Agricultural watermanagement ELSEVIER
Agricultural Water Management 32 (1997) 111-129
as a stochastic
a, W.W. Wallender
a Biological and Agricultural
Engineering Department, University of California, Davis, Davis, CA 95616, USA b Hydrologic Science, and Biological and Agricultural Engineering Departments, University of California, Davis, Davis, CA 95616, USA
Accepted 10 June 1996
Abstract Spatial variability of evapotranspiration (ET) within irrigation intervals and the temporal vulgaris) were measured variability of spatially averaged ET of Yolano pink beans (Phaseolus and statistically modeled to improve irrigation management. Field experiments were conducted at the University of California, Davis, on the Yolo soil series (Typic Xerorthents) during the summer of 1992. Daily soil water measurements were taken at 27 locations along a transect at 10 m intervals with a neutron probe. Changes in soil moisture over the 165 cm profile gave field-measured ET. Neutron probe measured ET was 20% less than the production function estimated ET, but both had similar pattern along the row. The trend of decreasing ET with distance down the row became increasingly steep as the growing season progressed. This trend was caused by a similar trend in soil water imposed by a furrow irrigation. After removing the trend with a first-order difference, ET was not spatially correlated (95% confidence interval) at 10 m. Therefore, each sample taken at a 10 m spacing provided maximum new information because it was not predictable from its neighbor. However, ET was temporally correlated (95% confidence interval at lag 1) and characterized by an autoregressive moving average (l,l> model. Therefore, ET could be predicted one day in advance. For the field conditions studied, neutron probe measurements can be used to estimate daily ET for about 44 days in the middle of the growing season, when measured crop ET was greater than 1.4 mm, and at an interval of 6 days at the beginning of the cropping season. 0 1997 Elsevier Science B.V. Keywords:
* Corresponding author. Veihmeyer Hall, University
Beans; Soil moisture
Hydrologic Science, and Biological and Agricultural of California, Davis, Davis, CA 95616, USA.
037%3774/97/$17.00 0 1997 Elsevier Science B.V. All rights resewed. PII SO378-3774(96)01267-X
Wurer Management 32 (1997) II l-129
1. Introduction Accurate estimates of evapotranspiration (ET) are required for irrigation scheduling and effective water management. However, factors which affect ET such as temperature, humidity, solar radiation, wind speed; crop growth stage and type; and soil hydraulic and physical properties vary both in space and time. ET can be seen as an integrated response to all these factors, and a major contributor to irrigation requirement. Therefore, it would be helpful to measure and statistically describe the variability in ET caused by these factors in preparation for the management of irrigation systems. Spatial variability of soil physical and hydraulic properties including soil water tension (Saddiq et al., 1985; Yeh et al., 1986; Burden and Selim, 1989), particle size distribution, available water, bulk density (Gajem et al., 1981; Vauclin et al., 1983), soil water content (Davidoff and Selim, 1988), and infiltration rate (Vieira et al., 1981; Bautista and Wallender, 198.5; Tarboton and Wallender, 1989; Childs et al., 1993) have been studied intensively. In addition to spatial variability, temporal variability of water content and infiltration have also been investigated (Jaynes and Hunsaker, 1989; Or and Hanks, 1992; Childs et al., 1993). The effects of spatially variable soil properties on irrigation management (Wallender and Rayej, 1987), and crop yield (Warrick and Gardner, 1983; Letey, 198.5), have received less attention. A few studies have reported field measured variability of evaporation from bare soils (Evett and Warrick, 1987; and Lascano and Hatfield, 1992), leaving an opportunity to investigate temporal and spatial variability of ET from a vegetated field. Although physical and hydraulic properties of soil vary widely, they may be spatially correlated. Knowing the spatial correlation structure, fewer measurements are required to map soil properties across the field. After the initial investment to quantify the spatial correlation, future sampling is less intensive and less costly. Similarly, spatial variability in ET represents the variability in irrigation requirement. Temporal correlation structure limits data to the most recent ET history and is useful in forecasting the next irrigation. It is hypothesized that irrigation scheduling cost will be less and accuracy will be greater if the spatial and temporal correlation structure of ET is known. The purpose of the study was to measure and statistically model the spatial variability of ET within irrigation intervals and the temporal variability of spatially-averaged ET of pink dry beans (Phaseolus vulgaris).
The semi-variogram is the most commonly used method for determining the spatial dependence of regionalized variables, however, it can not be used to identify the stochastic mechanism of ET in time. Therefore, autocorrelation and partial autocorrelation functions were used for space and time series assuming second order stationarity (Salas et al., 1988). Let [xi, x~+~] be a pair of ET measurements at i and i + h in space or time separated by a vector h (lag). Each xi, is a realization of the random variable Xi, where i is the fixed position in space or time. The set of random variables (Xi, i within the space or
32 (1997) Ill-129
time domain of interest) is called a random function and is said to be second order stationary if: (i) the expected value E[Xi] exists and is the same within the space or time domain: E[Xi]
(ii) the covariance for each pair of random variables space or time, and depends on h:
cov( h) = Stationarity
of the covariance
[Xi, Xi+h] exists, is the same in
- m*. implies
of the variance.
The autocorrelation function expresses the degree of dependency among neighboring observations. It is a process of self comparison expressing linear correlation between an equally spaced series and the same series at a specified lag or separation. Let x0, xi, x2,. . . , xN_ , be a realization of a stationary stochastic process, then the population autocorrelation function ( p(h)) can be defined as the quotient of the population autocovariance, cov( xi,xi+ h) and variance, var( xi>:
where xi is the value of variable at the ith location in space or time, and h is the space or time lag. Since, the series observed during the experiment is just one particular realization (out of an infinite set of realizations) of a stochastic process produced by the underlying probabilistic mechanism, the population autocorrelation function (Eq. 3) can be estimated using the sample autocorrelation function, r(h): N-h C r(h)=
iF,( ‘i - ‘1’ where X is the sample mean. The 95% confidence band function, r(h)9.5%, is given by Box and Jenkins (1970):
where q is the assumed order of the process beyond which autocorrelation is not significantly different than zero, rj is the autocorrelation coefficient at lag j and n is the number of observations in a chosen space or time series. The autocorrelation function is diagnostic of the moving average process. These processes do not have any space or time dependence. Therefore, the value of a variable at its current location or time can be
estimated from a purely random space or time steps. 2.2. Partial autocorrelation
Water Management 32 (1997) III-129
series using the weighted sum of the values at previous
The partial autocorrelation function is another way of representing the space or time dependent structure of a series. It is useful for diagnosing the order of autoregressive processes. Autoregressive (AR) processes have space or time dependence, i.e., the value of a variable at its current location or time depends on the values at previous space or time steps. Therefore, the idea of autocorrelation, that measures the correlation of variables separated by assigned lag, can be extended to that of the correlation where dependence on the intermediate terms has been removed. Mathematically, it can be defined as: ~k(k)=corr(x,,x,_,/x,_,,...,x,-,+,)
where &(k) is the correlation between x, and x,_~ excluding the effects of x,_ ,““, x~_~+ ,; and k is the distance or time lag between measured quantities. For example, the measured ET series is represented by the AR(2) model: I + 442)
x, = 4,(2)x,-
x,-~ + a,
in which a, is the time independent (uncorrelated) series, which is independent of x,, and it is also normally distributed with mean zero and variance o,*; c#J,(~) and &(2) are the autoregressive parameters. For the AR(2) model, estimators of autoregressive parameters are given below: r,(l 442)
If the measured ET follows a first-order AR process, then the term xI is correlated with only because both are correlated with xt_ 1 (x, depends on x,_ , and later on x,_ 2, Eq. 7). For AR(I) process, Eqs. (8) and (9) become J,(2) = r, and &(2) = 0, respectively, and &(2) is described as the sample partial autocorrelation coefficient of order two, that measures the excess correlation between x, and x,_~ that is not accounted for by r, . In general, for an AR process of order k, the partial autocorrelation coefficient $k(k) is a measure of the linear association between pi and pj_k (autocorrelation at lag j and j - k) for j I k. It is the kth autoregressive coefficient and &k(k) for k = 1, 2,... is the partial autocorrelation function. The lag j autocorrelation for an AR(k) process can be written as: X r_2
32 (1997) 111-129
coefficient of the AR(k) model. Eq. (10) vhere +j(k) is the jth autoregressive constitutes a set of linear_equations, that can be written in terms of the sample partial autocorrelation function, +k(k), as: 1
or 4, = P;‘pk,
k = 1,2,...
Thus, the sample partial autocorrelation function can be determined by successively applying Eq. (12). The 95% confidence band for the sample partial autocorrelation function, c$,(k)95%, is given by Bartlett (1946): c$k(k)95%=Orf:T
where n is the number of observations
in a series.
3. Field experiment and water balance The field experiment was conducted at the University of California, Davis during the summer of 1992. The soil is a uniform Yolo clay loam (Typic Xerorthents) with no layering within top 135 cm. The plot was approximately 265 m long and 16 m wide (Fig. 1). A pre-irrigation was applied on June 1 (day 153) to enhance germination. On June 10 (day 163), 20 rows of Yolano pink beans (Phaseoh vulgaris) were planted on 35 cm beds spaced at 80 cm. The ninth row from the west edge of the experimental plot was selected as a test row to avoid boundary effects. The day after sowing, 27 aluminum access tubes were installed to a depth of 180 cm at 10 m intervals in the middle of the bed. During the growing season (June 10 to Sept. 8), three irrigations were applied at
Fig. 1. Schematic
of field experiment
W. W. Wullender/Agricuirurcrl
32 (1997) Ill-129
fixed intervals of 21 days (July 1, day 183; July 22, day 204; and August 12, day 225). Constant inflow rates were maintained in the two furrows adjacent to the test row to eliminate boundary and wetted perimeter effects on infiltration during each irrigation. The water was turned off as soon as advance was completed in order to develop a gradient in infiltration amount along the furrow. However, water was ponded at the downstream end of the furrow. Because water was turned off just after completion of the advance phase, applied irrigations were not sufficient to bring soil moisture to the field capacity. Average deficit depths; and deficit area were 0.004, 0.075 and 0.046 m; and 20, 100 and lOO%, respectively, for the first, second and third irrigation. The water table depth was in excess of 20 m (Parlange et al., 1992). Volumetric moisture content was monitored using a Campbell Pacific Nuclear Probe model 503DR. The neutron probe was field calibrated by taking three probe readings each at 15, 30, 60, and 90 cm depths. One soil moisture sample was then taken with a SCS Madera sampler at the corresponding depths. This was repeated at distances of 30, 90, 150, and 210 m along the row. Soil samples of known volume were oven dried for 24 h at a temperature of 105°C to obtain the volumetric water content. The sampled volumetric water content varied from 17% to 35%. The linear calibration curve between ratio (count/standard count) and volumetric water content had a standard error of estimate of 0.43% and a coefficient of determination, rz = 0.89. The neutron gauge detects randomly decaying thermalised neutrons emitted by a radio-isotope source and variability within the reading is unavoidable. Therefore, a study was conducted to determine the standard deviation of measured soil water storage. The probe readings with a 32 second count time per reading were taken at the depths of 15, 30, 60, 90, 105, 120, and 150 cm. Measurements at these depths were repeated 30 times. For each set of measurements, soil water storage in the 165 cm profile was estimated using the developed field averaged calibration curve. The mean and standard deviation of stored soil water were found to be 513.0 mm and 1.4 mm, respectively (Fig. 2). The standard deviation of measured soil water is indicative of neutron probe measurement error range. Therefore, neutron probe can not be used to determine change in moisture content of less than 1.4 mm.
Fig. 2. Neutron probe measured soil moisture storage variability.
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Daily soil moisture readings were taken at 15, 30, 60, 90, 120, and 150 cm depths at all 27 locations between 7:15 A.M. and 9:45 A.M. beginning June 13 and ending September 8. Probe readings at 15, 30, 60, 90, 120, and 150 cm depth represented average soil moisture content for depth intervals from 0 to 22.5, 22.5 to 45, 45 to 75, 75 to 105, 105 to 135 and 135 to 165 cm, respectively. During the experiment the tubes were read in a systematic pattern, i.e., measurements were started at upstream end tube (0 m> and ended at the downstream tube (260 m). Precautions were taken to minimize disturbance of soil and plants adjacent to the tubes. 3.1. Daily water balance Evapotranspiration from the cropped field for any given estimated using the conservation of mass equation: Em- I,j= sWi-i,j-
and time was
SW is the soil moisture depths in the profile, I is where ET is the evapotranspiration, the depth of irrigation, R is the rainfall depth assuming no rainfall variability at the field scale. The term R was neglected in Eq. (14) because it did not rain. DP is the deep percolation, and i and j are, respectively, time and location index. Deep percolation at location j on day i - 1 can be estimated from Darcy’s law: DPi_,.j=
in which k is the hydraulic conductivity of soil and $ is the matric potential. Several studies (Nielsen et al., 1973; and Ahuja et al., 1988) reported that during the vertical redistribution of soil water at depths greater than 0.5 m, the hydraulic gradient may be taken as minus unity such that DP = k. For Yolo clay loam soil, Larue et al. (1968) estimated hydraulic conductivity as a function of soil water for depth intervals of 30-60, 60-90,90120,120- 150, and 150- 180 cm. The hydraulic conductivity for the 150- 180 cm layer at a soil moisture content of 33% (average measured soil moisture at the 150 cm depth on June 13, 1992) was about 0.2 mm/day. Therefore, during the growing season under the assumption of unit hydraulic gradient, the likely upper limit of average deep percolation from the bottom of the control volume (135 to 165 cm layer) was 17.4 mm. Wright (1990) compared ET measured by a weighting lysimeter to that measured by soil water balance using neutron probe and concluded that large errors in the water balance method occurred if the depth of the profile measured by neutron probe did not exceed the depth of wetting due to irrigation. Therefore, in this study, wetting front depth was determined by comparing soil moisture profiles corresponding to before and a day after irrigation event. For each irrigation event, wetting front depth was less than 100 cm. Parlange et al. (1992) used daily neutron probe measurements to estimate daily evaporation from bare soil (Yolo clay loam) assuming deep percolation from the bottom of control volume (105 cm) was negligible. They compared estimated evaporation with that measured by floating and weighing lysimeters and found 1:l correlation between the measured and estimated evaporation. In keeping with the findings of Parlange et al.
Water Management 32 (1997) 111-129
(1992) and because the hydraulic gradient was generally below unity, it was assumed that the daily drainage at the 165 cm depth was negligible. Daily soil water storage, SW, at each location along the furrow was estimated by integrating the volumetric water content readings for the 165 cm profile. Without irrigation or rainfall (it was neglected in Eq. (14) because it did not rain), ET was estimated from the change in soil water storage with time (Eq. 14). However, on the day of an irrigation event, estimated ET was assumed equal to the potential crop ET because changes in soil water content on these days reflected ET as well as infiltration (two unknowns in Eq. 14). Daily ET at each access tube location along the row was estimated from June 13 to Sept. 7.
4. Results and discussion Prior to the first irrigation the soil surface was dry in appearance. Soil moisture contours for the 165 cm profile were drawn using the plotting software SURFER (Version 4). SURFER uses the inverse distance squared technique for interpolating irregularly spaced measurements to a specified grid size. Soil water down to 15 cm was lost to evaporation and transpiration which caused a gradient in the soil moisture with depth (Fig. 3). The July 1 irrigation erased the gradient near the upstream end where intake opportunity time and thus infiltration was sufficient to elevate near surface water content to that at and below 30 cm (Fig. 4). Because the water was cutoff when it reached the field end, less water infiltrated at the downstream end leaving a water content gradient with depth. Beyond 160 m, sandier soil, observed at the 150 cm depth,
-150 -165 0
ROW ( ml
Fig. 3. Soil moisture profile along the furrow prior to the first irrigation
on July 1.
W.W. W&lender / Agricultural
32 (1997) 11 l-129
-150 -166 0
ALONG THE ROW ( ml
Fig. 4. Soil moisture profile along the furrow on July 2 after first irrigation.
retained less water. By July 12 (11 days after irrigation) high evaporation from the wet soil surface near the upstream end and increasing transpiration resulted in a moisture gradient with depth similar to the condition before irrigation (Fig. 5). Root water extraction at the 45 cm depth decreased water content from 0.32 to 0.30.
-30 -45 ?
El 0 -105 -120
THE ROW ( ml
Fig. 5. Soil moisture profile along the furrow on July 12, 11 days after the first irrigation.
Fig. 6. Daily evapotranspiration
Water Mrrnagemrnt 32 (1997) II l-129
along the furrow from June 12 to the first irrigation
on July 1.
Daily ET before the first irrigation (13 June, day 165, through 30 June, day 182) was stochastic in time and space and there were no obvious trends (Fig. 6). After the first irrigation (Fig. 7), ET increased as atmospheric water demand and the canopy developed. ET did not decrease along the row in response to the first irrigation induced trend in soil moisture content in the top 30 cm depth, because sufficient moisture was available for root water extraction below 30 cm (pre-planting irrigation on June 1). Shortly after the second irrigation, ET decreased with time indicating limited water availability in the root zone (Fig. 8). There was no trend in soil moisture or ET along the row prior to the third irrigation and water content was below 24% down to 150 cm. The third irrigation caused a gradient in water content (O-30 cm) from 35% at the upstream end to 27% near the downstream end followed by a large increase at the last station caused by ponding. ET tracked the water content trend along the row (Fig. 9). This was also confirmed by high correlation (0.84) between the third irrigation infiltrated depth and cumulative ET for the last interval (12 Aug. through 7 Sept). Daily ET decreased
Fig. 7. Daily evapotranspiration
along the furrow from July 1 to the second irrigation
on July 22.
N.S. Raghuwanshi, W.W. Wallender/Agricultural
Fig. 8. Daily evapotranspiration
Water Management 32 (1997) III-129
along the furrow from July 22 to the third irrigation
on August 12.
with time and approached the early season rate. Daily soil moisture content readings showed strong time stability, i.e., locations with high water content one day would have high water content the next day. This time stability was not found in daily measured ET, suggesting ET is not a function of soil moisture alone but also depends upon crop and climatic factors. For example, if the crop is stressed prior to an irrigation it may not recover even though sufficient water is available following the irrigation. 4.1. Spatial analysis Cumulative ET during the intervals depicted in Fig. 6, Fig. 7, Fig. 8 and Fig. 9 as well as for the season was calculated for each location along the run (13 June through 30 June, ETl; 1 July through 21 July, ET2; 22 July through 11 Aug., ET3; 12 Aug. through
Fig. 9. Daily evapotranspiration
along the furrow from August 12 through harvest on September
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200 I 0
a I! 150 ALONG
Fig. 10. Spatial evapotranspiration for (a) the chosen time intervals June 13 through June 30 (ET]), July 1 through July 22 (ET2), July 23 through August 12 (ET3) and August 13 through September 7 (ET4), and (b) measured (June 13 through September 7, TET) and production function estimated seasonal evapotranspiration (ETPF).
7 Sept., ET4; and 12 June through 7 Sept., TET) and plotted as space series (Fig. lOa, Fig. lob). An independent estimate of ET was used to verify the accuracy of neutron probe measured ET. Therefore, bean yield for a 2 m transect was measured adjacent to all 27 access tube locations. Seasonal ET at each access tube location was estimated using a relative production function and measured yield. The following relative production function was derived from the results of an earlier field experiment on the same crop and soil (Tosso, 1978): Y ET = 1.27Ymm J%ax
where Y is the bean yield in g/m2, Y,,, is the maximum bean yield (419.4 g/m’>, ET (41.6 is the evapotranspiration in cm and ET,,, is the maximum evapotranspiration cm). On average, estimated ET using the relative production function was 6.45 cm
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(20%) greater than the neutron probe measured ET (Fig. lob). Also, seasonal ET estimated using a daily water balance based model was 5.26 cm greater than the probe measured ET (Raghuwanshi and Wallender, 1996). This could have been due to (1) error in neutron probe readings at 15 cm, where changes in surface moisture contents were greater than measured, (2) use of the same calibration curve for all the measurement depths. However, measured and estimated ET had a similar local variation and a decreasing trend along the run, (Fig. lob). Evett et al. (1993) also reported 38% error in ET estimate using neutron probe measurements for a period of 16 days, however, error in ET estimate decreased to 8% when ET was computed by combining time domain reflectometry (TDR) measurements for the surface to 40 cm layer and neutron probe measurements below 40 cm in a 200 cm profile. The TDR is a technique to measure the high frequency electrical properties of material. In soil applications TDR is used to measure the dielectric constant. Water has a dielectric constant of 80 as contrasted with values of 2 to 5 for soil solids. Thus a measure of the dielectric constant of soil is a good measure of its water content. Cumulative ET (dominated by evaporation) along the row for the first time interval (ETl) had no trend and variability decreased beyond 200 m. ET variability increased dramatically for subsequent time intervals (Fig. 1Oa) most likely with increased variability in root development, root water extraction, plant response, and soil moisture availability. Also, a trend of decreasing ET along the row became more prominent for the last time interval (ET4) and was reflected in the seasonal ET (TET) series. The ET3, ET4 and TET series had greater ET than the other series at the field end because infiltration was increased by greater ponding time. In the beginning of the growing season higher contribution of evaporation as compared to transpiration resulted in low ET standard deviation, but because ET was low, the coefficient of variation was relatively large (Table 1). Thereafter, ET increased until maturity near the end of interval three (before the third irrigation) and then declined during the fourth interval. Standard deviation increased at slightly lower rate as the mean during intervals two and three and jumped dramatically during the last interval due to the moisture induced trend. Similarity in water application as well as nearly full canopy cover for the second and third intervals likely caused CVs to be nearly equal for those two periods. The large standard deviation caused the coefficient of variation (CV) for the ET4 series to exceed the CVs for ET2 and ET3 and caused a large standard deviation for seasonal ET. The moisture trend (in the O-30 cm profile, the third irrigation resulted in a moisture gradient from 35% at the upstream end to 27% near the Table 1 Descriptive
of spatial evapotranspiration
Mean (mm) Std. Dev. (mm) Coeff. of Variation (%) Minimum (mm) Maximum (mm) Range (mm)
12 4 33 2 18 16
65 12 19 36 85 49
108 16 15 66 137 71
67 31 46 17 140 123
251 33 13 186 307 121
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downstream end followed by a large increase at the last station caused by ponding) caused a larger range of ET during the fourth interval reflecting crop response to water deficit. 4.2. Autocorrelation
With our experimental design, spatial correlation of ET can only be considered in the intra furrow direction. The autocorrelation function from lag 0 to lag 13 (0 to 130 m) for the original ET series (ETl, ET2, ET3, ET4, and TET) was calculated using Eq. (4) and the 95% confidence bands were calculated using Eq. (5). Lag 0 autocorrelation is always plus 1 as it is correlation of the variable with itself. In general, as the lag spacing increases covariance and thus autocorrelation decays. The autocorrelation function of ETl, ET2, and ET3 series fell within the 95% confidence band and there was no significant correlation between neighbors. The ET4 and TET series exhibited significant correlation at lag 1 (10 m) and thereafter fell within the 95% confidence band. However, trend or drift (decrease in ET with distance) in the data in ET4 and TET, as shown in Figs. (10a and lob) may have violated the assumption of second order stationarity (Eqs. 1 and 2). Stationarity of a series along a transect depends on the sampling interval as well as the length of transect (Sisson and Wierenga, 1981). A non-stationary series can be transformed to a stationary series using transformation schemes, e.g., log transformation, first order difference, etc. In the case of ET4 and TET series trend was removed using the first order difference transformation as follows:
(17) in which subscript j represents location. The autocorrelation function for transformed ET4, and TET series were calculated for the lag 0 to lag 13 (0 m to 130 m) along with the corresponding 95% confidence band in a similar manner as described above. The autocorrelation functions decreased as lag increased. Lag one autocorrelation values were -0.0190, 0.1679, -0.1390, -0.2710 and -0.1289 for ETl, ET2, ET3, ET4, and TET series, respectively. If lag one autocorrelation is negative then the function oscillates. The autocorrelation function showed no periodicity or significant correlation at any lag for the chosen spatial ET series. Thus, each observation taken at 10 m interval was independent from observations at previous locations and each sample gave the maximum new information. Therefore, in order to identify the order and spatial correlation structure of ET along the row, a sample interval of less than 10 m is required. This observation is consistent with the Bautista and Wallender (1985) finding for infiltration. 4.3. Partial autocorrelation
The partial autocorrelation functions corresponding to the same five series were calculated for lag 0 to lag 13 (0 to 130 m), using Eq. (12) and 95% confidence bands using Eq. (13). None of the chosen series revealed significant correlation at 95%
” 200 DAY
Water Management 32 (1997) 111-129
Fig. 11. Spatially averaged evapotranspiration (SAET) for measurements and standard deviation of the measured soil water.
intervals of 1, 3, 6, 14, and 21 days
confidence interval for any lag except ET4 and TET series (lag 1, 10 m). This was caused by the presence of a trend. Therefore, again first order transformed series were used to calculate the PACF. Partial autocorrelation functions for earlier transformed and untransformed series fell within the 95% confidence bands, indicating no spatial correlation at any lag. The partial autocorrelation functions decayed faster than the corresponding autocorrelation function, which is common in autoregressive processes. It is impossible to determine the order of process in the absence of significant correlation and again sample spacing of less than 10 m is required to achieve this goal.
In preparation for time series analysis, daily ET was calculated as the average ET of the 27 locations along the run (Fig. 6 through 91, and the results are plotted as time series (SAET, Fig. 11). Due to small changes in water content associated with single day ET and the limited precision of the neutron probe, especially near the surface, the water balance method has usually been restricted to measurement of ET over several day periods (Carrijo and Cuenca, 1992). Therefore, the water balance was also performed for 3, 6, 14, and 21 days interval. Daily neutron probe measurements resulted in negative ET particularly at the beginning and at the end of the growing season, whereas, the 3 day interval resulted in negative ET only at the beginning of the cropping season. Measurement intervals of 6 days always gave ET greater than 1.4 mm (standard deviation of probe measured soil water storage, Fig. 11). Therefore, the neutron probe can be used to estimate ET at an interval of 6 days at the beginning of cropping season and even on a daily basis for about 44 days (194 to 237 day) in the middle of the growing season, when estimated ET is greater than 1.4 mm. Partial ET series (194 to 237 day), ET values above 1.4 mm, was used to determine the temporal correlation structure of daily measured ET.
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Fig. 12. Autocorrelation function (ACF) for spatially average daily evapotranspiration and corresponding 95% confidence band.
(SAET, day 194 to 237)
In order to remove the trend and satisfy second order stationarity criteria, partial SAET series (comprising of daily ET values from day 194 to 237) was transformed as difference of order one. The autocorrelation was negative at lag 1 for SAET series which caused oscillation (Fig. 12). The autocorrelation function at lag 1 (1 day) was significantly correlated (95% confidence level), and decayed slowly indicating a dominant autoregressive component. Actual order and nature of this series, however, would be determined using its partial autocorrelation function.
4.6. Partial autocorrelation
Similar to autocorrelation function, the estimated partial autocorrelation function for SAET series showed significant correlation at lag 1 (1 day), Fig. 13. The partial
95% CONFIDENCE BAND
Fig. 13. Partial autocorrelation 194 to 237) and corresponding
functions (PACF) for spatially 95% confidence band.
autocorrelation also exhibited slow decaying indicates the presence of a moving average slow decay pattern for both autocorrelation indicating a presence of both autoregressive the underlying stochastic process can be spatially averaged evapotranspiration series,
32 (1997) 111-129
dominated by damped exponentials, which component. The SAET series depicted a and partial autocorrelation functions, thus and moving average components. Hence, characterized by ARMA(l,l) model for SAET.
Evapotranspiration of a furrow irrigated bean crop, estimated from change in soil moisture using a neutron moisture probe, was stochastic in space and time. The trend of decreasing ET along the row appeared and steepened as the growing season progressed because irrigation and moisture distribution were non-uniform. In addition to the trend, there was local spatial variation in ET along the 265 m row. The mean seasonal ET estimated from the neutron probe measurements was about 20% less than the ET estimated using the production function. This difference may be caused by bias in the probe readings at the 15 cm depth and use of the same calibration curve for all measurement depths. The chosen spatial series was not significantly (95% level) correlated at 10 m. Thus, each sample was independent from its neighbor and provided maximum new information. However, sample spacing of more than 10 m could be used to estimate spatial ET, if the sampling points constitute a large sample. But, to identify the order and spatial correlation structure of ET along the row, a sample spacing of less than 10 m is required. The neutron probe can be used to estimate spatially averaged ET on a daily basis for about 44 days in the middle of the growing season, when measured crop ET is more than 1.4 mm (the neutron probe measured standard deviation of soil moisture storage), and at an interval of 6 days at the beginning of the cropping season. Similarly, except for the first 21 days, the probe can be used to estimate ET at an interval of 3 days. This suggests that the neutron probe measurements could be used to estimate ET at any interval, if the change in measured moisture content exceeds the standard deviation of probe measured soil moisture storage. Spatially averaged daily ET series (day 194 to 237) showed significant correlation at lag 1 (1 day). On the basis of autocorrelation and partial autocorrelation functions, this series can be characterized by an autoregressive moving average (1,l) model. The temporal dependence can be used to forecast ET one day in the future. Spatial variability of total ET within the chosen time interval reflects the variability in irrigation requirement, since ET is the major contributor to irrigation requirement. Using the water balance equation and forecasted ET, irrigation requirement can be predicted one day in advance. Therefore, stochastic irrigation scheduling can be performed knowing the spatial variability of ET, and the temporal correlation structure of ET. Knowing the irrigation requirement one day in advance, farmers could order the required deliveries, and optimize furrow irrigation designs (flow rate and cutoff time).
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Acknowledgements The authors would like to thank Hajime Ito, Sophie Lamacq, Ken Tarboton and Mike Mata for their help in conducting the field experiment. The authors are also thankful to the Rotary International for sponsoring graduate studies.
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