A Stochastic Precipitation Generator Conditioned on ENSO Phase: A Case Study in Southeastern South America

Transcription

1 15 AUGUST 2000 GRONDONA ET AL A Stochastic Precipitation Generator Conditioned on ENSO Phase: A Case Study in Southeastern South America MARTIN O. GRONDONA* Instituto Nacional de Tecnología Agropecuaria, Castelar, Argentina GUILLERMO P. PODESTÁ Rosenstiel School of ine and Atmospheric Science, University of Miami, Miami, Florida MARIO BIDEGAIN Dirección Nacional de Meteorología, Montevideo, Uruguay MONICA MARINO AND HUGO HORDIJ Servicio Meteorológico Nacional, Buenos Aires, Argentina (Manuscript received 15 ober 1998, in final form 9 June 1999) ABSTRACT Stochastic precipitation generators can produce synthetic daily rainfall series with statistical characteristics similar to those of historical data. Typically, parameters of precipitation generators have been fit using all historical data for a given period. This approach, however, fails to capture differences in the precipitation process associated with an El Niño Southern Oscillation (ENSO) signal. Stochastic precipitation generators conditioned on the ENSO phase were developed to address this problem. Precipitation models with a range of parameterization schemes were tested in six locations in central-eastern Argentina and western Uruguay (southeastern South America), an important agricultural region with a clear ENSO precipitation signal in ober ch. Conditional precipitation models (occurrence, intensity, or both) were superior to simple models in 24 of the 36 locations/ months analyzed. Graphic diagnostics showed that conditional occurrence models successfully captured differences in the number and persistence of wet days among ENSO phases. Similarly, conditional intensity models improved noticeably the agreement between theoretical and empirical distributions of daily rainfall amounts. Conditional precipitation generators can be linked to other process models (e.g., crop models) to derive realistic assessments of the likely consequences of ENSO-related variability. Conditional stochastic precipitation generators, therefore, can be useful tools to translate ENSO forecasts into likely regional impacts on sectors of interest. 1. Introduction Many interactions between crops and weather are nonlinear, and simple statistical approaches usually fail to assess the effect of climate variability on agricultural production risk. Dynamic, process-level crop simulation models provide alternative tools to assess climate impacts on agriculture and to evaluate the outcomes of a * Current affiliation: Zeneca Semillas S.A.I.C., Balcarce Research Station, Balcarce, Argentina. Corresponding author address: Guillermo Podestá, University of Miami RSMAS-MPO, 4600 Rickenbacker Cswy., Miami, FL gpodesta@rsmas.miami.edu wide range of management decision alternatives (Boote et al. 1996). These models typically require inclusion of weather data. Although historical weather sequences can be used, getting long-term daily weather data is laborious and costly at best and, in some cases, impossible. More fundamentally, the use of observed sequences gives a solution based on only one realization of the weather process (Richardson 1981). A thorough risk assessment should explore the range of outcomes of the process of interest if equally likely weather sequences, with the same properties as the observed series, were used (Richardson 1981). A solution to this problem is the use of stochastic weather generators, which can produce synthetic daily weather series with statistical characteristics similar to those of historical data. Several approaches have been proposed for the generation of daily precipitation, temperature, and solar ra American Meteorological Society

2 2974 JOURNAL OF CLIMATE VOLUME 13 diation series (the weather variables required by most crop simulation models); a few examples include the work of Bruhn et al. (1980), Richardson (1981), Geng et al. (1985), and Racsko et al. (1991). Hutchinson (1987) reviewed methods for generating daily weather series. Stochastic weather generators are frequently included as modules of crop simulation packages (Pickering et al. 1994). Recent reviews of commonly used generators can be found in Wallis and Griffiths (1995), Johnson et al. (1996), and Semenov et al. (1998). This work focuses on the stochastic generation of precipitation sequences. Stern and Coe (1984), Woolhiser (1992), and Lettenmaier (1995) reviewed approaches to this topic. The El Niño Southern Oscillation (ENSO) phenomenon, the result of a two-way interaction between the ocean and the atmosphere in the tropical Pacific Ocean, is the major single source of climate variability on seasonal-to-interannual scales in many parts of the world (Trenberth 1997a). The ENSO phenomenon involves two extreme phases: warm events (associated with warm sea surface temperature anomalies in the tropical Pacific) also known as El Niño years, and cold events, referred to as La Niña or El Viejo years. Years that do not fall in the extreme phases are labeled as neutral. ENSO events influence precipitation regimes in southeastern South America, the geographic focus of this study. Ropelewski and Halpert (1987) found a tendency for enhanced precipitation in warm ENSO events during ember through ruary of the following year. During cold events, Ropelewski and Halpert (1989) reported below-normal precipitation: of the 19 cold events in the series they analyzed, 16 were associated with dry June ember seasons. Pisciottano et al. (1994) found higher than average rainfall in Uruguay during warm ENSO events, especially from ember to the following uary. In contrast, cold events tended to have lower than average rainfall from ober to ember. Ropelewski and Halpert (1996) expanded their earlier results but emphasized shifts in the probability distribution of precipitation amounts among ENSO phases. Díaz et al. (1998) confirmed previously reported associations between ENSO and precipitation in southern Brazil and Uruguay, but they also explored regional modulation of the ENSO signal by sea surface temperature anomalies in the Atlantic Ocean. Grimm et al. (2000) analyzed ENSO precipitation signals in southeastern South America and discussed the association between precipitation and circulation anomalies. Typically, the parameters of stochastic weather generators have been fit unconditionally (Wilks 1989). That is, they have been estimated using all historical data for a given period (e.g., a month). However, if a period shows an ENSO-related signal such as enhanced or decreased aggregated rainfall, the characteristics of the precipitation process (and thus the parameters of precipitation models) must differ among ENSO phases. Indeed, Vila and Grondona (1996) found that the number of wet days during the last quarter tends to be higher in central-eastern Argentina during warm ENSO events. Bidegain and Caffera (1989) reported a similar finding for Uruguay, for the month of July of warm events. Moschini et al. (1996) found higher probability of wet days following dry days and a lower chance of dry spells during ember ember of warm events in central Argentina. The objective of this study is to investigate parameterization of stochastic weather generators conditional on ENSO phase. That is, model parameters are estimated separately for warm and cold ENSO events, and neutral years. The hypothesis is that models conditioned on ENSO should capture differences in the precipitation process among ENSO phases, thus providing a more realistic assessment of agricultural risks associated with ENSO-related climate variability. The approach is tested using precipitation records from central-eastern Argentina and western Uruguay, one of the major agricultural regions in the world (Hall et al. 1992). Wilks (1989) proposed the conditional parameterization of precipitation models, although not in the context of ENSO-related variability. He fitted models with separate parameters for dry, near-normal, and wet months. Katz and Parlange (1993, 1996) explored daily precipitation models conditional on a simplified monthly index of large-scale atmospheric circulation. Both Wilks (1989) and Katz and Parlange (1993) noted that unconditional models produced distributions of monthly total precipitation in which moderate amounts were overrepresented and more extreme amounts were underrepresented (i.e., dispersion was underestimated). The failure of stochastic models to reproduce the observed variance of monthly or seasonal precipitation was recently investigated by Katz and Parlange (1998). Woolhiser (1992) suggested that large-scale climate phenomena without annual periodicities (such as ENSO) might be at least partly responsible for the typical underestimation of the variance of monthly and annual precipitation in simulated series. He proposed the use of the Southern Oscillation index to perturb the parameters of a daily stochastic precipitation model. Woolhiser et al. (1993) used this approach to detect ENSO effects in daily precipitation regimes in the southwestern United States. In the following sections, unconditional stochastic precipitation models and their parameterization are described. Then, various parameterization schemes conditional on the ENSO phase are introduced. The conditional approach is tested using precipitation data from six meteorological stations in Uruguay and central Argentina. To assess whether the conditional models offer an improvement over simpler alternatives, various diagnostics are presented for the occurrence and intensity processes, and for the interannual variance of monthly total precipitation. Finally, frequency distributions of simulated monthly precipitation totals are examined to

3 15 AUGUST 2000 GRONDONA ET AL illustrate the consequences of conditional parameterization. 2. Unconditional stochastic precipitation generators This section provides background on stochastic precipitation models to motivate the conditional parameterization proposed in section 3. In most available models, the generation of precipitation involves two components: 1) the occurrence process (i.e., the sequence of dry or wet days), and 2) the intensity process (i.e., the sequence of precipitation amounts on wet days). A wet day is defined as one on which a measurable amount of precipitation (in this study, 0.1 mm) occurs. a. Occurrence process A simple model of the occurrence process is a twostate, first-order kov chain. This model has been used in many situations with remarkable success to describe observed rainfall occurrence (Gabriel and Neumann 1962; Richardson 1981; many references in Hutchinson 1987; Wilks 1989; Katz and Parlange 1993). The simple structure of a first-order kov model enables the analytical determination of many of its properties, facilitating its subsequent application (Katz and Parlange 1998). For example, it is fairly simple to calculate the probabilities of long dry spells (Coe and Stern 1982). In a first-order kov chain process, the probability of a wet (dry) state on day t depends only on the precipitation state on day t 1. Here J t is defined as the state indicator random variable for day t: J t 0 if day t is dry; J t 1 for a wet day. This model is characterized by the four transition probabilities: P ij Pr{J t j J t1 i}, i, j 0, 1. (1) As transition probabilities P 00 P 01 P 10 P 11 1, only two parameters are needed to define the firstorder, two-state kov process. An alternative formulation of this model is in terms of parameters d, the first-order autocorrelation coefficient, also referred to as the persistence parameter, and, the probability of a wet day (Katz and Parlange 1993). These parameters are related to the transition probabilities and to one another as d corr(j t1, J t) P11 P 01, (2) P01 P01 Pr{Jt 1}. (3) P01 P10 1 d The main limitation of the two-state, first-order kov approach is the accurate simulation of long runs of dry or wet sequences (also known as clustering) at some locations (Racsko et al. 1991; Lettenmaier 1995). When first-order models have been found inappropriate, an alternative has been to increase the order of the kov chain (Jones and Thornton 1993; Katz and Parlange 1998). That is, probabilities are conditioned on the most recent k (k 1) periods. This approach, however, requires a larger number of parameters: a kth-order, twostate kov process is defined by 2 k parameters. Parameter estimates for high-order kov chains are not very reliable, particularly when rainfall records are short (Hutchinson 1987). To improve the representation of clustering, a number of models based on continuous point processes have been proposed. References to these models are given by Georgakakos and Kavvas (1987) and Lettenmaier (1995). Because our ultimate interest is on impact analysis and the development of decision support tools, we adopted first-order kov chains as a reasonable compromise among simplicity, adequacy of fitting, and ease of stochastic simulation. b. Intensity process Many statistical distributions have been proposed to model the intensity process, that is, the distribution of rainfall amounts on wet days. Examples include the lognormal, cubic root normal, mixed exponential, kappa, gamma, and Weibull distributions (Bruhn et al. 1980; Richardson 1981; Stern and Coe 1984; Hutchinson 1987; Woolhiser 1992). This work adopts the frequently used gamma distribution, given by the expression (Wilks 1989) 1 (y/) exp(y/) f (y t Jt 1), (4) () for y t 0, 0 and 0, where y t denotes the rainfall amount on wet day t. This distribution is characterized by, the shape parameter, and, the scale parameter. The mean and variance of the daily amount of rain (on wet days) are E(Y t Jt 1) and (5) 2 2 var(y J 1). (6) t t To obtain parsimonious models, common assumptions are that daily rainfall is independent of the occurrence process and is also independent of rainfall amounts in previous days (Woolhiser 1992). Katz and Parlange (1998) explored model extensions that do not require these assumptions. c. Seasonal variation of model parameters A usual approach to deal with seasonal variation in precipitation is to divide the year in nonoverlapping periods and fit models to each period (Woolhiser 1992). Here, parameters are estimated for each month of the year. This approach implicitly assumes stationarity, not only within a month but also between years. Because low-frequency or interannual components such as longterm climate trends or ENSO events violate this assumption, changes in the distribution of precipitation

4 2976 JOURNAL OF CLIMATE VOLUME 13 should be expected. The following section describes how to capture ENSO-related changes in the probability distribution of rainfall via model parameters conditioned on ENSO phase. 3. Precipitation generators conditioned on ENSO phase A stochastic precipitation generator is now introduced in which parameters of the occurrence and intensity processes are conditioned on ENSO phase. That is, model parameters are estimated separately for warm and cold ENSO events, and neutral years. The basic model assumes a first-order, two-state kov rainfall occurrence process and a two-parameter gamma distribution for the intensity process. Such a model requires only a relatively small number of parameters, while it still accounts for the most important statistical properties of precipitation series (Katz and Parlange 1993). More complicated models, involving larger numbers of parameters, may not be viable for the conditional approach; the relatively small number of ENSO events in the historical data would introduce considerable uncertainty in estimates of a larger set of parameters. Nevertheless, Katz and Parlange (1998) noted that simple conditional models resembled more complicated extensions of unconditional models. Parameter estimates for both the intensity and occurrence processes can be obtained using either the method of moments or maximum likelihood. Although the former is simpler, the maximum likelihood procedure is adopted, as its estimators have better known asymptotic properties than estimators based on the method of moments. a. Conditional model parameterization The basic model can be conditionally parameterized with various degrees of complexity. A model fully conditioned on ENSO phase involves, for each month, a separate set of parameters for each ENSO phase: i, d i, i, and i denote the parameters for each of the ENSO phases, i W, N, C (for warm, neutral, or cold). This model (hereafter referred to as full ) requires a total of 12 parameters for each month (four parameters three ENSO phases). Fitting full models would imply that ENSO influences all parameters of the rainfall model, and there is no common structure or behavior among phases. In contrast, an unconditional model (hereafter, simple ) with a single set of parameters for all ENSO phases implicitly indicates no ENSO signal on rainfall distribution. A compromise between these extremes is to assume that some model parameters are common for all ENSO phases. These models will be referred to as restricted because restrictions are imposed for the sake of statistical parsimony and ease of interpretation. Two restricted models can be formulated for the occurrence process: common and conditional d, and vice versa. Similarly, two restricted models are possible for the intensity process: common and conditional, and vice versa. The combination of a full model, two restricted models, and a simple model for each process (occurrence and intensity) defines a total of 16 possible parameterization alternatives. Fortunately, it is not necessary to evaluate all these combinations. Because the log-likelihood of the joint process can be expressed as the sum of the log-likelihoods of the occurrence and intensity processes, the maximization can be performed separately, thus simplifying the optimization procedure. That is, the best joint model is the one with the best parameterization alternative for each process. The Akaike information criterion (AIC; Akaike 1974) is used to select the best occurrence and intensity models. Briefly, the AIC is the likelihood ratio statistic penalized by twice the degrees of freedom (Katz 1981). Katz (1981) and Gregory et al. (1993) discuss the use of the AIC to estimate the order of kov chains. The best model will be the one minimizing the AIC. According to the AIC, the best model fit will have a low dimension if there are few data, and the dimension will increase as more information becomes available (Buckland et al. 1997). Alternative criteria such as the Bayesian information criterion (Katz 1981) may yield more parsimonious models than the AIC. b. Monthly statistics and induced models A helpful way of interpreting changes in the parameters of the conditional precipitation models is to convert them into terms of monthly total precipitation (Katz and Parlange 1993). Any model proposed should reflect the main characteristics of the observed series at aggregated levels. The number of wet days in a month is a random variable. Aggregated moments, at a monthly level or any other time period, can be easily derived (e.g., see Katz and Parlange 1993, 1996, 1998). Let N(T) denote the number of wet days during period T (T denotes both a given period and its duration). The mean number of wet days is given by T t t1 E[N(T)] E J T. (7) Similarly, although not as straightforward as before, the variance of the number of wet days is var[n(t)] var T Jt t1 [ ] 1 d 1 d T (1 ) T 2d 2 1 d (1 d) 1 d T(1 ). (8) 1 d

5 15 AUGUST 2000 GRONDONA ET AL The mean and variance of the total precipitation amount in period T, S(T), also can be easily derived using Eqs. (5) and (6) (Katz and Parlange 1993, 1996). These values are S(T) T T and (9) [ ] ] 1 d 1 d 1 d 1 d T S(T) T (1 ) T 2d 2 1 d (1 d) [ 2 2 S(T) T 1 (1 ). (10) The separate conditional models (here, one for each ENSO phase) can be combined into an induced (or overall) model (Katz and Parlange 1993). The mean and variance of monthly precipitation totals for the induced model are 3 S(T) i S(T)i i1 w, (11) S(T) i S(T)i i S(T)i S(T) i1 i1 w w [ ]. (12) Note that the mean for the induced model [Eq. (11)] is a weighted average of the conditional means. In this case, the weights w i are the climatological relative frequencies of each ENSO phase. In contrast, the induced variance [Eq. (12)] is not simply a weighted average of conditional variances, but it reflects the variation in the conditional means as well (Katz and Parlange 1996). 4. Application of conditional precipitation generators a. Data series The generation of precipitation conditional on ENSO phase is tested using relatively long (60 65 yr) series of daily precipitation for six locations in central-eastern Argentina (Pergamino, Pilar, and Santa Rosa), and western Uruguay (Paysandú, Mercedes, and Estanzuela). Station locations are indicated in Fig. 1. The longest precipitation series encompass the period from 1 July 1931 to 31 ember 1996 (about 65.4 yr). To improve parameter estimation, all available data were used, so the number of years in the record may vary slightly for each month and location. Furthermore, some series had missing values; months with more than three daily precipitation values missing were excluded from the analyses. The stations analyzed are in areas of agricultural importance (Hall et al. 1992). The study region shows an east west gradient in both annual rainfall and range of the seasonal precipitation cycle (Prohaska 1976). Annual precipitation is highest at the eastern stations: median totals are 1142, 1055, and 973 mm for Paysandú, Estanzuela, and Mercedes. The annual precipitation cycle in these stations (shown only for Estanzuela in Fig. FIG. 1. Location of stations analyzed in Argentina and Uruguay. The inserts show the annual cycle of median monthly precipitation totals at the three Argentine stations and in Estanzuela, Uruguay. 1) is typical of regimes with maritime influence; it has a fairly small range but shows two maxima in fall and spring (see also Díaz et al. 1998). The western stations, Pilar and Santa Rosa, show drier conditions: median annual totals are 738 and 623 mm, respectively. In these stations, the annual cycle has a clear maximum in late spring and summer and a marked winter minimum (Fig. 1). Pergamino represents a transition between both regimes, with a median annual precipitation of 937 mm and a less-marked winter minimum. Literature reports (see the introduction) and preliminary exploration of the data suggest that the ENSO signal on rainfall in southeastern South America is most marked during late spring and summer. This is the period, thus, in which potential improvements associated with ENSO-conditional parameterization are most likely to be detected. For this reason, and for the sake of brevity, subsequent analyses focus on the period from ober to ch. b. ENSO signal on precipitation An ENSO phase (warm, neutral, or cold) is first assigned to each analyzed month (ober ch) in the historical precipitation records. There are several alternative definitions of ENSO events (Trenberth 1997b). Here, events were categorized according to an index developed by the Japan Meteorological Agency (JMA). The JMA ENSO index is based on a 5-month running mean of spatially averaged SST anomalies in the region of the tropical Pacific Ocean between 4N 4S and W. If index values between July and June of the following year are 0.5C or greater for at least six consecutive months (including the quarter ober ember, considered as the typical peak of ENSO-related anomalies), the period is categorized as a warm event. Similarly, if the index is 0.5C or lower for at least six consecutive months (including ober ember),

6 2978 JOURNAL OF CLIMATE VOLUME 13 TABLE 1. Warm and cold ENSO events between 1931 and 1995, as defined by the JMA index. Note that an ENSO year encompasses the period between Jul of the listed year and Jun of the following year. For example, the warm event of 1982 includes the period between Jul 1982 and Jun Years not listed are considered as neutral. ENSO phase Warm Cold Years 1940, 1951, 1957, 1963, 1965, 1969, 1972, 1976, 1982, 1986, 1987, , 1942, 1944, 1949, 1954, 1955, 1956, 1964, 1967, 1970, 1971, 1973, 1974, 1975, 1988 the year is categorized as a cold event. The rationale for the July June period is that this corresponds with the agricultural year or cropping cycle in southeastern South America. The JMA index is based on observed data from 1949 to the present. For years prior to 1949 (and going back to 1868), the index was derived from reconstructed monthly mean SST fields, estimated using an orthogonal projection technique (Meyers et al. 1999). Identified ENSO events are shown in Table 1. From July 1931 to TABLE 2. Median monthly precipitation totals by month and ENSO phase. A p value for a Kruskal Wallis test of differences in central tendency of total precipitation among ENSO phases is shown for each month and location. Month Warm Neutral Cold K WP Estanzuela Mercedes Paysandú Pergamino Pilar Santa Rosa FIG. 2. Box plots of monthly precipitation ( ) by ENSO phase for Pergamino, Argentina. The filled circles in the center of the box indicate the median. The box bounds the central half of the data; its width, therefore, corresponds to the interquartile range (IQR). The whiskers encompass 1.5 times the IQR on each side of the median, or the maximum and minimum values in the distributions. Outliers (values beyond the whiskers) are shown as open circles. June 1996, the JMA definition identifies 12 warm events and 15 cold events. ENSO effects on precipitation regimes are illustrated by boxplots of monthly rainfall totals by ENSO phase for Pergamino (Fig. 2). The central tendency of monthly Pergamino precipitation for ember and ember tends to be lower during cold events than during warm or neutral years (Fig. 2). Median rainfall totals by month and ENSO phase are presented in Table 2 for all locations. The ember ember period shows the most consistent differences in median precipitation among ENSO phases (Table 2): Kruskal Wallis tests for ember (ember) show significant differences (P

7 15 AUGUST 2000 GRONDONA ET AL TABLE 3. AIC values for simple and best occurrence and intensity models. Cond. stands for conditional. For example, cond. refers to a restricted intensity model with scale parameter conditioned on ENSO phase and common shape parameter. A 2 value and associated p value are shown as guidance on whether simple models should be rejected in favor of conditional models. Month Occurrence AIC simple AIC best Best model 2 p Intensity AIC simple AIC best Best model 2 p Full Full Full Cond. d Cond. d Full Cond. d Cond. d Estanzuela Mercedes Paysandú Pergamino Pilar 0.01 Santa Rosa Full ) for all stations but Pilar (Paysandú). The lack of significance for other months may be a consequence of the higher interevent variability, typical of the weaker ENSO signal in extratropical regions (Kumar and Hoerling 1997). Perhaps more striking than differences in central tendency is the difference among ENSO phases in the spread of precipitation totals for ember, ember, and ch in Pergamino (Fig. 2). Rainfall distributions are very narrow during cold events and much wider during neutral or warm events, suggesting a more consistent response to cold ENSO events. c. Model selection Four alternative models were explored for both the occurrence and intensity processes: a simple or unconditional model (two parameters per process), two restricted models (four parameters), and a full conditional model (six parameters). For each process, the model considered best was the one for which the AIC was minimized. AIC values for the simple model for each station, month, and process are shown in Table 3. When one or more conditional models had lower AIC values than the simple model (considered as the benchmark),

8 2980 JOURNAL OF CLIMATE VOLUME 13 the model with the lowest AIC value was selected as best; this model is listed in Table 3. To assess whether the simple model should be rejected in favor of a conditional model, a 2 value was calculated as twice the difference of the log-likelihoods for the best and simple models. This statistic and its associated p value are listed in Table 3. The p value is provided for guidance but should not be interpreted in the strict sense of hypothesis testing because the implications of selecting first the best of a range of alternative models and then comparing it to the simple model are unclear. Conditional precipitation models (occurrence, intensity, or both) were identified as an improvement over simple models in 24 of the 36 location month combinations analyzed. In ember and ember, conditional models were better than simple models in five of the six locations. In ober, uary, and ruary, conditional models were better in four locations. These results suggest a regionally consistent ENSO signature on precipitation. A concern, however, is the issue of multiplicity, which arises when several tests are conducted simultaneously and the test statistics are correlated (e.g., because of spatial correlation among locations). In this case, it is not possible to ascribe an overall level of significance to the results (Zwiers 1987). The high number of significant tests may have resulted, for example, from a combination of chance and spatial correlation among stations. It is difficult to assess how much independent support each location is providing to the hypothesis of a regional ENSO precipitation signal. Some of the stations are relatively far from one another (especially in Argentina), and they represent different precipitation regimes. To rigorously investigate whether the apparent coherence of the observed results is an artifact of the spatial correlation of the precipitation series, a full temporal spatial modeling approach would be necessary. This, however, is not feasible with the data density available. Empirical approaches have been proposed to deal with the multiplicity problem (Livezey and Chen 1983; Zwiers 1987; Wilks 1996, 1997). These approaches rely on computer intensive resampling or permutation techniques to build an empirical distribution for a test statistic. The attractive of these procedures is that they do not require that the covariance structure of data be explicitly modeled and estimated (Wilks 1997). In our case, a resampling procedure to estimate overall significance of the model comparisons might be based on reshuffling the ENSO phases among years in the data series for a given month (ignoring temporal correlation initially). If the same ENSO phase is assigned to all locations when phases are reshuffled, the spatial correlation among stations is preserved. The number of locations for which conditional models are identified as superior would serve as the test statistic. By repeating many times the shuffling of ENSO phases, an empirical distribution of the test statistic can be derived for a given month and a critical value can be determined for a given confidence level. If the number of locations in the original data showing conditional models as superior is higher than the critical value, overall results can be considered not to be due to chance and spatial correlation. A further level of complexity would consider the possible existence of temporal correlation. In this case, more complicated resampling schemes would be required (e.g., see Wilks 1997). Both occurrence and intensity conditional models seem to be comparably supported by the results: 15 conditional occurrence models and 17 conditional intensity models were identified as better than the simple models. Conditional occurrence models were identified as best in five of the six locations in ember and in four locations in ember. On the other hand, conditional occurrence models were best at only one location in uary and at none in ch. A full model was selected in four cases. Restricted models with conditional and common d were selected in seven cases. A restricted model with conditional d was selected in four cases. A likely interpretation is that ENSO clearly tends to influence the number of rain days in some months and possibly affects also the persistence of wet spells. An ENSO influence on the persistence of wet spells alone seems less frequent. Persistent wet spells are frequently associated with frontal passages; it is unclear if the number of fronts is influenced by ENSO, although, to our knowledge, there is no information on this. For the intensity process, ember is the month with the largest number of locations (four) showing conditional intensity models as best. These results, together with those for occurrence, are consistent with a clear ENSO signal in ember ember (Pisciottano et al. 1994). In every month, at least two stations show conditional intensity models as best. The majority of cases (15 of 17) for which conditioned models were best corresponded to restricted models with conditional parameter and common parameter. Wilks (1989) assumed such a restricted model for a study of North American stations, and it seems to be appropriate for the current study region as well. Full intensity models or models with conditional were each selected in only one case. 5. Performance of conditional models Various diagnostics are presented in this section to assess the performance of ENSO-conditional precipitation generators from the point of view of practical applications. Potential improvements over simple models should be balanced against increased model complexity and data requirements. For brevity, only representative results are presented, although examples will be selected from various stations and months.

9 15 AUGUST 2000 GRONDONA ET AL a. Diagnostics for the occurrence process The distribution of runs of dry days (dry spells) can be used as a diagnostic of the performance of simple and conditional occurrence models. Dry spells are of major concern for agricultural risk. The distribution of dry (or wet) spell duration for a first-order, two-state kov model is geometric (MacDonald and Zucchini 1997). The probability of observing a dry spell k days k1 long, therefore, is given by P(K k) P 00 (1 P 00 ), where P 00 is the dry dry transitional probability [Eq. (1)]. The cumulative distribution is F(k) P(K k) k 1 P 00 ; thus the probability of observing a dry spell k longer than k days is P(K k) P 00. The theoretical and empirical distributions of dry spell lengths longer than k days are compared via quantile quantile (Q Q) plots. In such plots, quantiles of the theoretical distribution are plotted against corresponding quantiles of the empirical distribution. If both distributions are similar, points should fall along a 1:1 line. To estimate the empirical distribution of dry spell lengths, runs are not truncated by the end of the month. The Q Q plots are shown for ember in Mercedes (Fig. 3). Empirical and theoretical distributions are estimated and plotted separately for each ENSO phase. The Q Q plot for a simple occurrence model is shown in Fig. 3a. For cold (warm) events, most points fall above (below) the 1:1 line. This indicates that the simple occurrence model clearly underestimates (overestimates) the probability of observing dry spells longer than k days during cold (warm) events. For all ENSO phases, the first-order kov model shows a reasonable fit for shorter runs but generally has difficulties describing the frequency of longer (k 15 days) dry runs. This is probably because inadequacies in model parameters are magnified for longer runs, as the dry spell length (k) is an exponent in the computation of the cumulative probabilities. The lack of model fit can be attributed partly to changes in the underlying transitional probability from one month to the other, as long runs frequently occur in more than one month. An occurrence model with conditional was identified as best for Mercedes/ember (Table 3). When this model is used (Fig. 3b), points for all ENSO phases lie much closer to the 1:1 line. That is, the theoretical and empirical distributions of dry spell lengths agree much more closely. This suggests that the conditional models successfully capture differences in the number and persistence of wet days among ENSO phases. The conditional approach significantly improved the description of the distribution of dry spells longer than days. Nevertheless, the first-order kov occurrence models (simple or conditional) still have problems describing the frequencies of longer dry runs. Alternatives to improve the distribution of long dry runs are increasing the order of the kov process (Katz and Parlange 1998) or modeling directly the distribution of dry/wet length runs (Racsko et al. 1991). FIG. 3. Q Q plots of theoretical and observed distributions of dry spell lengths longer than k days for in Mercedes, Uruguay. Points are plotted separately for each ENSO phase. A 1:1 line is shown to facilitate visualization of the agreement between the empirical and theoretical distributions: (a) simple occurrence model, (b) Conditional occurrence model (conditional ). b. Diagnostics for the intensity process Graphic diagnostics (Q Q plots) similar to those used for the occurrence process can be used to compare the performance of the simple and conditional intensity models. In this case, empirical quantiles of daily precipitation amounts for a given month (computed sepa-

10 2982 JOURNAL OF CLIMATE VOLUME 13 rately for each ENSO phase) are plotted against theoretical quantiles (derived from a gamma distribution) for 1) unconditional and 2) ENSO-conditional intensity models. A Q Q plot of daily precipitation amounts on wet days for uary in Paysandú is shown in Fig. 4. As before, theoretical and empirical quantiles are estimated and plotted separately for each ENSO phase. In Fig. 4a, theoretical quantiles are estimated from a simple model. The points for warm (cold) events lie mostly above (below) the 1:1 line. This suggests that the simple model underestimates (overestimates) daily precipitation amounts during warm (cold) events. The best intensity model for Paysandú/uary is a restricted model with conditional parameter. When theoretical quantiles are computed using the conditional model, points for all ENSO phases lie much closer to the 1:1 line, indicating a noticeable improvement in the agreement between the theoretical and empirical distributions (Fig. 4b). Admittedly, both simple and conditional models still fail to describe extreme daily precipitation values for all ENSO phases, generally underestimating (overestimating) the frequency of high values for warm (cold) events. As discussed below, the inadequate description of extreme daily rainfall amounts may affect other diagnostic quantifies, such as the dispersion of monthly precipitation totals. A possible solution may be the contamination of simulated distributions with a few extreme daily rainfall values; this is the topic of ongoing work. c. Diagnostics for monthly precipitation totals When evaluating the conditional precipitation models, it is important to consider not only their performance at daily levels, but also how well they reproduce aggregated quantities such as monthly precipitation totals. Because daily precipitation observations are directly fitted, the mean monthly total precipitation should be closely reproduced by both the simple and conditional models (Katz and Parlange 1996). Therefore, this is not an interesting diagnostic of model performance. Instead, attention is focused on the variance of monthly precipitation totals. Stochastic models tend to underestimate the observed ( interannual ) variance of monthly rainfall totals (Wilks 1989; Woolhiser 1992; Katz and Parlange 1996, 1998). This phenomenon, termed overdispersion, has been attributed to either inadequacies in the model (e.g., an incorrect order for the kov process) or to lowfrequency fluctuations not captured in the parameter estimation (Gregory et al. 1993; Katz and Parlange 1998). Models conditioned on ENSO should capture (at least partly) interannual differences in the characteristics of precipitation in months/locations with a clear ENSO signal. Conditional models, therefore, should reproduce a higher proportion of the observed interannual variance. FIG. 4. Q Q plot of the theoretical and observed distributions of daily precipitation amounts on wet days for in Paysandú, Uruguay. Points are plotted separately for each ENSO phase. A 1:1 line is shown to facilitate visualization of the agreement between the empirical and theoretical distributions: (a) simple intensity model, (b) Conditional intensity model (conditional ).

11 15 AUGUST 2000 GRONDONA ET AL FIG. 5. Observed and modeled standard deviations of monthly precipitation totals for the 24 months/locations for which conditional models were identified as better than simple models. Theoretical standard deviations are derived from simple models (filled triangles) and induced models (open circles) resulting from the mixture of separate conditional models. A 1:1 dashed line is shown. Standard deviations of monthly precipitation totals were derived from the daily stochastic models for both the simple [Eq. (10)] and induced models [Eq. (12)]. These values were compared to the corresponding standard deviations of observed monthly precipitation totals (for months with less than four missing daily values). Results are displayed in Fig. 5 for months/locations for which conditional models were best. Most points fall below the 1:1 line, indicating that modeled standard deviations for both simple and conditional models are lower than the observed values. Differences between empirical and modeled values seem to get larger as monthly variability increases. On average, standard deviations for simple models are 83% of the empirical values. This is comparable to values of about 80% 85% in California (Katz and Parlange 1993). Points for conditional models are slightly closer to the 1:1 line, but the improvement is only minor: standard deviations for conditional models are, on average, about 87% of the observed values. Despite previously shown improvements in the description of the occurrence and intensity processes, conditional models apparently fail to correct substantially the overdispersion phenomenon. To explore the reasons for the remaining overdispersion, the month of ember in Pergamino is analyzed in detail. This month/location has a clear ENSO signature (Fig. 2), a large empirical standard deviation of monthly precipitation totals (79.5 mm), and considerable overdispersion; the standard deviation for the induced conditional model is 63.6 mm, or about 80% of the empirical value. The variance of monthly precipitation totals can be decomposed into two terms: the variance of the number of wet days and the variance of a sum of daily intensities [Katz and Parlange 1998, their Eq. (2)]. The observed standard deviation of the number of wet days for Pergamino/ember is 3.1 days. The corresponding value derived from the induced model is 2.9 days (about 94% of the empirical value). Although some overdispersion remains, the conditional occurrence model seems to be reproducing reasonably well the variability in the number of wet days. Another possible reason for the overdispersion is the model for the intensity process. The large standard deviation of monthly precipitation totals for Pergamino/ ember is associated with two values that stand out from the main distribution, which has a median of 88 mm (histogram not shown). These values correspond to 1940 (449.3 mm), a warm ENSO event, and 1936 (345.9 mm), a neutral year. The influence of these extreme values can be assessed through the biweight standard deviation, a resistant (i.e., less influenced by outliers) estimator of dispersion (Lanzante 1996). The biweight standard deviation of monthly precipitation totals for Pergamino/ember is 63.1 mm, or about 79% of the classic standard deviation. When daily records are examined, it is apparent that the large ember totals for 1936 and 1940 are associated with a few days with very large precipitation. In 1940, daily rainfall for 6 and 28 ember was and mm, respectively. These two days, thus, accounted for 54% of the very large monthly rainfall. In 1936, the precipitation on 28 ember was mm (66% of the monthly total). These values are the three highest daily precipitation amounts for ember for the historical Pergamino record used here ( ). To explore the influence of extreme daily rainfall amounts on monthly totals and their dispersion, the top 0.2% daily precipitation values were excluded from the Pergamino/ember series. Four days were excluded: those listed above, plus 10 ember 1963 (105.9 mm). After the exclusion, the recomputed monthly standard deviation was 58.1 mm, or about 73% of the original standard deviation. That is, the exclusion of only a few extreme daily precipitation amounts caused a 27% decrease in the standard deviation of monthly totals. This example illustrates the large influence that a few extreme daily precipitation values may have on monthly totals and their variance. In previous sections, it was noted that extreme daily values are not well described by stochastic precipitation models, whether simple or ENSO conditioned. The overdispersion phenomenon, therefore, can be at least partly attributed to inadequate modeling of the daily intensity process. 6. Simulated monthly rainfall distributions Previous sections demonstrated the advantages of conditional precipitation models for some locations and months. Here, the consequences of ENSO-conditional parameterization are illustrated through simulated distributions of monthly precipitation totals. Daily precipitation series are simulated for Pergamino/ember using 1) a simple model and 2) a model with conditional

12 2984 JOURNAL OF CLIMATE VOLUME 13 FIG. 6. Empirical density of simulated monthly precipitation totals for in Pergamino, Argentina. Conditional models were used to generate 400 yr of synthetic precipitation values for each ENSO phase. Densities are plotted separately for warm (long dashes), neutral (dots) and cold (solid gray line) phases. Also shown is the empirical density for 1200 rainfall values simulated with a simple model (solid black line). persistence and scale parameters. For each ENSO phase, 400 yr are simulated. Simulated daily values are then aggregated into monthly totals. Distributions of monthly precipitation totals are shown as empirical density plots, which can be interpreted as smoothed histograms (Fig. 6). A nonparametric kernel density smoother with bandwidth obtained via biased cross validation is used (Venables and Ripley 1997). As the simple model has a single set of parameters, it is represented by a single line, which describes the distributions of monthly precipitation totals for 1200 simulated yr. The distributions simulated by the conditional model clearly separate by ENSO phase (Fig. 6). The distribution for cold events has both lower central tendency and spread than those for the other phases. The distribution for warm events mostly overlaps that for neutral years, although there is a larger proportion of higher values. The simulated distributions are consistent with historical observations (cf. ember panel in Fig. 2), indicating that the conditional models succeed in capturing the differential nature of precipitation processes among ENSO phases. In contrast, the unconditional model underrepresents the frequency of both low and high monthly precipitation totals, as previously noted by Wilks (1989) and Katz and Parlange (1993). Conditional stochastic precipitation generators, therefore, offer clear advantages for assessing climatic risk to agriculture during extreme ENSO phases. A fundamental problem in using observed weather sequences to estimate the climate risk on agriculture is that the historical data represent only one realization of the weather process (Richardson 1981). Stochastic generators can provide a larger number of equally likely weather sequences, but one needs to verify that the simulated series are consistent with observed data. If the FIG. 7. Q Q plot of historical and simulated monthly precipitation totals for in Pergamino, Argentina (open circles). An empirical 90% envelope (dashed lines) is constructed by drawing from the simulated data samples of length equal to and with the same phase proportions as the historical series model. historical observations fall outside an envelope of variability of simulated values, then the stochastic generator is not producing realistic results (i.e., historical data are not a likely realization of the process described by the generator). To verify this, quantiles of monthly precipitation totals for the Pergamino/ember historical series are plotted against corresponding quantiles from a mixed simulated density of monthly totals (Fig. 7, open circles). The mixed simulated density is obtained by combining the simulated densities for each ENSO phase (Fig. 6) with mixing weights equal to the relative climatological frequencies of each phase (0.19, 0.58, and 0.23 for warm, neutral, and cold phases). A 90% envelope is obtained by randomly selecting, from the 1200 yr of simulated data for all ENSO phases, 100 simulated series of length equal to, and with the same phase proportions as the historical series. Quantiles corresponding to 5% and 95% are extracted from the sorted 100 simulated series and are plotted (Fig. 7, dashed lines). If the Q Q points for the historical data fell outside the 90% envelope, this would suggest a possible lack of adequacy of the theoretical model. Instead, the observed monthly precipitation totals are consistent with the theoretical model. That is, the historical data can be one realization of the weather process described by the model. 7. Discussion Typically, parameters of stochastic weather generators have been fit unconditionally, that is, using all historical data for a given period. This approach, however, fails to capture differences in the precipitation process associated with an ENSO-related signal. To address this problem, stochastic precipitation generators conditioned on ENSO phase were developed. A range of precipi-