Agricultural and Forest Meteorology, 31 (1984) 143--157 143 Elsevier Science Publishers B.V., Amsterdam --Pr inted in The Netherlands

THE USE OF SURFACE RESISTANCE--SOIL MOISTURE RELATIONSHIPS IN SOIL WATER BUDGET MODELS

D.J. SHERRATT* and H.S. WHEATER

Imperial College, London SW7 (Gt. Britain)

(Received July 1, 1983; revision accepted October 18, 1983)

ABSTRACT

Sherratt, D.J~ and Wheater, H.S., 1984. The use of surface r e s i s t ance lo i l moisture relationships in soil water budget models. Agric. For. Meteorol., 31: 143--157.

An important feature of a soil water budget is the reduction of transpiration from a canopy below the rate of atmospheric demand with increasing soil dryness. Commonly, an empirical relationship between the ratio of actual evaporation (AE) to potential evapo- ration (PE) and soil water storage is adopted. Alternatively the Penman--Monteith equation can be used with a specified relationship between surface resistance and soil water storage.

Using actual evaporation rates determined from instrumented soil water profiles, a relationship between surface resistance and soil water storage can be inferred, and results are presented for different crops and soil-types in the United Kingdom. These results are compared with the surface resistance values implicit in the performance of two layer soil moisture models adopting an empirical AE/PE relationship with soil moisture deficit. The performance of the two approaches with respect to soil moisture estimation is compared.

INTRODUCTION

The aerodynamic and energy budget methods for calculating evaporation from an open water surface were synthesised by Penman (1948) who derived a combination equation based only on readily available meteorological data. This equation has been extensively used to estimate evaporation from a short green crop adequately supplied with water, i.e., potential evaporation (PE), by inclusion of the appropriate albedo in the net radiation term.

To represent the decrease in the ratio of actual evapotranspiration (AE) to potential evapotranspiration with increasing moisture stress, Penman (1949) introduced an empirical drying curve relating potential soil moisture deficit (calculated from the balance of precipitation and potential evaporation), to actual soil moisture deficit. The latter is defined by Penman with respect to field capacity; a datum which implies that soil moisture deficit can occur only through evaporation loss from the soil profile.

Subsequently, Monteith (1965) produced a more physically based descrip- tion of evaporation from vegetated surfaces by the inclusion of surface and aerodynamic resistances, describing the physiological and morphological

* Present address: Instem Computer Systems, Stone, Staffordshire, Gt. Britain.

0168-1923/84/$03.00 © 1984 Elsevier Science Publishers B.V.

144

features of the vegetation cover:

pCp ARn + - - - ( V P D ( z ) )

ra ~. (AE) =- (1) (r a)

A + T 1 +

where AE is the actual evaporation rate (kg m-2 s-~), Rn is the net radiation (W m- 2), VPD (z) is the vapour pressure deficit (mbar) at reference height z, Cp and p are the specific heat of air (J kg- 1 K - 1 ) and density of air (kg m- 3 ), respectively and X is the latent heat of vaporization of water (Jkg-1); r s represents the surface resistance of the crop (s m- 1 ) and ra the aerodynamic resistance (s m- 1 ) to height z. A and 7 are the slope of the saturation vapour pressure curve (mbar K-1) and the psychrometric constant (mbar K -1 ), respectively.

The aerodynamic resistance in the equation allows surfaces of varying roughness to be specified in the calculation, and under conditions of approximate neutral atmospheric stability may be approximated by (Monteith, 1965)

[ln (z - - d ) / z o ]2 (2) ra = k Z u ( z )

where z0 is the roughness length and d the zero plane displacement of the vegetation.

Surface resistance simulates control of water flow from leaf to atmosphere by the stomata; as the soil dries, plants cannot extract sufficient water to satisfy the atmospheric demand and the stomata close, increasing the resistance to moisture flow from the plant.

Penman's potential evapotranspiration formula and drying curve concept have been incorporated into a standard method of regional evaporation and soil moisture deficit calculation by Grindley (1970) of the U.K. Meteorolog- ical Office, which has been widely used in the U.K. for agricultural and hydrological calculations, and similar methods have been used extensively worldwide (e.g., Doorenbos and Pruitt, 1974). A subsequent Meteorological Office model (MORECS) developed by Wales-Smith (personal communi- cation, 1977), incorporated Monteith's modifications to the evaporation calculation but retained the drying curve concept, specifying a constant minimum value of stomatal resistance to calculate PE. However, a revised version of MORECS developed by Thompson et al. (1981) replaced the drying curve by a relationship between stomatal resistance and soil moisture.

In this paper, the performance of the drying curve concept is compared with the rs--soil moisture relationship for three field sites under grass. The rs--soil moisture relationships inferred from model optimization are com- pared with published relationships and the relative merits of the two approaches are discussed.

145

~atnfall

e l f AL>~)

AE ~ (PE, BL/:~AX)

fTt

Fig. 1. Two-layer soil moisture budget model (after Wales-Smith, personal communi- cation, 1977).

MATERIALS AND METHODS

Models tested

To compare drying curves with rs--soil moisture relationships, a similar model structure is required for both. For consistency with current U.K. practice, a two-layer representation was adopted (Fig. 1). Rainfall is assigned to the top layer, capacity A M A X , from which extraction occurs at a rate determined by potential evaporative demand. Extraction from the b o t t o m layer cannot occur until the top layer is exhausted; similarly recharge to the bo t tom layer only occurs when the top layer is filled to capacity. In both models, extraction from the b o t t o m layer depends on the moisture content of that layer. In model I this is specified as A E = PE* (BL /BMAX) , where PE is calculated from eq. 1 with rs set to a minimum value (rs(mln)) as defined in Table I. These values of rs(min) are as suggested by Thompson et al. (1981) for a dense green crop and make no allowance for soil evapor- ation. BL is the moisture content of the bo t tom layer, and B M A X the maximum capacity of the bo t tom layer. This relationship, which produces a

TABLE I

rs(min) for pasture (after Thompson et al., 1981)

Month Jan. Feb. Mar. Apr. May June July Aug. Sep. Oct. Nov. Dec. rs(min)(sm-1) 50 50 50 50 45 40 40 40 40 45 50 50

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linear decline in AE:PE ratio with increasing soil moisture stress, is identical to drying curves in the models of Wales-Smith (personal communicat ion, 1977), Calder et al. (1983) and Doorenbos and Pruitt (1974) and similar to drying curves in the empirical models of Shaw (1963), Baler and Rober tson (1966) and Ritchie {1972).

In model II, the effect of soil water stress on bo t tom layer extraction is included directly in the calculation of AE using the variable rs, where rs = f (BL/BMAX). Surface resistances of individual leaves may be measured directly and combined to give a canopy estimate. Alternatively, surface resistance of complete canopies may be estimated from vapour pressure, temperature and windspeed measurements at several levels above the canopy. In principle it should be possible to deduce r~---soil moisture relationships from these techniques. However, Stewart and Thom (1973) found for forest experimental data that derived r s values, although dependent on time of year and time o f day, could not be positively related to soil moisture data.

Monteith (1965) suggests a method of calculating rs from estimates of AE and PE from eq. 1 if

pCp ARn + (YPD(z))

ra ~.(PE) =

A + 3 ` ( 1 + rs(min)] ra ]

where rs(mm ) is defined as the minimum r s value, pertaining when the plant is not under moisture stress and potential evapotranspiration occurs. Then:

r~ = 1 - - + 1 r a + ~ - - r ~ ( ~ n ) ( 3 ) 3'

Alternatively, if the evaporation rate of intercepted water E i is considered (i.e., r~ = 0), f rom eq. 1, then

pCp ,SRn + (VPD(z))

ra XEi =

A + 3 `

and

r~ = r, 1+ --1 (4) AE

Unfortunately this approach is sensitive to errors in the specification of r~, which itself is uncertain due to inherent assumptions of atmospheric stability in eq. 2. However, using this method, relationships have been found between soil moisture and surface resistance; Szeiez and Long (1969) quote examples from Denmark and East Africa, whilst a recent study in the U.K. has been

147

made by Russell (1980). In the present paper the relationship between r~ and B L / B M A X is similar to that proposed by Thompson et al. (1981) after Russell (1980), given in eq. 5

rs = rs(m~) ]3.5 (1 B-~AX)+exp(O.2(B~LX 1))] (5)

with rs(min ) specified as in Table I; Thompson et al. (1981) include a soil resistance term in the definition of rs(mm ) .

Russell (1980) derived relationships between r s and soil moisture from neutron probe and climate data at two barley and two pasture sites in loam and clay soils near Nott ingham in the English Midlands. Using Monteith's (1965) method (eq. 4) r s was calculated from estimates of A E and E i. A E was estimated from changes in soil moisture using a graphical method devised by McGowan (1973), to separate drainage from evaporation in the soil profile. Ei was calculated using Monteith 's (1965) formula with r s = 0, and ra estimated from a formula proposed by Thom and Oliver (1977)

ra 4.72 In (Z/Zo)2

1 ÷ 0 .54u

which purports to include a correction for buoyancy errors.

Soil moisture data

Soil moisture data for comparative testing were taken from the Institute of Hydrology soil moisture data bank of neutron probe measurements (Gardner, 1981). Three grassland sites were used:

(1) The Grassland Research Station, Berkshire. A loamy chalk soil with a medium-high available water capacity. Soil moisture data are available to 1 m depth from 1974 to 1976, meteorological data provided from an on-site meteorological station.

(2) Stoke-on-Term, Shropshire. A sandy-loam medium available water capacity soil with soil moisture data to 1.5 m depth available from 1974 to 1976 and meteorological data f rom Shawbury meteorological station 15 km away.

(3) Thetford, Norfolk. A podsolic low available water capacity soil in a forest clearing with soil moisture data to 2.4 m depth available from 1974 to 1976 and meteorological data available from Santon Downham meteoro- logical site 5 km away.

At each site data were available in the form of volumetric moisture con- tents at 0.1-m intervals. Additionally at Thetford, soil water potential data were available from tensiometer measurements at 0.1- and 0.2-m intervals (Cooper, 1980).

148

RESULTS

Soil moisture deficit calculation

The soil moisture models are formulated in terms of soil moisture deficit (SMD), which is conventionally defined with respect to a profile "field capaci ty" value. As discussed by Wheater et al. (1982), this may lead to errors in apportioning soil moisture change between evaporation and drainage. For this paper, soil moisture deficits were derived from a deficit calculation procedure designed to minimize this ambiguity. A pre-requisite is the definition of the depth from which evaporation is occurring. As the profile dries, this may be done by identification of a zero flux plane from soil water potential data (Wellings and Bell, 1980), or from variation of layer moisture content with time (McGowan, 1973). In this s tudy the zero flux plane method was used for the Thetford site for which soil water potential data were available and for the other two sites in the absence of potential data the procedure after McGowan (1973) was adopted.

This analysis is based on layer moisture totals and for both methods the soil profile was split into discrete layers defined by the depths of neutron probe readings, normally at 0.1-m intervals. The analysis was performed separately for individual years and at the beginning of the calculation, in the winter months, it is assumed that a deficit does not exist. In an exceptionally dry winter such as 1975--1976, the assumption of zero deficit may not be valid, however, in most years at some point in the winter months this criterion will be satisfied and subsequent calculations will be correct.

Before a zero flux plane develops or during profile re-wetting, the absence of a zero flux plane precludes the direct estimation of actual evaporation. It is therefore assumed that evaporation occurs from the top layer of the profile (usually defined by neutron probe readings at 0.1 m depth), which is the region of greatest root density and, during wet conditions, of greatest soil water potential. A moisture content decrease within the surface layer between two successive moisture readings is compared with the overall balance of rainfall and evaporation in the same period. Consider a calculation between two successive reading dates, A and B; if (0 A -- OE) ~ (PDB --PDA ), where 0 is the moisture content of the surface layer, PD is the potential deficit and 0 A > 0B, then slow drainage is assumed to have occurred. Specifying the change in deficit during this period as 0A --0B produces a spurious value since both evaporation and drainage will be present. There- fore, the change in actual deficit due to evaporation alone is equated with the change in potential deficit. If, however, (0 n --OB) ~ (PDB --PDA ), then the change in deficit is given by OA -- 0~.

Consider a calculation between two reading dates, C and D, when a zero flux plane is present and the depth of evaporation is positively identified at 0.3 m depth. The change in deficit is equated with moisture content within the layers in the upper 0.3 m of the profile. If the increase in deficit is

149

greater than the change in potential deficit, it is assumed that either runoff or direct recharge has occurred; slow drainage could not have taken place from above a zero flux plane.

When rainfall exceeds potential evaporation and the profile re-wets, reduction in deficit is calculated from increases in moisture content in layers with a residual deficit.

Hence, the procedure is not based on a single profile depth relative to an absolute field capacity value, but a depth that increases as root extraction proceeds down the profile, with deficit calculation based on a dynamic datum.

Model fi t t ing

Both models I and II were fi t ted to derived deficits by optimizing A M A X and CMAX, where CMAX = A M A X + B M A X , to minimize the root mean square error between the "observed" and predicted deficits. This was achieved by a univariate method of optimizing A M A X for ratios of A M A X / CMAX, varying from 0.05 to 0.55. Table II contains a summary of the results; at each ratio of A M A X / C M A X the average opt imum CMAX and the

T A B L E II

S u m m a r y resul ts o f m o d e l o p t i m i z a t i o n

Site A M A X / C M A X Model I Model II

RMS RMS CMAX error CMAX error ( m m ) ( ram) ( m m ) ( m m )

T h e t f o r d 0 .05 40 4.8 35 5.9 0 .15 40 4.9 35 6.0 0 .25 39 5.3 33 6.2 0 .35 38 5.5 31 6.3 0 .45 38 5.8 31 6.3 0 .55 36 6.0 30 6.4

S toke 0 .05 93 7.1 70 7.2 0 .15 80 7.3 68 8.1 0 .25 76 7.6 66 8.1 0 .35 73 7.6 64 8.2 0 .45 71 7.7 62 8.2 0 .55 68 7.9 61 8.3

Grass land 0 .05 140 7.3 115 8.6 Research S t a t i o n 0 .15 124 7.5 100 8.2

0 .25 117 8.0 96 8.9 0 .35 112 8.2 94 9.3 0 .45 109 8.7 91 9.7 0 .55 106 9.2 90 10.1

Average er ror at bes t f i t ra t ios 6.4 7.2

150

average root mean square error for the period of data is presented. For model I, the drying curve model , the minimum average root mean square error occurs consistently at the 0.05 ratio. For model II the results are similar, except at the Grassland Research Station, for which the minimum error is at the 0.15 ratio. Two conclusions are important in the context of this paper:

(1) The drying curve model (model I) produces consistently better fits than model II.

(2) The average opt imum CMAX values associated with a given ratio of AMAX/CMAX are larger in model I optimizations.

To explain the inferior fit and parameter change associated with model II, several features have been investigated:

(1) The validity of the rs--SMD relationship used in model II is evaluated by comparison with an rs--SMD relationship specified using AE estimates derived from soil moisture data.

(2) The form of the rs--SMD relationship implied in model I is compared with that used in model II.

(3) The drying curve implied by the r~--SMD relationship in model II is compared with the drying curve inherent in model I.

2000 [ ( o

°

1600

1200 T E

8OO

I

400

0

2000

1600

T~" 1200

E

800

o

°

o °

o o i

IO

(c )

2000

o

o 1600

o

o _~- 1200

o --

o 800 o o o

o

o o o

o o o 400

oo o o o o ° oo

° ~ ~ o o i

20 30 40 50 60 0

SMD (turn)

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20 40 60 80 100 i 20 i40

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F i g . 2 . S u r f a c e r e s i s t a n c e v s . s o i l m o i s t u r e d e f i c i t . ( a ) T h e t f o r d , ( b ) G r a s s l a n d R e s e a r c h

S t a t i o n , ( c ) S t o k e - o n - T e r m .

151

r~--SMD relationships from field data

rs--SMD relationships were derived from the field sites by calculating rs from eq. 3 using actual evaporation estimates obtained from the soil profile analysis and potential evaporation calculated from eq. 1. The rs values were then related to the deficits obtained from the deficit calculation procedure. The specification of ra in eq. 1 was from eq. 2 with z0 and d taken as 0.1 and 0.6 o f the vegetation height respectively, r~¢min) in eq. 3 was taken as in Table I. Data from periods of rainfall were excluded from the analysis to exclude confusion of AE and E i during conditions of a partially wet canopy.

Plots of r s vs. SMD (Fig. 2) show a trend of increasing r~ with SMD, but with considerable scatter. Hence, quantitative relationships cannot be defined with confidence. The derivation of r~, using eq. 3 is dependent on the AE :PE ratio and the aerodynamic resistance. Figure 3 illustrates the scatter associated with the AE :PE--SMD relationship; this may be removed by plotting r~ against AE:PE ratio. This also provides standardisation between sites, allowing r~--AE :PE relationships for all sites to be plotted on the same axes (Fig. 4). Residual scatter in Fig. 4 results from variation in ra (due to variable windspeed) at a given AE : PE ratio. Although the points for different sites lie on a similar curve, the larger r~ values, which fall on a steeper curve for the Thetford site (average windspeed 1.4 m s -1) contrast with smaller r~ values falling on a shallower curve at Stoke (average wind- speed 3.4 m s- 1 ).

Comparison with previous work

Maximum r s values presented here contrast with those of Russell (1980) who observed maximum rs values of 200 s m -1 , on which Thompson et al. (1981) base eq. 5. Szeicz and Long (1969) also quote smaller maximum r s values than observed in Fig. 4; 400 s m- 1 in pasture sites in Denmark. Russell

1.0

0.8

0.6 m

0.4

0.2

I

0 ' 2'0 4'0

0

0 0 0

00 0

0

o 0

0

0

I L I I

60 80

SMD (ram)

0 0

0 0 0 0

O0 o 0 0

000 ° ° 0 0 0 °%

0 0 °°

o 0 0 0 0 0

0 o o 0

O0

0 0 0 0 O0 O 0 O0

0 0

I I I t J

I00 120 140

Fig. 3. AE/PE vs. soi l mo i s ture def ic i t , Grassland Research Stat ion.

152

1600

T E

1200

8O0

400

0 • • i

[.0

, A ,A • • Am • I.* •

4 .ILIA A Aol I • AAII

r i I i I I i t I I I i I i

0.9 0.8 0.7 0.6 0.5 0.4 0.3

AEIPE

I I I

0.2 0. I

Fig. 4. Surface resistance values (from soil profile estimation of AE)vs. AE:PE ratio. (o) Thetford, (A) Grassland Research Station, ( + ) Stoke-on-Term.

(1980), however, was restricted to AE :PE ratios of > 0.5 in pasture, which corresponds to an rs of 1 5 0 - - 2 0 0 s m -1 in the present analysis. Szeicz and Long (1969) were similarly limited to AE :PE ratios of > 0.37, correspond- ing to approximately 300 s m- 1 in the present analysis. The results here are therefore in broad agreement (over a comparable range) with those presented elsewhere.

r s values implied by the drying curve model

The r s values implied in model I may be derived from eq. 3 with PE/AE obtained from the optimized drying curve when the top layer is exhausted. Results of this analysis (again with rainfall days excluded) are presented in Fig. 5. For model I, the abscissa may be calibrated in terms of AE :PE or BL :BMAX. Considering the abscissa as AE :PE, the results from soil profile measurements of AE are compared. In contrast, considering the abscissa as BL/BMAX, the relationship used in model II can be compared, and it is seen that these latter results fall on a significantly lower curve (Fig. 5). The reduction in AE :PE, produced by the increasing magnitude of r s , will there- fore be less pronounced over a wide range of BL :BMAX, than that implied in model I.

The drying curve implied by the rs--SMD relationship in model H

An alternative comparison of evaporation control implied by models I and II is achieved by expressing the rs--SMD relationship in model II as a

153

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o_~o o

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0~-~ , ~ . ~ ~em~o

b ~ . . ~ a ~

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154

standard drying curve. This may be done by rearrangement of eq. 2 after Szeicz and Long (1969), to give

ra (l+~)+r~(m~n) AE/PE = (6)

r a 1 + - t - r s

To simplify the comparison individual daily values of ra were replaced by an average, calculated from eq. 2 with average annual windspeeds, and using mean month ly temperature. Figure 6 illustrates the curve derived from eq. 6 compared with the linear AE :PE reduction implied in model I.

The conclusions derived from Fig. 5 are confirmed by this analysis; the reduction of AE :PE ratio in model II with increasing SMD is shallower than in model I, until BL :BMAX is approximately 0.2, when it becomes much steeper than the linear relationship.

Explanation for the inferior simulations by model H

The reason for the poorer fit of model II compared with model I may be explained by the above conclusions if Fig. 7 is examined. The figure is a graphical representation of the simulation of "observed" deficits at the Grassland Research Station by models I and II, in 1975. Both models produce a similar fit f rom day 220 to day 360, underestimating deficits between days 210 and 240. However, significant discrepancy between the two simulations may be observed during the buildup of deficit (days 130-- 190), producing overestimation of "observed" deficits by model II. Early

r o

L

~,0 0.8 0,6 0.4 O.E 0.0

BLIBMZIX

Fig. 6. Comparison of Model I drying curve with Model II relat ionship (derived using averaged data). ( ) Model I, (o) Model II (derived from rs funct ion) .

150

155

° o o°

9©

E ~3

0 60 120 180 240 300 560

Day No. (1975)

Fig. 7. Observed and s imula ted soil moisture deficits, Grassland Research Station, 1975. (o) Observed, ( - - - - - - ) Model I, ( ) Model II.

season overestimation is observed at most sites (although the graphical simu- lations are not presented here) and may be explained by the delay in AE:PE reduction inherent in model II.

Similarly, the reduct ion in magnitude of op t imum CMAX at a given AMAX/CMAX ratio observed in model II simulations, evident from Table II may be explained as a correction for the higher evaporation rate that is implied by the model II "drying curve".

DISCUSSION AND CONCLUSIONS

A simple two layer moisture model based on a linear "drying curve" relating AE :PE ratio to a measure of SMD has been compared with a similar model incorporating an rs--SMD relationship. The "drying curve model" (model I) produced bet ter fits to observed deficits, although this may be due to inappropriate specification of the rs--SMD relationship rather than inherent defects in the concept itself.

The comparison of relationships between r~ and soil moisture state must be viewed with some reservations. Systematic errors could arise due to underestimation of groundlevel rainfall and enhanced wetting around the neutron probe access tube. Both of these effects would lead to underesti- mation of AE which would imply that the estimated r s values are slightly high (5% increase in AE would reduce r~ by approximately 6% at an AE :PE ratio of 0.1 and by 10% at an AE:PE ratio of 0.8). However, due to the

156

negative feedback associated with soil moisture models, model soil par- ameters are unlikely to be significantly influenced (Wheater, 1977).

Nevertheless, the results imply significantly higher r s values at high SMD conditions in comparison with Russell (1980). A direct comparison with MORECS (Thompson et al. (1981)) cannot be made, since a number of procedural differences exist between MORECS and the models tested. In particular, the inclusion of soil evaporation in MORECS has the effect of increasing Fs(min) by 10% during the period in which peak soil moisture deficits occur. However, after allowing for these effects the results here still suggest significant discrepancies between the implied r s relationships and the MORECS version.

Determination of rs--soil moisture relationships is difficult. The most appropriate method of obtaining such a relationship (from eq. 3) relies only on estimates of AE from the water balance of the soil profile, r a from eq. 2 and PE from eq. 1. The method is sensitive to errors in the specification of ra, the calculation of which is subject to buoyancy errors, particularly at low windspeeds. A more satisfactory method of r S determination is based on instantaneous measurements of windspeed, temperature and humidi ty profiles. Such measurements are difficult to obtain and could not easily be a t tempted on the widespread basis needed for the specification of rs--SMD variation with crop and soil type.

Despite the differences in method and form of AE :PE reduction with increasing soil moisture stress, the differences in model performance were not large. Figure 3 illustrates the wide scatter in observed AE:PE--SMD relationships, in spite of which, the models give a reasonable fit to observed data. It appears that negative feedback in soil moisture budget models renders them insensitive to the form in which the AE:PE reduction is specified; more important is the specification of the parameters of the reduction model (viz. the A M A X and CMAX in the "drying curve").

The results therefore indicate that despite the theoretical attractions of specifying AE :PE reduct ion in terms of rs, for practical application on a regional scale a two-parameter linear drying curve may be at present more appropriate. The specification of soil moisture model parameters for a given soil and land use is likely to be of greater significance than the form of AE :PE reduct ion employed.

A C K N O W L E D G E M E N T S

This work was supported by a Natural Environment Research Grant studentship for D.J. Sherratt, in cooperat ion with J.P. Bell, Institute of Hydrology, Wallingford. The assistance of John Bell and his colleagues, in particular Cate Gardner and David Cooper who provided data for this s tudy, is gratefully acknowledged.

157

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