Investigation of the Thermodynamic Component of Penman’s Method for Estimating Evaporation

1Makdessi, M, 2A. Rahman, 3G. Kibria 1Sinclair Knight Merz, 2University of Western , 3Sydney Catchment Authority, E-Mail:[email protected]

Keywords: Evaporation; Penman Method; Water balance; Thermodynamics.

EXTENDED ABSTRACT This paper presents modifications to the Penman The study presented here was initially Method which make it suitable for use with high commissioned by the Sydney Catchment quality solar radiation measurements. These Authority (SCA) in 2004, in response to the modifications involved exclusion of some empirical independent recommendations for the equations reported to have errors of up to ±20%, improvement of evaporation estimates being used and resulted in a statistical improvement of general in SCA water balance models. A study evaporation estimation. by Sinclair Knight Merz (SKM) independently reviewed the WATHNET model used by the The study also produces a set of solar radiation SCA, and suggested improvement of the basic constants calibrated based on the results of the best “Pan Conversion” evaporation estimates used in evaporation estimates. These constants are useful the model. Best estimates suggest that evaporation substitutes for the frequent gaps in measured solar can cause as much as 14 billion litres per month radiation data, and allow consistency for long (14Ml/month) to be lost from water supply periods of estimation. in Sydney in the worst of conditions. This accounts for as much as 20% for worst Surprisingly, it was discovered that the semi- months of all water lost from the main water empirical approximations of solar radiation terms supply reservoirs in Sydney, such as Warragamba cause an average annual error of only 23mm, () and Woronora Dam. compared to the estimates derived from real These estimates are based on observations of measured solar radiation data. historical Pan Evaporation data. This level of improvement may be useful where Evaporation is the largest loss component of accuracy of evaporation estimates is an absolute water balance in open water reservoirs. However, priority, or where vast water surfaces mean that it is difficult to estimate evaporation with any evaporation rate equates to the loss of a very large reasonable certainty. The Penman Method is volume of water. The findings can also benefit universally acknowledged as one of the best cases where the resolution of evaporation estimates physically based evaporation estimation methods is critical, by improving accuracy for shorter time to date (Penman, 1948, 1956). Use of this method steps. on a continual basis is a difficult task due to limited data availability of a fairly large number Two conclusions were drawn from this study which of variables. These variables are a crude can benefit the use and understanding of the representation of the thermodynamic and Penman Method in general. The first was that the aerodynamic properties of an open water surface. semi-empirical equations suggested by some texts such as Thompson (1999), which are commonly Focusing on the thermodynamic component of the used to approximate many of the inputs into the Penman Method, the study sought to reduce errors Penman Method, though seemingly far fetched and associated with the estimation of solar energy flux based on very limited data, do not result in any into a water surface, particularly in Sydney. In acute errors in the annual total evaporation most applications of the Penman Method, a series estimate. of semi-physical and semi-empirical equations are used to derive net energy flux based on a solar The second conclusion drawn from this study was radiation constant, air and water temperature that the Pan Conversion Method of estimating readings, and a cloudiness coefficient, as the input evaporation has a strong tendency to over estimate variables. evaporation by as much as 7% if compared to the best Penman Method estimates made in this study.

1867 on the subject. This standing places the “Energy 1. INTRODUCTION Budget” evaporation estimate as the closest thing to a true “measure” of evaporation loss rates. The This paper primarily focuses on the Penman results of such studies are very useful as Method for estimating evaporation (Penman, 1948; benchmarks for the calibration of relatively 1956). It has undergone widespread use worldwide simpler evaporation estimation methods, such as proving it to be a robust and comprehensive the Penman Method. method of estimating evaporation losses from open water surfaces. 2. BACKGROUND While there are many methods of estimating open This paper is part of a much wider study water evaporation losses based on various theories, addressing the many ways of improving the Penman Method encompasses several of the evaporation estimates. The study was most fundamental physical evaporation principles. commissioned by the Sydney Catchment Authority It can be generalised as a combination of two key (SCA) in 2004, in response to independent principles, aerodynamics and thermodynamics. recommendations for the improvement of Thermodynamics cover the requirement of energy evaporation estimates being used in SCA models. as a driving force of vaporisation. While this may be a commonly known scientific fact, the Given the severe drought conditions currently distinction between vaporisation and evaporation afflicting Sydney’s water supply, the importance is less commonly known. In laboratory conditions of accurate water balance modeling has increased (no wind agitation), vaporisation and condensation by orders of magnitude. The results of such models would occur in equilibrium, with the volume of have much further reaching influence. Forecasting vapour at any given time being dependant upon the Sydney’s water supply is necessary to give temperature of the system. Hence for evaporation appropriate notice of shortage, and support for to take place the second essential component is political approval of alternative measures of water introduced, aerodynamics. supply, such as desalination. Aerodynamics encompasses the mechanisms of An old adage rightfully states that a model can vapour transport from a water surface. Reducing only ever be as accurate as the information put into vapour pressure above a water surface maintains a it. Hence, the notorious margins of error associated vapour pressure gradient in favour of vaporisation. with evaporation estimates used in any water It is the net vaporisation - the difference between balance study can translate into and even procreate the rate of vaporisation and condensation - which larger errors within water resources modeling defines the true rate of evaporation. results. According to Penman Method, evaporation fails to In addressing evaporation estimation, this paper cease when wind speed is zero. However, given addresses only a small part of the problem in the that the method does not account for convection hope of contributing to more accurate water currents by any direct means, this oversight is not balance modeling capabilities in the future. Further entirely incorrect. In fact, the Penman Method’s still it focuses on only one of the many possible combination of aerodynamics and thermodynamics avenues for improving evaporation estimates. This has been proven to match the best estimates of is firstly by suggesting the Penman Method as a evaporation with remarkable accuracy. standard practice for estimation of evaporation in Sydney, but more specifically, improving the The individual components of the Penman Method Thermodynamic component of the Penman are by no means original with regards to the Method by adopting real solar radiation physical principles of evaporation. Aerodynamic measurements. evaporation theory precedes the Penman Method, appearing in earlier stand-alone aerodynamic 3. SITE AND DATA estimation methods. Thermodynamics too is a firmly established evaporation science in its own right. Evaporation estimates derived from This study required solar radiation measurements thermodynamic studies of large water bodies are at the earth’s surface on a daily basis, which was collectively known as “Energy Budget” obtained from the Australian Bureau of evaporation estimates. To date this is the most Meteorology, measured at Observatory Hill, Sydney. accurate method of estimating evaporation losses from open water bodies, a fact openly acknowledged by leading academic publications The study period covered January 1996 to May 2001 as this was the only common period of the

1868 available data across each of the input variables in Data was obtained from two sources, the Sydney the Penman Method, with priority given to solar Catchment Authority, and the Australian Bureau of radiation data. Gaps in data were filled with Meteorology. The SCA provided data that covered appropriate statistical methods based on recent hydrometric variables for Warragamba and trends, as well as seasonal or yearly trends (which Woronora, as well as “Class A” Pan evaporation at ever was deemed most appropriate to match the Prospect. nature of variability for each type of data). 4. METHODOLOGY A critical component of this study was the The Penman Method itself can be modularised standard measure of “Class A” Pan evaporation. based on the “intended” purpose of each of the For comparison purposes, Class-A pan estimates equations in the method. Each “module” can be and Penman estimates at the same location would individually assessed on how well it achieves its be ideal. purpose, and improved if possible. Class-A Pan evaporation data was available in best As mentioned earlier, the Penman Method is record at , in the center of the considered to be a physically based method. This Sydney metropolitan area. For consistency all is true in that it is a relationship between hydro-meteorological variables were collected for evaporation and hydro-meteorological variables, this location, as best as possible. Hydrometric based on physical principles. However this variables however are not available at Prospect assumes that these variables, such as long and such as is required for this study. To substitute, short wave solar radiation etc, can be measured hydrometric variables (e.g. water temperature etc.) accurately. In most cases, it is either not possible, were assessed for Warragamba and Woronora or more often, not practical to measure these , two large water bodies with sufficient variables. In their places semi-empirical formulas spatial, altitudinal, and volumetric differences to are used, and this is where most of the weakness in represent a variety of open water bodies in Sydney. the Penman Method currently lies. Comparisons suggested very little variation between the hydrometric variables of the two dams. Hence the differences were deemed This study maintained the physical theory shell, insignificant and hydrometric variables for and addressed each internal component Warragamba were used in conjunction with the individually. In this way the Penman Method Hydro-meteorological variables for Prospect to maintains its general structure, remaining true to create the study data set. its base theory. Changes to each component are verified by the effect on the overall method’s performance. This provides some independence Further attempts were made to assess the spatial between the improvement and verification variability of all the hydro-meteorological processes. For example, a change in one variables used in this study. This would have been component might seem beneficial, however the useful if it could have proved that all variables performance of the overall method will suffer if it were suitably applicable to , is a poor change, or if it somehow clashes with the Sydney’s primary water supply reservoir. fundamental theory. Examples of evidently poor However, no useful patterns of spatial variability changes are negative estimates, inability to could be determined for any of the variables maintain the Priestly Taylor relationship (Priestly between any of the 3 sites (Warragamba, and Taylor, 1972), and irregular or Woronora, and Prospect). Particularly, and disproportionate behaviour with regards to short somewhat unfortunately, no conclusions could be term and seasonal trends. drawn regarding the applicability of Prospect atmospheric variables to the Warragamba site, other than to say that both typically follow These semi-empirical additions are not truly part Sydney’s weather patterns with minor random of the Penman Method. They are simply a means spatial variability. The study area could not simply of estimating some of the inputs into the method. be moved to Warragamba (where both hydrometric and meteorological data are available), since the The Penman Method can be generally represented “Class A” evaporation pan was only available at by the following equation (Thompson, 1999): Prospect, and the Penman estimates needed to be comparable to this as much as possible. This was ⎡ Δ ⎤⎡γ ⎤ EEE=+ (1) accepted as a forced limitation to the study, ona⎢Δ+γγ⎥⎢ Δ+ ⎥ imposed by a lack of all necessary data at any ⎣ ⎦⎣ ⎦ single location.

1869 In this equation, Eo is the evaporation rate in on much more rudimentary atmospheric variables, cm/day; Δ is the slope of saturated vapour pressure which are easily obtained in long record. per temperature; γ is the psychometric constant; En is the evaporation rate due to net all wave radiation Some of the most recent texts (Thompson, 1999), (cm/day); and Ea is the evaporation due to quote empirical equations to estimate the variables aerodynamic forces (cm/day). It is easy to see how in Equation 3. Some of these equations have the method is a direct combination of two parts: purported confidence intervals of ±20%. While thermodynamics (term with En part of Equation 1), this is commendable for an empirical equation, it and aerodynamics (term with Ea part of Equation must be remembered that being empirical, these 1). Each term expands to create a large hierarchy equations are not necessarily applicable to of equations. It should be noted here that the different climates and locations. Furthermore, Penman Method adopted in this study was taken accumulation of errors throughout the Penman from Thompson (1999) in that Dalton’s wind Method greatly magnifies any uncertainties. For function (Dalton 1802, as cited in Thompson instance, if all 7 of the standard Penman input 1999) is recommended, and is used throughout this variables have a 20% error, elementary statistics study to be consistent with Thompson (1999). suggest a joint probable error as high as 79%.

This paper addresses the term En (evaporation This paper presents a method that could identify resulting from net allwave solar radiation). This and reduce the errors associated with the empirical term is derived simply from net allwave solar means of approximating the solar radiation 2 energy flux (Qn) (calories/cm /day), the density of variables of Equation 3. water at a given temperature (ρ) (g/cm3), and the latent heat of vaporisation of water Lv (calories / The key term in this equation is Qs (gross solar g): radiation). One of the best empirical methods of estimating Qs is presented in Thompson (1999) Q (Equation 4). This method uses solar radiation E = n (2) n ρ constants quoted by Dunne & Leopold (1978), and Lv reduces them for atmospheric and cloud related losses based on an empirical formula. These Further expansion of the net allwave solar energy atmospheric constants represent the average daily term (Qn) brings the thermodynamic component to solar radiation energy projected onto the earth by the point where data input is required: the Sun for given latitudes and given month (average daily radiation for each month). =− − QQQQn s rs lw (3) =−−2 QIso()0.803 0.340 C 0.458 C (4) In Equation 3, Qs is the gross solar radiation, Qrs is the solar radiation reflected off a water surface, In Equation 4, solar radiation arriving at the earth and Qlw represents the net longwave energy, which (Io) is reduced by 19.7% (on account of does not play a significant role in the process of atmospheric deflection and absorption), and then evaporation, and hence is subtracted from the gross reduced further as a polynomial function of energy flux. These three terms are actually a cloudiness coefficient C. Cloudiness coefficient is simplification of the Energy Budget theory. The available from the Australian Bureau of three terms considered in Equation 3 are those Meteorology, and is a measure of relative which are most influential on evaporation. To this cloudiness on a scale of 0 to 8. Cloudiness point, the thermodynamics part of the Penman coefficient is generally acknowledged as a ratio of Method appears completely physical. It is beyond measured daylight hours to maximum daylight this point that the method is affected by the use of hours for a given day in the year. Such a measure empirical relationships. is clearly subjective, since clouds vary in height, consistency and density. Rather than measuring each of the variables in Equation 3 on a daily basis, most applications of In this study, the entire Qs term was replaced by the Penman Method instead use equations which actual daily measurements of solar radiation can estimate each of the variables based on measured at Observatory Hill, Sydney. Radiation empirical relationships. The temptation lies in that measured at the earth’s surface inherently accounts obtaining such specific radiation measurements for atmospheric and cloud losses. Hence, Qs can be requires expensive and very sensitive equipment. defined as radiation arriving at a water surface. Empirical relationships on the other-hand can approximate them with reasonable accuracy based

1870 The term Qrs benefits equally from this change, are cloud related, and presenting them as a fraction since it is simply Qs multiplied by a constant comparable against the conventional cloudiness fraction albedo (α), which represents radiation coefficient. It is known at this point that reflected off an open water surface at zenith angle. atmospheric losses account for approximately 20% Albedo is a physical characteristic of water and of gross radiation. Therefore the gross radiation only varies with angle and possible surface time series is multiplied at every ordinate by a agitation. These variations were assessed in the factor of 0.8. This was then superimposed over a wider study, but are outside the immediate scope plot of measured solar radiation on “clear days” of this paper. (no cloud):

Finally the term Qlw remains. This is addressed by Adjusted D&L Solar Values Measured Solar Radiation

Thompson (1999), again using a complex semi- 900 Adjusted D&L values. physical equation. While no alternatives could be 800 rationally proposed to better this equation, some 700 important observations were made. 600 500

400

300 Firstly, Qs was replaced by the measured solar Io (Cal/cm2/day) 200 radiation data (Im) and run through the Penman 100

Method. It became immediately obvious that the 0 Q equation began to act disproportionately to the 23/12/88 7/05/90 19/09/91 31/01/93 15/06/94 28/10/95 11/03/97 24/07/98 6/12/99 19/04/01 1/09/02 lw Date other components of the radiation balance (Equation 3). For cloudier days in the study period, Figure 1. 80% of gross daily radiation, plotted Equation 3 was resulting in a negative net solar against solar radiation measurements for cloudless radiation as a result of the disproportion. In-depth days. analyses of the measured solar radiation (Im) revealed that the inherent cloud losses of the Figure 1 shows the two time series are almost radiation data did not match those implied by the identical (except for the flat-line period of missing cloudiness coefficients measured by the Bureau of data). This is the definition of “clear day” radiation Meteorology. In simple terms, the measure of used throughout the study. This time series was then statistically “ironed out” to reduce the cloud in the Qs term and Qlw term were not consistent. To reconcile this difference a new influence of short term irregularities, creating an measure of cloudiness was “extracted” from the average annual clear day radiation pattern for solar radiation data. Sydney. Establishing the “clear day” radiation pattern was This new measure of relative cloudiness was an important step in deducing radiation deficit. For termed “Radiation Deficit Cloudiness” throughout it was then very evident that for a given day of the the study, and denoted as Crd. It was called this year, the difference between the predicted “clear- because, rather than being derived from missing day” radiation for that day, and the actual measure sunlight for a given day, as is traditionally the of solar radiation on that day, could safely be case, it was derived from the missing radiation, or attributed to cloud losses. This solar radiation “radiation deficit”. deficit, expressed as a fraction was plotted against conventional cloudiness coefficient in Figure 2. Radiation arriving at the earth’s atmosphere is defined with acceptable accuracy by Dunne & Measured Cloudiness Vs Radiative Deficit Cloudiness Leopold’s (1978) constants. These are a function 12 Measured C of radiation travel through space and the earth’s 10 Rad. Def. C position relative to the sun. These can be assumed 8 sufficiently constant and accurate for the purposes 6

4 of this study. Dunne & Leopold’s daily averages Factor Cloudiness for each month were applied at mid month and 2 interpolated for days in between to create a 0 smoother daily pattern. 1/10/2000 21/10/2000 10/11/2000 30/11/2000 20/12/2000 9/01/2001 Date

At this stage, there were two daily radiation time Figure 2. Conventional Cloudiness Coefficient series, one arriving at the earth’s atmosphere free vs. Radiation Deficit Cloudiness Coefficient. of losses (Dunne & Leopold’s 1978 radiation constants), and one measured at the earth’s surface The similarities are compelling, proving that the after atmospheric and cloud related losses. It was radiation difference was successfully reconciled. then simply a case of identifying the losses which Despite the similarities, and assuming that

1871 radiation deficit (being a direct measurement) had study period before and after the amendments, become the more accurate representation of cloud plotted against the raw Class-A Pan readings. activity, it was clear that the conventional Ammended Penman Penman vs Pan Original Penman cloudiness coefficient does fail to represent the 250 occasional cloud “event”. "Class A" Pan 200 Evaporation was estimated once again using the 150 amended Penman Method on a daily basis for the study period. Amendments include using measured 100 Monthly Eo (mm) Monthly Eo daily solar radiation in place of Qs, and “radiation 50 deficit” cloudiness coefficient in place of 0 conventional cloudiness coefficient in the Qlw Oct-95 Mar-97 Jul-98 Dec-99 Apr-01 Sep-02 equation. Date Figure 3. Monthly evaporation estimates for The evaporation estimates made post amendment standard Penman Method, amended Penman could then be considered as the better estimate for Method, and raw Class-A Pan readings. the study period, since they were based on real solar radiation inputs, with less empirical The coincidence between the Penman Method approximations. The results are presented in evaporation pattern for the study period, and the Section 5. raw Class-A pan readings is remarkable, as shown in Figure 3. While raw Class-A pan readings are It is not always practical in routine application of by no means the most reliable evaporation the Penman Method to obtain measured solar estimates, it must be remembered that they and the radiation data. This study encountered difficulty in Penman Method are two completely different obtaining such data as a homogenous time series. methods of estimating evaporation. Other studies It would be even more difficult if the data was such as that by Mosner & Aulenbach (2003) required for a specific time period. Therefore it covering a variety of evaporation methods show would be very useful if the findings of this study that, even amongst the most popular evaporation could be instilled into the standard Penman estimation methods, this level of correlation is Method, so the method itself can improve even in unusual. the absence of measured solar radiation. Table 1. Result summary of comparisons between amended Penman Method, standard Penman In the wider study, many options of achieving this Method, and raw Class-A Pan readings. were tested, the most obvious of which would be to adjust the empirical constants in the equations to Total Depth mm better re-create the results of this study. However, the option which maintained the best short term Standard Penman 7,591 accuracy was instead to adjust Dunne & Leopold’s Amended Penman 7,739 solar radiation constants. Class A Pan 8,276

5. RESULTS The standard and amended Penman Methods in Figure 3 seem to be very similar. Table 1 Since there is no real “answer” to the actual rate of illustrates that the total difference between the two evaporation loss, the benefits of the changes made methods over the study period was 148mm. This is to the Penman Method can only be expressed and an annual average of 23mm, which is surprisingly evaluated in relative terms. low, considering how the standard Penman Evaporation estimates were calculated using the Method approximates net solar radiation with standard Penman Method as presented by empirical equations. Unfortunately there are no Thompson (1999) as a control. The same process evaporation estimates of this level of accuracy was repeated using the Penman Method amended against which these results can be validated. according to this study. Figure 1 showed that a The substitute solar constants created for use in the large gap of about 3 years existed in the measured absence of solar radiation data is able to emulate radiation data. For this time period, the standard the results gained using real solar radiation data to Penman Method was used with amended solar within 3.4mm per annum. radiation constants. Dunne & Leopold (1978) solar radiation constants Estimates were calculated on a daily time step, and are denoted as I . The substitute solar radiation the results were summarised into monthly total o constants created in this study are shown in Table evaporation. Figure 3 compares the results for the 2 as Isub. The units for these constants are

1872 calories/cm2/day, and represent average daily Overall, the Penman Method appears to be a sound radiation for each month of the year. method for estimating evaporation. Even with the use of empirical approximations instead of real Table 2. Solar radiation constants designed to data, the method gives very reasonable results. emulate the use of measured solar radiation when used in the standard Penman Method. 7. ACKNOWLEDGMENTS Month I sub 1 1031 This study was undertaken as part of an honours 2 922 thesis with the full support and generous 3 735 contributions of academic staff at the University of 4 554 Western Sydney. Thanks also to the Sydney 5 416 Catchment Authority for permitting the publication 6 337 of this study. The contribution by Sinclair Knight 7 363 Merz is also acknowledged. 8 478 9 661 8. REFERENCES 10 836 Blaisdell, E. A., 1998, Statistics in Practice, 11 940 Second Edition, Harcourt, Orlando. 12 1054 Chow, V. T., 1964, Handbook of Applied It must be pointed out that while these values are Hydrology, McGraw-Hill, New York. based on Dunne & Leopold’s (1978) values, they Mosner, M. S. and Aulenbach, B. T., 2003, do not maintain the same meaning. Dunne & Comparison of methods used to estimate lake Leopold’s values actually represent solar radiation evaporation for a water budget of Lake reaching the earth’s atmosphere at given latitude Seminole, South-Western Georgia and North- and given time of year. The substitutes presented Western Florida, U.S. Geological Survey, in Table 2 however are notional, and were Georgia. developed only for the purposes on maintaining some consistency in Penman estimates across a Penman, H. L., 1948, Natural Evaporation from time period where gaps are likely to occur in Open Water, Bare Soil, and Grass, Proc R. measured solar radiation data. Soc, London. Penman, H. L., 1956, Estimating Evaporation, 6. CONCLUSIONS Trans Am Geophysics Union, U.S. The error incurred by the use of semi-empirical Priestly, C. H. B. and Taylor, R. J., 1972, On the equations to approximate the radiation terms in the Assessment of Surface Heat Flux and Penman Method seems not to be as dramatic as Evaporation Using Large Scale Parameters, claimed by publications in the past. According to Monthly Weather Review. the findings of this study, use of these semi- empirical radiation equations will cause the Sinclair Knight Merz, 2003, Independent Review standard Penman Method to underestimate of SCA’s Water Supply Systems Model evaporation by little over 2% compared to WATHNET, Internal SCA Document, SKM, estimates using real radiation data. Malvern (VIC). For most applications, such a level of Singh, V. P., 1992, Elementary Hydrology, improvement is not required. Hence the standard Englewood Cliffs, NJ : Prentice Hall. Penman Method would suffice. For studies of Stephen and Hoy, 1977, Field Study of Lake water bodies with large water surfaces, or where Evaporation, Australian Government higher levels of accuracy are sought in a water Publishing Services, Canberra. budget, the findings of this study may prove to be valuable. Thompson, S. A., 1999, Hydrology for Water Management, A. A. Balkema, Rotterdam. A second conclusion to be drawn from this study is that the Pan Conversion Method of estimating Viessman, W. Jr., Lewis, G. L. and Knapp, J. W., evaporation currently used by water resources 1989, Introduction to Hydrology, Third authorities has a tendency to over estimate Edition, HarperCollins, New York. evaporation by as much as 7% if compared to the Ward, R. C. and Robinson, M., 2000, Principles of post amendment Penman Method estimates made Hydrology, 4th Edition, McGraw-Hill, in this study. Berkshire, England.

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