4.1 Methodology

4.1.1 Model Description

The soil-water balance model uses a simple accounting scheme to predict soil-water storage, evaporation, and water surplus. Surplus is precipitation which does not evaporate or remain in soil storage and includes both surface and sub-surface runoff. The conservation of mass equation for soil-water can be written as follows:

(4.1)

In Equation 4.1, 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 also not considered in these computations, but this probably does not introduce significant error for a study in Texas. Willmott et al., 1985, describe a simple scheme that could be included to account for snow melt.

At first glance, it would seem that the most natural spatial unit to use in a soil-water balance would be a soil map unit, but these map units have very irregular shapes and a wide range of sizes. Because climate data also play an important role in the soil-water balance, the cells generated when climate data are interpolated onto regular grids are a judicious choice for use as the modeling units in the soil-water balance. Climate data interpolated onto 0.5 grid boxes are used in this study.

A major source of uncertainty in evaluating Equation 4.1 is estimating the evaporation. Estimation of evaporation is based upon knowledge of potential evapotranspiration, water-holding capacity of the soil, and a moisture extraction function. These concepts and a method for evaluating Equation 4.1 are described below. Special consideration of the potential evapotranspiration concept is provided in Section 4.2.

4.1.2 Description of input data

4.1.2.1 Climate data

Global data sets of mean monthly temperature and precipitation interpolated to a 0.5° grid were obtained from an anonymous File Transfer Protocol (ftp) server to the University of Delaware (climate.geog.udel.edu). These data are from the "Global Air Temperature and Precipitation Data Archive" compiled by D. Legates and C. Willmott. The 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).

4.1.2.2 Water-holding capacity data

Global estimates of "plant-extractable water capacity" have recently become available on a 0.5° grid (Dunne and Willmott, 1996). As used in this report, the term plant-extractable water capacity is equivalent to available water-holding capacity. One reason given for developing this global database was to eliminate the need for assuming spatially invariant plant-extractable water capacity in soil-water balance computations made over large areas. Information about sand, clay, organic content, plant rooting depth, and horizon thickness was used to estimate the plant-extractable water capacity. Figure 4.1 shows the distribution of this parameter throughout Texas. The global average for this parameter is 86 mm while the average in Texas is 143 mm.

4.1.2.3 Open Water Evaporation Estimates

Estimates of open water evaporation based upon pan evaporation measurements were provided by Alfredo Rodriguez at the Texas Water Development Board (TWDB, 1995). The data consist of monthly average gross reservoir evaporation estimates for one degree quadrangles in and around Texas. Monthly data for 1940 to 1990 are available in 75 quadrangles thoughout Texas and monthly data for 1971-1990 in an additional 28 quadrangles at the border of Texas. Mean monthly values were computed from these data and used for estimates of potential evaporation in the soil-water balance calculations. Figure 4.2 shows the one degree quadrangle index map, shaded to indicate where data are available. Figure 4.3 shows mean annual reservoir evaporation. As an alternative, a global radiation data set described in the next section has recently become available that facilitates making potential evaporation estimates using the Priestley-Taylor equation. This method was also considered for use in the soil-water balance computations. An insightful comparison of these two methods for estimating potential evapotranspiration is described in Section 4.2.

A global radiation data set recently made available makes using the Priestley-Taylor method a feasible option for estimating potential evapotranspiration in large scale studies. These data are described by Darnell et al., 1995, and were obtained by anonymous ftp to cloud.larc.nasa.gov. The data set includes longwave and shortwave radiation flux estimates 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, describe advances in input data and flux estimation algorithms that improve the ability to assess the radiation budget on a global scale. Input data improvements have come from the International Satellite Cloud Climatology Project (ISCCP) and the Earth Radiation Budget Experiment (ERBE). Using this satellite data, the radiation budget components that cannot be measured directly are estimated independently using physical approaches that have been validated against surface observations. According to Darnell et al., 1995, longwave flux estimates fall within +/- 25 W/m² of surface measurements while Whitlock et al., 1995, estimate the accuracy of shortwave estimates to be within +/- 20 W/m² of surface measurements. For comparison, the energy required to evaporate 1 mm/day of water is about 30 W/m². In this study, net radiation (equivalent to net shortwave + net longwave) is used.

4.1.3 Water-holding capacity of the soil

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 capacity 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. 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 4.1 gives some typical values for available water-holding capacity.

Table 4-1 : Typical Values for Soil-water Parameters by Texture*

Texture Class	Field Capacity	Wilting Point	Available Capacity
Sand	0.12	0.04	0.08
Loamy Sand	0.14	0.06	0.08
Sandy Loam	0.23	0.10	0.13
Loam	0.26	0.12	0.14
Silt Loam	0.30	0.15	0.15
Silt	0.32	0.15	0.17
Silty Clay Loam	0.34	0.19	0.15
Silty Clay	0.36	0.21	0.15
Clay	0.36	0.21	0.15

*Values obtained from ASCE, 1990, Table 2.6, p.21.

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. The water-holding capacity is denoted with w* and the current level of moisture storage in the soil is denoted by w [mm]. 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.

4.1.4 Estimating Actual Evapotranspiration

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

(4.2)

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 no longer controlled by meteorological conditions. Below this critical point, there is a 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 situations when daily climate data are used. A function in which the ratio of evapotranspiration to potential evapotranspiration is proportional to the current moisture level, f = w/w*, has been applied when budgeting with monthly climate values and this function is used here. This function predicts lower evapotranspiration rates than a function with a breakpoint which may offset (to some degree) the low rainfall intensities resulting from the use of monthly averaged rainfall.

There are drawbacks to using simple soil moisture extraction functions. Indices based on a function of soil moisture alone, do not account for the effects of vegetation. Mintz and Walker, 1993, cite field studies that show f may vary with potential evapotranspiration for a given soil wetness and f 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 determine the relationship between plant transpiration and potential evapotranspiration is to correlate f with satellite-derived indices of vegetation activity so that f will reflect plant growth stage and the spatial vegetation patterns. Gutman and Rukhovetz (1996) investigate this possibility. Using their approach still requires an estimate of potential evapotranspiration to get actual evapotranspiration.

4.1.5 Budgeting soil moisture to yield surplus

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. In dry locations, the mean potential evaporation for a given month may be higher than the mean precipitation and budgeting with monthly values may yield zero surplus, even though there is some observed runoff. For this reason, the use of daily values is preferred over monthly values when feasible, yet daily budgeting still does not adequately describe storm runoff that occurs when the precipitation rate exceeds the infiltration capacity of the soil. One drawback to using daily data is that it is difficult to interpolate daily rainfall over space. For the statewide study undertaken here, the use of daily data was deemed too cumbersome.

Equation 4.3 describes how soil moisture storage is computed.

if w_i < w*

S_i = w_i - w* and set w_i = w* if w_i > w* (4.3)

In Equation 4.3, 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 evapotranspiration, S_i is the surplus in a given day, f is the soil-moisture extraction function and w* is the water-holding capacity. 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. When evaluating Equation 4.3, if w_i drops below zero, then w_i is set equal to 0.01; if w_i > w*, then the surplus for that day is w_i-w* and w_i is set equal to w*. The soil-moisture extraction function f =w/w* was used for this study.

4.1.6 Balancing Soil Moisture

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 (w₁) is then set equal to the soil moisture at time N+1 (w_N+1) and the budget is re-computed until the difference (w₁ - w_N+1) is less than a specified tolerance.

4.2 Potential Evapotranspiration

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. Two potential evapotranspiration estimates were considered for this study, gross reservoir evaporation estimates from pan coefficients and estimates made using the Priestley-Taylor equation. As discussed later, the gross reservoir evaporation estimates are considered to be better than the Priestley-Taylor estimates for use in the soil-water budget calculations.

4.2.1 Potential evaporation vs. potential evapotranspiration

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 and 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 impedance 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 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 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 Priestley-Taylor method exhibits weak performance at arid sites is related to this ambiguity because the assumptions under which the expressions were derived break down. This is particularly relevant to West Texas and is the main reason why evaporation estimates derived from pan coefficients are considered more applicable for the type of computations being made in this study. A comparison of the two methods is described in Section 4.2.3.3.

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.

4.2.2 Penman combination method

Two requirements for evaporation to occur are an energy input and a mechanism for the transport of water vapor away from the 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 radiation absorbed less longwave radiation 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 involves a vapor transport coefficient times the vapor pressure gradient between the saturated surface and an arbitrary measurement height. Determination ofthe vapor transport coefficient 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 1948, 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:

(4.4)

where E_r = and . R_n is net radiation [W m^-2], l_v is latent heat of vaporization [J kg^-1], r_w is density of water [kg m^-3], K(u) is a mass transfer coefficient, e_s 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 (E_r) and turbulent mass transfer (E_a). 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).

4.2.3 Simpler Methods

Two simpler methods that are much easier to apply than forms of the Penman equation were considered in this study, a pan coefficient approach and the Priestley-Taylor method.

4.2.3.1 Pan coefficients

Evaporation pans are commonly used to estimate open water evaporation from nearby lakes and reservoirs. The rate of evaporation is estimated by measuring the change in water level with time. Lake evaporation is estimated by multiplying the pan evaporation by a pan coefficient. Typical values of the pan coefficient range from 0.67 to 0.78 in Texas, so the measured evaporation from the pan is higher than that from the lake surface. Pan coefficients vary with location and season. The development of gross reservoir evaporation estimates used in this study is described by TWDB, 1995. As discussed in Section 4.2.1, open water evaporation and potential evapotranspiration are often of similar magnitude, justifying the use of open water evaporation estimates in soil-water budget calculations.

4.2.3.2 Priestley-Taylor Method

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 over land and ocean 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 (aerodynamic) term in the Penman equation to go to zero. Both the Slatyer-McIlroy and the Priestley-Taylor definitions consider 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 (a) times Penman's radiation term.

(4.5)

Equating this expression to the combination equation reveals that the advection term must be a constant fraction of the radiation term if a is a constant.

(4.6)

(4.7)

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 a. The fact that a is greater than one indicates that true advection-free conditions do not exist. Since 1972, several other researchers have confirmed that a values in the range 1.26-1.28 are consistent with observations under similar conditions. Some researchers have found significantly lower values for the a 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 a 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 a coefficient may be required in arid climates (ASCE, 1990). Based on arid sites studied in ASCE, 1990, a value of a=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.

4.2.3.3 Comparison of Pan and Priestley-Taylor Methods

Figure 4.4, Figure 4.5, and Figure 4.6 are maps of mean annual temperature, mean annual net radiation distribution, and mean annual potential evapotranspiration made using the Priestley-Taylor method. A comparison between Figure 4.4 which shows the mean annual Priestley-Taylor potential evapotranspiration and Figure 4.3 which shows the gross reservoir evaporation is quite revealing. It is clear that the highest values of reservoir evaporation are in West Texas with a decreasing trend moving eastward. The converse is true for the Priestley-Taylor estimates where the lowest values occur in West Texas with an increasing trend towards East Texas. The reason for the non-intuitive, low potential evapotranspiration estimates from the Priestley-Taylor method in West Texas is that radiation and temperature data that were measured under non-potential conditions have been used. The Priestley-Taylor estimates are proportional to the net radiation (Figure 4.5) at the earth's surface. In wetter areas of East Texas, there is more water on the land surface and in the atmosphere to absorb incoming solar radiation and this results in higher net radiation values. In addition, greater cloud cover and water vapor in the atmosphere trap a larger percentage of the longwave radiation emitted from the earth. Spatial variation of surface albedo (fraction of incident shortwave radiation reflected) also contributes to this trend because drier, less vegetated areas in West Texas tend to have higher albedos. In addition to spatial trends caused by moisture variation, net radiation values increase from north to south because of the earth's shape and its tilt relative to the sun. The spatial patterns in Priestley-Taylor potential evaporation shown in Figure 4.6 reflect the spatial patterns of temperature and net radiation in Figures 4.4 and 4.5.

Because the net radiation at the earth's surface is directly related to the wetness of the area, it may be a better surrogate for actual evapotranspiration than potential evapotranspiration. In Section 5.23 a map of Bowen ratios for Texas is computed. As discussed in Section 5.3.3, use of net radiation and temperature data, along with a map of Bowen ratios may be an alternative approach to estimating evaporation that eliminates the use of the difficult potential evapotranspiration concept.

Figure 4-4: Mean Annual Temperature in Texas from Legates and Willmott Climatology

Figure 4-5: Mean Annual Net Radiation Estimates from the ERBE Program

In terms of absolute magnitude, the statewide average reservoir evaporation is much higher 1690 mm year^-1 than the Priestley-Taylor estimate 1120 mm year^-1; however, the values in East Texas are more comparable because the lowest reservoir evaporation estimates and highest Priestley-Taylor estimates both occur here. Looking at the results of the next section, differences in the spatial and temporal distribution between the two potential evaporation estimates make a big difference in the resulting surplus.

4.3 Results

The results from the soil water balance are monthly estimates of evaporation, surplus, and soil moisture in each 0.5 grid cell covering the State. Figure 4.7 shows the mean annual surplus estimated from two separate calculations, the first using the Priestley-Taylor potential evapotranspiration method and the second using the reservoir evaporation as potential evapotranspiration. Using the Priestley-Taylor potential evapotranspiration method yields an average of 85 mm year^-1 of surplus across the State while the use of the reservoir evaporation method yields 42 mm year^-1 and the observed runoff (78.4 mm year^-1 from Section 5) is somewhere between these two estimates. A major problem is that this soil-water balance model predicts zero runoff for much of the State even though it is known that some runoff occurs in these areas. The time distribution of precipitation, actual evaporation, soil moisture, and surplus for two cells are shown in Figure 4.8. In the cell on the left, the water-holding capacity (162.5 mm) is never reached, but for the cell on the right the water-holding capacity (91 mm) is exceeded during seven months out of the year and surplus is generated.

The effects of that the water-holding capacity estimate has on soil-water budget can be seen in Figures 4.9 and 4.10. Figure 4.9 shows the mean annual soil moisture [mm] and Figure 4.10 shows the mean annual soil moisture divided by the water holding capacity. The differences in Figure 4.9 and 4.10 occur where the soil water-holding capacity has a limiting effect on evaporation relative to surrounding cells.

4.4 Summary

The rudimentary soil-water balance approach used in this study provides a qualitative sense of how precipitation is partitioned between runoff, evaporation, and soil moisture storage. The surplus and soil moisture values computed with this model are interpreted better as indexes of relative wetness rather than absolute estimates because none are calibrated against measured values. Use of a monthly time step, a simplified representation of soil and plant hydrology, the ambiguity in applying the potential evapotranspiration concept to dry areas, and the errors in estimating potential evapotranspiration are major limitations of this model. The model time step cannot account for storm runoff, an important mechanism for runoff generation. The soil-water balance model is an incomplete hydrology model because it is very difficult to calibrate against observed values. Coupling a soil-water balance model with measured runoff is the only realistic way to derive accurate runoff estimates. A simplified coupling of the soil-water balance model to a surface runoff model was achieved in a recent study to develop a GIS-based water planning tool for the Niger River Basin in West Africa (Maidment et al., 1996; http: // www.ce.utexas.edu / prof / maidment / GISHydro / africa / africa.htm). This model was calibrated for monthly flows but not validated. A more detailed approach to this type of study could be taken by implementing a continuous stream flow simulation model with daily time stepping (or less); however, implementing this type of model on a region the size of Texas is a formidable task.