## 1. Introduction

Weather generators have been used successfully for a wide array of applications. They became increasingly used in various research topics, including more recently, climate change studies. They can generate series of climatic data with the same statistical properties as the observed ones. Furthermore, weather generators are able to produce series for any length of time. This allows developing various applications linked to extreme events, such as flood analyses.

Weather generators can also be used with downscaling approaches to create local-scale climate scenarios from the global circulation models’ (GCMs) outputs. Indeed, the GCMs are limited for regional analysis by their low spatial resolution. Their results cannot be directly used on a local scale. For this reason, weather generators represent an attractive approach to generate precipitation and temperature time series on the watershed using the output of these models. Weather generators for assessing the effect of climate change can also generate series of climatic data based on climate change scenarios from GCMs.

Weather generators may be classified from a statistical point of view in two main categories. The first one is parametric such as the Richardson-type Weather Generator (WGEN) and the Long-Ashton Research Station Weather Generator (LARS-WG) (Semenov and Barrow 1997), developed respectively in the United States and Europe. WGEN generates precipitation occurrences using a first-order two-state Markov chain, while the precipitation amounts on rainy days are simulated by commonly used frequency distributions such as the gamma or exponential distributions. The precipitation occurrences are modeled in LARS-WG as alternate wet and dry series. The length of each sequence is chosen randomly from a semiempirical distribution. For the wet days, the precipitation amounts are generated using other semiempirical precipitation distributions.

The second type of weather generators uses nonparametric methods (Brandsma and Buishand 1997; Buishand and Brandsma 2001). This technique does not require any assumption or parameter estimation and uses only the resampling of a weather variable’s vector on a day of interest from the historical data by conditioning on the simulated values of previous days. Resampling is done from the *k* nearest neighbors of these values using a weighted Euclidean distance.

Most of the existing weather generators are used at a single site independently of the others and thus ignoring the spatial dependence exhibited by the observed data. Some models, such as space–time models (Bardossy and Plate 1992; Bogardi et al. 1993) have been developed to regionalize the weather generators. In these models, the precipitation is linked to the atmospheric circulation patterns using conditional distributions and conditional spatial covariance functions. Another multisite model called the nonhomogeneous hidden Markov model (Bellone et al. 2000; Hughes and Guttorp 1994a, b; Hughes et al. 1999) uses an unobserved weather state to link the large-scale atmospheric measures and the small-scale spatially discontinuous precipitation field. Although these two models preserve the relevant statistical information, they are complicated to implement and are unable to adequately reproduce the observed correlations. Buishand and Brandsma (2001) developed a multisite approach based on the nearest-neighbor resampling. This approach does not deal with climate change, as the simulated series are based on the repetition of historical segments of data.

Wilks (1998) developed an approach based on serially independent but spatially correlated random numbers. For a network of *k* stations, a collection of *k*(*k* − 1)/2 empirical curves must be developed for all possible pair stations, for each precipitation process and for each month. These curves link the random number correlations and the precipitation amount or occurrence correlations for every station pair and month. Even if this approach is able to take into account the spatial dependence of climatic data, it has the drawback of being computationally intensive. This approach also involves the matrix of interstation correlations, which is often not positive definite and computation is therefore not feasible. Moreover, for large station networks this matrix becomes large and difficult to handle and all the coefficients of interstation correlations are not necessarily significant.

This article proposes an approach for multisite generation of weather data based on the concept of spatial autocorrelation. The methodology was applied to the Péribonca River basin and surrounding area, in the Canadian province of Quebec. The next section provides a definition of spatial autocorrelation. Section 3 describes the multisite approach. The dataset used for this study is presented in section 4, and the results obtained in the studied basin are presented and discussed in section 5.

## 2. Spatial autocorrelation definition

Spatial autocorrelation is a correlation among values of a single variable in reference to their geographical adjacency (Griffith 2003). In other words, it is a correlation of a variable with itself in a geographic space. Spatial autocorrelation is an important feature for geographical dependence. It is analogous to the serial autocorrelation approach, which indicates the strength of the correlation between a single variable with itself at different times. Serial autocorrelation is usually employed to simulate and forecast series exhibiting autoregressive characteristics.

Spatial dependence can arise because there are processes or phenomena that connect different locations and constrain the observations in a given place to be similar to those in a nearby area. Thus, the spatial autocorrelation determines the strength with which the occurrence of a process in a location affects other locations that are spatially contiguous. For example, the occurrence of a weather system may affect precipitations at several weather stations, and the opening of a freeway interchange may affect the prices of housing in nearby locations (Odland 1988).

The most interesting aspect of spatial autocorrelation is the weighing scheme, which highlights the neighbors influencing the location of interest and ignores or reduces the influence of other locations that do not interact with it. Neighboring geographical values around an observation may be defined by a spatial weight matrix. Literature provides several functions to express the geographic contiguity between locations (Odland 1988; Ullah and Giles 1991, 237–289; Anselin 1980; Murdoch et al. 1993).

It should be noted that contiguity between observations is not limited to the geographical distance but can also mean other information describing the relation between locations. According to Odland (1988), the flexibility in defining the weights makes spatial autocorrelation statistics a useful means of investigating alternative hypotheses about relations among places. Tobler (1970) mentioned in the first law of geography that “everything is related to everything else, but near things are more related than distant things.” Thus, the spatial weights can represent distance metrics, as well as hypotheses about the spatial processes, which produce the data.

*I*(Moran 1950; Odland 1988; Griffith 2003), presented below:

*x*denotes the observed value at location

_{i}*i*,

*is the average of the*x

*x*over the

_{i}*n*locations, and

*w*is the spatial weight between two locations

_{ij}*i*and

*j*. Here

*w*takes a nonzero value if the two locations are neighbors and zero otherwise. In the matrix form of the right-hand part of Eq. (1), all weights are stored in the spatial weight matrix. Usually, the weights

_{ij}*w*may be in a row-standardized form, which means that all weights in a row sum up to 1 and by convention

_{ij}*w*= 0.

_{ii}As defined by Moran’s *I*, the spatial autocorrelation differs from the Pearson product moment correlation coefficient expression by using a single variable and the weight parameters. Therefore, the Pearson coefficient describes the relation between two given variables **X** and **Y**, while Moran’s *I* indicates the dependence among values of a single variable taking into account their geographical locations. In this paper, the spatial autocorrelation will also be known as Moran’s *I* or *I*.

Moran’s *I* lies approximately between −1 and 1 (Griffith 2003). When the values are uncorrelated and arranged randomly over space, the Moran’s *I* is equal to its expected value [−1/(*n* − 1)] (Moran 1950; Cliff and Ord 1981), *n* is the total number of locations. Moran’s *I* larger than the expected value represents positive spatial autocorrelation, which means that geographically nearby values tend to be similar and Moran’s *I* smaller than the expected value represents negative spatial autocorrelation, which means that the values tend to be dissimilar.

*C*(Odland 1988; Griffith 2003), which is defined using another measure of covariation:

*C*values are between 0 and 2. The spatial autocorrelation is positive if

*C*is lower than 1, negative if

*C*is between 1 and 2, and null if

*C*is equal to 1.

## 3. Methodology

### a. Unisite weather generator

*x*(

_{t}*i*) at site

*i*on day

*t*. A uniform [0, 1] random number

*u*(

_{t}*i*) is drawn and compared with a critical probability, which is equal to one of the transitional probabilities, depending on whether the previous day was wet or dry:

*υ*(

_{t}*i*), independent from the previous one, is then used to simulate the synthetic precipitation amounts by inverting a probability distribution function of the precipitation amounts. The exponential distribution has frequently been used as a first approximation of the distribution of rainfall amounts (Todorovic and Woolhiser 1974) as well as the gamma distribution (Katz 1977). In the case of an exponential distribution, the cumulative density function can be defined as

*λ*(

*i*) is the parameter describing the exponential distribution function at site

*i*. Therefore, the precipitation amounts

*r*(

_{t}*i*) at site

*i*can be computed as

### b. Proposed multisite generation approach

The proposed approach aims at using, in the weather generator, spatially autocorrelated random numbers whose spatial autocorrelations will reproduce the daily spatial autocorrelations in the synthetic precipitation processes similar to those of observations to which the weather generator has been fitted. For a given day, the single variable used in Moran’s *I* [Eq. (1)] represents the values observed at the set of stations on that day. Note that the observed daily spatial autocorrelations to be reproduced are the averages of the daily spatial autocorrelations observed between the selected weather stations over their shared recording years. If all stations are wet or dry, the value 1 is assigned to the spatial autocorrelation of occurrence. This is also the case for the precipitation amounts if all stations are dry. The total of the random numbers for each of the occurrence and the amount time series is equal to the total number of stations used in the analysis.

#### 1) Spatially autocorrelated random numbers model

*n*spatially autocorrelated synthetic time series of occurrences and amounts at

*n*stations. The spatial moving average process may be expressed as (Cliff and Ord 1981; Cressie 1993)

**V**(

*n*, 1) is a vector of

*n*spatially autocorrelated random numbers to be used for

*n*locations; 𝗪(

*n, n*) is a weight matrix;

**u**(

*n*, 1) is a vector of

*n*independent and uniformly [0, 1] distributed random numbers;

*γ*is the moving average coefficient. The extreme eigenvalues of the weight matrix establish the range of this coefficient, which is ](−1/

*w*

_{max}); (−1/

*w*

_{min})[, where

*w*

_{max}is the maximum positive eigenvalue and

*w*

_{min}is the largest negative eigenvalue in absolute value (the reversed bracket mean that the extremes (−1/

*w*

_{max}) and (−1/

*w*

_{min}) are not included in this range).

The spatial moving average process contains a vector **u**(*n*, 1) of *n* independent and uniformly [0, 1] distributed random numbers as well as the part that creates the spatial dependence between the **u**(*n*, 1) components. Hence, the spatial dependence of the resulting random numbers, components of the vector **V**(*n*, 1), depends on the *γ* value. In other words, for different values of this coefficient one can obtain vectors **V**(*n*, 1) of *n* random numbers exhibiting different degrees of spatial autocorrelation noted *I*** _{V}** computed by Moran indicator [Eq. (1)].

It is important to note that the random numbers resulting from the spatial moving average process are not necessarily uniformly distributed, nor necessarily lie between 0 and 1, whereas the weather generator requires the uniform [0, 1] random numbers to be used in Eqs. (4) and (6), as noted above. Therefore, a transformation is needed to turn these spatially autocorrelated numbers into uniformly [0, 1] distributed ones preserving the required spatial autocorrelation. This transformation is carried out using the cumulative distribution function of the spatially autocorrelated numbers. In fact, for each value of *γ*, 1000 vectors **V**(*n*, 1) were generated and a histogram is plotted using all the spatially autocorrelated numbers. The cumulative frequency distribution is thus computed by using a cumulative sum of frequencies from the histogram. Such a sample size was considered to ensure that the result is affected as little as possible by the noise. The polynomial that fits the cumulative distribution function is used to find the cumulative probabilities of the spatially autocorrelated numbers. These values that occur between 0 and 1 will be used in the weather generator [(4) and (6)] to model the occurrence and amount processes at each station.

Another issue that arises is what values of the coefficient *γ* should be used in Eq. (7) to obtain vectors of random numbers that will be fed in the weather generator to simulate the synthetic precipitation processes with daily spatial autocorrelations similar to those observed. Finding these values of the coefficient *γ* is the focus of the following analysis.

#### 2) Moving average coefficients estimation

Using different values of the moving average coefficient *γ*, one can obtain vectors **V**(*n*, 1) enclosing *n* spatially autocorrelated random numbers with different spatial autocorrelations, indicated by *I*** _{V}** values, and thus the weather generator can model precipitation time series also displaying different spatial autocorrelations. Accordingly, the first step for the approach is to empirically establish the relationship between

*γ*values and the

*I*

**values of the resulting random numbers from the spatial moving average process. In fact, this operation consists in varying**

_{V}*γ*over its range ](−1/

*w*

_{max}); (−1/

*w*

_{min})[ and calculating Moran’s

*I*

**values of the resulting random numbers. Such a relationship is required for both occurrence and amount processes.**

_{V}*n*components of

**V**(

*n*, 1) will be used for each of the

*n*stations. This operation yields another curve linking the

*I*

**values computed for the used random numbers and the**

_{V}*I*

**values of the resulting synthetic precipitation occurrences. Here**

_{X}*I*

**indicates the spatial autocorrelation computed by Moran’s**

_{X}*I*for a vector

**X**of

*n*values of precipitation occurrence

*x*generated at

*n*stations:

*I*

**values are in relationship with the**

_{V}*γ*values. As was the case with the precipitation occurrences, a curve is also established between the

*I*

**values of the used random numbers and the**

_{V}*I*

**values of the resulting synthetic precipitation amounts. Here**

_{R}*I*

**indicates the spatial autocorrelation computed by Moran’s**

_{R}*I*for a vector

**R**of

*n*values of precipitation amount

*r*generated at

*n*stations:

It should be noted that the used *γ* values for the precipitation amount process will give a full vector **V**(*n*, 1) with nonzero components from the spatial moving average process. Since each component of this vector will be used in the weather generator to produce the precipitation amount at each station, the simulated amounts will thus be different from zero. Therefore, the generated occurrence process will be used to change the precipitation amount at each dry station to zero. The spatial autocorrelations for the precipitation amounts as defined above are thus computed after this operation.

The next step is to find which values of coefficient *γ* should be used to generate synthetic time series with daily spatial autocorrelations similar to those of the observations for occurrence and amount processes. For each process, both relationships discussed above, namely, (*I*** _{V}**,

*γ*) and (

*I*

**,**

_{X}*I*

**) or (**

_{V}*I*

**,**

_{R}*I*

**), will be used. One should identify on the one hand the**

_{V}*I*

**values of the random numbers linked to the spatial autocorrelations observed for this process (**

_{V}*I*

**or**

_{X}*I*

**values), and on the other hand, the**

_{R}*γ*values linked to these

*I*

**values. However, this procedure results in relationships between the**

_{V}*γ*values and spatial autocorrelations for each process (

*I*

**,**

_{X}*γ*) or (

*I*

**,**

_{R}*γ*), which can substitute for the two relationships mentioned above. Therefore, one can use these relationships directly to find the

*γ*values that correspond to the observed values of

*I*

**and**

_{X}*I*

**. Note that such relationships can be established for each month of the year and the spatial autocorrelation values can be averaged over 1000 repetitions for high stability and low noise.**

_{R}#### 3) Data generation

Once *γ* values linked to the observed daily spatial autocorrelations of both precipitation occurrences and amounts are calculated from the monthly curves (*I*** _{X}**,

*γ*) and (

*I*

**,**

_{R}*γ*), the generation phase starts. In fact, the spatial moving average model is reused with the selected

*γ*values to give the spatially autocorrelated random numbers that will be inserted in the weather generator [Eqs. (4) and (6)] to produce synthetic meteorological time series exhibiting daily spatial autocorrelations similar to those of the observed ones. The length of the synthetic time series is specified by the user.

The flowchart in Fig. 1 summarizes this approach. It illustrates the parameter estimation and the generation phases for a given month. In this figure, *m* is the total number of *γ* values taken from its range and *l* is the total number of days in a given month.

## 4. Description of dataset

The Péribonca River basin (and surrounding area) (Fig. 2) was used to test the approach. This watershed is located north of the Lac-Saint-Jean, in the Canadian province of Quebec, an area characterized by a wet climate, relatively cool summers and snow precipitations from November to April. The territory under study extends in latitude from 50°44′00″ to 48°37′12″N and in longitude from 72°35′35″ to 71°10′06″W. Seven stations were selected for the analysis: Péribonca (P), Hémon (H), Bonnard (B), Chute-du-Diable (Cd), Chute-des-Passes (Cp), St-Léon-de-Labrecque (Sl), and Normandin CDA (N). The location of these stations is provided in Fig. 2 and in Table 1. Table 2 gives the observed numbers of wet days per month per station and Table 3 shows the monthly average amounts of precipitation observed at each station. The data for each station are averaged over its recorded period.

The daily spatial autocorrelation values of occurrences and amounts depend on the series length employed. The user may opt for a 30-yr duration, as recommended by the World Meteorological Organization (WMO). For the selected stations, such a period is not always available, with the recording durations varying from 17 to 57 yr as indicated by Table 1. Nevertheless, the computation of the daily spatial autocorrelations was carried out using a shared recorded period for all stations, which is 14 yr starting from 1963 to 1976. The weather generator parameters were calculated monthly at each station over its recorded period (Table 1) using the maximum likelihood estimator. The simulation period can be run for any length of time. The results presented are for a 50-yr simulation period.

Each element *w _{ij}* indicates the weight of the pair stations (

*i*,

*j*) and by convention

*w*= 0. The stations used are classified as mentioned above. Thus, the nonzero elements of the first row represent the weights between the Péribonca location and the remaining stations Hémon, Bonnard, Chute-du-diable, Chute-des-passes, St-Léon-de-Labrecque, and Normandin CDA respectively. The second row contains the weights between the Hémon station and the other stations according to the order noted above and so on. It appears that the weights of the pair stations including Bonnard or Chute-des-passes stations stored in the third and the fifth columns, respectively, are low compared to the others, which means that Bonnard and Chute-des-passes, which are actually located far from the remaining stations, are not deemed their nearest neighbors according to the inverse square distance method. The third and the fifth rows related to Bonnard and Chute-des-passes stations, respectively, indicate that the pairs of these stations, which are the closest, have the most important weights.

_{ii}Although these techniques based on geographical information perform well in representing the arrangement of the weather stations in the watershed, it is more general to select the spatial weights that describe the accommodating hypotheses about the relations among weather stations as mentioned by (Odland 1988). In fact, according to the tests carried out for this study, the low weight values assigned to some pairs of stations by the methods mentioned above result in a loss of correlation (Pearson product moment correlation) observed between the stations that compose those pairs (e.g., the correlations between Bonnard station and the remaining stations excluding Chute-des-passes).

## 5. Results and discussion

As mentioned above, the extreme eigenvalues of the weight matrix establish the range of the moving average coefficient *γ*, which is ]−1 ; 4.6795[ for the used weight matrix. The coefficients *γ* that are out of this range may not be in perfect relationship with the spatial autocorrelations of the resulting precipitation processes. Consequently, the largest daily spatial autocorrelations may not be adequately reproduced. This problem can be solved using the iterated spatial moving average process to increase the spatial autocorrelations of the resulting random numbers, as well as those of the synthetic precipitation processes. One can use two or more successive moving average processes.

*γ*values and

*I*

**values of the occurrence series are established and represented adequately by a three-order polynomial function such as**

_{X}*a*,

*b*,

*c*, and

*d*are empirical parameters that describe the relationship, determined by a regression fit for each month. The three-order polynomial function has been chosen because it produces the best fit to a dataset. Such a curve is established for each month of the year. Figure 3 gives this relationship for June. Similar curves were obtained for the other months. The curve presented in Fig. 3 can be adequately represented by the following equation:

*γ*value to be used in the moving average process is 3.19.

The *γ* values linked to all observed values of *I*** _{X}** for each month may be extracted from these curves to generate the spatially autocorrelated random numbers from the moving average process. These random numbers will be used in the weather generator to produce synthetic precipitation occurrences with daily spatial autocorrelations similar to those observed between the seven stations.

Figure 4 illustrates the observed versus simulated monthly numbers of rainy days at all stations for January, April, July, and October. The observed data are averaged over the recorded period for each station and the simulated ones are averaged over the simulated period, which is 50 yr. Similar results are obtained for the other months. Clearly, the monthly numbers of rainy days are adequately reproduced. The multisite approach succeeded in modeling the occurrence processes using the weather generator at the seven stations and for all months.

Figures 5a and 5b illustrate the joint probabilities of the dry and wet states respectively for each pair of stations and each month. It appears that the joint probabilities of the occurrence process are also well reproduced by the multisite generator (**R**^{2} = 0.8936 for the dry state and **R**^{2} = 0.8681 for the wet state).

The performance of the multisite approach in reproducing the observed spatial dependence between the seven stations on each day was also investigated. Figure 6 shows the observed and simulated daily spatial autocorrelations of precipitation occurrences. The observed ones are averaged over the shared recorded period between the seven stations and the simulated ones are averaged over the simulation period. Therefore, 365 points representing the coordinate pairs are plotted in this graph. This figure indicates that the spatial autocorrelations, exhibited by the seven stations on each day, are adequately reproduced by the multisite approach (**R**^{2} = 0.8508).

*γ*values and

*I*

**values of precipitation amounts are also represented by a three-order polynomial function such as**

_{R}*e*,

*f*,

*g*, and

*h*are parameters provided by a regression fit for each month. Such a curve will also be established for each month. Figure 7 shows such a relationship for January. Similar curves were established for the other months.

The *γ* values linked to all observed values of *I*** _{R}** for each month can be found from these curves to generate the spatially autocorrelated random numbers from the moving average process. These random numbers will be used in the weather generator to produce synthetic precipitation amounts on the wet days with daily spatial autocorrelations similar to those observed.

Figure 8 illustrates the total monthly precipitations at the seven stations for February, June, September, and December. The observed and simulated total monthly precipitations are generally in good agreement. A similar result is obtained for the other months. Figure 9 provides a scatterplot of observed and simulated daily spatial autocorrelations of precipitation amounts. It appears that the daily spatial autocorrelations of precipitation amounts are satisfactorily reproduced over the year (**R**^{2} = 0.757). However, the daily spatial autocorrelations of the occurrence process are better reproduced than those of the amount process because the generation of amounts is influenced by the spatial autocorrelations of the occurrence process generated before.

The multisite approach adequately reproduces the daily spatial dependence of both precipitation occurrences and amounts, as well as the total monthly precipitations and the monthly numbers of rainy days. However, it would be interesting to investigate how this approach reproduces the precipitation variation. Figure 10 illustrates the observed and simulated daily precipitation standard deviations for January, April, August, and October. The standard deviations are computed for all days of each month over the 50-yr simulation period. This figure shows that the multisite approach successfully reproduces the standard deviations. Similar results are obtained for the remaining months.

It will also be interesting to investigate if the approach reproduces the Pearson correlations of occurrences and amounts between every pair of stations. Figure 11 shows the interstation correlations between daily precipitation occurrences for all pairs of stations and months. The observed ones are calculated over the shared recorded period between each pair of stations, and the simulated ones are calculated over the simulation period. In this figure, it seems that some of the larger interstation correlations are underestimated. Figure 12 illustrates the interstation correlations between daily precipitation amounts for all pairs of stations and months. In this case, the interstation correlations seem to be overestimated. This behavior of the synthetic interstation correlations is expected because the multisite approach did not try to reproduce exactly the observed interstation correlations despite the good result obtained for the occurrences (**R**^{2} = 0.5555 for the occurrences and **R**^{2} = 0.1259 for the amounts).

To improve this result, monthly weight matrices representing the monthly interstation correlations of precipitation between all pairs of stations are used for each month. Taking the spatial weights to the second power for the occurrences and the third power for the amounts was necessary to optimize the results. Figures 13 and 14 show these results. It can be seen that the monthly weight matrices improve the estimation of the interstation correlations for the precipitation processes, thus highlighting the importance of selecting the proper weight matrix (**R**^{2} = 0.7989 for the occurrences and **R**^{2} = 0.791 for the amounts). The remaining aberrations can be attributed to the sample size, which is 14 yr, used to calibrate the proposed approach. Note that these interstation correlations have been reproduced automatically in the synthetic time series by reproducing the daily spatial autocorrelations.

The spatial moving average process used for this multisite approach consists of simple computations that do not require spatial weight matrix operations. Therefore, the proposed approach does not present any disadvantage for a large dataset. Moreover, the spatial weight matrix tends to be sparse, which facilitates the computations and does not need a lot of memory. From a geographic point of view, for a large dataset, the spatial autocorrelation becomes more interesting to investigate because the spatial patterns are more likely to develop over an extended region. Using the proposed approach, the Moran’s I summarizes a complete spatial dependence into a single number and only 12 relationships have to be developed to carry out the multisite generation for each of the occurrence and amount process for all months. This number does not increase if the number of stations increases.

Because the multisite approach proposed in this paper uses the spatial autocorrelation concept that summarizes a complete spatial dependence among the set of stations into a single number, it appears interesting to compare its results to those obtained with spatially correlated random numbers used in Wilks (1998). In this second case, a number of *k*(*k* − 1)/2 curves have to be developed for a network of *k* stations to reproduce the interstation correlations observed between each pair of stations in each month.

For the Peribonca River basin, 504 empirical curves have been developed for all pairs of stations, for both precipitation occurrences and amounts and for all months. Figure 15 shows an example of such empirical curve obtained for the Peribonca and Chute-du-diable stations in January to find which correlation of random numbers must be used to reproduce the precipitation occurrence correlation observed between these two stations in this month.

Figures 16 and 17 show the interstation correlations of daily precipitation occurrences and amounts, respectively, for all pairs of stations and months obtained using the spatially correlated random numbers. Since the aim of the 504 empirical curves is to reproduce the precipitation correlations observed between each pair of stations in each month, the result is relatively better than that of the proposed approach obtained automatically via the spatial autocorrelation concept (**R** ^{2}= 0.8948 versus 0.7989 for the occurrences and **R**^{2} = 0.8651 versus 0.791 for the amounts). However, comparing the observed versus simulated monthly spatial autocorrelations, Figs. 18 and 19 indicate that the spatial autocorrelations observed in the study area in each month are well reproduced by the proposed approach especially for the occurrences (**R**^{2} = 0.9701 versus 0.6302 for the occurrences and **R**^{2} = 0.8455 versus 0.7461 for the amounts). Note that the recorded periods used to calibrate the two approaches are different. Indeed, the proposed approach uses the shared recorded period between the set of stations, which is 14 yr while the spatially correlated random number approach uses the shared recorded periods between each pair of stations whose minimum is 14 yr.

Furthermore, the daily spatial autocorrelations are not reproduced by the spatially correlated random number approach because it works monthly to reproduce the interstation correlations. For the other statistical criteria presented above, the two approaches give practically similar results.

## 6. Conclusions

A multisite generation approach of daily precipitation data was presented. It is based on the notion of spatial autocorrelation, which is the correlation between values of a single variable through space. The aim of this approach is to introduce, in a weather generator, series of random numbers with spatial autocorrelations that can reproduce the observed daily spatial autocorrelations of precipitation processes, namely, occurrences and amounts.

The important aspect in the definition of the spatial autocorrelation is the weight matrix, which gives a score to identify neighboring observations. It should be noted that the values of Moran’s indicator depend on the weighting scheme. Thus, the observed daily spatial autocorrelations may be changed using alternative neighborhood criteria. This suppleness in establishing the weight matrix allows for the investigation of different hypotheses about the spatial pattern, and the differences in the obtained results can be a measure of the advantages of one hypothesis over the other.

The Péribonca River basin and surrounding area in the Canadian province of Quebec were used to test this regionalization approach. Relationships were found between the spatial moving average coefficients, used to generate spatially autocorrelated random numbers, and the spatial autocorrelations for the precipitation processes computed by Moran’s indicator. These relationships provide the moving average coefficients, which enable the observed daily spatial autocorrelations to be reproduced in the synthetic precipitation processes.

Results indicate that this approach is successful in reproducing the daily and monthly spatial autocorrelations between the seven stations used in the Péribonca watershed. A good agreement was also found between the observed and simulated monthly numbers of rainy days, the total monthly precipitation amounts, as well as the daily precipitation variance.

It is important to emphasize that using the appropriate weight matrix representing the degree of significance of the weather stations surrounding each observation allows the multisite approach to reproduce not only the daily spatial autocorrelations over the watershed, but also the monthly interstation correlations of precipitation processes between each pair of stations in each month.

A comparison has been done between the proposed approach using spatially autocorrelated random numbers and the approach of spatially correlated random numbers developed by Wilks (1998). While this latter leads to relatively more precision in reproducing the monthly interstation correlations, the monthly spatial autocorrelations are well reproduced by the proposed approach.

Finally, the spatial autocorrelation is a useful tool to describe spatial dependence of data in a geographic space. Using spatial autocorrelation for multisite generation of precipitation processes is successful. This approach has the advantage of being versatile and easy to implement, requiring only 12 empirical curves linking the moving average coefficient values to the spatial autocorrelation values of each of the precipitation occurrences and amounts and does not apply any restrictions on the spatial weight matrix. For a large dataset, the proposed approach remains easy to execute because the overall pattern in the data is summarized in a single statistic. Moreover, the spatial moving average process used in this paper consists of a simple computation process.

## Acknowledgments

This research was supported by the Natural Science and Engineering Research Council of Canada, Hydro-Québec, and the Ouranos Consortium on climate change through a collaborative research and development grant. Their support is gratefully acknowledged. We also gratefully acknowledge the anonymous reviewers for their valuable comments to improve an earlier version of the manuscript.

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## APPENDIX

### List of Variable Types

*x*Observed value at location_{i}*i**w*Spatial weight between two locations_{ij}*i*and*j**I*Moran value*n*Total number of locations*C*Geary value*x*(_{t}*i*) Daily precipitation occurrence at site*i*on day*t**u*(_{t}*i*) Uniform [0, 1] random number*υ*(_{t}*i*) Uniform [0, 1] random number*λ*(*i*) Parameter of the exponential distribution function at site*i**r*(_{t}*i*) Precipitation amount at site*i*on day*t***V**(*n*, 1) Vector of*n*spatially autocorrelated random numbers𝗪(

*n*,*n*) Weight matrix**u**(*n*, 1) Vector of*n*independent and uniformly [0, 1] distributed random numbers*γ*Moving average coefficient*w*_{max}Maximum positive eigenvalue of the matrix 𝗪(*n*,*n*)*w*_{min}Largest negative eigenvalue in absolute value of the matrix 𝗪(*n*,*n*)*I*Moran’s_{V}*I*computed for the vector**V**(*n*, 1)**X**Vector of*n*values of precipitation occurrence*x*generated at*n*stations*I*Moran’s I for the vector_{X}*X***R**Vector of*n*values of precipitation amount*r*generated at*n*stations*I*Moran’s_{R}*I*for the vector**R***m*Total number of*γ*values taken from its range*l*Total number of days in a given month

The Peribonca watershed, showing the location of meteorological stations. Elevations in meters.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

The Peribonca watershed, showing the location of meteorological stations. Elevations in meters.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

The Peribonca watershed, showing the location of meteorological stations. Elevations in meters.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Relationship between the moving average coefficients *γ* and Moran’s *I*** _{X}** computed for the precipitation occurrences at the seven stations for June.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Relationship between the moving average coefficients *γ* and Moran’s *I*** _{X}** computed for the precipitation occurrences at the seven stations for June.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Relationship between the moving average coefficients *γ* and Moran’s *I*** _{X}** computed for the precipitation occurrences at the seven stations for June.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed vs simulated monthly numbers of rainy days at all stations for January, April, July, and October. The observed data are averaged over the recorded period for each station, and the simulated ones are averaged over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed vs simulated monthly numbers of rainy days at all stations for January, April, July, and October. The observed data are averaged over the recorded period for each station, and the simulated ones are averaged over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed vs simulated monthly numbers of rainy days at all stations for January, April, July, and October. The observed data are averaged over the recorded period for each station, and the simulated ones are averaged over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Joint probabilities from observed and simulated occurrence process for each pair of stations and each month: (a) both stations are dry and (b) both stations are wet. The observed values are calculated over the shared recorded period between each pair of stations, and the simulated ones are calculated over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Joint probabilities from observed and simulated occurrence process for each pair of stations and each month: (a) both stations are dry and (b) both stations are wet. The observed values are calculated over the shared recorded period between each pair of stations, and the simulated ones are calculated over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Joint probabilities from observed and simulated occurrence process for each pair of stations and each month: (a) both stations are dry and (b) both stations are wet. The observed values are calculated over the shared recorded period between each pair of stations, and the simulated ones are calculated over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated daily spatial autocorrelations *I*** _{X}** of precipitation occurrences; 365 points representing the coordinate pairs are plotted in this graph. The observed data are averaged over the shared recorded period between the seven stations, and the simulated ones are averaged over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated daily spatial autocorrelations *I*** _{X}** of precipitation occurrences; 365 points representing the coordinate pairs are plotted in this graph. The observed data are averaged over the shared recorded period between the seven stations, and the simulated ones are averaged over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated daily spatial autocorrelations *I*** _{X}** of precipitation occurrences; 365 points representing the coordinate pairs are plotted in this graph. The observed data are averaged over the shared recorded period between the seven stations, and the simulated ones are averaged over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Relationship between the moving average coefficients *γ* and Moran’s *I*** _{R}** computed for the precipitation amounts at the seven stations for January.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Relationship between the moving average coefficients *γ* and Moran’s *I*** _{R}** computed for the precipitation amounts at the seven stations for January.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Relationship between the moving average coefficients *γ* and Moran’s *I*** _{R}** computed for the precipitation amounts at the seven stations for January.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated total monthly precipitations at the seven stations for February, June, September, and December. The observed data are averaged over the recorded period for each station, and the simulated ones are averaged over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated total monthly precipitations at the seven stations for February, June, September, and December. The observed data are averaged over the recorded period for each station, and the simulated ones are averaged over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated total monthly precipitations at the seven stations for February, June, September, and December. The observed data are averaged over the recorded period for each station, and the simulated ones are averaged over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated daily spatial autocorrelations *I*** _{R}** of precipitation amounts; 365 points representing the coordinate pairs are plotted in this graph. The observed data are averaged over the shared recorded period between the seven stations, and the simulated ones are averaged over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated daily spatial autocorrelations *I*** _{R}** of precipitation amounts; 365 points representing the coordinate pairs are plotted in this graph. The observed data are averaged over the shared recorded period between the seven stations, and the simulated ones are averaged over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated daily spatial autocorrelations *I*** _{R}** of precipitation amounts; 365 points representing the coordinate pairs are plotted in this graph. The observed data are averaged over the shared recorded period between the seven stations, and the simulated ones are averaged over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated daily precipitation standard deviations at the seven stations for January, April, August, and October. The observed data are averaged over the recorded period for each station, and the simulated ones are averaged over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated daily precipitation standard deviations at the seven stations for January, April, August, and October. The observed data are averaged over the recorded period for each station, and the simulated ones are averaged over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated daily precipitation standard deviations at the seven stations for January, April, August, and October. The observed data are averaged over the recorded period for each station, and the simulated ones are averaged over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated interstation correlations between daily precipitation occurrences for all pairs of stations and months. The observed ones are calculated over the shared recorded period between each pair of stations, and the simulated ones are calculated over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated interstation correlations between daily precipitation occurrences for all pairs of stations and months. The observed ones are calculated over the shared recorded period between each pair of stations, and the simulated ones are calculated over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated interstation correlations between daily precipitation occurrences for all pairs of stations and months. The observed ones are calculated over the shared recorded period between each pair of stations, and the simulated ones are calculated over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated interstation correlations between daily precipitation amounts for all pairs of stations and months. The observed ones are calculated over the shared recorded period between each pair of stations, and the simulated ones are calculated over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated interstation correlations between daily precipitation amounts for all pairs of stations and months. The observed ones are calculated over the shared recorded period between each pair of stations, and the simulated ones are calculated over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated interstation correlations between daily precipitation amounts for all pairs of stations and months. The observed ones are calculated over the shared recorded period between each pair of stations, and the simulated ones are calculated over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated interstation correlations between daily precipitation occurrences for all pairs of stations and months using monthly weight matrices. The observed ones are calculated over the shared recorded period between each pair of stations, and the simulated ones are calculated over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated interstation correlations between daily precipitation occurrences for all pairs of stations and months using monthly weight matrices. The observed ones are calculated over the shared recorded period between each pair of stations, and the simulated ones are calculated over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated interstation correlations between daily precipitation occurrences for all pairs of stations and months using monthly weight matrices. The observed ones are calculated over the shared recorded period between each pair of stations, and the simulated ones are calculated over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated interstation correlations between daily precipitation amounts for all pairs of stations and months using monthly weight matrices. The observed ones are calculated over the shared recorded period between each pair of stations, and the simulated ones are calculated over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated interstation correlations between daily precipitation amounts for all pairs of stations and months using monthly weight matrices. The observed ones are calculated over the shared recorded period between each pair of stations, and the simulated ones are calculated over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated interstation correlations between daily precipitation amounts for all pairs of stations and months using monthly weight matrices. The observed ones are calculated over the shared recorded period between each pair of stations, and the simulated ones are calculated over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Relationship between precipitation occurrence correlations and random number correlations for Péribonca and Chute-du-diable stations in January.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Relationship between precipitation occurrence correlations and random number correlations for Péribonca and Chute-du-diable stations in January.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Relationship between precipitation occurrence correlations and random number correlations for Péribonca and Chute-du-diable stations in January.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated interstation correlations between daily precipitation occurrences for all pairs of stations and months using the Wilks approach. The observed ones are calculated over the shared recorded period between each pair of stations, and the simulated ones are calculated over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated interstation correlations between daily precipitation occurrences for all pairs of stations and months using the Wilks approach. The observed ones are calculated over the shared recorded period between each pair of stations, and the simulated ones are calculated over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated interstation correlations between daily precipitation occurrences for all pairs of stations and months using the Wilks approach. The observed ones are calculated over the shared recorded period between each pair of stations, and the simulated ones are calculated over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated interstation correlations between daily precipitation amounts for all pairs of stations and months using the Wilks approach. The observed ones are calculated over the shared recorded period between each pair of stations, and the simulated ones are calculated over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated interstation correlations between daily precipitation amounts for all pairs of stations and months using the Wilks approach. The observed ones are calculated over the shared recorded period between each pair of stations, and the simulated ones are calculated over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated interstation correlations between daily precipitation amounts for all pairs of stations and months using the Wilks approach. The observed ones are calculated over the shared recorded period between each pair of stations, and the simulated ones are calculated over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated monthly spatial autocorrelations of precipitation occurrences using (a) the proposed approach and (b) the Wilks approach. The observed data are averaged over the shared recorded period between the seven stations, and the simulated ones are averaged over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated monthly spatial autocorrelations of precipitation occurrences using (a) the proposed approach and (b) the Wilks approach. The observed data are averaged over the shared recorded period between the seven stations, and the simulated ones are averaged over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated monthly spatial autocorrelations of precipitation occurrences using (a) the proposed approach and (b) the Wilks approach. The observed data are averaged over the shared recorded period between the seven stations, and the simulated ones are averaged over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated monthly spatial autocorrelations of precipitation amounts using (a) the proposed approach and (b) the Wilks approach. The observed data are averaged over the shared recorded period between the seven stations, and the simulated ones are averaged over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated monthly spatial autocorrelations of precipitation amounts using (a) the proposed approach and (b) the Wilks approach. The observed data are averaged over the shared recorded period between the seven stations, and the simulated ones are averaged over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Observed and simulated monthly spatial autocorrelations of precipitation amounts using (a) the proposed approach and (b) the Wilks approach. The observed data are averaged over the shared recorded period between the seven stations, and the simulated ones are averaged over the 50-yr simulation period.

Citation: Journal of Hydrometeorology 8, 3; 10.1175/JHM588.1

Location and recorded years of the used stations.

Observed numbers of wet days per month per station.

The monthly average amounts of precipitation observed at each station.