Ck T7 Tytk yy y m i

where N is the sample size and k is the time lag. The plot of rk versus k, i.e., the correlogram, may give an idea of the degree of persistence of the underlying time series, and it may be useful for choosing the type of stochastic model that may represent the series. When the correlogram decays rapidly to zero after a few lags, it may be an indication of small persistence or short memory in the series, while a slow decay of the correlogram is an indication of large persistence or long memory. The lag-one serial correlation coefficient r] is a simple measure of the degree of time dependence of a series. Generally, r] for annual flows is small but positive, although negative r]'s may occur because of sample variability. Large values of rx for annual flows can be found for a number of reasons including the effect of natural or man-made surface storage such as lakes, reservoirs, or glaciers, the effect of slow groundwater storage response, and the effect of nonstationarity. The estimators s2, g, and rk are biased (downward relative to the corresponding population statistics). Corrections for bias for these estimators have been suggested (Bobee and Robitaille, 1975; Yevjevich, 1972a; Fernandez and Salas, 1990).

In addition, the sample spectrum is another way of studying the variability of hydroclimatic series in the frequency domain (Yevjevich, 1972b). The sample spectrum g(fj) may be determined as gifj) = 2

1+2E Dkrk cosdnfjk)

where Dk is a smoothing function and m is the maximum number of lags considered. Figure 4 illustrates the autocorrelation function and the spectrum obtained for the time series of annual PDO indices for the period 1900-1999. The time series shows evidence of low-frequency components, which are manifested in a slow decaying and pseudoperiodic correlogram and a spectrum with visible high values at frequencies near 0.02 and 0.18 cycles per year.

When analyzing several time series jointly, cross-correlations may be important. The cross-correlation coefficient between series y® —A "<7) i and j, is determined as and _y, , t — 1,..., N for stations

The plot of r'Jk vs. k is the cross-correlogram. For n time series, the values of r\, i = ... ,n and j = 1,... ,n are elements of the lag-k cross-correlation n x n matrix Mk. Figure 5 is a graphical display of the lag-zero cross-correlation matrix obtained for the annual streamflows of 29 stations in the Colorado River system. For reference, station 1 is one of the farthest upstream site while station 29 is the farthest downstream site. The cross-correlation between stations 1 and 29 is large (of the order of 0.9) while the cross-correlation between stations 1 and 27 is small (the

Figure 5 Lag-zero cross-correlation matrix obtained for the annual streamflows of 29 stations in the Colorado River system. For reference station 1 is the furthest upstream site while station 29 is the furthest downstream site.

1979). One interpretation of the Hurst phenomenon has been to associate h — j with short memory models possessing short-term dependence structure, and h > \ with long memory models possessing long-term dependence. A number of models, including the autoregressive moving average (ARMA) processes, can have long-term dependence structure, yet asymptotically they give h = i. Furthermore, a stationary model with long-term dependence and h > \ is the fractional ARMA (FARMA) model (refer to Section 4 for definitions of ARMA and FARMA processes). Estimates of h can be useful for comparing the performance of alternative modeling strategies and estimation procedures. Statistical tests to determine whether a given time series exhibits the Hurst effect are also available (Mesa and Poveda, 1993).

Furthermore, drought-related stochastic properties are also important in modeling some hydroclimatic time series such as precipitation and streamflow. Consider a hydrologic time series yt, t — 1 N, and a demand level d (crossing level).

Assume that y, is an annual series and d is a constant (e.g., d = ay and 0 < a < 1). A deficit at any given time t occurs when y, < d. A consecutive sequence of deficits (until y, > d again) may be called a drought, and such a drought can be characterized by its duration L, its magnitude M, and its intensity / = M/L (Yevjevich, 1967). Because a number of droughts can occur in a given hydrologic sample, the maximum drought duration, magnitude, and intensity (in a given sample) have been indicators of the so-called critical drought and have been widely used in water resources studies.

Periodic (Seasonal) Statistical Properties

While overall stochastic properties of hydroclimatic time series, such as those previously defined above, may be determined from either annual series or for seasonal series as a whole, specific seasonal (periodic) properties may provide a better picture of the stochastic characteristics of certain hydroclimatic time series that are defined at time intervals smaller than a year such as monthly streamflow data. Let the seasonal time series be represented by yv v = 1,..., /V; t = 1, ..., to in which v is the year, r is the season, N is the number of years of record, and co is the number of seasons per year (e.g., to = 12 for monthly data). Then, for each season r one can determine a number of statistics such as the seasonal mean y%, variance si, coefficient of variation cvT, and skewness coefficient gz. Furthermore, the season-to-season correlation coefficient rk z may be estimated by

For instance, for monthly streamflows r14 represents the correlation between the flows of the fourth month with those of the third month. Likewise, for multiple seasonal time series, the lag-/: seasonal cross-correlation coefficient r\ T between the seasonal time series vi'l and y\'\. k for sites i and j, can be determined.

The statistics yt, sz, gz, and rk T may be plotted versus time r = 1,..., to to observe whether they exhibit a seasonal pattern. Fitting these statistics by Fourier series is especially effective with weekly and daily data (Salas et al., 1980). Generally, for seasonal streamflow series yt > sT although for some streams yx may be smaller than s% especially during the "low-flow" season. Furthermore, for intermittent streamflow series generally the mean is smaller than the standard deviation, i.e., yx < sT throughout the year. Likewise, values of the skewness coefficient gT for the dry season are generally larger than those for the wet season indicating that data in the dry season depart more from normality than data in the wet season. Values of the skewness for intermittent hydrologie series are usually larger than skewness for similar nonintermittent series. Seasonal correlations rk z for streamflow during the dry season are generally larger than those for the wet season, and they are significantly different than zero for most of the months. Figure 6 displays r, T, i.e., the lag-1 month-to-month correlations, for the monthly streamflows of the referred 29 stations of the Colorado River system. It may be observed that the correlations vary with the month, and with a few exceptions the correlation pattern for the entire system is similar. On the other hand, seasonal correlations for monthly precipitation are generally low or not significantly different from zero for most of the months (Roesner and

Figure 6 The lag-1 month-to-month correlations, i.e., rl T for the monthly streamflows of 29 stations of the Colorado River system.

Yevjevich, 1966), while for weekly, daily, and hourly precipitation they are generally significant and greater than zero.

Complex, long-term dependence (long memory) of seasonal flows may be evident when the correlations rk/I are significant and decay slowly as k increases beyond to seasons (beyond a year). These correlations are usually small or not significant for many streams, but in river systems such as the Nile River such seasonal correlations may persist for several years. Rivers that exhibit long-term correlation in seasonal flows will exhibit also long-term autocorrelation in the annual flows. In addition, some streamflow hydrographs such as daily and weekly hydrographs may possess directionality (nonreversibility), which means that some of their statistical properties change when direction of time is reversed. This is evident from the typical form of hydrographs in which the rising limb is shorter than the recession limb. In these cases, it is desirable that the mathematical models have such directionality attribute (Fernandez and Salas, 1986).

4 STOCHASTIC MODELS AND MODELING TECHNIQUES

A number of stochastic models and modeling schemes have been developed for simulation and forecasting of hydroclimatic processes. Some of the models are conceptually (physically) based, some others are empirical or transformed or adapted from existing models developed in other fields, while some others have arisen specifically to address some particular features of the process under consideration.

In general models for continuous time processes and models for short time scales such as hourly are more complex than models for larger time scales. Also some of the models have been developed specifically for precipitation while some others are for streamflow. Yet many of them are useful for both and for many other hydrocli-matic processes. We will illustrate here as a matter of introduction and subsequent reference, the family of autoregressive and moving average (ARMA) models and extensions and modifications thereof. These models have become quite popular for both simulation and forecasting of many hydroclimatic processes. However, many other stochastic models have been developed, some of them quite different than ARMA models, aimed at the specific process under consideration or the particular features (of the underlying process) one tries to address. For example, for intermittent processes such as daily rainfall, Markov chains and the discrete counterpart of ARMA models, i.e., discrete ARMA (DARMA), are available (e.g., Chang et al., 1984; Guttorp, 1995). Likewise, models with infinite memory such as the Fractional Gaussian noise (e.g., Mandelbrot and Van Ness, 1968) and shifting level models that are capable of simulating sudden shifts (e.g., Salas and Boes, 1980) are available.

Stochastic Models

Stationary Models. The family of ARMA models has been widely used for modeling hydroclimatic processes at various time scales. The ARMA(p, q) model is defined as (Brockwell and Davis, 1991)

p i yt = j" + E fyiyt-j - n) + £t - E 9j£H (5a)

where (i, the the 0's, and <r2(i:) are parameters of the model, p is the order of the autoregressive terms, q is the order of the moving average terms, B'z, = zt_h and

Particular models derived from (5) are the ARMA(p, 0) or AR( p) and the

ARMA(0, q) or MA(q) models. In addition, the fractional autoregressive moving average FARMA(p, d, q) model is defined as (Hosking, 1981; Montanari et al., 1997)

(j)(B)(\ -B)d(yt-fi) = d(B)et - 0.5 < d < 0.5 (7)

This model is capable of representing long-term dependence. The foregoing ARMA and FARMA models are stationary, hence their applications to modeling hydroclimatic time series require that the underlying data be stationary or be converted to stationary by some appropriate transformation.

These models have been generally applied to annual hydroclimatic data. Sometimes they have been applied to seasonal data after seasonal standardization. Likewise, they have been applied to daily data either after seasonal standardization or by separating the year into several seasons and applying different models to the daily series in each season. For example, Parlange et al. (1992) applied physically based concepts to the daily variations of soil moisture and found that it can be described by an AR(1) process. Properties of AR and ARMA models, such as the autocorrelation function, variance, and spectrum, and hydrologic applications may be found in Salas et al. (1980), Loucks et al. (1981), Bras and Rodriguez-Iturbe (1985), Salas (1993), and Hipel and McLeod (1994). Also Chu and Katz (1985) fitted AR and ARMA models to seasonal and monthly Southern Oscillation Index (SOI). Before fitting the models, the annual cycle was removed from the data in order to make them stationary. Xu and Storch (1990) used Principal Oscillation Pattern (POP) analysis to model monthly SOI data. They concluded that their POP scheme was superior over the ARMA scheme. Furthermore, Chu et al. (1995) applied a bivariate AR model for modeling jointly the seasonal SOI and a precipitation index in Florida. The fitted bivariate AR model was then used to forecast precipitation.

Periodic Models. A number of periodic and other nonstationary models such as the family of PARMA, ARIMA, and multiplicative PARMA models has been suggested in the literature for modeling seasonal hydroclimatic processes such as seasonal precipitation and streamflow series (Salas et al., 1980; Loucks et al., 1981; Salas, 1993; Hipel and McLeod, 1994). In particular, the PARMA(p,#) model is defined as and B'zv_T = zv T_,. When q = 0, the foregoing model becomes the well-known PARMA(/j, 0) or PAR( //). More specifically the PAR(l) model (also known as the Thomas-Fiering model) is likely one of the most widely used models in hydrology. In general low-order PARMA models have become popular for modeling seasonal hydroclimatic processes. Physically based or conceptual arguments of the underlying hydrologic cycle of a watershed or river basin justify the applicability of these models. For instance, Salas and Obeysekera (1992) showed that assuming that the precipitation input is an uncorrected periodic-stochastic process and under some p 1

where

<t>x{B) = 1 - cj>uBl - 4>2 xB2-----4>PiXBP

linear reservoir considerations for the groundwater storage, the stochastic model for seasonal streamflow becomes a PARMA(1,1) process. Chu et al. (1995) analyzed time series of seasonal and monthly SOI and fitted AR and ARMA models after the annual cycle was removed from the data. They also used ARMA models with seasonally varying coefficients.

In addition, ARIMA(jt?, d, q), multiplicative ARMA, and multiplicative ARIMA models have been applied for forecasting hydroclimatic processes (e.g., Salas et al., 1980; Hipel and McLeod, 1994), sampling groundwater levels (Ahn and Salas, 1997), and for detection and estimation of trends in climatological time series (e.g., Visser and Molenaar, 1995; Zheng and Basher, 1999). Furthermore, simulation of complex processes such as the Nile River monthly flows has been accomplished with multiplicative PARMA models (Salas et al., 1995). Also Lund et al. (1995) provide a general overview of the analysis and modeling of climatological time series having periodic correlation structure. They suggest a test for detection of periodic correlation and applied PARMA models for modeling such series. While the referred stationary and nonstationary models [e.g., models (5) and (8), respectively] are written for single-site or univariate series, their multisite or multivariate counterparts are also available (e.g. Salas, 1993; Hipel and McLeod, 1994).

Stochastic Models for Forecasting

A number of stochastic models have been widely applied for forecasting hydroclimatic processes such as precipitation and streamflow. Many of such models fall in the family of transfer function models. The general transfer function noise (GTFN) model may be written as y(B)(y,

where }'(B), oj(B), <5(B), 0(B), and 0(B) are polynomials in B of different orders [similar to those defined in Eq. (6)], xt is the exogenous variable such as precipitation or ENSO index, the yu's represent the means, r is the time delay, and e is the noise term. Some special cases (models) such as the ARMA, ARMAX, unit hydrograph type, multiple linear regression, and the Box-Jenkins transfer function noise models can be derived or simplified from (10). Equation (10) assumes single-site variables, but they are applicable to multisite variables if the variables are vectors and the parameters are matrices. Applications of many of these models can be found in Hipel and McLeod (1994). In addition, forecasting equations based on ARMA, ARMAX, and GTFN models can be written in sequential and recursive forms (e.g., using a Kalman filter). Furthermore, artificial neural networks (ANN) have emerged in the last decade as a useful technique for many modeling applications including forecasting (e.g., Hsu et al., 1995; Govindaraju and Rao, 2000). The application of many of these models, estimation procedures, and ANN algorithms for forecasting precipitation and streamflow are described in some detail in Valdes et al. (2001).

Modeling Schemes

Some specific models have been developed in the hydrologie field to address some unique features related to hydrologie and water resources problems. An example is the so-called disaggregation models (e.g., Valencia and Schaake, 1973). The failure of some traditional models such as the PAR(l) model to reproduce annual statistics (or upscale statistics) led to the development of disaggregation techniques. While the main intent of such disaggregation models has been to enable one to generate hydrologie sequences that can reproduce statistics at the annual and seasonal time scales, it has brought a major dimension into the capability of modeling complex hydrologie processes and complex hydrologie systems. Complex systems involve several sites, and the temporal and spatial mass balance requirements, often require the use of modeling schemes that may consist of an array of single-site, multisite, and temporal and spatial disaggregation models. While this requirement has been more evident in models constructed for simulation, the same is true for forecasting complex hydrologie systems. Furthermore, in order to facilitate the practical application of stochastic models for simulation of hydrological processes, software packages such as SPIGOT (Grygier and Stedinger, 1990) and SAMS (Salas et al., 2000) have been developed. Still, actual applications of such packages in real-world systems, especially for simulating complex hydrologie processes and complex water resources systems such as the Great Lakes system in North America or the Nile River system in Africa, may not be a straightforward application. Thus adjustments, modifications, additions, etc., may have to be made before a satisfactory or acceptable solution to the problem is attained.

Stochastic Modeling

Stochastic modeling of hydroclimatic processes may involve four major steps: model identification, parameter estimation, model testing, and model verification. By model identification is meant determining a specific model structure and the model order; for example, determining that the model for annual streamfiow series is an ARMA( 1,1) or determining that the model for daily rainfall is a simple Markov chain. Generally models that belong to the family of ARMA, ARIMA, and transfer function models are amenable for certain identification procedures based on autocorrelation, partial correlation, and cross-correlation analysis (Brockwell and Davis, 1991; Hipel and McLeod, 1994). However, model identification techniques are not available for some models or they are too complex, so instead a model of a certain type and order is applied to the particular hydroclimatic series at hand and its performance is judged by testing and verification. Some hydroclimatic processes such as streamfiow and soil moisture have been identified using physically based concepts and arguments (e.g. Salas and Smith, 1981 ; Parlange et al., 1992; Salas and Obeysekera, 1992).

Once a model is identified, its parameters may be estimated by a number of techniques such as the method of moments, least squares, and maximum likelihood, depending on the particular model and data at hand. Typical method of moments estimation procedures involve matching historical and population (model) first- and second-order statistics, although in some cases some other properties such as skew-ness and storage and drought related statistics have been used (e.g., Salas et al., 1980; Hipel and McLeod, 1994). In addition, recursive parameter estimation methods and filtering techniques have been used particularly for forecasting problems (e.g., Bras and Rodriguez-Iturbe, 1985). Furthermore, modeling of hydroclimatic time series either for simulation or forecasting generally requires that the underlying series be transformed to approximately normally distributed series (e.g., Salas, 1993; Hipel and McLeod, 1994). Thus parameter estimation is usually made in the transformed domain. Model testing procedures have been well developed for models within the ARMA, ARIMA, ARMAX, and transfer function type of models (e.g., Brockwell and Davis, 1991). Likewise, testing procedures are available for PARMA models (e.g., Salas et al., 1980; Salas, 1993; Hipel and McLeod, 1994). The tests usually involve diagnostic checks to verify whether the model residuals comply with the underlying assumptions of independence and normality (of the residuals). Since many models may comply with such requirements, a model selection criteria based on the Akaike Information Criteria (AIC) is available to discriminate and find a parsimonious model (Brockwell and Davis, 1991). On the other hand, model testing for some other models cannot be done based on analysis of residuals; so instead model testing is based on data generation experiments. In addition, model verification is usually needed beyond testing residuals depending on whether the modeling exercise is geared to simulation or forecasting. For instance, for simulation (data generation) one may like to test whether the model is capable of generating sequences that reproduce a number of storage and drought related historical characteristics. This is usually accomplished by Monte Carlo experiments. On the other hand, model verification for forecasting may involve examining whether the model is capable of estimating the hydrologic process under consideration one or more lead times in advance within a specified error criteria. This may be done by split sampling estimation and testing.

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