HydrologicalSciences -Journal- des Sciences Hydrologiques,40,5, October 1995

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Spatio-temporal precipitation modelling in rural watersheds

NICOLAS R. DALEZIOS Department of Agriculture, Crop and Animal Production, University of Thessaly, Pedion Areos, 38334 Volos, Greece

KAZIMIERZ ADAMOWSKI Department of Civil Engineering, University of Ottawa, Ottawa, Ontario, Canada KIN 6N5 Abstract The natural variability of precipitation in agricultural regions both in time and space is modelled using extensions of Box & Jenkins (1976) methodology based on the ARMA procedure. This broad class of aggregate regional models belongs to the general family of Space-Time Autoregressive Moving Average (ST ARM A) processes. The paper develops a three-stage iterative procedure for building a STARMA model of multiple precipitation series. The identified model is STMA (13). The emphasis is placed on the three stages of the model building procedure, namely identification, parameter estimation and diagnostic checking. In the parameter estimation stage the polytope (or simplex) method and three further classical nonlinear optimization algorithms are used, namely two conjugate gradient methods and a quasi-Newton method. The polytope method has been adopted and the developed model performed well in describing the spatio-temporal characteristics of the multiple precipitation series. Application has been attempted in a rural watershed in southern Canada. Modèles spatio-temporelles des précipitations dans les régions Résumé La variabilité spatio-temporelle des précipitations à l'intérieur d'une région agricole peut être modélisée en utilisant des extensions de la méthodologie de Box et Jenkins fondée sur des processus ARMA. Cette grande famille de modèles d'agrégation régionale appartient à la classe des processus STARMA (modèles autoregressifs de moyennes mobiles spatio-temporelles). Nous présentons ici la procédure itérative en trois étapes permettant de construire un modèle STARMA à partir d'un ensemble de séries pluviométriques. Nous avons identifié un modèle STMA (13). L'accent est mis sur les trois étapes de la construction du modèle, à savoir l'identification, l'estimation des paramètres et la validation. En ce qui concerne l'étape d'estimation la méthode du simplexe ainsi que des méthodes de gradient conjugué et quasi-Newton ont été utilisées. La méthode du simplexe semble bien adaptée et le modèle qui en résulte reproduit bien les caractéristiques spatio-temporelles de l'ensemble de séries pluviométriques utilisées. Une application a été réalisée dans une région rurale du Sud du Canada.

Open for discussion until I April 1996

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N. R. Dalezios & K. Adamowski

INTRODUCTION Precipitation is a space-time phenomenon and its variability is very important in simulation studies of the hydrological cycle. Moreover, precipitation is the most essential input to any modelling effort of the land phase of the hydrological cycle. There is an increasing research interest in precipitation studies mainly, among others, due to recent developments in climate variability and/or change and its impact on water resources (Houghton et al., 1992). The nonoptimization approach in simulation modelling is usually associated with the assessment of hydrological data or the quantification of the physical process and can be categorized into statistical and physical/deterministic models. Early simulation studies have been based on deterministic modelling in the form of empirical and/or conceptual methods. In order to account for uncertainties in either the data or the model and its parameters, several statistical models have been gradually introduced, which include regression and correlation techniques as well as probabilistic and stochastic methods. Stochastic models are developed from random data and allow for a memory effect. Thus, statistical properties of observed time series can be used in an attempt to identify the stochastic process. This identification approach consists of an a priori model postulation followed by acceptance or rejection of the model in terms of statistical tests. Time series analysis for precipitation modelling is a valuable step in water resources planning and management. The selection of models for the analysis of time series is essentially based on simulation and statistical decision theory (Salas et al., 1980). A flexible class of empirical models is the general family of autoregressive moving average (ARMA) processes (Box & Jenkins, 1976). These models have proved to be very useful in hydrological and meteorological analyses (Hipel et al., 1977), but, since they are univariate, are applicable only to single series of data. An alternative to univariate time series modelling is multivariate time series modelling (Anderson, 1958; Hannan, 1970), which is provided by linear stochastic difference equations. These models attempt to describe simultaneously a set of iV observable time series (Pfeifer & Deutsch, 1980; Mohamed, 1985; Adamowski etal., 1987). When these Ntime. series represent spatially located data, the interrelationships and the spatial correlation (Cliff & Ord, 1973) between the different spatial data sets can be taken into account and thus a better system description should result. The objective of this paper was to develop a spatio-temporal precipitation time series model from the general class of Space-Time Autoregressive Moving Average (STARMA) processes suitable for regional meteorological and hydrological analyses in rural watersheds. In constructing the appropriate dynamic stochastic STARMA model, a three-stage iterative procedure was followed, commonly referred to as the Box-Jenkins method (Box & Jenkins, 1976). At the parameter estimation stage of the model building procedure the classical polytope (or simplex) method was used (Nelder & Mead, 1965). Moreover, three further classical nonlinear parameter optimization algorithms were also employed for illustrative purposes, namely two conjugate gradient methods

Spatio-temporal precipitation modelling

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developed by Powell (1964) and Zangwill (1967) and a quasi-Newton (BFGS) method (Davidon, 1959; Fletcher & Powell, 1963). Applications were attempted in a rural and agricultural watershed. The paper is organized as follows: the next section explains the three-stage iterative STARMA model development procedure and describes the parameter optimization algorithms; the following section analyses and discusses the results in a rural watershed in southern Canada.

DEVELOPMENT OF THE STARMA MODEL The general form of the STARMA model (Martin & Oeppen, 1975; Pfeifer & Deutch, 1980; Mohamed, 1985; Adamowski et al, 1987) is given by: l

p

m

q

yit = ait+ I ZtpskLsyi(t^- E E ^ / V * ) s~0k=l

(!)

s=0k=l

where yit is the time series at time t and at site i, i = 1,2, ,,., N; p and q are the temporal orders of the AR and MA terms, respectively; / and m are the spatial orders of the AR and MA terms, respectively; A N

T N

Li LI^- it f=l(=l

t=ii=i

T

N

EE 7=1

where zit = yit — y with y being an estimate of the space-time grand mean given by: N

y

T

(5)

Arr L Ly^ Nl i=\ f=i

Space-Time Partial Autocorrelation Function (STPACF) The STPACF may be defined by extending the classical definition of Kendall & Stuart (1976) to space-time series. To compute the partial autocorrelations it is necessary to calculate the symmetric space-time autocorrelation matrix Cy (Fig. 3) (Adamowski et al., 1987; Mohamed, 1985). This matrix estimates the intercorrelations between all pairs of lagged variables WpZ/(f-g) a n d wij£j{t-k)' where h, s = 0, 1, ..., / are the spatial lags; g, k = 1, 2, ..., p are the time lags; and i, j = 1, 2, ..., N refer to the spatially located variables. These correlations may be expressed by the extended notation [rsksk\ and the matrix Cy may be computed by repeatedly solving the following equation: T

N

N

E E ' hgsk

t=v+l i=l

7=1

7=1 2

T

N

N

E E KwiA) v 7=1

m>

1

(6)



T

N

N

EE

f=i/=i

7=1

N. R. Dalezios & K. Adamowski

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where v = max(g,k) in t = v + 1 (equation (6)). Thus the space-time partial correlation between zit and wijsZjit_k), say \poosk is given by:

t oosk

-C. oosk ^oooo

(7)

^sksk)

where Coosk is the cofactor of the estimated STACF [roosk] in the correlation determinant j C\. It should be mentioned that in this method the order in which the autocorrelations are written is not significant, since each partial autocorrelation is computed by assigning certain fixed values to all other lagged terms. If a time series is nonstationary, temporal and/or spatial differencing is required to achieve stationarity. For example, the first order temporal (J) and spatial (5) difference operators are given by: w

Tyu = ya-y idit //((-l)

(8)

VsV/r =)>«-I>..v WJl M

k=0

h=1

r

s=0 s=l .... s=l r 0 0 0 1 r 0 0 1 1 ... r 0 on r 0 1 0 1 r 0111 ... r o m r 1101 r l m ... rnll

U00 r !l0l

r

llll

-

r

's=0 r

s=l

3=1 '

001p

1 Iff

s=0 9=0 h=0 r 0 0 0 0 h=l roioo h=2 r 1100

k==p

k=l

000p

r

lpOp

r

!lli

Cy=

g=p

h = 0

r

h

r

=l

0p00 • lpOO •

. h=l r i p 0 0 r )p01 r l p U

... r,pn

r

lplp

r

lplp

Fig. 3 The symmetric space-time autocorrelation matrix C .

STARMA processes are characterized by distinct STACFs and STPACFs (Martin & Oeppen, 1975; Pfeifer & Deutsch, 1980). Specifically, a STAR (/, p) process exhibits autocorrelations that decay in space and time and partial autocorrelations that cut off after p lags in time and / lags in space. Alternatively, a STMA (m, q) process exhibits autocorrelations that cut off after m lags in space and q lags in time and partials that decay exponentially both in space and time. Finally, the STARMA (/, p, m, q) models are characterized by STACF and STPACFs that both tail off in time and space.

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Parameter estimation The determination of the optimal values for the parameter in a system model is often of crucial importance both in the formulation of mathematical models for systems and in their subsequent use in simulation studies. Optimization subproblems are therefore associated with modelling studies. Many methods for solving nonlinear unconstrained problems have been suggested over the years. The objective of this work consisted of finding values for a set of parameters within a model which causes its behaviour to best approximate time series. Estimates of the parameters