Case Study of Monthly Regional Rainfall Evaluationby Spatiotemporal Geostatistical Method
Mohammad Karamouz, F.ASCE1; Sedigheh Torabi, Ph.D.2; and Shahab Araghinejad, Ph.D.3
Abstract: The rainfall time series as a spatiotemporal process requires suitable tools for prediction. In this paper, the application of akriging �geostatistic� method in modeling the point rainfall time series is presented in time and space. The two components of rainfall timeseries—deterministic trends and random components—are modeled using the kriging method. Sequential Gaussian and LU �lower andupper triangular matrix decomposition� simulation are used to simulate the random process of each station, both in space and time. Finally,simulated random components and deterministic trends are used to generate different realizations of rainfall time series at each grid point.Thirty-four years of monthly data of 34 rain gauges in the Zayandeh-rud river basin in the central part of Iran are utilized in this study tomodel and simulate rainfall data in space and time. A network of 8 by 8 km grids is used to represent the region of approximately 232 by224 km. The results will be useful in regional studies of climatic and hydrologic events such as droughts, as well as for assessment ofexceedance probability of rainfall in time and space. This information will be a basis for probabilistic forecasting of rainfall in this region.
DOI: 10.1061/�ASCE�1084-0699�2007�12:1�97�
CE Database subject headings: Rainfall; Evaluation; Case reports; Statistics; Predictions; Droughts.
Introduction
Geostatisics offers a collection of deterministic and statisticaltools for modeling spatial and/or temporal variability. The basicconcept of geostatistics is to determine any unknown value z as arandom variable Z with a probability distribution. One of thepractical methods in geostatistics is kriging, which is used widelyin the field of environment, mining, surveying, groundwater, andwater resources engineering studies. Simulation methods in geo-statistics such as sequential Gaussian and LU �lower and uppertriangular matrix decomposition� are also used to generate differ-ent realizations of rainfall processes �Deutsch and Journel 1998�.
Eynon and Switzer �1983� used the geostatistical method todetermine the space-time variations of the pollutant accumulationin the atmosphere. Bras and Rodrigues-Iturbe �1984� presentedthe use of kriging methods for spatial estimation of rainfall andpiesometric head. Haslett and Raftery �1989� studied the spatialand temporal variations of geophysical parameters. Goovaerts andSonnet �1993� used geostatistical methods to model the timevariation of spatial pattern of soil moisture capacity. Hohn et al.�1993� used these methods to calculate the population dynamicsin ecology.
1Professor, Center of Excellence for Infrastructure Engineering andManagement, Fanni College of Engineering, Univ. of Tehran, Tehran,Iran. E-mail: [email protected]
2Head of Water Allocation Office of WRMO and Visiting Professor,Univ. of Tehran, Iran.
3Visiting Assistant Professor, Iran University of Science andTechnology, Tehran, Iran; formerly, School of Civil Engineering, AmirKabir Univ., Tehran, Iran.
Note. Discussion open until June 1, 2007. Separate discussions mustbe submitted for individual papers. To extend the closing date by onemonth, a written request must be filed with the ASCE Managing Editor.The manuscript for this paper was submitted for review and possiblepublication on November 17, 2003; approved on January 12, 2006. Thispaper is part of the Journal of Hydrologic Engineering, Vol. 12, No. 1,
Recent advances in application of geostatistics in hydrologicstudies are found in Goovaerts �2000� for regional rainfall esti-mation, Merz and Blöschl �2004� for estimation of regional wa-tershed parameters, and Mouser et al. �2005� for hydrologic stud-ies of a watershed.
A generic problem of kriging can be considered as follows: nmeasurements of z data at locations with spatial coordinatesX1 ,X2 , . . . ,Xn are given, estimate the value of z at coordinate X0.Each coordinate has 2 or 3 dimensions.
A kriging estimator is simply a procedure or formula that usesdata to find a representative value or estimate of the unknownquantity such as
z0 = �i=1
n
�iz�xi� �1�
Thus the problem is reduced to selecting a set of coefficients�1 ,�2 , . . . ,�n. These coefficients are selected so that the estimatormeets the conditions of unbiasedness and minimum variance. Un-biasedness conditions mean that the average value of estimationerror must be zero, and the minimum variance condition impliesthat the mean square estimation error must be minimized.
A spatiotemporal kriging method was developed by Kyriakidisand Journel �1999� to demonstrate how point data could be ap-plied in the time-space domain. The same method was used byTorabi �2002� for regional analysis and prediction of droughts inthe central part of Iran. The method presented by Kyriakidis andJournel �1999� is extended and applied in this study.
In this paper, the method developed by Kyriakidis and Journel�1999� is used to model rainfall time series at each station. Thenthe spatial correlation of time series is assessed to obtain the timeand space correlation of the data network. Temporal trend modelsare determined by a parametric approach using time series pro-files at different stations. Temporal trend models are composed ofpartial time-dependent functions, which could be expressed byannual or seasonal periodic variations. Such patterns are station-
The components of the trend model are regionalized and theircross correlations are evaluated. Subtracting the trend from theoriginal series, a spatiotemporal model of residual time series isdeveloped to model the residual values. After determining tempo-ral covariance models at all stations, spatial models are developedfor interpolation of data at the adjacent stations. As a result, spa-tial and temporal patterns can be observed over a region. Simu-lation of the residual time series is limited to the synthetic gen-eration of realizations of a one-dimensional �1D� process bymeans of a spatiotemporal covariance model for each location.The spatiotemporal kriging method used in this study is explainedin the following sections. A case study and the results are pre-sented, followed by a summary and conclusion.
Spatiotemporal Trend
Variables such as rainfall are measured by gauges for many yearsbut usually are irregularly distributed over a region. This mayprovide rich temporal but poor spatial data, and therefore spatialcorrelations between measured variables are studied after the de-velopment of a temporal parametric trend model for the timeseries.
Stochastic rainfall Z�u� , ti� at each station u� and time ti isdetermined as follows:
Z�u�,ti� = m�u�,ti� + R�u�,ti� ti � T� u� � N �2�
where m�u� , ti�=space-time distribution of mean process at sta-tion u�; R�u� , ti�=stationary stochastic residuals of zero average;T�=time domain of observations at station u�; N=set of stations;and �=station number.
Deterministic variations at each station, u�, are modeled as thesum of K+1 temporal basic functions fk�ti� as follows:
m�u�,ti� = �k=0
K
bk�u��fk�ti�; i = 1, ¯ ,T� �3�
where bk�u��=coefficient of the kth function, fk�ti�, and f0�ti� istaken to equal 1. These functions usually consist of linear func-tions to show the long-term trend and the seasonal or annualperiodic components. The periodicity of the basic functions canbe obtained using spectral analysis �Kyriakidis and Journel 1999�.
Once the values of m�u� , t� are determined at each station,bk�u�� coefficients are regionalized using geostatistical methodsto develop a stochastic spatiotemporal model, M�u� , ti�. This pro-cess enables the estimation of temporal variations at points in thestudy area with no observed data. The correlation between eachcoefficient of the periodicity model is considered to estimate theregional coefficients.
Temporal variation models of m�u� , t� are developed indepen-dently at each station, u�, and the temporal spatial correlationsamong the components of the model are also calculated
M�s��u,t� = �k=0
K
bk�s��u�fk�t�; ∀ u � N, t � T �4�
The superscript s shows the realization of each variable. Suchsimulation is used to model the covariance matrix of randomfunctions RF�b0�u� ,b1�u� , ¯ ,bk�u��. It is also important for re-generating each correlation between different coefficients.
The coefficient vector bk�u�� can be calculated at each station
u�. In this case, individual simulations of principal components of
bk�u� are estimates for regionalizing the simultaneous observa-tions of these coefficients, as explained in the next section.
Principal Component Analysis
Spatiotemporal simulation is changed into individual simulationof independent factors obtained by principal component analysis�PCA� for K+1 set of coefficients of periodic trends �Kyriakidisand Journel 1999�. PCA transfers a group of dependent variablesbk�u� into the independent components at zero lag time in whicheach component is a linear combination of basic variables. Thecorrelation matrix, R= ��kk��, between K+1 coefficients of peri-odic trends is
�kk� =
1
n��=1
n
��bk�u�� − mBk� � �bk�u�� − mBk�
��
sBk. sBk�
�5�
where
mBk=
1
n��=1
n
bk�u�� �6�
and
sBk=�1
n��=1
n
�bk�u�� − mBk�2 �7�
Spectral analysis of the correlation matrix is written asfollows:
R = Q−1DQ �8�
where D is a diagonal �K+1�� �K+1� matrix, the diagonal mem-bers �dkk� are the eigenvalues of correlation matrix R, and Q isthe matrix of eigenvectors of the correlation matrix andQ−1Q=I, where I is a �K+1�� �K+1� unit matrix.
A set of K+1 independent factors is calculated as follows:
xk�u�� = �k�=0
K
qkk�bk��u�� − mBk�
sBk�
; k = 0, ¯ ,K �9�
where qkk� shows the elements of the matrix Q.The next step is the simulation of K+1 independent xk�u� vari-
ables and recalculating the K+1 simulated coefficients of periodictrends. This simulation is carried out through the following steps:
The K+1 realizations of xk�u�� corresponding to the K+1 ran-dom functions RFs�xk�u� ,u�D�, k=0, . . . ,K are generated inde-pendently. A simulation algorithm such as Gaussian may be ap-plied for this purpose �see Journel �1993�, Deutsch and Journel�1998� for details�. It is necessary to use variograms for simulat-ing and regionalizing a process. Two well-known variographymethods that are used in this study are Gaussian and sphericaltheoretical variograms, which have a better fit with the observedvariograms.
At each node, the simulated value for the kth coefficient of�s�
bk �uj� is re-estimated as follows:
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bk�s��uj� = �
k�=0
K
hkk�xk�s��uj�sBk�
+ mBk��10�
where hkk� shows the elements of matrix H, which is the inverseof matrix Q�H=Q−1�, and mBk�
and sBk�are the average and stan-
dard deviation of k�th coefficient of the trend model.
Spatiotemporal Residuals
As explained earlier, K+1 parametric time functions�fk�t�, k=0, . . . ,K� are used to represent the long-term variationpatterns. Seasonal rainfall variations are good examples of suchpatterns. Other variables such as those explaining the relationshipbetween the spatial climatic patterns are usually calculated andsimulated through a stochastic residual process. The model ofspatiotemporal residual process R�u , t� is developed in form of agroup of residual time series, which are spatially correlated. Thismeans that a time series is determined at each point. The residualvalues R�u� , ti� are obtained by subtracting the average variationsM�u� , ti� from rainfall values Z�u� , ti�. Therefore, residual timeseries models �R�u , t� , t�T� are determined by the following pro-cedure. The average of the residual time series �R�u� , ti�,i=1, . . . ,T�� is zero based on its properties, but its variance dif-fers from one station to another.
Simulation of spatiotemporal residuals R�u , t� consists of anumber of 1D simulations of spatially correlated time series. Thefirst step in model development for a residual time series is tostandardize n residual profiles for each station. The profiles arestandardized by dividing each value of R�u� , ti� by the standarddeviation of the residual profile SR�u�� at the same station asfollows:
R�u�,ti� =R�u�,ti�SR�u��
; i = 1, ¯ ,T� �11�
The standard deviation �scale� SR�u� represents the spatialvariations of the average spatiotemporal variation of M�u , t�. Toestimate the variance SR
2�u� of the residual profile R�u , t� at pointu with no measured data, it is necessary to regionalize the vari-ances SR
2�u� of n rain gauge stations.
The residual time series �R�u� , t� , t�T� is then decomposed toL+1 independent time series components with zero average asfollows:
where �ii�=Kronecker delta ��ii�=1 if i= i� and zero otherwise�;and CR�� ;ql�u���=correlogram of lth component of time series
Rl�u� , t�.The correlogram model CR�� ;ql�u��� is correlated in space
with parameter ql�u��, representing the lth correlogram of thebasic model where the parameter ql�u�� is estimated through the
variography of the Rl�u� , t�. ql�u�� with two basic parameters of
nugget effects and range. For simplification, it is assumed that all
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L+1 basic components of CR�� ;ql�u��� are of the same kind, forinstance spherical, Gaussian, or exponential correlogram models.
The above analysis represents a combination of L+1 indepen-
dent time series Rl�u� , t�. The covariance function CR�� ;ql�u��� atstation u� is calculated as follows:
CR��;q�u��� = �l=0
L
al�u��CR��;ql�u��� �15�
where al�u�= �wl�u��2 is a realization of the random variable Al�u�at point u, and ql�u� is a realization of random variable Ql�u� atthe same point. Usually, n components of correlogramCR�� ;q�u��� of the n standardized residual are produced indepen-dently at each station u�, and their parameters are spatially re-gionalized. This process requires regionalizing L+1 coefficients�Al�u� ,u�D�, l=0, . . . ,L and L+1 parameters, �Ql�u� ,u�D�,l=0, . . . ,L. Using the kriging method for these coefficients, onecould obtain the model of CR�� ;q*�u��� for the covariance func-tion of time series at any point u with no observed data.
Spatiotemporal residual simulation R�u , t� can be carried outafter regionalizing the model parameters. Residual spatiotemporalsimulation means to generate the observations of the standardresidual process, one series of the 1D process �r�u� , ti�,i=1, . . . ,T��, �=1, . . . ,n using special covariance models at eachpoint. Any stochastic simulation model can be applied to simulatethe residuals profiles, such as sequential Gaussian simulation orsimulation by Cholesky analysis of the covariance matrix LU�lower and upper triangular matrices method�. More explanationsof these methods are given in Deutsch and Journel �1998� andKyriakidis and Journel �1999�.
In this method the covariance matrix can be written as a pro-jection of two lower and upper triangular matrices, Lu and Uu, ineach period at point u� by Cholesky analysis as follows:
CRu= �CR��;q*�u��� = LuUu �16�
r�s��u,t� = Lu�u�s� �17�
where �u�s�= ���s��u , t�� is the vector of simulated values of stan-
dardized residual time series. Independent simulation of the
R�u , t� time series from one point u to another point u� causesspatial discontinuity in simulated observations, namely no datacould be generated between the points. The two simulated profilesmay have noticeable differences, even though the correlograms ofq*�u�� and q*�u�, parameters of the corresponding temporalcovariance models, are similar. Vector �u
�s� at point a must besimilar to vector �
u��s�
at a nearby point u� in order to prevent suchdiscontinuities.
Since there are no data at the nonmeasuring points, spatialcovariance is based on standardized residual data of n measuringstations around it. A set of T specific covariance functions at eachinterval is derived, one for each period ti, as follows:
CV�h;ti� � CR�h;ti� =1
n�h�ti
� r�u�,ti� · r�u� + h,ti� �18�
where n�h�ti=number of matched data points �r�u� , ti� ,rˆ
�u�+h , ti��, separated by vector h.The above expression can be simplified by identifying all co-
variances for each period with respect to the average covariance¯
Adjacent data r�u� ; ti� are not considered in the simulation of timeseries at point u during generation of different realization of spa-
tiotemporal residuals, R�u , t�. Therefore, large values of r�s��u ; ti�can be simulated after generating conditional small values ofr�u� ; ti�. This may cause discontinuity between the values ofsimulated time series at point u, with no observed data and aresidual profile at a given station.
One solution is to generate random numbers for spatiotempo-ral realizations of residuals, V�u , t�, conditioned on standardizedresiduals data, r�u� ; ti�, at each station. This condition causes thegeneration of large simulated residuals to be close to calculatedresiduals at each station.
Spatiotemporal residual simulation is carried out below. Notethat the term semivariogram is used instead of variogram. Semi-variogram is defined as half of the average degree of similaritybetween two values �Deutsch and Journel 1999�. The followingsteps are considered for modeling the residual time series of thisstudy:1. Estimate the standard deviation, SR�u��, of the residual time
series �r�u� , ti� , ti=1, . . . ,T�� at each gauging station andstandardize the obtained values.
2. Calculate the temporal semivariogram, �R�� ;q�u���, of stan-dardized residual time series �r�u� , ti� , ti=1, . . . ,T�� at all sta-tions.
3. Regionalize semivariogram parameters �al�u�� , l=0, . . . ,L�and �ql�u�� , l=0, . . . ,L� of the fitted models �R�� ;q�u��� and
4. Calculate the average of spatial covariance through time
Cv�h� from standardized residual data r�u� ; ti�.5. Attribute the location of the rain gauges to the nearest nodes
of the network and determine a random direction for theintersection of all N nodes in the domain D.
6. Generate a series of T independent realization of V�u , t�using the covariance model Cv�h�.
7. At each node u apply the estimated parameters �al*�u�� ,
l=0, . . . ,L� and �ql*�u�� , l=0, . . . ,L� of temporal semivari-
ograms for a standardized residual time series of�r�s��u , t� , t�T�, disregarding other nodes. Apply the specificvector of standard normal deviations ���s��u , t��, whose mem-bers are the spatiotemporal observation V�u , t� at each node.Multiply the standardized simulated time series by the esti-mated standard deviation at each node s*�u� obtained in step1 to obtain a realization of time series R�u , t�.
8. Repeat step 7 until all nodes u�D are covered.9. Generate another realization of R�u , t� by repeating steps 5
through 8.Rainfall realizations, �Z�s��u , t� , t�T�, are finally generated by
adding the trend model to the residual profiles. A simple rule is toadd the sth realization of the residual time series to the sth realiza-tion of the trend model from the average profile variation at pointu. The set of s alternatives with equal probabilities of simulatedrainfall realizations represent a stochastic model of the rainfall intime and space. This information can be used for the assessmentof risk and uncertainty or identifying the sampling points to beadded to the network.
Case Study
The proposed methodology has been applied to the Zayandeh-rud
and location of rain gauges
basin
river basin in Iran, originating from foothills upstream of
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Zayandeh-rud reservoir to Gavkhooni wet land some 300 kmdownstream of the reservoir. Zayandeh-rud basin has an area of32000 square kilometers with the density of one rain gauge per941 sq km �34 stations�. Thirty-four years of monthly historicaldata from October 1968 to September 2001 are considered in thisstudy. The study area and the location of gauging stations arepresented in Fig. 1.
The region is largely dependent on the Zayandeh-rud reservoirto provide the water supply for municipal, industrial, and a majorpart of agricultural demands. Therefore, the hydroclimatic assess-ment of the subbasin of the reservoir subbasin is very importantfor the entire basin.
The time series plot of monthly and mean monthly rainfalltime series in the Zayandeh-rud reservoir subbasin is presented inFig. 2. The annual average rainfall over the watershed is236.4 mm with a standard deviation of 253.7; the monthly rainfallaverage in the region is 19.7 mm with a standard deviation of39.2; the annual average rainfall over the subbasin of the reservoiris 566.4 mm with a standard deviation of 495.2; and the monthlyrainfall average in this subbasin is 47.2 mm with a standard de-viation of 39.2.
Rainfall data are converted into the standard normal values,called normal scores in this paper. This conversion was made bythe method of reserving the rank of normal scores �Deutsch andJournel 1998�. All of the analyses throughout this study are donein the Gaussian domain �environment�. The final simulated dataare transferred back into the original series by an inverse trans-formation.
The region is divided into a network of 29 by 28 grids con-taining 812 cells of 8 by 8 km. The entire 34 years of recordeddata from the 34 rainfall stations are used to simulate rainfall timeseries in the region.
Trend Models for Each Station
To determine the long-term components of temporal variations,spectral analysis is applied independently to each rainfall timeseries at each station. This analysis shows that the components ofa 12-month period �annual cycle� are more representative, whichis expected from rainfall data.
Temporal trend models at each rain gauge station u� are de-veloped to study the long-term trends such as linear or periodictrends in the observed rainfall values. These models are applied tothe monthly data.
Trend models for transformed rainfall profile at the gauging
Fig. 2. Monthly rainfall and mean monthly rainfall ti
stations are written as follows:
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m�u�,ti� = �k=0
3
bk�u�fk�ti� . ti � T� = b0�u�� + b1�u��ti
+ b2�u��cos2
12ti + b3�u��sin2
12ti �20�
where b0�u�� and b1�u�� are the intercept and the slope of lineartemporal trend model m�u� ; ti�, respectively, and b2�u�� andb3�u�� are related to the domain and phases of the periodic com-ponents of the temporal trend model.
A normal score time series and the periodic trend of a stationare presented in Fig. 3. As shown in this figure, no linear trendhas been observed for the rainfall profile. The analysis of allstations shows the same results, therefore only periodic compo-nents are used in the trend model and b0�u�� and b1�u�� are as-sumed to be zero. Coefficients bk�u�� are determined indepen-dently at each station by the ordinary least-squares method.Specific temporal trend models are fitted to each rainfall station asshown in Fig. 3.
The coefficients of sine and cosine coefficients, b2�u�� andb3�u��, for each station are shown in Fig. 4. Note that cosinecoefficients vary from −0.32 to −0.61 with an average of −0.49,and the sine coefficients vary from 0.66 to 0.99 with an averagevalue of 0.8. The higher values for the sine coefficient show thehigher relative weight of this term in the trend model. The stationswith the above-average cosine coefficient happen to have theabove-average sine coefficient as well, but the maximum valuesof coefficients did not occur at the same stations.
Once a deterministic temporal periodic model m�u� , ti� is de-veloped independently at each station, the trend coefficients arethen regionalized to develop a stochastic trend at nonmeasuringpoints by the kriging method.
ries in Zayandeh-rud reservoir subbasin �1968–2001�
Fig. 3. Time series and periodic model at Mahyar station
The PCA method was applied to the two sets of correlated trendcoefficients, b2�u�� and b3�u��, which were obtained by spectralanalysis. PCA generated two independent factors for these coef-ficients, and the results show that the estimated share of varianceis 29.5% for the first factor and 70.5% for the second. Each factorhas been generated independently, and the above ratios indicatethe relative importance of each factor.
The next step is to simulate the set of two factors �x1 and x2�independently with no lag time and regionalizing the simulated
Fig. 4. Variation of sine and cosine amplitude of 34 stations in regiovalues of sinusoidal and cosine coefficients.
Fig. 5. Spatial variation of first factor coefficients �top left�, generasemivariogram �bottom right�
coefficient of temporal periodicity in the study area. This set oftwo joint realizations leads to a corresponding event for a stochas-tic spatiotemporal trend, as explained in the next section.
Variography was done on the two sets of independent factors,as explained in the section on PCA. Because of the insufficientnumber of stations, the semivariograms in different directions arenot calculated and the two factors are modeled in form of semi-variograms, which are shown in Figs. 5 and 6. As it is shown inthese figures, Gaussian semivariogram model is fitted to the firstfactor and spherical semivariogram model is fitted to the second
of x- and y-axis are 8 km grids are 8 by 8 km. The right bars show
lization �top right�, Q-Q plot of simulated values �bottom left�, and
n units
ted rea
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factor. As shown in the variograms illustrated in Figs. 4 and 5,they may have unstable behavior at the distances more than therange �which is about 200 km in these figures�. The value of � isconsidered as a constant parameter, sill, after the range distanceonward. The semivariogram for the two factors x1 and x2 areexpressed as follows:
The first factor x1 is
�x1�h� = 0.65 + 0.69Gauss �h�
170 �21�
where Gauss�.� indicates a Gaussian semivariogram model, and�h� is an absolute value of the distance vector. This semivariogrammodel has an influence range of 170 km. This distance was ob-
Fig. 6. Spatial variation of second factor �top left�, generated rsemivariogram �bottom right�
Fig. 7. Periodic rainfall �in mm� at each station in February 20
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tained as the range of the variogram function between stations.The second factor, x2, is estimated by
�x2�h� = 0.1 + 1.4Sph �h�
170 �22�
where Sph�.� indicates a spherical semivariogram model. Thismodel has a range with a radius of 170 km. Gaussian sequentialsimulation is used to generate independent realization of the twofactors of corresponding data at 34 stations. Each of the semivari-ograms of the two factors in Figs. 5 and 6 shows the kth measuredfactor at 34 stations, sample semivariogram and model fittings, arealization obtained from direct sequential simulation, and theQ-Q �quantile-quantile� graph between this realization and the
ion �top right�, Q-Q plot of simulated values �bottom left�, and
realization of rainfall based on sequential Gaussian simulation
corresponding conditional realization. The Q-Q graph shows thestatistical similarity of the simulation and real values of the fac-tors. Each realization of the two factors is converted into an eventof the two trend factors. A set of two converted realizations indi-cates a common event of temporal trend coefficient in the studyarea
bk�s��uj� = �
k�=0
K
hkk�xk�s��uj�sBk�
+ mBk��23�
where hkk� are members of matrix H=Q−1, and hkk� and hkk� arethe average and standard deviation of kth simulated coefficient. Ateach node, an average number of 100 simulated values of coeffi-cients, bk
�s��uj�, are generated by Gaussian sequential simulation atnode u, estimated as follows:
bk�u� =1
S�s=1
S
bk�s��u�; u � D �24�
where the superscript indicates the sth simulated observation.Then an observation of temporal trend is estimated by using Eq.�23�.
An observation of a spatiotemporal trend is presented in theright diagram in Fig. 7 for the month of February. This month isselected because the spatial and temporal variations of rainfall aremore apparent throughout this region.
The results of this stage are the trend model of the temporalrainfall variation of 812 nodes at the region. Once the spatiotem-poral process is simulated, the next task is to simulate the residu-als. Note that the models are treated independently for the trendand the residual process.
Spatiotemporal Residual Simulation
The model of the spatiotemporal residual, R�u , t�, is developed asa set of spatially correlated residual time series. Spatiotemporalcovariance of the residual is obtained by deriving the residualtemporal profile at nonmeasuring points from specific covariancemodels at point u. The spatiotemporal simulation process is thencontinued by generating 1D time-varying stochastic profiles dataat nonmeasuring points.
Temporal Residual ModelsDeveloping parametric temporal covariance for the profiles at sta-tion u derives specific temporal covariances of the residual mod-els. The same as in the deterministic trend model, a spatial simu-lation is developed for the principal components of the model.The model parameters are interpolated spatially at nonmeasuringpoint u.
Based on the normality hypothesis for the normal scoresZ�u� , ti�, the distribution of this set of residual time series R�u , t�is transformed to a normal distribution. The average of each re-sidual rainfall time series is zero, but their variance may not beequal to one. The variance of residuals shows the local effect ofthe temporal trend component on the total profile used in thetemporal trend model. This effect shows a spatial variation in thisregion.
The standard deviation of residuals, SR�u��, is then calculatedat each station and the sample semivariogram of standard devia-tion is obtained as follows:
A kriging method was used by the above semivariogrammodel and sample standard deviation SR�u�� ,�=1, . . . ,34, to es-timate the value of the unknown standard deviation SR�u� at anonmeasuring point.
The semivariograms of the standardized temporal residualtime series are calculated at each of the 34 stations independentlyfrom other stations. Temporal semivariograms of the residualsample rainfall profile are presented in Fig. 8, including the fittedmodels for some selected stations. All temporal semivariograms,�, consist of a nugget effect and an exponential structure with avarying effective area of influence for the range parameters asfollows:
�R��,q�u��� = a0�u�� + �1 − a0�u���exp �
q1�u�� �26�
where Exp�.� indicates a semivariogram with an exponential
Fig. 8. Semivariograms of residual profiles for �a� Kouhpaye; �b�Eskandary; �c� Jafar-abad; and �d� Razveh stations
structure in a predetermined span of time �range�, q1�u��, at sta-
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tion �u��. Spatial deviations of semivariogram are used to region-alize the above parameters.
For the nugget effect a0
�a0�h� = 0.85 + 0.2Gauss �h�
50 �27�
For the effectiveness range parameter q1
�q1�h� = 0 + 1 � Sph �h�
70 �28�
Spatiotemporal Residual ModelThe temporal covariance function of residuals at each location,CR�� ,q*�u��, is a model of the covariance function that consists ofstandardized residual values �r�u , ti� , ti�T� at each location.
Any simulated time series of residuals is multiplied by thecorresponding standard deviation obtained from simple kriging atthe same location to obtain an observation of the residuals at eachnode. Realization of the standardized residuals is obtained at eachindividual node u. In this method, the r values are generated usingr data at the nearby gauging stations.
Fig. 9 shows the standard residual data for February 2000 ateach station. Also, a realization of standardized residual process,
R�u , t�, is presented in the left diagram of this figure for thismonth. Similar information for other months is presented in Kyri-akidis and Journel �1999� and Torabi �2002�.
Rainfall Realizations
The generated value of standardized residual series R�u , t� is con-verted into nonstandardized residuals, R�u , t�, by multiplying eachsimulated time series by the estimated standard deviation, S*�u� atnode u. Then the trend is simulated independently and the re-sidual series are added to the simulated normal scores trend,M�u , t�. The calculated value of events is finally converted intothe rainfall values by back transforming the standardized data ofall stations.
Discussion of Results
As explained earlier, geostatistical methods are used to regional-ize the limited data measured at all stations scattered over theregion. This section explains experimental application of the re-
Fig. 9. Standardized rainfall residuals �right� and
sults. One of the most important requirements in a region with
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J. Hydrol. Eng. 2007
limited data is data generation for nonmeasuring points. A timeseries of rainfall data is generated for a 10-year period by theproposed method. The results of spatiotemporal simulation of sixpoints, consisting of point A with grid coordinates �1,18�, point Bwith �2,18� coordinates, point C with �4,19� coordinates, point Dwith �3,16� coordinates, point E with �6,15� coordinates, andpoint F with �2,13� coordinates, shown in Fig. 10, are discussedin this section �coordinates are expressed in grids of 8�8 km�.These points are located in the Zayandeh-rud reservoir subbasinbecause of its hydroclimatic importance.
The average of 100 realizations generated in points A throughF is compared with the measured values of six adjacent stations.The Aznaveleh, Kalbali, Dameneh, Eskandari, Zayandeh-rud res-ervoir, and Chelgerd stations are considered as the adjacent sta-tions for points A through F, respectively, as shown in Fig. 10.The validation points near measuring stations were selected to testthe estimated rainfall series using the proposed algorithm.
As shown in Figs. 11�a–f�, the average of rainfall time seriesrealizations at the validation points are plotted along with therecorded monthly time series at the rain gauge stations in a10-year period from 1991 through 2000.
As shown in Figs. 11�a–f�, the simulated series has reproducedthe variability of historical data and shows a good match with theobserved data. About 92% of the time, simulated values at stationA reproduced the observed values with ±10% error. The values
tion of rainfall in region �left� for February 2000
Fig. 10. Location map of stations ��� and validation points ���
for the other stations, B through F, were 91, 85, 89, 90, and 94%respectively. The difference is partly due to the error associatedwith the disparity in the location of the observed and generateddata, as well as the effect of elevation variations. This range oferror is acceptable, and therefore developing a spatiotemporalmodel could be an effective tool in generating long-term series ofrainfall data.
Estimation of the regional probability distribution of rainfall atmonthly time intervals is another application of this study. As anexample, Fig. 12 shows the regional map of rainfall in the month
Fig. 11. Average of rainfall realization at location A and time seriesZayandeh-rud reservoir; and �f� Chelgerd
of February with 75% probability of nonexceedance. The figure
implies that in February the rainfall at each location of the regionis equal to or less than the values shown in the map with a prob-ability of 75%.
Summary and Conclusion
A spatiotemporal simulation model of rainfall data was developedby the geostatistical method in two distinct stages of trend andresidual simulation. One of the most important applications ofthis method is converting local and limited data into continuous
In this study, two components of the rainfall time series—deterministic periodicity and residuals—have been modeled at the34 rain gauges in the Isfahan region of Iran. The trend model wasestimated by spectral analysis. The coefficients of the trend modelhave been regionalized for 812 grid points over the region by thesequential Gaussian simulation method. The temporal correlationbetween residual time series was modeled by temporal vari-ograms. The characteristics of the temporal variograms �rangeand nugget effect� have been regionalized using the LU �lowerand upper triangular matrices� method. A 10-year time horizonwas considered for temporal simulation, and 10-year rainfall dataat 812 points of the region were obtained by combining the de-terministic and residual components. This procedure generates aunique realization of rainfall time series. Twenty five differentseed numbers were used through SG and LU simulation to pro-vide 25 realizations, which were then used for stochastic assess-ment of the rainfall variation in time and space.
The simulated rainfall time series at six points in the region,based on the realizations, were compared with data of six nearbystations, showing the application of the proposed methodology fordata generation. A regional map of monthly rainfall with a spe-cific probability of exceedance was plotted. This map shows an-other ability of the proposed method in the stochastic simulationof rainfall data and rainfall forecasting. It could be used for riskanalysis and planning of natural events such as droughts. Thestochastic modeling of spatiotemporal distributions of rainfall isuseful for water resources assessment, which is sensitive to space-time variation of rainfall. The important feature of the proposedframework is that the joint space-time assessment of uncertaintiesis considered in rainfall simulation to obtain the best estimate ofthe stochastic variation of rainfall in the region utilizing the entirerecorded data over the region.
Additionally, the results of space-time assessment could beconverted into the applicable tools for climatic and hydrologicanalysis of a basin. Because of the integrated space-time assess-ment used in this study, the results could be used for a wide rangeof problems in hydrological sciences such as:• Climate change analysis in the region based on long-term gen-
eration of different realizations;• Data generation for a specific part of the region that is suffer-
ing from shortage of data;
Fig. 12. Regional map for rainfall with 0.75 nonexceedanceprobability in February �right bar shows rainfall in mm�
• Probabilistic rainfall forecasting; and
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J. Hydrol. Eng. 2007
• Identifying vulnerable areas in the region during droughtperiods.The results of this study show the significant value of the
proposed method in assessment of climatic and hydrologicalevents and converting point data to distributed data in time andspace.
Acknowledgments
This paper was partially supported by the regional water board ofIsfahan. The contribution of managers and engineers of the re-gional water board of Isfahan, Mr. Torfeh, Mr. Asadi, and Mr.Heidarpour, as well as graduate students and computer staff Ms.Asgharzadeh, Ms. Parvini, Mr. Lotfi, Mr. Zolfagharpoor, and Ms.Hashemi, is hereby acknowledged.
Notation
The following symbols are used in this paper:a nugget parameter of semivariogram;
Bk stochastic realization of coefficients bk;bk�u�� kth coefficient of trend model at station u�;
C correlogram operator;Cov covariance operator;
D diagonal matrix of dii;E expected value;
Exp exponential semivariogram model;fk�t� kth time-dependent function of trend model;
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