- Email: [email protected]
Time series modeling of channel transmission losses
Agricultural water management Agricultural Water Management 29 (1996) 283-298
Time series modeling of channel transmission losses Tahir Hameed a, M.A. Marifio b,*, M.N. Cheema ’ ”Department r,fCivil and Environmental Engineering, University ofCalifornia, Davis, CA 95616, USA ’Department ofLand, Air, and Water Resources and Department ofCivil and Environmental Universi~ ofCalifornia, Davis, CA 95616, USA ’Punjab Irrigation and Power Department, Lahore, Pakistan
Engineering,
Accepted 18 April 1995
Abstract The transmission losses through a canal network depend on a number of physical, operational, and climatic factors. Some of these factors can be quantified but most of them are not easy to be incorporated into precise calculations. Due to climatic and other uncertainties, prediction of the reliable amount of losses in irrigation projects cannot be done confidently. Because of these uncertainties, the water loss series observed over a period of time can be treated as a stochastic process that can be modeled by time series techniques. In the present study, the transmission losses series observed at the Imperial Irrigation District, California, USA, is modeled as an autoregressive-integrated-moving average ( ARIMA) process. The ARIMA models are found to be plausible and appropriate models for such series. These are simple to develop and provide dependable forecasts. Reliable forecasts of canal transmission losses are useful in the efficient management of irrigation projects.
1. Introduction The correct estimation of transmission water losses from an irrigation network is vital for the proper management of the system. Water managers are forced to divert extra water in order to compensate for the losses and to ensure that the users get the water according to their requirements. Thus, for the effective operational planning and management of an irrigation system, a dependable forecasting of the losses is very important. Seepage and evaporation are the two dominant processes by which water is lost in the canals. Under arid conditions, water loss can be substantial; sometimes, even the total amount of water in a canal can be lost through highly permeable soils (Hathoot, 1984). Seepage rate from a channel depends on a number of factors such as type of construction * Corresponding author: Department of Land, Air, and Water Resources, University of California, Davis, CA 9.56 16, USA. 0378.3774/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDlO378-3774(95)01201-X
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material, channel geometry, depth of ground water, total time of canal operation, flow characteristics, sediment amount, and so forth (Kraatz, 197 1) . It is very difficult to incorporate the effects of all of these factors in any calculation. For example, the flow characteristics, which have a major impact on seepage rate, can be highly non-uniform in an irrigation system. Trout (1979) studied earthen channels of the Indus Basin (Pakistan) and found that the loss rate was very sensitive to the fluctuations in flow depths. It was also observed that losses varied widely and often were highly unpredictable. The flow non-uniformity can be caused by many factors, including a variable water demand and continuous control changes, i.e. water movement in and out of the channel network (Mitchell et al., 1990). Palmer et al. ( 1989) have observed that for high-performance irrigation projects, flows are often non-uniform. They have identified flexible deliveries as the major reason for such wavering flows. This how variability cannot be avoided as the flexibility in rate, duration, and timing is necessary for efficient water use by farmers (Replogle and Merriam, 1980). The demand of water is also very sensitive to the climatic conditions. Thus, varying flow characteristics coupled with climate variables complicate the seepage process and the precise prediction of water losses under such conditions becomes very difficult. Sediment rate can also be an important factor in seepage calculations. It has been estimated that the seepage rate from the Interstate Canal, Nebraska, USA, was reduced by 20% by the introduction of a small amount of 1000 p.p.m. sediment (Kraatz, 1991). The prediction of sediment rate in channels is itself a major task and its incorporation and subsequent water loss forecasting are not straightforward. The methods based on hydraulic conductivity and ponding as well as inflow-outflow methods have been found to be reasonable techniques for seepage calculations (Hotes et al., 1985). However, even these reasonable methods can give widely varying seepage rates. The difference of estimation by these techniques may vary from 10% to 50%, which shows the complex nature of the seepage process. Lane ( 1985) has developed an analytical solution to the governing equations of the transmission losses in ephemeral stream channels. He found that the complicated dynamics of the seepage process is very hard to be represented by equations. It is clear from the preceding reviews that the complex nature of the seepage process makes it very hard to be represented by simple relationships. From an operational management point of view, a complicated modeling of the seepage process is not required. Instead, simple models yielding better forecasts of channel transmission losses are needed for the optimal use of water resources. The other major component of transmission loss through an irrigation system happens via the evaporation process. For the case of lined channels, the evaporation component can be substantial in the total percentage loss. The evaporation process is very sensitive to climate conditions. Air temperature, sunshine hours, humidity, cloud cover, solar radiation and wind speed are some of the important factors that affect evaporation (Hameed et al., 1995). The dependence of evaporation on climate variables induces uncertainties in the evaporation process, making it probabilistic in nature. Thus, evaporation should be represented in an appropriate probabilistic environment. It is very difficult to ascertain the fraction of total losses attributable to evaporation or seepage, Thus, from an operational point of view, evaporation and seepage should be treated under one single heading, i.e. as the transmission losses in an irrigation system. AS noted
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earlier, the complexity and uncertainties in the losses makes the process probabilistic in nature. The theory of time series can provide useful models for such a stochastic process. Because most of the irrigation systems have already been developed, the challenge for engineers is to run these systems efficiently. With reasonably accurate measuring devices, the inflow-outflow approach can be used for the reliable collection of data for estimating the transmission losses in an irrigation system. Such records are routinely maintained in irrigation systems. Daily, weekly, or monthly records of reasonable lengths can easily be formed as time series. Time series methods can incorporate most of the uncertainties associated with transmission losses and the developed models can be utilized for reliable forecasts. The objective of the present study is to explore the possibility of the autoregressiveintegrated-moving average (ARIMA) process as a viable model option for transmission losses series. The model adequacy will then allow to proceed to the next step, where forecasts from such a model can be checked for their reliability.
2. ARIMA
theory
Data observed over time usually have strong correlations among neighboring observations. These autocorrelations in a series prohibit the use of standard statistical methods and in such cases the time series theory provide a solution for the appropriate modeling. Box and Jenkins ( 1976) popularized the ARIMA models. ARIMA models have proven to be very useful in the analysis of a number of time series (Salas et al., 1980; Pankratz, 1988; Shumway, 1988). The check for stationarity of the time series is the first step in the development of an ARIMA model. The weak stationarity of a time series requires that the series has a constant mean, variance, and autocovariance function whereas for the case of a strong form of stationarity, the entire probability distribution function for the process has to be independent of time (Shumway, 1988). Such a form is not practical, and a weak form of stationarity, in which only the mean, variance and autocovariance are independent of time, provides a sufficient theoretical framework for a reasonably accurate analysis. A non-stationary time series can either have a non-constant mean or a non-constant variance, or both. The term “non-constant!’ means that these statistics are dependent on time. A trend in a series produces a non-constant mean and such a series can be detrended (and subsequently made stationary) simply by differencing. The differencing can be continued until the series produces a constant mean and variance; however, for most practical cases a maximum difference order of 2 is sufficient. For the case of seasonal data, the level of series shifts regularly up and down from the mean and these seasonal trends can be removed by seasonal differencing. The seasonal differencing depends on the nature of the season over which the data are collected. For example, a difference order of 12 is needed for monthly data; 4 is for quarterly data; and so on. The relationship between observations at this lag (e.g. 12 for monthly data) can also be included in the final solution. Such seasonal relationships coupled with non-seasonal relationships provide a better explanation of the underlying process of a seasonal time series.
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The non-constant variance can be stabilized by a suitable transformation. The Box-Cox transformation (Pankratz, 1988) is a family of several transformations that is sufficient for most of the cases. The log and square root transformations are members of this family. A new series 5 is defined by the following relationship: 2,’
=I
zA- 1
(1)
h
in which 1 is a real number; and 5 is a non-stationary series. The A&MA model has autoregressive (AR) and moving average (MA) parts. The middle letter, “P’in ARIMA, means integrated and represents the order of differencing for a non-stationary series. If a difference order is zero, then such a series is already stationary and the resulting model is simply an ARMA model. The AR part got the name autoregressive because the dependent variables has a regression-type of relationship with its own previous values Z, , , Z, ?, . , taken as independent variables, as follows: Z,=4,Zz
I -~2Zr-*-...-~,~,-,,far
(2)
in which -4,) & ,..., 9, = the first, second,..., and p-order AR coefficients, respectively; a, is random shock error term or white noise, which is assumed to be normally and independently distributed with a zero mean and a constant variance. The AR process has to satisfy certain stationarity conditions. For example, for the first-order AR process [ AR( 1) 1, stationarity requires that the root of the process should lie outside of the unit circle or /&I < I. For AR( 2)) the following three conditions must be fulfilled: l&II<1 &2+&l<
1
&-&I<1 In the moving average (MA) process, the variable of interest is explained through a random shock error term. The moving average model of order q [MA(q) ] can be described as: z,=
-&a,_,
-&7-z-...-
+&/+a,
(3)
in which 5, H,,..., & is first, second, . . . . and q-order coefficients of the MA process, respectively. Its worthwhile to note that a,-k is not exactly the past value of Z, but the random shock part of Z,_,, which means that the MA process contains a part of the past value of Z, as well. Therefore, the MA process has an equivalent AR form and it can be proved that the MA process is equivalent to an infinite AR sequence (Cryer, 1986). Because of this characteristic, the stationarity conditions for an MA sequence are known as the invertibility conditions. For a MA( 1) model, invertibility requires that I 8, I < 1. The need for such a condition can be seen by a simple examination of the fact that a value of & more that 1 will assume that the recent observations have less influence on Z, than the further past ones. For a MA( 2) model, the following invertibility conditions are required:
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The autocorrelation (ACF) and partial autocorrelation functions (PACF) show the measures of the strength of the relationship between observations at different lags. The ACF and PACF of an observed series help in the identification of an ARIMA model. The comparison of sample ACF and PACF with the commonly occurring theoretical ACFs and PACFs leads to different possible choices. A variety of theoretical ACFs and PACFs are given in Box and Jenkins ( 1976) and Pankratz ( 1988). An ARIMA model is generally presented as ARIMA (p, d, q) in which p and q respectively represent the AR and the MA parts and d shows the difference order. The general ARIMA (p, d, q) can be defined as: 4(B) (1 -B)“Z,=
c+
B(B)&
(4)
in which C is a constant; B is the difference operator defined as B’Z, = Z,_ i; d is the difference order; &!?) = 1 - +,B -&B* - . . . - #Y’ the p-order AR operator; and B(B) = l0, B - 8,B2 - . . - 0&P = the q-order MA operator. The stationarity condition for ARIMA models requires that the roots of the characteristic equation of 4(B) = 0 lie outside of the unit circle. Similarly for invertibility, the roots of the characteristic equation e(B) = 0 must lie outside the unit circle (Pankratz, 1988). Parsimony is another important consideration in the development of ARIMA models. The higher order AR or MA models can be represented by ARMA models of fewer parameters. Box and Jenkins ( 1976) defined a good model as one having the fewest possible parameters. In the case of seasonal data, simple ARIMA models may be inappropriate and inadequate. Seasonal time series have relationships at lag s. The lag s depends upon the nature of data. For example, in a monthly observed series, the relationship at a lag of 12 is important and may not be neglected for the development of an adequate model. Such relationships can easily be represented by multiplicative seasonal ARIMA models which simultaneously include both seasonal and non-seasonal parts. The standard notation for such models is ARIMA(p, d, q) (P, D, Q)s and can be represented as: ~(B)~(EY)(l-B)d(l-B)~Z,=C+B(B)O(B”);a,
(5)
in which $( B’) = 1 - @?’ - +*,,B*’- . . - c#+~B~’ = the P-order seasonal AR operator and 0( B”) = 1 - f),B’- f&Bzs - . . . - 13,,$@” = the Q-order seasonal MA operator. The stationarity and invertibility conditions require that all the roots of the characteristic equation 4(B) = 0, I$( B”) = 0, 0(B) = 0, and 0( B”) = 0, lie outside of the unit circle in the complex plane. Details regarding roots of characteristic equations can be found in Box and Jenkins (1976). Box and Jenkins (1976) recommended a three-stage strategy for the development of ARIMA models. In the first stage, identification of the correlation structure within a series is done with the help of graphical representations of sample ACF and PACF. A variety of possible models, selected in the first stage, are then estimated in the second stage. The stationarity conditions are checked and insignificant parameters are dropped in the second stage. The Akaike Information Criteria (AIC; Shumway, 1988) and residual variances of different models can be used for the tentative selection of a particular model. The most appropriate model is then verified in the diagnostic-checking third stage. If the selected model proves to be adequate, then it may be used for forecasting purposes.
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3. Study area Monthly discharge data from January 197 1 to December 1990 recorded by the Imperial Irrigation District, Imperial, California, USA, are used for this study. The inflow-outflow technique is employed for the calculation of transmission losses in the district. The combined outflow at different outlets is subtracted from the total inflow and then a percentage loss series is constructed by the following relationship: Percentage
Transmission
Loss =
(Total Inflow - Total Outflow) x 1o. (6)
Total Inflow
The change of data into percentage series is necessary to make it standardized as the inflow discharge is not constant for each month. The transmission losses include every loss occurring during the transportation of water from inlet to the tail of the system. Data of the first 19 years (January 1971-December 1989; 228 months) are used for the model development and data from the last year (January-December 1990; 12 months) are utilized for the calibration of the final model.
4. Model development The percentage transmission loss series (LOSS) recorded at Imperial Irrigation District, Imperial, California is shown in Fig. 1. The series shows a cyclical behavior and the cycle repeats at a lag of 12 months, which is typical for the monthly observed data. Although some of the extreme transmission losses occurred in a relatively short period of 96-144 data points, still these do not seem to jeopardize the stability of the variance. The transformation of original data in such marginal cases is not advisable because the backtransformation can introduce a bias into the analysis (Salas et al., 1980). 22.2
17.4
E $12.6
0 rl 7.8
3.0 12
36
60
84
108
132
156
180
204
228
Time (months) Fig. I. Percentage
transmission
losses (LOSS)
series.
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1.00
-
-0.50
-
-1.00
-
I
I
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I
I
29 (I 996) 283-298
I
I
12
289
I 24
Lag (months) Fig. 2. Autocorrelation
function
( ACF) of LOSS series.
The ACF and PACF of LOSS series are plotted in Fig. 2 and Fig. 3. The ACF for the series shows significant cyclical peaks at lags 12, 24, and 36. These peaks do not die out quickly, meaning that the series has a seasonal non-stationarity which is the usual case for monthly data. The significant autocorrelations at half-seasonal lags 6, 18, and 30 can be due to strong seasonal variations and they often become insignificant after seasonal differencing and/or when seasonal AR and MA coefficients are estimated (Pankratz, 1988). These seasonal peaks can also be seen from the spectrum of the LOSS series in Fig. 4. This figure
1.00 -
.50
-
&
y.oo&
.
-0.50
-
-1.00
-
I
I II
I .
I ’ I I I . I , , I 1
I
1
I
I
I
I
I
I I 1
I
I
12
Lag (mo%) Fig. 3. Partial autocorrelation
function (PACF)
of LOSS series.
1
290
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162
54
a
I
I
. 048
. 096
-144
. 192 Frequency
Fig. 4. Spectrum of LOSS series
shows that two frequencies, 0.084 and 0.165, are significant which correspond to the periods of 12 and 6. respectively. The ACF and PACF of seasonally differenced (5 = 12) LOSS series are shown in Fig. 5 and Fig. 6. It is clear from Fig. 5 that the significant autocorrelations at lags 6, 12,. . . have disappeared. The slow decaying pattern has also vanished and only big spikes at lags 12 and 24 are left. Such a pattern of big spikes is an indication of the need for a seasonal MA term. In addition to these peaks, the autocorrelation at lag 2 is quite high and the r-value at this lag is 1.43. A r-value of 1.25 or higher at the identification stage normally goes beyond I
1.00
-
.
t::_JJ
-0.50
i -1.oot
I
I
I
I
I
I
,
I 24
12
Lag (months) Fig. 5. ACF of differenced
loss series
(s = 12)
I
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291
l-oo’ -1.001 ,
,
,
(
,
I
I
12
Lag Fig. 6. PACF of differenced
I
24 (months)
loss series (s = 12)
2 in the estimation stage (Pankratz, 1988) and therefore such a correlation if not taken care of, can cause the underfitting of the model. The autocorrelation at lag 1 is not significant and only has a f-value of 0.30. The autocorrelations at lags 3,4, and 5 have also respective low r-values of 0.90, 0.70, and 0.70. This kind of autocorrelation structure is ambiguous and such patterns are sometimes formed only because of sampling errors. A safe procedure in such circumstances is to ignore this pattern altogether and perform the identification of the first tentative model by considering only the remaining patterns. The remaining pattern in this case is only the significant spikes at lags 12,24,. . . for which a seasonal MA process may be sufficient. The ARIMA( O,O,O) (O,l, 1) i2 could possibly be an appropriate model for such a structure and the model can be estimated with the help of the ASTSA computer package (Shumway, 1988) as follows: LOSS,-LOSS,_,,=a,,-0.65a,,_,,
(7)
Standard error of the coefficient 0.65 = 0.054 t-value for the coefficient 0.065 = 12.03 Residual variance = 4.50 AIC = 2.52 The ACF and PACF of residuals from this model (Model 7) are shown in Fig. 7 and Fig. 8. It is clear from these plots that the autocorrelations at lags 1, 2, and 3 are quite high and the respective t-values at these lags are also greater than 2. The significant correlations at lower lags ( 1, 2, 3) cannot be taken as sampling error and therefore should not be neglected. Such correlations in the residuals show the inadequacy of the model under consideration. The warning signal of possible trouble in Fig. 5 and Fig. 6 was not a false alaim and for the proper model the patterns in these figures, neglected in the first model (Model 7), have to be incorporated. However, the residual autocorrelation structure (Fig. 7 and Fig. 8) is not obscure and indicates relatively easier options.
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1
1.00 -
.50
-0.50
-l.OOy
-
-
I
I
I
I
I
I
I
12
I 24
Lag (months) Fig. 7. ACF of residuals from Model 7.
The autocorrelation in Fig. 7 shows a decaying pattern which is the indication for the inclusion of a non-seasonal AR term. Normally, a second-order autoregressive [ AR( 2) ] process can be adequate for such patterns. Sometimes a lower order AR or mixed ARMA processes can also be sufficient. Pursuing these indications, the following ARIMA models can be considered as possible choices.(i) ARIMA( l,O,O)(O,l,l),,; (ii) ARIMA(2,0,0)(0,1,1),,; (iii) ARIMA( 1,O, 1) (0, 1,l) , *These three models can also respectively be described as:
1.00
.50
e d .oo e.
-0.50
-1.00
I
,
I
I
I
I
12 Fig. 8. PACF of residuals from Model 7.
I
I
24 Lag (months)
T. Harmed era/. /Agricultural Table 1 Estimation
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293
results of final models
Model 10 Parameter
AR(l)
MA(l)
MA( 12)
Parameter value Standard error t-value Variance Akaike Information
0.83 1 0.105 7.890 4.228 2.479
0.659 0.140 4.710
0.749 0.047 15.810
Parameter
AR(l)
AR(2)
MA( 12)
Parameter value Standard error t-value Variance Akaike Information
0.185 0.066 2.730 4.234 2.481
0.159 0.067 2.360
0.742 0.048 15.420
Criteria (AK)
Model 9
Criteria (AK)
(l-(b,‘~>(Z,-Z,-,2)=~2,-~12’~2,_,2 (l-~,"~-~,"B*)(Z,-Z,-,*)
=%,-~,,ll%_,,
(1-~,“‘B)(Z,-Z,~,,)=(1-0,“‘B)(a,,-e,,”’u,,_,,)
(8) (9) (10)
in which 4, = the AR( 1) parameter or the first parameter of the AR( 2) process; 42 = the second parameter of the AR(2) process; 8i = the MA( 1) parameter; and & = the seasonal MA( 12) parameter. Models 8-10 are then estimated and checked for adequacy. Model 8 failed the autocorrelation test of residuals. The significant autocorrelation at lower lags (not shown) showed that the model is inadequate and inappropriate. This model is therefore rejected. The estimation results from the ASTSA package for the remaining two models (Models 9 and 10) are contained in Table 1. Both of these models passed the adequacy and residual tests. In such marginal cases (Models 9 and lo), the residual variance and AIC can give some indication for a better model selection. The variance and AIC are also given in the Table 1. It is clear that the variances and AIC of these two models do not vary significantly. However, the rapid decaying pattern of autocorrelation in Fig. 7 coupled with a marginally better variance and AIC favor the ARIMA( 1 ,O,1) (O,l, 1) i2 model (Model 10) and therefore, this model is finally selected for the time series LOSS. The ACF of residuals of Model 10 are plotted in Fig. 9, which is clear from any significant peaks at lower lags. A higher correlation at lag 11 may be due to the seasonal nature of the data. Although the data are seasonally differenced, seasonal effects are not always cleared completely and therefore such correlations at higher lags can safely be neglected. Model 10 is then checked for the consistency of its inherent behavior. Such a check is performed by deleting some of the last data points and the same model (one with full data) is again estimated from the reduced data set (Pankratz, 1988). The number of points to be
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1-oar----.50
-
q.00.
5
-0.50
t
-l.OOC, 2
,
,
,
4
6
8
, 10
,, 12
Lag (months) Fig. 9. AFC of residuals from Model IO.
deleted depends upon the total number of data points in the series. In the present study, the last 28 months are dropped and the first 200 months are used. The previously selected is again estimated from this subset of the data. The comparison ARIMA(l.O,l)(0,1.l),~ of’the parameters for the full and reduced data sets is shown in Table 2. The estimated parameters are almost the same, which shows that the underlying long-term behavior of the LOSS series was consistent with the final selected Model 10. The observed and fitted Model IO series is shown in Fig. IO. The realized and the fitted series do match reasonably, which Table 2 Comparison of final model
IOand
model from reduced data of 200 months
Model l0:ARIMA(1.0.1)(0,1,1)12 Paramete1
AR(l)
MA(l)
MA( 12)
Parameter value
0.83 I
0.659
Standard crro,
0.105
0.140
0.047
t-value
7.890
4.710
lS.XIO
Variance
4.228
Akaike Information Criteria (AK)
2.479
MA( 12)
Model from reduced data of 200 mon.: ARIMA(
0.749
I ,O, I ) (0, I, I ) I2
P:lmmcter
AR(I)
MA(l)
Parameter value
0.800
0.63 I
Srandard errs
0.137
0.174
0.730 0.052
t-vuluc
5.831
3.623
14.1 I6
Variance
4.28 I
Akaike Information Criteria ( AIC)
2.498
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I
%OSS
295
Fit
3 12
36
60
84
108
132
156
180 204 228 Time (months)
Fig. 10. Observed LOSS series and fitted series from Model 10.
strengthens the appropriateness of Model 10 in modeling the data under consideration. The forecasting capability of an ARIMA model is the ultimate criterion for its selection and capability. The last 12 observations (January-December 1990), which were left for model calibration, are then compared with the forecasted values from Model 10. These forecasts are obtained by the one-step-ahead forecasting procedure. In this procedure, the first forecast is started from a reference point (in this study the reference point is December 1989 and the first forecast is January 1990) and the subsequent forecast is attained from the first forecast. The procedure continues until the required number of forecasts are obtained. Such a procedure yields the so called “bootstrap” forecasts. This procedure allows the incorporation of the more recently observed values in the model, update the model parameters, and start the forecast from the newly observed last data point. Such additions offer two advantages: first, the model parameters are updated, and second, the future forecasts are better than those obtained without new additions. Also, because ARIMA models are better suited for short-term forecasts (Box and Jenkins, 1976) ; the incorporation of recently observed values strengthens the reliability of the forecasts. Table 3 shows a comparison of observed and forecast values from Model 10. The forecast error is the difference between observed and forecast values for the respective months. It is clear that the forecasts track the seasonal pattern of the series appropriately. The forecasts from Model 10 were then compared with those from Model 9 (not shown) and the root mean square error from Model 10 were found to be marginally better than that from Model 9, thus verifying the correct selection of Model 10. Table 3 also contains 95% confidence bounds for the forecasts. It can be seen that all the observed values of percentage transmission loss lie within those bounds. These excellent forecasts prove that the ARIMA models can successfully be used for reasonable predictions of transmission losses. Those forecasts may be invaluable for efficient water management. In on-demand irrigation systems, water is supplied to farmers according to their needs. These systems help in better management of water resources as they do not allow the
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Table 3 Observed and forecast values of transmission Month no.
229 230 231 232 233 234 23.5 236 237 238 239 240
Observed value
14.69 11.47 8.26 7.42 8.49 9.75 9.01 9.5 I IO.51 II.04 I I .30 12.98
losses from Model 10
Forecast value
16.S3 12.66 8.48 7.36 8.93 10.25 9.82 9.50 10.85 IO.95 14.26 15.42
Root mean squared error is 1.316.Maximum
Water Management 29 (1996) 283-298
95% Confidence bounds Lower
Upper
12.50 8.57 4.35 3.20 4.76 6.06 5.62 5.30 6.64 6.14
20.56 16.75 12.61 11.52 13.11 14.44 14.02 13.71 15.06 15.16 18.47 19.64
10.05 11.21
Forecast error
- 1.84 - 1.19 - 0.22 0.06 - 0.44 - 0.50 -0.81 0.0 1 - 0.34 0.09 - 2.96 - 2.44
observed error is 2.96.
wastage of water. The correct water application also avoids the possibility of water logging and salinity. Such systems heavily depend on the correct forecasting of water demands and thus the accurate prediction of transmission losses is also very important. ARIMA models have proven to be plausible models for channel transmission losses. They provide accurate forecasts and thus can be very useful for the efficient management of irrigation districts.
5. Conclusions Channel transmission losses depend on a number of factors. Operational limitations and climatic variables are some of the important factors that induce uncertainties. A physical model, which can incorporate all of the factors and accurately predict channel losses, is not easy to be developed. Even the development of a complicated model needs to have forecasts of independent variables in order to predict transmission losses. For operational purposes, such complexities are normally avoided and in practice, methods based on simple average of past estimated values of losses are usually used for the future calculation of water needs. The methods based on such calculations do not include the time-dependent behavior of losses and are naturally inadequate. Due to their inherent shortcomings, the use of current methods can result in shortage or wastage of water resulting from underprediction or overprediction of channel losses. The importance of a new, simple method with an ability to provide reasonably accurate forecasts cannot be overemphasized. The uncertainties associated with channel transmission losses make it a stochastic process and such a process can be modeled by time series techniques. ARIMA models in the context of time series have proven to be very useful for many cases and these models are found to be plausible models for the transmission losses in an irrigation network. The appropriate ARIMA model provides excellent forecasts for all of the post-data observed values of the
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percentage transmission loss series. All of the monthly observed values that are compared with the corresponding forecasts fall within the 95% confidence limits of the forecasts for those respective months. These forecasts also accurately track the seasonal pattern of the data and the forecasts errors are very small. ARIMA models can be developed with relative ease with the help of a good computer software package. Such models have a great potential for reflecting the effects of complex conditions and generating a dependable level of accuracy with a lesser amount of computing effort. Forecasts from an ARIMA model can be very useful for determining proper water allocations. Furthermore, such forecasts aid in the optimum management of any irrigation system in general and a demand irrigation system in particular.
Acknowledgements We thank J.P. Silva, Manager of the Imperial Irrigation District, Imperial, California, for providing the data used in this study. The research leading to this report was conducted at the University of California, Davis, CA, and was supported by the Agricultural Research Service under Cooperative Agreement 4350-H.
References Box, G.E.P. and Jenkins, G.M., 1976. Time Series Analysis: Forecasting and Control. Holden-Day, San Francisco, CA, 548 pp. Cryer, J.D., 1986, Time Series Analysis. Duxbury Press, Boston, MA, 286 pp. Hameed, T., Marifio, M.A. and Shumway, R.H., 1995. Evapotranspiration-transfer-function-noise modeling. J. brig. Drain. Eng., ASCE, 121(2): 159-169. Hathoot, H.M., 1984. Total losses from trapezoidal open channels. Bull. Int. Comm. brig. Drain., 33(2): 81-84. Hotes, F.L., Kruse. E.G., Christopher, J.N., Niaz, S. and Robinson, A.R., 1985. Irrigation canal seepage and its measurements: A state-of-the-art review. In: C.G. Keyes and T.J. Ward (Editors), Development and Management Aspects of Irrigation and Drainage Systems. Proceedings of the ASCE Specialty Conference, San Antonio, TX, pp. 93-105. Kraatz, D.B., 1991. Irrigation Canal Lining, FAO Irrigation and Drainage Paper No. 2, Food and Agriculture Organization of the United Nations, Rome, 170 pp. Lane, L.J., 1985. Estimating Transmission Losses. In: C.G. Keyes and T.J. Ward (Editors), Development and Management Aspects of Irrigation and Drainage Systems, Proceedings of the ASCE Specialty Conf., San Antonio, TX, pp. 106-I 13. Mitchell, K.C., James, L.C., Elgar, S. and Pitts, M.J., 1990. Characterizing cyclic water-level fluctuations in irrigation canal. J. brig. Drain. Eng., ASCE, Il6(2): 261-272. Palmer, J.D., Clemmens, A.J. and Dedrick, A.R., 1989. Several sources of nonuniformity in irrigation delivery flows. J. lrrig. Drain. Eng., ASCE, 115(6): 920-937. Pankratz. A., 1988. Forecasting with Univariate Box-Jenkins Models: Concepts and Cases. Wiley, New York, 562 pp. Replogle, J.A. and Merriam, J.L., 1980. Scheduling and management of irrigation water delivery systems. Proceedings of the Second National Irrigation Symposium, Irrigation Challenges of the 80’s. American Society of Agricultural Engineers, ASAE Publication No. 6-81, pp. 112-126. Salas, J.D., Delleur, J.W., Yevjevich, V. and Lane, W.L., 1980. Applied Modeling of Hydrologic Time Series. Water Resources Publications, Littleton, CO, 484 pp.
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Shumway, R.H.. 1988. Applied Statistical Time Series Analysis. Prentice-Hall, Englewood Cliffs, NJ, 379 pp. Trout. T.J., 1979. Factors Affecting Losses from Indus Basin Irrigation Channels. Water Management Technical Report No. SO. Colorado State University, Fort Collins. CO, 201 pp.
Time series modeling of channel transmission losses Tahir Hameed a, M.A. Marifio b,*, M.N. Cheema ’ ”Department r,fCivil and Environmental Engineering, University ofCalifornia, Davis, CA 95616, USA ’Department ofLand, Air, and Water Resources and Department ofCivil and Environmental Universi~ ofCalifornia, Davis, CA 95616, USA ’Punjab Irrigation and Power Department, Lahore, Pakistan
Engineering,
Accepted 18 April 1995
Abstract The transmission losses through a canal network depend on a number of physical, operational, and climatic factors. Some of these factors can be quantified but most of them are not easy to be incorporated into precise calculations. Due to climatic and other uncertainties, prediction of the reliable amount of losses in irrigation projects cannot be done confidently. Because of these uncertainties, the water loss series observed over a period of time can be treated as a stochastic process that can be modeled by time series techniques. In the present study, the transmission losses series observed at the Imperial Irrigation District, California, USA, is modeled as an autoregressive-integrated-moving average ( ARIMA) process. The ARIMA models are found to be plausible and appropriate models for such series. These are simple to develop and provide dependable forecasts. Reliable forecasts of canal transmission losses are useful in the efficient management of irrigation projects.
1. Introduction The correct estimation of transmission water losses from an irrigation network is vital for the proper management of the system. Water managers are forced to divert extra water in order to compensate for the losses and to ensure that the users get the water according to their requirements. Thus, for the effective operational planning and management of an irrigation system, a dependable forecasting of the losses is very important. Seepage and evaporation are the two dominant processes by which water is lost in the canals. Under arid conditions, water loss can be substantial; sometimes, even the total amount of water in a canal can be lost through highly permeable soils (Hathoot, 1984). Seepage rate from a channel depends on a number of factors such as type of construction * Corresponding author: Department of Land, Air, and Water Resources, University of California, Davis, CA 9.56 16, USA. 0378.3774/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDlO378-3774(95)01201-X
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material, channel geometry, depth of ground water, total time of canal operation, flow characteristics, sediment amount, and so forth (Kraatz, 197 1) . It is very difficult to incorporate the effects of all of these factors in any calculation. For example, the flow characteristics, which have a major impact on seepage rate, can be highly non-uniform in an irrigation system. Trout (1979) studied earthen channels of the Indus Basin (Pakistan) and found that the loss rate was very sensitive to the fluctuations in flow depths. It was also observed that losses varied widely and often were highly unpredictable. The flow non-uniformity can be caused by many factors, including a variable water demand and continuous control changes, i.e. water movement in and out of the channel network (Mitchell et al., 1990). Palmer et al. ( 1989) have observed that for high-performance irrigation projects, flows are often non-uniform. They have identified flexible deliveries as the major reason for such wavering flows. This how variability cannot be avoided as the flexibility in rate, duration, and timing is necessary for efficient water use by farmers (Replogle and Merriam, 1980). The demand of water is also very sensitive to the climatic conditions. Thus, varying flow characteristics coupled with climate variables complicate the seepage process and the precise prediction of water losses under such conditions becomes very difficult. Sediment rate can also be an important factor in seepage calculations. It has been estimated that the seepage rate from the Interstate Canal, Nebraska, USA, was reduced by 20% by the introduction of a small amount of 1000 p.p.m. sediment (Kraatz, 1991). The prediction of sediment rate in channels is itself a major task and its incorporation and subsequent water loss forecasting are not straightforward. The methods based on hydraulic conductivity and ponding as well as inflow-outflow methods have been found to be reasonable techniques for seepage calculations (Hotes et al., 1985). However, even these reasonable methods can give widely varying seepage rates. The difference of estimation by these techniques may vary from 10% to 50%, which shows the complex nature of the seepage process. Lane ( 1985) has developed an analytical solution to the governing equations of the transmission losses in ephemeral stream channels. He found that the complicated dynamics of the seepage process is very hard to be represented by equations. It is clear from the preceding reviews that the complex nature of the seepage process makes it very hard to be represented by simple relationships. From an operational management point of view, a complicated modeling of the seepage process is not required. Instead, simple models yielding better forecasts of channel transmission losses are needed for the optimal use of water resources. The other major component of transmission loss through an irrigation system happens via the evaporation process. For the case of lined channels, the evaporation component can be substantial in the total percentage loss. The evaporation process is very sensitive to climate conditions. Air temperature, sunshine hours, humidity, cloud cover, solar radiation and wind speed are some of the important factors that affect evaporation (Hameed et al., 1995). The dependence of evaporation on climate variables induces uncertainties in the evaporation process, making it probabilistic in nature. Thus, evaporation should be represented in an appropriate probabilistic environment. It is very difficult to ascertain the fraction of total losses attributable to evaporation or seepage, Thus, from an operational point of view, evaporation and seepage should be treated under one single heading, i.e. as the transmission losses in an irrigation system. AS noted
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earlier, the complexity and uncertainties in the losses makes the process probabilistic in nature. The theory of time series can provide useful models for such a stochastic process. Because most of the irrigation systems have already been developed, the challenge for engineers is to run these systems efficiently. With reasonably accurate measuring devices, the inflow-outflow approach can be used for the reliable collection of data for estimating the transmission losses in an irrigation system. Such records are routinely maintained in irrigation systems. Daily, weekly, or monthly records of reasonable lengths can easily be formed as time series. Time series methods can incorporate most of the uncertainties associated with transmission losses and the developed models can be utilized for reliable forecasts. The objective of the present study is to explore the possibility of the autoregressiveintegrated-moving average (ARIMA) process as a viable model option for transmission losses series. The model adequacy will then allow to proceed to the next step, where forecasts from such a model can be checked for their reliability.
2. ARIMA
theory
Data observed over time usually have strong correlations among neighboring observations. These autocorrelations in a series prohibit the use of standard statistical methods and in such cases the time series theory provide a solution for the appropriate modeling. Box and Jenkins ( 1976) popularized the ARIMA models. ARIMA models have proven to be very useful in the analysis of a number of time series (Salas et al., 1980; Pankratz, 1988; Shumway, 1988). The check for stationarity of the time series is the first step in the development of an ARIMA model. The weak stationarity of a time series requires that the series has a constant mean, variance, and autocovariance function whereas for the case of a strong form of stationarity, the entire probability distribution function for the process has to be independent of time (Shumway, 1988). Such a form is not practical, and a weak form of stationarity, in which only the mean, variance and autocovariance are independent of time, provides a sufficient theoretical framework for a reasonably accurate analysis. A non-stationary time series can either have a non-constant mean or a non-constant variance, or both. The term “non-constant!’ means that these statistics are dependent on time. A trend in a series produces a non-constant mean and such a series can be detrended (and subsequently made stationary) simply by differencing. The differencing can be continued until the series produces a constant mean and variance; however, for most practical cases a maximum difference order of 2 is sufficient. For the case of seasonal data, the level of series shifts regularly up and down from the mean and these seasonal trends can be removed by seasonal differencing. The seasonal differencing depends on the nature of the season over which the data are collected. For example, a difference order of 12 is needed for monthly data; 4 is for quarterly data; and so on. The relationship between observations at this lag (e.g. 12 for monthly data) can also be included in the final solution. Such seasonal relationships coupled with non-seasonal relationships provide a better explanation of the underlying process of a seasonal time series.
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The non-constant variance can be stabilized by a suitable transformation. The Box-Cox transformation (Pankratz, 1988) is a family of several transformations that is sufficient for most of the cases. The log and square root transformations are members of this family. A new series 5 is defined by the following relationship: 2,’
=I
zA- 1
(1)
h
in which 1 is a real number; and 5 is a non-stationary series. The A&MA model has autoregressive (AR) and moving average (MA) parts. The middle letter, “P’in ARIMA, means integrated and represents the order of differencing for a non-stationary series. If a difference order is zero, then such a series is already stationary and the resulting model is simply an ARMA model. The AR part got the name autoregressive because the dependent variables has a regression-type of relationship with its own previous values Z, , , Z, ?, . , taken as independent variables, as follows: Z,=4,Zz
I -~2Zr-*-...-~,~,-,,far
(2)
in which -4,) & ,..., 9, = the first, second,..., and p-order AR coefficients, respectively; a, is random shock error term or white noise, which is assumed to be normally and independently distributed with a zero mean and a constant variance. The AR process has to satisfy certain stationarity conditions. For example, for the first-order AR process [ AR( 1) 1, stationarity requires that the root of the process should lie outside of the unit circle or /&I < I. For AR( 2)) the following three conditions must be fulfilled: l&II<1 &2+&l<
1
&-&I<1 In the moving average (MA) process, the variable of interest is explained through a random shock error term. The moving average model of order q [MA(q) ] can be described as: z,=
-&a,_,
-&7-z-...-
+&/+a,
(3)
in which 5, H,,..., & is first, second, . . . . and q-order coefficients of the MA process, respectively. Its worthwhile to note that a,-k is not exactly the past value of Z, but the random shock part of Z,_,, which means that the MA process contains a part of the past value of Z, as well. Therefore, the MA process has an equivalent AR form and it can be proved that the MA process is equivalent to an infinite AR sequence (Cryer, 1986). Because of this characteristic, the stationarity conditions for an MA sequence are known as the invertibility conditions. For a MA( 1) model, invertibility requires that I 8, I < 1. The need for such a condition can be seen by a simple examination of the fact that a value of & more that 1 will assume that the recent observations have less influence on Z, than the further past ones. For a MA( 2) model, the following invertibility conditions are required:
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The autocorrelation (ACF) and partial autocorrelation functions (PACF) show the measures of the strength of the relationship between observations at different lags. The ACF and PACF of an observed series help in the identification of an ARIMA model. The comparison of sample ACF and PACF with the commonly occurring theoretical ACFs and PACFs leads to different possible choices. A variety of theoretical ACFs and PACFs are given in Box and Jenkins ( 1976) and Pankratz ( 1988). An ARIMA model is generally presented as ARIMA (p, d, q) in which p and q respectively represent the AR and the MA parts and d shows the difference order. The general ARIMA (p, d, q) can be defined as: 4(B) (1 -B)“Z,=
c+
B(B)&
(4)
in which C is a constant; B is the difference operator defined as B’Z, = Z,_ i; d is the difference order; &!?) = 1 - +,B -&B* - . . . - #Y’ the p-order AR operator; and B(B) = l0, B - 8,B2 - . . - 0&P = the q-order MA operator. The stationarity condition for ARIMA models requires that the roots of the characteristic equation of 4(B) = 0 lie outside of the unit circle. Similarly for invertibility, the roots of the characteristic equation e(B) = 0 must lie outside the unit circle (Pankratz, 1988). Parsimony is another important consideration in the development of ARIMA models. The higher order AR or MA models can be represented by ARMA models of fewer parameters. Box and Jenkins ( 1976) defined a good model as one having the fewest possible parameters. In the case of seasonal data, simple ARIMA models may be inappropriate and inadequate. Seasonal time series have relationships at lag s. The lag s depends upon the nature of data. For example, in a monthly observed series, the relationship at a lag of 12 is important and may not be neglected for the development of an adequate model. Such relationships can easily be represented by multiplicative seasonal ARIMA models which simultaneously include both seasonal and non-seasonal parts. The standard notation for such models is ARIMA(p, d, q) (P, D, Q)s and can be represented as: ~(B)~(EY)(l-B)d(l-B)~Z,=C+B(B)O(B”);a,
(5)
in which $( B’) = 1 - @?’ - +*,,B*’- . . - c#+~B~’ = the P-order seasonal AR operator and 0( B”) = 1 - f),B’- f&Bzs - . . . - 13,,$@” = the Q-order seasonal MA operator. The stationarity and invertibility conditions require that all the roots of the characteristic equation 4(B) = 0, I$( B”) = 0, 0(B) = 0, and 0( B”) = 0, lie outside of the unit circle in the complex plane. Details regarding roots of characteristic equations can be found in Box and Jenkins (1976). Box and Jenkins (1976) recommended a three-stage strategy for the development of ARIMA models. In the first stage, identification of the correlation structure within a series is done with the help of graphical representations of sample ACF and PACF. A variety of possible models, selected in the first stage, are then estimated in the second stage. The stationarity conditions are checked and insignificant parameters are dropped in the second stage. The Akaike Information Criteria (AIC; Shumway, 1988) and residual variances of different models can be used for the tentative selection of a particular model. The most appropriate model is then verified in the diagnostic-checking third stage. If the selected model proves to be adequate, then it may be used for forecasting purposes.
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3. Study area Monthly discharge data from January 197 1 to December 1990 recorded by the Imperial Irrigation District, Imperial, California, USA, are used for this study. The inflow-outflow technique is employed for the calculation of transmission losses in the district. The combined outflow at different outlets is subtracted from the total inflow and then a percentage loss series is constructed by the following relationship: Percentage
Transmission
Loss =
(Total Inflow - Total Outflow) x 1o. (6)
Total Inflow
The change of data into percentage series is necessary to make it standardized as the inflow discharge is not constant for each month. The transmission losses include every loss occurring during the transportation of water from inlet to the tail of the system. Data of the first 19 years (January 1971-December 1989; 228 months) are used for the model development and data from the last year (January-December 1990; 12 months) are utilized for the calibration of the final model.
4. Model development The percentage transmission loss series (LOSS) recorded at Imperial Irrigation District, Imperial, California is shown in Fig. 1. The series shows a cyclical behavior and the cycle repeats at a lag of 12 months, which is typical for the monthly observed data. Although some of the extreme transmission losses occurred in a relatively short period of 96-144 data points, still these do not seem to jeopardize the stability of the variance. The transformation of original data in such marginal cases is not advisable because the backtransformation can introduce a bias into the analysis (Salas et al., 1980). 22.2
17.4
E $12.6
0 rl 7.8
3.0 12
36
60
84
108
132
156
180
204
228
Time (months) Fig. I. Percentage
transmission
losses (LOSS)
series.
T. Hutneed et al. /Agricultural
1.00
-
-0.50
-
-1.00
-
I
I
Water Manapnent
I
I
I
29 (I 996) 283-298
I
I
12
289
I 24
Lag (months) Fig. 2. Autocorrelation
function
( ACF) of LOSS series.
The ACF and PACF of LOSS series are plotted in Fig. 2 and Fig. 3. The ACF for the series shows significant cyclical peaks at lags 12, 24, and 36. These peaks do not die out quickly, meaning that the series has a seasonal non-stationarity which is the usual case for monthly data. The significant autocorrelations at half-seasonal lags 6, 18, and 30 can be due to strong seasonal variations and they often become insignificant after seasonal differencing and/or when seasonal AR and MA coefficients are estimated (Pankratz, 1988). These seasonal peaks can also be seen from the spectrum of the LOSS series in Fig. 4. This figure
1.00 -
.50
-
&
y.oo&
.
-0.50
-
-1.00
-
I
I II
I .
I ’ I I I . I , , I 1
I
1
I
I
I
I
I
I I 1
I
I
12
Lag (mo%) Fig. 3. Partial autocorrelation
function (PACF)
of LOSS series.
1
290
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162
54
a
I
I
. 048
. 096
-144
. 192 Frequency
Fig. 4. Spectrum of LOSS series
shows that two frequencies, 0.084 and 0.165, are significant which correspond to the periods of 12 and 6. respectively. The ACF and PACF of seasonally differenced (5 = 12) LOSS series are shown in Fig. 5 and Fig. 6. It is clear from Fig. 5 that the significant autocorrelations at lags 6, 12,. . . have disappeared. The slow decaying pattern has also vanished and only big spikes at lags 12 and 24 are left. Such a pattern of big spikes is an indication of the need for a seasonal MA term. In addition to these peaks, the autocorrelation at lag 2 is quite high and the r-value at this lag is 1.43. A r-value of 1.25 or higher at the identification stage normally goes beyond I
1.00
-
.
t::_JJ
-0.50
i -1.oot
I
I
I
I
I
I
,
I 24
12
Lag (months) Fig. 5. ACF of differenced
loss series
(s = 12)
I
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et al. /Agricultwal
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291
l-oo’ -1.001 ,
,
,
(
,
I
I
12
Lag Fig. 6. PACF of differenced
I
24 (months)
loss series (s = 12)
2 in the estimation stage (Pankratz, 1988) and therefore such a correlation if not taken care of, can cause the underfitting of the model. The autocorrelation at lag 1 is not significant and only has a f-value of 0.30. The autocorrelations at lags 3,4, and 5 have also respective low r-values of 0.90, 0.70, and 0.70. This kind of autocorrelation structure is ambiguous and such patterns are sometimes formed only because of sampling errors. A safe procedure in such circumstances is to ignore this pattern altogether and perform the identification of the first tentative model by considering only the remaining patterns. The remaining pattern in this case is only the significant spikes at lags 12,24,. . . for which a seasonal MA process may be sufficient. The ARIMA( O,O,O) (O,l, 1) i2 could possibly be an appropriate model for such a structure and the model can be estimated with the help of the ASTSA computer package (Shumway, 1988) as follows: LOSS,-LOSS,_,,=a,,-0.65a,,_,,
(7)
Standard error of the coefficient 0.65 = 0.054 t-value for the coefficient 0.065 = 12.03 Residual variance = 4.50 AIC = 2.52 The ACF and PACF of residuals from this model (Model 7) are shown in Fig. 7 and Fig. 8. It is clear from these plots that the autocorrelations at lags 1, 2, and 3 are quite high and the respective t-values at these lags are also greater than 2. The significant correlations at lower lags ( 1, 2, 3) cannot be taken as sampling error and therefore should not be neglected. Such correlations in the residuals show the inadequacy of the model under consideration. The warning signal of possible trouble in Fig. 5 and Fig. 6 was not a false alaim and for the proper model the patterns in these figures, neglected in the first model (Model 7), have to be incorporated. However, the residual autocorrelation structure (Fig. 7 and Fig. 8) is not obscure and indicates relatively easier options.
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1
1.00 -
.50
-0.50
-l.OOy
-
-
I
I
I
I
I
I
I
12
I 24
Lag (months) Fig. 7. ACF of residuals from Model 7.
The autocorrelation in Fig. 7 shows a decaying pattern which is the indication for the inclusion of a non-seasonal AR term. Normally, a second-order autoregressive [ AR( 2) ] process can be adequate for such patterns. Sometimes a lower order AR or mixed ARMA processes can also be sufficient. Pursuing these indications, the following ARIMA models can be considered as possible choices.(i) ARIMA( l,O,O)(O,l,l),,; (ii) ARIMA(2,0,0)(0,1,1),,; (iii) ARIMA( 1,O, 1) (0, 1,l) , *These three models can also respectively be described as:
1.00
.50
e d .oo e.
-0.50
-1.00
I
,
I
I
I
I
12 Fig. 8. PACF of residuals from Model 7.
I
I
24 Lag (months)
T. Harmed era/. /Agricultural Table 1 Estimation
Water Management 29 (1996) 283-298
293
results of final models
Model 10 Parameter
AR(l)
MA(l)
MA( 12)
Parameter value Standard error t-value Variance Akaike Information
0.83 1 0.105 7.890 4.228 2.479
0.659 0.140 4.710
0.749 0.047 15.810
Parameter
AR(l)
AR(2)
MA( 12)
Parameter value Standard error t-value Variance Akaike Information
0.185 0.066 2.730 4.234 2.481
0.159 0.067 2.360
0.742 0.048 15.420
Criteria (AK)
Model 9
Criteria (AK)
(l-(b,‘~>(Z,-Z,-,2)=~2,-~12’~2,_,2 (l-~,"~-~,"B*)(Z,-Z,-,*)
=%,-~,,ll%_,,
(1-~,“‘B)(Z,-Z,~,,)=(1-0,“‘B)(a,,-e,,”’u,,_,,)
(8) (9) (10)
in which 4, = the AR( 1) parameter or the first parameter of the AR( 2) process; 42 = the second parameter of the AR(2) process; 8i = the MA( 1) parameter; and & = the seasonal MA( 12) parameter. Models 8-10 are then estimated and checked for adequacy. Model 8 failed the autocorrelation test of residuals. The significant autocorrelation at lower lags (not shown) showed that the model is inadequate and inappropriate. This model is therefore rejected. The estimation results from the ASTSA package for the remaining two models (Models 9 and 10) are contained in Table 1. Both of these models passed the adequacy and residual tests. In such marginal cases (Models 9 and lo), the residual variance and AIC can give some indication for a better model selection. The variance and AIC are also given in the Table 1. It is clear that the variances and AIC of these two models do not vary significantly. However, the rapid decaying pattern of autocorrelation in Fig. 7 coupled with a marginally better variance and AIC favor the ARIMA( 1 ,O,1) (O,l, 1) i2 model (Model 10) and therefore, this model is finally selected for the time series LOSS. The ACF of residuals of Model 10 are plotted in Fig. 9, which is clear from any significant peaks at lower lags. A higher correlation at lag 11 may be due to the seasonal nature of the data. Although the data are seasonally differenced, seasonal effects are not always cleared completely and therefore such correlations at higher lags can safely be neglected. Model 10 is then checked for the consistency of its inherent behavior. Such a check is performed by deleting some of the last data points and the same model (one with full data) is again estimated from the reduced data set (Pankratz, 1988). The number of points to be
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1-oar----.50
-
q.00.
5
-0.50
t
-l.OOC, 2
,
,
,
4
6
8
, 10
,, 12
Lag (months) Fig. 9. AFC of residuals from Model IO.
deleted depends upon the total number of data points in the series. In the present study, the last 28 months are dropped and the first 200 months are used. The previously selected is again estimated from this subset of the data. The comparison ARIMA(l.O,l)(0,1.l),~ of’the parameters for the full and reduced data sets is shown in Table 2. The estimated parameters are almost the same, which shows that the underlying long-term behavior of the LOSS series was consistent with the final selected Model 10. The observed and fitted Model IO series is shown in Fig. IO. The realized and the fitted series do match reasonably, which Table 2 Comparison of final model
IOand
model from reduced data of 200 months
Model l0:ARIMA(1.0.1)(0,1,1)12 Paramete1
AR(l)
MA(l)
MA( 12)
Parameter value
0.83 I
0.659
Standard crro,
0.105
0.140
0.047
t-value
7.890
4.710
lS.XIO
Variance
4.228
Akaike Information Criteria (AK)
2.479
MA( 12)
Model from reduced data of 200 mon.: ARIMA(
0.749
I ,O, I ) (0, I, I ) I2
P:lmmcter
AR(I)
MA(l)
Parameter value
0.800
0.63 I
Srandard errs
0.137
0.174
0.730 0.052
t-vuluc
5.831
3.623
14.1 I6
Variance
4.28 I
Akaike Information Criteria ( AIC)
2.498
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I
%OSS
295
Fit
3 12
36
60
84
108
132
156
180 204 228 Time (months)
Fig. 10. Observed LOSS series and fitted series from Model 10.
strengthens the appropriateness of Model 10 in modeling the data under consideration. The forecasting capability of an ARIMA model is the ultimate criterion for its selection and capability. The last 12 observations (January-December 1990), which were left for model calibration, are then compared with the forecasted values from Model 10. These forecasts are obtained by the one-step-ahead forecasting procedure. In this procedure, the first forecast is started from a reference point (in this study the reference point is December 1989 and the first forecast is January 1990) and the subsequent forecast is attained from the first forecast. The procedure continues until the required number of forecasts are obtained. Such a procedure yields the so called “bootstrap” forecasts. This procedure allows the incorporation of the more recently observed values in the model, update the model parameters, and start the forecast from the newly observed last data point. Such additions offer two advantages: first, the model parameters are updated, and second, the future forecasts are better than those obtained without new additions. Also, because ARIMA models are better suited for short-term forecasts (Box and Jenkins, 1976) ; the incorporation of recently observed values strengthens the reliability of the forecasts. Table 3 shows a comparison of observed and forecast values from Model 10. The forecast error is the difference between observed and forecast values for the respective months. It is clear that the forecasts track the seasonal pattern of the series appropriately. The forecasts from Model 10 were then compared with those from Model 9 (not shown) and the root mean square error from Model 10 were found to be marginally better than that from Model 9, thus verifying the correct selection of Model 10. Table 3 also contains 95% confidence bounds for the forecasts. It can be seen that all the observed values of percentage transmission loss lie within those bounds. These excellent forecasts prove that the ARIMA models can successfully be used for reasonable predictions of transmission losses. Those forecasts may be invaluable for efficient water management. In on-demand irrigation systems, water is supplied to farmers according to their needs. These systems help in better management of water resources as they do not allow the
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Table 3 Observed and forecast values of transmission Month no.
229 230 231 232 233 234 23.5 236 237 238 239 240
Observed value
14.69 11.47 8.26 7.42 8.49 9.75 9.01 9.5 I IO.51 II.04 I I .30 12.98
losses from Model 10
Forecast value
16.S3 12.66 8.48 7.36 8.93 10.25 9.82 9.50 10.85 IO.95 14.26 15.42
Root mean squared error is 1.316.Maximum
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95% Confidence bounds Lower
Upper
12.50 8.57 4.35 3.20 4.76 6.06 5.62 5.30 6.64 6.14
20.56 16.75 12.61 11.52 13.11 14.44 14.02 13.71 15.06 15.16 18.47 19.64
10.05 11.21
Forecast error
- 1.84 - 1.19 - 0.22 0.06 - 0.44 - 0.50 -0.81 0.0 1 - 0.34 0.09 - 2.96 - 2.44
observed error is 2.96.
wastage of water. The correct water application also avoids the possibility of water logging and salinity. Such systems heavily depend on the correct forecasting of water demands and thus the accurate prediction of transmission losses is also very important. ARIMA models have proven to be plausible models for channel transmission losses. They provide accurate forecasts and thus can be very useful for the efficient management of irrigation districts.
5. Conclusions Channel transmission losses depend on a number of factors. Operational limitations and climatic variables are some of the important factors that induce uncertainties. A physical model, which can incorporate all of the factors and accurately predict channel losses, is not easy to be developed. Even the development of a complicated model needs to have forecasts of independent variables in order to predict transmission losses. For operational purposes, such complexities are normally avoided and in practice, methods based on simple average of past estimated values of losses are usually used for the future calculation of water needs. The methods based on such calculations do not include the time-dependent behavior of losses and are naturally inadequate. Due to their inherent shortcomings, the use of current methods can result in shortage or wastage of water resulting from underprediction or overprediction of channel losses. The importance of a new, simple method with an ability to provide reasonably accurate forecasts cannot be overemphasized. The uncertainties associated with channel transmission losses make it a stochastic process and such a process can be modeled by time series techniques. ARIMA models in the context of time series have proven to be very useful for many cases and these models are found to be plausible models for the transmission losses in an irrigation network. The appropriate ARIMA model provides excellent forecasts for all of the post-data observed values of the
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percentage transmission loss series. All of the monthly observed values that are compared with the corresponding forecasts fall within the 95% confidence limits of the forecasts for those respective months. These forecasts also accurately track the seasonal pattern of the data and the forecasts errors are very small. ARIMA models can be developed with relative ease with the help of a good computer software package. Such models have a great potential for reflecting the effects of complex conditions and generating a dependable level of accuracy with a lesser amount of computing effort. Forecasts from an ARIMA model can be very useful for determining proper water allocations. Furthermore, such forecasts aid in the optimum management of any irrigation system in general and a demand irrigation system in particular.
Acknowledgements We thank J.P. Silva, Manager of the Imperial Irrigation District, Imperial, California, for providing the data used in this study. The research leading to this report was conducted at the University of California, Davis, CA, and was supported by the Agricultural Research Service under Cooperative Agreement 4350-H.
References Box, G.E.P. and Jenkins, G.M., 1976. Time Series Analysis: Forecasting and Control. Holden-Day, San Francisco, CA, 548 pp. Cryer, J.D., 1986, Time Series Analysis. Duxbury Press, Boston, MA, 286 pp. Hameed, T., Marifio, M.A. and Shumway, R.H., 1995. Evapotranspiration-transfer-function-noise modeling. J. brig. Drain. Eng., ASCE, 121(2): 159-169. Hathoot, H.M., 1984. Total losses from trapezoidal open channels. Bull. Int. Comm. brig. Drain., 33(2): 81-84. Hotes, F.L., Kruse. E.G., Christopher, J.N., Niaz, S. and Robinson, A.R., 1985. Irrigation canal seepage and its measurements: A state-of-the-art review. In: C.G. Keyes and T.J. Ward (Editors), Development and Management Aspects of Irrigation and Drainage Systems. Proceedings of the ASCE Specialty Conference, San Antonio, TX, pp. 93-105. Kraatz, D.B., 1991. Irrigation Canal Lining, FAO Irrigation and Drainage Paper No. 2, Food and Agriculture Organization of the United Nations, Rome, 170 pp. Lane, L.J., 1985. Estimating Transmission Losses. In: C.G. Keyes and T.J. Ward (Editors), Development and Management Aspects of Irrigation and Drainage Systems, Proceedings of the ASCE Specialty Conf., San Antonio, TX, pp. 106-I 13. Mitchell, K.C., James, L.C., Elgar, S. and Pitts, M.J., 1990. Characterizing cyclic water-level fluctuations in irrigation canal. J. brig. Drain. Eng., ASCE, Il6(2): 261-272. Palmer, J.D., Clemmens, A.J. and Dedrick, A.R., 1989. Several sources of nonuniformity in irrigation delivery flows. J. lrrig. Drain. Eng., ASCE, 115(6): 920-937. Pankratz. A., 1988. Forecasting with Univariate Box-Jenkins Models: Concepts and Cases. Wiley, New York, 562 pp. Replogle, J.A. and Merriam, J.L., 1980. Scheduling and management of irrigation water delivery systems. Proceedings of the Second National Irrigation Symposium, Irrigation Challenges of the 80’s. American Society of Agricultural Engineers, ASAE Publication No. 6-81, pp. 112-126. Salas, J.D., Delleur, J.W., Yevjevich, V. and Lane, W.L., 1980. Applied Modeling of Hydrologic Time Series. Water Resources Publications, Littleton, CO, 484 pp.
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Shumway, R.H.. 1988. Applied Statistical Time Series Analysis. Prentice-Hall, Englewood Cliffs, NJ, 379 pp. Trout. T.J., 1979. Factors Affecting Losses from Indus Basin Irrigation Channels. Water Management Technical Report No. SO. Colorado State University, Fort Collins. CO, 201 pp.