Efficient stochastic generation of multi-site synthetic precipitation data

Transcription

1 Journal of Hydrology (2007) 345, available at journal homepage: Efficient stochastic generation of multi-site synthetic precipitation data F.P. Brissette *, M. Khalili, R. Leconte Department of Construction Engineering, École de technologie supérieure, Université du Québec, 1100 Notre-Dame West, Montréal, Qc, Canada H3C 1K3 Received 31 October 2006; received in revised form 6 June 2007; accepted 30 June 2007 KEYWORDS Precipitation; Weather generator; Multi-site weather generator Summary Although weather generators have been used for a long time, recent years have seen a surge in interest in them as potential downscaling tools for climate change impacts studies. Weather generators can only generate weather data at a single point, or independently at several points, whereas many climate change impact studies require information at the basin scale. Such studies would require the coherent generation of weather data at several locations over a basin. The literature on multi-site weather data generation is very thin, with most of the work based on the approach described by Wilks (Wilks, D.S., Multi-site generalization of a daily stochastic precipitation generation model. J. Hydrol. 210, ). This approach demands significant work to set up, and may suffer from ill-defined correlation matrices due to noise in the observed data. In addition, multi-site generation must address the complex problem of spatial intermittence, in which precipitation amounts depend on neighbouring stations being wet or dry. This likely explains why multi-site generation, despite its obvious advantages, has not been widely used. This paper presents an algorithm for the efficient stochastic generation of multi-site precipitation data following the Wilks approach. The effect of noise on the performance of the algorithm is examined, and the results indicate that it performs very well, even with excessive noise added to the data. The algorithm is fast, simple, easy to implement, and significantly simplifies the generation of multi-site precipitation data for impact studies of climate scenarios. The spatial intermittence problem is dealt with by linking average precipitation to an index that describes the distribution of precipitation occurrence at the basin scale. ª 2007 Elsevier B.V. All rights reserved. * Corresponding author. Tel.: ; fax: address: fbrissette@ctn.etsmtl.ca (F.P. Brissette) /$ - see front matter ª 2007 Elsevier B.V. All rights reserved. doi: /j.jhydrol

2 122 F.P. Brissette et al. Introduction Weather generators are computer models that produce time series of meteorological data that have similar statistical properties as that of observed data. Precipitation, minimum and maximum temperature as well as solar radiation are the most commonly modelled variables, but weather generators can produce time series of any meteorological data as long as the observed time series is long enough to be analyzed in a statistically significant manner. The appealing property of weather generators is their ability to rapidly produce time series of infinite length, thus permitting impact studies of rare occurrences of meteorological variables. In theory, as long as the statistics of the observed data are well described, the use of synthetic time series with physical models (such as crop models or hydrology models) allows for the direct modelling of very large return period events such as floods or droughts, which would otherwise have to be extrapolated. Interest in weather generators surfaced in the 1980s, with the main focus being on agriculture and crop models. The past decade has seen a sharp and renewed increase in interest in weather generators, linked to their potential use in climate change studies. Of the many potential impacts linked to climate change, the increased frequency of extreme events is perhaps the most significant. Most of what is known about future climate is derived from the various coupled dynamic climate models operated by various agencies throughout the world. The computing costs of running these global climate models severely limits both the model resolution and the length of the time series produced by the models under different emission scenarios. Since most impact studies require localized weather data, with the addition of long-time series, in the case of rare events, direct outputs from climate models cannot be used directly for such tasks. The different techniques that use climate model data to produce weather data locally are collectively known as downscaling methods. There now exists extensive literature on downscaling, and many good reviews of downscaling methods can be seen in Kidson and Thompson (1998) Von Storch et al. (1993) and Wilby et al. (1998a,b), among others. For the purposes of this paper, downscaling methods at the daily time step can be grouped under 2 categories. The first category brings together techniques that rely on the direct use of climate model outputs. This category groups regional climate models and all statistical methods. The regional climate models use global model results as boundary conditions to a smaller and higher resolution domain, whereas the statistical methods attempt to extract linear or non-linear relationships between outputs from the global climate model and weather data at the local scale. The second category essentially comprises weather generators. Downscaling with a weather generator demands that the parameters governing the weather generation process be linked globally (and not on a daily basis) to results from the global model. Whereas all methods from the first category are limited to the length of the time series generated by climate models, weather generators can produce time series of any length, which is a significant advantage for the impact studies of rare events. It should be emphasized that a 1000-year time series in the case of a weather generator will not represent the climate into the next millennium, but would rather represent 1000 years of climate statistically similar to the period over which the weather generator parameters were established (typically years for climate change modelling). Weather generators There are currently many weather generators in existence, with the three main types being parametric, semi-parametric/empirical and non-parametric. Most parametric weather generators follow the WGEN model (Richardson, 1981; Richardson and Wright, 1984), where precipitation occurrence is modelled with a Markov chain, and precipitation amounts follow a given distribution function generally, the one-parameter exponential distribution, twoparameter gamma function or three-parameter mixedexponential distribution. The best-known semi-parametric/empirical weather generator is LARS-WG (Semenov and Barrow, 1997). In LARS-WG, precipitation occurrence and amounts are represented by histograms fitted to data. In the case of precipitation amounts, the 11 bin histograms in fact represent a 23-parameter distribution function (12 bin location parameters and 11 magnitude parameters). The last type is essentially a resampling approach (Brandsma and Buishand, 1997; Buishand and Brandsma, 2001). Most other existing weather generators are based on the first two types presented above (for example, Hayhoe, 2000; Corte-Real et al., 1999). Comparisons between different weather generators and their ability to reproduce the statistical behaviour of observed data at different stations have been carried by Semenov et al. (1998) and Qian et al. (2002), among others. A review of weather generators can be found in Wilks and Wilby (1999). Weather generators and climate change Compared to the abundant published literature on statistical downscaling, there has been far less work done on how to use weather generators for climate change impact studies. Considering the relatively disappointing performance of statistical downscaling techniques when it comes to precipitation, it appears that the conditioning of weather generators to climate change studies should warrant more interest. The LARS-WG software has the ability to simulate future climate with the use of monthly DP and DT incremental factors derived from the Hadley Center climate model. However, the semi-empirical structure of LARS- WG does not easily lend itself to more complex conditioning. Bardossy and Plate (1992), Bogardi et al. (1993), Corte-Real et al. (1999), Brandsma and Buishand (1997), Qian et al. (2002) and Wilby et al. (2002) have all presented work on the conditioning of weather generators on circulation patterns or synoptic scale atmospheric information. However, for many climate change impact studies, information about weather parameters must be known at several locations, and not just at a single site. For example, water

3 Efficient stochastic generation of multi-site synthetic precipitation data 123 resource studies must generally be performed at the watershed scale. For all but the smallest watersheds, singlesite precipitation will not adequately represent the basin hydrology. Flooding related to convective precipitation may be linked to a much localized portion of the watershed, and so using single-site precipitation for an entire watershed (or several unrelated single-site precipitations) may result in a severe overestimation or underestimation of potential impacts. For these reasons, it can easily be argued that a consistent multi-site generation of weather data should be an important feature of weather generators for climate change impact studies, and particularly for precipitation. While temperature and solar radiation may indeed be relatively easily extrapolated from a single site by taking into account geographical coordinates, elevation and slope aspect, precipitation, which can rapidly vary in space and in time, constitutes a much bigger challenge. All the above reasons explain why this paper focuses on multi-site precipitation. Multi-site generation of precipitation data Despite the importance of accurately representing the spatial variability of precipitation for impact studies, little work has been devoted to the multi-site generation of precipitation. As discussed in Qian et al. (2002), over the past 20 years, there have been a few attempts made to include multi-site generation into stochastic weather generators, but such attempts have however had little success in reproducing observed joint statistics. Wilks (1998, 1999) presented a first method that adequately reproduced the main statistics of multi-site precipitation data. This method has been also been used by Qian et al. (2002) and Khalili et al. (2004). To date, this is the only method that has been successful in that regard. An exception would be the resampling method of Brandsma and Buishand (1997), in which they isolate multi-site precipitation data from the historical records based on the closest similarity with atmospheric circulation indices. This method cannot however produce precipitation events larger than those on historical record, and is therefore not particularly conducive to climate change impact studies involving extreme precipitation. Based on serially independent, but spatially correlated, random numbers, the Wilks method works well with parametric weather generators such as WGEN (Richardson, 1981). Despite its potential usefulness, we are not aware of the Wilks method having been used in any work other than those of Qian et al. (2002) and Khalili et al. (2004). There may be several reasons for that. The method involves significant work as it involves producing nðn 1Þ=2 empirical curves, corresponding to all possible pairs taken from n stations. Mathematical problems may likely arise due to the pairwise construction of correlation matrices, resulting in non-positive-definiteness problems. Lastly, the length and depth of the Wilks papers allow no more than a brief description of the mathematical details linked to the method. This paper aims to improve on the Wilks approach by presenting a more efficient computing approach, and by discussing problems of noise in correlation matrices, and how to deal with such ill-defined matrices mathematically. The mathematical process is described in detail in order to ensure that readers from different backgrounds can easily reproduce the results presented in the paper. Finally, the problem of spatial intermittence, discussed in detail by Wilks (1998) and Bardossy and Plate (1992), is dealt with by conditioning the precipitation amounts on the spatial distribution of precipitation occurrence. The Wilks approach The Wilks approach consists in driving the stochastic weather generator with serially independent and spatially correlated random numbers. Each set of serially independent random numbers is then used to generate precipitation occurrence and amounts at a particular station. In essence, Wilks approach concentrates on the bi-variate description of the precipitation field at given rain gauges, expressed by the correlation matrices of both precipitation occurrence and amounts at weather stations. Let n be the number of precipitation recording stations and m the length of the synthetic precipitation time series to be generated. C is the observed correlation matrix of precipitation occurrence or amounts: r 1;2 r 1;n 1 r 1;n 6 r 2;1 1 r 2;n 1 r 2;n 7 C ¼ r n 1;1 r n 1;2 1 r n 1;n r n;1 r n;2 r n;n 1 1 There are several methods that can be used to generate spatially correlated random fields (that have the same covariance/correlation matrix). For relatively small size arrays (which would be the case for weather stations mathematically a small size array would be n <10 3 ), the simplest approach is through matrix decomposition (Fang and Tacher, 2003). For a positive-definite matrix (its implications will be discussed in greater detail later), the Cholesky factorization of the correlation matrix C results in an upper triangular matrix R so that C = R 0 R. The Cholesky factorization is a standard function of most mathematical computing packages. Multiplying the lower triangular matrix obtained from the Cholesky factorization (R 0 )bya[n,m] matrix of random, normally distributed numbers N, results in a field with the same covariance/correlation structure as the observed data. The R 0 N product can be presented in greater detail as 2 3 R 0 1; R 0 2;1 R 0 2; R 0 n 1;1 R 0 n 1;2 R n 1;n R 0 n;1 R 0 n;2 R 0 n;n 1 R 0 n;n 2 3 N 1;1 N 1;2 N 1;m 1 N 1;m N 2;1 N 2;2 N 2;m 1 N 2;m ð2þ N n 1;1 N n 1;2 N n 1;m 1 N n 1;m 5 N n;1 N n;2 N n;m 1 N n;m 7 5 ð1þ

4 124 F.P. Brissette et al. These correlated random numbers are then fed into the weather generator. Precipitation occurrences X t are generated from a Markovian process, using conditional probabilities obtained from the observed data (following the Richardson (1981) approach). If a 1st order Markov chain is used, conditional probabilities p00 (probability that a dry day is followed by another dry day) and p10 (probability that a wet day is followed by a dry day) must be defined. For higher order Markov chains, additional transitional probabilities need be defined, but the process remains the same. If the uniform distribution random number exceeds a critical probability p c, the following day is wet, and vice-versa if it is below p c. Mathematically, this is expressed as p c ¼ p 00 if X t 1 ¼ 0 ð3aþ p 10 if X t 1 ¼ 1 X t ¼ 0 if U t 6 p c ð3bþ 1 otherwise A problem arises in that the normally distributed random numbers generated using Eq. (1) are normally distributed whereas p 00 and p 11 Eqs. (3a) and (3b) require a uniform random number. The simplest solution is to transform the normally distributed random numbers into uniform ones using the cumulative distribution function. Alternatively, p 00 and p 11 could be transformed into normally distributed numbers. A second problem arises in that the generated occurrences are less correlated than the observed ones. In other words, in order to generate occurrences with the same correlation as the observed ones, it is necessary to use random numbers that are more correlated than the occurrences. Wilks (1998, 1999) used empirically-derived curves for each pair of stations to relate the needed correlation of random numbers to the observed correlation of occurrences. Computing-wise, the approach is not efficient since nðn 1Þ=2 curves (corresponding to all possible station pairs) must be constructed for each monthly period. This process is also difficult to automate and incorporate into a weather generator. To reduce the amount of work linked to this process, Khalili et al. (2004) explored the possibility of using regional curves instead of all possible station pairs. Despite some improvements, their approach still retains some of the computational downsides of the Wilks approach. This paper presents numerical approaches to solve this seemingly simple, but work-intensive problem. Essentially, a correlation matrix of serially independent random numbers C R must be defined, that produces (through a Markov process) a correlation matrix of synthetic occurrences or quantities C x,syn identical to the correlation matrix of observed occurrences or quantities C x,obs. This can be seen as an optimization problem, where a matrix C R that minimises a function relating C x,syn and C x,obs must be found. One such possible function is f ¼ Xn i;j¼1 jc x;obs C x;syn j For n stations, the symmetrical matrix C R contains nðn 1Þ 2 initially unknown elements corresponding to the summation of all the elements above the diagonal. Practically, this can be tackled with classic optimization methods. However, all such methods will disrupt the initial correlation matrix, ð4þ possibly in a manner inconsistent with the physics of the correlations, likely resulting in a non-positive-definite matrix, and thus not solvable with the Cholesky factorization. Correlation matrices and positive definiteness True correlation matrices must, by definition, be positivedefinite. This implies mathematically that all the eigenvalues of C are real and non-negative. Conversely, this is equivalent to saying that the matrix determinant (and all its minors) must be positive. Even real world measured correlation matrices may sometimes not respect this criterion due to excessive noise or to a few outliers (Rebonato and Jäckel, 2000). Programs that estimate correlations on a pairwise basis are likely to yield correlation matrices that are not positive-definite. Wothke (1993) discusses several aspects of the problem in detail. Linear systems of equations and correlation matrices are often by nature ill-conditioned. An ill-conditioned system is one in which small perturbations result in large outcome changes. Very little noise may result in a negative eigenvalue, particularly if small eigenvalues are already present, as would be the case for large matrices including little uncorrelated data, and for large arrays of weather stations. It may be possible to compute all correlation coefficients simultaneously using a Maximum Likelihood or Bayesian approach, but this would greatly complicate the approach, and would badly slow down the algorithm. It would be desirable to keep the pairwise computation of correlation coefficients. In the Wilks (1998) approach, this problem was encountered with the random number correlation matrix used in the generation of precipitation amounts. It can be shown however, that this problem is essentially linked to the initial conditioning of the precipitation amounts correlation matrix and to noise added during the generation of the nðn 1Þ=2 empirical curves. If the original correlation matrix of precipitation amounts is badly conditioned, then the random number correlation matrix needed in the Wilks approach will also be badly conditioned. The reverse is also true. In our experience, we see that while the correlation matrices of precipitation occurrence tend to be positivedefinite, those of precipitation amounts tend to be more badly conditioned. In order to circumvent the problem, Wilks (1998) and Qian et al. (2002) modelled the pairwise correlations of random numbers as a function of site separation using an exponential equation involving horizontal distance (instead of directly using the values from the nðn 1Þ=2 empirical curves). While this smoothes the data sufficiently to induce positive definiteness, it can also result in an unacceptable smoothing of the solution. In the case of mountainous watersheds, information about altitude and slope aspect might be needed to prevent unacceptable smoothing. In many mountainous watersheds, weather stations are often concentrated in lower elevation areas. As such, site separation functions would be biased toward correlations observed between these stations. Smoothing with such functions could result in spurious correlations between nearby stations at different elevations that would otherwise not be correlated as much. They would have obtained similar results with a direct smoothing of the correlation matrix of

5 Efficient stochastic generation of multi-site synthetic precipitation data 125 observed precipitation amounts (sufficient smoothing of the observed correlation matrix of precipitation amounts would have resulted in a semi-positive-definite random number correlation matrix). In addition to this problem, iterative numerical algorithms (such as the one proposed here) may very well start with a positive-definite matrix and reverse this condition during the iterative process. This is in fact very likely if the initial matrix initially contains small eigenvalues. If the initial correlation matrix of observed precipitation amounts is initially badly conditioned, the problem will almost certainly appear. Since a true correlation matrix can only be non-positive-definite as a result of noise and errors in the data, it is critical that this problem be dealt with while preserving as much information as possible. Rebonato and Jäckel (2000) present a general methodology to ensure the creation of a valid correlation matrix from an initially ill-defined one. The procedure involves the diagonalization of the matrix and the replacements of all negative eigenvalues with a small positive value. With D as the diagonal matrix of eigenvalues and M as the matrix of column eigenvectors, the modified positive-definite correlation matrix C r is then computed as C r ¼ MDM 1 ð5þ Strictly speaking, this operation results in a covariance matrix, since the diagonal elements will be slightly different from 1. Preferably C r should be normalized using the standard equation: C r C r ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð6þ diagðc r ÞdiagðC r Þ 0 where diag(c r ) is the column vector of the diagonal elements of C r. This step can be avoided if a decision is made to work with the covariance matrix instead of the correlation matrix. Alternatively, the eigenvalues could be renormalized to the proper variance prior to the application of Eq. (5). This approach may seem empirical at first sight, but it must be realized that negative eigenvalues cannot exist in a true correlation matrix, and that their removal has a physical sense in addition to allowing a subsequent Cholesky factorization of the modified matrix. The computation of eigenvalues adds to the computational burden, but for weather station networks, the number of stations for multi-site generation in most realistic cases should be a few dozens at most, and in most cases, less than 20 for watershed-based studies. Continental-size watersheds may certainly contain a few hundred stations, but stations as close as a few hundred kilometres away (and much closer in mountainous watersheds) may not exhibit any correlation at all. In any event, by today s standards, a few hundred stations remains a relatively easy computational problem, especially considering that the correlation structure of the random numbers C R needs to be computed only once. Every subsequent weather generation will use the same matrix. A first algorithm was constructed using the basic simplex optimization method. The nðn 1Þ 2 elements of the upper triangle of the random number correlation matrix were the unknown values to be evaluated, with Eq. (3) as the minimizing function. The starting point for the iterative process was the observed correlation matrix of rainfall occurrence or quantity. The algorithm worked quite well with up to six stations. With a higher number of stations, it became significantly slower and likely to converge on a local minimum instead of on the true solution. A brute force algorithm was then developed to ensure robustness and convergence for any number of stations. Again using the observed correlation matrix of rainfall occurrence or quantity as a starting point, random numbers were generated and fed through a Markov process to generate synthetic occurrence or quantity. The generated correlation matrix of this synthetic data will initially be too small (since the needed correlation of random numbers must be higher than the initially used observed correlations of occurrence and amounts). The starting correlation matrix used to initiate the scheme is then modified using the following perturbation equation, where subscript i denotes the iterative step: C Riþ1 ¼ C Ri þ gðc x;obs C x;syn Þ ð7þ g is a convergence criterion whose value affects the convergence speed. Setting a high value results in rapid convergence at the expense of precision. A value of 0.1 was found to yield a reasonable convergence speed, while giving a precision greater than 10 3 for each synthetic correlation coefficient. Dataset In order to examine the performance of the algorithm and the effect of noise present in the correlation matrix, it is necessary to work with an initially noise-free correlation matrix. The only way to produce such a matrix is to create a synthetic dataset. A synthetic array of 20 weather stations was randomly distributed over a synthetic digital elevation model, as shown in Fig. 1. The inter-station correlation q between occurrence and amounts was generated with the following function: Algorithms Figure 1 Network of virtual stations. The 20 stations are labelled A to T.

6 126 F.P. Brissette et al. q ¼ e a 0DH a 1 a 2 DZ þ a 3 Nð0; 1Þ ð8þ where DH and DZ represent absolute differences in horizontal and vertical distances, respectively in km and m. The first part of the equation is similar to that used by Wilks (1998), to which the second term was added to take into account the elevation difference. The third term is Gaussian noise, which is the result of sampling bias, difference in measuring equipment, local effects, missed data and other errors linked to precipitation measurement. Although Eq. (8) is physically based, its form is not of great importance since the goal is only to create synthetic correlation matrices with characteristic similar to observed one. Eq. (8) creates larger correlations for close stations and progressively smaller ones for more distant stations (distance- or elevation-wise). In Fig. 1, stations C and D are the most correlated, while stations B and T are the least correlated. To simplify the analysis and presentation of results, no seasonal variability of the parameters of Eq. (8) was introduced. Table 1 represents the constant values used to generate the correlation matrices for both occurrences and amounts. Different values assigned to the a 3 noise factor results in the correlation matrices being non-positive-definite (a 3 = 0.08 for the occurrence and a 3 = for the precipitation amounts. This will result in correlation matrices of random numbers that are also non-positive-definite. Constant values Table 1 Value of constants (Eq. (8)) for calculation of correlation matrices of precipitation occurrence and amounts Constant values in Eq. (8) a 0 a 1 a 2 a 3 Occurrence , 0.025, 0.08 Amount , Varying amounts of noise were used, which explains the multiple values of coefficient a 3. used in Eq. (8) result in positive correlation values varying from nearly 0 to about 0.9. In order to create synthetic rainfall data that closely respects the above correlation structure, rainfall parameters must be specified. The precipitation occurrence will follow a 1st order Markov process with transition probabilities p00 and p10. The transition probabilities were specified following the exact procedure of Richardson (1981) using 0.25 mm as a precipitation threshold for a wet day. The rainfall amounts distribution function is modelled with the simple exponential distribution with distribution and cumulative distribution functions: fðxþ ¼ke kx ð9aþ FðxÞ ¼1 e kx ð9bþ where k =1/l with l the sample mean. Parameter k was modelled as a simple function of elevation z: k = z. All relevant parameters for the 20 virtual stations are presented in Table 2. The selected parameters result in higher elevation stations being wetter, and easterly stations being dryer. It is important to note that precipitation parameters (occurrence and amount) must be consistent with the synthetic correlation matrix in order to avoid physically impossible situations. Specifying a very high correlation between 2 stations having significantly different precipitation parameters would constitute such a situation. Driving those 2 stations with identical random numbers would still result in a low correlation of occurrence and amounts. Two stations driven with identical random numbers will normally always have correlations of occurrence and amounts smaller than 1, unless all other parameters (transition probabilities and distribution function) are also equal. In theory, this would be the case for 2 stations that are located very close to each other, but in fact, even if 2 rain gauges were placed 1 m apart, the two would not exactly register the same amounts, and, in some cases, the same occurrence. This synthetic dataset does not address the spatial intermittence problem as precipitation amounts are not linked in any way to the precipitation occurrence at neighbouring stations. The spatial intermit- Table 2 Characteristics of virtual stations: X, Y and Z coordinates, 1st order Markov process transition probabilities p10 and p00 and exponential distribution parameter k A B C D E F G H I J Y (km) X (km) Z (m) p p k K L M N O P Q R S T Y (km) X (km) Z (m) p p k

7 Efficient stochastic generation of multi-site synthetic precipitation data 127 tence problem will be discussed in reference to observed data in a subsequent section. Synthetic data: Results and discussion In Figs. 2 6, the term observed refers to the target synthetic occurrence and precipitation amounts correlation Figure 5 Left: observed correlation of occurrence vs. generated correlation using the stochastic generator, for all station pairs. Right: observed correlation of occurrence vs. the needed random number correlation, for all station pairs. Both graphs computed with large random noise in the correlation matrix (a 3 = 0.08). Figure 2 Left: observed correlation of occurrence vs. generated correlation using the stochastic generator, for all station pairs. Right: observed correlation of occurrence vs. the needed random number correlation, for all station pairs. Both graphs computed without noise (a 3 = 0). Figure 3 Left: observed correlation of amounts vs. generated correlation using the stochastic generator, for all station pairs. Right: observed correlation of amounts vs. the needed random number correlation, for all station pairs. Both graphs computed without noise (a 3 = 0). Figure 4 Left: observed correlation of amounts vs. generated correlation using the stochastic generator, for all station pairs. Right: observed correlation of amounts vs. the needed random number correlation, for all station pairs. Both graphs computed without noise (a 3 = 0) in the occurrence correlation matrix and random noise (a 3 = 0.045) for the precipitation amounts correlation matrix. Results are nearly identical to that obtained with noise (a 3 = 0.025) in the occurrence correlation matrix. Figure 6 Left: observed correlation of amounts vs. generated correlation using the stochastic generator, for all station pairs. Right: observed correlation of amounts vs. the needed random number correlation, for all station pairs. Both graphs computed with large random noise (a 3 = 0.08) in the occurrence correlation matrix and random noise (a 3 = 0.045) for the precipitation amounts correlation matrix. Compare with results from Fig. 4. matrices. This data is not actual observed field data, but rather, it aims to represent such data. The term generated refers to the actual synthetic precipitation data generated using spatially correlated random numbers. The correlation matrices (occurrence and amounts) of the generated precipitation are aimed at reproducing the observed ones. Figs. 2 and 3 present results in the absence of external noise (a 3 = 0 for both occurrence and amounts), for the generation of occurrence and precipitation amounts, respectively. The left figure presents the observed correlation compared to the generated one for all 190 station pairs. The 45 C perfect fit line is shown. The figures on the right present the correlation of random numbers needed to generate the observed correlation. At each iteration step, 10,000 days were generated for each station. This high number of days was chosen in order to eliminate random effects that would be linked to a simulation length that is too short. Total CPU time for the algorithm was 10 s for the occurrence and 26 s for the precipitation amounts on a modest mobile 2.2 GHz Pentium processor running MATLAB v.6. The computational cost of the proposed approach is negligible considering that it allows the user to completely bypass

8 128 F.P. Brissette et al. the construction of nðn 1Þ 2 inter-station curves and subsequent modelling of pairwise correlations using distance separation functions, as used by Wilks (1998). Results indicate that in the absence of noise, it is possible to exactly recreate the observed correlations for both occurrence and precipitation amounts (Figs. 2 and 3). For both occurrences and precipitation amounts, the random number correlation matrix was never rendered non-positive-definite in all the iteration steps. The figures on the right indicate that the random number correlation must indeed be larger than the correlation of occurrence and amounts. They also show in this case that there is a general regional trend present in the relationship although significant local variability is involved. Figs. 4 6 present results in the presence of variable noise in the correlation structure of both occurrence and precipitation amounts. Fig. 4 presents the results for precipitation amounts in the case where no noise was present in the occurrence correlation matrix, but where noise was added to the correlation matrix of precipitations amounts (a 3 = 0.045). In this case, the added noise renders the correlation matrix non-positive-definite. This calls for the Rebonato and Jäckel (2000) procedure in order to start the iteration scheme (since the scheme starts with the observed correlation matrix) and for all subsequent steps. Fig. 4 indicates that although the general trend is well replicated, it is now impossible to exactly retrieve the observed correlation matrix of precipitation amounts. This should not however be interpreted as a flaw in the algorithm, but rather, as a normal consequence of starting with an ill-defined correlation matrix. Since the initial non-positive-definite matrix cannot by definition be the true correlation matrix, the generated correlation matrix (although not equal to the observed one) should be considered as the best possible candidate (the closest positive-definite matrix). In our test cases, the diagonalization process took less than 5% of the total algorithm time. Fig. 5 presents occurrence results with large noise added to the observed occurrence matrix (a 3 = 0.08). Such noise results in an absolute error in the correlation coefficients of up to 0.22, and renders some coefficients negative. This a case where the observed correlation matrix is very grossly incorrect, and would likely only occur in a network of stations with significant errors present in the data (large amount of missing data or incorrect date stamp in some stations). The correlation matrix yields 4 negative eigenvalues, and is significantly non-positive-definite (compared to only one very close to zero negative eigenvalue in the preceding case). The algorithm does a good job of reproducing the observed correlation. Once again, the scatter should be interpreted in the context of starting with pairwise computed correlation coefficients that result in a matrix that cannot be a true correlation matrix. A slight bias can be observed for higher correlation coefficients, in which case generated coefficients are lower than observed ones. This is the result of the strong diagonalization needed to ensure the positive definiteness of the random number correlation matrix that diminishes larger values proportionally more than smaller ones. Fig. 6 presents a last test case with the same amount of large random noise as Fig. 5 for the generation of occurrences (a 3 = 0.08), and with significant noise in the correlation matrix of precipitation amounts (a 3 = 0.045). It is interesting to note that the generated precipitation amounts exhibit a very similar behaviour to that observed in Fig. 4. This indicates that the performance for the generation of precipitation amounts is largely independent of the performance for the occurrence generation process. The generation of precipitation amounts using perfectly correlated occurrences (Fig. 4), or with large noise (Fig. 6) are very similar. There is however much more scatter in the correlation of random numbers needed, which indicates that the algorithm can compensate for the generation of flawed sequences of occurrence. As such, a good performance in the generation of precipitation amounts will not necessarily mean that occurrences have been correctly simulated. Since most impact studies require that both occurrences and amounts be properly described, care must be taken to ensure that both are adequately generated. Observed data: Results and discussion The performance of the algorithm is tested on an array of eight stations located within or close to the Peribonka River Figure 7 Chute-du-diable watershed, southernmost subwatershed of the Peribonka River, with locations of weather stations. See Table 3 for station coordinates and elevation.

9 Efficient stochastic generation of multi-site synthetic precipitation data 129 Table 3 Main characteristics of weather stations labelled 1 8 in Fig. 7 ID Name Longitude Latitude Elevation 1 Péribonka Saint-Prime Hémon Bonnard Chute-du-diable Chute-des-Passes Saint-Léon Normandin Figure 9 Joint precipitation at stations 1 and 3, observed (left hand side) and generated (right hand side). Basin, in Quebec, Canada (Fig. 7). The stations main characteristics are presented in Table 3. Data used in this work covered the period, although there is only a 14- year window during which all stations were simultaneously active. Fig. 8 presents results for the generation of occurrence for all months and all station pairs. The algorithm is able to compute correlation matrices of random numbers from which precipitation occurrences can be generated with correlation matrices identical to those observed (up to the error tolerance specified in the algorithm). The near perfect fit indicates that the observed matrices are well conditioned. Problems arise when trying to generate precipitation amounts. The algorithm fails to converge as many of the elements of the correlation matrix of random numbers approach unity. The problem can be better described by looking at Fig. 9, which shows a joint precipitation plot for stations 1 and 3 for the month of January. This Figure shows both dry and wet day precipitation amounts. The graph on the left (observed data) clearly indicates that precipitation amounts are much likely to be larger if there is rainfall occurring at a neighbouring station than when it only rains at a single site. This leads to the spatial intermittence problem discussed before. Unless this problem is addressed, precipitation amounts are generated using the same precipitation distribution, whether or not precipitation is occurring at neighbouring stations. This leads to what is observed on the right hand graph of Fig. 9. Since large precipitation amounts at a single site result in a lower correlation of precipitation amounts, the algorithm compensates by increasing the random number correlation. In some cases, this is a lost cause, and driving both stations with identical random numbers will still result in correlations of precipitation amounts being smaller than the one observed. Even if the observed correlation is reproduced, the generated precipitation field would be totally different from the one observed. A simple solution to this problem would be to compute the correlation matrix using only precipitation data when there is joint occurrence at each pair of stations. In this case, the algorithm is able to reproduce the observed correlation of precipitation amounts (Fig. 10) up to the algorithm tolerance. The monthly correlation matrices of precipitation amounts were mostly well conditioned, but some rediagonalization was needed to avoid problems with non-positive-definite matrices. Problems were minor because the array was relatively small, and there was severe data quality control. However, this would still lead to inadequate precipitation fields with some single-site precipitation (conditional to a neighbouring station being dry) being too high. As mentioned by Wilks (1998): Failure to address this point leads to unrealistically sharp transitions between wet and dry portions of the spatial domain. In Figure 8 Left: observed correlation of occurrence vs. generated correlation using the stochastic generator, for all months and station pairs, for the Peribonka River Basin stations. Right: observed correlation of occurrence vs. the needed random number correlation, for all months and station pairs. Figure 10 Left: observed correlation of amounts vs. generated correlation using the stochastic generator, for all months and station pairs, for the Peribonka River Basin stations. Right: observed correlation of amounts vs. the needed random number correlation, for all months and station pairs. Correlations were computed using only data with joint occurrence for all station pairs.

10 130 F.P. Brissette et al. other words, reproducing bi-variate dependence between stations is necessary but not sufficient to adequately reproduce precipitation fields. Spatial properties of precipitation fields must also be accounted for. To tackle this problem, Wilks (1998) used a mixed-exponential distribution and conditioned the generator to link the choice of the scale parameters from the mixed distribution (small or large) to the occurrence process. This was done while preserving the unconditional precipitation mean. However, from Wilks account, this approach did not produce dependence between the synthetic precipitation amounts and occurrences that is as strong as in the observations. One of the drawbacks of Wilks approach is that it is applied on a pairwise basis whereas it would be preferable to use information at the basin scale. In other words, the precipitation amount at a single station would be expected to be related to occurrence at the basin scale. To study the dependency between precipitation amounts and occurrence at the basin scale, an occurrence index K is defined and calculated at each station m and for each day where precipitation at this station is non-zero: K m ¼ O C0 U C 0 ð10þ where O is the occurrence vector for every station but station m, and C is a vector containing the correlation between the occurrence at station m and every other station. For example, if the correlation vector between occurrences at station 1 and six others on the basin is equal to C = [ ], and the occurrence vector at other stations (on a day where station m is rainy) is equal to O = [ ], the occurrence index will be equal to: ( )/( ) = 2.2/3.2 = U is a vector of unit values the same length as vector C. The occurrence index describes the spatial distribution of precipitation occurrences at the basin scale from the viewpoint of station m. K values will vary between 0 (just one station displaying precipitation) and 1 (all stations having precipitation). The correlation of the occurrence vector is a weighting function that takes into account the fact that an occurrence at a highly correlated station is more significant than with a station that is not very correlated. While a weight vector could be defined based on station distance for example, the use of observed correlation values makes sense since they are physically-based and readily available. Fig. 11 displays the seasonal link between the precipitation mean and occurrence index. To ensure the computation of a meaningful value of K, a minimum of 6 active stations (out of 8) was specified for each day. This restricted the amount of usable data, and a seasonal analysis was preferred in order to reduce data scatter. Even in this case, we have less than 1000 days to work with (about 10 years for each season), which should be considered a minimum for such an analysis. K values were separated into 9 approximately uniform classes (to insure a minimal amount of data within each class) and precipitation averages were computed for each class. A second degree polynomial function was fitted to the data. A more complex function could have been used since a careful examination of Fig. 11 for the spring and fall seasons reveal that the fitted function is Figure 11 Seasonal dependency of mean precipitation and occurrence index (Eq. (10)) for station 1. non-monotonic at low K values. A strong relationship exists between the occurrence index and the average precipitation for all seasons and all stations. Fig. 12 shows that rainfall amounts follow an exponential distribution when classified according to the value of the occurrence index. This graph uses four classes of occurrence values (instead of nine as in Fig. 11) to allow for a better representation of the exponential distribution within each class. This allows a straightforward definition of a multi-exponential distribution to describe global rainfall amounts at each station: DðxÞ ¼ Xn i¼1 a i ð1 e k ix Þ with Xn i¼1 a i ¼ 1 ð11þ The parameters are all readily defined based on a best-fit line drawn from data in Fig. 11, with n being the number of classes (9 in this case), the a i coefficients being equal Figure 12 Histograms of rainfall distribution as a function of occurrence index for the months of June, July and August (JJA) for station 1.

11 Efficient stochastic generation of multi-site synthetic precipitation data 131 to the ratio of the number of rainfall events in each class to the total number of events, and the k i being equal to the inverse of the precipitation mean for each class. In order to verify that the above distribution fits the data satisfactorily, its cumulative distribution function (CDF) was plotted in Fig. 13 with the exponential CDF and the empirical CDF obtained from the data. For this example, the multi-exponential CDF fits the data better, which at first should not appear surprising since it contains more parameters than the exponential distribution. However, this distribution function was derived from a small subset of the data, and a best-fit line was used to reduce the observed scatter. For other periods and stations, the exponential distribution (fitted to the entire dataset at each station) performs slightly better. Overall, both functions perform similarly in fitting the observed data. Moreover, it should be reemphasized that the purpose of the multi-exponential function is not to better fit the data observed at each station, but to allow a coherent generation of multi-site precipitation amounts. The k i parameters in Eq. (11) were adjusted to the observed precipitation mean. The adjustment is minor and is attributable to errors introduced by the data fitting shown in Fig. 11. Multi-exponential distribution functions based on the occurrence index were derived for each station and each season, and precipitation occurrence and amounts were simulated for a period of 100 years. Fig. 14 indicates that the precipitation mean is well reproduced, with small deviations occurring as a result of the stochastic nature of the process. Inter-station correlations for precipitation amounts are now perfectly reproduced, indicating that the algorithm works well and that any convergence problem linked to spatial intermittence is now solved. To further validate the approach and ensure that it deals appropriately with the spatial intermittence problem, the continuity ratio defined by Wilks (1998) was computed for all stations pairs: Figure 14 Comparison of synthetic to observed precipitation (left) and inter-station correlation (right), for all seasons, stations (left) and station pairs (right). continuity ratio ¼ Eðx ijx i > 0 & x j ¼ 0Þ ð12þ Eðx i jx i > 0 & x j > 0Þ For each station pairs, it defines the ratio of the precipitation mean at station i depending on whether station j is wet or dry. The ratio will be small for highly correlated stations (where there is a link between occurrence and amounts), and large for stations that are not correlated (no link between occurrence and amounts). Fig. 15 shows the result for the exponential and multi-exponential functions. The left figure shows the case where the precipitation amounts are generated without taking into account the occurrences at neighbouring stations. Despite the fact that inter-station correlations are perfectly reproduced, the precipitation field is not properly defined. The figure on the right shows the results using the multi-exponential distribution, which links precipitation amounts with the occurrence index (Eq. (10)) of the occurrence field. The results are markedly better, and show little bias. There is however, still much scatter in the data, and there are many reasons that may explain this situation. An accurate definition of the relationship between precipitation amounts and occurrence index would require longer time series. Because of the scatter shown in Fig. 11, the best-fit lines may introduce small errors at high and low occurrence index values (while preserving the overall mean), which may result in relatively much larger errors for the computation of the continuity ratios. A different choice of weights in the computation of the occurrence would affect the results. While the use of observed correlation as weights is a good starting point, it Figure 13 Cumulative distribution of exponential and multiexponential distribution, for the months of June, July and August (JJA) for station 1. For the sake of clarity, only 1 out of 20 observed data points is shown on the graph. Figure 15 Comparison of synthetic to observed continuity ratio for all station pairs and all seasons, for the exponential (left) and multi-exponential (right) distributions.