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A scale model of multivariate rainfall time series
Journal of ELSEVIER
Journal of Hydrology 168 (1995) 1 15
Hydrology
[1]
A scale model of multivariate rainfall time series D a v i d A. Chin Department of Civil Engineering, University of Miami, 1251 Memorial Drive, Coral Gables, FL 33124-0630, USA
Received 9 March 1994; revision accepted 3 November 1994
Abstract
A multivariate time-series model that uses a factor-analytic approach is shown to provide an effective description of both monthly and annual rainfall in south Florida. In the case of monthly rainfall, the scale model shows that deviations from monthly means are caused primarily by large-scale phenomena that have temporal structure. These sort of phenomena are not accounted for by using conventional contemporaneous ARMA models. In the case of annual rainfall, the majority of variance is associated with random normally distributed largescale phenomena that do not have temporal structure.
1. Introduction
Rainfall time series generally display 12 m o n t h circular stationarity, and therefore the objective in modeling monthly and annual aggregates differs in that monthly models attempt to describe the within-year variability, while annual models attempt to describe the long-term variability. In multivariate analyses of rainfall, each variate represents the rainfall at a single location, and classical multivariate time-series models (e.g. Salas et al., 1980; Bras and Rodriguez-Iturbe, 1985) are generally designed to preserve the auto and crosscorrelations via autoregressive or moving average ( A R M A ) models. In this paper, a so-called scale model is presented that views the rainfall at individual locations as a linear combination o f a few independent regional-scale processes, plus small-scale processes that are correlated with the rainfall at individual locations. The question of what constitutes a regional-scale process, or conversely what constitutes a small-scale process, is determined by the spacing between rainfall measurements. A regional-scale process is defined as a time series that is significantly correlated with rainfall measurements at several stations, and therefore the spatial scale of this phenomenon is on the order of the size of the region containing the correlated measurement locations. 0022-1694/95/$09.50 © 1995 - Elsevier ScienceB.V. All rights reserved SSDI 0022-1 694(94)02668-8
D,A. Chin / Journal of Hydrology 168 (1995) 1-15
Similarly, a small-scale process is defined as a time series that is significantly correlated with the measurements at only a single location, and therefore the spatial scale of this process is on the order of the size of the region that is closest to the measurement station. The separation of regional-scale processes from small-scale processes is conveniently done using factor analytic techniques, with the subsequent development of time-series models for the extracted factors (processes). The application of factor analysis to modeling multivariate hydrologic time-series is a novel approach, however, such an approach has been previously advocated for multivariate time series analysis in the field of psychology (Anderson, 1963). Previous applications of factor analysis to the study of rainfall have generally been concerned only with the segregation of rainfall into various fundamental factors (e.g. Beaudoin and Rousselle, 1982), while the time-series structure of the extracted factors have not traditionally been explored. The main advantage of the proposed scale model over conventional multivariate rainfall models is that fundamental (linearly independent) regional-scale rainfall processes are easier to relate to specific geophysical phenomena, and therefore provide insight into the forcing mechanisms affecting the rainfall. A second advantage of the proposed scale model over the more conventional multivariate ARMA model is that the scale model provides a more defensible approach to modeling rainfall time series where the temporal and spatial structure of the rainfall are related.
2. Scale model versus A R M A models
The conventional approach to multivariate ARMA modeling is described by Salas et al. (1980), Bras and Rodriguez-Iturbe (1985), and Salas (1993). The conventional methodology for fitting multivariate ARMA models to rainfall measurements requires an assumption of the order of the AR and MA components of the model, followed by parameter estimation to preserve auto and cross-correlation matrices (Bras and Rodriguez-Iturbe, 1985). Parameter estimation techniques tend to become quite complex as the order of the ARMA model increases, and therefore Salas (1993) has proposed using so-called contemporaneous ARMA (CARMA) and periodic ARMA (PARMA) models to describe multivariate rainfall time series. The use of CARMA and PARMA models, in lieu of generalized ARMA models, are justified whenever lagged cross-correlations between rainfall measurements at different stations are not significant. The proposed scale-model is also a contemporaneous model, and must therefore be contrasted with the contemporaneous models described by Salas et al. (1980) and Salas (1993). The conventional approach using CARMA models consists of two main steps. In the first step, ARMA models are fitted to each of the p seasonally adjusted rainfall time series, and then the random residual time series at each of the p stations are estimated. The second step is to compute the zero-lag correlation matrix of the random residuals at each of the p stations, and use standard decomposition techniques to express the random residual at each station as a linear combination ofp independent random time series. The limitation of the CARMA approach is that it a priori assumes that the structured temporal
D.A. Chin / Journal of Hydrology 168 (1995) 1-15
fluctuations at the individual stations are uncorrelated with the structured temporal fluctuations at all other stations. In effect, this means that structured temporal fluctuations at different stations are not tied together, and are therefore modeled as small-scale phenomena. Realistically, this is probably not a sound assumption. Since only the correlation between random residuals are accounted for in contemporaneous models, a second assumption inherent in the C A R M A model is that regional-scale phenomena are all random. Again, this assumption is questionable in most cases and is seldom validated. The proposed scale model addresses the aforementioned limitations of conventional contemporaneous models by first segregating the spatially correlated (regional-scale) rainfall fluctuations from spatially uncorrelated (smallscale) rainfall fluctuations. The temporal structure of both the large-scale and small-scale processes are then investigated separately.
3. The scale model
The scale model of rainfall has the following form Yt = Iz + A x t
-t- ¢;t
(1)
where Yt is the rainfall vector;/z is the mean o f y t ; A is a matrix accounting for the contribution of each regional-scale process to the rainfall vector; xt is the vector time series of regional-scale processes; and Et is a vector time series of small-scale processes. The functional form of the scale model is identical to the factor model (e.g. Morrison, 1990), where A is the factor loading matrix, xt is the vector of common factors, and et is the vector of specific variates. The scale model differs from factor model in that classical univariate time series models are used to describe the 'factor processes, thereby producing a forecasting and simulation tool and, in addition, definite spatial scales are associated with the factor processes. In cases where there are m regionalscale processes (factors) and p rainfall time series, then Yt,/~ and Et, are p-element vectors, xt is an m-element vector, and A is a p x m matrix. If the contemporaneous covariance matrix of the rainfall measurements are given by YT, then it can easily be shown from Eq. (1) (Morrison, 1990) that , ~ = A A T + fir
(2)
where g' is the p-vector of variances of the small-scale processes, Et. The representation given in Eq. (2) clearly partitions the variance in the rainfall record into a portion that is described by the regional-scale processes, xt, and a portion described by smallscale processes, Et. A second relationship resulting from the scale model is that the matrix A is the covariance matrix between the measured rainfall, Yt, and the regionalscale process, xt, therefore A =cov
(ytxtT)
(3)
Since the regional-scale time series, xt, are generally required to have a mean of zero and a variance of one, it follows directly that the correlation between the p measured time series and m regional-scale processes is given by the m × p correlation matrix C,
4
D.A. Chin / Journal of Hydrology 168 (1995) 1-15
where C = A-1/ZA
(4)
and A is a diagonal matrix whose diagonal elements are equal to the variances of measured time series, Yr. This correlation matrix is critical in assessing the spatial scale of any rainfall process. For example, a regional-scale process is correlated with several elements ofyt, while a local-scale process is significantly correlated with only a single element o f y t. A useful property of the scale model is that if the coefficient matrix, A, is multiplied by any orthogonal matrix T, then the partitioning of the variance explained by the regional-scale processes, xt, and the independent variates, et, are unaffected (Morrison, 1990). What changes is the correlation matrix, C, which is transformed into C' where C' = C ( r - l ) T
(5)
The transformation matrix, T, provides a useful mechanism for transforming the regional-scale processes such that the correlation matrix becomes more polarized, thereby making interpretation of processes clearer. Criteria for selecting the transformation matrix are described extensively in factor analysis texts (e.g. Morrison, 1990). The parameters to be estimated in the scale model are A and ~, and there exists a variety of methods that have been proposed for their evaluation. The most common method of estimating A and fit is the maximum-likelihood method which assumes that Yt is a random multinormal variate, in which case the sample covariance matrix, S, is described by a Wishart probability density function. The parameters A and ~ are then estimated by maximizing a log-likelihood function derived from the Wishart distribution. A complicating factor in applying this estimation method to multivariate time series is that such series are not necessarily random and therefore may not be consistent with the randomness assumption. Anderson (1963) addressed the problem of applying factor analysis methods to general multivariate (psychological) time series and concluded that temporal correlations would probably not be a serious limitation in using standard factor-analytic techniques to estimate A and ~. This proposition is intuitive for cases where the length of the time series is much longer than the temporal correlation length scales of the rainfall process.
4. Identification of scale model
Specification of the scale model by Eq. (1) requires that the number of regionalscale processes, m, be determined. Knowing this number of independent processes, the rotated factor loadings, A T, and the specific variance vector, ~, are then computed by conventional factor-analysis techniques such as by the maximum-likelihood method. Steps to identify the model parameters are given below. (1) Specify the number of regional-scale processes, m, to be equal to one, then use the maximum-likelihood estimation procedure to determine the factor-loading matrix, A, and the specific variance matrix, ~.
D.A. Chin / Journal of Hydrology 168 (1995) 1-15
5
IORL~DO
ATLANTIC DCEAN
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Fig. 1. Hydrologic basins in the South Florida Water Management District.
6
D.A. Chin / Journal of Hydrology 168 (1995) 1-15
(2) Use the quartimax criterion (Morrison, 1990) to rotate the factor loading matrix, A, and thereby maximize the correlations given by the elements of the rotated correlation matrix C . (3) Compute the chi-square statistic associated with the hypothesis that the covariance matrix of the m-factor model is equal to the measured covariance matrix (Morrison, 1990). Requiring the null hypothesis to be supported with 95 % confidence indicates when the assumed order of the factor model is justified. If the null hypothesis is not supported, then the number of factors is increased by one, and the computations are started again from Step (1). If the null hypothesis is supported, then the order of the scale model, m, and the computed factor model parameters A and ~ are accepted. (4) Determine the statistical characteristics of the regional-scale time series, xt, and small-scale time series, et, extracted in the previous step. These time series are analysed for randomness and normality, using conventional statistical analyses. If any of these time series turn out to be non-random, then conventional univariate ARMA time series models are developed for each of these linearly independent time series. The battery of statistical analyses conducted on each of the independent regional-scale and small scale time series are particularly illuminating in associating the identified regional-scale processes with specific large-scale geophysical processes.
5. Model apphcafion Water-resource management in south Florida is governed by the South Florida Water Management District (the 'District'), which manages an area of approximately 44 000 km 2. This region is shown in Fig. 1, and is composed of 12 hydrologic basins. In this study, the validity of using the scale model to describe rainfall in the 12 hydrologic basins is investigated.
5.1. Monthly rainfall model 5.1.1. Local analysis The temporal variation in the average monthly rainfall in each of the 12 hydrologic basins are shown in Fig. 2. Monthly rainfall is clearly a non-stationary process. Analysis of the temporal structure in each basin is preceded by standardization according to the relation
Yij--
W i j - - VV~i
Si
,
i = 1,...,12,
j = 1,...,N
(6)
where Yij is the standardized rainfall in month i and year j, Wij is the transformedmonthly rainfall, I~i is the mean of Wig in the month i, and Si is the standard deviation of W O. in month i. Transformation of the measured rainfall time series (prior to standardization) is generally required due to a relation between the mean and variance of the time series. Square-root transformations are the most common
D . A . Chin / Journal o f H y d r o l o g y 168 (1995) 1 - 1 5 25.0
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for monthly rainfall. The time-series model appropriate for describing the standardized-monthly rainfall may have either constant or seasonal parameters, and both models have been proposed in the past (Salas et al., 1980). The constant parameter model views the standardized rainfall, Yij, as being a stationary time series, while the variable parameter model recognizes the inherent variability in the statistical character of rainfall for different months. Previous analyses (Chin, 1993) have shown that the standardized-rainfall series are non-stationary and that time-series models with monthly varying parameters are appropriate. The correlation between rainfall in consecutive months was investigated and shown to be significant only between the low-rainfall months of December and January. As a reasonable approximation, time-series models for all months were developed independently, thereby neglecting correlations between the rainfall in adjacent months. Statistical analyses of individual-month standardized rainfall show that the fluctuations in each month are almost all random and normally distributed. Exceptions to this rule occur in April and October, where least-squares regression analysis indicates that a significant linear downtrend occurs in many basins. Specifically, UKR, KRB, UEC, COL and CAL basins (Fig. 1) show significant linear downtrends in April while UEC, LEC, EAA, ENP, WC 1 and WC3 basins show significant linear downtrends in October. The timeseries model appropriate for describing the standardized series y/j is therefore of the form y(k) = m~k)Tj + cl k) + al~ ) (7) ij
where k is the basin,
ml k) and c~k) are the slope and intercept of the trend in basin k in
8
D.A. Chin / Journal of Hydrology 168 (1995) 1-15
month i, Tij is the time step corresponding to thejth year, and al(~) is a random normal deviate with a mean of zero and a standard deviation of al k). Clearly the parameters of the time-series model are ml k), c~k), and al k). It should be noted that in most basins ml k) and elk) are equal to zero and the standardized-monthly rainfall varies randomly from year to year. The model described by Eq. (7) may be conveniently written in the vector form Yij -~- mi Tj q- c i q- aij
(8)
This model describes the standardized rainfall in each of the 12 basins. The CARMA model that would be extracted by the procedure described in Salas et al. (1980) is then given by (9)
Yij : m i T j q- cl q- Prhj
where r/i) is a vector of independent random variates, and P is the matrix resulting from the Choleski decomposition of the covariance matrix of the correlated random fluctuations a O. The CARMA model given in Eq. (9) is based on the assumption that structured temporal fluctuations within basins are uncorrelated between basins or, in other words, correlations between the rainfall in various hydrologic basins are accounted for only by random processes. Viewed in the context of physical space, the conventional CARMA model (Eq. 9) considers all structured temporal fluctuations to be small-scale (local), while regional-scale processes are assumed to be random. The validity of these assumptions are investigated in the following sections.
5.1.2. Identification of scale model The scale model of the standardized rainfall is of the form
Yij = AFij + eij
(10)
Table 1 Correlation of m o n t h l y rainfall with regional-scale processes Basin UEC UKR KRB LEC COL EAA CAL ENP LOK WC1 WC2 WC3 Explained variance
First process 0.80 0,71 0.87 0,81 0,79 0,85 0,79 0.76 0.93 0.88 0.78 0.88 67 %
Second process
Third process
0.11 -0.30 -0.29 0.52 -0.17 -0,05 -0.34 0.14 -0.21 0.39 0.51 0.31
-0.25 -0,13 -0.20 -0.02 0.52 0.03 0.15 0.42 -0.09 -0.09 -0.05 0.25
11%
5%
D.A. Chin I Journal of Hydrology 168 (1995) 1-15
where A is a parameter matrix, F u is the vector of regional-scale processes, and e u is a vector of small-scale processes. Following the model identification steps described earlier, the chi-square statistic was used to determine how many regional scale processes are necessary to adequately represent the covariance matrix of the rainfall measurements. Requiring the null hypothesis to be supported within a 95% confidence interval, reveals that a three-factor model is appropriate, and the correlations between the three regional-scale time series and the standardized-rainfall time-series are shown in Table 1. It is clear that the first process is correlated with the rainfall in all basins and reflects a global-scale process, while the second and third regional-scale processes reflect smaller scale processes. The large-scale rainfall processes are identified from the model parameters A and if' via the relation (11)
Fi s = A T ( A A x + ff/-l)Yij
where y,j is the standardized rainfall time series. The statistical character of each of the three linearly independent regional-scale rainfall processes will now be investigated separately. The first regional-scale time series (process) accounts for approximately 67 % of the total variance. The mean for each month is plotted in Fig. 3, and the statistics of the first process for each month are shown in Table 2. F r o m the statistics shown in Table 2 it is clear that in almost all months this (dominant) global-scale process consists of essentially random normally distributed fluctuations, with the exceptions of April, October, and November where there are significant trends. The trend is downward in 1.0
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D.A. Chin / Journal of Hydrology 168 (1995) 1-15
10
Table 2 Statistics o f the first r e g i o n a l - s c a l e p r o c e s s in the m o n t h l y r a i n f a l l Month
Slope a
Lag a (months)
Exceedence probabilities Q-statistic
Jan. Feb. Mar. Apr. a May Jun. Jul. Aug. Sep. Oct. Nov. Dec.
-
2 -
-0.017 -0.022 0.020 -
1 -
0.20 0.22 0.55 0.71 0.74 0.61 0.77 0.63 0.82 0.02 0.24 0.53
x 2 statistic 0.22 0.63 0.10 0.82 0.33 0.11 0.91 0.49 0.16 -0.63 0.33 0.33
a O n l y statistically s i g n i f i c a n t q u a n t i t i e s s h o w n .
April and October, while the trend is upward in November. Viewed collectively, these results indicate a global-scale (District-wide) decrease in April and October rainfall, with a global-scale increase in November rainfall. This is tantamount to a 'shifting of the seasons'. The statistics of the first global-scale process support, the following model
FD) = p!l) + (miTj q_ Cij) _1_aij
(12)
where i is the month index, j is the time index, and the parameters of the model are /~/(1), mi, ci, and the variance of the random normal deviate, aij. The fact that this global-scale process has temporal structure, at least for three of the 12 months, violates the assumption that regional-scale fluctuations in standardized rainfall do not have any temporal structure. In this sense, the scale model appears to be an appropriate model, presenting a fundamentally different picture of the rainfall process than the CARMA-type model identified earlier. Considering the second regional-scale process, which accounts for approximately 11% of the total variance, the monthly variation in the mean is plotted in Fig. 3, and the cyclical (semi-annual) nature of the variation in the monthly mean is clearly evident. The mean-removed (process) is random and normally distributed, and therefore the second regional-scale process may be described by the following model
F[2)
=/~t
!2) q- aij
(13)
where p//2) are the seasonal means. The third regional-scale process accounts for approximately 5% of the total variance, and the variation in the monthly mean of the third factor is shown in Fig. 3, where the annual periodicity in the mean is clearly evident. As in the case of the second regional-scale process, the mean-removed residuals are random and normally distributed. The third factor is therefore
D.A. Chin / Journal of Hydrology 168 (1995) 1-15
Table 3 Variance explained by
the three-scale model of m o n t h l y rainfall
Basin
Explained variance (%)
UEC UKR KRB LEC COL EAA CAL ENP LOK WC 1 WC2 WC3
72 62 88 93 91 72 77 78 91 93 88 93
Average
83
described by the following model F~3) = ~!3) + aij
(14)
where ~{3) are seasonal means. To assess the overall ability of the three-scale model to describe the rainfall in each of the 12 hydrological basins, the explained variance in each of the hydrologic basins were computed and these results are shown in Table 3. Clearly the model performs best in the KRB, LEC, COL, WC1, WC2 and WC3 basins where the three scalemodel explains more than 88% of the total variance. On the other hand, the three scale model performs worst in the UKR basin, where only 62% of total variance is explained. The model performs adequately in the remaining basins where 72-78% of the variance is explained. In summary, it has been shown that fluctuations in standardized monthly rainfall are driven primarily by regional-scale phenomena, which account for an average of 83% of the variance in the 12 hydrologic basins. These regional-scale fluctuations are explicit components of the scale model, and are effectively extracted using factor analytic techniques. In a statistical sense, it has been shown that three regionalscale processes are adequate to account for the important covariances in the measured monthly rainfall. The primary regional-scale process has been shown to have structured temporal fluctuations, indicating that the scale model is more appropriate than the conventional CARMA model proposed by Salas et al. (1980). 5.2. Annual rainfall model 5.2.1. Local analysis The average annual rainfall in the District for each of the 12 hydrologic basins are
D.A. Chin / Journal of Hydrology 168 (1995) 1-15
12
Table 4 A v e r a g e a n n u a l rainfall in h y d r o l o g i c basins Basin
R a i n f a l l (cm)
CAL COL EAA ENP KRB LEC LOK UEC UKR WC1 WC2 WC3
133 134 138 131 119 147 120 131 130 150 151 144
given in Table 4. There is obviously significant variability between basins. The statistics of the annual rainfall time-series at each of the 12 basins are shown in the Table 5, and from these results, it appears that in 8 of the 12 basins the (Box-Cox) transformed annual rainfall is random and normally distributed, while in EAA and LOK there is a significant downward trend, and in LEC and UKR there appears to be some temporal structure that can be described by AR(3) and AR(2) models. The statistical results shown in Table 5 further indicate that the log-transform appears to be most appropriate for transforming the annual rainfall into Gaussian distribution.
Table 5 Statistics o f a n n u a l rainfall series (1941-1990) Basin
UEC UKR KRB LEC COL EAA CAL ENP LOK WCI WC2 WC3
Box-Cox transform
Z - 1.0 Z -1'° Log Z Log Z Z -°'5 Log Z Log Z Log Z Log Z Log Z Log Z Z -°'5
Slope a
_ -
-0.0048 -
-0.0035 -
" O n l y statistically significant q u a n t i t i e s shown.
Lag a (years)
2 1, 3 10 1 1 -
Exceedence p r o b a b i l i t i e s Q-statistic
X2-statistic
0.42 0.01 0.50 0.01 0.43 0.27 0.77 0.06 0.89 0.10 0.15 0.33
0.78 0.68 0.91 0.87 0.18 0.91 0.63 0.49 0.13 0.54 0.49 0.34
D.A. Chin / Journal of Hydrology 168 (1995) 1-15
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Table 6 Correlation of annual rainfall with regional-scale processes Basin
First process
UEC UKR KRB LEC COL EAA CAL ENP LOK WC1 WC2 WC3
0.87 0.75 0.88 0.86 0.88 0.88 0.81 0.79 0.91 0.92 0.86 0.94
Explained variance
74%
Second process 0.11 0.10 0.30 -0.46 -0.01 0.13 0.20 -0.15 0.42 -0.21 -0.33 -0.26 7%
5.2.2. Identification o f scale model F a c t o r analysis i n d i c a t e s t h a t two regional-scale processes are sufficient to describe the a n n u a l rainfall. T h e c o r r e l a t i o n s b e t w e e n the r e g i o n a l - s c a l e processes a n d the a n n u a l rainfall time-series a r e s h o w n in T a b l e 6. T h e first r e g i o n a l - s c a l e process is highly c o r r e l a t e d with the rainfall in all basins, reflecting a global-scale process t h a t a c c o u n t s for a p p r o x i m a t e l y 74% o f the t o t a l variance. T h e s e c o n d regional-scale process, w h i c h a c c o u n t s for 7 % o f the v a r i a n c e is c o r r e l a t e d with the rainfall at a fewer n u m b e r o f stations. Statistical a n a l y s e s indicate t h a t b o t h factors are r a n d o m a n d n o r m a l l y d i s t r i b u t e d , w i t h a m e a n o f zero a n d a v a r i a n c e o f unity. Therefore, these processes are d e s c r i b e d b y the f o l l o w i n g simple m o d e l s
Table 7 Variance explained by the two-scale model of monthly rainfall Basin
Explained variance
UEC UKR KRB LEC COL EAA CAL ENP LOK WC1 WC2 WC3
76% 55% 83% 95% 82% 56% 67% 52% 99% 85% 83 % 95%
Average
77%
D.A. Chin / Journal of Hydrology 168 (1995) 1-15
14
F) 2) = a !z) .1
(16)
where ~ are the annual factors, and ai are standard normal deviates. It is noteworthy that the form of the annual-rainfall model is identical to the monthly-rainfall model, except that the seasonal-mean term is missing. The performance of the scale-model in describing the annual rainfall fluctuations and the 12 hydrologic basins may be measured by the percentage of variance explained in each of the basins by the proposed model. These percentages are shown in Table 7, which indicates that the two regional-scale processes explain an average of 77% of the variance.
6. Summary and conclusions The effectiveness of using a multivariate scale model to describe rainfall in south Florida has been investigated. The scale model has the general form Yij = A F i j + eij
(17)
where yij is measured (or standardized) rainfall in season i and year j, A is a parameter matrix, Fij is a vector of regional-scale processes that show significant correlation with the rainfall at multiple locations, and eij is a vector describing localized fluctuations. The results presented in this paper show that the scale model provides an effective description of the monthly and annual rainfall in South Florida. In the case of monthly rainfall, the scale model indicates that deviations from monthly means are caused primarily by regional-scale phenomena, which have temporal structure. This result is contrary to the basic assumptions of the contemporaneous multivariate approach suggested by Salas et al. (1980), which assumes that nonrandom temporal fluctuations are not spatially correlated. The results of the present study therefore show that the conventional (CARMA) modeling approach should not be applied. The scale-modeling approach explored in this study presents an attractive alternative. In the case of annual rainfall, it has been shown that the majority of variance is associated with regional-scale phenomena that are random and normally distributed. The contemporaneous multivariate model would also be valid in this case.
Acknowledgments The investigation reported in this paper was supported by the United States Department of the Interior under Grant No. 14-08-001-G2012, and also by the South Florida Water Management District under Grant No. C91-2237. Lin Li of the University of Miami assisted in conducting many of the statistical analyses, Dr. David Maidment of the University of Texas provided many useful comments.
D.A. Chin / Journal of Hydrology 168 (1995) 1-15
15
References Anderson, T.W., 1963. The use of factor analysis in the statistical analysis of multiple time series. Psychometrika, 28(1): 1-25. Beaudoin, P. and Rousselle, J., 1982. A study of space variations of precipitation by factor analysis. J. Hydrol., 59: 123-138. Box, G.E.P. and Cox, D.R., 1964. An analysis of transformations. J. R. Stat. Soc., 26:211-252. Bras, R.L. and Rodriguez-lturbe, I., 1985. Random Functions and Hydrology. Addison-Wesley, Reading, MA. Chin, D.A., 1993. Analysis and prediction of South-Florida rainfall. Tech. Rep. No. CEN-93-2, Department of Civil and Architectural Engineering, University of Miami, Coral Gables, FL. Morrison, D.F., 1990). Multivariate Statistical Methods, 3rd edn. McGraw-Hill, New York. Salas, J.D., 1993. Analysis and modeling of hydrologic time series. In: D.R. Maidment (Editor), Handbook of Hydrology. McGraw-Hill, New York. Salas, J.D., Delleur, J.W., Yevejevich, V. and Lane, W.L., 1980. Applied Modeling of Hydrologic Timeseries. Water Resources, Littleton, CO. Wei, W.W., 1990. Time Series Analysis. Addison-Wesley, Redwood City, CA.
Journal of Hydrology 168 (1995) 1 15
Hydrology
[1]
A scale model of multivariate rainfall time series D a v i d A. Chin Department of Civil Engineering, University of Miami, 1251 Memorial Drive, Coral Gables, FL 33124-0630, USA
Received 9 March 1994; revision accepted 3 November 1994
Abstract
A multivariate time-series model that uses a factor-analytic approach is shown to provide an effective description of both monthly and annual rainfall in south Florida. In the case of monthly rainfall, the scale model shows that deviations from monthly means are caused primarily by large-scale phenomena that have temporal structure. These sort of phenomena are not accounted for by using conventional contemporaneous ARMA models. In the case of annual rainfall, the majority of variance is associated with random normally distributed largescale phenomena that do not have temporal structure.
1. Introduction
Rainfall time series generally display 12 m o n t h circular stationarity, and therefore the objective in modeling monthly and annual aggregates differs in that monthly models attempt to describe the within-year variability, while annual models attempt to describe the long-term variability. In multivariate analyses of rainfall, each variate represents the rainfall at a single location, and classical multivariate time-series models (e.g. Salas et al., 1980; Bras and Rodriguez-Iturbe, 1985) are generally designed to preserve the auto and crosscorrelations via autoregressive or moving average ( A R M A ) models. In this paper, a so-called scale model is presented that views the rainfall at individual locations as a linear combination o f a few independent regional-scale processes, plus small-scale processes that are correlated with the rainfall at individual locations. The question of what constitutes a regional-scale process, or conversely what constitutes a small-scale process, is determined by the spacing between rainfall measurements. A regional-scale process is defined as a time series that is significantly correlated with rainfall measurements at several stations, and therefore the spatial scale of this phenomenon is on the order of the size of the region containing the correlated measurement locations. 0022-1694/95/$09.50 © 1995 - Elsevier ScienceB.V. All rights reserved SSDI 0022-1 694(94)02668-8
D,A. Chin / Journal of Hydrology 168 (1995) 1-15
Similarly, a small-scale process is defined as a time series that is significantly correlated with the measurements at only a single location, and therefore the spatial scale of this process is on the order of the size of the region that is closest to the measurement station. The separation of regional-scale processes from small-scale processes is conveniently done using factor analytic techniques, with the subsequent development of time-series models for the extracted factors (processes). The application of factor analysis to modeling multivariate hydrologic time-series is a novel approach, however, such an approach has been previously advocated for multivariate time series analysis in the field of psychology (Anderson, 1963). Previous applications of factor analysis to the study of rainfall have generally been concerned only with the segregation of rainfall into various fundamental factors (e.g. Beaudoin and Rousselle, 1982), while the time-series structure of the extracted factors have not traditionally been explored. The main advantage of the proposed scale model over conventional multivariate rainfall models is that fundamental (linearly independent) regional-scale rainfall processes are easier to relate to specific geophysical phenomena, and therefore provide insight into the forcing mechanisms affecting the rainfall. A second advantage of the proposed scale model over the more conventional multivariate ARMA model is that the scale model provides a more defensible approach to modeling rainfall time series where the temporal and spatial structure of the rainfall are related.
2. Scale model versus A R M A models
The conventional approach to multivariate ARMA modeling is described by Salas et al. (1980), Bras and Rodriguez-Iturbe (1985), and Salas (1993). The conventional methodology for fitting multivariate ARMA models to rainfall measurements requires an assumption of the order of the AR and MA components of the model, followed by parameter estimation to preserve auto and cross-correlation matrices (Bras and Rodriguez-Iturbe, 1985). Parameter estimation techniques tend to become quite complex as the order of the ARMA model increases, and therefore Salas (1993) has proposed using so-called contemporaneous ARMA (CARMA) and periodic ARMA (PARMA) models to describe multivariate rainfall time series. The use of CARMA and PARMA models, in lieu of generalized ARMA models, are justified whenever lagged cross-correlations between rainfall measurements at different stations are not significant. The proposed scale-model is also a contemporaneous model, and must therefore be contrasted with the contemporaneous models described by Salas et al. (1980) and Salas (1993). The conventional approach using CARMA models consists of two main steps. In the first step, ARMA models are fitted to each of the p seasonally adjusted rainfall time series, and then the random residual time series at each of the p stations are estimated. The second step is to compute the zero-lag correlation matrix of the random residuals at each of the p stations, and use standard decomposition techniques to express the random residual at each station as a linear combination ofp independent random time series. The limitation of the CARMA approach is that it a priori assumes that the structured temporal
D.A. Chin / Journal of Hydrology 168 (1995) 1-15
fluctuations at the individual stations are uncorrelated with the structured temporal fluctuations at all other stations. In effect, this means that structured temporal fluctuations at different stations are not tied together, and are therefore modeled as small-scale phenomena. Realistically, this is probably not a sound assumption. Since only the correlation between random residuals are accounted for in contemporaneous models, a second assumption inherent in the C A R M A model is that regional-scale phenomena are all random. Again, this assumption is questionable in most cases and is seldom validated. The proposed scale model addresses the aforementioned limitations of conventional contemporaneous models by first segregating the spatially correlated (regional-scale) rainfall fluctuations from spatially uncorrelated (smallscale) rainfall fluctuations. The temporal structure of both the large-scale and small-scale processes are then investigated separately.
3. The scale model
The scale model of rainfall has the following form Yt = Iz + A x t
-t- ¢;t
(1)
where Yt is the rainfall vector;/z is the mean o f y t ; A is a matrix accounting for the contribution of each regional-scale process to the rainfall vector; xt is the vector time series of regional-scale processes; and Et is a vector time series of small-scale processes. The functional form of the scale model is identical to the factor model (e.g. Morrison, 1990), where A is the factor loading matrix, xt is the vector of common factors, and et is the vector of specific variates. The scale model differs from factor model in that classical univariate time series models are used to describe the 'factor processes, thereby producing a forecasting and simulation tool and, in addition, definite spatial scales are associated with the factor processes. In cases where there are m regionalscale processes (factors) and p rainfall time series, then Yt,/~ and Et, are p-element vectors, xt is an m-element vector, and A is a p x m matrix. If the contemporaneous covariance matrix of the rainfall measurements are given by YT, then it can easily be shown from Eq. (1) (Morrison, 1990) that , ~ = A A T + fir
(2)
where g' is the p-vector of variances of the small-scale processes, Et. The representation given in Eq. (2) clearly partitions the variance in the rainfall record into a portion that is described by the regional-scale processes, xt, and a portion described by smallscale processes, Et. A second relationship resulting from the scale model is that the matrix A is the covariance matrix between the measured rainfall, Yt, and the regionalscale process, xt, therefore A =cov
(ytxtT)
(3)
Since the regional-scale time series, xt, are generally required to have a mean of zero and a variance of one, it follows directly that the correlation between the p measured time series and m regional-scale processes is given by the m × p correlation matrix C,
4
D.A. Chin / Journal of Hydrology 168 (1995) 1-15
where C = A-1/ZA
(4)
and A is a diagonal matrix whose diagonal elements are equal to the variances of measured time series, Yr. This correlation matrix is critical in assessing the spatial scale of any rainfall process. For example, a regional-scale process is correlated with several elements ofyt, while a local-scale process is significantly correlated with only a single element o f y t. A useful property of the scale model is that if the coefficient matrix, A, is multiplied by any orthogonal matrix T, then the partitioning of the variance explained by the regional-scale processes, xt, and the independent variates, et, are unaffected (Morrison, 1990). What changes is the correlation matrix, C, which is transformed into C' where C' = C ( r - l ) T
(5)
The transformation matrix, T, provides a useful mechanism for transforming the regional-scale processes such that the correlation matrix becomes more polarized, thereby making interpretation of processes clearer. Criteria for selecting the transformation matrix are described extensively in factor analysis texts (e.g. Morrison, 1990). The parameters to be estimated in the scale model are A and ~, and there exists a variety of methods that have been proposed for their evaluation. The most common method of estimating A and fit is the maximum-likelihood method which assumes that Yt is a random multinormal variate, in which case the sample covariance matrix, S, is described by a Wishart probability density function. The parameters A and ~ are then estimated by maximizing a log-likelihood function derived from the Wishart distribution. A complicating factor in applying this estimation method to multivariate time series is that such series are not necessarily random and therefore may not be consistent with the randomness assumption. Anderson (1963) addressed the problem of applying factor analysis methods to general multivariate (psychological) time series and concluded that temporal correlations would probably not be a serious limitation in using standard factor-analytic techniques to estimate A and ~. This proposition is intuitive for cases where the length of the time series is much longer than the temporal correlation length scales of the rainfall process.
4. Identification of scale model
Specification of the scale model by Eq. (1) requires that the number of regionalscale processes, m, be determined. Knowing this number of independent processes, the rotated factor loadings, A T, and the specific variance vector, ~, are then computed by conventional factor-analysis techniques such as by the maximum-likelihood method. Steps to identify the model parameters are given below. (1) Specify the number of regional-scale processes, m, to be equal to one, then use the maximum-likelihood estimation procedure to determine the factor-loading matrix, A, and the specific variance matrix, ~.
D.A. Chin / Journal of Hydrology 168 (1995) 1-15
5
IORL~DO
ATLANTIC DCEAN
UKR
FT. PIF..RCE
KRB
UEC
STUART
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_VEST PALM ~trACH
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0
i
10
20
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30
Fig. 1. Hydrologic basins in the South Florida Water Management District.
6
D.A. Chin / Journal of Hydrology 168 (1995) 1-15
(2) Use the quartimax criterion (Morrison, 1990) to rotate the factor loading matrix, A, and thereby maximize the correlations given by the elements of the rotated correlation matrix C . (3) Compute the chi-square statistic associated with the hypothesis that the covariance matrix of the m-factor model is equal to the measured covariance matrix (Morrison, 1990). Requiring the null hypothesis to be supported with 95 % confidence indicates when the assumed order of the factor model is justified. If the null hypothesis is not supported, then the number of factors is increased by one, and the computations are started again from Step (1). If the null hypothesis is supported, then the order of the scale model, m, and the computed factor model parameters A and ~ are accepted. (4) Determine the statistical characteristics of the regional-scale time series, xt, and small-scale time series, et, extracted in the previous step. These time series are analysed for randomness and normality, using conventional statistical analyses. If any of these time series turn out to be non-random, then conventional univariate ARMA time series models are developed for each of these linearly independent time series. The battery of statistical analyses conducted on each of the independent regional-scale and small scale time series are particularly illuminating in associating the identified regional-scale processes with specific large-scale geophysical processes.
5. Model apphcafion Water-resource management in south Florida is governed by the South Florida Water Management District (the 'District'), which manages an area of approximately 44 000 km 2. This region is shown in Fig. 1, and is composed of 12 hydrologic basins. In this study, the validity of using the scale model to describe rainfall in the 12 hydrologic basins is investigated.
5.1. Monthly rainfall model 5.1.1. Local analysis The temporal variation in the average monthly rainfall in each of the 12 hydrologic basins are shown in Fig. 2. Monthly rainfall is clearly a non-stationary process. Analysis of the temporal structure in each basin is preceded by standardization according to the relation
Yij--
W i j - - VV~i
Si
,
i = 1,...,12,
j = 1,...,N
(6)
where Yij is the standardized rainfall in month i and year j, Wij is the transformedmonthly rainfall, I~i is the mean of Wig in the month i, and Si is the standard deviation of W O. in month i. Transformation of the measured rainfall time series (prior to standardization) is generally required due to a relation between the mean and variance of the time series. Square-root transformations are the most common
D . A . Chin / Journal o f H y d r o l o g y 168 (1995) 1 - 1 5 25.0
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for monthly rainfall. The time-series model appropriate for describing the standardized-monthly rainfall may have either constant or seasonal parameters, and both models have been proposed in the past (Salas et al., 1980). The constant parameter model views the standardized rainfall, Yij, as being a stationary time series, while the variable parameter model recognizes the inherent variability in the statistical character of rainfall for different months. Previous analyses (Chin, 1993) have shown that the standardized-rainfall series are non-stationary and that time-series models with monthly varying parameters are appropriate. The correlation between rainfall in consecutive months was investigated and shown to be significant only between the low-rainfall months of December and January. As a reasonable approximation, time-series models for all months were developed independently, thereby neglecting correlations between the rainfall in adjacent months. Statistical analyses of individual-month standardized rainfall show that the fluctuations in each month are almost all random and normally distributed. Exceptions to this rule occur in April and October, where least-squares regression analysis indicates that a significant linear downtrend occurs in many basins. Specifically, UKR, KRB, UEC, COL and CAL basins (Fig. 1) show significant linear downtrends in April while UEC, LEC, EAA, ENP, WC 1 and WC3 basins show significant linear downtrends in October. The timeseries model appropriate for describing the standardized series y/j is therefore of the form y(k) = m~k)Tj + cl k) + al~ ) (7) ij
where k is the basin,
ml k) and c~k) are the slope and intercept of the trend in basin k in
8
D.A. Chin / Journal of Hydrology 168 (1995) 1-15
month i, Tij is the time step corresponding to thejth year, and al(~) is a random normal deviate with a mean of zero and a standard deviation of al k). Clearly the parameters of the time-series model are ml k), c~k), and al k). It should be noted that in most basins ml k) and elk) are equal to zero and the standardized-monthly rainfall varies randomly from year to year. The model described by Eq. (7) may be conveniently written in the vector form Yij -~- mi Tj q- c i q- aij
(8)
This model describes the standardized rainfall in each of the 12 basins. The CARMA model that would be extracted by the procedure described in Salas et al. (1980) is then given by (9)
Yij : m i T j q- cl q- Prhj
where r/i) is a vector of independent random variates, and P is the matrix resulting from the Choleski decomposition of the covariance matrix of the correlated random fluctuations a O. The CARMA model given in Eq. (9) is based on the assumption that structured temporal fluctuations within basins are uncorrelated between basins or, in other words, correlations between the rainfall in various hydrologic basins are accounted for only by random processes. Viewed in the context of physical space, the conventional CARMA model (Eq. 9) considers all structured temporal fluctuations to be small-scale (local), while regional-scale processes are assumed to be random. The validity of these assumptions are investigated in the following sections.
5.1.2. Identification of scale model The scale model of the standardized rainfall is of the form
Yij = AFij + eij
(10)
Table 1 Correlation of m o n t h l y rainfall with regional-scale processes Basin UEC UKR KRB LEC COL EAA CAL ENP LOK WC1 WC2 WC3 Explained variance
First process 0.80 0,71 0.87 0,81 0,79 0,85 0,79 0.76 0.93 0.88 0.78 0.88 67 %
Second process
Third process
0.11 -0.30 -0.29 0.52 -0.17 -0,05 -0.34 0.14 -0.21 0.39 0.51 0.31
-0.25 -0,13 -0.20 -0.02 0.52 0.03 0.15 0.42 -0.09 -0.09 -0.05 0.25
11%
5%
D.A. Chin I Journal of Hydrology 168 (1995) 1-15
where A is a parameter matrix, F u is the vector of regional-scale processes, and e u is a vector of small-scale processes. Following the model identification steps described earlier, the chi-square statistic was used to determine how many regional scale processes are necessary to adequately represent the covariance matrix of the rainfall measurements. Requiring the null hypothesis to be supported within a 95% confidence interval, reveals that a three-factor model is appropriate, and the correlations between the three regional-scale time series and the standardized-rainfall time-series are shown in Table 1. It is clear that the first process is correlated with the rainfall in all basins and reflects a global-scale process, while the second and third regional-scale processes reflect smaller scale processes. The large-scale rainfall processes are identified from the model parameters A and if' via the relation (11)
Fi s = A T ( A A x + ff/-l)Yij
where y,j is the standardized rainfall time series. The statistical character of each of the three linearly independent regional-scale rainfall processes will now be investigated separately. The first regional-scale time series (process) accounts for approximately 67 % of the total variance. The mean for each month is plotted in Fig. 3, and the statistics of the first process for each month are shown in Table 2. F r o m the statistics shown in Table 2 it is clear that in almost all months this (dominant) global-scale process consists of essentially random normally distributed fluctuations, with the exceptions of April, October, and November where there are significant trends. The trend is downward in 1.0
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D.A. Chin / Journal of Hydrology 168 (1995) 1-15
10
Table 2 Statistics o f the first r e g i o n a l - s c a l e p r o c e s s in the m o n t h l y r a i n f a l l Month
Slope a
Lag a (months)
Exceedence probabilities Q-statistic
Jan. Feb. Mar. Apr. a May Jun. Jul. Aug. Sep. Oct. Nov. Dec.
-
2 -
-0.017 -0.022 0.020 -
1 -
0.20 0.22 0.55 0.71 0.74 0.61 0.77 0.63 0.82 0.02 0.24 0.53
x 2 statistic 0.22 0.63 0.10 0.82 0.33 0.11 0.91 0.49 0.16 -0.63 0.33 0.33
a O n l y statistically s i g n i f i c a n t q u a n t i t i e s s h o w n .
April and October, while the trend is upward in November. Viewed collectively, these results indicate a global-scale (District-wide) decrease in April and October rainfall, with a global-scale increase in November rainfall. This is tantamount to a 'shifting of the seasons'. The statistics of the first global-scale process support, the following model
FD) = p!l) + (miTj q_ Cij) _1_aij
(12)
where i is the month index, j is the time index, and the parameters of the model are /~/(1), mi, ci, and the variance of the random normal deviate, aij. The fact that this global-scale process has temporal structure, at least for three of the 12 months, violates the assumption that regional-scale fluctuations in standardized rainfall do not have any temporal structure. In this sense, the scale model appears to be an appropriate model, presenting a fundamentally different picture of the rainfall process than the CARMA-type model identified earlier. Considering the second regional-scale process, which accounts for approximately 11% of the total variance, the monthly variation in the mean is plotted in Fig. 3, and the cyclical (semi-annual) nature of the variation in the monthly mean is clearly evident. The mean-removed (process) is random and normally distributed, and therefore the second regional-scale process may be described by the following model
F[2)
=/~t
!2) q- aij
(13)
where p//2) are the seasonal means. The third regional-scale process accounts for approximately 5% of the total variance, and the variation in the monthly mean of the third factor is shown in Fig. 3, where the annual periodicity in the mean is clearly evident. As in the case of the second regional-scale process, the mean-removed residuals are random and normally distributed. The third factor is therefore
D.A. Chin / Journal of Hydrology 168 (1995) 1-15
Table 3 Variance explained by
the three-scale model of m o n t h l y rainfall
Basin
Explained variance (%)
UEC UKR KRB LEC COL EAA CAL ENP LOK WC 1 WC2 WC3
72 62 88 93 91 72 77 78 91 93 88 93
Average
83
described by the following model F~3) = ~!3) + aij
(14)
where ~{3) are seasonal means. To assess the overall ability of the three-scale model to describe the rainfall in each of the 12 hydrological basins, the explained variance in each of the hydrologic basins were computed and these results are shown in Table 3. Clearly the model performs best in the KRB, LEC, COL, WC1, WC2 and WC3 basins where the three scalemodel explains more than 88% of the total variance. On the other hand, the three scale model performs worst in the UKR basin, where only 62% of total variance is explained. The model performs adequately in the remaining basins where 72-78% of the variance is explained. In summary, it has been shown that fluctuations in standardized monthly rainfall are driven primarily by regional-scale phenomena, which account for an average of 83% of the variance in the 12 hydrologic basins. These regional-scale fluctuations are explicit components of the scale model, and are effectively extracted using factor analytic techniques. In a statistical sense, it has been shown that three regionalscale processes are adequate to account for the important covariances in the measured monthly rainfall. The primary regional-scale process has been shown to have structured temporal fluctuations, indicating that the scale model is more appropriate than the conventional CARMA model proposed by Salas et al. (1980). 5.2. Annual rainfall model 5.2.1. Local analysis The average annual rainfall in the District for each of the 12 hydrologic basins are
D.A. Chin / Journal of Hydrology 168 (1995) 1-15
12
Table 4 A v e r a g e a n n u a l rainfall in h y d r o l o g i c basins Basin
R a i n f a l l (cm)
CAL COL EAA ENP KRB LEC LOK UEC UKR WC1 WC2 WC3
133 134 138 131 119 147 120 131 130 150 151 144
given in Table 4. There is obviously significant variability between basins. The statistics of the annual rainfall time-series at each of the 12 basins are shown in the Table 5, and from these results, it appears that in 8 of the 12 basins the (Box-Cox) transformed annual rainfall is random and normally distributed, while in EAA and LOK there is a significant downward trend, and in LEC and UKR there appears to be some temporal structure that can be described by AR(3) and AR(2) models. The statistical results shown in Table 5 further indicate that the log-transform appears to be most appropriate for transforming the annual rainfall into Gaussian distribution.
Table 5 Statistics o f a n n u a l rainfall series (1941-1990) Basin
UEC UKR KRB LEC COL EAA CAL ENP LOK WCI WC2 WC3
Box-Cox transform
Z - 1.0 Z -1'° Log Z Log Z Z -°'5 Log Z Log Z Log Z Log Z Log Z Log Z Z -°'5
Slope a
_ -
-0.0048 -
-0.0035 -
" O n l y statistically significant q u a n t i t i e s shown.
Lag a (years)
2 1, 3 10 1 1 -
Exceedence p r o b a b i l i t i e s Q-statistic
X2-statistic
0.42 0.01 0.50 0.01 0.43 0.27 0.77 0.06 0.89 0.10 0.15 0.33
0.78 0.68 0.91 0.87 0.18 0.91 0.63 0.49 0.13 0.54 0.49 0.34
D.A. Chin / Journal of Hydrology 168 (1995) 1-15
13
Table 6 Correlation of annual rainfall with regional-scale processes Basin
First process
UEC UKR KRB LEC COL EAA CAL ENP LOK WC1 WC2 WC3
0.87 0.75 0.88 0.86 0.88 0.88 0.81 0.79 0.91 0.92 0.86 0.94
Explained variance
74%
Second process 0.11 0.10 0.30 -0.46 -0.01 0.13 0.20 -0.15 0.42 -0.21 -0.33 -0.26 7%
5.2.2. Identification o f scale model F a c t o r analysis i n d i c a t e s t h a t two regional-scale processes are sufficient to describe the a n n u a l rainfall. T h e c o r r e l a t i o n s b e t w e e n the r e g i o n a l - s c a l e processes a n d the a n n u a l rainfall time-series a r e s h o w n in T a b l e 6. T h e first r e g i o n a l - s c a l e process is highly c o r r e l a t e d with the rainfall in all basins, reflecting a global-scale process t h a t a c c o u n t s for a p p r o x i m a t e l y 74% o f the t o t a l variance. T h e s e c o n d regional-scale process, w h i c h a c c o u n t s for 7 % o f the v a r i a n c e is c o r r e l a t e d with the rainfall at a fewer n u m b e r o f stations. Statistical a n a l y s e s indicate t h a t b o t h factors are r a n d o m a n d n o r m a l l y d i s t r i b u t e d , w i t h a m e a n o f zero a n d a v a r i a n c e o f unity. Therefore, these processes are d e s c r i b e d b y the f o l l o w i n g simple m o d e l s
Table 7 Variance explained by the two-scale model of monthly rainfall Basin
Explained variance
UEC UKR KRB LEC COL EAA CAL ENP LOK WC1 WC2 WC3
76% 55% 83% 95% 82% 56% 67% 52% 99% 85% 83 % 95%
Average
77%
D.A. Chin / Journal of Hydrology 168 (1995) 1-15
14
F) 2) = a !z) .1
(16)
where ~ are the annual factors, and ai are standard normal deviates. It is noteworthy that the form of the annual-rainfall model is identical to the monthly-rainfall model, except that the seasonal-mean term is missing. The performance of the scale-model in describing the annual rainfall fluctuations and the 12 hydrologic basins may be measured by the percentage of variance explained in each of the basins by the proposed model. These percentages are shown in Table 7, which indicates that the two regional-scale processes explain an average of 77% of the variance.
6. Summary and conclusions The effectiveness of using a multivariate scale model to describe rainfall in south Florida has been investigated. The scale model has the general form Yij = A F i j + eij
(17)
where yij is measured (or standardized) rainfall in season i and year j, A is a parameter matrix, Fij is a vector of regional-scale processes that show significant correlation with the rainfall at multiple locations, and eij is a vector describing localized fluctuations. The results presented in this paper show that the scale model provides an effective description of the monthly and annual rainfall in South Florida. In the case of monthly rainfall, the scale model indicates that deviations from monthly means are caused primarily by regional-scale phenomena, which have temporal structure. This result is contrary to the basic assumptions of the contemporaneous multivariate approach suggested by Salas et al. (1980), which assumes that nonrandom temporal fluctuations are not spatially correlated. The results of the present study therefore show that the conventional (CARMA) modeling approach should not be applied. The scale-modeling approach explored in this study presents an attractive alternative. In the case of annual rainfall, it has been shown that the majority of variance is associated with regional-scale phenomena that are random and normally distributed. The contemporaneous multivariate model would also be valid in this case.
Acknowledgments The investigation reported in this paper was supported by the United States Department of the Interior under Grant No. 14-08-001-G2012, and also by the South Florida Water Management District under Grant No. C91-2237. Lin Li of the University of Miami assisted in conducting many of the statistical analyses, Dr. David Maidment of the University of Texas provided many useful comments.
D.A. Chin / Journal of Hydrology 168 (1995) 1-15
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