Evaluation of an Hourly Empirical Method Against ASCE PM (2005), for Hyper-Arid to Subhumid Climatic Conditions of the State of California

Abstract

Accurate estimations of reference evapotranspiration (ETo) are critical for hydrologic studies, efficient crop irrigation, water resources management and sustainable development. The evaluation of an empirical method was carried out to estimate hourly ETo, utilizing short-wave radiation and relative humidity as a surrogate of vapor pressure deficit (VPD), calibrated under semi-arid conditions and validated for different climatic regimes (hyper-arid, arid, subhumid) using American Society of Civil Engineers Penman–Monteith (ASCE PM) (2005) values as a standard, for the state of California. For hyper-arid climatic conditions, the empirical method resulted in underestimation and had coefficient of determination (R²) values of 0.88–0.95 and root mean square error (RMSE) values of 0.062–0.115 mm h⁻¹. Hyper-arid climatic conditions correspond to lower R² and different relations between the vapor pressure deficit (VPD) and the relative humidity function (1/lnRH) that the empirical method utilizes. For the other climatic regimes (arid, semi-arid, subhumid), the empirical method performed satisfactorily. The RMSE was calculated for groups of empirical estimates corresponding to various wind velocity values, and it was satisfactory for >99% of wind speed values (u₂). The RMSE was also calculated for grouped values of the estimates of the empirical method corresponding to observed VPDs and was satisfactory for >97% of all observed values of VPD, except for hyper-arid stations (59% of u₂ and 60% of all observed values of VPD).

1. Introduction

ETo is important for efficient irrigation, conservation of water resources, mitigation of climate change impacts, hydrologic studies, etc. This importance is reflected in the numerous models that have been proposed in the scientific literature.

Combination methods use a theoretical background [1,2,3] and need data, which are not always available, to estimate evapotranspiration rates. The theoretical background of these methods is a key component of their ability to predict evapotranspiration rates with much higher accuracy, compared to other methods that lack such a background. Therefore, they are applicable to many different climates and altitudes. The PM method was evaluated [4] against lysimeter data, using data from hyper-arid, arid, semi-arid, subhumid, humid and high altitude climatic regimes, and it was concluded that they were sufficiently accurate. The ASCE PM (2005) [3], (Walter et al., 2005) method standardizes the calculations of the method so that the results of the calculations, the reference evapotranspiration (ETo) rates, can be universally comparable between researchers, and this method is widely accepted by the scientific community.

Empirical methods are easier to use and are more flexible in the data they require [5,6,7,8,9,10,11,12]. Fewer input variables, on the other hand, combined with the lack of a theoretical background, limit their ability to produce accurate estimations. When they are applied to climatic regimes or locations other than those for which they were calibrated, the error increases. The evaluation of a method for a different climatic regime or location should be carried out before its application. The ASCE PM (2005) method is the standard method against which empirical methods are evaluated and can only yield reliable estimates under reference conditions.

Many authors have evaluated empirical methods against the PM method [13,14,15,16,17,18,19,20,21,22,23] in order to determine the reliability of these methods for purposes like irrigation scheduling, hydrological sustainable development of natural resources, establishing trends for ETo, etc. Empirical methods have also been evaluated against measurements by various researchers [24,25,26,27]. The evaluation of these methods contributes to the establishment of a sustainable relationship between human activities, i.e., agriculture and conservation of natural resources. Ref. [28] investigated the driving forces for the estimation of potential evapotranspiration (PET) in hyper-arid conditions of NW China using the Hargreaves–Samani method [29], as PET’s driving forces are very important in mitigating the impacts of climate change. Ref. [30] evaluated 20 different empirical models for the hyper-arid climatic conditions of the State of Qatar. Empirical models are subject to limitations, and the investigation of their accuracy is important for crop growth, water resources management, etc. It is important that the evaluations against PM methods be carried out in a reference environment; otherwise, systematic cumulative errors occur [31].

Significant agricultural activity in California [32] and the variety of local climatic regimes [33] require an evaluation of the methods that estimate evapotranspiration to assess their validity. Numerous researchers have conducted studies that evaluate empirical methods that estimate evapotranspiration for the state of California [34,35]. The network of agrometeorological stations in the California Irrigation Management System (CIMIS) provides data under a reference environment that is necessary for the evaluation of empirical methods against the ASCE PM (2005) method.

Ref. [36] compared daily vs. hourly time steps of the ASCE PM equation and suggested that the efficiency of precision agriculture can benefit from hourly estimations of reference ETo. There are a few empirical methods that estimate ETo on an hourly basis. Ref. [37] (Chatzithomas and Alexandris, 2015) proposed an empirical method that estimates hourly rates of reference ETo under semi-arid climatic conditions, calibrated with CIMIS–Davis data and validated with CIMIS and Agricultural University of Athens (AUA) agrometeorological station data. The empirical method does not require wind speed or temperature data; therefore, it is suitable for application when wind speed and temperature data are missing or are questionable. The empirical method can be utilized for estimating the crop water needs on a finer time scale (hourly time step) to improve irrigation efficiency. This can be achieved using fewer measurements, namely Rs and RH, resulting in reduced costs for instruments. The empirical method uses a function of relative humidity as an aerodynamic-related term, which is increasingly important when conditions become drier. It relies on short-wave radiation and relative humidity measurements to estimate hourly rates and avoid overestimation (5–6% on average) of ETo rates, due to the inclusion of net radiation as an input when daily sums are under consideration [38].

In this paper, we evaluate the empirical method for hyper-arid, arid, semi-arid and subhumid locations in the state of California and the contribution of the aerodynamic-related term to the accuracy of the method in drier climatic conditions than the calibration conditions.

2. Materials and Methods

2.1. Methods

As mentioned earlier, ETo measurements are a difficult and technically challenging way to quantify the amount of water vapor losses from a cultivated surface. They require skilled personnel and specialized instruments that are both costly and difficult to maintain. Therefore, the use of indirect ways to estimate the ETo rate is imperative for agriculture, water resources management, etc. For the purposes of this study, two methods have been selected.

2.1.1. ASCE PM (2005) Method

As a reference method, the ASCE-PM (2005) [3] (Walter et al., 2005) method for the hourly rate of ETo for a short crop (12 cm) is used. The following equation is used:

$$\mathrm{E}{\mathrm{T}}_{\mathrm{o}}={\displaystyle \frac{\Delta \left({\mathrm{R}}_{\mathrm{n}}-\mathrm{G}\right)}{\mathsf{\lambda }\left(\Delta +\mathsf{\gamma }\left(1+{\mathrm{C}}_{\mathrm{d}}{\mathrm{u}}_2\right)\right)}}+{\displaystyle \frac{\mathsf{\gamma }\left(\frac{37}{273.16+{\mathrm{T}}_{\mathrm{a}}}\right){\mathrm{u}}_2\left({\mathrm{e}}_{\mathrm{s}}-{\mathrm{e}}_{\mathrm{a}}\right)}{\left(\Delta +\mathsf{\gamma }\left(1+{\mathrm{C}}_{\mathrm{d}}{\mathrm{u}}_2\right)\right)}}$$

(1)

where ΕΤ_ο is the reference evapotranspiration in mm h⁻¹; R_n, the net radiation flux density in Mj m⁻² h⁻¹; G, the soil heat flux density in Mj m⁻² h⁻¹; u₂, the average hourly wind speed at 2 m in m s⁻¹; T, the average hourly temperature at 2 m in °C; C_d, 0.24 for daytime and 0.96 for nighttime; ${\mathrm{e}}_{\mathrm{s}}$, the saturation vapor pressure of the atmosphere at T_a in kPa; e_a, the actual vapor pressure of the atmosphere in kPa; $\Delta$, the slope of the saturation vapor pressure curve in kPa °C⁻¹; $\mathsf{\gamma }$, the psychrometric constant in kPa °C⁻¹; and λ, the latent heat of vaporization in MJ kg⁻¹.

2.1.2. Empirical Method

The hourly rate of ETo was also estimated using the method of Ref. [37] (Chatzithomas and Alexandris, 2015), hereafter referred to as the empirical method. The measured independent variables of the empirical equation are Rs (short-wave radiation) and RH (%). Theoretical daylight hours (N) are also required and are calculated from the station’s location (latitude) and the day of the year (Julian day, 1–365). The method accounts for daylight hours (N) as a fraction of the 24 h period (fN); for example, a daylight duration (N) of 12 h would give an fN value of 12/24 = 0.5.

The equation is

$$E{T}_o=\left\{\begin{array}{c}0.200382744+0.000411692R_s-0.002353982RH+\\ 0.0002321\left(R_s^{1+fN-f{N}_{min}}\right)\left({\displaystyle \frac{1}{\ln \left(RH\right)}}\right), R_s\ge 0\\ 0, R_s<0\end{array}\right.$$

(2)

where ETo the hourly rate of ETo in mm h⁻¹, Rs the incoming short-wave (solar) radiation in W m⁻², RH the relative humidity %, fN is a function of the theoretical daylight hours as a fraction of the total (24 h) and fNmin is the minimum theoretical daylight hours of the year as a fraction of 24 h.

2.2. Statistical and Climatic Indices

2.2.1. Coefficient of Determination

$$R^2={\displaystyle \frac{\sum \left(y-y_{av}\right)\left(\widehat{y}-{\widehat{y}}_{av}\right)}{\sqrt{\sum {\left(y-y_{av}\right)}^2{\left(\widehat{y}-{\widehat{y}}_{av}\right)}^2}}}$$

(3)

where $y$ is the ASCE PM value, $y_{av}$ is its average, $\widehat{y}$ is the estimation of the empirical method, and ${\widehat{y}}_{av}$ is its average.

2.2.2. Slope

The slope of the regression line has the following equation:

$$\mathrm{s}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}={\displaystyle \frac{\sum \left(\mathrm{y}-{\mathrm{y}}_{\mathrm{a}\mathrm{v}}\right)\left(\widehat{\mathrm{y}}-{\widehat{\mathrm{y}}}_{\mathrm{a}\mathrm{v}}\right)}{\sum {\left(\mathrm{y}-{\mathrm{y}}_{\mathrm{a}\mathrm{v}}\right)}^2}}$$

(4)

The same symbols as those in Equation (3) are used.

2.2.3. Intercept

The intersection of the regression line and Y axis has the following equation:

$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{p}\mathrm{t}={\mathrm{y}}_{\mathrm{a}\mathrm{v}}-\mathrm{s}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\times {\mathrm{x}}_{\mathrm{a}\mathrm{v}}$$

(5)

The same symbols as above (3) are used.

2.2.4. Root Mean Square Error (RMSE)

Root mean square error is a weighted measure of the error with the same units as the dependent variable:

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{1}{\mathrm{n}}}\sum {\left(\mathrm{y}-\widehat{\mathrm{y}}\right)}^2}$$

(6)

The same symbols as above are used.

Ref. [39] proposed a value of 0.073 mm h⁻¹ as the upper limit for an acceptable estimation for the RMSE. This limit is used in this study as an optimum value for the performance of the empirical method. RMSE values up to ~90 W m⁻², or 0.13 mm h⁻¹ for hourly values, were considered reasonable by other researchers [40]. This value is considered as the upper limit for the RMSE, and for values between 0.073 mm h⁻¹ and 0.13 mm h⁻¹, further research is suggested.

2.2.5. Index of Agreement (IoA)

According to this index, perfect agreement between the empirical method and ASCE (2005) corresponds to 1, while total disagreement corresponds to a value of zero. The IoA was introduced by [41].

$$\mathrm{I}\mathrm{o}\mathrm{A}=1-{\displaystyle \frac{{\sum \left(\widehat{\mathrm{y}}-\mathrm{y}\right)}^2}{\sum {\left(\left|\widehat{\mathrm{y}}-{\mathrm{y}}_{\mathrm{a}\mathrm{v}}\right|+\left|\mathrm{y}-{\mathrm{y}}_{\mathrm{a}\mathrm{v}}\right|\right)}^2}}$$

(7)

where the symbols have the same meaning as above (3).

2.2.6. The Aridity Index (AI)

The aridity index (AI) is a climatic index used to assess the dryness of a geographical location [42]. The AI was calculated as the ratio of yearly values of precipitation (nominator, mm) and ETo (denominator, mm), according to the following equation:

$$\mathrm{A}\mathrm{I}={\displaystyle \frac{\mathrm{P}}{\mathrm{E}{\mathrm{T}}_{\mathrm{o}}}}$$

(8)

Based on the value of the aridity index, dryness is classified as hyper-arid (AI ≤ 0.05), arid (0.05 < AI ≤ 0.20), semi-arid (0.20 < AI ≤ 0.50) and subhumid (0.50 < AI ≤ 0.75).

2.3. Data

2.3.1. Data Source: CIMIS Network

CIMIS (California Irrigation Management Information System) is operated by the Water Use and Efficiency Branch of the Division of Statewide Integrated Water Management, California Department of Water Resources (DWR). The network currently includes 145 active stations, with data being collected every minute and stored and processed at the CIMIS headquarters. Hourly data (e.g., 1300 h) represent measurements from the preceding 60 min (1200–1300 h). Data transmission from the stations to the server, located in Sacramento, is performed hourly; the server polls each station to collect the data. If a station does not respond, the system proceeds to the next station, and once all stations have been polled, the server attempts to reconnect with any unresponsive stations. All measurements are obtained from stations maintained under standardized conditions, with grass 8–15 cm in height.

2.3.2. Quality Control

Quality Control (QC) is performed for the collected data, which are then flagged accordingly. Programmed calculations are stored along with the flagged data. QC checks include outliers, data with values 2 or 3 standard deviations from the long-term station mean, missing data, sensor malfunctions, data with no meaning (negative solar radiation values), etc. Data are freely accessible to the public, provided that users are registered. Values of RH equal to or less than 4% were excluded (four records of Indio 2 station).

2.3.3. Meteorological Data

All available monthly data from 139 stations were downloaded from the database and summed to yearly values. A total of 26,364 records with monthly data were used. For each of the 139 stations of the CIMIS network, yearly averages were calculated for reference evapotranspiration and for all the meteorological parameters. California is characterized by a wide variety of microclimates [33]. The calculated averages from the 139 stations of the CIMIS network were used for the selection of stations representative of the observed variety of microclimates in the state of California. Based on these calculated averages, ten (10) CIMIS meteorological stations were selected (see

Table 1. Meteorological stations (CIMIS) studied for this paper. In total, 83,807 records (hourly) from 10 meteorological stations were analyzed. The selection was based on the criteria described in the right-hand column of the table. The AI (yearly precipitation divided by reference evapotranspiration) is given for all the stations. Stations 221 (Cadiz Valley), 68 (Seeley) and 136 (Oasis) are under hyper-arid climatic regimes.

Stn Id	Name	Long.	Lat	Elev (m)	Year	# of Records	AI (P/ETo)	Remarks
200	Indio 2	33.75	−116.25	12	2008	8235	0.00	hyper-arid
190	Five Points	36.38	−120.23	82	2005	8759	0.10	arid
80	Fresno	36.82	−119.74	103	2000	8783	0.20	arid
32	Colusa	39.23	−122.02	16	2000	8254	0.34	semi-arid
12	Durham	39.61	−121.82	130	2001	8664	0.45	semi-arid
83	Santa Rosa	38.40	−122.80	24	2011	8471	0.74	subhumid
221	Cadiz Valley	34.51	−115.51	47	2011	8741	0.03	high VPD (hyper-arid)
140	Twitchell Island	38.12	−121.66	−1	2000	8318	0.23	high wind speed (semi-arid)
68	Seeley	32.76	−115.73	12	2011	7679	0.04	high ETo (hyper-arid)
136	Oasis	33.52	−116.16	4	2005	7903	0.03	max temp (hyper-arid)

). Years with complete records were preferred as much as possible. From the ten (10) CIMIS meteorological stations, six (6) were selected according to the value of the aridity index (AI), with values of the AI ranging from 0.00 to 0.74. The AI values of the stations chosen cover the range of observed AI values in a representative way. The AI for the Davis station of the CIMIS network was calculated to be equal to 0.33 for the whole period of its operation (1982–April 2015).

Four (4) CIMIS meteorological stations were selected according to maximum average VPD value (vapor pressure deficit of the atmosphere, 2.3 kPa average value for Cadiz station, Stn Id 221), maximum average u₂ value (wind speed at 2 m above ground surface, 3.4 m s⁻¹ average wind speed value for Twitchell Island station, Stn Id 140), maximum average ETo value (1838 mm year⁻¹ average value for Seeley station, Stn Id 68) and maximum average air temperature value (23.1 °C average air temperature value for Oasis station, Stn Id 136). In total, 83,807 hourly records were downloaded for all ten stations.

3. Results and Discussion

In total, 83,807 hourly records from 10 stations of the CIMIS network were used to evaluate the estimations of the empirical method for hyper-arid, arid, semi-arid and subhumid climatic conditions across the state of California.

The hourly rate of ETo was calculated (83,807 values) with the empirical method. The CIMIS values of the hourly rate of ETo according to the ASCE PM (2005) method were used. Yearly totals of hourly ETo estimates were calculated for both the empirical and the ASCE PM (2005) method. Additionally, the absolute difference between the two methods (empirical total − ASCE PM total), the relative difference expressed as a percentage of the ASCE PM (2005) yearly total, and four statistical indicators (R², RMSE, slope and IoA) were computed (see

Table 2. Annual totals for the empirical method and ASCE PM (2005) were calculated, along with their difference, their difference as a percentage (%) of the ASCE PM (2005) annual total, and the statistical indices of the empirical method against the ASCE PM (2005) for each of the 10 stations of the CIMIS network (83,807 hourly records). The statistics in the last row were calculated using all (83,807) data.

Station	Emp (mm yr−1)	PM (mm yr−1)	Emp-PM	(Emp-PM)%	R2	RMSE	Slope	ΙοA
Indio 2	1825.8	2019.4	−193.7	−9.6%	0.94	0.095	0.71	0.957
Five Points	1502.2	1479.6	22.6	1.5%	0.96	0.046	0.90	0.989
Fresno	1389.4	1422.0	−32.6	−2.3%	0.97	0.038	0.99	0.993
Colusa	1189.2	1233.7	−44.5	−3.6%	0.97	0.037	0.99	0.993
Durham	1477.6	1350.9	126.8	9.4%	0.97	0.040	0.98	0.991
Santa Rosa	999.3	980.1	19.3	2.0%	0.98	0.033	1.11	0.992
Cadiz Valley	2108.1	2141.9	−33.8	−1.6%	0.88	0.115	0.60	0.919
Twitchell Island	1368.1	1434.1	−66.0	−4.6%	0.97	0.041	0.92	0.991
Seeley	1877.6	1916.4	−38.9	−2.0%	0.93	0.088	0.75	0.968
Oasis	1749.3	1755.7	−6.4	−0.4%	0.95	0.062	0.85	0.983
Average	1548.7	1573.4
Total (83,807 records)	15,486.6	15,733.7	−247.1	−1.6%	0.95	0.060	0.88	0.978

).

The values (83,807) calculated with the empirical method and those calculated with the ASCE PM (2005) method were summed, as shown in the last row of Table 2. The total sum across all 10 stations was 15,486.6 mm for the empirical method and 15,733.7 mm for the ASCE PM (2005) method. The difference was found equal to −247.1 mm or −1.6% of the sum of the estimations of the ASCE PM (2005) method for all 10 stations. In Figure 1, we can see the cumulative plot of the hourly values of both methods. The statistical indices for the 83,807 hourly estimations of both methods were also calculated. The coefficient of determination (R²) was found equal to 0.95, the RMSE was found equal to 0.060 mm h⁻¹, and the slope of the regression line between the values of the hourly ETo estimations of the empirical method and the values of the hourly ETo estimations of the ASCE PM (2005) method was found equal to 0.88. The IoA was 0.978, indicating strong model performance. The RMSE is also satisfactory. The lines of the cumulative values of the empirical method and of the ASCE PM (2005) show satisfactory agreement for arid, semi-arid and subhumid conditions. In the case of hyper-arid climatic conditions, however, the cumulative values diverge (−247.1 or −1.6%). Cumulative values under non-extreme climatic regimes suggest strong agreement, while in extreme conditions (i.e., hyper-arid), these values present differences that can affect long-term planning for irrigation schedules. While the two methods show satisfactory agreement in non-hyper-arid environments, hyper-arid conditions require further detailed analysis to assess and mitigate these deviations.

3.1. Stations with Hyper-Arid Climatic Regimes

Four CIMIS stations (all located in the southern and most arid part of California and characterized by similar climatic conditions), out of the ten used for this study, were characterized by hyper-arid conditions. Indio 2 had AI = 0.00, Cadiz Valley had AI = 0.03 and the highest yearly average VPD value, Seeley had AI = 0.04 and the highest annual ETo value, and Oasis had AI = 0.03 and the highest annual average temperature. We examined the four stations as a group because these four represent the most arid part of California.

The sums of the values estimated with both the empirical method and the ASCE PM (2005) method were in good agreement for all four stations. Indio 2 had the highest underestimation, 9.6%, and the minimum underestimation was observed for Oasis station with 0.4%. The minimum R² value, 0.88, was observed for Cadiz Valley station, while Oasis station had the best R² value, 0.95 (see Figure 2). Oasis station had the minimum RMSE value (0.062 mm/h) between the two methods. The maximum value for RMSE for the hyper-arid stations was observed at Cadiz Valley station and was equal to 0.115 mm/h. The slopes of the regression lines between the two methods (see Figure 2) were all below 1, showing a consistent underestimation, with Cadiz Valley station having the minimum value equal to 0.60. The maximum value of the slope of the regression line, equal to 0.85, is for Oasis station, again indicating underestimation on the part of the empirical method. The values of the IoA ranged from 0.919 for Cadiz Valley station and 0.983 for Oasis station.

While the value of the IoA and the yearly totals of the values of the two methods for the four hyper-arid stations of the CIMIS network indicate a good agreement, the values of the slopes of the regression lines are small, and the RMSE values, although smaller than 0.13 mm h⁻¹, are the highest, in comparison with all the other stations, and, with the exception of Oasis, are not satisfactory. Radiation methods perform well in humid climates [43,44,45] and tend to produce underestimations in arid climates [45]. The main reason for this is that the higher temperatures and lower moisture content of the atmosphere increase the significance of the aerodynamic term of the PM method in hyper-arid climates compared to more humid climates. Vapor pressure deficit (VPD) and wind speed (u₂) are the two important factors of the aerodynamic term of the PM method that determine the performance of radiation methods in hyper-arid climates.

Wind speed values are the measure of the capacity of the atmosphere to remove water vapor above the evaporating/transpiring surface. They influence the estimations of the ETo method, when present, and tend to create deviations from the standard ASCE PM (2005) method when measured values are different from those measured in the calibration of the empirical method. The deviation tends to become more important as the dryness increases because air masses tend to be less saturated and, therefore, have higher VPDs.

Since the empirical method does not take into account the values of u₂, its coefficients reflect the observed values of u₂ during calibration (Davis, 2000) (average u₂ value of 2.52 m s⁻¹). Higher observed u₂ values in the four hyper-arid stations would contribute to an underestimation of the hourly ETo values. In the same way, lower observed u₂ values in the four hyper-arid stations would contribute to an overestimation of the hourly ETo values. The average hourly values of wind speed for the three hyper-arid stations (Indio 2, Seeley, Oasis) are below the average hourly value of wind speed for Davis, year 2000 (2.52 m s⁻¹), and one station (Cadiz) has a higher yearly average wind speed (2.73 m s⁻¹). Cadiz station’s yearly sum of the hourly estimations of the empirical method was very close (−1.6%) to that of the ASCE PM (2005) yearly sum. The biggest underestimation was for Indio 2 station, which had an average hourly wind speed value lower (2.33 m s⁻¹) than that for Davis station (2.52 m s⁻¹). Ref. [46] found that ETo is least sensitive to u₂. Ref. [47] argued that combination methods are less sensitive to u₂. The findings of this research agree with [46,47].

3.2. Investigation of the VPD − 1/ln(RH) Relation

We calculated the values of slope, intercept and R² for all the hourly records of the four stations (8235 for Indio 2, 8741 for Cadiz, 7679 for Seeley and 7903 for Oasis), for the hourly records of the warm part of the year (80 < DOY < 267) for each station, for all the hourly values of the warm part of the year of the four stations (80 < DOY < 267, 15,011 records), for the hourly records of the cold part of the year (remaining part of the year) for each of the four stations and for all the values of the cold part of the year of the four stations (17,547 records, see

Table 3. Values of slope, intercept, R² and RMSE between the VPD and 1/ln(RH) values for the four hyper-arid stations (Indio 2, Cadiz, Seeley, Oasis, first four rows of the table) and for all of the hourly values of the four stations (last row). The total of the hourly values (column 1) was divided into the warm period (column 2) and cold period (column 3) for all four stations and for all the hourly values of the four stations. The calculations were repeated for values grouped on a daily basis (Column 4).

	All Hourly Values (1)			Warm Period (2)			Cold Period (3)			Daily Values (4)
	Slope	Intercept	R2	Slope	Intercept	R2	Slope	Intercept	R2	Slope	Intercept	R2
Indio 2	0.0239	0.2343	0.61	0.0235	0.2284	0.61	0.0327	0.2304	0.66	0.0365	0.2192	0.93
Cadiz	0.0256	0.2671	0.51	0.0249	0.2654	0.46	0.0359	0.2569	0.49	0.0471	0.2388	0.88
Seeley	0.0175	0.2425	0.45	0.0181	0.2337	0.56	0.0266	0.2390	0.46	0.0290	0.2296	0.88
Oasis	0.0248	0.2263	0.67	0.0236	0.2263	0.61	0.0338	0.2205	0.80	0.0325	0.2162	0.96
All stations	0.0249	0.2395	0.52	0.0253	0.2314	0.54	0.0339	0.2357	0.54	0.0366	0.2262	0.91

).

The calculated R² values ranged from 0.45 (Seeley station, all hourly values) to 0.80 (Oasis station, cold period). The R² value for all the hourly values (32,558) was found to be equal to 0.52. For both the warm (80 < DOY < 267) and the cold period (remaining part of the year), R² was calculated to be equal to 0.54. Slope values ranged from 0.0175 (Seeley station, all hourly values) to 0.0359 (Cadiz station, cold period). Smaller slope values were calculated for the warm part of the year, and higher values of the slope were calculated for the remaining (cold) part of the year. During the cold part of the year, the range of VPD is smaller, while the range of RH is not affected as much. Intercept values ranged from 0.2205 (Oasis station, cold period) to 0.2671 (Cadiz station, all hourly values).

For saturated atmospheric conditions (VPD = 0), the 1/ln(RH) term takes its minimum value (0.2174). As VPD values increase, so does the value of the 1/ln(RH) term, giving higher values for the intercept of the regression line between VPD and 1/ln(RH). The maximum intercept value is for Cadiz station (intercept = 0.2671), which also has the highest average yearly VPD value compared to the other 3 stations (and also to all of the 10 CIMIS stations used for this study). The minimum intercept value is for Oasis station (intercept = 0.2205), which also has the lowest average yearly VPD value of the four (drier) stations (see Table 3).

Values of R² were much higher when grouped on a daily basis. On a daily basis, values of actual vapor pressure are more or less the same [48], so the change in the VPD is due to the change in air temperature, which determines the saturation vapor pressure of the atmosphere. RH then follows the change in air temperature [45].

We investigated the assumption that the actual vapor pressure of the atmosphere during the day remains more or less constant, as [48] suggested. We examined the calculated R² values for all four arid stations (Indio 2, Cadiz, Seeley, Oasis) and found that Indio 2 station had 63 days with R² less than 0.90, Cadiz station had 146 days with R² less than 0.90, Seeley station had 118 days with R² less than 0.90 and Oasis station had 31 days with R² less than 0.90. The same calculation for Davis station gave two days with R² less than 0.90. During those two days, we had abrupt changes in weather patterns that gave big shifts in the values of the actual vapor pressure.

We used 24-hourly records of DOY 97, year 2008, from Indio 2 (see Figure 3) to calculate the R² between the values of VPD and the values of the inverse of the natural logarithm of RH. When we used the average of the 24 observed actual vapor pressure values for that day (DOY 97) for the Indio 2 station, equal to 0.91 kPa, to calculate the vapor pressure deficit and the RH; the R² value was found equal to 1. We repeated the calculation using the observed 24-hourly values of the actual vapor pressure, and we found R² equal to 0.18.

During that day, a continuous supply of moisture to the atmosphere was recorded, as shown by the rising curve of actual vapor pressure in Figure 2, until approximately 0800 to 0900 in the morning. The value of the actual vapor pressure was equal to about 1.05 kPa for the rest of the day. After 1800 h, when actual vapor pressure reached a daily minimum, it continued to increase until the end of the day (2400 h). Increases in air humidity during the night are not normally associated with evapotranspiration. Evapotranspiration during the night has been reported by [49] and was associated with wind speeds greater than 6 m s⁻¹ at 3 m above ground, as the wind is the only turbulence source during the night. During DOY 97, year 2008, for Indio 2 station, the mean wind speed was calculated from the hourly values of wind speed and found equal to 4.6 m s⁻¹ at 2 m above ground, which equals 4.99 m s⁻¹ at 3 m above ground. The conversion of the wind speed from 2 m to 3 m was performed in accordance with the guidelines given by [45]. Therefore, the diurnal pattern of air temperature only partly accounts for the variation in the vapor pressure deficit of the atmosphere in such cases. As the diurnal variation in the actual vapor pressure becomes larger, it reduces the R² value between the vapor pressure deficit of the atmosphere and the inverse of the natural logarithm of the RH of the atmosphere. As a result, the model estimates a smaller portion of the variation in the dependent variable, which results in reduced accuracy of the estimations of the empirical method.

The empirical method uses 1/ln(RH) to introduce the aerodynamic contribution to ETo. As climatic conditions change from semi-arid to hyper-arid, the empirical method adapts satisfactorily. The term contributes to this behavior despite the different climatic regime under which it was calibrated. Hyper-arid climatic conditions are the driest, so we investigated arid, semi-arid and, for completeness of the research, subhumid conditions.

3.3. Stations with Arid, Semi-Arid and Subhumid Climatic Regimes

From the remaining six out of the ten CIMIS stations selected, two are classified as arid (Five Points station, 8759 records, AI = 0.10; Fresno station, 8783 records, AI = 0.20), three as semi-arid (Colusa station, AI = 0.34, 8254 records; Durham station, AI = 0.45, 8664 records; Twitchell Island station, AI = 0.23, 8318 records, also the station with the highest wind speed value) and one as subhumid (Santa Rosa station, AI = 0.74, 8471 records).

We calculated the values of the hourly ETo estimations of the empirical method for all 51,249 hourly records of the six CIMIS stations, and we compared them with the values of the hourly ETo estimations of the ASCE PM (2005). The sum of the values of the empirical method for all six stations was found equal to 7925.9 mm, and it overestimated the ASCE PM (2005) totals, equal to 7900.3, by 25.6 mm or 0.32%. The average difference of the total of the empirical method for each station from the total sum of the values of the hourly ETo estimations of the ASCE PM (2005) method was equal to 4.1 mm per year. The overestimation of the empirical method was small and acceptable for the sum of the values for all six stations (51,249 hourly records).

We also compared the sum of the values for each station. The empirical method produced overestimations in three stations (Five Points, Durham, Santa Rosa), with overestimations ranging from 22.6 mm to 126.8 mm, and underestimations in the remaining stations (Fresno, Colusa, Twitchell Island), with underestimations ranging from 33.6 mm to 66.0 mm. Both underestimations and overestimations were observed at arid and semi-arid stations. The over- or underestimations between the two methods could not be attributed to the climatic regime of the stations when the yearly sums of the hourly ETo estimations of the two methods were compared. The differences were small and were considered satisfactory.

We calculated R² between the values of the two methods for each station (see Table 2, Figure 4). The R² values ranged from 0.96 (Five Points station, arid) to 0.98 (Santa Rosa station, subhumid) and were considered satisfactory; the RMSE values showed a minimum of 0.033 mm h⁻¹ (Santa Rosa station, subhumid) and a maximum of 0.046 mm h⁻¹ (Five Points, arid). All RMSE values were below the threshold of 0.073 mm h⁻¹ set by [39]. The RMSE values calculated were small and satisfactory for the empirical method. The slope of the regression lines between the two methods for the remaining stations was calculated (see Figure 4, Table 2).

The minimum slope value of the regression line was equal to 0.90 (Five Points station, arid), and the maximum value was equal to 1.11 (Santa Rosa station, subhumid). These values, however, were not reflected in the annual totals for the two methods, which were almost identical. The scatter plots of the estimations of the two methods confirm the agreement between the two methods. The IoA was also calculated for the two methods. The maximum value of the IoA was equal to 0.993 (Fresno station, arid, and Colusa station, semi-arid), and the minimum value of the IoA was equal to 0.989 (Five Points station, arid). The IoA values calculated for the estimations of the two methods were satisfactory.

We plotted (see Figure 5) the hourly values of the two methods for DOYs 170–175 (19 June until 24 June) for four stations with arid (Five Points, Fresno) and hyper-arid (Indio 2, Oasis) climatic conditions. The plotted values showed a good agreement for the arid stations and an underestimation for the hyper-arid Indio 2 station; Oasis station (hyper-arid) is more satisfactory.

3.4. Evaluation of the Relation of the Values of u₂ and VPD, and the RMSE Values Between the Two Methods, for All the Climatic Regimes

The accuracy of the estimations was investigated in relation to the aerodynamic-term-related variables (wind speed, VPD) because of their significance for dry climatic conditions. The range of the observed values (83,807 hourly values for each parameter) was equal to 0–14.8 m s⁻¹ for the wind speed and 0–9.23 kPa for the VPD. Wind speed and VPD values were grouped into intervals of 1 m s⁻¹ for u₂ and of 1 kPa for the VPD. RMSE was calculated for the respective values of the two methods, which correspond to each group of u₂ and VPD. Intervals different from 1 m s⁻¹ or 1 kPa were introduced when the RMSE approached the limit of 0.073 mm h⁻¹ as suggested by [39] or 0.13 mm h⁻¹ set by [40]. We repeated this procedure for the hourly values of the hyper-arid stations (Indio 2, Cadiz, Oasis, Seeley, 32,558 records), for the hourly values of the arid stations (Five Points, Fresno, 17,542 hourly records), for the hourly values of the semi-arid stations (Colusa, Durham, Twitchell Island, Santa Yanez, 34,010 hourly records) and for the hourly values of the subhumid station (Santa Rosa, 8471 hourly records). We then calculated the percentage of the estimations that yield RMSE values lower than 0.073 mm h⁻¹ and lower than 0.13 mm h⁻¹ (see

Table 4. Wind speed and VPD values with RMSEs lower than the limits set in this study (0.073 mm/h, 0.113 mm/h) for hyper-arid, arid, semi-arid and subhumid climatic conditions, with the percentages of the total number of values. The first column gives the percentage of all the records (83,807) used for this study.

		u2
RMSE (mm h−1)	All hourly records	Hyper-arid	arid	semi-arid	subhumid
0.073	(5 m s−1) 94%	(2.2 m s−1) 60%	(6 m s−1) 99%	100%	100%
0.13	(8 m s−1) 99%	(5.3 m s−1) 94%	100%	100%	100%
		VPD
RMSE (mm h−1)	All hourly records	Hyper-arid	arid	semi-arid	subhumid
0.073	(2.5 kPa) 87%	(2.1 kPa) 59%	4.5 (kPa) 98%	(3 kPa) 97%	100%
0.13	(7.5 kPa) 100%	(4.1 kPa) 86%	100%	100%	100%

) for each variable and for each climatic regime.

The RMSE of the estimations between the two methods was below 0.073 mm h⁻¹ [39] for 94% of the hourly ETo estimations, and 99% were below 0.13 mm h⁻¹ [40], when all hourly records (83,807 records) were considered. For arid, semi-arid and subhumid climatic regimes, the RMSE was below the 0.073 mm h⁻¹ threshold in 97–100% of the cases for both wind speed and VPD. In all these climatic regimes, the empirical method exhibited satisfactory RMSE values.

Approximately 60% of the empirical method’s estimations had RMSE values smaller than 0.073 mm h⁻¹, and 86–94% of the estimations had RMSE smaller than 0.13 mm h⁻¹ for hyper-arid stations (32,558 hourly records). The performance of the empirical method for the hyper-arid stations was not satisfactory. Further research on the uncertainty of the aerodynamic-related term for hyper-arid climatic conditions is needed.

4. Conclusions

The empirical method [37] (Chatzithomas and Alexandris, 2015) was evaluated for hyper-arid, arid, semi-arid and subhumid climatic regimes against the ASCE PM (2005) method using four statistical indices (RMSE, R², slope, IoA). The CIMIS database provided all data (83,807 records) used for this study. Ten (10) stations were selected based on their climatic regime and the values of meteorological parameters related to the aerodynamic term of the PM equation. The RMSE value of 0.073 mm h⁻¹ was set as a threshold for evaluating the method.

The statistical indices for all the 83,807 hourly values of the two methods were satisfactory. The empirical method overestimated the sum compared to ASCE PM (2005) by 3.2%. The RMSE value was satisfactory and below the limit of 0.073 mm h⁻¹.

In the four most arid stations (min AI value, maximum VPD, maximum ETo, maximum temperature), the empirical method showed the highest RMSE values (0.062 mm h⁻¹ to 0.115 mm h⁻¹), the smallest slope values and consistent underestimation for all four stations. The relation of VPD and 1/ln(RH), which the empirical method takes into account, was investigated. The VPD − 1/ln(RH) relation, utilized by the empirical method, reflects the semi-arid climatic conditions under which it was calibrated. Very arid climatic conditions correspond to different VPD − 1/ln(RH) relations. Varying moisture content of the atmosphere during the day reduces the coefficient of determination between VPD and 1/ln(RH), therefore reducing the accuracy of the empirical method. Low values of the coefficient of determination between the two variables were observed in hyper-arid conditions. Further research is recommended for hyper-arid climatic conditions. The performance for semi-arid climatic conditions was satisfactory as expected. The RMSE values were 0.037 mm h⁻¹ and 0.041 mm h⁻¹. The uncertainty related to arid climatic conditions was satisfactory (RMSE 0.038–0.046). Both RMSE values were below 0.073 mm h⁻¹.

The aerodynamic term of the method contributes to the adaptation to climatic conditions much drier than originally calibrated for and is adequate for ETo assessments within the limits set for the examined data. For extremely dry conditions, a recalibration of the equation would be recommended. All stations used for the evaluation were in agricultural areas where hourly estimates of ETo could be useful for precision agriculture. The use of only two measurements for the estimations is cost-effective, and this method can also be used for quality checks on data with questionable temperature or wind speed data. Further research is recommended.