# Stochastic multi-site generation of daily weather data

- First Online:

DOI: 10.1007/s00477-008-0275-x

- Cite this article as:
- Khalili, M., Brissette, F. & Leconte, R. Stoch Environ Res Risk Assess (2009) 23: 837. doi:10.1007/s00477-008-0275-x

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## Abstract

Spatial autocorrelation is a correlation between the values of a single variable, considering their geographical locations. This concept has successfully been used for multi-site generation of daily precipitation data (Khalili et al. in J Hydrometeorol 8(3):396–412, 2007). This paper presents an extension of this approach. It aims firstly to obtain an accurate reproduction of the spatial intermittence property in synthetic precipitation amounts, and then to extend the multi-site approach to the generation of daily maximum temperature, minimum temperature and solar radiation data. Monthly spatial exponential functions have been developed for each weather station according to the spatial dependence of the occurrence processes over the watershed, in order to fulfill the spatial intermittence condition in the synthetic time series of precipitation amounts. As was the case for the precipitation processes, the multi-site generation of daily maximum temperature, minimum temperature and solar radiation data is realized using spatially autocorrelated random numbers. These random numbers are incorporated into the weakly stationary generating process, as with the Richardson weather generator, and with no modifications made. Suitable spatial autocorrelations of random numbers allow the reproduction of the observed daily spatial autocorrelations and monthly interstation correlations. The Peribonca River Basin watershed is used to test the performance of the proposed approaches. Results indicate that the spatial exponential functions succeeded in reproducing an accurate spatial intermittence in the synthetic precipitation amounts. The multi-site generation approach was successfully applied for the weather data, which were adequately generated, while maintaining efficient daily spatial autocorrelations and monthly interstation correlations.

### Keywords

Weather generatorMulti-sitePrecipitationTemperatureSolar radiation### List of symbols

*A*matrix (3,3) whose elements are defined from lag 0 and lag 1 serial and cross-correlation coefficient matrices of observed residuals

*B*matrix (3,3) whose elements are defined from lag 0 and lag 1 serial and cross correlation coefficient matrices of observed residuals

*F*spatial exponential cumulative distribution function

*I*Moran value

*l*total number of days in a given month

*m*total number of

*γ*_{Tmax},*γ*_{Tmin}or*γ*_{Sr}values taken from their range*M*_{0}matrix of lag 0 serial and cross-correlations

*M*_{1}matrix of lag 1 serial and cross-correlations

*n*total number of locations

*r*_{t}(*k*)synthetic precipitation amount at site

*k*on day*t*- SDI
spatial dependence indicator

*u*_{Tmax}(*n*, 1)vector of n independent and normally distributed random numbers used for maximum temperature

*u*_{Tmin}(*n*,1)vector of n independent and normally distributed random numbers used for minimum temperature

*u*_{Sr}(*n*,1)vector of n independent and normally distributed random numbers used for solar radiation

*v*_{t}(*k*)uniform [0, 1] random number

*V*_{Tmax}(*n*, 1)vector of

*n*spatially autocorrelated random numbers used for maximum temperature*V*_{Tmin}(*n*,1)vector of

*n*spatially autocorrelated random numbers used for mimimum temperature*V*_{Sr}(*n*,1)vector of

*n*spatially autocorrelated random numbers used for solar radiation*w*_{ij}spatial weight between two locations

*i*and*j**W*(*n*,*n*)weight matrix

*w*_{max}maximum positive eigenvalue of

*W*(*n*,*n*)*w*_{min}largest negative eigenvalue of

*W*(*n*,*n*) in absolute value*X*single variable

*x*_{i}observed value at location

*i*- \( \bar{x} \)
average of the

*x*_{i}over*n*locations- \( \bar{X}_{k} \left( j \right) \)
mean of temperature or solar radiation

*χ*_{p,k}(*j*)matrix (3,1) of maximum temperature (

*j*= 1), minimum temperature (*j*= 2) and solar radiation (*j*= 3) residuals for day*k*of year*p**λ*_{t}(*k*)inverse of the precipitation mean at site

*k*on day*t**σ*_{k}(*j*)standard deviation of temperature or solar radiation

*ɛ*_{p,k}(*j*)matrix (3, 1) of independent standard normal random numbers

*N*[0,1] for day*k*of year*p*- \( \rho_{{{{\upchi}}_{i,0\;} {{\upchi}}_{j,0} }} \)
lag 0 cross-correlation coefficient between the residuals of variable

*i*and the residuals of variable*j*- \( \rho_{{{{\upchi}}_{i,0\;} {{\upchi}}_{j, - 1} }} \)
lag 1 cross-correlation coefficient between the current residuals of variable

*i*and the previous residuals of variable*j*- \( \rho_{{{{\upchi}}_{i,0\;} {{\upchi}}_{i, - 1} }} \)
lag 1 serial correlation of variable

*i**γ*_{Tmax}moving average coefficient used for maximum temperature

*γ*_{Tmin}moving average coefficient used for minimum temperature

*γ*_{Sr}moving average coefficient used for solar radiation