An analytical model of rainfall interception by forests GashInterceptionModel.pdf

7/27/2019 An analytical model of rainfall interception by forests GashInterceptionModel.pdf

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Quart. J . R . M e t . SOC.1979), 105, pp. 43-55551.573 556.13 3

An analytical model of rainfall interception by forestsBy J. H. C. GASHInstitute a Hydrolog y, W a lingford, Oxfordshire

(Received 22 December 1977; revised 2 June 1978)SUMMARY

The description of the evaporation of rainfall intercepted by forests in terms of a regression of evapora-tion losson incident rainfall is discussed and some of the assumptions implicit in that method are re-examined.The two major factors which control the evaporation of intercepted rainfall are identified. These are: (i) theamount of time that the canopy spends saturated during rainfall and the evaporation rate applicable underthese conditions; and (ii) the canopy saturation capacity and the number of times this store is emptied, bydrying out after the cessation of rainfall. A model is then constructed which is conceptually similar to theRutter model, but which replaces that model's numerical approach with an analysis by storm events. Theevaporation from a saturated canopy during rainfall is estimated from the Penman-Monteith equation; theevaporation after rain has ceased, the effect of small storms insufficient to saturate the canopy, wetting-upthe canopy and evaporation from the trunks are added as separate terms. The model has been tested againstdata from Thetford Forest in East Anglia, with satisfactory agreement between observation and estimation.It is suggested that the model may be capable of making useful estimates of the evaporation of interceptedrainfall, solely from rainfall measurements.

1 . INTRODUCTIONThere have been many previous studies of the interception an d ev aporatio n o f rainfallby forests. The results of these studies have frequently been expressed in the form of em-pirical regression equations between interception loss and rainfall : tha t is, as functions ofthe form

I = a P , + b .where I is the depth of water intercepted and lost by evaporation, the interception loss;PG,he gross rainfall incident on the canopy; and a and b are regression coefficients. Such anequation can be used either to describe sets of storm da ta, or, if it is assumed there is onlyone rainfall event per day, to describe daily interception loss as a function of daily grossrainfall. T he reviews of Zinke (1967) an d Blake (1975) contain exam ples of this appro ach.Helvey and Patric (1965) reviewed results from studies in hardwood forests in the easternUnited States and concluded that two regression equations, one for winter and one forsummer, were adequate to describe the loss from all hardwood forests in that region.However, the ap proach has been criticized (e.g. Jackson 1975) for tak ing n o account of suchvariables as rainfall intensity and duration, and the interval between storms. A fur thercriticism is tha t the results are empirical and can be extrapolated only t o similar forests inareas of climate similar to tha t w here the original d at a were collected.In contrast to the regression equation approach, Rutter er al. (1971) have constru cted aphysically based computer model, which uses inputs of rainfall and the meteorologicalvariables controlling evaporation, t o calculate a run ning water balance o f a forest canop y o fknown structure, thereby producing a n estimate of the interception loss. T he m odel has beentested against measu rements from several forests (R utter et al. 1975) an d must be consideredthe most rigorous m ethod for estimating interception presently available. The model does,however, have practical disadvantages : it requires hourly meteorological data, which areseldom available, and a complex computer programme, which is time consuming to con-struct and operate.

43


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44 J . H . C . GASHThis paper re-examines some of the assumptions implicit in the regression equationappro ach. A cknowledging these assumptions, a model is then defined w hich, while retainingsome of the simplicity of the regression equatio n, includes m uch of the fundam ental physi-

cal reasoning explicit in the Rutter model.2. THEORYF REGRESSION EQUATIONS

For any storm large enough to saturate the canopy, Horton (1919) was the first toexpress interception loss asI = sdE d t + S . (2)

where E is the evaporation rate of intercepted water d uring rainfall; S , the canopy capacity(the amount of water on the canopy when rainfall and throughfall have ceased and thecanopy is saturated ); t, the duration of rainfall; and evapo ration from trunk s is neglected.If we now separate evapo ration before and after satura tion of the can opy:

where t is the time taken for saturation of the canopy to occur.Defining a mean ev aporation rate, 1, rom a saturated canopy during rainfall byI? = ( l / (t - ) ) L : E d t

R = ( l / ( t - t$R d t ,

(4)with rainfall similarly described by a mean rainfall rate:

we have( 5 ), - P ; , = R ( t - t )

where PA is the am ount of rain necessary to saturate the canopy. Assuming there is no d ri pfrom the canopy before saturation, PA is also given by(l-p-p,)P& = s+ E d t . ( 6 )1

where, following Rutter et al. (1971, 1975), p is the free throughfall coefficient, i.e. theproportion of rain which falls through the canopy without striking a surface, and pr, hepropo rtion which is diverted to the trunks as stemflow.Substituting Eqs. (4) and ( 5 ) into Eq. (3)

Substituting for PA from Eq. ( 6 ) and rearranging:I = (E/R)P,+ ( s +1 d t ) (1 -(E/K)(l- p - pJ- > (7)


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RAINFALL INTERCEPTION 45Eq uating the coefficients of Eq s. ( 1 ) an d (7), the coefficients of th e regression equ ation canbe identified as:

a = E/R, and b


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46 J. H . C. GASH2. The logarithmic dependence of the 'drip-rate' on th e degree of cano py sa tur atio n,observed by Rutter et al. (1971), is sufficiently sensitive th a t there is in effect virtually n odr ip from the canopy during wetting-up; an d the amou nt of water on the canopy a t the end

of a storm is quickly reduced (within 20 to 30 minutes) to S , the m inimum value necessaryfo r saturation, independent of the initial value when rain ceased.The effect of these assumptions in the model can best be assessed by its application toreal data and by analysing the sensitivity of the results to the various parameters. Theirconsequence, however, is in effect to separa te the dependence of interception loss on 'forest'parameters, such as storage, S, from meteorological variables, such as evaporation rate. Bymaking the dependence of interception loss on these parameters explicit, rather thanimplicit as they a re in a regression equation, it is hpped t ha t results ca n be m ore readilyextrapolated between forests and climatic regions.( b ) Evaporation from a saturated canopy

Consider a series of n storms each large enough to saturate the canopy and eachsepara ted by a sufficient period for the canopy to dry. If Eq. (3) is applied to each storm thetotal interception loss is given by

If E is now redefined byj = 1

while similarly R is given by

o requation (8) can be written as

( c ) Wetting-up the canopyFrom Eq. (6) the interception loss for the period before saturation is i r k d l =

J o( I - - , ) P b-S. Substituting this into Eq. (9) for all n storms,

F o r sm all storm s insufficient to saturate the canopy completely, I = (1 - p - p , ) P , . Includ-ing these (say) m storms, the total interception loss is


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RAINFALL INTERCEPTION 47(d'l Evaporation f rom the trunks

Emulating the Rutter model, where the trunk water balance is conceived as beingseparate from the canopy water balance, the proportion of the rain which is directed to thestem is remaved from the incident rainfall before the canopy water balance is considered.The stemflow for each storm with rainfall greater than S J p , is described by stemflow =p t P c - S , , S , being the trunk water store, which must be filled before any stemflow isapparent at the base of the trunks.

Evaporation from trunks is likely to be small compared with evaporation from thecanopy (Rutter and Morton (1977) calculate it to be typically 1 to 5 % of the above-canopyvalue during rainfall) and it is neglected here. Interception loss from trunks is then only theamount of water which remains on the trunks after stemflow has ceased. If the assumptionis made that the interval between storms is sufficiently long to allow all the water in thetrunk store to evaporate, then the interception loss from the trunks is given by

when out of the total ( n + m ) storms there are q storms above the critical rainfall, S , /p , ,necessary to fill the trunk store.

Equation (1 1) can now be rewritten as

( e ) The rainfall necessary to saturate the canopyEvaporation from partially wet canopies and its description in terms of the Monteith-

Penman equation has been the subject of recent discussion (Shuttleworth 1976; Monteith1977). However, Shuttleworth (1978) has demonstrated that the somewhat arbitrary assump-tion made by Rutter et al. (1971) leads to a theoretically reasonable description of thepartially wet canopy. That assumption is that evaporation from the canopy is described byE = (C/S)E, , where E,, is the evaporation which would occur from a totally wet canopy andC is the amount of water on the canopy.

This is applied to conditions on the canopy before saturation is reached, with thefurther assumption that mean evaporation and rainfall rates also apply. The evaporationduring this period is then given by E = (Cis). Assuming there is no water dripping fromthe canopy before saturation, the rate of change of water on the canopy can be described bydC/dt = (1 - p - p , ) R - E , or K / d t = ( I -p-pt)R-(E/S)C, which has the solution

orThe time taken for saturation to occur is therefore

t' = -(S/E)In{l-(E/R)(l-P-p,)-'}.Following the assumption that the mean rainfall rate applies, at the point when the canopyreaches saturation the rainfall is given by P& = Rt'; therefore


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48 J . H. . GASH

and Eq. (12) can be rewritten with PA considered as a constant for the period:

4. APPLICATION OF THE MODEL TO THETFORDORESTATA( a ) Site and instrumentation

Thetford Forest is a pine forest of over 200 km2 situated in East Anglia. Observationswere made in a stand of Scots pine (Pinus sylvestris L.) planted in 1931 and now thinned to adensity of some 800 trees per hectare.Details of the instrumentation have been given previously by Gash and Stewart (1977).Interceptionloss was determined from the difference between gross rainfall, throughfall andstemflow. Throughfall was measured with a random grid of 24 raingauges which weremeasured and rearranged within an area 30 m x 15 m every two weeks.

Gross rainfall was measured both above the canopy and in an adjacent clearing. Gashand Stewart found good agreement between the two gauges and used the above-canopygauge in their analysis. That gauge is also used in this analysis. In March 1976 the gauge waschanged from a resolution of 0.05mm per tip to 0.25mm per tip.

Two automatic weather stations (Strangeways 1972) were mounted on a tower abovethe forest. Measurements of net radiation, temperature, wet bulb depression and windspeedwere made every five minutes and recorded on magnetic tape. Readings from the twostations were later averaged to produce a single set of hourly average values. The measure-ments were made throughout 1975 and 1976; during that period the weather stations wererecalibrated four times.

(b ) The fore st structure parametersThe canopy capacity, S, is defined as the amount of water present on the canopy in

conditions of zero evaporation, when throughfall has ceased. S for the Thetford site hasbeen determined to be 0.8mm (Gash and Morton 1978) with an estimated error of about

The free throughfall coefficient,p , was estimated to be 0.32 from an analysis of smallstorms, or 0.26


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RAINFALL INTERCEPTION 49The theory requires that &' and R should be calculated from those hours when the

canopy is saturated; however, in practice, to determine whether or not this condition issatisfied requires the drawing up of an hourly water balance, as in the Rutter model. Tomaintain the simplicity of the model it is therefore necessary to approximate this conditionby selecting a rainfall rate, above which it is considered the canopy will always be saturated.Once the canopy is saturated, the rainfall rate necessary to maintain saturation is E / ( l-p - p J , typically 0*3mmh-'; if the canopy is completely dry the rainfall necessary toachieve saturation in the hour is P& typically 1.3 mm. As a compromise, the threshold rateof 0.5 mm h-' was chosen as representing saturated conditions. It is accepted that this willinclude some hours of rainfall when the canopy is only partially wet before saturation andwill exclude some hours when the canopy is saturated during light rainfall. The followingprocedure was therefore adopted :

For each four-week period, hours with P, 2 0.5mm were isolated and the evapora-tion calculated using the Monteith-Penman equation as in the Rutter model :

1.

1E = (AR,+pc ,D/r , ) (A+y) - ' ,where c p s the specific heat of air at constant pressure (Jkg-'K-l); E, the evaporation rate(kgm-'s-'); r,, the aerodynamic resistance (sm-'); R,, the net radiation (W m-');D, the vapour pressure deficit (mb); y , the psychrometric constant (mbK-I); A, the slopeof the saturation vapour pressure curve at air temperature (mbK-'); and p , the density ofair (kgm-j).The aerodynamic resistance, as in the Rutter model, was calculated from the expressionr , = (l/k2u){ln(z-d)/z,)2, with k = 0-41 and u the windspeed at height z(ms-') , and theassumption that d = 0.75h and zo = O*lOh, when h is the vegetation height (m) and z =h + 2 (m).Under the assumption that hours with P, 2 0-5mm represent saturated canopyconditions, these hourly values of E were then averaged over the four-week period to givean estimate of 17.2 . Similarly, for the same hours P, was extracted and, making the same assumption,averaged to give 8.

3. These values were then used to calculate PI; from Eq. (13) for each four-weekperiod, using the following values of the other parameters (Gash and Morton): S = 0.8 mm,

Making the further assumption that there is only one storm per rainday, raindayswere divided into those with PG3 PA and those with P G < Pt.These two sets of dailyrainfall amounts were then summed to give 1 , and 1 G j .

p = 0 . 3 2 , ~ ~0.016.4.

n mj = 1 j = 1

5 . The number of raindays, n, with P, 2 P b was noted.6. The number of raindays, q, with PG 2 St/pr was noted, and the rainfalls of

The procedure provided sufficient nformation for the interception loss to be calculatedraindays with P, < St/prwere summed.from Eq. (14).

(d) ResultsThe calculated values of E , R and PA for each four-week period are given in Fig. 1.

The error bars represent standard deviations of the mean hourly values of I? and w ; henumbers of.hours data used are also shown.


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50

2

IEE 1 -a"0

J. H . C. GASH

. .. . . 9 . . . * . . .r

6

1

There is no significant correlation between E and R in e ither 1975 or 1976, the coeffi-cients of determination, R2, eing 0.178 and 0.008 respectively.The cumulative totals of observed and estimated interception loss are sho wn in Fig. 2.The maximum difference between observed a nd estimated loss is 16 mm. Th e final totals ofobserved and estimated loss are 245 and 257 mm respectively; the m odel therefore over-estimates the interception loss by 12 mm, or 5 % of the o bserved loss.The calculations were made over four-week periods t o give a b etter app reciation of th eperformance of the m odel and variability of the estimates. There is, however, n o reason whya single calculation cann ot be m ade o n the d ata .When calculations are performed for the whole period the inputs to the model are:j = n j = m

j = 1 j = 1E = Oa19mmh-'; K = 1.38mmh- ' ; 1 G j= 822.1mm; 1 G j = 73.7mm; and


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RAINFALL INTERCEPTION 51

c---1975-----.,-19764 -w ee k periods

Figure 2. Cumulative totals of measured (continuous line) and estimated (broken line) interception loss,for 21 four-week periods during 1975 and 1976.

1975 1976

Figure 3. Each block shows from left to right: total rainfall, with measured interception loss in black;estimated interceptionloss, using the parameters derived for each period (plain); and the estimated intercep-tion loss, using the mean parameters derived from the whok data set (hatched).


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J. H . C.GASH2j = n t + n - q

j = 11 p G j = 46.5mm; n, m and q being 132, 161 and 157 respectively. This givesPA = 1.35mm.

The interception loss is then made up as shown in Table 1. The rainfall and the observedand estimated interception loss for each period are also shown in Fig. 3, together with theinterception loss estimated using the mean parameters derived for the whole period, butapplied to each individual period. In both cases a regression of observed on calculatedinterception loss gave a line whose slope did not differ significantly from unity, and anintercept which did not differ significantly from zero.TABLE 1. THE OMPONENTSOF THE TOTAL INTERCEPTION LOSS,CALCULATEDFOR THE WHOLE PERIOD O F 84

WEEKS OF AVAILABLE DATA DURING 1975 AND 1976Component of interception loss Analytical form Value (mm)

Small stormsWetting-up the canopy 12.5(~ / l~ ) ip c j - p ~ ) 87.4vaporation from saturation untilrainfall ceases I =Evaporation after rainfall ceases nS 105.6Evaporation from trunks m t n - , y

j = 1qS,+Pt c pa, 2.9

Total interception loss 257.3

5. ERROR NALYSISThe error in the model prediction due to any variable, X , isestimated as 61 = (ail8X ) d X .

If the variables are independent, the resultant errors may then be summed quadratically togive the total error in the estimate.

The percentage errors in the two stemflow terms are large, but the absolute magnitudeof evaporation from trunks is, in this case, so small that the errors can be neglected. For thepurposes of this analysis the error in the measurement of rainfall is also neglected. Theremaining variables are E , S nd (1 - p - p J .

(a) The error i n EThe error in E will depend on the accuracy of the measurements, in particular that of

wet bulb depression, and also on the validity of the assumptions made in determining r, .These assumptions, which are discussed by Rutter et a/. (1975), take account of atmosphericstability (Thom e t al. 1975), and of the effects discussed by Thom (1971, 1972), but essen-tially estimate the parameters d and zo from empirical relationships.E , the mean evapora-tion rate, has been calculated under the assumption that the evaporation for hours whenP, > 03mm is equivalent to the evaporation for all hours of rainfall onto a saturatedcanopy. This assumption is obviously an approximation to the truth. However, an inde-pendent estimate of Eis available for the same site in 1972 and 1973, from direct measure-ments of evaporation rate made using the Bowen ratio technique when the canopy wassaturated (Stewart 1977). Periods when the available energy was less than 20 W m-'(0.03mm h -') were omitted from the analysis due to large percentage errors in the measure-ment technique at low energy levels; but a total of 245 twenty-minute periods gave a mean


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RAINFALL INTERCEPTION 53evaporation rate of 129.1 Wm-2, equivalent to 0*19mmh-'. This agreement with theestimate of E reported here suggests that the assumptions made in the derivation of E arenot seriously in error. Eis relatively insensitive o changes in the arbitrarily selected thresholdof PG 2 0*5mmh-'. Changing the threshold to 0.25 and to I-Ommh-' changed Efrom0*19mmh-' to 0.21 and 0*17mmh-', respectively. An error of +15% was thereforearbitrarily assigned to E and the consequent error in the interception loss, given by 61 =(aZ/aE)6E,was & 14.9 mm, or k6% of the total loss.

(b) The error in SThe canopy capacity is assumed to have been determined to within &10% (Gash and

Morton). The error due to S, given by 61 = ( a l / a S ) S , is f 9.4 mm, or +4 % of the totalloss.

( c ) The error in (1- p - p t )The difference between the two values derived for p (0.26 and 0-32,Gash and Morton)

suggests that the error in this parameter may be relatively large. However, an error off 0% in p , which covers both values, is equivalent to an error of f 14% in (1- p - p t )and results in an error in the interception loss, given by 61 = {al/a(l - p - p , ) } 6 ( 1 - p - p t ) ,of 8.9 mm, or f % of the total loss.

This analysis ignores error due to the assumption of equivalence between rainday andrainfall event. The assumption was made partly for simplicity but also to avoid the need forany subjective assessment of what constitutes a rainfall event. Disregarding this, the errorresulting from all the other variables considered is 19.7 mm, or 8% of the total.

6. DISCUSSIONND CONCLUSIONSThe model described in this paper in effect separates the interception loss resulting from

evaporation during rainfall, from that which results from the structure of the canopy.Following some simplifying assumptions, this is accomplished analytically rather thannumerically as in the Rutter model, although conceptually the models are similar. Theagreement between observation and estimation appears to be sufficiently good, in th,e casestudied, to justify the additional assumptions and simplifications which have been made.Further tests will, however, be required before these assumptions can be considered gener-ally justifiable.

As it has been applied here, the model suffers from the same disadvantage as the Ruttermodel, in that it requires an input of hourly meteorological data. Although rainfall andrainfall duration are measured at climatological and agrometeorological stations in manyareas, the hourly values of the meteorological parameters required to calculate the evapora-tion are rarely measured above forests, except in research projects. There are some sixtymeteorological stations in the United Kingdom where hourly observations are made, butthese are generally associated with aviation or marine activities and therefore tend to be atconsiderable distances, and different altitudes, from the major areas of forest. These aremainly inland, upland regions. In any practical application of the model it will therefore benecessary to use empirical values of at least E and possibly R .

Little is known about the seasonal, areal or altitudinal variation of A!?, although somepreliminary information can be obtained from Fig. 1, where a seasonal variation is indica-ted. However, it should be noted that in an analysis of a similar parameter for a site in thesouth of England during 1966 and 1967, Rutter and Morton (1977) found no such markedseasonal trend, which may therefore be associated with the anomalous meteorological


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54 J. H. .GASHconditions during the summers of 1975 and 1976. A preliminary study of evaporation froman upland forest in central Wales has given a value for E of 0.12 mm h-', which compareswith the value of 0.19mmh-1 obtained for Thetford Forest. These two sites representextremes of climate likely to be found in forested areas of Great Britain; so if this value forcentral Wales is correct, it is likely that the variation in I? may be sufficiently small foradequately accurate estimates of evaporation to be made, with little regard for variation inE. The sensitivity of the interception loss to E will vary with rainfall climate, being moresensitive in regions of long rainfall duration. However, in the case studied, Thetford Forest,to obtain an accuracy of f 15% in interception loss (equivalent to approximately 5 % in netrainfall) would, with the estimated accuracy in the other variables, require l? to be estimatedto an accuracy of 2 37%. This relative insensitivity to E is illustrated in Fig. 3, where esti-mates made using the values of E derived for each individual period can be compared withthose derived using fixed means for the whole period. The consequent possibility of usingempirical estimates of E to estimate interception loss solely from rainfall data, suggests thatthe problem of estimating the evaporation of rainfall intercepted by forests may have arelatively simple solution.

Thom and Oliver (1977) have suggested that total regional evaporation (interceptionplus transpiration) may be calculated from daily measurements made at climatological andagrometeorological stations, using a version of the Penman equation, modified by theintroduction of two additional factors. The first is related to the surface roughness of thevegetation, while the second represents the combined effect of the average surface resistanceof the vegetation and the duration of canopy wetness. In practice this latter parameterrequires an independent estimateof the interception loss (Gash 1978), which, it was sugges-ted, might be obtained from the extrapolation of previous catchment analyses or measure-ments of interception loss. However, the model presented here allows, in the case of forests,the attractive possibility of obtaining this required estimate of rainfall interception over thesame period and for the same site as those being used to calculate the transpiration.

ACKNOWLEDGMENTS1 am indebted t o N. F. Cowell and C. R. Lloyd for their help in the collection and

analysis of the Thetford data and to W. J. Shuttleworth for his constructive criticism at allstages in the preparation of this paper. Thanks are also due to Professor P. G. Jarvis,Professor A. J. Rutter, J. B. Stewart and A . S . Thom for their helpful comments during thefinal stages of preparation and to referees for their helpful criticism of the first draft of thispaper.The work described in this paper was carried out with funds provided by the Departmentof the Environment under contract number DGR 480135.

REFERENCEThe interception process. In Prediction in catchment hy drol-ogy, Eds. T. G. Chapman and F. X. Dunin, AustralianAcademy of Science, Canberra, 59-81.Comment on the paper by A. S . Thom and H . R . Oliver, OnPenman's equation for estimating regional evaporation,Quart . J . R . M e t . SOC.,104,532-533.An application of the Rutter model to the estimation of theinterception loss from Thetford Forest, J. Hydro l . , 38,The evaporation from Thetford Forest during 1975, Ib id . ,

Blake, G. J . 1975

Gash, J . H . C . 1978

Gash, . H . C. and M orton, A . J .Gash, J. H. C. and Stewart, J. B.

197849-58.1977 35, 385-396.


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RAINFALL INTERCEPTION 55Helvey, J. D. and Patric, J. H.Horton, R. E.Jackson, I. J.Leyton, L., Reynolds, E.R. C.and Thompson, F. B.Monteith, J. L.Rutter, A . J.Rutter, A. J., Kershaw, K.A.,Robins, P. C. andMorton, A. J.Rutter, A . J. and Morton, A. J.Rutter, A . J., Morton, A. J. andRobins, P. C.Shuttleworth, W. J.

Stewart, J. B.Strangeways, I. C.Thom, A. S.

Thom, A. S.and Oliver, H. R.Thom, A . S.,Stewart, J. B.,Oliver, H. R. ndGash, J. H. C.Zinke, P. J.

19651919I9751967

1977I975

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1978197719721971I972I9771975

1967

Canopy and litter interception of rainfall by hardwoods ofeastern United States, Water Resour. Res., 1, 193-206.Rainfall interception. M o n . Weath. Rev. , 47, 603-623.Relationships between rainfall parameters and interceptionby tropical forest, J . Hydrol., 24, 215-238.Rainfall interception in forest and moorland, Int. Symp onForest Hydrology, Eds. W. E. Sopper and H. W. Lull,Pergamon Press, Oxford, 163-1 78.Resistance of a partially wet canopy: whose equationfails?, Boundary Layer M et., 12, 379-383.Chapter 4. The hydrological cycle in vegetation. In Vegeta-t i o n and rhe atmosphere Vol. I , Ed. J. L. Monteith,Academic Press, London, I 11-1 54.A predictive model of rainfall interception in forests.I: Derivation of the model from observations in aplantation of Corsican Pine, Agr. Met . , 9, 367-384.A predictive model of rainfall interception in forests.111: Sensitivity of the model to stand parameters andmeteorological variables, J . Appl. Ecol., 14, 567-588.A predictive model of rainfall interception in forests.11: Generalization of the model and comparison withobservations in some coniferous and hardwood stands,Ibid., 12, 367-380.Experimental evidence for the failure of the Penman-Monteith equation in partially wet conditions, BoundaryLayer Met., 8 , 81-99.A simplified one-dimensional theoretical description of thevegetation-atmosphere interaction, Ibid., 14, 3-27.Evaporation from the wet canopy of a pine forest, WaterResour. Res., 13, 915-921.

Automatic weather stations for network operation, Weather,27 , 4 0 3 4 0 8 .Momentumabsorption by vegetation, Quarr.J. R. Me t . SOC.,97,414-428.Momentum, mass and heat exchange of vegetation, Ibid.,98, 124-134.On Penmans equation for estimating regional evaporation,Ibid., 103, 345-357.Comparison of aerodynamic and energy budget estimatesof fluxes over a pine forest, Zbid., 101, 93-105.Forest interception studies in the United States,InternationalSymposium on Forest Hydrology, Eds. W. E. Sopper

and H. W. Lull, Pergamon Press, Oxford, 137-161.

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