Effective Rainfall Estimation Methods

EFFECTIVE RAINFALL ESTIMATION METHODS

By Avinash S. Patwardhan,1 John L. Nieber,2 and Eldon L. Johns,3

Member, ASCE

ABSTRACT: Numerous methods for estimating effective rainfall have been proposed in the past, including: direct measurement techniques; empirical methods; and soil water balance methods. The best estimates of effective rainfall can be obtained by conducting soil water balance computations. A soil water balance model (SWBM) for estimating effective rainfall was used to test the accuracy of the United States Department of Agriculture Soil Conservation Service (USDA-SCS) and the Hershfield effective rainfall estimation methods for a well-drained soil and for a poorly drained soil. Estimates of mean annual monthly effective rainfall by the USDA-SCS and estimates of mean annual growing season effective rainfall by the Hershfield method were found to compare closely with estimates from the SWBM for the well-drained soil but not for the poorly drained soil. Effective rainfall estimates by these two methods for either soil condition did not compare well with the SWBM estimates for annual events with return periods higher than the mean annual event.

INTRODUCTION

In the design and operation of irrigation systems it is becoming increasingly important to account for the contribution made by natural rainfall in crop production. Natural rainfall can contribute significantly in meeting consumptive use requirements of crops, provided the knowledge of effective rainfall is available. In the context of this paper, the term effective rainfall is taken to mean the portion of total rainfall that assists in meeting the consumptive use requirements of growing crops. This definition contrasts with the conventional definition used in hydrology where the term effective rainfall means that portion of total rainfall that contributes to runoff.

Numerous methods exist for estimating effective rainfall, including: direct field monitoring techniques; empirical techniques (equations, tables, charts); and soil water balance methods. The direct field monitoring techniques and the empirical techniques received most of the attention in terms of research and applications prior to the 1970s. As computational facilities have become increasingly accessible to researchers and practitioners the soil water balance methods have increased in popularity (Saxton et al. 1974; Sands et al. 1982). Detailed reviews of methods used in estimating effective rainfall have been presented by Dastane (1974) and Patwardhan et al. (Unpublished report, 1986).

Soil water balance techniques can best characterize effective rainfall for a given location. The technique accounts for all necessary components of the soil water balance (rainfall, runoff, infiltration, evapotranspiration, and

'Formerly Grad. Res. Asst., Dept. of Agric. Engrg., Univ. of Minnesota. Presently Agric. Engr., Aqua Terra Consultants, Mountain View, CA 94043.

Assoc. Prof., Dept. of Agric. Engrg., Univ. of Minnesota, 1390 Eckles Ave., St. Paul, MN 55108.

3Hydr. Engr., Bureau of Reclamation, Engrg. and Res. Center, Denver Federal Center, Denver, CO 80225.

Note. Discussion open until September 1, 1990. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on April 20, 1988. This paper is part of the Journal of Irrigation and Drainage Engineering, Vol. 116, No. 2, March/April, 1990. ©ASCE, ISSN 0733-9437/90/0002-0182/ $1.00 + $.15 per page. Paper No. 24519.

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deep percolation) and therefore has the flexibility to adapt to various climatic and soil conditions.

There are at least three potential uses for soil water balance models with respect to the estimation of effective rainfall. These include: (1) Real-time estimation of effective rainfall for operation of irrigation systems; (2) design of irrigation systems with the consideration of effective rainfall; and (3) evaluation of methods currently used for the prediction of effective rainfall.

The purpose of this paper is to assess the predictive accuracy of two empirical effective rainfall estimation methods. These methods include the USD A Soil Conservation Service method (USDA-SCS: "Irrigation" 1967) and the Hershfield method (1964). A soil water balance model was used in the performance of the assessment.

EFFECTIVE RAINFALL ESTIMATION METHODS

All effective rainfall estimation methods are based on representations and varying degrees of simplification of the hydrologic cycle. The processes included are: rainfall; irrigation; interception; infiltration and runoff; evapo-transpiration; redistribution (downward) of soil water; and deep percolation.

The equation for the conservation of mass (water) in the soil profile is expressed as:

Ay = R + IR - {I + Q + ET + DP) (1)

where AV = the change in soil water storage (in mm of water); R = rainfall (mm); IR = irrigation amount (mm); / = the interception loss (mm); Q = runoff (mm); ET = evapotranspiration (mm); and DP = deep percolation (mm). All of these quantities occur over a time period Af.

Effective rainfall (ER) is defined in the model as being that portion of total rainfall that infiltrates into the soil profile and does not contribute to deep percolation. This can be expressed as

ER = R - (7 + Q + DP) (2)

Hershfield Nomograph Hershfield (1964) developed a nomograph for estimating mean annual

growing season ER using mean annual seasonal rainfall, mean annual seasonal ET, and net irrigation application amount. The method assumes that the irrigation application is uniform throughout the growing season for each time of irrigation. The nomograph was developed using soil water balance calculations based on 50 years of daily rainfall data for 22 weather stations representing every geographical region of the contiguous United States. The method also provides for the estimation of seasonal ER for frequencies other than the mean annual value.

Soil Conservation Service Method The Soil Conservation Service effective rainfall estimation method (USDA-

SCS) is described in detail in USDA-SCS ("Irrigation" 1967) and appears in several other publications (Pair et al. 1975). The USDA-SCS ER method was developed with water balance calculations based on 50 years of weather data from 22 stations within the United States. The 22 stations used in the USDA-SCS method are identical to those used by Hershfield (1964). The

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TABLE 1. Reference Sources for Equations Used for Developing SWBM

Hydrologic process

0) Interception Evapotranspiration Runoff Deep percolation

Reference of equation used (2)

Jong and Cameron (1979) Smith and Williams (1980) Smith and Williams (1980) Williams et al. (1985)

USDA-SCS method is presented in tabular form and is intended to be used for estimating mean annual monthly effective rainfall based on mean annual monthly rainfall and mean annual monthly evapotranspiration. The method does not explicitly account for geographical location, soil type, nor for event frequency.

The upper limit of mean annual monthly ER is the mean annual monthly ET according to the USDA-SCS definition. This upper limit condition can be expressed as

ER^ET (3)

To estimate monthly ER for frequencies other than the mean annual condition (return period of approximately 2 years) the corresponding return period monthly rainfall is used (USDA-SCS: "Irrigation" 1967). Since monthly ET does not vary significantly from year to year it is not necessary to determine ET on a frequency basis; a mean annual value is sufficient.

Soil Water Balance Model The soil water balance model (SWBM) is physically based, but employs

the simple hydrologic process equations that have been used in models such as a field scale model for chemical, runoff, and erosion from agricultural management systems (CREAMS) (Knisel 1980) and simulator for water resources in rural basins (SWRRB) (Williams et al. 1985). The variables on the right-hand side of Eq. 1 are simulated using equations and methods that have been incorporated in models developed for other purposes (Richardson and Wright 1984; Knisel 1980; Williams et al. 1985). The reference sources for the equations used to simulate all of the hydrologic variables represented in Eq. 1, except for the rainfall and irrigation components, are summarized in Table 1. The reader should refer to these references for details on the rational basis for the selected equations and the method of applying the equations. A detailed description of the SWBM is presented by Nieber and Pat-wardhan (1988).

Time series of weather variables including rainfall, solar radiation, and air temperature were generated using the model WGEN developed by Richardson and Wright (1984). This model generates the weather variables on a daily basis. The model accounts for the cross correlation of weather variables and employs a stochastic model with cross-correlated and location-dependent parameters.

Irrigation was implemented in the model through the addition of irrigation water to the soil profile whenever a specified level of soil water deficit was reached. The net irrigation amount was set so that the soil water deficit was reduced to zero on the day of irrigation. The specified level of soil water deficit at which irrigation occurs is an input parameter for the model.

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METHOD OF ANALYSIS

Since the period of interest for estimating effective rainfall is the growing season, the water balance calculations were limited to the period (for the northern hemisphere) beginning April 1 and ending October 31 of each year. The soil water status on April 1 is therefore a model input parameter. Rainfall data and weather variables used in estimating ET are not usually readily available for time intervals less than one day. Therefore, to allow the model to be widely used, a daily time step was selected as the basis for all model computations.

The soil water balance model was applied to 50 years of weather data synthesized for each of 22 locations in the United States using a model for generating daily weather variables (WGEN model) (Richardson and Wright 1984). Each of these 22 locations were in the vicinity of one of the 22 stations evaluated in the USDA-SCS and Hershfield methods. Only data for the period beginning on April 1 and ending on October 31 were used in the simulations. It was assumed that the soil profile was at field capacity on April 1 of each year.

Water balance calculations were performed for two distinct soil conditions. One soil condition represented a highly permeable, well-drained soil with a curve number of 42 (antecedent moisture condition /). The other soil represented a low permeability, poorly drained soil with a curve number of 90 (antecedent moisture condition /). The simulations with the SWBM yielded monthly rainfall, monthly ET, and monthly ER for each of the seven months of the 50 years at each of the 22 stations.

In all comparisons of the SWBM results to effective rainfall estimates by the USDA-SCS and Hershfield methods the net irrigation input was set to 76.2 mm. To derive ER estimates using the USDA-SCS method, values of monthly rainfall and monthly ET derived from the SWBM simulations were used as input. To derive ER estimates using the Hershfield method, values of seasonal rainfall and seasonal ET derived from the SWBM simulations were used as input.

Two measures of effective rainfall were used in comparisons of the SWBM and the USDA-SCS methods. The first is the mean annual monthly ER amount. The SWBM estimate of mean annual monthly ER was calculated by averaging the SWBM monthly ER results over 50 years and 22 stations for each month. The USDA-SCS estimates of mean annual monthly ER were determined using the mean annual monthly rainfall and mean annual monthly ET from the SWBM calculations as input.

The second measure of effective rainfall is the monthly effective rainfall. The monthly ER values calculated by the SWBM for the 50 years and 22 stations contain the variations in ER resulting from event frequency and climate. The USDA-SCS method does not account explicitly for event frequency nor for climatic variations due to location. To test the adequacy of the USDA-SCS method to account for event frequency and location dependence, the USDA-SCS method was applied with the monthly rainfall and monthly ET computed with the SWBM as input. The computed values from the SWBM yielded 50 annual monthly values for each of the 22 stations for the months of April-October. Thus, there were 1,100 data values for each of the seven months considered in the analysis. The SWBM ER predictions

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used in the comparison included both the unmodified predictions and those predictions modified using Eq. 3.

Two measures of effective rainfall were used in comparison of the SWBM and Hershfield methods. Both measures are for growing season effective rainfall. The growing season was assumed to begin on April 1 and end October 31. The first measure is referred to as the mean annual seasonal effective rainfall. This measure is the average of seasonal ER over all 50 years and 22 stations. The second measure was calculated by averaging the seasonal effective rainfall over all 50 years for each of the 22 stations, and will be referred to as the station average mean annual growing season ER. This second measure provided an assessment of the sensitivity of mean annual seasonal ER to mean annual seasonal rainfall.

To calculate the mean annual growing season ER with the Hershfield method, the mean annual growing season rainfall and ET calculated by the SWBM were used as input to the Hershfield nomographs. To calculate the station average mean annual growing season effective rainfall with the Hershfield method, the SWBM estimates of station average mean annual rainfall and ET were used as input to the Hershfield nomographs.

The USDA-SCS method was also used to derive estimates of growing season effective rainfall. The USDA-SCS estimate of seasonal effective rainfall was calculated by summing the seven USDA-SCS estimates of monthly effective rainfall in any given year for each location. The USDA-SCS estimate of mean annual growing season effective rainfall was derived by averaging the USDA-SCS estimate of seasonal effective rainfall over the 50 years and 22 stations. The USDA-SCS estimate of station average mean annual growing season effective rainfall was then derived by averaging the USDA-SCS estimate of seasonal effective rainfall over the 50 years for each station.

RESULTS AND ANALYSIS

The mean annual monthly rainfall, mean annual monthly ET, and the mean annual monthly effective rainfall calculated by the SWBM and estimated by the USDA-SCS method are presented as histograms in Fig. 1 for the well-drained soil and in Fig. 2 for the poorly drained soil. Under the conditions assumed in the analysis it is seen that for both soil conditions the mean annual monthly ER is a substantial part of mean annual monthly rainfall.

From Fig. 1 it can be seen that for the months of April, May, September, and October the mean annual monthly ER, as estimated by the SWBM model, exceeds the USDA-SCS estimates and the mean annual monthly ET. This means that rainfall from a given month might be effective in the next month. In essence, it is carryover soil water. Application of Eq. 3 to the £7? predictions for the months of April, May, September, and October results (not shown) in fair agreement between the SWBM and the USDA-SCS estimates.

For the months of June, July, and August the SWBM estimates of mean annual monthly ER exceed the results given by the USDA-SCS method but are less than the mean annual monthly ET.

For the poorly drained soil, the USDA-SCS estimates of mean annual monthly ER exceed the SWBM estimates for all the months as seen in Fig. 2. The USDA-SCS estimates are significantly higher than the SWBM estimates. After application of Eq. 3 to the SWBM estimates, there is no change

186


vmm RAIN

IASV\M ET

P777J ER-SWBM

frSSH ER-SCS

APRIL MAY JUNE JULY AUG SEPT OCT

FIG. 1. Comparison of SWBM and USDA-SCS Predictions of Mean Annual Monthly Effective Rainfall. SWBM Predictions Were Derived for Case of Well-Drained Soil {CN = 42)

Ezza R A I N

KW?I ET

E a ER-SWBM

^ ER-SCS

APRIL MAY JUNE JULY AUG SEPT OCT

FIG. 2. Comparison of SWBM and USDA-SCS Predictions of Mean Annual Monthly Effective Rainfall. SWBM Predictions Were Derived for Case of Poorly Drained Soil (CN = 90)

in the SWBM estimates of mean annual monthly effective rainfall. The estimates of mean annual effective rainfall did not change since these values were all less, than the corresponding mean annual monthly ET.

Monthly ER will depend on the weather conditions that prevail in a given year as well as on the climatic characteristics of a given location. Due to this dependence, it was considered necessary to examine the versatility of the USDA-SCS method in estimating monthly ER for each of the seven months in the 50-year sequences for each of the 22 geographical locations.

Comparisons of the SWBM and the USDA-SCS monthly effective rainfall estimates for the months of June-August are presented in Figs. 3 and 4 for the well-drained soil condition and in Fig. 5 for the poorly drained soil condition. There are 1,100 data points on the plot for each month. The results in Fig. 3 are for the unmodified SWBM predictions and the results in Fig. 4 are for the SWBM predictions modified with Eq. 3. The results in Fig. 5 are for both the unmodified and modified SWBM predictions, because for


jr 1 2 3

SOS (102 mm) July

"0 1 2 3 SCS (102 mm)

October

FIG. 3. Comparison of SWBM and USDA-SCS Predictions of Monthly Effective Rainfall. SWBM Predictions Were Derived for Case of Well-Drained Soil (CN = 42). SWBM Predictions Are Not Limited by Monthly ET

the poorly drained soil condition there was no difference between the two predictions.

As seen in Fig. 3, the USDA-SCS estimates of monthly effective rainfall fall below the SWBM predictions. The USDA-SCS estimates are closest to the SWBM predictions for small monthly rainfall amounts, but diverge from the SWBM predictions as monthly rainfall increases. The maximum difference between the unmodified SWBM predictions and the USDA-SCS estimates for the seven months shown was found to be 170 mm in August.

The modification of the SWBM predictions with Eq. 3 produced a somewhat better agreement with the USDA-SCS estimates, as seen in Fig. 4. The maximum difference between the modified SWBM predictions and the USDA-SCS estimates for the seven months shown was found to be 46 mm in July.

In contrast to these results for the well-drained soil condition, the results for the poorly drained soil condition shown in Fig. 5 illustrate that the SWBM


Mr r i i i

/ -

-

> 1 2 3

SCS(102 mm) June

1 2 3 SCS (102 mm)

October

FIG. 4. Comparison of SWBM and USDA-SCS Predictions of Monthly Effective Rainfall. SWBM Predictions Were Derived for Case of Well-Drained Soil (CN = 42). Upper Limit of SWBM Predictions Is Monthly ET Values

predictions are generally less than the USDA-SCS estimates. The maximum difference between the SWBM predictions and the USDA-SCS estimates was found to be - 8 0 mm in July.

The mean annual growing season effective rainfall estimates using the SWBM, USDA-SCS, and the Hershfield methods are presented in Table 2. The USDA-SCS and Hershfield estimates are close as expected. Two predictions by the SWBM are given in the table. The first does not consider constraints on the ER estimate, whereas the second considers the constraint given by Eq. 3. It is seen that the SWBM predictions exceed the USDA-SCS estimates for the well-drained soil condition, whereas for the poorly drained soil condition the SWBM predictions are less than the USDA-SCS estimates. These comparisons are consistent with the comparisons presented earlier for the mean annual monthly ER estimates.

The dependency of mean annual seasonal effective rainfall on mean annual


1 2 SOS (102 mm)

Ap r i l

3 0 1 2 SCS(102 mm)

May

3 0 1 2 SCS (102 mm)

June

3

1 2 SOS (102 mm)

July

0 1 2 SOS (102 mm)

August SCS (10 mm) September

FIG. 5. Comparison of SWBM and USDA-SCS Predictions of Monthly Effective Rainfall. The SWBM Predictions Were Derived for Case of Poorly Drained Soil (CN = 90). Graphs Apply to Unmodified and Modified SWBM ER Predictions

TABLE 2.

Soil type (D

Well-drained soil (CN = 42) Poorly drained soil (CN = 90)

Seasonal Effective Rainfall

Seasonal Effective Rainfall (mm)

SWBM" (2)

429 163

SWBM" (3)

330 163

SCS (4)

309 308

Hershfield (5)

322 322

"Unmodified. 'Modified by constraining estimates with Eq. 3,

190


o SWBM (CN-42) + SWBM (CN-90)

R a i n f a l l (mm)

FIG. 6. Comparison of SWBM, USDA-SCS, and Hershfield Predictions of Station Average Mean Annual Growing Season Effective Rainfall

seasonal rainfall is illustrated in Fig. 6. Here, one prediction of mean annual seasonal ER is given for each of the 22 stations. As defined previously, this mean annual growing season value is referred to as station average seasonal ER. Estimates presented are for the SWBM (well-drained soil and poorly drained soil), the USDA-SCS method, and the Hershfield method. Both the unmodified and the modified (by Eq. 3) SWBM estimates are given for the well-drained soil. The unmodified and modified SWBM estimates for the poorly drained soil are essentially identical and so only one plot is given for that soil condition.

In Fig. 6 the USDA-SCS estimates and the Hershfield estimates are seen to be in close agreement. This agreement is to be expected since both methods were derived from the same data sets and with a water balance modeling procedure assuming a well-drained soil condition.

The SWBM predictions are seen to be sensitive to soil drainage condition. For the well-drained condition, the unmodified SWBM predictions of seasonal ER are well above the USDA-SCS estimates. With correction of the SWBM predictions for the well-drained soil condition, the SWBM predictions are in fairly good agreement with the USDA-SCS estimates. For the poorly drained soil condition, the unmodified SWBM predictions fall well below the USDA-SCS and Hershfield estimates. This finding is a verification of an important conclusion made by Dastane (1974). His conclusion was that the USDA-SCS method is applicable to areas having soils with characteristically high water intake rates relative to the characteristic rainfall intensity.

The USDA-SCS method and the nomograph developed by Hershfield (1964) ignore soil type. It can be further noted from Table 2 that seasonal ER, as predicted by the USDA-SCS and Hershfield methods, are in close agreement. This result is expected as USDA-SCS method and the Hershfield nomograph use identical 22 locations and 50 years of weather data for estimating ER.


On a conceptual basis the discrepancies in results of ER as estimated by SWBM and USDA-SCS methods could be attributed to the following:

1. The USDA-SCS method is not sensitive to soil type. 2. The USDA-SCS method does not directly account for carryover soil water. 3. The USDA-SCS method does not account explicitly for event frequency

nor for location dependent climatic characteristics.

Similar reasons can be outlined for the dissimilar mean annual seasonal ER results obtained by the SWBM method and the Hershfield nomograph. Also, the soil water balance computations used for developing the Hershfield nomograph used a constant daily ET value (Hershfield 1964). The method thus ignores the dynamic nature of climatic conditions, which have a direct influence on daily crop water use. Using constant daily ET values can lead to either overestimation or underestimation of ER, depending on the period of growing season.

As new restrictions on the use of quality water resources develop, in the future it will become increasingly important to more accurately estimate the design ER for situations specific to frequency of event and to location. In addition, the operation of irrigation systems will require the real-time estimation of ER on different time scales including weekly, monthly, and seasonal. To provide estimates for these quantities it will be necessary to develop new prediction tools. Established ER estimating methods including the USDA-SCS method and the Hershfield method will not be sufficient.

To meet future requirements of effective rainfall estimation, the new prediction tools should be based on the use of computers that perform the many calculations involved in solving the equation for the water balance of an irrigated field. With the ready availability of microcomputers, there is little excuse to limit oneself to the employment of charts and tables alone.

In the formulation of the new prediction tools, the mathematical representation of the water balance components can range widely in complexity. Of course, the simplest representation will facilitate faster computation and a minimum of data input. A more complex representation may provide more accurate results, but this will only occur when an extra effort is made to provide the model with high-quality data for an expanded input data set.

The computer-based water balance model is only one part of the overall strategy for prediction of effective rainfall. In addition, the analyst needs to make maximum use of available field experiences and observations to temper simulated results.

SUMMARY AND CONCLUSIONS

A soil water balance model (SWBM) for estimating effective rainfall is applied in evaluating the accuracy of two established effective rainfall estimation methods. The two methods evaluated are the USDA-SCS method ("Irrigation" 1967) and the Hershfield (1964) method.

The predictions of mean annual monthly ER by the USDA-SCS method and the SWBM are in fairly good agreement for well-drained soil conditions. However, the USDA-SCS method overpredicts ER for the case of poorly drained soils when compared to the SWBM method. This result is in agreement with the comments made by Dastane (1974) regarding the conditions


for applicability of the USDA-SCS method. The conditions stated by Dastane are that the USDA-SCS method is applicable to areas having soils with characteristic intake rates that are high relative to the characteristic intensity of local rainfall. Estimates of seasonal ER by the Hershfield method and the USDA-SCS method are well above those given by the SWBM for poorly drained soils, but estimates by these two methods are nearly the same as those from the SWBM for well-drained soil conditions.

When using the USDA-SCS method to estimate monthly ER for various frequency events it is found that there are large discrepancies between the USDA-SCS estimate and the SWBM prediction. This is because the USDA-SCS method was not developed to specifically handle frequencies other than that for the mean annual condition.

Both USDA-SCS and Hershfield methods average soil type, climatic conditions, and soil water storage to estimate ER. This averaging limits the usefulness of these methods for irrigation planning for specific locations. New effective rainfall prediction tools are needed to more closely simulate the dynamic nature of soil water and weather conditions in the field. The SWBM presented in this paper is an example of a relatively simple model that will fill this need for new tools. Since the model is physically based and accounts for the dynamic nature of soil water and weather changes on a daily basis it should more accurately represent actual field conditions than either the USDA-SCS or Hershfield methods.

APPENDIX. REFERENCES

de Jong, R., and Cameron, D. R. (1979). "Computer simulation model for predicting soil water content profiles." Soil Sci., 128(1), 41-48.

Dastane, N. G. (1974). "Effective rainfall in irrigated agriculture." Irrigation and Drainage Paper No, 25, Food and Agr. Organization, United Nations, New York, N.Y.

Hershfield, D. M. (1964). "Effective rainfall and irrigation water requirements." J. oflrr. and Drain. Div., ASCE, 90(2), 33-47.

"Irrigation water requirements." (1967). Tech. Release No. 21, United States Dept. of Agr., Soil Conservation Service, Washington, D.C.

Knisel, W. G., ed. (1980). "CREAMS. A field-scale model for chemicals, runoff, and erosion from agricultural management systems." Conservation Res. Report No. 26, U.S. Dept. of Agric, Washington, D.C.

Nieber, J. L., and Patwardhan, A. S. (1988). "Estimation of effective rainfall." Research project completion report, Contract No. USDI-5-CR-81-07420, U.S. Dept. of Interior, Bureau of Reclamation, Feb., Denver, Colo.

Pair, C. H., et al. (1975). Sprinkler irrigation. Fourth Ed., The Sprink. Irrig. Assoc, Silver Spring, Md.

Richardson, C. W., and Wright, C. A. (1984). "WGEN: A model for generating daily weather variables." ARS-8, U.S. Dept. of Agr., Agric. Res. Service, Belts-ville, Md.

Sands, G. R., Moore, I. D., and Roberts, C. R. (1982). "Supplemental irrigation of horticultural crops in humid regions." Water Resour. Bulletin, 18(5), 831-839.

Saxton, K. E., Johnson, H. P., and Shaw, R. H. (1974). "Modeling evapotrans-piration and soil moisture." Trans, of the American Society of Agric. Engrs., 17(4), American Soc. of Agric. Engrs., 618-621.

Smith, R. E., and Williams, J. R. (1980). "Simulation of the surface water hydrology." CREAMS. A field-scale model for chemicals, runoff, and erosion from agricultural management systems, Chapter 2, Knisel, W. G., ed., 13-35.

Williams, J. R., Nicks, A. D., and Arnold, J. G. (1985). "Simulator for water resources in rural basins," / . ofHydr. Engrg., ASCE, 111(6), 970-986.


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