and lagged correlations between (standardized, conditional on precipitation occurrence) maximum and minimum temperatures at the K locations. Again, to the extent that these correlations vary appreciably and systematically in space they can be parameterized as functions of station separation. 4. Weather generator parameters as functions of seasonal forecasts Briggs and Wilks (1996) presented a procedure to estimate climatological statistics for a broad range of

198 JOURNAL OF HYDROMETEOROLOGY VOLUME 3 subseasonal variables, conditional on seasonal forecast probabilities, by bootstrapping (Efron and Tibshirani 1993) the observed climatological record consistent with the forecast probabilities. Instead of weighting each datum equally, this procedure computes conventional climatological statistics using weights equal to the probabilities specified in the forecasts. For sufficiently simple statistics it is straightforward to make the computations analytically (Briggs and Wilks 1996; Crowley 000). Let N B, N N, and N A be the number of years in a climatological record for a given location and season in which either the temperature or precipitation was below, near, or above normal, respectively, as determined using Eq. (1). Note that these three subsample sizes need not be equal, and will tend not to be equal if the terciles q 1/3 and q /3 of the climatological distribution f CL are defined using only a subset of the data (e.g., only the years 1961 90). Imagine a bootstrapping procedure in which some large number L resamples are taken from this record with replacement according to the probabilities in a seasonal forecast {p B, p N, p A }. On average there will be p B L bootstrap samples from the below-normal years, and p B L/N B bootstrap samples from a given below-normal year, with the corresponding numbers for near- and above-normal years computed analogously. Let X be some climatological statistic of interest. The bootstrapped expected value (i.e., climatological average, conditional on the forecast) of this statistic is then [ NB NN 1 pl B pl (B) N (N) boot i i L L i 1 NB i 1 NN N A pl A (A) xi i 1 NA E [X] lim x x ] [ ] L p p p x x x LN N N NB NN NA B (B) N (N) A (A) i i i B i 1 N i 1 A i 1 (B) (N) (A) px B pnx pa x, (7) (B) where, for example, x i is the statistic of interest from the ith below-normal year. a. Precipitation generator parameters The unconditional probability of a rain day is produced by Eq. (7) when the statistic X is the proportion of wet days in the ith season of each of the three (B, N, A) precipitation season types. Because this probability exhibits an appreciable annual cycle, separate forecast-conditional estimates of are made for each calendar month comprising a season. For example, for the June July August (JJA) season, Eq. (7) would be applied three times, with X being, in turn, the proportion of wet days in June, July, and August, but stratified in each case by the seasonal precipitation being below, near, or above normal for the entire JJA period. FIG. 1. Contour plot showing dependence of the precipitation occurrence autocorrelation [Eq. (3b)] for Jul in Ithaca, NY, as a function of seasonal precipitation forecast probabilities for JJA. It is possible to carry out the calculation of in Eq. (7) as needed for particular seasonal forecasts of interest. However, the computed changes in as functions of the seasonal precipitation forecast probabilities {p B, p N, p A } are linear, that is, planar in any two of the three forecast probabilities. Using p B and p A to define a forecast, (p B, p A) b0 bp B B bp, A A (8) where b 0, b B, and b A are response parameters specific to particular locations, seasons, and each of the three months within a season. Equation (8) follows from Eq. (7), which expresses the unconditional mean as a linear combination of the conditional means because the weights contain a linear dependency, for example, p N 1 p B p A. In particular, b 0 x (N), b B x (B) (N) x, and b A x (A) x (N). The autocorrelation parameter for daily precipitation occurrence d can also be estimated using Eq. (7) by computing the conditional expectations of the two transition probabilities using Eq. (), and then applying Eq. (3b) [the same procedure can be used to compute the conditional expectation of, using Eq. (3a), with identical results to those just described]. The variation of these estimates as functions of the forecast d are fully specified by quadratic surfaces above the (p B, p A ) plane, d(p B, p A) b0 bp B B bbbpb bapa baapa bbapbp A. (9) For example, Fig. 1 shows a contour plot of this surface for July in Ithaca, New York, as a function of JJA sea-

APRIL 00 WILKS 199 sonal precipitation forecasts. This plot is representative of the remainder of the data considered below, in that the absolute magnitudes of the variations in d are rather small, particularly in the central part of the triangle where nearly all operational forecasts are located. For practical purposes, d will be assumed independent of the forecasts for each location and month. This assumption yields essentially identical results in the validation exercises described in section 5. Since parameters of the mixed exponential distributions [Eq. (4)] are fit iteratively using a maximum likelihood procedure, the simple averaging in Eq. (7) is not applicable. Rather, raw bootstrap samples of nonzero daily precipitation amounts, for each month within a season, are assembled consistent with a forecast {p B, p N, p A }, and used to fit the distribution parameters, 1, and. Estimates for the mixing parameter are the least consistent (exhibit the most interstation random variation) of the three parameters within a network of stations, and in practice good results are achieved by assuming it to be constant for all locations in a domain for a given month (Wilks 1999b). In the data considered here the mixing parameter does not vary as a function of the forecasts. Variations of the two mixed exponential means, 1 and, as functions of the forecasts are essentially planar, although not exactly as is. For the data introduced below, R for 1 in planar regressions of the form of Eq. (8) are typically around 95% and, for the less important, are generally closer to 75%. Specifications of the form of Eq. (8) are nevertheless used for 1 and in the following, with separate planes fit for each location and for each of the three months within each season. b. Temperature generator parameters Recall that temperature means will be specified as quadratic functions of the date t, within each 3-month season, and that separate mean functions are calculated for maximum and minimum temperatures on wet and dry days. In each of these four cases, Eq. (7) can be applied to the temperature data for each date within a season, with different quadratic functions of date then being produced for each seasonal temperature forecast {p B, p N, p A }. Each of the three parameters for these parabolas can be fully specified as planar functions of two of the three forecast probabilities: (t p B, p A) b 0(p B, p A) b 1(p B, p A)t b (p B, p A)t ( p p ) 0 B B A A ( 0 BpB Ap A)t ( 0 BpB Ap A )t. (10) Again, because of the relationship between unconditional and conditional expectations, and the linear dependence among the forecast probabilities, the parameters in Eq. (10) can be found by fitting the quadratic (N) regressions 0 0 t 0 t (B) (N) t,[ t t ] B B t B t (A) (N), and [ t t ] A A t A t. Separate regression coefficients are computed for each location, each season, and each of the four temperature mean functions. Dependence of the four temperature standard deviation functions on the seasonal temperature forecasts can be approached through a simple extension of Eq. (7). For each location, date within a season, and for each combination of maximum and minimum temperature on wet and dry days, the raw weighted temperature standard deviation, N p B B (B) ŝ(t p B, p A) [x i (t) (t p B, p A )] N i 1 B N p N N (N) i B A N N i 1 N 1/ p A A (A) i N i 1 [x (t) (t p, p )] A [x (t) (t p, p )], (11) B A is calculated. Here (t p B, p A ) indicates the mean temperature on day t, conditional on the seasonal temperature forecast as specified by Eq. (10). The seasonal variations of these sample standard deviation estimates are then represented as quadratic functions of the date, the parameters of which are themselves described by quadratic surfaces in p B and p A : (t p B, p A) b 0(p B, p A) b 1(p B, p A)t b (p B, p A)t ( 0 BpB BBpB ApA AApA BApp) B A ( 0 BpB BBpB ApA AApA BApp)t B A ( 0 BpB BBpB ApA AApA BApp)t B A. (1) c. Spatial coherency parameters and other correlations The weather generator parameters whose dependence on seasonal forecasts have just been described all pertain to individual locations. The extension to coherent multiple-site weather generation described in section 3b requires estimation of spatial correlation functions for both precipitation occurrences and amounts. Investigating the dependence of these correlations on seasonal forecasts is complicated by the fact that particular pairs of locations may not necessarily be assigned identical forecasts during a particular season. However, the existence of such dependence can be investigated by computing these correlations for subsets of years when both

00 JOURNAL OF HYDROMETEOROLOGY VOLUME 3 FIG.. Distributions (over 5 locations and 1 seasons) of relative frequencies (in 10 000 synthetic realizations each) of seasonal precipitation category outcomes, as functions of forecast probabilities constrained according to the CPC convention of p A /3 p B. The ideal 1:1 line is gray. members of a station pair experienced the same type (below, near, or above normal) of season. For the stations in New York State investigated here, these correlations differ only slightly in aggregate and will be assumed in the following to be independent of the seasonal precipitation forecasts. The two sets of precipitation correlations are represented in the following as simple functions of pairwise horizontal station separation. Good fits to the precipitation occurrence correlations were obtained using functions of the form c k,l 1 1, (13) k,l while the precipitation amount correlations were modeled as k,l exp( 3 c k,l ). (14) Here the subscripts k and l refer to two locations separated by a horizontal distance c k,l. Both Eqs. (13) and (14) yield positive definite correlation matrices (Cressie 1993). Separate functions were fit for each of the 1 months, reflecting the fact that the horizontal scales of precipitation processes are much larger in winter than summer. The single-site temperature autoregression [Eq. (5)] requires estimation of five correlations: one simultaneous correlation between the two standardized temperature variables and four lagged correlations (e.g., Wilks 1995). The dependence of these correlations on the seasonal temperature forecasts is similar to the dependence of the precipitation parameter d on the seasonal precipitation forecasts [Eq. (9)]: the dependence can be described completely by parabolic surfaces above the p B p A plane, but the magnitudes of these variations are small enough for the correlations to be regarded as being independent of the seasonal temperature forecasts. Separate correlations are calculated for each of the 1 months. Extending Eq. (5) to multiple stations requires also that interstation simultaneous and lagged correlations be calculated. These were also found to be essentially independent of the seasonal temperature outcome, and therefore regarded as independent of the temperature forecasts. Monthly values of the spatial temperature correlations are used to define the multisite temperature autoregression as described in Wilks (1999b). The spatial dependence for each of these correlations was modeled by functions in the form of Eq. (14), with separate parameters 3 for each of the 1 months. 5. Recovery of seasonal statistics a. Data The foregoing development will be illustrated using daily temperature and precipitation observations from 1951 to 1996 over the same network of 5 locations in New York State as in Wilks (1998). They are distributed across an area of approximate dimension 500 km (east west) by 100 km (north south), centered near 4 N, 76 W. All 5 stations report precipitation data, while only 11 report temperatures. All are morningobserving Cooperative Observer stations. The simultaneous observation time simplifies, but is not necessary for, calculation of the spatial correlations (Wilks

APRIL 00 WILKS 01 FIG. 3. As in Fig., but for seasonal temperature outcomes as functions of seasonal temperature forecast probabilities. 1999b). Results described in this section pertain to all 1 (January February March through December January February) seasons. Tercile boundaries [Eq. (1)] are computed for each season based on data from the 1961 90 period, assuming Gaussian distributions for seasonal temperature and gamma distributions for seasonal precipitation. b. Consistency with forecast categories One fundamental aspect of the performance of daily weather generators, conditioned on seasonal forecast probabilities, is that the proportion of synthetic outcomes in each of the three seasonal categories should agree with the forecast probabilities. That is, over a large number of seasons simulated with a daily weather generator conditioned on a seasonal forecast, approximately p B 100% of the synthetic seasonal values should fall below the lower tercile q 1/3, p A 100% should fall above the upper tercile q /3, and p N 100% should fall between the two terciles. Note again that the original forecasts are assumed to be well calibrated, so the question addressed here is the extent to which calibration of the seasonal forecast probabilities is carried forward to the simulated daily series. Figure shows the results for seasonal precipitation, summarizing 10 000 realizations each for the 300 combinations of the 5 stations and 1 seasons. The forecast combinations have been selected subject to the CPC constraint that p B /3 p A. For each location, season, and forecast probability, the proportion of seasonal precipitation outcomes in each of the three categories has been tabulated. These are displayed in Fig. as boxplots for each forecast probability. The panels in Fig. are in the form of reliability diagrams, in that relative frequencies of the outcomes are plotted as functions of the forecast probabilities, and the ideal 1:1 line has been included for reference (gray). The agreement between forecast probabilities and outcome relative frequencies is generally good, and indeed compares favorably with the reliability of the forecasts themselves (Wilks 000a; Wilks and Godfrey 000, 00). Figure 3 shows the corresponding results for temperature forecasts at the 11 stations in the network that report temperature data. For these simulations the precipitation forecast has been specified as p B p N p A 1/3 (recall that daily temperature simulations are conditioned on series of simulated daily precipitation occurrences). Again the daily weather generators yield distributions of seasonal outcomes that are consistent with the proportions specified by each forecast. A quite different result is found when conditioning the weather generator parameters on seasonal forecasts whose largest probability is p N. Results corresponding to those in Figs. and 3, but for forecasts where p A p B (1 p N )/ yield essentially no differences in the distributions of outcomes as functions of the forecast probabilities: the relative frequencies of each of the three outcomes are near 1/3, regardless of the forecast (figure omitted). This disappointing result is unfortunately quite consistent with the performance to date of seasonal forecasts for which p N is the largest probability. Experience with these near-normal forecasts has shown that they generally do not resolve differences between future seasonal outcome relative frequencies and the

0 JOURNAL OF HYDROMETEOROLOGY VOLUME 3 FIG. 4. Comparisons of weather-generator simulated vs calculated mean seasonal precipitation for (a) a dry forecast, (b) a near-normal forecast, (c) a wet forecast, and (d) the climatological forecast; over 300 combinations of stations and seasons. The 1:1 line is drawn for comparison. climatological probabilities (Wilks 000a; Wilks and Godfrey 000, 00). c. Moments of the seasonal distributions A useful diagnostic check for a daily weather generator is a comparison of its synthetic seasonal statistics with the seasonal statistics of the observations to which it has been fit (Gregory et al. 1993; Wilks and Wilby 1999). Figure 4 compares seasonal mean precipitation as simulated by daily weather generators (horizontal) with corresponding analytical calculations (Wilks 000b) based on the climatological seasonal means and the seasonal forecast probabilities (vertical). Results for four example forecasts are presented: (a) a dry forecast (p B 0.6, p N p A 0.), (b) a normal forecast (p N 0.6, p B p A 0.), (c) a wet forecast (p A 0.6, p B p N 0.), and (d) the climatological forecast (p B p N p A 1/3). In each case (5 stations 1 seasons), there is very little scatter around the 1:1 line, indicating very good portrayal of the seasonal mean by the weather generators. The slight scatter evidently results from the small errors induced by assuming planar response of the two mixed exponential distribution parameters 1 and to the seasonal forecast probabilities: the mean precipitation in this simple weather generator depends only on the number of days simulated (i.e., length of

APRIL 00 WILKS 03 FIG. 5. As in Fig. 4, for standard deviations of seasonal precipitation. the season), the probability of a wet day, the average precipitation amount on wet days (e.g., Wilks and Wilby 1999), and the expressions for (p B, p A ) [Eq. (8)] recover these probabilities exactly. Figure 5 shows the corresponding results for standard deviations of seasonal precipitation, which describe the interannual variability of the seasonal precipitation totals. Panels (a), (c), and (d) exhibit the commonly observed overdispersion phenomenon (e.g., Katz and Parlange 1998; Wilks and Wilby 1999); that is, the seasonal variations as simulated by the daily generators are smaller on average than their counterparts in the observations. In contrast, the interannual variations of seasonal precipitation implied by the daily generator for the near-normal forecast (b) are larger than the relatively small calculated values, which is a consistent result with this modification of the weather generator parameters producing too many dry and wet seasons (cf. section 5b). The results for seasonal temperature statistics are substantially the same as those shown in Figs. 4 and 5 for precipitation, and the corresponding figures have been omitted for brevity. Recovery of the seasonal temperature means is essentially exact (that is, there is essentially no scatter around the 1:1 line; less scatter than in Fig. 4), which results from the mean functions in Eq. (10). Similarly, the daily generator underrepresents interannual variability of seasonal temperature for the

04 JOURNAL OF HYDROMETEOROLOGY VOLUME 3 FIG. 6. Average return periods for (a) 1-day and (b) -day area-averaged precipitation, calculated from the observed data 1951 96 (circles); and 10 000 synthetic years corresponding to dry (p B 0.50, p N 0.35, p A 0.15) and wet (p B 0.15, p N 0.35, p A 0.50) forecasts (light solid lines), and the climatological (p B p N p A 1/3) forecast (heavy solid line). Dashed ( incoherent ) lines show corresponding return periods for the climatological forecast when the spatial correlations among the 5 time series are set to zero. cool, warm, and climatological forecasts, as might have been expected. The daily generators also overrepresent the reduced interannual temperature variability for the near-normal forecast, which is again consistent with the results in section 5b. 6. Two examples Generating long records of synthetic weather series consistent with actual or possible seasonal forecasts will be of interest mainly for simulating weather-dependent quantities derived from the series, rather than examining the synthetic series per se. In some cases single-station simulations will be sufficient, but a much broader range of environmental models can be brought to bear when spatially coherent series are available. This section describes two simple and idealized, but instructive, examples. a. Summer extreme precipitation Occurrence of an extremely large precipitation amount in a localized area can be an important event, but very large precipitation amounts simultaneously over a watershed of substantial area has major implications for downstream flooding. While the New York stations considered in this paper are not part of a single watershed (they contribute to the Hudson, Delaware, Susquehanna, and St. Lawrence Rivers), they will be treated in this section as if they composed a single watershed, with each station contributing equally to an estimate of the watershed total precipitation. Consider the average return period R for an event of magnitude X. The return period is conventionally understood to be the average time separating events of magnitude X or larger, and can be defined in terms of the frequency distribution of the annual maxima of X, 1 R(x), (15) 1 F(x) where F(x) is the cumulative distribution function for the annual maximum of X in each of a sample of n years, estimated here as rank(x i ) F(x i ) ; i 1,...,n. (16) n 1 In this section the event of interest X is the daily precipitation during the JJA season averaged over the entire watershed, so that large values of X require large daily precipitation amounts simultaneously at a substantial fraction of the 5 stations. The 46 years (1951 96) of observed summer precipitation provide the data-based estimate for the distribution in Eq. (16). Also considered are n 10 000-yr realizations of spatially coherent daily precipitation corresponding to a dry forecast (p B 0.50, p N 0.35, p A 0.15), a wet forecast (p B 0.15, p N 0.35, p A 0.50) and the climatological (CL) forecast (p B p N p A 1/3). Figure 6a shows the results for average return periods between and 1000 yr. Circles indicate observed data values, which (because of the spatial averaging) are smaller than but similar in magnitude to the corresponding single-station statistics in Wilks and Cember (1993). The bold curve shows results for the

APRIL 00 WILKS 05 CL forecast, and the light solid lines indicate the dry and wet forecasts. Results for the CL forecast agree reasonably well with the observed data, while simulated series based on the forecasts show substantially altered probabilities for the extreme outcomes. Also shown in Fig. 6a are the results for the CL forecast, but with zero spatial correlation among the 5 simulated precipitation series (incoherent CL). In this case very large precipitation amounts occur simultaneously at multiple stations with rather low probability, and the relationship to the observed extreme statistics is quite poor. Figure 6b shows the corresponding results for the largest precipitation total in two consecutive days. Here the agreement with observed values is not particularly good for the longer return periods as a result of the assumption in the weather generator that consecutive daily nonzero precipitation amounts are uncorrelated. While this assumption is reasonable in most instances, the largest multiple-day precipitation amounts in this region occur because of the persistence of a single generating event. For example, the largest areally averaged precipitation amounts indicated in Fig. 6 derive from Tropical Storm (previously Hurricane) Agnes, during 3 June 197. The watershed average precipitation for June was 6. cm (46-yr return value in Fig. 6a), but in addition the area-average precipitation on 3 June was 5.3 cm (yielding 46-yr return value in Fig. 6b of 11.5 cm). b. Winter snowpack water equivalent Another quantity of hydrological significance is the snowpack water equivalent (SWE), or liquid-equivalent water content of the snow and ice on the ground. The dynamics of SWE result from the interacting influences of precipitation amount, temperature effects on the form (liquid or frozen) of precipitation, and temperature (and other) effects on the rates of melting and sublimation. Variations in SWE are accordingly expected to be sensitive to variations in winter temperature conditions described by seasonal forecasts. SWE is not routinely measured at the Cooperative Observer stations but will be modeled as a function of daily precipitation and temperature using the degreeday model of Carr (1988), which specifies 0.366 cm of snowmelt for each degree Celsius of average daily [(T max T min )/] temperature above 0 C. This relationship was developed in Ontario, Canada, and has been found to be useful for quality-control of measured SWE data in the northeastern United States (Schmidlin et al. 1995). The following simple model of SWE dynamics will be used: SWE(t 1) ppt(t), T(t) 0 C (17a) min ] [ T (t) T max(t) SWE(t) max 0, SWE(t 1) 0.366T(t) ppt(t), 0 T(t) (17b) T max(t) T max(t) max[0, SWE(t 1) 0.366T(t)], T(t). (17c) On days when the average temperature is below freezing [Eq. (17a)], no snowmelt occurs and all precipitation (ppt) is added to the SWE from the previous day. On days when the minimum temperature is above freezing (i.e., the average Celsius temperature is greater than half of T max ), snowmelt occurs according to the Carr model, and any precipitation that may occur is not added to the SWE [Eq. (17c)]. On days when the average temperature is above freezing but the minimum temperature is below freezing [Eq. (17b)], snowmelt occurs according to the degree-day model, and a fraction of the precipitation depending on the relationship of the maximum and minimum temperatures is added to the SWE. Note that in Eq. 17b, T max T min, and T min /T max 0, so that Eq. (17b) is equivalent to Eq. (17a) for T 0 C, and is equivalent to Eq. (17c) for T min 0 C. Figure 7 shows extreme-value statistics for SWE during the December January February (DJF) season, areally averaged over the 11 stations that have both temperature and precipitation data. Figure 7a shows results for the maximum annual watershed SWE during DJF, and Fig. 7b shows results for average SWE on 8 February at the end of this season. Circles show modeled SWE using the 45 winters (1951/5 1995/96) of observed temperature and precipitation data. As would be expected, these are broadly consistent with but smaller in magnitude than the single-station extremes in Wilks and McKay (1994). The solid lines show results for 10 000-yr spatially coherent weather generator simulations using combinations of dry or cool (p B 0.50, p N 0.35, p A 0.15), and wet or warm (p B 0.15, p N 0.35, p A 0.50) forecasts, or the climatological (p B p N p A 1/3) forecast (CL). Simulations based on the CL forecast (heavy line) reproduce reasonably the results calculated from observed data. The combinations cool/dry and warm/wet nearly compensate in terms of the SWE extremes. In contrast, the warm/dry and cool/wet combinations produce orderof-magnitude changes in extreme SWE return periods. The dashed lines in Fig. 7 show corresponding results

06 JOURNAL OF HYDROMETEOROLOGY VOLUME 3 FIG. 7. As in Fig. 6, but for (a) maximum DJF simulated snowpack water equivalent (SWE), and (b) simulated 8 Feb SWE. Cool or dry forecasts indicate p B 0.50, p N 0.35, p A 0.15, and warm or wet forecasts indicate p B 0.15, p N 0.35, p A 0.50. for spatially independent CL forecasts, which yield SWE extremes that are very much too light. 7. Conclusions Forecasts of atmospheric conditions in future seasons have the potential to be of substantial value in a variety of weather-sensitive decision contexts. One impediment to their wider adoption is the mismatch in temporal scale between the forecasts, which sketch the uncertainty in the upcoming seasonal averages, and many models that can be used to support decision making, which are driven by weather data at a daily time step. This paper has presented a method to adjust the parameters of daily time series models for weather data called weather generators in a way that is consistent both with the observed climate of a location and seasonal forecasts in the format that is currently available operationally. It was found that only a subset of the weather generator parameters are sensitive to changes in the seasonal climate implied by the forecasts. For the precipitation submodel, only the unconditional probability of precipitation on a given day and parameters controlling the mean precipitation on wet days are sensitive to variations in the seasonal precipitation forecast. In the data considered here, only the daily temperature means and standard deviations were found to be sensitive to variations in the seasonal temperature forecasts. Correlations controlling time dependence of simulated weather series at individual locations varied to a much smaller degree, and in practice the unconditional climatological correlations could be used. Similarly, the correlations required to construct spatially coherent networks of weather generators were not sensitive to different forecast probabilities, and unconditional values for these parameters could be used as well. Before using the procedure described here for other regions, the validity of these results should be checked. While the examples in section 6 have assumed the same seasonal forecasts throughout the domain, insensitivity of the spatial correlation functions to the seasonal forecasts implies that real applications could involve different forecast probabilities at different sites. While subseasonal statistics consistent with a particular seasonal forecast can be estimated easily through bootstrapping (Briggs and Wilks 1996), for simple statistics equivalent analytic calculations such as Eqs. (7) and (11) are faster, more accurate, and no more difficult to implement. These calculations can be carried out on an as-needed basis, or (as has been done here) once and for all, through summary functions. First-moment statistics (mean fraction of wet days, and mean precipitation amounts and temperatures) are planar functions of the forecast probabilities [Eqs. (8) and (10)], while second-moment statistics (standard deviations and correlations) are quadratic surfaces [e.g., Eqs. (9) and (1)]. Independence of the precipitation and temperature forecasts has been assumed, in the sense that daily precipitation parameters have been adjusted using only the seasonal precipitation forecasts, and daily temperature parameters have been adjusted using only the seasonal temperature forecasts. However, Briggs and Wilks (1996) found examples of daily temperature and precipitation statistics that are sensitive to both seasonal precipitation and temperature outcomes. Refinement of the procedure to account for such interactions, which could be accomplished by extending Eqs. (7) and (11) to nine rather than three terms each, is left for future

APRIL 00 WILKS 07 work. In addition, the procedure could be easily extended to include other variables such as windspeed and dewpoint temperature (Parlange and Katz 000) by increasing the dimension of the vector autoregression in Eq. (5) and its multisite counterpart. Finally, note again that use of the procedures described here is predicated on the seasonal forecasts being well calibrated ( reliable ). That is, relative frequencies of the event outcomes, conditional on the forecast probabilities, need to be essentially equal to the forecast probabilities in order for these procedures to be valid. Limited experience to date has found notable deficiencies in the calibration of seasonal forecasts (Wilks 000a; Wilks and Godfrey 000, 00). Both the science and practice of seasonal forecasting continue to improve, but the forecast products should be used carefully. Acknowledgments. I thank Jery Stedinger for helpful comments and observations. 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