a hydrometeorological model for basin-wide seasonal evapotranspiration

WATER RESOURCES RESEARCH, VOL. 35, NO. 11, PAGES 3409-3418, NOVEMBER 1999

A hydrometeorological model for basin-wide seasonal evapotranspiration Nelson Luis Dias • and Akemi Kan

Paranti Meteorological System (SIMEPAR), Curitiba, Brazil

Abstract. A new methodology is proposed to capture the seasonal behavior of evapotranspiration from precipitation and streamflow data and to develop hydrometeorological evapotranspiration models tailored for each basin. The water budget method for determining evapotranspiration is downscaled to periods between 15 and 160 days that occur between well-marked hydrological recessions. Using these uneven time periods, the error associated with the unknown soil moisture storage is minimized, whereas groundwater storage changes are estimated by means of a classical linear groundwater reservoir whose time constant is obtained by recession analysis. This seasonal water budget (SWB) method is able to reproduce the seasonal signal of evapotranspiration even when it is absent from the precipitation and streamflow records. The estimates are also compatible with calculated monthly net radiation. By selecting short enough water budget periods it is possible to check the relationship between SWB evapotranspiration estimates and net radiation, Penman and Priestley-Taylor potential evaporation, precipitation minus outflow, water vapor deficit, and basin storage. The ratio of SWB evapotranspiration to an upper limit value represented by either net radiation or potential evaporation is well correlated with precipitation minus outflow, water vapor deficit, or both but is very poorly related to basin storage. The calculated regressions lead to a family of hydrometeorological evapotranspiration monthly (HEM) models fitted to the basins in question, in a way analogous to the calibration of rainfall-runoff models. In the two watersheds where the methodology was applied the HEM models were able to preserve mass, with total accumulated differences no larger than 0.25 mm d -• and root-mean- square errors of the order of 0.7 mm d -•.

1. Introduction

The calculation of reliable estimates of basin-wide evapotranspiration remains one of the most difficult questions for the science of hydrology. Still, many engineering applications of physical hydrology require evapotranspiration estimates. For instance, virtually all methods for estimating streamflow from rainfall require either an index of soil moisture and, consequently, antecedent evapotranspiration or a conceptual model of the basin's hydrologic processes.

Evapotranspiration estimates are particularly important in water budget studies to investigate the quantity of interbasin transfers, as checks on consumptive use estimates, and as a means of removing climatic variability from monthly or annual flow records to try to detect trends in streamflow; soil moisture and evapotranspiration estimates may also be useful in ecolog- ical studies and water resources planning [Alley, 1984; Church et al., 1995; Domokos and Sass, 1990]. They can also be used in distributed hydrologic modeling with geographical information systems [Mendes, 1996].

Another case in point is the treatment of streamflow data. For example, the building of "cascades" of reservoirs in several important watersheds in Brazil has created the necessity of

1Also at Setor de Tecnologia, Univesidade Federal do Paran•t, Cu- ritiba, Brazil.

Copyright 1999 by the American Geophysical Union.

Paper number 1999WR900230. 0043-1397/99/1999 WR900230509.00

extending the natural streamflow series to the periods after the construction of the dams. This, in turn, requires the calculation of the water balance (at least on the monthly timescale) of each reservoir and an estimate of the evapotranspiration that would have occurred under natural conditions from the areas now

flooded by the reservoirs. Central to the problem is the evaluation of most of the terms

of the water budget of a watershed. New methods for the measurement or estimation of hydrologic variables which are not routinely monitored, such as evapotranspiration and soil moisture, are an important step for the development of hydrology as a science [Klemeg, 1986]. For example, Mroczkowski et al. [1997] used independent data sets of stream chloride, groundwater levels, and percent of hillslope saturated to help identify conceptual model limitations that could not be re- vealed by a split-sample test of streamflow alone; Findell and Eltahir [1997] used 14 years of soil moisture measurements at 19 stations in Illinois to establish empirically the relationship between soil moisture and ensuing summer precipitation. Since the available routine data used in hydrology is insuffi- cient to identify all of the underlying physical processes, one is often faced with a situation where several different models

yield similar fits and prediction errors and still indicate entirely different types of basin response to precipitation [Alley, 1984].

In trying to deal with the problem in terms of climatic classification and agricultural planning, Thornthwaite [1948] intro- duced the concept of potential evapotranspiration, and a methodology for monthly water balances that is still useful today [Arnell, 1992]. Thornthwaite [1948, p. 55] noted that

3409

3410 DIAS AND KAN: HYDROMETEOROLOGICAL EVAPOTRANSPIRATION MODEL

Q(n mday -•)

)t(days)

Figure 1. Determination of the beginning and the end of a water budget period from analysis of hydrograph recessions.

The catalogue of climatic elements consists of those that are customarily measured and usually includes temperature, precipitation, atmospheric humidity and pressure, and wind velocity,... But the sum of the climatic elements that have been under ob-

servation does not equal climate. One element conspicuously missing from the list is evaporation.

Since then, evapotranspiration measurements have evolved considerably. Energy budget, eddy correlation, and mean pro- file methods produce "point" or very local values over spatial scales of <1 km [Brutsaert, 1986], but their routine use in networks to obtain basin-wide estimates is still uncommon, although progress is being made to develop robust and cost- effective flux-measuring systems able to operate continuously [Blanford and Gay, 1992; Stannard et al., 1994; Baldocchi et al., 1996; Verkaik, 1998]. Alternative, independent estimates which are not based on surface layer micrometeorological measurements have also been obtained for different temporal and spatial scales by such diverse means as isotopic methods [Ma- garitz et al., 1990], atmospheric boundary layer soundings [Brut- saert and Mawdsley, 1976; Brutsaert and Sugita, 1991; Hipps et al., 1994; Mecikalski et al., 1997] and streamflow recession analysis [Daniel, 1976].

Even when reliable routine estimates of basin-wide evapotranspiration become available, however, it will still be neces- sary to work with historical records measured under the conditions described by Thornthwaite [1948]. In this work we try to address the problem of generating reliable monthly evapotranspiration estimates from the available time series of precipitation, streamflow, and meteorological variables measured at manually operated stations. Any solution that can be offered is necessarily limited by the nature of the available data. We neglect groundwater losses and assume that long-term evapotranspiration is equal to precipitation minus runoff. However, instead of making only long-term estimates, we estimate basin storage by assuming a relationship with recession flows, thus obtaining seasonal estimates over timescales that vary from 15 to 160 days. By sampling those estimates for the shortest periods between recessions available and relating them to monthly means of meteorological variables, we build monthly models of evapotranspiration tailored for each basin.

2. Seasonal Water Budget (SWB) Method Consider the instantaneous water budget of a watershed

dS/dt = P - Q - E, (1)

where S is total basin storage as surface water, soil moisture, and groundwater, P is precipitation; Q is outflow; and E is evapotranspiration. We use the term soil moisture to mean the

sum of soil, intermediate vadose, and capillary water above the water table and groundwater to mean phreatic water in the zone of saturation [Linsley et al., 1975, Figures 1-1 and 6-1; Bear, 1972, Figure 1.1.2]. We assume that Q is equal to the gaged flow at the outlet and that groundwater gains or losses are negligible. Integrating over a time interval At yields

(Sf- Si)/At = (P) - (Q) - (E), (2)

where Sf and S i are the final and initial basin storages and angle brackets indicate a time average. Since both S and E are unknown, to close the water budget one must use an additional equation or else work over time periods so large that the left side of (2) becomes negligible. Now consider a streamflow recession during which it is assumed that P = 0; then

dS/dt = -(Q + E) _< -Q. (3)

If an empirical relation between storage and outflow holds, its form and parameters can be estimated by inspection of recession data. For a linear reservoir,

Q = S/T, (4)

where T is a time constant of the recession. Integrating the differential inequality (3) with (4) gives

Q(t + At) _< Q(t)e -at/r, (5)

so T can be determined by drawing the straight line which is the upper envelope of a Q(t) versus Q(t + At) plot.

Brutsaert [1982, pp. 244-247] proposed the use of recession analysis and a nonlinear flow-storage relation to estimate E on a daily timescale. However, it has been found since then that recession flows are only sensitive to the evapotranspiration from vegetation of the riparian zone, where the roots are in touch with the water table; farther away from the stream, vegetation and groundwater seem to be essentially uncoupled [Zecharias and Brutsaert, 1988]. This means that S in (4) cannot be taken to represent total basin storage until soil moisture has been depleted and E has been reduced considerably, a situation that can only happen after a long recession. Therefore we will apply the mean water budget (2) over periods At larger than 15 days.

To obtain T, periods of at least 15 consecutive days of decreasing flows were analyzed. The water budget periods, on the other hand, are at least 15 days long, at whose end points the streamflow (Qi, Qf) is <1 mm d-•; for water budget periods longer than 60 days, a final value of 2.0 mm d -• is accepted. We assume that by the end of such a period most of the water is stored as groundwater and that evapotranspiration is negligible; then, using Qi and Qf (the last flow of the previ- ous and current recession) in (4), Si and Sf can be estimated, and the average water budget (2) can be solved for (E) (see Figure 1).

Application of this method produces a sequence of irregularly spaced water budget periods and their corresponding estimates. Such a sequence is unwieldy for water resources studies where the basic time unit is the month. In order to

obtain monthly averages, the seasonal water budget estimate for month i is assigned the value

Ill(El} q- nc(Ec) + Ilr(Er} (El) = Ill + Ilc + Ilr ' (6)

where the subscripts l, c, and r stand for the water budget periods to the left, within (center), and to the right of the

DIAS AND KAN: HYDROMETEOROLOGICAL EVAPOTRANSPIRATION MODEL 3411

ml mc mr ',• •:• >1< •

i 1 2 months ,-1 'l I I • i+ nl nc nr

Figure 2. Definition of monthly evapotranspiration from water budget estimates.

current month's limits (see Figure 2), whose lengths are m l, rn c, and mr. The numbers of days within month i corresponding to the water budget periods are hi, nc and nr.

We call the (Ei) estimates seasonal because it is not uncommon that a water budget period as defined here spans several months, in which case n c equals the number of days of month i. The longer the water budget periods encompassing or neigh- boring a given month, the farther (E i) will be from the actual monthly evapotranspiration. To take this effect into account in an objective way, we define the equivalent length of the (Ei) estimate as

m/rtl + mcnc + mrnr ß (7) /gi--

The reason for defining rt i in (7) is to have an objective con- fidence index for the associated evapotranspiration estimate; thus large n• values indicate poor estimates because of long water budget periods. Because the exact mathematical form is not so important, a simple weighted average was adopted.

3. Application of the SWB Method The SWB method was applied to two watersheds in the state

of Paranti, Brazil (Figure 3). The Jangada River basin is located in the south of the state at 26ø30'S, 50ø20'W. Mean altitude is 920 m; the drainage area to the gaging station is 1055 km 2. Vegetation is mainly secondary forest (Pinus elliottii and Pinus taeda) with some remaining native forest (Araucaria angustifolia). Winter and summer are well defined, but there are no wet and dry seasons, with precipitation being distributed more or less uniformly throughout the year. Soil is a mixture of sandy clay, clay, and silty clay.

The Cinzas River basin is located in the north of the state, at 23ø30'S, 50øW. Mean altitude is 540 m; the drainage area is 5622 km 2. Summer is the wet season, and vegetation cover is mostly pasture, with a few reforested areas in the south of the basin. Soil is mostly clay; soil types were obtained from Min- istry of Agriculture [Empresa Brasileira de Agropecudria (EMBRAPA), 1981] soil maps; vegetation types were derived from local inspections and U.S. Geological Survey (USGS) digital maps from the EROS Data Center (available on the World Wide Web at http://edcintl.cr.usgs.gov).

The Jangada River basin has three nearby meteorological stations with records spanning different periods between 1961 and 1993. The data available during a common period of op- eration (1975-1990) were used to produce, for each season, univariate linear regressions for temperature, specific humidity, wind speed, and sunshine duration between the reference station at Porto Uni•o (27 km from the Jangada River) and the other two, located farther away. The linear regressions were used to fill the periods when data were not recorded at Porto Uni•o, thus generating a single homogeneous record of mean daily meteorological data for 1961-1993 at this station. The monthly means (used in the hydrometeorological evapotrans-

10 o-

40 ß . BRAZIL

CINZA8 RIVER

BASIN

JANOADA RIVER

BASIN

.85 ß .60 ß .75 ø .70 ø .85 ø .60 ø • -500 .45 ß .40 ß .35 ø

Figure 3. Location of the Jangada and Cinzas catchments.


._, 60

-o 40 E E 20

0

E 10 E

0

.-, 20 E E •i' 10

5

o Ol 03 05 07 09 11 Ol 03 05 07 09 11

Month

Figure 4. Seasonal estimates of evapotranspiration (E} from water budget and hydrograph recession analysis for Cinzas River, 1990-1991.

piration models described in section 4) that are generated by this procedure agree quite well with the observed ones. In the Cinzas River basin a single weather station has been operating continuously since 1957 and provided the same kind of data. A comparison of the records of temperature, sunshine duration, and relative humidity shows that the Jangada River basin is colder, more humid, and less sunny than the Cinzas River basin.

In both basins the records of precipitation and streamflow are longer than these meteorological series: The Jangada River streamflow time series started in 1945, and data were available until 1991, whereas the Cinzas River streamflow time series started in 1931 and continued until 1991. Mean daily precipitation from the rain gage networks was calculated by a com- puterized version of the Thiessen method and then monthly means were produced. A standard linear recession analysis for both basins produced the time constants of the recessions: 17 (Jangada) and 28 days (Cinzas).

An algorithm for identifying streamflow recessions accord- ing to the criteria described above was written and applied to both streamflow series to define the water budget periods. At

the end points of each water budget period At we calculated S i and S s with (4) and obtained (E) with (2). The process is depicted for the Cinzas River during 1990-1991 in Figure 4.

The (E} estimates cannot be independently assessed: The SWB method is an approximate method for deriving evapotranspiration estimates at time resolutions finer than 1 year. (Its advantage is that it is simple enough to be widely and easily applicable and still is based on a few sound concepts such as recession analysis and an equivalent linear groundwater reservoir.) However, (1) It is possible to verify the relative importance of the storage term (Ss - Si)/At compared with (P) - (Q). If this term correctly captures changes in basin storage, it should often be a substantial fraction of precipitation minus streamflow. (2) The relative quality of the monthly evapotranspiration estimates can be assessed by the equivalent lengths n• (see (6) and (7) and the discussions on their meanings). (3) The overall consistency of the method with the known climatological patterns of precipitation, streamflow, radiation, and potential evapotranspiration can be established.

To check point 1, in Figure 5 we have plotted, for the irregularly spaced water budget periods, the quantity IASI/ [At((P} - (Q})] against the length of the water budget periods At. Naturally, the longer the period, the less important the storage term becomes. The solid curve, which envelops most of the points, highlights this fact. However, particularly for periods shorter than, say, 50 days, the storage term can represent as much as 50% of (P} - (Q}. It is exactly those periods that generate monthly evapotranspiration estimates with the small- est equivalent lengths. These estimates resemble "monthly values" most closely. Clearly, just as in the actual water budget of a catchment, the storage term based on recession analysis is significant in the SWB method.

To check point 2, we computed regularly spaced (monthly) time series of precipitation (P) and streamflow (Q) for each basin. Monthly time series of (E) were obtained from the irregularly spaced seasonal estimates by means of (6). As an index of relative quality of the estimate, for each month the equivalent length n i was calculated from (7). Figure 6 shows the relative frequency of n i for the Cinzas River basin; a similar pattern was found for the Jangada River. It shows clearly that the SWB estimates cannot be interpreted directly as monthly values: Although the mode is in the interval 30-40 days, well

0.6

0.5

0.4

0.3

0.2

0.1

Cinzas River basin (1961-• 980) Tendency envelope

ß ß ß

0 50 100 150 200

n (days)

Figure 5. Relative importance of the storage term as a function of the water budget length in (2) for the Cinzas River.

25

20

=e 15

._

_• lO

' Cinzas River basin (1961 980)

0 150 200 50 1 O0

Equivalent length

Figure 6. Histogram of equivalent length of the {E} estimates for the Cinzas River basin.


' Jan•lada •lver'basir• (19•5-19.•0) ' ' precipitation -- --e- -.

streamflow -.-m -. SWB evapotranspiration .... •,---

"0

"-ll' '• ...... e'"

""' ...... •.. ,a. .... a- .... •', ,• .... ".•-•. '•'-•,. ..•::2•.'" •.•

...,I....... • / "'-•.. • ...•' •... ß ..,..ffi/ ' .....

0 I I I I I I I I I I I I Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

' Cihzas hiver'basir[ (19•1-19•0) ' ' precipitation -- --e- ..

streamflow -.-m.-. SWB evapotranspiration .... •,---

',.. •.. 't, /0' I,. •"- "'1,. ,'" A"

'l.-....m ..... •, ..... .•. ..... 4•-' ..m "•... ß ..... m•. ..-•' .... •."'"

I I I I I I I I I I I I

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Month

Figure 7. Mean monthly climatology of precipitation, streamflow, and seasonal water budget (SWB) evapotranspiration estimates at the Jangada and Cinzas basins.

over half span more than 40 days. This indicates that seasonal rather than monthly patterns are more likely to be correctly captured.

Nonetheless, the method seems to be able to capture the long-term mean monthly behavior: In Figure 7 we show the climatology of monthly means of P, Q, and E for the Jangada and Cinzas catchments. Although precipitation and streamflow scarcely show any seasonality in the Jangada River, the SWB estimates correctly follow the march of the seasons, with lower values in winter than in summer. In the Cinzas River basin,

precipitation has a marked seasonal pattern, also followed by SWB evapotranspiration. In both basins it is remarkable that the climatologies of SWB evapotranspiration, derived solely from precipitation and streamflow data, show smoother patterns than those of precipitation and streamflow themselves.

Finally, to check point 3, further confirmation of the ability of the method to capture the mean monthly behavior can be obtained by comparing the SWB estimates with net radiation and potential evaporation calculated independently from meteorological data. The mean daily meteorological data from the weather stations were used to derive daily values of net radiation R,, Penman [1948] potential evaporation E v, and Priestley and Taylor's [1972] potential evaoporation E s. In all cases, net radiation was estimated with Angstr6m-Prescott's equation for solar radiation, Brutsaert's [1975] equation for downwelling longwave radiation, and an empirical factor to account for its increase due to cloudiness [Brutsaert, 1982, pp. 128-144]. The parameters a and b for Angstr6m-Prescott's equation were obtained by Wrege et al. [1997] for several meteorological stations in Paranti State; Table 1 shows the parameters used in both basins for the calculation of net radiation.

Since its introduction by Thornthwaite [1948], the concept of potential evaporation has been the subject of several interpre- tations [Morton, 1983; Granger, 1989; Nash, 1989]. The Penman evaporation formula is akin to the evaporation from a small saturated surface surrounded by natural, usually unsaturated,

Table 1. Parameters Used to Compute Solar Radiation With •qstr6m-Prescott's Equation and Net Radiation Watershed a b Albedo Emissivity

Jangada 0.18 0.47 0.15 0.97 Cinzas 0.24 0.51 0.15 0.97

surfaces; therefore it is possible for it to become even larger than net radiation due to energy advection from drier, warmer surroundings. The Priestley-Taylor evaporation formula, on the other hand, is constrained to produce values lower than net radiation except under extremely hot temperatures and is akin to the upper limit evaporation/evapotranspiration from large saturated surfaces.

The climatologies of monthly net radiation, evapotranspiration, Penman potential evaporation, and Priestley-Taylor potential evaporation, all given in W m -2, are shown in Figure 8. In the Jangada River basin the annual variations of net radiation are larger than those of SWB evapotranspiration, with the implication that during May, June, July, and August the mean monthly sensible heat flux (H) is negative (directed downward). Although somewhat surprising for the subtropical latitudes of the Jangada basin, persistent negative heat fluxes are not uncommon during fall and winter [Morton, 1976]. Be- cause of the prevailing high relative humidity and moderate temperatures, (Ep) and (Es) are usually very close to each other and below net radiation. This may be equally surprising, given that (LE) > (R,) in wintertime, where L is the latent heat of the evaporation of water. However, it is known that the Priestley-Taylor equation does not perform well when H < 0, in which case the value of its constant a (usually assumed to be 1.26) would have to be increased for it to match the energy budget equation [Dias, 1992; Eichinger et al., 1996]. Penman's equation underestimates actual evapotranspiration during winter as well. Bearing in mind that the Jangada catchment is covered with forests, it is likely that the usual wind function used in the aerodynamic term of Penman's equation (originally derived for short grass and open water) underestimates the surface roughness. In the Cinzas River basin, on the other hand, SWB evapotranspiration is always less than (R.), (LEs), and (LEp).

4. Hydrometeorological Evapotranspiration Monthly Models

As seen in section 3, the time series of monthly (Ei) derived from the SWB method cannot fully capture the monthly behavior of evapotranspiration. For example, in Figure 9 the SWB evapotranspiration time series for the Cinzas River basin remains flat for several months at a time because the equivalent length n i for those months is very large, and the corre-


200

150

E 100

.... Jang•da !•iver basin'(197'5-19•0) ' '

(LE) ---•--- -

%. ., ø,•f:...•" -' ' 50 '%• '" • .... • ß ....."

"'õ'"'""'"""a ......... er .......... Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

.... Cin'zas •iver basid('•96•-'•9&• ' '

/L• ...... a ...... ' (L/: ---'•---

[3 .... "x ...... •"'• ß ,,, .,.. / /' -

"•; ?/ iZ[/'

ß ,.•,..,.,..., 'x', / /' ß .. ".,,• , •' ,,

•",, ..... 15 ......... e/ ,•o ',•, o,,,'

i i i i i i i I I dct I I Jan Feb Mar Apr May Jun Jul Aug Sep Nov Dec Month

Figure 8. Mean monthly climatology of net radiation, SWB evapotranspiration E, Penman potential evaporation Ep, and Priestley-Taylor potential evaporation E• for the Jangada and Cinzas basins.

sponding (E•) estimates were derived essentially from a single water budget period.

Evapotranspiration can be estimated from atmospheric data on scales from 1 hour to 1 month, e.g., with the help of soil- vegetation-atmosphere transfer (SVAT) schemes (hourly [No- ilhan and Planton, 1989]), evapoclimatonomy (daily [Nicholson et al., 1997]), the complementary relationship (daily [Brutsaert and Stricker, 1979], hourly [Parlange and Katul, 1992], and monthly [Morton, 1983]), or an empirical relation between actual evapotranspiration (E) and an upper limit (E•uv) of the form

(E)/(Esup) = /3, (8)

where (Esup) can be potential evaporation or solar or net radiation and/3 is assumed to depend on soil moisture availability [Brubaker and Entekhabi, 1996; Entekhabi et al., 1992; erago, 1996]. Lettau and Baradas [1973] used (8) with solar radiation for (Esup) and assumed/3 to depend on precipitation minus outflow from the basin. Guetter et al. [1997] used Pen- man's potential evaporation for (Esup) and assumed/3 to be a linear function of total basin storage in a large watershed in southern Brazil. Inherent limitations of (8) are the need to assume a functional dependence of/3 (say, on soil moisture) and the ambiguity in the definition of potential evaporation [Brutsaert, 1982, pp. 243-244].

To examine possible relationships between/3 and standard hydrometeorological parameters, we have selected those SWB evapotranspiration estimates with equivalent lengths -<45 days and assumed they are representative of monthly evapotranspiration. These values were used in (8), with (Esup) taken either as net radiation, Penman potential evaporation, or Priestley- Taylor potential evaporation to assess the dependence of/3 on mean basin storage (as an index of water availability), mean water vapor pressure deficit, and precipitation minus outflow. With an independently derived data set of evapotranspiration, the validity of (8) can be checked rather than assumed before it is applied, say, in a water budget or monthly rainfall-runoff model such as Thornthwaite's [1948] or Lettau and Baradas' [1973]. Fitting an empirical relationship between SWB evapotranspiration and hydrometeorological variables for a given basin is similar to the calibration of a rainfall-runoff model, providing a tool that can be used locally in water resources planning and design.

To calculate mean monthly basin storage, an arbitrary initial

null value for S• is assumed, and then (2) is applied for regularly spaced At (1 month), with Sf = S•+• from the SWB monthly evapotranspiration time series. Thus mean storage is simply

(Si+l) '- (Si "Jr- Si+•)/2. (9)

The empirical relationships were investigated in both dimen- sional and nondimensional form:

(E)/(Esup) = Co + ci((P) - (Q)) + c2((e*a) -- (ea)) + c3((S)),

( <e>h ( <ea> (E)/(Esup) = Co + ci 1 - (p) / + C 2 'l -- (e*a),]

(•_o)

'Jr- C 3 1 -- S---•/' (11) where Sma x is the amplitude (maximum to minimum) of basin storage calculated from a monthly water budget with the SWB evapotranspiration estimates, e a is water vapor pressure, and

' ' ' Cin•as R•verb•asin ('1981-'1991i ' (E) (HEM) ....... (E) (SWB• (• (HEM• (• (SWB)

6 ,

2

0

lOO

-lOO

-200

-300

Figure 9. Evapotranspiration and basin storage for the Cin- zas River basin for 1981-1992. Solid lines represent SWB estimates; dotted lines are hydrometeorological evapotranspiration monthly (HEM) estimates ((Esup) = (Rn); independent variables are (P) - (Q) and (e'a) - (ea)).

81 82 83 84 85 86 87 88 89 90 91

Year


Table 2. Best Evapotranspiration Models for the Jangada River Basin

r 2, BIAS, RMSE, TAD, Period (Esup) Independent Variables % s mm d -1 mm d -1 mm d -1

Year (En) (ea*) - (ea) 46.22 0.3908 -0.179 0.901 -0.089 (P) - (Q) and (e'a) - (ea) 46.24 0.3819 -0.173 0.898 -0.081

(E,) (ea*) -- (ea) 51.72 0.3621 --0.241 0.839 --0.184 (P)- (Q) and (e'a) - (ea) 53.46 0.3466 -0.203 0.800 -0.124

(Es) (ea*) -- (ea) 47.81 0.5197 --0.163 1.002 --0.041 (P)- (Q) and (e'a) - (ea) 47.86 0.5078 --0.174 1.005 --0.058

Summer (Rn) (P)- (Q) and (e'a) - (ea) 45.25 0.1325 -0.204 0.613 -0.173 (P) -- (Q), (ea*) -- (ea) , and (Si) 45.43 0.1322 -0.152 0.608 -0.229

(Ep) (P)- (Q) and (e'a) - (ea) 54.22 0.1454 --0.223 0.622 --0.195 (P) -- (Q), (ea*) -- (ea) , and (Si) 54.35 0.1451 -0.174 0.616 -0.252

(Es) (P)- (Q) and (e'a) - (ea) 44.28 0.1536 --0.216 0.609 --0.169 (P) -- (Q), (ea*) -- (ea) , and (Si) 44.32 0.1536 -0.193 0.605 -0.236

Winter (Rn) (P)- (Q) and (e'a) - (ea) 31.14 0.4223 --0.231 0.891 --0.173 (P) - (Q), (ea*) - (ea), and (Si) 34.62 0.4115 -0.497 1.010 -0.229

(Ep) (P)- (Q) and (e'a) - (ea) 43.42 0.3681 --0.220 0.763 --0.195 (P) - (Q), (ea*) - (ea), and (Si) 47.89 0.3533 -0.482 0.904 -0.252

(Es) (P)- (Q) and (e'a) - (ea) 30.93 0.5637 --0.234 0.642 --0.169 (P) - (Q), (ea*) - (ea), and (Si) 33.63 0.5526 -0.296 1.043 -0.236

Calibration period is 1975-1984, and validation period is 1985-1990. Period indicates whether the regression used all the data (yearly) or only that from cold or warm months. Independent variables show the set of variables that produced the best results. Finally, r 2 is the coefficient of determination for the regression, s is the standard error of estimate, RMSE is root-mean-square error, and TAD is total accumulated difference.

e* is saturation water vapor pressure. For each basin and choice of either (Rn) , (Ep), or (Es) to take the place of (Esup) , (10) and (11) represent seven regression equations each, de- pending on various combinations of one, two, or three independent variables. Moreover, we have tried regressions using data from all months, "cold" (April-September) months, and "warm" (October-March) months. Altogether, this amounts to 14 combinations x 3 (Esup)variables x 2 basins x 3 regression periods (yearly, cold, and warm) - 252 regressions.

Data from both basins were split into a calibration period and a validation period. Selected "best" results are shown in Table 2 for the Jangada River and Table 3 for the Cinzas

River. The criteria for choosing those best cases were (1) the coefficient of determination attained in the calibration and (2) simultaneously low values of the standard error of estimate s in the calibration and the root-mean-square error (RMSE) in the validation.

The best regression equations provide a means to use hydrological ((P) - (Q) and (S)) and meteorological ((ra) , (ea) , and (Rn)) data to estimate actual evapotranspiration from the watershed on a monthly basis using (10) or (11). We shall refer to the family of models described by (10) and (11) as hydrometeorological evapotranspiration monthly models (HEM) because they use both hydrological and meteorological

Table 3. Best Evapotranspiration Models for the Cinzas River Basin

r 2 , Period (Esup) Independent Variables % s

BIAS, RMSE, TAD, mmd -1 mmd -1 mm d -1

Year (Rn) (P)- (Q) 38.85 0.2074 (P)- (Q) and (e*•) - (ea) 42.98 0.1973

(Ep) (P)- (Q) and (e'a) - (ea) 52.43 0.2041 (P) - (Q), (ea*) - (ea), and (Si) 52.47 0.2040

(Es) (P)- (Q) and (e'a) - (ea) 38.51 0.2410

(1 _ •) and (1 _ e%) 13.70 0.2862 Summer (Rn) (P) - (Q) 44.42 0.1492

(P)- (Q) and (e'a) - (ea) 44.74 0.1449 (Ep) (P)- (Q) and (e'a) - (ea) 51.94 0.1512

(1 _ •) and (1 _ e%) 13.58 0.2028 (Es) (P) - (Q) 43.28 0.1666

(P)- (Q) and (e'a) - (ea) 43.40 0.1620 Winter (Rn) (P) - (Q) 65.97 0.1982

(P)- (Q) and (e'a) - (ea) 69.00 0.1826 (Ep) (P)- (Q) and (e'a) - (ea) 73.82 0.1905

(1 _ •) and (1 _ e•) 35.50 0.2989 (Es) (P) - (Q) 64.55 0.2454

(P)- (Q) and (e'a) - (ea) 68.80 0.2221

0.029 0.738 0.055 --0.147 0.698 --0.048 --0.034 0.755 0.009 --0.224 0.810 --0.137 --0.082 0.717 --0.012 -0.044 0.705 -0.182

-0.092 0.766 0.003 -0.044 0.795 -0.015

0.035 0.819 0.056 0.026 0.806 -0.012

-0.032 0.780 0.060 -0.001 0.802 0.026

0.111 0.715 0.003 0.055 0.659 -0.015 0.189 0.743 0.056 0.095 0.642 -0.012

0.196 0.787 0.060 0.122 0.702 0.026

Calibration period is 1961-1980, and validation period is 1981-1991.


2.5 , ' Clnza• River baslr[ (1961-1986) Calibration period ß

2

ß

1,$ ß

1L ß e ee ß • i .• ._.t_..•_ - •.

o,• - •'_•-••'"-' ..' 0 I I I I I I

0 2 4 6 8 10

(P)- (Q) (mm day '1)

' ' Cinz',,s River'basin (1'961-198'0) ' Calibration period ß

e e • ee e • ß ee ß ee..-

ie ß

' ' ' Cinz='s River'basin i1961-1680) ' Calibration period ß

ß ß eee ß

' ' ' ' o'o ;o ;o ' • ' 'o ;o ' ;o ;o 200 400 600 800 1 0 1 0 1 0 -20 - 0 0 1 30 60

(e;)- (ea) (Pa) ($) (mm)

Figure 10. Evapotranspiration as a fraction of net radiation versus (P) - (Q), (e'a) -- (ea), and (Si) in the Cinzas River basin. Data points are from all months of the year for which n i < 45 days.

data. They also constitute a practical way to investigate the actual dependence, if any, that may exist between (E) and the independent variables used in the regressions. For example, in all cases shown in Tables 2 and 3, c2 < 0. This shows that actual evapotranspiration responds negatively to water vapor deficit, the opposite of the behavior predicted by Penman's equation.

In the Jangada River, (e'a) - (ea) usually explains most of the variance, followed by (P) - (Q); the opposite occurs at the Cinzas River. The fact that (P) - (Q) figures so often among the best regression models is not at all surprising, since the SWB evapotranspiration estimates are derived from precipitation and streamflow data. However, the exercise is not redundant since (P) - (Q) data are being used to validate the models over distinct periods in an analogy to rainfall-runoff modeling. In fact, it is apparent from our results that the use of hydrological data may improve monthly evapotranspiration models.

Another concern might be that the use of dimensionless quantities on the left sides of (10) and (11) introduce spurious correlations with the right side variables. This could be particularly serious when (Esup) = (Ep), the Penman estimate, since the latter is known to depend strongly on the aridity of the environment through the water vapor deficit term. This problem does not threaten either Priestley-Taylor evaporation (Es) or net radiation (Rn). Therefore one cannot ascertain whether using the Penman equation in (Esup) will produce robust models, particularly in more arid regions. In the case of the Jangada River basin, where better correlation coefficients are found

when (Esup) = (Ep), the region is rather humid throughout the year, (Ep) and (Es) are always close in magnitude, and evapotranspiration estimates do not depend much on the choice for (Esup); still, the higher correlation coefficients when Ep is used are probably due to the presence of the water vapor deficit term on both sides of the regression models.

Although, as mentioned above, many models of the monthly water budget assume that/3 depends on basin storage, we have found SWB storage to be very weakly correlated with (Ei)/(Esup): It never appears independently in Tables 2 and 3; its coefficient of determination in univariate regressions (not shown) is never above 6%, and the root-mean-square error in model validation is always very large (see section 5). The weakness of the SWB relation with (Ei)/(Esup) does not imply that basin storage and soil moisture are not important factors determining evapotranspiration; as shown in section 5, the SWB basin storage timescale is seasonal rather than monthly. We conjecture that

(P) - (Q) is a better descriptor of storage variations within a month than is (Si).

One can gain an idea of the HEM regressions in the univariate case in Figure 10 where all-year data of (E)/(R,•) are plotted against (P) - (Q), (e'a) -- (ea), and (St) for the Cinzas River basin.

S. HEM Model Validation and Discussion

To validate the HEM models, one cannot compare the HEM and SWB estimates directly because their timescales are different. This is seen in Figure 9, where the damped nature of the SWB time series reflects water budget periods much longer than 1 month. Therefore we selected again only those months in the validation period with n i -< 45 days (the same criterion used in the selection of data points for model calibration) and calculated from those data the bias (BIAS) and RMSE. It is also possible to calculate the total accumulated difference (TAD) between the SWB and HEM estimates for the whole validation period, as well as to plot the monthly accumulated values, to gain an idea of how well the fitted models preserve mass. This is done in Figure 11 for the Cinzas River ((Esup) -- (R,,); independent variables are (P) - (Q) and (e'a) - (ea)). Similar results were obtained for the Jangada River. The independent variables used in the HEM models are the same for Figures 9 and 11.

Tables 2 and 3 also give the values of BIAS, RMSE, and

12000 ' ' ' Cinias Ri[tor b•si. ({ 981-• 991)' ' 11000 (E) (HEM) ....... -I

10000

9000

8000

'•' 7000 '-' 6000

• 5000

4000

3000

2000 1000

o ' ' ' ' ' ' ' 81 82 83 84 86 89 90 91

Year

Figure 11. Accumulated SWB and HEM monthly evapotranspiration for the Cinzas River during the validation period.


TAD for the best HEM models. Notice that when data are split into warm and cold months, the corresponding model involves two sets of regression coefficients on the same independent variables; this is why the TAD values, which are related to the whole validation period, appear repeated in summer and winter.

The HEM estimates are far from accurate: Even in calibra-

tion, the standard error of estimate s usually amounts to 20% or more of (Esup). The RMSE values in the validation periods are of the order of 0.7 mm d -•, again not a very small value. On the other hand, what the HEM models provide is the correct monthly behavior, effectively "disagregating" the seasonal SWB estimates to the monthly timescale, and in this they seem rather successful (Figure 9). What is equally important is the ability of the HEM models to preserve mass approximately.

In Figure 9 we have plotted the basin storage (S) calculated from (2) and (8) for both SWB and HEM evapotranspiration estimates. The SWB time series of basin storage displays the same damped behavior of the corresponding evapotranspiration series and tends not to depart too much from its initial arbitrary value of zero. The HEM values of basin storage, on the other hand, fluctuate much more, both because they capture the monthly behavior of evapotranspiration and because their accumulated values remain above and below the SWB

totals during several years (see Figure 11), reflecting marked interannual variations of basin storage.

6. Conclusions

Water budget studies at the watershed level are always difficult because of the incompleteness of hydrological data. Evapotranspiration, soil moisture, and groundwater time series are almost never existent. This fact leads to the use of models

with unverified assumptions that may induce erroneous conclusions about the basin's response to meteorological forcings such as precipitation and net radiation. While scientifically sound approaches require experimental efforts to measure more hydrological variables continuously and the use of de- tailed models of the soil-plant-atmosphere system, there still exists a demand for more complete and accurate analyses of historical data sets and to generate evapotranspiration monthly time series from the limited information available in the precipitation, streamflow, and meteorological daily data.

Motivated by this practical need, we have tried to interpolate the water budget equation to periods shorter than 1 year. This can be done by calculating the budget over irregularly spaced time periods, whose end points are the ends of relatively long hydrologic recessions. By assuming that a linear flow-storage relation holds during each recession, the time constants for the recessions are easily estimated. This allows storage terms to be estimated, rather than neglected, in the water budget: Storage changes can be significant (up to 50% of (P) - (Q)) for the shorter water budget periods. The corresponding seasonal water budget (SWB) evapotranspiration estimates have typical timescales longer than 1 month; however, their mean monthly behavior captures the net radiation signal even when it is not evident in the parent precipitation and streamflow data. The monthly SWB means also exhibit the correct order of magnitude when compared to net radiation and Penman and Priest- ley-Taylor evaporation estimates.

From the shortest SWB periods and meteorological data it is possible to derive simple hydrometeorological evapotranspiration monthly (HEM) models with fair accuracy (a standard error of ---20% of the net radiation monthly values, or 0.7 mm

d-•). These HEM models produce the expected monthly behavior while approximately conserving mass over validation periods of 10 or more years. Since the HEM parameters are obtained locally by regression, the HEM models are more likely to capture local behavior than models employing "uni- versal" a priori constants or relationships. In this they resemble classical rainfall-runoff models that are calibrated for specific catchments. Still, for two climatically different basins analyzed here, some uniform features were found, the most important being a negative relationship between actual evapotranspiration and water vapor deficit.

Acknowledgments. The authors wish to thank Wilfried Brutsaert and an anonymous reviewer, whose comments and suggestions con- tributed to improve the original version considerably.

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(Received December 22, 1998; revised July 19, 1999; accepted July 22, 1999.)

a hydrometeorological model for basin-wide seasonal evapotranspiration

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