Seann M. Reed



This document summarizes the soil-water budget methodology as applied to a study of the water resources of the Niger River basin. This work is part of the FAO/UNESCO Water Balance of Africa project.

Basin-averaged values of "surplus" water were estimated for 167 sub-basins of the Niger River basin for two scenarios: long term average climatic conditions and conditions during a historical period (short term) representative of dry conditions, July 1983 to Dec. 1990. The phrases long term and short term will be used throughout the remainder of this document to refer to these two cases. Surplus is water which does not evaporate or remain in soil storage and includes both surface and sub-surface runoff.

surplus = precipitation - evaporation - (change in storage)

Part I of this report to FAO/UNESCO describes the methodolgy and results for soil-water budgeting. Part II provides a technical description of how computations were made and includes a list of files and programs and there locations on the project CD-ROM. Part II has not been converted to HTML and has not been placed on our HTTP server because the tedious details are not expected to be of general interest (contact the author if you would like more details). A budgeting exercise created for the GIS in Water Resources class is available for those who wish to see how the calculations work.

Despite numerous uncertainties associated with simple soil-water budget model like the one used in this study, many researchers have applied this type of model to problems ranging from catchment scale studies to the global water balance and climate change scenerios (Thornthwaite, 1948; Budyko, 1956; Manabe, 1969; Mather, 1978; Alley, 1984; Willmott et al., 1985; Mintz and Walker, 1993; Mintz and Serafini, 1992). The popularity of this approach in hydrologic studies is most likely due to its simplicity. The simple bucket model used here requires minimal input data : precipitation, temperature, net-radiation, and soil-water holding capacity. For catchment scale applications, the soil-water budget approach for predicting runoff has shown reasonable agreement with measured annual flows, but has not shown good agreement with monthly flows (Mather, 1978). Although the soil-water budget roughly accounts for the soil storage effects on the time distribution of surface and sub-surface runoff, it has not demonstrated accurate prediction of monthly flow volumes.

For the Niger Basin project, a surface water routing model takes computed surplus as input and optimizes overland flow and streamflow loss parameters to predict monthly flows. For more information about the surface water routing model, link to Ye's Map-based Surface Water Modeling page.

Three difficult questions involved with simple soil-water budget calculations that are discussed to in this document are how to deal with situations when inadequate daily data are available and only monthly data can be used, how to derive spatially distributed estimates of climatic variables from point calculations, and how to estimate potential evaporation. Section IIc discusses time stepping (daily vs. monthly). Section V describes an approach for generating climate surfaces from point data and explores the sensitivity of water surplus calculations to two methods of estimating potential evaporation. Sections I-IV provide background material and a detailed description of the soil-water budget methodology. Sections VI briefly describes results.


Where detailed data about soil layers, depth to groundwater, and vegetation are not available, hydrologists have often resorted to simple bucket models and budgeting schemes to model near-surface hydrology. The soil-water budget is a simple accounting scheme used to predict soil-water storage, evaporation, and water surplus. A typical budgeting time step is one day. Surplus is the fraction of precipitation that exceeds potential evapotranspiration and is not stored in the soil. The simple model used here does not distinguish between surface and subsurface runoff, so surplus includes both. For the Niger project, the main purpose of calculating the water budget is to estimate surplus, which serves as input to groundwater and surface water flow models. With this in mind, the basic equation for calculating surplus is:


In Equation 1, S is surplus, P is precipitation, E is evaporation, w is soil moisture, and t is time. Horizontal motion of water on the land surface or in the soil is not considered by this model. Snow melt was ignored in the water budget computations for this study because temperatures throughout the study region remain above freezing throughout the year.

A major source of uncertainty in evaluating Equation 1 is estimating the evaporation. Estimation of evaporation is based upon knowledge of the potential evapotranspiration, available water-holding capacity of the soil, and a moisture extraction function. The concepts of water-holding capacity and the method for evaluating Equation 1 are discussed here, followed by a more detailed discussion of the potential evapotranspiration concept in Section III.


In order to calculate the soil-water budget, an estimate of the soil's ability to store water is required. Several terms are used by soil scientists to define the water storage capacities of soils under different conditions. The field capacity or drained upper limit is defined as the water content of a soil that has reached equilibrium with gravity after several days of drainage. The field capacity is a function of soil texture and organic content. The permanent wilting point or lower limit of available water is defined as the water content at which plants can no longer extract a health sustaining quantity of water from the soil and begin to wilt. Typical suction values associated with the field capacity and wilting point are -10 kPa (-0.1 bars) and -1500 kPa (-15 bars) respectively. Like water content, field capacity and permanent wilting point are defined on a volume of water per volume of soil basis. Given these two definitions, the water available for evapotranspiration after drainage ( or the available water-holding capacity ) is defined as the field capacity minus the permanent wilting point. Table 1 gives some typical values for available water-holding capacity.

Table 1: Water Retention Properties for Agricultural Soils

(Values Taken from ASCE,1990, Table 2.6, p. 21.)

Texture ClassField Capacity Wilting PointAvailable Capacity
Sand0.12 0.040.08
Loamy Sand0.14 0.060.08
Sandy Loam0.23 0.100.13
Loam0.26 0.120.15
Silt Loam0.30 0.150.15
Silt0.32 0.150.17
Silty Clay Loam0.34 0.190.15
Silty Clay0.36 0.210.15
Clay0.36 0.210.15

For budgeting calculations, it is useful to know the total available water-holding capacity in a soil profile. This value is typically expressed in mm and can be obtained by integrating the available water-holding capacity over the effective depth of the soil layer. A one meter soil layer with a uniform available water-holding capacity of 0.15 has a total available water-holding capacity of 150 mm. For the remainder of this paper, the term water-holding capacity means total available water-holding capacity in units of mm. Soil-water storage (mm) is denoted by w and water-holding capacity is denoted with w*. A large water-holding capacity implies a large annual evapotranspiration and small annual runoff relative to a small water-holding capacity under the same climatic conditions.


To estimate the actual evapotranspiration in the soil-water budget method many investigators have used a soil-moisture extraction function or coefficient of evapotranspiration beta which relates the actual rate of evapotranspiration to the potential rate of evapotranspiration based on some function of the current soil moisture content and moisture retention properties of the soil.

E = beta * PE (2)

Dyck, 1983, Table 1, (reprinted in Shuttleworth, 1993, Table 4.4.6) provides a summary of some moisture extraction functions used by different investigators. Mintz and Walker, 1993, Figure 5, also illustrates several moisture extraction functions. Many researchers agree that soils show the general pattern of behavior that moisture is extracted from the soil at the potential rate until some critical moisture content is reached when evapotranspiration is not longer controlled by meteorological conditions. Below this critical point, there is a linear decline in soil moisture extraction until the wilting point is reached. This type of behavior is illustrated by Shuttleworth, 1993, Figure 4.4.3 p. 4.46 and Dingman, 1994, Figure 7-21. Shuttleworth, 1993, notes that the critical moisture content divided by the field capacity is typically between 0.5 and 0.8. The type of moisture extraction function just described is commonly applied to daily potential evaporation values. A simpler function, beta = w/w*, has been applied to monthly values.

There are several drawbacks to using simple soil moisture extraction functions. Mintz and Walker, 1993, cite field studies that show beta may vary with potential evapotranspiration for a given soil wetness and beta may also vary with leaf-area index. In addition, the spatial variation of water-holding capacity is difficult to determine. A new and possibly better approach to determining the relationship between plant transpiration and potential evapotranspiration is to correlate beta with satellite-derived indices of vegetation activity so that beta will reflect plant growth stage and the spatial vegetation patterns. Gutman and Rukhovetz, 1996, investigate this possibility. Their approach still requires an estimate of potential evapotranspiration.


Soil-water budget calculations are commonly made using monthly or daily rainfall totals because of the way data are recorded. Computing the water balance on a monthly basis involves the unrealistic assumption that rain falls at constant low intensity throughout the month, and consequently surplus estimates made using monthly values are typically lower than those made using daily values. Particularly in dry locations, the mean potential evaporation for a given month may be higer than the mean precipitation, yet there is observed runoff, and budgeting with monthly values may yield zero surplus. For this reason, the use of daily values is preferred over monthly values when feasible. In this study, daily rainfall records were provided by FAO for a number of stations in the Niger basin; however, the spatial coverage of these stations is sparse in some areas, and it is difficult to interpolate daily rainfall over space.

Because of the difficulty in estimating daily rainfall at our regularly spaced computational points, and because a consistent basis for comparison is needed between the short term case (for which daily data are available) and the long term case (only monthly data are available), monthly data were used in this study. However, the use of monthly data does not yield enough surplus to match observed river flows in some areas of the Niger basin even with the assumption of zero overland flow and stream losses. To resolve this problem, a modification was made to the commonly used bucket model in which it is typically assumed that no surplus is generated until the soil is completely saturated with water; this assumption is not consistent with situations where the rainfall rate exceeds the infiltration rate of the soil. In our modified model, a fraction of the precipitation is extracted and declared runoff, before remaining precipitation is passed to the soil. This extraction of precipitation is represented by the P*alpha term in Equation 3 below. This scheme generates more runoff, enough to satisfy mass balance constraints in most areas of the Niger basin. Without this term, the model will predict zero runoff during many months of the year when some observed streamflow actually occurs. The runoff extraction term roughly accounts for "event" or "quick" flow that cannot be modeled using monthly averaged values.

Equation 3 describes how soil moisture storage is computed.


In Equation 3, w(i) is the current soil moisture, w(i-1) is the soil moisture in the previous time step, P is precipitation, PE is potential evaporation, alpha is the runoff extraction function, and beta is the soil-moisture extraction function. With monthly data, computations are made on a quasi-daily basis by assuming that precipitation and potential evapotranspiration for a given day are equal to their respective monthly values divided by the number of days in the current month. Several conditions apply when evaluating Equation 3: If w(i) drops below zero, then set w(i) equal to 0.01; if w(i)>w* where w* is the water-holding capacity, then the surplus for that day is w(i)-w*+P*alpha.
The soil-moisture extraction function beta=w/w* was used for this study. Since there is no precedent for the use of a runoff extraction function (alpha), the formulation of this function was more speculative and deserves further study. In the mean time, a simple relationship, alpha = w/w*, was used and yielded enough surplus to satisfy mass balance constraints in the surface flow routing model.


If the initial soil moisture is unknown, which is typically the case, a balancing routine is used to force the net change in soil moisture from the beginning to the end of a specified balancing period (N time steps) to zero. To do this, the initial soil moisture is set to the water-holding capacity and budget calculations are made up to the time period (N+1). The initial soil moisture at time 1 (w(1)) is then set equal to the soil moisture at time N+1 (w(N+1)) and the budget is re-computed until the difference (w(1) - w(N+1)) is less than a specified tolerance.


The soil-water budget is most easily applied at single points in space dictated by the location of climate stations where water-holding capacity can be measured or estimated, but the result of these point calculations must be interpolated over space in order to get a surplus volume. An alternative approach taken in this study was to use pre-computed climate, net radiation, and water-holding capacity grids, augmented with climate station measurements, to calculate the soil-water budget at each point on a 0.5 by 0.5 grid. Using a single value for precipitation, temperature, and net radiation in each 0.5 cell seems reasonable at the monthly time scale. The water-holding capacity may vary considerably within each 0.5 cell, and the value used in budgeting calculations is only an average property of the cell.


One aspect of the soil-water budget that involves significant uncertainty and ambiguity is estimating potential evapotranspiration. Just the concept of potential evapotranspiration is ambiguous by itself, as discussed in the next section. Due to limited meteorological data, two simple methods for estimating potential evapotranspiration were considered for the Niger basin study, the Priestley-Taylor and Thornthwaite methods. For the short term simulation (July 1983 to December 1990), a global net radiation data set obtained from NASA facilitated making potential evapotranspiration estimates using the Priestley-Taylor method. For reasons discussed later in this paper, the Priestley-Taylor method is considered superior to the Thornthwaite approach; however, it is simpler to apply the Thornthwaite approach to long term average conditions and to selected historical periods because the global net radiation data used in this study are only available from July 1983 to June 1991. It would be nice to have consistent methods for estimating potential evapotranspiration over different time periods so that fair comparisons can be made. Because the Thornthwaite method is more easily applied over different historical time periods, determining whether there are significant differences between predicted runoff using the Priestley-Taylor and Thornthwaite methods is an important question. The conclusion is that there are significant differences and the Priestley-Taylor approach is better. For this study, the average net radiation over the eight year period when net radiation data were available was taken to be the long term average net radiation. Both of the Priestley-Taylor and Thornthwaite methods perform poorly in arid regions and the significance of this is briefly discussed.


Thornthwaite, 1948, first used the concept of potential evapotranspiration as a meaningful measure of moisture demand to replace two common surrogates for moisture demand temperature and pan evaporation. Potential evapotranspiration refers to the maximum rate of evapotranspiration from a large area completely and uniformly covered with growing vegetation with an unlimited moisture supply. There is a distinction between the term potential evapotranspiration and potential evaporation from a free water surface because factors such as stomatal impedence and plant growth stage influence evapotranspiration but do not influence potential evaporation from free water surfaces.

Brutsaert, 1982, notes on pp. 214 and 221 the remarkable similarity in the literature among observations of water losses from short vegetated surfaces and free water surfaces. He poses a possible explanation that the stomatal impedance to water vapor diffusion in plants may be counterbalanced by larger roughness values. Significant differences have been observed between potential evapotranspiration from tall vegetation and potential evaporation from free water surfaces. The commonly used a value of 1.26 in the Priestley-Taylor equation was derived using observations over both open water and saturated land surfaces. For the most part, the term potential evapotranspiration will be used predominantly in this paper and, as used, includes water loss directly from the soil and/or through plant transpiration.

An additional ambiguity in using the potential evapotranspiration concept is that potential evapotranspiration is often computed based on meteorological data obtained under non-potential conditions (Brutsaert, p. 214). In this study, temperature and net radiation measurements used for calculating potential evapotranspiration in dry areas and for dry periods will be different than the values that would have been observed under potential conditions. The fact that the Thornthwaite and Priestley-Taylor methods have exhibited weak performance at arid sites is related to this ambiguity because the assumptions under which the expressions were derived break down. Poor performance in arid regions is highly relevant to the Niger Basin study because of large arid areas in the northern part of the basin. This problem will be addressed a bit further during the detailed discussions of each method.

Although not used directly in this study, a brief review of the widely used Penman equation serves as a good starting point for discussing the estimation of potential evapotranspiration.


Two requirements for evaporation to occur are an energy input and a mechanism for the transport of water vapor away from saturated surface. In light of this, two traditional approaches to modeling evaporation are an energy budget approach and an aerodynamic approach. With the energy budget approach, the net radiation available at the surface (shortwave absorbed plus longwave emitted) must be partitioned between latent heat flux and sensible heat flux, assuming that ground heat flux is negligible. This partitioning is typically achieved using the Bowen ratio which is the ratio of sensible heat flux to latent heat flux. Approximating the Bowen ratio typically requires measurements of temperature and humidity at two heights. The aerodynamic approach typically involves a vapor transport coefficient times the vapor pressure gradient between the saturated surface and an arbitrary measurement height. Determination of the vapor transport coefficient typically requires measurements of wind speed, humidity, and temperature. Brutsaert, Chow et al., and Dingman, present equations for calculating the Bowen ratio and vapor transport coefficients. Without simplifying assumptions, energy budget and the aerodynamic methods require meteorological measurements at two levels.
In 1944, Penman combined the energy budget and aerodynamic approaches. Penman's derivation eliminates the need for measuring water surface temperature; only the air temperature is required. The resulting equation is as follows:

Er = (5)
Rn is net radiation (typical units are W/m2), lv is latent heat of vaporization (J/kg), row is density of water (kg/m3), K(u) is a mass transfer coefficient, es is saturated vapor pressure at air temperature, and e is the actual vapor pressure.

The Penman equation is a weighted average of the rates of evaporation due to net radiation (Er) and turbulent mass transfer (Ea). Provided that model assumptions are met and adequate input data are available, various forms of the Penman equation yield the most accurate estimates of evaporation from saturated surfaces. The "Evapotranspiration and Irrigation Water Requirements Manual," ASCE, 1990, offers a performance comparison of twenty popular methods for estimating potential evaporation. The top six rated methods in ASCE, 1990, are forms of the Penman equation (p.249).


Due to lack of data, it is not feasible to use the complete Penman equation to make potential evaporation estimates for the Niger Basin study. Because of their simplicity, the Thornthwaite and Priestley-Taylor methods for estimating potential evaporation are widely used in regional and global scale climatic water budget studies. In the Niger Basin project, the only input data currently available with adequate spatial coverage are temperature and net radiation. While other simple temperature and/or radiation methods are available and considered briefly later in this report, the Thornthwaite and Priestley-Taylor methods are the focus of this study.



A new global radiation data set makes using the Priestley-Taylor method a feasible option for estimating potential evapotranspiration at the scale of this study. Surface longwave and shortwave radiation flux estimates are available for a 96 month period extending from July 1983 to June 1991. The data are given on the ISSCP equal-area grid which has a spatial resolution of 2.5 at the equator. Darnell et al, 1992, assert that recent (last decade) advances in input data and flux estimation algorithms have greatly improved the ability to assess the radiation budget on a global scale. Improvements in the input data come from the International Satellite Cloud Climatology Project (ISCCP) and the Earth Radiation Budget Experiment (ERBE). According to Darnell et al., 1995, longwave flux estimates are accurate to within +/- 25 W/m2 and Whitlock et al. estimate the accuracy of shortwave estimates to be +/- 20 W/m2. The energy required to evaporate 1 mm/day of water is about 30 W/m2.


Global data sets of mean monthly temperature and precipitation interpolated to a 0.5 by 0.5 grid were obtained from Cort Willmott at the University of Delaware. These data are from the "Global Air Temperature and Precipitation Data Archive" compiled by D. Legates and C. Willmott. These monthly precipitation estimates were previously corrected for gage bias. Data from 24,635 terrestrial stations and 2,223 oceanic grid points were used to estimate the precipitation field. The climatology is largely representative of the years 1920 to 1980 with more weight given to recent ("data-rich") years (Legates and Willmott, 1990).

Daily precipitation and temperature estimates for stations around the globe are available on a "Global Daily Summary" CD-ROM produced by the National Climatic Data Center in Asheville, NC. The period of record for these data is 1977-1991 although not all stations have records for this entire period. The density of climate stations is quite high in some parts of the Niger River Basin but low in other areas. In addition, daily rainfall data for 191 stations in Niger and 41 stations in Mali were provided on floppy disk by FAO. The period of record for these stations varies, but the most recent date with available records from these stations is December 31, 1990. A more detailed discussion of the data is provided in Part II of this report.


A grid of water-holding capacity estimates was provided by FAO for the Niger Basin project. Values in this grid were compiled by an expert at FAO using information from the CD-ROM Digitized Soil Map of the World (FAO, UNESCO, 1974-1981). There 6 unique values of water-holding capacity (mm) in this grid: 0, 10, 30, 75, 125, and 200. The value zero is assigned to water bodies.



1. Creating precipitation surfaces

Monthly time series tables of precipitation and potential evaporation for the period July 1983 to December 1990 were estimated at each point on a regular mesh of 0.5 degree cells. This computational mesh was selected because long term mean monthly estimates of rainfall and temperature at these points were obtained from C.J. Willmott at the University of Delaware. The mean monthly values from Willmott were used in conjunction with thiessen polygons to create 0.5 surfaces of precipitation and potential evaporation for each month in the short-term study period. The basic methodology is described here and more details are provided in Part II of this report.
The initial approach for creating monthly precipitation surfaces was as follows: (1) Created thiessen polygons based on selected precipitation stations and used these polygons to associate each 0.5 computational cell with a given station. (2) Calculate the monthly value in cell i and historical month m with the following equation:

The subscript i indicates cell index where precipitation is being estimated and g indicates value at the gage. The superscript "mon" indicates a monthly average value obtained from the Legates and Willmott climatology. If the mean value at the gage is zero, then the ratio is set equal to one. The idea is to try to say something about the spatial distribution of rainfall in a historical month based on a spatial distribution of mean monthly rainfall created by an expert in rainfall interpolation. It turns out that this method yields poor results for dry months because the ratio of the mean value at a cell i to the mean value at the gage can be excessively high or low when dealing with small rainfall totals. To rectify this problem in a second round of calculations, the ratio of mean annual values at cell i to mean annual values at the gage were used. Using annual values, there is one adjustment ratio for each cell instead of 12 and the new equation is as follows.


An attempt to distribute daily rainfall values in a similar manner to the monthly values was made, but in locations where our precipitation gage network is sparse, this involves the poor assumption that individual rainfall events occur over very large areas. Thus, daily calculations made using this approach were not put to use as input to the flow simulation model. One approach to dealing with this problem would be to use a library of dimensionless daily time series distributions to distribute monthly rainfall totals for each cell throughout a month; however, because the merits of adopting this approach are unclear and because it involves a large degree of complication, it was not pursued.


1. Radiation Methods
a. Priestley-Taylor

In 1972, C.B. Priestley and R.J. Taylor showed that, under certain conditions, knowledge of net radiation and ground dryness may be sufficient to determine vapor and sensible heat fluxes at the Earth's surface. When large land areas (on the order of hundreds of kilometers) become saturated, Priestley and Taylor reasoned that net radiation is the dominant constraint on evaporation and analyzed numerous data sets to show that the advection or mass-transfer term in the Penman combination equation tends toward a constant fraction of the radiation term under "equilibrium" conditions. According to Brutsaert, 1982, Slatyer and McIlroy, 1961, first defined the concept of equilibrium evaporation as a state that is reached when a moving air mass has been in contact with a saturated surface over a long fetch and approaches vapor saturation -- thus causing the advection term in the Penman equation to go to zero. Both Slatyer-McIlroy and Priestley-Taylor considered the radiation term in the Penman equation to be a lower limit for the evaporation from a moist surface. The form of the evaporation equation developed by Priestley and Taylor is as follows, a constant (alpha) times Penman's radiation term.
Equating this expression to the combination equation reveals that the advection term must be a constant fraction of the radiation term if alpha is a constant.

Using micro-meteorological observations over ocean surfaces and over saturated land-surfaces following rainfall, Priestley and Taylor came up with a best-estimate of 1.26 for the parameter alpha. The fact that alpha is greater than one indicates that true advection-free conditions do not exist. Since 1972, several other researchers have confirmed that alpha values in the range 1.26-1.28 are consistent with observations under similar conditions. Some researchers have found significantly lower values for the alpha coefficient, but these coefficients were found for different types of surfaces (i.e. tall vegetation or bare soil as opposed to grass and open water). There have also been indications that the alpha coefficient may exhibit significant seasonal variation (Brutsaert, p. 221).

Priestley-Taylor estimates have shown good agreement with lysimeter measurements for both peak and seasonal evapotranspiration in humid climates; however, the Priestley-Taylor equation substantially underestimates both peak and seasonal evapotranspiration in arid climates. The advection of dry air to irrigated crops is likely to be greater in arid climates because large saturated areas are rare, resulting in a more dominant role of the advection term. A higher alpha coefficient may be required in arid climates (ASCE, 1990). Based on arid sites studied in ASCE, 1990, a value of alpha=1.7-1.75 seems more appropriate for arid regions. Shuttleworth, 1993, states that the Priestley-Taylor method is the "preferred radiation-based method for estimating reference crop evapotranspiration." Shuttleworth, 1993, notes that errors using the Priestley-Taylor method are on the order of 15% or 0.75 mm/day, whichever is greater, and that estimates should only be made for periods of ten days or longer.

b. Other Radiation Methods

The Jensen-Haise, FAO-24 Radiation, and the Turc method are all classified as radiation-based methods in ASCE, 1990, but the FAO-24 Radiation Method and the Turc method require basic information beyond temperature and net radiation mean humidity and mean wind speed for FAO-24 and mean humidity for the Turc method. The Jensen-Haise method does receive an overall ranking higher than the Priestley-Taylor method in ASCE 1990, Table 7.20; however, the Priestley-Taylor equation outranks the Jensen-Haise equation for humid climates. Predicted surplus volumes are much more sensitive to the accuracy of the potential evaporation estimate in humid climates than arid climates; therefore, the Priestley-Taylor approach is considered a better choice than Jensen-Haise for our application. It is also noteworthy that the Jensen-Haise, FAO-24 Radiation, and the Turc method all use the incoming shortwave radiation rather than net radiation as an input. In theory, net radiation is a better indicator of potential evaporation than shortwave radiation because variations in albedo and cloudiness have already been taken into consideration.

2. Temperature Methods
a. Thornthwaite Empirical Approach

Thornthwaite (1948, Wilm et al., 1944) developed an empirical equation for estimating potential evapotranspiraion from a reference grass surface that requires only mean monthly temperature and day length estimates as input. The regression equation was developed using data from lysimeter and small watershed water balance experiments at several sites scattered throughout the United States. Thornthwaite recognized that there is a more direct physical relationship between potential evaporation and net radiation than between potential evaporation and temperature, but foresaw correctly that sufficient radiation measurements or accurate calculations to reliably estimate potential evaporation would be difficult to come by for many years to come. Using temperature as a surrogate for net radiation involves the implicit assumptions that albedo is constant, the rate of evapotranspiration is not influenced by advection of moist or dry air, and that the Bowen ratio is constant. These conditions do not exist in arid and semi-arid regions except during short periods after regional rain storms (ASCE, 1990). The comparison and ranking of potential evaporation estimating methods in the ASCE Manual (Table 7.18) clearly shows the poor performance of the Thornthwaite methodology in arid climates.

Mintz and Walker discuss the fact that the Thornthwaite equation was developed for temperatures measured under potential conditions and only represents "true" potential evaporation when there is no soil moisture stress. Application in locations with soil moisture stress results in an "apparent" potential evaporation. The energy balance at the land surface and thus the surface temperature are altered under non-potential conditions. Mintz and Walker observe that dry areas have higher temperatures than wet areas at the same latitude and season and develop an empirical equation that relates the potential air temperature to the measured air temperature. Based upon measured air surface temperatures, the Thornthwaite method will overestimate the "true" potential evaporation in arid regions. Mintz and Walker assert that the largest difference between calculated "true" and "apparent" potential evaporation occurs in the central Sahara where "apparent" is 6.2 mm/day in July and the "true" is 5.5 mm/day.

Willmott et al., 1985 summarize the Thornthwaite evaporation equations as follows.
Potential evapotranspiration (PE) in (mm/month) without adjustment for day length is computed with:

where T is mean surface air temperuture in month i (°C) and I is the heat index defined in Equation 13 below. The exponent a in Equation 12 is a function of the heat index (I).


Monthly estimates of potential evapotranspiration calculated with Equation 12 need to be adjusted for day length because 30 day months and 12 hour days were assumed when this relationship was developed.
The adjusted potential evaporation accounting for month length and daylight duration is given by


where APE is in (mm/month), d is length of the month in days, and h is the duration of daylight in hours on the fifteenth day of the month.

During the course of this investigation, a FORTRAN code for computing water budgets called "WATBUG" obtained from Cort J. Willmott at the Universtiy of Delaware (Willmott, 1977) has been very helpful. The WATBUG code allows for daily or monthly budgeting and includes subroutines for balancing, computing day length given latitude, and calculating potential evaporation with the Thornthwaite equation. All or parts of the WATBUG routine have been used in this investigation, either in their original or modified form. An Avenue script for use with monthly data that uses the budgeting scheme of Equation 3 has also been written.

b. SCS Blaney-Criddle

This empirical temperature-based method requires only mean monthly temperatures and an estimate of the monthly percentage of annual daytime hours. Based upon the results in ASCE, 1990, Tables 7.18 and 7.19, this method outperforms the Priestley-Taylor method in all months and in the peak month for arid regions but exhibits poor performance in humid regions.

3. Results: Thornthwaite Vs. Priestley-Taylor

Figure 1 shows the spatial distribution of monthly average (90 months) potential evapotranspiration (PE) predicted using the two methods. ( Click here to view all of Figure 1.; Zoom in on top map , and bottom map in Figure 1.) There are clear differences in the PE spatial variations. The Priestley-Taylor PE estimates show decreasing trend from high values in the humid south portions to lower values in the arid north. The Thornthwaite estimates exhibit no consistent trend. On a basin average, the Thornthwaite PE (136.8 mm/month) is higher than the Priestley-Taylor PE (114.1 mm/month). Figure 2 shows the variation in monthly PE throughout the year (7 year average) for the cells identified in Figure 2. ( Click here to view all of Figure 2.; View top , middle , and bottom charts in Figure 2 respectively.) Moving south to north, seasonal trends of higher PE in the summer show up. This is related to the fact that most of the rainfall occurs in the summer and relatively less rainfall occurs during other parts of the year as you move north. Net radiation is higher in wet months because more solar radiation is absorbed. Moving south to north, Thornthwaite PE becomes larger relative to Priestley-Taylor PE. This is the reason that the Thornthwaite method actually yields higher average surplus despite having a higher basin-average PE (Figure 3). Figure 3 shows a comparison of mean annual surplus values computed from the soil-water budget using the two different estimates for PE. ( Click here to view all of Figure 3.; Zoom in on top , middle , and bottom maps in Figure 3 respectively.) Because the Priestley-Taylor PE is higher in the south where all the rainfall occurs, the actually evaporation estimated using this method is greater and the average surplus generated is smaller.
The surplus results presented in Figure 3 were computed using a slightly different methodology than that described in Equation 3. This earlier analysis used the methodology of C.J. Willmott written in the WATBUG FORTRAN code. The methodology can be summarized as follows:

if (P-PE) < 0 (16)
if (P-PE) > 0 (17)
If wi > w*, then surplus for that day is wi-w* and wi is set equal to w*. In addition to the fact that it does not inclued the rainfall extraction term, Equation 16 differs from Equation 3 because it involves the assumption that if precipitation is less than potential evaporation then all precipitation used to satisfy the potential demand is consumed immediately, before having a chance to enter the soil. This generates less surplus than the approach described in Equation 18 which is the same as Equation 3 without the rainfall extraction term.

Variables in Equation 18 are subject to the same constraints as in Equation 3.
As of 1992-93, Mintz and Walker used the Thornthwaite approach for PE calculations because they were skeptical about our knowledge of global net radiation, either from direct measurements or calculations, and thus didn't envision the application of the Penman, Budyko, or Priestley-Taylor approaches directly on a global basis. Mintz and Walker indicate that the Thornthwaite and Priestley-Taylor methods are consistent to within +/- 0.5 mm/day or 15 mm/month in any given month. However, Figure 2 indicates differences of up to 50 mm/month or more in Cell 3. Recall that the accuracy of the of the radiation fluxes are about +/-20-25 W/m2 which corresponds to about 0.7 mm/day or 21 mm/month. Shuttleworth puts the accuracy of the Priestley-Taylor method at about 0.75 mm/day.

It is clear from the discussion of errors above that potential evaporation estimation errors can easily be on the order of 1 mm/day. This seemingly small error can yield significant differences in surplus volumes, and surplus calculations are especially sensitive to the spatial pattern of potential evapotranspiration estimates. The seasonal pattern of PE may also play a significant role in surplus generation, but no definite conclusions have been made from this brief study.

Three reasons for favoring the Priestley-Taylor method over the Thornthwaite method are (1) net radiation has a more direct physical relationship to evaporation, (2) the Priestley-Taylor method receives a higher ranking than the Thornthwaite method in comparison to lysimeter estimates (ASCE, 1990), and (3) the spatial pattern of Priestley-Taylor PE estimates in the Niger basin are more sensible. The range of errors in net radiation estimates (in units of mm/day) are of similar magnitude to the differences in Priestley-Taylor and Thornthwaite PE.

Estimated basin mean surplus volume is very sensitive to method of calculating PE. The ratio of Thornthwaite basin mean surplus to Priestley-Taylor basin mean surplus is 1.7. The formulation of Equation 3 should be less sensitive to PE than the formulation of Equations 16 and 17.


Several figures are included here to try to illustrate some of the results. Figure 4 illustrates the mean annual surplus for the long term and short term cases.( Click here to view all of Figure 4.; Zoom in on top map , and bottom map in Figure 1.) Figure 5 shows the conversion of cell surplus values to watershed average values. Figure 6 illustrates the computed time series of surplus for several cells, and Figure 7 illustrates all components of the soil-water balance at a cell. ( Click here to view all of Figure 7.; Zoom in on precipitation , potential evaporation , evaporation , soil moisture , and surplus .) The names of the coverages and data files containing surplus results are provided in Part II, Section II of this report.


Alley, W.M., "On The Treatment Of Evapotranspiration, Soil Moisture Accounting, And Aquifer Recharge In Monthly Water Balance Models," Water Resources Research, 20,1137-1149,1984

ASCE, Evapotranspiration and Irrigation Water Requirements, Jensen, M.E., R.D. Burman, and R.G. Allen (editors), ASCE Manuals and Reports on Engineering Practice No. 70, 1990.

Brutsaert, W., Evaporation into the Atmosphere: Theory, History, and Applications, D. Reidel Publishing Company, Dordrecht, Holland, 1982.

Chow, V.T., D.R. Maidment, and L.W. Mays, Applied Hydrology, McGraw-Hill, Inc., New York, NY, 1988.

Darnell, W.L., W.F. Staylor, S.K. Gupta, N.A. Ritchey, and A.C. Wilber, "Seasonal Variation of Surface Radiation Budget Derived from ISCCP-C1 Data," J. Geophysical. Res., 97, 15741-15760, 1992.

Darnell, W.L., W.F. Staylor, S.K. Gupta, N.A. Ritchey, and A.C. Wilber, "A Global Long-term Data Set of Shortwave and Longwave Surface Radiation Budget," GEWEX News, 5, No.3, August 1995.

Dingman, S.L., Physical Hydrology, Prentice Hall, Inc., Englewood Cliffs, NJ, 1994.

Dyck, S., "Overview on the Present Status of the Concepts of Water Balance Models," IAHS Publ. 148, Wallingford, 3-19, 1983.

Gutman, G., and L. Rukhovetz, "Towards Satellite-Derived Global Estimation of Monthly Evapotranspiration Over Land Surfaces," Adv. Space Res., 18, No. 7, 67-71, 1996.

Legates, D.R., and Willmott, C.J., "Mean Seasonal and Spatial Variability in Gauge-Corrected, Global Precipitation," International Journal of Climatology, 10, 111-127, 1990.

Maidment, D.R., McKinney, D., Lindner, R., Olivera, F., Reed, S., Zichuan, Y., "Water Balance of the Niger Basin : Interim Report," Prepared for the United Nations Food and Agricultural Organization and UNESCO by the Center for Research in Water Resources University of Texas at Austin, July 1995.

Manabe, Syukuro, "Climate and the Ocean Circulation: I. The Atmospheric Circulation and the Hydlrology of the Earth's Surface," Monthly Weather Review, 97, No. 11, Nov. 1969.

Mather, J.R., The Climatic Water Budget, Lexington Books, 1972.

Mintz, Y., and Serafini, Y.V., "A Global Monthly Climatology of Soil Moisture and Water Balance," Climate Dynamics, 8, 13-27, 1992.

Mintz, Y., and G.K. Walker, "Global Fields of Soil Moisture and Land Surface Evapotranspiration Derived from Observed Precipitation and Surface Air Temperature," J. Applied. Meteor., 32, 1305-1334, 1993.

NCDC, "Global Daily Summary," 1977-1991, CD-ROM, Asheville, NC.

Penman, H.L., "Natural Evaporation from Open Water, Bare Soil and Grass," Proc. R. Soc. London, Ser. A, 193, 120-145, 1948.

Priestley, C.H.B., and R.J. Taylor, "On the Assessment of Surface Heat Flux and Evaporation Using Large-Scale Parameters," Monthly Weather Review, 100, No. 2, 81-92, February 1972.

Shuttleworth, J.W., "Evaporation," Handbook of Hydrology, D.R. Maimdent Editor, McGraw-Hill, Inc., 1993.

Thornthwaite, C.W., "An Approach Toward a Rational Classification of Climate," Geographical Review, 38, 55-94, 1948.

Whitlock et al., "First Global WCRP Shortwave Surface Radiation Budget Data Set," Bull. Amer. Meteor. Soc., 76, 6, 905-922, 1995.

Wilm, H.G., C.W. Thornthwaite, E.A. Colman, N.W. Cummings, A.R. Croft, H.T. Gisborne, S.T. Harding, A.H. Hendrickson, M.D. Hoover, I.E. Houk, J. Kittredge, C.H. Lee, C.G. Rossby, T. Saville, and C.A. Taylor, "Report of the Committee on Transpiration and Evaporation, 1943-44," Transactions, American Geophysical Union, 25, 683, 1944.

Willmott, C.J., C.M. Rowe, and Y. Mintz, "Climatology of the Terrestrial Seasonal Water Cycle," Journal of Climatology, 5, 589-606, 1985.

Willmott, C.J., "WATBUG: A FORTRAN IV Algorithm for Calculating the Climatic Water Budget," Publications in Climatology, 30, 2, 1977.