Stochastic multi-site generation of daily weather data

  • Malika Khalili
  • François Brissette
  • Robert Leconte
Original Paper

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 generator Multi-site Precipitation Temperature Solar 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 γTmax, γTmin or γSr values taken from their range

M0

matrix of lag 0 serial and cross-correlations

M1

matrix of lag 1 serial and cross-correlations

n

total number of locations

rt(k)

synthetic precipitation amount at site k on day t

SDI

spatial dependence indicator

uTmax (n, 1)

vector of n independent and normally distributed random numbers used for maximum temperature

uTmin (n,1)

vector of n independent and normally distributed random numbers used for minimum temperature

uSr (n,1)

vector of n independent and normally distributed random numbers used for solar radiation

vt(k)

uniform [0, 1] random number

VTmax(n, 1)

vector of n spatially autocorrelated random numbers used for maximum temperature

VTmin(n,1)

vector of n spatially autocorrelated random numbers used for mimimum temperature

VSr(n,1)

vector of n spatially autocorrelated random numbers used for solar radiation

wij

spatial weight between two locations i and j

W(n,n)

weight matrix

wmax

maximum positive eigenvalue of W(n, n)

wmin

largest negative eigenvalue of W(n, n) in absolute value

X

single variable

xi

observed value at location i

\( \bar{x} \)

average of the xi over n locations

\( \bar{X}_{k} \left( j \right) \)

mean of temperature or solar radiation

χp,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

λt(k)

inverse of the precipitation mean at site k on day t

σk(j)

standard deviation of temperature or solar radiation

ɛp,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

γTmax

moving average coefficient used for maximum temperature

γTmin

moving average coefficient used for minimum temperature

γSr

moving average coefficient used for solar radiation

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Malika Khalili
    • 1
  • François Brissette
    • 2
  • Robert Leconte
    • 2
  1. 1.Department of Civil Engineering and Applied MechanicsMcGill UniversityMontrealCanada
  2. 2.École de technologie supérieureQuebec UniversityMontrealCanada

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