A sensitivity analysis of the Penman-Monteith actual evapotranspiration estimates

Journal of Hydrology, 44 (1979) 169---190

169

© Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

[3] A SENSITIVITY ANALYSIS OF THE PENMAN--MONTEITH ACTUAL EVAPOTRANSPIRATION ESTIMATES

KEITH BEVEN

Institute of Hydrology, Wallingford, Oxon OX10 8BB (Great Britain) (Received January 10, 1979; revised and accepted May 21, 1979)

ABSTRACT

Beven, K., 1979. A sensitivity analysis of the Penman--Monteith actual evapotranspiration estimates. J. Hydrol., 44: 169--190. The evapotranspiration component, based on the Penman--Monteith equation, of the distributed Syst~me Hydrologique Europ~en (SHE) model of catchment hydrology is briefly described. The importance of evapotranspiration predictions to the operation of the model is stressed. However, such predictions may be expected to be in error and ideally it would be desirable to predict the error variance associated with estimates of evapotranspiration. This is not yet possible but the sensitivity of such estimates to errors in input data and estimated model parameter values can be investigated. It is shown that for dry canopy conditions at three sites within a broadly humid temperate region, the sensitivity of Penman--Monteith estimates of evapotranspiration to different input data and parameters is very dependent on the values of the aerodynamic and canopy resistance parameters that introduce the influence of vegetation type into the predictions. For forest surfaces in particular, the evapotranspiration predictions are highly sensitive to values of the canopy resistance, so that accurate estimation of this parameter is important.

INTRODUCTION

This paper reports a study of the sensitivity of the Penman--Monteith equation for estimating actual evapotranspiration rates to errors in parameter values and input data. The study has been carried out as part of continuing work on the Syst~me Hydrologique Europ~en (SHE), a deterministic distributed hydrologic modelling system which is being jointly developed by the Danish Hydraulics Institute, SOGREAH (France) and the Institute of Hydrology (U.K.). A brief outline of SHE is given here to place the present study in context. The SHE model is physically-based in that it has been developed from equations of flow for the processes of overland and channel flow, unsaturated and saturated subsurface flow. Finite-difference methods are used to obtain solutions to these non-linear flow equations. With present computer limitations, it is not economically viable to produce an operational model fully three-dimensional in space (see Freeze, 1978) that would allow the required accuracy of discretisation in both horizontal and vertical planes. On the

170

assumption that in the unsaturated zone vertical flow is far more important than lateral flow, the model has been rationally simplified such that independent one-dimensional unsaturated flow components of varying depth are used to link a two-dimensional groundwater flow component and a twodimensional surface-water flow component (Fig. 1). This structure has important implications for the modelling of other components of the hydrological cycle. It is simple, for example, to model spatial variations in evapotranspiration rates due to changes in moisture conditions and vegetation assemblages between different grid squares of the model. Moreover, it becomes possible to attempt to model the effects of localised vegetation and land-use changes over time in some more rigorous physically-based manner than has been hitherto feasible using lumped catchment models. The evapotranspiration component of SHE is necessarily of particular importance since in many catchments in varied climatic zones, evapotranspiration is the dominant hydrological process in volume or water balance terms. Even when the response to storm rainfall of a catchment may be little affected by evapotranspiration during the period of a storm, losses to the atmosphere may be extremely important between storm periods insetting up the initial soil-moisture conditions prior to the next event. In developing the SHE model it has been explicitly recognised that, for a model that is expected to have widespread application, it must always be the

R~modelzone

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Fig. 1. A diagrammatic representation of the Syst~me Hydrologique Europ~en (SHE) model.

171 aim to make maximum use of the (often limited) data that will be available for a given site. Thus it was felt that no clear-cut decisions as to the most suitable evapotranspiration model could be taken independent of the application. Thus flexibility has been retained in the model programming, with a hierarchy of evapotranspiration components requiring different levels of data availability and parametric input, but consequently with different degrees of theoretical acceptability. At the top of this hierarchy, as the most complex and physically realistic model considered for the purposes of SHE, is the Penman--Monteith combination equation (Monteith, 1965) for the prediction of actual evapotranspiration rates. However, the choice of an evapotranspiration model is not simple. The complexities of the processes of evapotranspiration and its interaction with soil moisture are paramount. The current generation of predictive models of these processes, including the Penman--Monteith equation, are broadly physically-based, but remain simplistic and subject to important limiting assumptions. The parameters of these models exhibit significant variations in both space and time, and whereas there are now a reasonable number of studies in which parameter values have been derived from measurements of evapotranspiration, studies of the p r e d i c t i o n of parameter values in other situations have been notably lacking. The evapotranspiration component shares these problems with other model components, but in the case of evapotranspiration the problems are compounded by the scale of variations in time and space, and the physical scale of measurement studies, which are both small relative to catchment scales of interest. Yet, the importance of accurate predictions of evapotranspiration rates cannot be underestimated. The use of a distributed model in itself goes some way towards diminishing these problems but, in accepting the limitations of available evapotranspiration models, an assessment of the effects of errors in both the measurement of input data and the estimation of parameter values becomes of particular importance. The aim of such a sensitivity analysis is to make clear what range of accuracy is required for an input variable or parameter value, and consequently where the greater effort should be expended in measurement or model calibration. The acceptability of measurement or extrapolation techniques, and the confidence in model predictions should reflect the interpretation of sensitivity estimates reported from studies such as that in this paper. THE PENMAN--MONTEITH EVAPOTRANSPIRATIONEQUATION The Penman--Monteith equation is a one-dimensional single-source model of the evapotranspiration process. The assumptions on which it is based are described in the development by Monteith (1965). Within the limitations of these assumptions actual evapotranspiration (Ea) is predicted b y : E a

~---

A s A + Pcp(qw, TD - q)/r a ~.[As + (Cp/~.)(1 + rc/ra)]

(kg m -2 s -l ~ m m

s-I )

(I)

172 where

Cp P A RN G

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(qW,TD SHD

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= latent heat of vapourisation of (= 2.47.106 J kg -l) water = specific heat of air at constant pressure (= 1.01 103 J kg -l °C -I) (= 1.2 kg m -2) = density of air = available energy given by A = (W m -2) RN - G = net radiation measured at the reference height, z (W m -2) denotes the sum of energy fluxes into the ground, to adsorption by photosynthesis and respiration and to storage between ground level and z (W m -2) = saturated specific humidity at dry-bulk temperature, TD (kg kg- ~) = specific humidity deficit (kg kg -~) = derived from measurements of TD and wet~bulb temperature depression, DEP = slope of the specific humidity/ temperature curve between the air temperature TD and the surface temperature of the (kg kg -1 °C-1) vegetation T s = aerodynamic resistance to the transport of water vapour from the surface to the reference level z (s m -l) and = (Monteith) canopy resistance to the transport of water from some region within or below the evaporating surface to the surface itself, and is expected to be a function of the stomatal resistance of individual leaves. Under wet-canopy conditions rc = 0 (s m -x)

Eq. 1 continues the one-dimensional vertical structure of SHE and assumes further that all evapotranspiration within the complex soil--vegetation canopy system takes place from a single representative source layer. Stewart (1979) describes the development and assumptions underlying the Penman--Monteith equation and its relationship to other similar evapotranspiration models. The

173

meteorological data required for the model are values of A (or RN if G is small), TD and DEP (= TD -- TW ) where Tw is a wet-bulb temperature. The parametric data required are values of ra and rc. PARAMETRIC AND METEOROLOGICAL INPUT DATA

This study has been designed to allow consideration of two vegetation types and a range of meteorological conditions within a broadly temperate maritime climatic regime. Data from three meteorological sites maintained by the Institute of Hydrology, Wallingford, have been used, spaced across England and Wales from west to east (Fig. 2). At each site one or two automatic weather stations (AWS), as developed by the Institute of Hydrology (Strangeways and McCulloch, 1965), collect data on incident solar radiation, net radiation, air temperature, wet-bulb temperature, wind run and direction, and rainfall on a 5-min. time base. The original records are routinely validated and processed into the mean hourly values that are used in the present study. The input data required by the Penman--Monteith equation can all be supplied by the standard AWS measurements with the exception of G, which has been neglected in the sensitivity analysis. Values of A s and SHD are calculated from dry-bulb (TD) and wet~bulb (Tw) temperature measurements from equations given in Appendix A. The characteristics of each measurement site are as follows.

Fig. 2. A location diagram for the three meteorological stations used in this study. 1 = Carregg Wen, Plynlimon; 2 = G r e n d o n U n d e r w o o d ; and 3 = Thetford Forest.

174

S i t e 1 - - P l y n l i m o n , Carreg Wen. This station lies at an altitude of 575 m

A.O.D. and is the highest of the meteorological stations in the Institute of Hydrology's experimental catchments at Plynlimon, Wales. Vegetation at the site is an upland grassland community. Site 2 - - G r e n d o n U n d e r w o o d . The Grendon Underwood site lies in the River

Ray experimental catchment in the Oxford clay vale. The altitude of the site is 67 m A.O.D. and the vegetation is short grass pasture. Site 3 - - T h e t f o r d Forest. The site at Thetford Forest, Norfolk, was main-

tained as part of the Institute of Hydrology's Thetford Forest evaporation project. The site is at an altitude of 47 m A.O.D. and is above the canopy of Scots and Corsican pine (see Stewart and Thom, 1973).

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175

Seven months of fully processed data were available for all three sites, covering the period May--November 1976, and including the abnormally dry British summer of that year. The calculation of evapotranspiration estimates and the sensitivity analysis described below was carried out for all hours with positive net radiation and a complete set of satisfactory meteorological measurements, giving 2406 hr. for the Plynlimon site, 2056 hr. for Grendon Underwood and 2191 hr. for Thetford Forest over the 7-month period. The meteorological input to the present study for the three sites is summarised and compared in terms of mean monthly values and mean hourly values over the complete study period in Figs. 3 and 4. Fig. 3 shows some interesting differences between the sites. The lowland sites are consistently warmer than the upland Plynlimon site, with similar mean monthly temperature curves. However, the central Grendon Underwood site has much lower mean net radiation than the Thetford site, close indeed to the Plynlimon site (which 400

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176 must reflect the nature of the surface over which the measurements were made) b u t the wet-bulb depression and consequently the specific humidity deficit are both higher. Fig. 4 demonstrates the mean diurnal pattern of change at the sites and shows h o w dry-bulb temperatur6 wet-bulb depression and consequently A s and SHD all continue to rise throughout the afternoon following the peak in net radiation at 12h00 m GMT. The mean temperature characteristics of the two lowland sites correspond quite closely, with the upland site having lower air temperatures and wet-bulb depression. The parametric data required essentially consists of the two resistance coefficients r a and re. Both resistances are expected to vary over time in some complex way. One form in which a variable ra may be calculated is from: [ln((z ra =

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where u = mean wind speed z = reference height of the anemometer d = zero plane displacement Zo = roughness length k = v o n Karman's constant

(m s -l) (m) (m) (m) (= 0.41)

This form was originally used by Penman and Long (1960), Monteith and Szeicz (1962) and Van Bavel (1966). However, there is evidence that b o t h z0 and d themselves vary with wind speed (e.g., Szeicz and Long, 1969), and also that the improvement in model predictions b y allowing ra to vary with wind speed, as opposed to using a constant value may be small (Calder, 1977). For the purpose of the present study, constant values have been used, chosen to be representative of grass and forest vegetation surface. The values used were 46 s m -1 for the former surface (Thorn and Oliver, 1977), and 4 s m -1 for the latter (Calder, 1977). Values of rc are known to have a diurnal variation (see, e.g., Van Bavel, 1967; Szeicz and Long, 1969; Stewart and Thorn, 1973; Tan and Black, 1976) and there is evidence that they may also show important seasonal variation (Calder, 1977). Diurnal variations in mean hourly Values of rc relative to an assumed mid-day value for the grass and forest surfaces have been assumed on the basis of measurements given in Szeicz and Long (1969), and Stewart and Thom (1973) as shown in Fig. 5. The relative diurnal distribution has been assumed constant over the s t u d y period and for both surfaces. Changes in surface resistance as a result of a w e t canopy following rainfall (see, e.g., Stewart, 1978) have also been ignored for present purposes.

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SOURCES OF MODEL ERROR VARIANCE

We must accept that applications of the evapotranspiration c o m p o n e n t of the SHE model will be subject to errors (perhaps significant) in both input data and parameter estimation. Such errors must result in an error variance associated with the estimates of evapotranspiration. At least four sources of error contributing to the estimation error m a y be identified.

I n s t r u m e n t error It is expected t h a t every measurement that forms the input data to the model will be subject to error, such that:

rni(t) = m i ( t ) + E l ( t ) where ~ni(t ) is the ith recorded measurement at time t, mi (t) is the true measurement, and E i ( t ) is an error term. In general E i will be a random variable, but may be biased due to errors in instrument settings, etc. This bias may show a trend over time due to instrument drift, with sudden changes as the instruments are reset.

178 S i t e errors

When applying the evapotranspiration model over a catchment area we may have meteorological measurements at only one site (possibly outside the area of interest) which m u s t be taken as representative of the whole catchment. Where there are satisfactory measurements at more than one site, we must use some interpolation procedure to estimate evapotranspiration at points within the catchment. In either case, evapotranspiration estimates may be in error relative to the true amounts from the whole catchment, or an elemental grid square. This may be particularly true under advective conditions (see, e.g., McNaughton, 1976), and for measurement sites poorly situated in respect of vegetational changes in the area. Site errors may exhibit bias as well as a random c o m p o n e n t and m a y show correlation in both space and time. M e a s u r e m e n t m o d e l errors

Consider now the measurement (estimation) of evapotranspiration in the field. Errors in the field estimates will be due to both instrument errors and measurement model errors when the measurement model (say the water balance equation or Bowen ratio estimation) is inappropriate to the situation in which it is used and does n o t conform to the true nature of the physical process. Such errors may have important consequences for the later developmert of error variance in the evapotranspiration component, since the parameter values used will be based on estimates from field experiments. Again it may be expected t h a t measurement model errors m a y have bias as well as a random component, with a dependency structure over time. M o d e l l i n g errors

Where a model such as the Penman--Monteith equation is being used to fit measured evapotranspiration rates, it is (for this example) always possible to alter the parameter values to fit the measured rates exactly at each t i m e step. However, it may then be extremely difficult to predict the changes in parameter values over time to again obtain a good fit under any given conditions. In the general application of the model, parameter values must be predicted a priori and it is expected t h a t the model will subsequently be in error. Due to diurnal and seasonal variations in evapotranspiration this will be particularly true when estimates of average parameter values are the best that are available. Errors may then involve diurnal and seasonal time dependency. Considerable reductions in apparent model error variance m a y be achieved by optimising functions for expected parameter changes against measured evapotranspiration rates (see Calder, 1977). However, the number of such studies is so small as to make general application impossible at present. At present we have little or no knowledge of the characteristics of the

179 error variance associated with the estimates of evapotranspiration. On the basis of a consideration of the contributing sources of variance, we cannot assume that the individual error terms are stationary, independent and nonautocorrelated (with the possible exception of pure instrument error). Ultimately it is desirable to predict the evolution of the error variance around estimates of cumulative evapotranspiration as it changes over time, and the significance of the expected errors in the overall predictions of the model. This aim cannot yet be achieved but it is possible, using the techniques of sensitivity analysis, to make an assessment of the relative importance of the primary operational errors in input data and parameter values. This is the aim of the present study. MODEL SENSITIVITY ANALYSIS Consider first the sensitivity of an estimate of actual evapotranspiration, Ea, calculated using the Penman--Monteith equation to changes in a parameter or input data variable, Pi, where in general:

Ea = f(Pl,P2,P3,... ,PN) where N is the number of parameters and input data variables. Then:

E a + A E a = f ( p l + A p l , p 2 + A p 2 , . . . , PN+APN)

(4)

Expanding eq. 4 in Taylor series and ignoring second-order' terms and above leads to: AEa = ~~Ea Apl + aEa Ap2 + . . . + ~Ea -Ap N ~Pl ~P2 aPN

(5)

where the differentials aEa/aPi define the sensitivity of the estimate to each parameter or variable. These sensitivity coefficients are themselves sensitive to the relative magnitudes of E a and the Pi and following McCuen (1974) a non-dimensional relative sensitivity may be defined as: Si -

~Ea

Pi

(6) ~Pi Ea Si now represents t h a t fraction of the change in Pi that is transmitted to change Ea, i.e. an Si value of 0.1 would suggest t h a t a 10% increase in Pi may be expected to increase Ea by 1%. Negative coefficients would indicate t h a t a reduction in Ea will result from an increase in Pi. Note t h a t both sensitivity coefficients may vary from time step to time step being dependent on the current value of all the Pi and the value of Ea. Eq. 6 remains sensitive to the magnitudes of E a and Pi. In particular, the relative sensitivity coefficients Si may n o t be a good indication of the significance of the Pi if either E a or Pi tend to zero independently, or if the range of values taken by Pi is small in relation to its magnitude. Parameter sensitivity

180

analyses of other evaporation and evapotranspiration models have been published by McCuen (1974), Saxton (1975), and Coleman and DeCoursey (1976). The sensitivity coefficients for the Penman--Monteith equation used in this study are as follows: aEa RN aRN E a

SRN

aE a r a

Sra -

ara

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1 1 + pcpSHD/AsRNr a 1 1 + (As+Cp/X) raX

-

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TO

=

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TD

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aAs/aTD ] - A s + Cp(1 +rc/ra) aE a D E P SDEP . . . . aDEP Ea

(9)

'J

(10)

DEPpcp(aSHD/aDEP)

{raAsR N + p c p S H D }

(11)

where the differentialsaAs/aTD, a S H D / a T D and a S H D / a D E P are derived from the functional relationships of Appendix A and are given in full in Appendix It D E P rather than T w was chosen for use in the sensitivityanalysis because the value of T w includes an implicit dependence on TD. D E P m a y be considered as independent of TD (although the consequent calculation of atmospheric humidity is not). The analysis of the sensitivityof individual parameters and variables can be extended to an analysis of the error variance in the Pi values, but only under restrictiveassumptions. Following Hahn and Shapiro (1967) for a system:

Z = h(xl,x2,... ,XN) in which the parameters and variables, x, are uncorrelated, the error variance of mean system performance is given by:

N N E[var(z)] = ~ (ah/axi) 2 var(xi) + (ah/axi) (a2h/axi2)p3(xi) i=l i=1

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181 differentials are evaluated at their expected values. This expression was used by Coleman and DeCoursey (1976) to explore the error variance of several models arising from instrument measurement errors alone, assuming that those errors are independent, that the second term of the RHS of eq. 12 is negligible relative to the first and using some average sensitivity differential determined by simulation. Hahn and Shapiro (1967) extended the analysis to variables that are crosscorrelated. However, in b o t h cases there are problems in application for the present study, due to the lack of knowledge a b o u t the distribution of the residuals, and the effects of possible autocorrelation structures on the error variance of cumulative evapotranspiration rates. In the face of other sources of error, variance due to instrumentation errors when the instruments are working normally, may be expected to be relatively small. RESULTS OF MODEL SENSITIVITY ANALYSIS

Sensitivity coefficients for a grass surface Sensitivity coefficients, as defined b y eqs. 7--11 have been calculated on an hourly basis using data from the three meteorological stations and parameters representative of a grass surface as given in the section "Parametric and meteorological input data" above. These sensitivity coefficients exhibited considerable variation over time. In particular values during the dusk to dawn periods, when calculated values of Ea are small, were erratic. This is reflected in the mean hourly values of the sensitivity coefficients for the grass surface over the full study period plotted in Fig. 6 together with actual evapotranspira tion values predicted by the Penman--Monteith equation. The predicted evapotranspiration increases in value from .the upland Plynlimon site in the west to the Thetford site in the east. The mean hourly sensitivity coefficients all exhibit a diurnal variation, b u t not necessarily following the pattern of change in the calculated evapotranspiration rate. The differences in mean sensitivity coefficients between sites are generally smaller than the differences in evapotranspiration rates relative to the diurnal range, and the relationship between the sites is also different with the Thetford (higher evapotranspiration) site generally lying between the other t w o on the sensitivity plots. In the case of Src, the values are uniformly negative as befits its operation in the denominator of the Penman--Monteith equation. However, the pattern of diurnal change in Src is not consistent between the sites becoming more negative during the day at the Carreg Wen site and less at the Grendon Underwood site. The magnitude of Sr c suggests that the influence of the canopy resistance in reducing predicted evapotranspiration below some "potential" value is restricted, as would be expected for grass surfaces where the noon-tide values of rc/ra under the conditions of the analysis are close to unity.

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The changes in Sr a between positive and negative values are a result of the occurrence of ra in both numerator and denominator of the Penman-Monteith equation so that the bulk sensitivity to ra will depend on the relative importance of the radiation and aerodynamic terms in the numerator. It is clear from Fig. 6 that the aerodynamic term has greater significance during early and late parts of the day when net radiation is low, and that it is relatively more important at the Grendon Underwood site, there suppressing the mean values of Sr a to below zero throughout the day. For all the sensitivity coefficients, values for the mid-day hours, when evapotranspiration is highest, are relatively stable. Cmsidering daytime values alone, the evapotranspiration estimates are most sensitive to RN, with all the other coefficients generally falling below a mean value of 0.5 close to midday. This confirms that for grass surfaces under these conditions, the radiation term is generally dominant over the aerodynamic term in the prediction equation. Given the diurnal variability of the sensitivity coefficients, mean mid-day values ( 1 2 h 0 0 m - - 1 3 h 0 0 m) have been used to investigate the seasonal change in the sensitivity of evapotranspiration rates to parameters and input data. The mean values together with their standard deviations are plotted for the three sites in Fig. 7 and show that seasonal variation in the sensitivity coefficients is small relative to the change in predicted actual evapotranspiration. The sensitivity to RN remains the highest, while those to the resistance coefficients Sra and Src show the greatest difference between sites. In both cases, the Thetford and Grendon sites plot closely together while the Plynlimon site, where the sensitivity to rc is particularly high, differs.

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Sensitivity coefficients for a forest surface The sensitivity calculations were repeated using parameters representative of a forest surface, as given in the Section "Parametric and meteorological input data". The mean hourly estimated evapotranspiration and sensitivity coefficients are shown in Fig. 8. Evapotranspiration rates are not generally mrnS-'

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Fig. 8. Mean hourly values o f predicted actual evapotranspiration and sensitivity coefficients for a forest canopy. Mean of hours with positive net radiation, May--November, 1976. Site key as Fig. 3.

184

as high as those for the grassland. [Note that dry-canopy conditions are assumed throughout. For a discussion of the significance of the evaporation of intercepted water to total losses of forests see Stewart (1977)i] The sensitivity coefficients show the same erratic nature during the dusk to dawn period but during the day show more regular diurnal pattern in the mean. It is interesting to note that SRN is n o w quite low with a diurnal distribution having higher morning values, whereas STD and S D E P are n o w much higher with higher values in the afternoon. Values of Sra remain small, though with less scatter than for grass, but those for Src are uniformly high at all three sites with mean daytime values greater than 0.9, rising further during the afternoon. Mean mid-day values have again been used to demonstrate the seasonal drift in the sensitivity coefficients (Fig. 9). Only STD shows any marked seasonal change in this case, but the relative significance of the value of Src and SDE P is reinforced. These results for the forest canopy suggest that given the low aerodynamic resistance of the forest, the aerodynamic term of the combination equation dominates the radiation term but that the estimated evapotranspiration rates are significantly controlled by the canopy resistance under dry-canopy conditions. mmS-' 0~2 I.O

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DISCUSSION

The results of the previous section show that within a broadly humid temperate climate, .although the three sites differ considerably in the range of meteorological conditions experienced, and consequently in the predicted

N

185 evapotranspiration rates, the sensitivity coefficients do n o t differ significantly between sites. It would appear t h a t vegetation type, as introduced through the resistance parameters of the P e n m a n - M o n t e i t h equation, has a much more significant effect. This is n o t reflected in the sensitivity coefficients for the aerodynamic resistance since it appears in both numerator and denominator of the P e n m a n - M o n t e i t h equation and the sensitivity values are consequently generally low. However, the sensitivity to the specific h u m i d i t y deficit of the aerodynamic term, which is closely reflected by SDEP,will also show the influence of the value of r a. However, it is the values of Src that show the greater difference between grass and forest surfaces, being especially high in the case of the forest, when a 10% error in rc will result in a 9% or more error in predicted Ea. This assumes particular importance since values of rc may be difficult to estimate accurately. It is accepted that values of rc may be complex functions of other variables that have yet to be established satisfactorily (see, e.g., Jarvis 1976; Tan and Black, 1976; Calder, 1977). Further runs of the sensitivity analysis program with different values of ra and rc were made to evaluate the importance of the two resistance parameters. The data used were mean mid-day 12h00m--13h00 m August values of the input variable at the central Grendon Underwood site. The results are shown In Fig. 10 (note t h a t dry-canopy conditions were again assumed throughout). On the basis of the mean mid-day wind speed for August 1 at the site and assuming that (from Maki, 1975): z = h + 2 (m);

z = 0.1 h (m);

and

d = 0.7 h (m)

where h is mean vegetation height, an equivalent height scale has been added to the r a scales of Fig.10. Fig. 10 allows the regions of ra and rc for which the prediction of evapotranspiration is particularly sensitive to net radiation RN, dry-bulb temperature, TD, wet-bulb depression, DEP, or canopy resistance, rc, to be identified. Although specific values of the sensitivity coefficients will change, the evidence of this paper suggests that these regions, representing the changing balance between the aerodynamic and radiation terms of the combination equation, will be representative of a wide range of humid temperate areas under the summer daytime conditions when evapotranspiration rates are highest. The results presented in this paper may be compared with those of other sensitivity analyses of combination-type evapotranspiration equations, although in all the published cases available, only the prediction of " p o t e n t i a l " or water surface evapotranspiration rates has been considered. This is equivalent to the case of high r a and zero rc in the Penman--Monteith equation. Both McCuen (1974) and Saxton (1975) defined a relative sensitivity coefficient similar to that used here and analysed different forms of the Penman equation. The results of Saxton, who used daily data from Iowa for surfaces of corn and grass, were similar to those obtained for a grass surface in this study. McCuen analysed the Penman equation for evaporation from openwater surfaces for a number of stations over a wide range of climates in the

186 ?

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Fig. 1 0 . T h e c h a n g e in p r e d i c t e d e v a p o t r a n s p i r a t i o n a n d s e n s i t i v i t y c o e f f i c i e n t s w i t h r a a n d r e for m e a n A u g u s t m i d - d a y c o n d i t i o n s at t h e G r e n d o n U n d e r w o o d site. ( R N -- 2 6 2 . 3 9 W m - 2 ; T D -- 2 2 . 5 5 ° C ; D E P -- 6 . 9 6 ; v a l u e o f u = 2 . 5 2 m s -1 u s e d in c a l c u l a t i o n o f h - s c a l e . )

U.S.A. Mean summer meteorological conditions only were used. The results suggested that radiation, humidity and temperature could all show very high sensitivity coefficients. Coleman and DeCoursey's (1976) definition of the sensitivity coefficient was based on a different form of eq. 6, where the approximate differential is evaluated using an (undefined} perturbation of the variable concerned. A long period (16 yr.) of daily U.S. Weather Bureau data from Oklahoma was used. The sensitivities defined in this way are not directly comparable but Coleman and DeCoursey's results showed that the Penman equations for water surfaces analysed were equally sensitive to air temperature, relative humidity and solar radiation; the sensitivity balance between these variables changing over the year. Air temperature and solar radiation were the most important in the summer. Thus comparison with the present results reinforces the view that the introduction of the influence of vegetation directly into the prediction of sensitivity, changes the sensitivity balance markedly. Sensitivity to equation parameters was not analysed in any of these three previous studies.

187 CONCLUSIONS

The most important conclusion of this w o r k is that the sensitivity of Penman - M o n t e i t h estimates of actual evapotranspiration to different input variables and parameters is more dependent on the values of the aerodynamic and canopy resistance parameters that introduce the influence of vegetation t y p e into the equation, than on climatic difference between sites in a broadly humid temperate area. Dependent on these parameters, all the input variables exhibit high daytime sensitivity coefficients during a summer period when evapotranspiration rates are high. Thus accurate measurement of these variables is essential to accurate predictions of evapotranspiration. This will generally not be difficult with the temperature variables TD and DEP b u t the energy balance term, which has here been assumed to be adequately characterised by measured net radiation, R N , will in general involve other energy fluxes (see eq. 1) that may be difficult to quantify. However, such considerations may be reduced in significance by losses in accuracy due to errors in the estimation of the equation parameters r a and rc. Under conditions when the values of these parameters are markedly different (e.g., for forest surfaces) evapotranspiration estimates will be particularly sensitive to the canopy resistance rc. It has already been noted that this parameter may n o t be easy to estimate accurately. However, for vegetation surfaces that allow significant amounts of interception of rainfall, evapotranspiration rates may be highest of all under wetcanopy conditions that have been neglected in the present study. The use of a model to predict evaporation from intercepted rain water [such as that of Rutter et al. (1975)] which may take place at a "potential" rate (rc = 0) will reduce the overall sensitivity to values of rc. This will be examined further in a later paper analysing the joint interception- -evapotranspiration--unsaturated zone c o m p o n e n t of the SHE model. ACKNOWLEDGEMENTS This study would n o t have been possible without the careful work of the staff of the Institute of Hydrology concerned with the collection and processing of the automatic weather station data. I should especially like to thank Richard Harding and Dave Woolhiser who helped in the early stages of the study, and John Stewart and Howard Oliver who provided valuable comments on earlier drafts of this paper. Our colleagues on the SHE project at the Danish Hydraulics Institute and SOGREAH also contributed useful discussions. The paper is published with the permission of the Director of the Institute of Hydrology. APPENDIX A Given input data values of dry-bulb temperature, TD, and wet-bulb depres-

188

sion DEP = (TD - TW) in °C the values of the slope of the specific h u m i d i t y / temperature curve, AS, and the specific h u m i d i t y deficit SHD = {qw, T,., - q required in the Penman--Monteith equation (eq. 1) are calculated as follows. Let: x = TD/s- 3

(A-l)

then the saturation vapour pressure at dry-bulb temperature, 8VPTD is calculated from: SVPTD = 0.003x 4 + 0.063x 3 + 0.776x: + 5.487x + 17.044

(A-2)

and DSVPT D

-

dSVPTD - (0.012x 3 + 0.189x 2 + 1.552x + 5.487)/5 dT

(A-3)

Then: AS = (0.622 D S V P T D P / 1 . 0 0 4 5 ) / ( P / 1 . 0 0 4 5

- 0.378SVPTD)2

(A-4)

where P is atmospheric pressure in mbar. The saturated specific h u m i d i t y at dry-bulb temperature, qw,TD, is calculated from: 0.622 qw, T D =

(A-5)

P/1.0045SVPTD - 0.378

and the actual specific h u m i d i t y from: q = qw, Tw - K ( T D - TW)

where qw, Tw is the saturated specific h u m i d i t y at wet-bulb temperature calculated by substituting Tw into eqs. A-l, A-2 and A-5, and K is the screen psychrometric constant. APPENDIX B To complete the sensitivity analysis of the Penman- -Monteith equation, it is necessary to calculate: aAs/aTD,

aSHD/aTD

and

aSHD/aDEP

given the relationships of Appendix A. Then: aA S ~TD

1 2c B2 [0.622A ( d S V P T D / d T D ) ] - "-~ ( - 0 . 3 7 8 ( d S V P T D / d T D )

where A = P/1.0045;

B = A - 0.378SVPT D;

and

d S V P T D / d T D = (0.036x 2 + 0.278x + 1.552)/25

C = 0.622 DSVPTDA

189

and x, dSVPTD/dTD axe d e f i n e d in A p p e n d i x A, using TD a n d D E P as t h e i n p u t variables o f i n t e r e s t s u c h t h a t Tw = TD - DEP, t h e n : 0SHD_ 0 TD

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REFERENCES Calder, I.R., 1977. A model of transpiration and interception loss from a spruce forest in Plynlimon, central Wales. J. Hydrol., 33: 247--265. Coleman, G. and DeCoursey, D.G., 1976. Sensitivity and model variance analysis applied to some evaporation and evapotranspiration models. Water Resour. Res., 12(5): 873-879. Freeze, R.A., 1978. Mathematical models of hillslope hydrology. In: M.J. Kirkby (Editor), Hillslope Hydrology, Wiley, Chichester. Hahn, G.J. and Shapiro, S.S., 1967. Statistical Models in Engineering. Wiley, New York, N.Y. Jarvis, P.G., 1976. The interpretation of the variations in leaf water potential and stomatal conductance found in canopies in the field. Philos. Trans. R. Soc. London, Ser. B, 273: 593--610. Maki, T., 1975. Wind profile parameters of various canopies as influenced by wind velocity and stability. J. Agric. Meteorol. (Tokyo), 31(2): 61--70. McCuen, R.H., 1974. A sensitivity and error analysis of procedures used for estimating evaporation. Water Resour. Bull., 10(3): 486--498. McNaughton, K.G., 1976. Evaporation and advection II Evaporation downwind of a boundary. Q.J.R. Meteorol. Soc., 102: 193--202. Monteith, J.L., 1965. Evaporation and environment. Symp. Soc. Exp. Biol., 19: 205--234. Monteith, J.L. and Szeicz, G., 1962. Radiative temperature in the heat balance of natural surfaces. Q.J.R. Meteorol. Soc., 88: 496--507. Penman, H.L. and Long, I.F., 1960. Weather in wheat. Q.J.R. Meteorol. Soc., 86: 16. Rutter, A.J., Morton, A.J. and Robins, P.C., 1975. A predictive model of rainfall interception in forests, II. Generalisation of the model and comparison with observations in some coniferous and hardwood stands. J. Appl. Ecol., 12: 367--380. Saxton, K.E., 1975. Sensitivity analysis of the combination evapotranspiration equation. Agric. Meteorol., 15: 343--353. Stewart, J.B., 1977. Evaporation from the wet canopy of a pine forest. Water Resour. Res., 13(6): 915--921. Stewart, J.B., 1979. Using the combination equation to estimate evaporation: a note on the relationships between the principle forms. Agric. Meteorol. (submitted). Stewart, J.B. and Thom, A.S., 1973. Energy budgets in pine forests. Q.J.R. Meteorol. Soc., 99: 154--170.

190

Strangeways, I.C. and McCulloch, J.S.G., 1965. A low priced automatic hydrometeorological station. Bull. Int. Assoc. Sci. Hydrol., 10(4): 57--62. Szeicz, G. and Long, I.F., 1969. Surface resistance of crop canopies. Water Resour. Res., 5(3): 622--633. Tan, C.S. and Black, T.A., 1976. Factors affecting the canopy resistance of a Douglas-fir forest. Boundary-Layer Meteorol., 10: 475--488. Thorn, A.S. and Oliver, H.R., 1977. O n Penman's equation for estimating regional evaporation. Q.J.R. Meteorol. Soc., 103: 345--357. Van Bavel, C.H.M., 1966. Potential evaporation: the combination concept and its experimental verification.Water Resour. Res., 2: 455--467. Van Bavel, C.H.M., 1967. Changes in canopy resistance to water loss from alfalfainduced by soil water depletion. Agric. Meteorol., 4: 165--176.