Evaporation from open water is among the most rigorously studied problems in hydrology. Robert E. Horton, unbeknownst to most investigators on the subject, studied it in great detail by conducting experiments and heuristically relating his observations to physical laws. His work furthered known theories of lake evaporation, but it appears that it was dismissed as simply empirical. This is unfortunate because Horton's
century-old insights on the topic, which we summarize here, seem relevant for contemporary climate-change-era problems. In rediscovering his
overlooked lake evaporation works, in this paper we (1) examine several of his publications in the period 1915–1944 and identify his theory sources for evaporation physics among scientists of the late 1800s, (2) illustrate his lake evaporation formulae, which require several equations, tables, thresholds, and conditions based on physical factors and assumptions, and (3) assess his evaporation results over the continental U.S. and analyze the performance of his formula in a subarctic Canadian catchment by comparing it with five other calibrated (aerodynamic and mass transfer) evaporation formulae of varying complexity. We find that Horton's method, due to its unique variable vapor pressure deficit (VVPD) term, outperforms all other methods by

The problem of accurate lake or open water evaporation estimation has been a
subject of scientific inquiry, in the modern sense of combined experimental
and theoretical study, for the past 4 centuries. Factors that control
evaporation have been investigated since the time of Edmund Halley (1687), with rapid progress in theories of thermodynamics, aerodynamics (turbulence theory), and molecular kinetics (kinetic theory of gases) that led to a better understanding of evaporation due to wind's influence, convection, and
diffusion. Brutsaert's (1982, chap. 2) treatise on

Robert E. Horton, a pioneer in hydrology and well regarded for his contributions to areas of hydrology like infiltration, overland flow, and river geomorphology, is not usually considered a fundamental contributor to the field of evaporation. However, unbeknownst to most in mainstream evaporation theory, tucked away in his home-based experimental catchment beside a pond, Horton conducted rigorous experiments and theoretical work on open water evaporation from the 1910s until the end of his career (circa 1945). In particular, in 1917, he published a set of formulae for estimating evaporation (including within-lake variations in evaporation) based on physical laws which he believed were more robust than the then existing methods. The subheading to the title of his first 1917 paper claims the following:

Empirical Statement Based on Physical Law Agrees with Observed Facts and Is Held To Be an Improvement Over Existing Formulas (Horton, 1917a).

He held the view that his equation was superior to other known methods for the following decades, even in the face of rapid developments in evaporation theory in that period (e.g., see Horton, 1934). After we examined several of Horton's papers and reports related to evaporation from lakes and pan evaporimeters (or, simply, pans) from 1917 to 1944 (the year before his death), we noted that he derived his formula theoretically, but since the values of the coefficient in his formula were not easily available, and his formula resembles other empirically derived formulae, several investigators may have dubbed it as simply empirical (see Rohwer, 1931). However, Horton's nuanced understanding of the boundary layer physics of his time (turbulence theory, horizontal vapor transport via laminar flow, convective transfer of vapor, and wind and vapor blanket characteristics), and the sound premise of his work based on molecular kinetics, reveal the potential of his work to offer new insights for an improved formulation of evaporation. The theory behind his work is illustrated in Sect. 2. After evaluating Horton's evaporation formulae (in Sect. 3), we find that his claim of having developed an improved method not only stands to be true in his time but also holds great contemporary value, and it is unfortunate that it has been largely overlooked or forgotten. Therefore, in this paper, we examine his evaporation work from the perspective of contemporary theories and those of his time to highlight his ingenious perceptual, experimental, and theoretical insights into the subject. We revisit his claims, replot his figures with recent data, simplify the use of his experimental tables (by converting them to parametric forms), assess his method's ability to generalize across wide-ranging conditions, and show the relevance of his method for contemporary large-scale evaporation problems.Hydrologists need no introduction to some of Horton's contributions like
infiltration theory, overland flow, and geomorphological laws, but what may
not be widely known is that he published an estimated 200 papers and reports, and of these, only about 80 works (mostly single authored;

About 80 of Horton's contributions were provided by Hall (1987) and curated
by the American Geophysical Union (AGU) Virtual Hydrologist Project (see Foufoula-Georgiou (Foufoula-Georgiou, 2021; Folse, 1929). A more complete list of Horton's works was collated by Elizabeth Clark, which includes

About a dozen of Horton's papers and reports are related to his evaporation
method and supporting ideas, but one can gain a full understanding of his published contributions on lake evaporation from five key publications (i.e., Horton, 1917a, 1927, 1934, 1941b and 1943b). Horton's evaporation method was first introduced in Horton (1917a), as part of a three-paper series (Horton, 1917a, b, c) in

The various above-mentioned works related to lake evaporation have been cited sparingly which shows that they were largely overlooked. They have not been collectively examined in any previous work to our knowledge, and in the few citations to them, the value and sophistication of the method was not recognized. Horton's lake evaporation equation received some attention in
Chow's (1964)

For a context of the works preceding Horton's time, interested readers are
directed to an excellent contribution by Grace Livingston, published as eight pieces in

Horton's evaporation method was apparently developed and used in New York, Michigan, and Chicago (see Horton, 1927), but, in the same time period, many similar efforts were underway throughout the United States (presumably in other countries too). Worth highlighting are the following three works: First, there is the thermodynamic approach, using Le Châtelier's principle applied to energetics. This approach, which was undertaken in California at the Scripps Institute of Oceanography and California Institute of Technology, led to the energy balance solution of lake evaporation and the Bowen (1926) ratio. Subsequent works by others that picked up on this work are summarized in a succinct compendium by McEwen (1930) and a historical summary by Lewis (1995). Second, there is a review of mass-transfer-based and energy-balance-based evaporation studies on Lake Hefner, resulting from collaboration between several U.S. agencies, including the Geological Survey, Department of Navy, Bureau of Ships, Navy Electronics Laboratory, Department of Interior, Bureau of Reclamation, Department of Commerce, and Weather Bureau (USGS, 1954). Third, there is a statistical attack on the problem led by geophysicist John. F. Hayford, who notably spent over 2000 h developing a superior method, including a mammoth effort by 41 persons, who collectively spent some 32 000 h on this work (Folse, 1929, p. 7). The method uses the temperature and humidity of the preceding day to calculate the following day's evaporation, and includes a large system of equations with many free parameters, which is optimized to minimize error (for more details, see Folse, 1929). It was developed for the Great Lakes, and did perform reasonably well there, but generalized poorly in other lakes and did not gain wider attention (see the critical review by Bernard, 1936). These highlight some of the various independent efforts dedicated to calculating evaporation around the time when Horton's method was developed.

Citations provided in Horton's work show that he relied on the works of several European scientists for the concepts related to the physics of
evaporation. He did examine several empirical equations developed in the U.S.
(see Horton, 1934), but he does not appear to have followed the works conducted by Bowen and Cummings (Bowen, 1926). Perhaps this is because Bowen's works appeared in

A molecular kinetics view of evaporation is fundamental to Horton's approach, and he developed this view mainly from John Dalton's (1802) theories and experiments on evaporation of water and other chemicals. Dalton's (1802) work was, in fact, the only work that Horton directly cited when he first published his evaporation paper (Horton, 1917a), though, with a closer look at his later papers (Horton, 1927, 1933), it does appear that he developed his method by building upon multiple works. It appears that Horton studied the following works: he consulted Thomas Stevenson's (1882) work on wind speed variation by height, while conducting his own experiments on the role of wind on evaporation (see Horton, 1927); he referred to the work of Geoffrey Ingram Taylor and William Napier Shaw (1918), for the role of turbulence and the vapor blanket (Horton, 1934); he drew from Napier Shaw's work in

Horton's references also included American textbooks, particularly the following two: Allen Risteen's (1896)

Before we delve into the details of the evaporation equation, the following quote from Horton contextualizes how he supposedly viewed his evaporation formula:

A rational equation may be defined as one which can be derived directly from fundamental principles, which fits all the experimental data and which represents the physical conditions correctly throughout the entire range of their occurrence and hence is valid outside the range of experimental observation. (Horton, 1941a)

Some fundamental principles that he alluded to in his evaporation formula are related to thermodynamics (i.e., work done in phase changes and latent heat), and they include references to geometric proofs of these principles from the perspective of kinetic theory drawn from Risteen (1896), as discussed in Horton (1934). More importantly, the premise of Horton's fundamental principles in his evaporation method is the kinetic theory of gases (Loeb, 1934), which he explicitly stated in Horton (1917a). His molecular kinetics view of evaporation is best captured by the following quote:In a mixture of air and water-vapor there is a certain number of vapor molecules per unit volume. When there is wind the air and vapor are swept along together at a rate depending on the pressure-gradient. This, as in case of hydraulic flow, is independent of the total pressure. At a given vapor-pressure the same amount of vapor is carried by the wind per unit of time and per unit of volume of air, whether the number of air molecules per unit volume is large or small. (Horton, 1934)

His first paper on evaporation (Horton, 1917a) does not discuss the thermodynamic perspective, but his derivation of the various parts of the
evaporation equation does use the underlying principles, as exemplified in
the following quote:

[Latent heat] comprises of two elements: (1) Internal work in overcoming molecular attractive forces which, in general, including viscosity and surface-tension, increase as the temperature decreases, and the latent heat of internal work also increases as the temperature decreases; (2) the external latent heat, which measures the work done by the emitted vapor in expanding against the external pressure, decreases slightly as the pressure on the liquid surface decreases with decreased boiling temperature, but the total latent heat increases slowly as the temperature decreases. (Horton, 1934)

He examined these thermodynamic factors to identify the role of pressure in impacting vapor emission and vapor removal. While pressure does affect vapor emission rates due to external latent heat, it is negligible, so the impact of pressure on evaporation can be attributed to vapor removal (somewhat like a proof by elimination).Vapor return is controlled by wind action (which is nonlinear) and the vapor pressure of the overlying air or the vapor blanket, i.e., a thin layer of vapor just above the water surface analogous to viscous sublayer in open channel flow. The characteristics and role of the vapor blanket is discussed separately and in more detail in Sect. 3.4.

Vapor removal, as previously stated, happens due to diffusion, wind action, and convection.

Horton's conception of evaporation via diffusion is perhaps drawn from
Dalton's (1802) original work, which is the only reference he cites when he
first published his lake evaporation formula in Horton (1917a). Dalton
posited the following:

Evaporation […] is caused by

According to the contemporary evaporation literature (see Brutsaert, 1982), wind can have two effects, namely (1) turbulence transfer of vapor away from the surface and (2) advective (bulk fluid mass) transport due to mean horizontal wind. In Horton's work, wind action is considered separately as a bulk exhaustion process that removes vapor at a maximum rate equal to the rate of vapor emission. The rate of wind action in Horton's work is based on Dalton's observation, as follows:

[Dalton] found that a strong wind made the amount of evaporation

It may help the reader to first disambiguate the term convection, as it is
sometimes used interchangeably with advection (e.g., convection–dispersion
equation/advection–dispersion equation). Convection normally refers to heat
transport via vertical plumes in fluids when wind shear is overcome by
thermally driven buoyant production of kinetic energy, while advection normally refers to the transport of quantities (heat or matter) due to the mean horizontal flow of wind (see Hess, 1979; Stull, 1988; and Eagleson, 1970). Horton's usage of the term convection does share similarities with the common parlance in turbulence theory pertaining to heat transport, i.e., convection happens due to expansion from surface air heating and vapor addition, which causes a reduction in density (as the bulk dry air is heavier than moist air) that results in instability. Convective plumes are fed and sustained by laminar wind that feeds moisture horizontally into it and continues until the buoyant force overcomes the shear force due to horizontal wind. It is sustained until the moisture available to feed the plume is depleted. This conceptualization of convection is not clearly described in Horton's evaporation papers, but we inferred it from the following quote in his paper (Horton, 1933) on columnar vapor drift (a mechanism of evaporation):

In the eerie morning hours […] vapor columns present a spectral appearance as they travel slowly over the water surface, resembling sheeted ghosts or white-robed whirling Dervishes walking on the water. […] Obviously columnar vapor drift [also amorphous vapor drift] is a visualization of convective vapor removal from a water surface during evaporation. […] A vapor column forms wherever a sufficient degree of instability develops through the warming of a layer of air close to the water surface and through the accumulation of water vapor (which is lighter than air) therein. A vapor column is fed by horizontal flow of air and vapor toward it close to the water surface. Apparently it grows until its feeding area encounters another area from which the vapor has already been exhausted or until the frictional resistance of horizontal flow balances the vertical convective forces. (Horton, 1933)

Horton regarded convection as a rheologic system, i.e., a flow process with solid and fluid characteristics, typically in response to forces (in the case of evaporation, as pressure over a unit elemental area). In the following quote, his view of convection as a rheologic system is clearly stated:The ordinary, vertically convective system […] may be considered hydrodynamically as a rheologic or flow system, resembling the flow through a vertical pipe connecting two reservoirs, with lower pressure in the upper reservoir. This may be called the tubular type of vertical convection. (van Vliet and Horton, 1949)

Horton's understanding of the primary processes that control evaporation.

Here, the colored balloons represent evaporation aided by vapor removal due to diffusion (purple), wind action (gray), and convection (red). Diffusion can be upward or downward in direction, where upward (positive) is evaporation, and downward (negative) is condensation (see Sect. 3.5). The gray balloon (wind action) depends on the wind speed about 1 ft (0.3 m) away from the surface of the water, and it is governed by an inverse exponential law (see Sect. 3.2) and can happen during the day or night, though it is accentuated during the day when wind speed is higher. The red balloon (convection) depends on the temperature deficit across a vertical gradient and a laminar wind that accompanies vapor removal (see Sect. 3.2.3), and it occurs predominantly during the night (Horton, 1917a) when the water is warmer than air due to its higher heat memory (i.e., specific heat capacity).

In what follows, we illustrate Horton's evaporation equations, their theoretical basis (using direct quotes where possible), correction factors and tables (as parametric equations), and provisional values of coefficients with appropriate units.

If

Pan evaporation (

The evaporation capacity (

The maximum rate of evaporation which can be produced by a given atmospheric environment from a unit area of wet surface exposed parallel with the wind, the surface having at all times a temperature exactly equal to that of the surrounding air. (Horton, 1919a)

For small water bodies, particularly those with shallow depths, in the absence of water surface temperature data, when the lag between water and air temperature is negligible, Eq. (1b) can be used. Over pans, an area factor and the variability in the vapor blanket thickness should be taken into account (discussed in Sect. 3.3 and 3.4) but can be ignored over large lakes.Lake evaporation (

The inclusion of

The wind factor,

In the case where warm days are followed by cool nights, convective vapor removal may be important. Convective vapor removal happens more readily in the nighttimes than in the day times. When surface winds are suppressed by inversion, and when water temperature is higher than that of air, evaporation may be dominated by convection, so an alteration of the formula for

One familiar with the combined equation of Penman may recognize that Horton's approach to adjusting the wind term with a convective term bears some resemblance to the physics represented in the combined equation which uses a harmonic mean-like weighting, wherein the psychrometric constant accounts for the role of pressure (the aerodynamic term), and the slope of the saturation vapor pressure curve accounts for the role of temperature (the energetics term), together forming the combination method. Similarly, Horton's assumption that convection is caused by a combined effect of calm wind and temperature gradient appears to be logically related to part of the physics represented by the flux Richardson number (

Though the rationale behind

Most existing evaporation formulas are in error in that they involve a linear factor for wind correction such that wind effect apparently increases indefinitely as the wind velocity increases. It has been proved experimentally, and is indicated by physical considerations, that since the wind can do no more than to remove the water vapor as fast as it is emitted from the liquid surface, there is a maximum or limiting value of the wind factor corresponding to each water surface temperature. (Horton, 1917a)

Other investigators followed Dalton's (1802) suggestion and included a wind correction factor that assumes the form[…] with the same evaporating force, a strong wind will double the effect produced in a still atmosphere. (Dalton, 1802, 581–584)

The removal of vapor by wind corresponds to a condition of natural exhaustion to which the inverse exponential law commonly applies. (Horton, 1917a)

The theoretical basis for such a view appears in some detail in Horton (1934), as follows:In accordance with the Dalton formula, with the form of wind factor hitherto commonly used, the rate of evaporation increases indefinitely as the wind velocity is increased. This is obviously incorrect, since the rate of evaporation cannot in any event exceed the rate of vapor emission, and the latter is not affected by wind velocity in the absence of waves and spray. There must be for each water-surface temperature a maximum rate of evaporation, which rate cannot be increased by further increase in the wind velocity. (Horton, 1917a)

The rationale for the wind factor can be understood by considering the extremes. Evaporation is at its maximum rate when wind speed is high (i.e., evaporation happens at double the rate as compared to still air, as Dalton observed), i.e.,The limitations of Dalton's (1802) evaporation work were well known before Horton's time. For example, it has been noted that Dalton's (1802) observations were for the month of August only, and the evaporation estimated using his equation was found to be imprecise in other summer months (Soldner, 1807). Quantifying the influence of wind on evaporation seems to have had some attention in a few other works, as evident from the following quote from Brutsaert (1982):

[Soldner's] perceptive remarks notwithstanding, during the next half century, apparently little progress was made as regards the effect of the air stream. […] Schübler's [1831] data obtained during 1826 at Tübingen […] showed that evaporation of a water surface exposed to wind was 1.7 times larger than that of a sheltered surface in summer, and 4 times larger in winter. (Brutsaert, 1982)

Nearly a century after Schübler, Kennedy (1933) revisited the topic. It appears that Horton was not aware of Kennedy's or Soldner's works; he seems to have relied solely on Dalton's observations.Condensation or dew rarely occurs on windy nights […] experiments were made to determine the effect of wind on the condensation of moisture on the surface of cans containing ice and water, and mixtures of ice and salt. (Horton, 1917a)

In a paper 17 years later, Horton (1934) discussed the role of condensation, revisiting experimental results in conjunction with the properties of his equation, and he writes the following:It is evident that wind – except a slight wind – does not affect the rate of vapor-emission and return by diffusion but it does increase the rate of mechanical removal of newly emitted vapor. Consequently it appears that wind tends to decrease condensation instead of increasing it. Horton (1934)

Horton felt quite strongly about improper usage of pan data.

The land-exposed evaporation pan appears to be about the poorest device humanly contrivable for the purpose of determining the evaporation losses from broad water surfaces. (Horton, 1917a)

But it is important to note that Horton did not advocate for not using pan data. The use of pan data as a proxy for lake evaporation is justified after due consideration of various factors that cause lake and pan evaporation to differ from each other, namely (1) humidity corrections, (2) rim height and depth effects, (3) vapor blanket formation and exhaustion characteristics governed by meteorological factors (wind speed), and (4) temperature difference between pan and lake surface (especially important in the case of large lakes). Used correctly, pan evaporation can be a good proxy or a validation to cross-check actual lake evaporation. The wind speed at ground level has to be corrected considering the pan diameter (Horton (1927) conducted ingenious experiments on wind that circumvented the need for wind tunnels.

For the purpose of determining the effect of wind on evaporation, experiments were carried out at the author's laboratory, using pails filled close to the rim, and suspended so as to swing freely from a rotating frame. […] These experiments and studies served to determine the coefficients in the formula. (Horton, 1927)

Wind factor (While using pan evaporation to calculate lake evaporation, an area factor is required (Horton, 1927, p. 162) to cross-check their respective values. The area factor,

If the air and water temperatures are equal, then the correction factor

[The author] deduced a rational expression for area-factor based on the
assumption that near the windward edge of a broad water-surface an unknown fraction

The vapor blanket is conceptually similar to a viscous sub-layer in open channel flow and is formed due to the existence of a laminar flow layer which horizontally transports moisture in the downwind direction, which leads to its growth in height. The horizontal variation in the vapor blanket height, which is of the order of a few meters, is critical when estimating pan evaporation. Pans have a poorly formed vapor blanket because of their small size, as even weak winds can remove the laminar layer before it is fully formed. Once pan evaporation is corrected for the formation and disturbance of the vapor blanket layer, their use for lake evaporation can be readily justified (Horton, 1927). In the case of both pans and lakes, the vapor blanket characteristics are the same (both are governed by meteorologic factors), but over pans the variation in evaporation over the variable thickness of vapor blanket is more important, while over large lakes they can be ignored as the area involved is small. It is important to account for the effect of the vapor blanket during both daytime (when it is slightly larger) and nighttime conditions (see the example problem in Horton, 1917a).

Understanding the process of vapor blanket formation and accurately quantifying its development and disturbance from the windward fringe of the lake to the leeward side can be considered as one of the main theoretical breakthroughs in Horton's evaporation work. The reason for it being an important breakthrough is that it explains why pans and large lakes have different evaporation rates. It provides a basis for ignoring the vapor blanket thickness variation in large lakes, and it explains why it would be a big mistake to ignore it from pans.

Horton derived an expression (see Eq. 5 below) to capture where, when, and
how much the evaporation rate varies across the lake (or pan) surface.
Assuming a strip of unit width, the horizontal distance of the vapor blanket before its thickness becomes constant (

Though Horton does not provide the steps to derive Eq. (5), derivations for analogous problems which resemble this equation, as solved by Horton and others, may provide some insight. For convenience of reference, one such derivation by Horton (1927, p. 63) and how it can be interpreted for the derivation of Eq. (5) is given in the Supplement. Some examples of viscous sub-layer problems in open channel flow are given in Horton et al. (1936).

Another useful formula Horton provides is one for calculating evaporation (

In most cases, the vapor blanket thickness is only a few millimeters, and it is related to wind velocity. Horton (1943b) presents an equation for the vapor blanket thickness given by Taylor and Shaw (1918). Though Horton's reference has the same title as that provided in reference, the year specified by Horton (i.e., 1934) could have been a typo, and the correct reference is likely to be the one given here. After inspecting Taylor's papers from 1934 and conducting a cursory search of his bibliography for similar titles, we did not find the equation Horton provided. From Horton (1943b), the vapor blanket thickness (

Horton is among the few hydrologists to rigorously examine the role of the vapor blanket in lake evaporation. So, to conclude this section, a brief synopsis of some of the other studies conducted by other investigators may aid the readers in pursuing further research in this direction. Horton's source for the idea of vapor blanket and its contributions to evaporation rates could perhaps be the Slovenian scientist Jožef Štefan (1882).

The fact that the amount of evaporation from a basin is proportional not to the surface content but rather to the square root of this surface content leads to the result that evaporation from large water basins is proportionally smaller compared to the evaporation from a small basin. Let us also add that this is true not only for diffusion-driven evaporation but also for convection-driven evaporation.

The characteristics of the vapor blanket have been studied in only a few
other works, to our knowledge. Sutton (1934) and Vercauteren (2011) have
considered the shape of the vapor blanket in the windward edge, but its
properties with respect to evaporation (and with regards to turbulence,
convection, etc.) over lakes were not explored. Millar's (1937) apparently rigorous study of the vapor blanket was not accessible to us (we were unable to obtain a copy of the paper), but a summary is provided in a United States Geological Survey (USGS) report (1954; see the chapter on “Mass Transfer Studies” by Marciano and Harbeck), which shows Millar's equations. They indeed seem to resemble Štefan's work on diffusion. Finally, there is an indirect reference to the vapor blanket in Peter Eagleson's textbook on

To understand the role of vapor removal and diffusion, for convenience we can consider a general form of Eqs. (1) and (2). Ignoring convection, inserting

It is evident that in order to determine the effect of change in barometric pressure on evaporation, other things equal, its effect on vapor removal by diffusion, which is always present, and its effect on vapor removal by wind-action, must be considered separately. This may readily be accomplished by the use of an evaporation formula published some years ago. (Horton, 1934)

An inverse relationship between diffusion and pressure was first proposed by Thomas Tate (1862) and later derived by Štefan (1882). IfThe relation of barometric pressure to convective vapor removal has apparently not been studied. Since convection is, in general, not present when there is strong wind-action, it will not be considered here. (Horton, 1934)

The precision that went into Horton's experimental measurements is quite remarkable. He performed detailed experiments on the melting of snow, considering dozens of physical variables measured at 10–20 min intervals (Horton, 1915). These experiments and his earlier study on evaporation from snow (see Horton, 1914) resemble his later experiments on condensation (see Horton, 1917a). He designed his own instruments to measure minimum and maximum daily temperatures of water surface and a geometrical approach for snow temperature (Horton, 1919b). To cross-check his daily snow measurements, he made additional measurements at an accuracy of one-fifth of a degree at hourly intervals to cross-check the diurnal (min and max) daily snow temperature readings (Horton and Leach, 1934). He used graphical methods to calculate vapor pressure and humidity, which give values to within 1 %–2 % accuracy (Horton, 1921). Some evaporation measurements to cross-check his evaporation calculation (see Horton, 1927, 150–155) were made to approximately one-hundredth of an inch (0.00254 cm) precision.

High-latitude lakes are quite important in the context of accelerated Arctic warming (Smith et al., 2005), as the region is besprinkled with numerous tiny lakes, where the mean evaporation for each lake may vary appreciably due to the variability in the vapor blanket thickness (Eqs. 5–8), which means that the role of the vapor blanket cannot be ignored. In the domain of Canada and Alaska alone, there are over 13 million lakes measured at Landsat resolution (

Performance metrics (

We tested Horton's evaporation equation on Baker Creek in subarctic Canada, where 30 min meteorological data were available as measured over the lake and near the lake (see Spence and Hedstrom, 2018, for the data description and measurement heights). For vapor pressures of air measured in either location, the difference in evaporation was slight. To evaluate the performance of Horton's equation, following Singh and Xu (1997), we selected five other equations that resemble Horton's equation, namely Konstantinov, Dalton, Meyer, Rohwer, and Penman. Note that the Penman equation referred to here is not the combined equation but only a part of the combined equation (aerodynamic) provided in Penman's original work (Penman, 1948). The general forms of the equations are given in Table 3 (Sect. 4.3). We calibrated each of them by treating all the coefficients as free parameters, preserving only the structure of the equation. Most empirical Dalton-type formulas do not include a temperature deficit term, except for a few that are of the type of Konstantinov (1968).

The actual vapor pressure of the air is one of the most important variables, but it is difficult to obtain. To understand how robust or how prone to error these methods are when it comes to this variable, in addition to using observed measurements available for the test site, we calculated actual vapor pressure as a function of solar geometry, diurnal temperature range, and seasonal precipitation (see Bennett et al., 2020 and Bohn et al., 2013). The data for this calculation were drawn from our previous work (Vimal et al., 2019).

We used a bootstrap approach to obtain the mean (

Table 1 shows that Horton's method is substantially more accurate than the other methods, and this is seen consistently across timescales and sample sizes. This seems to also be true when considering other classes of models (radiation based, temperature based, or a combination), as evidenced by relative performances reported in Tan et al. (2007). The only exception seems to be artificial neural network (ANN) models, which appear to have the potential to be marginally superior to Horton's, going by their relative performance, but they require sufficient site-specific data and tuning.

Surprisingly, Horton's method outperforms other methods, even when using the estimated input vapor pressure (Table 2) and even if the results of Horton's equation from Table 2 (estimated actual vapor pressure) are compared with the five methods from Table 1 (local measurements). It must be noted that previous studies have shown that vapor pressure near water bodies (e.g., coastal regions) has a large bias and uncertainty (see Bohn et al., 2013), which makes the result even more surprising. A reason for the poorer performance of other methods could be that we estimated wind velocity at various heights by back-calculating, using Eq. (3b), and the bisection method previously mentioned (Sect. 3.2). Another reason could be the dependence of the vapor pressure measurement on the observation height for some, even if not all, of the other methods. Konstantinov's equation depends on wind speed at ground height (which is same as Horton's method), and uses more input variables related to temperature, and yet does not perform better. We do not draw bald conclusions directly from Tables 1 and 2 before testing under multiple catchments and lakes of wide-ranging meteorological conditions. However, if this result holds across various locations and regions, as we will show in Sect. 4.5 more generally, then, taken together, we can arrive at a few conclusions. (1) Horton's formula is robust against over-fitting of errors, making it more physically based. (2) The variable vapor pressure deficit (VVPD) term, unique to Horton's evaporation formula, is a better control on evaporation than VPD.

Performance metrics (

We use the term generality to mean the following connotations: (1) parameter certainty, i.e., how relatively unchanging the parameters in the calibrated equations are across wide-ranging conditions, time averages (the mean of evaporation is considered when time averaging, so the effect of time in parameters is ignored), and record lengths; (2) how well it performs in wide-ranging conditions across various meteorological conditions and altitudes; and (3) how well it performs over continental scales, which follows from both (1) and (2). The ability of a method to generalize across such conditions shows that the method is not an empirical fit but has a rational or physical basis.

If the parameter values are unchanging or have only a slight variability, then they can be assumed to possess a physical meaning which does not need
site-specific tuning (or calibration). Such unchanging values are termed
constants, and identifying such constants is common in physics. Of the three connotations of generality we are interested in, parameter certainty is the most important. In all the six methods we compared, there were 17 parameters, and each one was tuned for each of the 1650 bootstrapped samples, using a vectorized approach (see Sect. 4.1 for a breakdown of the sample size and record lengths). The tuned parameters are summarized in Table 3 (shown below). To make their comparison straightforward, the time unit of reference observation was kept identical to the native resolution, e.g., daily or monthly evaporation values were averaged into units of millimeters per 30 min, which allows us to compare values of parameters across methods and timescales. Some outliers in the parameter values were found (possibly due to errors in the data) but were removed using the same criteria (10th percentile) for all six methods each considered independently. The last column here shows normalized values of variability (

Parameter uncertainty comparison between six evaporation formulas (mean

Among all the parameters, parameter

To our knowledge, there is no other evaporation formulation that captures the role of

Horton claimed that his method was rational (physical) in that it is robust to conditions outside of which it was used (Horton, 1927, p. 159), which the other empirical methods of his time were not (e.g., the method by Carpenter and Fitzgerald; see Fitzgerald, 1886), as most were tuned for local conditions. Horton investigated the role of condensation rates, evaporation from snow surfaces (Horton, 1914), temperature deficits, and wind speed in high altitude and polar regions (Horton 1934). In a Central Snow Conference paper (Horton, 1941b), he comments on the processes involved in evaporation from snow that includes independent variables that depend on latitude and altitude, which were not known with certainty. When lake surfaces are partially covered with ice, he recommends using a weighted average of lake water and ice temperatures for partially frozen lakes. The role of the thickness of ice on the air–water temperature relationship was observed, i.e., thicker ice brings the air and water temperature closer. Additional factors that influence evaporation under such conditions could be the percentage, intensity, and duration of laminar and turbulent air flow, which depend on latitude and elevation (Horton, 1943a, b), and also other physical factors due to snow and ice, that is, (A) the area exposed to air (vs. projected area from snow surface) due to the influence of snow porosity may increase evaporation and (B) the disproportionate distance of air temperature from ice temperature (as opposed to water temperature). Horton (1934) suggested that these additional factors may require a separate treatment.

Horton used the data of 112 pan evaporimeters over the continental U.S. and plotted precipitation, evaporation, and runoff into one figure sliced by longitudes (see the figure in Horton, 1943a, b). We replotted his chart together with land surface model results simulated at over 200 000 model grid locations over the continental U.S. by Livneh et al. (2013). We aggregated the model results in the same way as Horton did by 2

Comparison of continental U.S. 2

The difference in evaporation is substantial in the Great Lakes region
(between longitudes of

While this paper highlights a century-old method, we do not fail to
recognize that advancements in evaporation theories of the last century have been stellar. One needs to only look at the number of numbers (mostly dimensionless) that are used to represent the physics that control evaporation, including Dalton, Reynolds, Prandtl, Taylor, Karman, Stanton, Schmidt, (flux) Richardson, Péclet, Nusselt, Sherwood, and Raleigh (see Pasquill, 1942, Hess, 1979, and Brutsaert, 1982, for an introduction to many of these developments). Besides the fields of aero-, thermo- and hydrodynamics, where most of these numbers emerged, there have been also great strides forward in the kinetic theories of evaporation (see Gerasimov and Yurin, 2018). One can argue that progress would lead to the unification of these numbers into a smaller set. Nevertheless, in the quest for the smaller set, among the candidate numbers, we believe two of Horton's core contributions discussed in this paper could be considered for their fundamental relevance to lake evaporation estimation. (1) The ratio

Among the five equations we evaluated, Meyer's, Rohwer's, and Penman's equation shapes and results differ but only slightly. Expectedly, Konstantinov's (1968) method, which draws additional information from a temperature-deficit term, in addition to VPD and wind (as done in other methods), has the second-highest complexity and performs the second best, while Dalton's method (the simplest one) is the poorest. What we have shown here suggests that Horton's equation can indeed replace these other methods. A question that begs to be answered here is whether Horton's evaporation equation for lakes should be preferred over the Penman (combination) equation, especially in the context of continental-scale land surface modeling? Before answering this question, it is worth noting that Penman's formula is, in part, adapted from Rohwer's (1931) formula, who, in turn, in his work commented on Horton's evaporation formula, saying the following:

From a theoretical standpoint [Horton's] formula is worthy of consideration, but, as the values of the constants in the formula have not been definitely determined, the practical value of the formula is small. (Rohwer, 1931)

Our answer to this question from this study is that it could be for the following reasons.The relationship between pan and actual evaporation is a topic of great
importance today in the wake of accelerated climate warming. There is
unanimous consensus that pan evaporation is reducing globally, while in a
warming climate the opposite is generally expected, which is known as the
evaporation paradox (Roderick and Farquhar, 2002). A friendly introduction to the topic is given in Singh (2016; see Sect. 42.2.3). This paradox is explained by evaporation observations in larger scales across sites of variable moisture availability, considering how energy is redistributed between latent and sensible heat based on moisture availability. This paradox is considered to be resolved by Bouchet's (1963) principle of complementarity, which shows the relationship between pan, actual, and theoretical evaporation. Morton (1994), Szilagyi et al. (2017), Brutsaert and Yeh (1970), and Brutsaert (1982, 2015) further extended the work by Bouchet (1963). In studies that involve pans, including several that are related to the evaporation paradox, pan evaporation calculations are often done with a static pan correction parameter, but as Horton expressed very clearly (see the quote in Sect. 3.2.2), it would be quite wrong to use a simple, static pan correction parameter (Horton, 1917a). The explicit role of the vapor blanket has been ignored in these studies, except perhaps indirectly (as moisture availability is related to atmospheric humidity, which influences vapor blanket characteristics). A table in Maidment (1992; Table 4.3.1; see Sect. 4 on “Evaporation” by Shuttleworth) taken from Doorenbos and Pruitt (1977) provides a quasi-quantitative guidance on pan correction as a function of humidity values and a scale similar to the Beaufort wind force scale (i.e., light, moderate, and strong winds). However, Horton's quantitative treatment and physical explanations for the differences in evaporation rates from the pan to lake precedes Doorenbos and Pruitt (1977) by half a century and is more nuanced and rational (i.e., has a strong physical basis) and is quantitative. Furthermore, Horton's insights on the vapor blanket's physical properties (Sect. 3.4) and the area factor,

Horton's century-long forgotten works on lake evaporation seem to have great contemporary value for the theoretical insights they offer and for their relevance in modeling lakes of all sizes. The fine-scale precision afforded by Horton's law-of-the-wall-type equation (Eq. 5) and Eqs. (6) and (7) for the vapor blanket characteristics, credited to Jožef Štefan and Horton, appear to be essential to estimate the evaporation in small lakes and pans and using pan evaporation as a proxy for large lakes. From these equations, considering the importance of the Horton ratio (

As a closing note, to entertain the

We hope that the various observations and conclusions drawn here to highlight the value of Horton's lake evaporation works will be developed further. We also hope that this serves to rekindle the interest of readers in (re-)discovering Horton's contributions to lake evaporation, in addition to his broader published and unpublished works.

The data sets used in this article are publicly available (see Sects. 1, 3.2.3, 4.1, and 4.5), and the codes are available upon request to the corresponding author. Horton's updated bibliography (with full references and notes) is available upon request to the corresponding author. Titles and years of the same are provided in the Supplement.

The supplement related to this article is available online at:

SV conceptualized the study, performed simulations, and prepared the paper. VPS conducted the literature search, designed the simulations, interpreted Horton's derivations, and edited the paper.

The contact author has declared that neither they nor their co-author have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the special issue “History of hydrology” (HESS/HGSS inter-journal SI). It is not associated with a conference.

The authors would like to acknowledge the following individuals: (1) Anna Bonazzi, from the UCLA Germanic Languages department, for the translation of a German text (Štefan, 1882) cited here, for proofreading, and for sharing some ideas which improved this work; (2) Elizabeth Clark (formerly at UCLA), for inspiring Fig. 2 and for sharing her collection of 135 of Horton's titles; (3) Eric Sheppard, UCLA, for inspiring the work on scientific knowledge production in a graduate class and also for providing numerous comments on a broader version of this work; (4) Nikolai Mikuszeit (see Vimal and Mikuszeit, 2021), who offered his feedback via stack overflow and provided a superior solution for Horton's wind velocity height correction; (5) Keith Beven, Lancaster University, handling Editor, who provided numerous comments to improve the paper; (6) Dennis Lettenmaier, UCLA, for sharing some interesting perspectives and articles by Horton.

This paper was edited by Keith Beven and reviewed by Thomas McMahon and Adriaan J. (Ryan) Teuling.