United States 

Department of 

Agriculture 

Forest Service 

Southern 

Research Station 

Research Note 

SRS-6 

Estimating Actual Evapotranspiration 

for Forested Sites: Modifications to 

the Thornthwaite Model 

Randall K. Kolka and Ann 2'. Wolf 

September 1998 

Abstract 

A previously coded version of the Thornthwaite water balance model 

was used to estimate annual actual evapotranspiration (AET) for 29 

forested sites between 1900 and 1993 in the Upper Great Lakes area. 

Approximately 8 percent of the data sets calculated BET in error. Errors 

were detected in months when estimated AETT was greater than potential 

evapotranspiration. Annual climate variability led to errors in accrued 

soil water storage, which led to errors in the calculation of AET. Two 

hydrologically justifiable modifications were made to correct errors 

resulting from the use of the original coded Thornthwaite model. The first 

modification allows for soil water accumulation throughout the year, 

whereas the second allows for oversaturated soil conditions in the spring. 

The modified program appears to be robust for any temperate climate 

condition. 

Keywords: Hydrology, potential evapotranspiration, Upper Great Lake 

States, water balance. 

Introduction 

The most difficult parameter to measure when calculating a 

site's water balance is actual evapotranspiration (AET), 

which is a function of precipitation, temperature, solar 

radiation, soil water storage, wind, canopy and understory 

interception, and growth rates. Few methods for measuring 

AET directly are available. Although field studies using 

lysimeters and air-monitored tents have been somewhat 

successful in measuring AET in apcultural or open 

situations, AET cannot be measured directly within forested 

systems by any practical field method (Brooks and others 

199 1). Because of their size and complex surface dynamics, 

trees are not easily measured by either of these methods. 

One approach for estimating AET on a watershed basis is to 

use paired watersheds where differences in streamflow 

Randal1 K. Kolka, Research Soil Scientist, USDA Forest Service, Center 

for Forested Wetlands Research, Savannah River Ecology Lab, Aiken, SC 

29802; Ann T. Wolf, Consultant, Connect Computer Company, 

Minneaplis, MN 55439. 

following clearcutting are attributed to changes in BET 

(Hornbeck and others 1970). Unfortunately, the scale, cost, 

availability of sites, and the time necessary to implement this 

approach are not appropriate for site-specific studies. 

At the site or stand scale, the method commonly used to 

estimate MT is a water balance that recognizes the 

relationship between potential evapotranspiration (PET) and 

AET. The results of equations used to calculate PET, and, 

thus, AET, are indices, not absolute values. The water 

budget method uses soil water storage in conjunction with 

PET and precipitation to estimate mT (Brooks and others 

199 1). A number of methods have been developed to 

estimate PET, including the Thornthwaite equation 

(Thornthwaite and Mather 1955), which is based on 

temperature and day length. The traditional approach has 

been to estimate PET using long-term monthly averages of 

temperature and precipitation as well as latitude (to 

determine day length), which are easily accessible data. 

Thornthwaite and Mather (1957) used PET, soil water 

storage, and precipitation to calculate a monthly site-specific 

water balance from which they could estimate AET. Simply, 

AET is the sum of monthly change in soil moisture storage 

(_+) and precipitation. Many people have coded 

Thornthwaite's approach for computer calculation, but one 

of the first to do so was P.E. Black of Syracuse University 

(Black 1966). I)r. Black's progmm calculates a site's water 

balance using monthly temperature and precipitation, soil 

water storage, and latitude. Crigal and Bloom ( 1985) later 

modified Black's program to better estimate runoff for sites 

in Minnesota, Modifications stemmed from the comparison 

of observed and estimated ratios of runoff to precipitation 

for six watersheds in eastern and northern Minnesota. The 

program tended to underestimate runoff. Modifications 

were made in the evapotranspiration portion of the program 

to better approximate the measured runoff. For this paper, 

we will refer to the Grigal and Bloom (1985) modified 

program as the "original program.'" 
We used the original progrm to estimate annual AET for 29 

sites located in the Uppr Great Lakes arm. As one 

environmena variable, AET has been used to predict 

amual aspen (Popukus tremulot.des Mich.) height grow^ 

(kary and others 1997). Instead of long-tern average 

climatic conditions, we interpolakd monthly cEmatic data 

for the life of individual trees on each site. We ran 2,726 

site-year combinations, and our evaluation of the results 

showed that, in some cases, annual ~T was greater than 

PET, which is a physical impossibility. A search of 

wcwenees where AET > PET showed that 229 of 2,726, or 

8.4 percent of annual results, had been calculated in error. 

Black (196Cj) discussed these types but gave no remedy. 

Although such errors are not prevalent when using long-term 

averages (fig, la), large variability in monthly precipitation 

during 3 consecutive months, i.e., wet-dry-wet or dry-wetdry, leads to such errors (fig. I b). When circumstances lead 

to such a variable annual climatograph as is shown in figure 

LOU I 

Jan Feb Mar Apr Mag Jun Jul -tug Sep Oct Uov Dee 

Month 

200 

-%3- PET 9 

-+- Precipitation 

R 

jd \ I 

o* n-- i'" "' , ' b-c- .Jan Feb 'Jar 4pr CIJV Jun Jul lug Sep Oct \or Dei 

Month 

Figure I-(a) Comparison of long-term average annual precipitation and 

potential evapotranspiration (PET) (b) with those that are typically 

encountered when analyzing single years of data. 

lb, the program incomctly calcdates the recovery of soil 

water. hors in the original progfam are not in calculating 

AET but in the method used to detemirre accumulated soit 

water. In this swdy, we modified the progam to prwrly 

measure soil water gains and losses for all climatic 

condiGons encountered. 

Program Modifications and Results 

Two successive modifications were needed to fully correct 

for errors in the original program. Before discussing those 

mdifications, however, some parameters need to be defined 

(Black 1966): 

I. PET = potential evapotranspiration calculated with the 

Thornthwaite equation. 

2. P-PET = precipitation less the potential evapotranspiration. 

3. ACPWL, = accumulated potential water loss, which is the 

amount of soil water lost when PET exceeds P; i.e., there is 

less precipitation than potential evapotranspiration. In the 

calculation of AET, ACPWL is not a factor until P-PET 

becomes negative. To detennirre the ACPWL for a 

particular month, the previous month's ACPWL and the 

current month's P-PET are summed. In the original program, 

ACPWL becornes 0 after a month in which PET < F? 

4. STRGE = soil storage; this is the maximum soil storage 

at field capacity (ACPWL = 0). When below field capacity 

(ACPWL < 0), STRGE is a function of both maximum soil 

storage and ACPWL. 

5. DELTA = the difference between STRGE in successive 

months when it is less than maximum. When DELTA is 

negative, then AET < PET i.e., soil moisture is limiting 

evapotranspiration. When DELTA is positive, then AET = 

PET. 

6. AET = actual evapotranspiration. This is the sum of 

available precipitation for the month 2 the change in 

STRGE. Men DELTA is positive, AET = PET. When 

DELTA is negative, AET = precipitation for the month + the 

absolute value of DELTA. 

The original program assumes that once P-PET becomes 

negative (ACPWL becornes negative), it can not become 

positive until the end of the hydrologic season. Our 

modification assumes that soil water deficits are 

accumulated throughout the year regardless of the positive 

or negative value of P-PET. As in the original program, we 

begin to accumulate potential water loss when P-PET 

becomes negative. However, if P-PET then becomes 
positive we simply continue to accumulate ACPW until it 

reaches zero or the hydrologic season ends. At values 

grater than zero, ACPW does not enter into AET 

calculations (AET = PET). Hydrologically, the scenario we 

are simarlating is more natural than that simulated by the 

original program. In the original program, 1 mm of excess 

precipitation (P-PET) following a month with a large soil 

water deficit will allow PET to equal AET. However, the 

following month's soil deficit should reflect the large deficits 

already present in the soil. If one considers the soil as a 

bucket, the bucket is half full in the first month (soil storage 

deficit). In the second month, water is taken out 

(evapo&anspiration) at the same rate as it is being put in 

(precipitation); thus, AET = PET, and the excess 

prwipitation ( 1 mm) is added to storage. In the third month, 

the bucket does not begin full, as assumed in the original 

model; it has only 1 mm more water than it did at the end of 

the first month. After the first modification, we found that 

197 of 2,726 data sets, or 7.2 percent, had still erroneously 

calculated AET. 

We investigated the data sets with errors remaining after the 

first modification. In all cases, STRGE never fell to field 

capacity in the spring; it dropped from saturated (soil water 

+ snow water) to undersaturated (soil water), never 

equilibrating to maximum capacity. After assessing the siteyear combinations in our data and those in the literature 

(Thornthwaite and Mather 1957, Black 1966, Grigal and 

Bloom 1985), we found all those that calculated AET < PET 

fell to the exact value of maximum storage sometime in the 

spring before ACPW began to accumulate. When 

temperatures are > - 1 OC (Eq. l), the original program 

calculates STRGE with the following equation: 

S~CE = 10 [log MSW - (o.525/(MSW'03") x ACPW)] ( 1) 

where 

STRGE = soil water storage, 

MSW = the maximum soil water, and 

ACPWL = the absolute value of accumulated potential water 

loss. 

Equation 1 was derived from soil storage tables given in 

Thornthwaite and Mather (1957). When ACPW = 0, 

STRCE is equal to maximum soil storage; and when 

ACPWL < 0, soil water storage is less than maximum. In 

most situations, ACPWL = 0 for some period after mean 

monthly air temperatures are > -I "C (when PET begins). 

For the few circumstances encountered here, ACPWL began 

to accumulate in the same month that PET began. The 

original program calculates AET in error for these situations 

because it calculates changes in storage as soon as ACPWL 

begins. When ACPWL begins directly after a month when 

snow is still accumulating (temperature < - 1 "C), the original 

program calculates the change in storage from an 

oversaturated condition (soil water + snow water) to an 

undersaturated condition, not from maximm soil water to 

undersaturated conditions. Our second modification 

recognizes these conditions and simply calculates ACPWL 

from maximum soil storage instead of oversaturated soil 

storage. As with the first modification, the second more 

nearly simulates natural conditions. Meltwater is not 

transpired or evaporated to a great extent, especially at rates 

greater than potential; it either infiltrates, raising the soil to 

field capacity, or is lost as runoff. With the second 

modification, 100 percent of the site- year data calculated 

AET < PET. 

Examples of program output for three selected sites show the 

results of modifications (tables 1, 2, and 3). The 

modifications did not change AET estimates for years where 

climate data followed long-term patterns; although the 

second modification does affect how STRGE (soil storage) 

is accumulated (table 1). The second data set calculated 

AET > PET using the original prograrn, but AET < PET 

after the first and second modifications (table 2). Values for 

P-PET changed from negative in June to positive in the wet 

month of July. Thus, using the original program, ACPWL 

began from zero in August. This error leads to zero storage 

in the original program and a subsequent overestimation of 

AET. The first modification corrects this error by 

continuing to accumulate potential water loss. Again, the 

change is justified hydrologically because the soil does not 

fully recover to maximum soil water conditions if PET = 

AET in the previous month. In table 2, August does not 

begin with zero ACPWL as assumed in the original program 

but with the sum of soil water losses and gains from all 

previous months. The third data set estimates AET 

incorrectly for both the original and the initially modified 

version of the program (table 3). Soil water storage is never 

at the exact maximum value. The DELTA value for April is 

the difference between soil storage in March and April. The 

large negative DELTA was calculated in error and led to an 

overestimate of AET; it actually should have been the 

difference between maximum soil storage and soil storage 

for April (table 3). The final modification allows for a 

reliable estimation of AET. 
Table 1-Model calculation of AIET in normal climatic year, The original program, first modification, and second 

mdification calculate AET < PET ( imurn soil storage = 153 latitude = 46.6" N, year = 190.1) 

Monrhs 

and years Temp PPT PET P-PET ACPWL STRGE DELTA AET 

Original hogram 

Jan 

Feb 

Mar 

APf 

May 

Jun 

Jul 

Aug 

Sep 

Oct 

Nov 

Dec 

Year 

First Modification 

Jan 

Feb 

Mar 

AP~ 

May 

Jun 

Jul 

Aug 

S~P 

Oct 

Nov 

Dec 

Year 

Second Modification 

Jan 

Feb 

Mar 

Apr 

May 

Jun 

Jul 

Aug 

S~P 

Oct 

Nov 

Dec 

Year 

TEMP = Temperature; PPT = Precipitation, PET = potential evapotranspiration; P-PET = precipitation less potential evapotranspiration; ACPWL = accumuIated potential water loss; STRGE = soil storage; DELTA = the difference between STRGE in successive months when it is less than maximum; AET = actual 

evapotranspiration. 
Table 2-Model calculation of AET in abnormal climatic year. The original program calculates AET > PET in 

August, while the fwst and second modification calculate M3T < PET for all months (maximum soil storage 

= 153 mmq latitude = 46.6" N, year = 1903) 

Months 

and years Temp PPT PET P-PET ACPWL STRGE DELTA BET 

Jan 

Feb 

Mar 

Apr 

May 

Jun 

Jul 

Aug 

Sep 

Oct 

Nov 

Dec 

Year 

First Modffication 

Jan 

Feb 

Mar 

AP~ 

May 

Jun 

Jul 

Aug 

Sep 

Oct 

Nov 

Dec 

Year 

Second Modification 

Jan 

Feb 

Mar 

AP~ 

May 

Jun 

Jul 

Aug 

S~P 

Oct 

Nov 

De€ 

Year 

TEMP = Temperature; PPT = Precipitation, PET = potential evapotranspiration; P-PET = precipitation less potential evapotranspiration; ACPWL = accumulated potential water loss; STRGE = soil storage; DELTA = the difference between STRGE in successive months when it is less than maximum; AET = actual 

evapotranspiration, 
Table %Model calculation of AET in abnormal climtic year. The original program and first modification calculate 

AE=T > PET in April while the second modification calculates AET < PET for all months (m~irnum soil storage = 177 

rnw latitude = 47,7"N, year = 1926) 

Monrhs 

and years Temp PPT PET P-PET KPWL STRCE DELTA AET 

Original &@gram 

Jan 

Feb 

Mar 

AP~ 

May 

Jwn 

Jul 

Aug 

S~P 

Oct 

Nov 

Dec 

Year 

First Modification 

Jan 

Feb 

Mar 

Apr 

May 

Jun 

Jul 

Aug 

S~P 

Oct 

Nov 

De€ 

Year 

Second Modification 

Jan 

Feb 

Mar 

AP~ 

May 

Jun 

Jul 

Aug 

S~P 

Oct 

Nov 

Dec 

Year 

TEMP = Temperature; PPT = Precipitation, PET = potential evapotranspiration; P-PET = precipitation less potential evapotranspiration; ACPWL = accumulated potential water loss; STRGE = soil storage; DELTA = the difference between STRGE in successive months when it is less than maximum; AET = actual 

evapotranspiration. 
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