Modelling daily net radiation of open water surfaces using land-based meteorological data

Accurate quantification of net irradiance of open water ( R n water ) is of paramount importance for the estimation of open water evaporation, which is critical for the efficient management of water resources. Alternatively, model estimates of R n water are often used when quality measurements of R n water are not readily available for the water storage of interest. A Daily Penman, Monteith, Equilibrium Temperature Hargreaves-Samani (DPMETHS) model has been developed for the estimation of R n water using land-based meteorological data. The DPMETHS model is a spreadsheet-based iterative procedure that computes R n water using daily land-based meteorological measurements of solar irradiance ( R s land ), minimum and maximum air temperatures ( T min and T max ), minimum and maximum relative humidity ( RH min and RH max ) and average wind speed ( U land ). In this study, the DPMETHS model was evaluated using daily R n water in-situ measurements acquired from 5 sites in both hemispheres, representing very different climatic conditions. Results showed reasonable model performance at all 5 sites, with the coefficient of determination ( r 2 ) values greater than 0.85 and root mean square error (RMSE) values ranging from 0.60 MJ∙m -2 for Stratus Ocean (East Pacific Ocean) to 1.89 MJ∙m -2 for Midmar Dam (South Africa). The results of this study suggested that the DPMETHS model can be reliably used to estimate R n water for a wide range of climatic conditions. The performance of the DPMETHS model depends on the representativeness of the land-based meteorological data to the weather conditions above the open water surface. The DPMETHS model is user-friendly with minimal computational and data requirements that allows easy data handling and visual inspection.


INTRODUCTION
The high temporal and spatial variability of rainfall in semi-arid regions such as South Africa results in water resources being not uniformly distributed throughout the region (Mukheibir and Sparks, 2003). To ensure water security at various times of the year, water is stored in reservoirs (McJannet et al., 2013;Spears et al., 2016). However, significant amounts of water may be lost from open water storages to the atmosphere as water vapour, and this phenomenon is referred to as open water evaporation (Schulze, 2011;McJannet et al., 2008). Within this context, accurate quantification of open water evaporation is of paramount importance for efficient management of water resources, as water scarcity posed by climate change advances in the semi-arid of South Africa (Everson, 1999;Savage et al., 2004;Mengistu and Savage, 2010;Schulze, 2011;Savage et al., 2017).
Energy balance models are the most accurate methods for estimating open water evaporation, after the direct measurements, and are often used as a reference method against which other methods are compared (Finch, 2001). The energy balance techniques for estimating open water evaporation require either measurements or estimates of net irradiance of open water (R n water ) (McJannet et al., 2008;Zheng, 2014). Measurements of R n water are monitored by net radiometers mounted above water storage. Net radiometers are expensive, requiring regular calibration and maintenance to attain accurate measurements (Dong et al., 1992;Kjaersgaard et al., 2007;Savage and Heilman, 2009;Carmona et al., 2017;Myeni et al., 2020). Consequently, R n water measurements are often not readily available for the water storage of interest, especially in developing countries (McJannet et al., 2013;Zheng, 2014). Alternatively, the lack of R n water data above water bodies could be solved by using models that estimate R n water from land-based meteorological data (McJannet et al., 2013;McMahon et al., 2013). The models used to estimate R n water from land-based meteorological data vary in their level of accuracy, complexity and data input requirements (Wang and Liang, 2009). McJannet et al. (2008) stressed that R n water should be determined from models that are universally applicable and relatively easy to utilise with minimal data input requirements, to improve the estimation of open water evaporation.
The modified Penman-Monteith model of McJannet et al. (2008) utilizes basic land-based meteorological data to estimate R n water required for the computation of open water evaporation. The modified Penman-Monteith model was implemented in Microsoft Excel by Savage et al. (2017) to incorporate the daily solar radiation estimation model introduced by Hargreaves and Samani (1982), which utilizes daily minimum and maximum air temperature to gap-fill missing solar irradiance data (the spreadsheet is available on request). The Daily Penman, Monteith, Equilibrium Temperature Hargreaves-Samani (DPMETHS) model of Savage et al. (2017) estimates daily open water evaporation from the land-based meteorological data. This model utilises the concept of equilibrium temperature to estimate water-body temperature of the water storage using an iterative procedure to obtain the wet-bulb temperature (Savage, 2017). The estimated water-body temperature is essential for computing outgoing infrared irradiance from the water surface L u water .
For operational purposes, such as water resources management, irrigation management and hydrologic studies, where nearreal time estimates of evaporation are needed, the DPMETHS model seems to be a promising model for estimating open water evaporation due to its user-friendliness and minimal data input requirements. However, rigorous validation of the DPMETHS model for different climatic conditions using an extended period of in-situ measurements collected from different sizes of water storages is required to improve the confidence of the estimates of the open water evaporation . Within this context, validation of the procedure to estimate R n water using the DPMETHS model is critical, since R n water is one of the key drivers of open water evaporation (McJannet et al., 2008). Consequently, poor estimation of R n water using the DPMETHS model could result in significant errors in estimating open water evaporation, leading to inefficient management of water resources. Therefore, the estimates of R n water from the DPMETHS model need to be tested for suitability against in-situ measurements of R n water collected from water storages from different climatic conditions before the model could be utilised with confidence to estimate R n water for open water evaporation. Therefore, the main aim of this study was to evaluate the performance of the DPMETHS model to estimate R n water using land-based meteorological data from a nearby weather station. In this study, the procedure of the DPMETHS model to estimate daily R n water was evaluated using daily R n water in-situ measurements acquired from 5 sites in both hemispheres, representing very different climatic conditions.

Study site description
Data scarcity of R n water is the major challenge that hinders the evaluation of newly developed models for estimating R n water in most countries (Wang and Liang, 2009;McMahon et al., 2013;Savage et al., 2017). Five sites that represent different climatic conditions were selected for the evaluation of the DPMETHS model. The site characteristics, record period and available daily data from each site are presented in Table 1. The choice of the duration of the R n water measurements at each site was based on the availability of quality radiative flux measurements using a 4-component net radiometer mounted above the open water surface, and the corresponding land-based daily meteorological data.

Description of the DPMETHS model for computing net irradiance for open water
The model description provided by McJannet et al. (2008) forms the basis of the daily time-step DPMETHS spreadsheet-implemented model of Savage et al. (2017). The DPMETHS model computes R n water (MJ m -2 ) using daily measurements of solar irradiance (R s land , MJ•m -2 ), maximum and minimum air temperature (T a , °C), minimum and maximum relative humidity (RH, %) and average wind speed (U land , m•s -1 ) from a nearby landbased weather station. The estimates of R n water are calculated from the solar irradiance reaching the water surface (R s water ) minus rR s water based on the reflection coefficient of the water surface (r water ) and net outgoing infrared irradiance (L d water − L u water ). The net infrared irradiance is calculated from T a at 09:00, the estimated daily-average water temperature and a cloudiness factor, following the procedure of De Bruin (1982). The model assumes that the land-based meteorological data represent climatic conditions over open water surfaces and thus, R s land = R s water .
Then R n water is calculated from: where r water is approximately 0.08 (Finch and Hall, 2001) and L d water is calculated from: . exp . (2) where σ = 4.9 × 10 -9 MJ•m -2 •K -4 is the modified for daily timescale Stefan-Boltzmann constant. The cloudiness factor (C f ) is determined using the procedure presented by Jegede et al. (2006): if R s land /R s clear ≤ 0.9, then: where: where G sc is the solar constant (0.0820 MJ•m -2 •min -1 ), d r is the inverse relative distance from the earth to the sun, Ω is the sunset hour angle (rad), ϕ is the latitude (rad) and δ is the solar declination (rad), where d n In Eq. 1, L u water is given by: where T water (°C) is the temperature of the water surface. The L u water may be approximated using a Taylor series expansion at T a as: . ) where the factor 0.97 corresponds to the emissivity of water (McJannet et al., 2008), T a is the land-based daily averaged air temperature (°C) at a reference height of 2 m and T water i-1 is the average water temperature of the previous day (°C).
The daily-average water temperature on day i, T water i (°C), is calculated from T water i-1 , a water-body time constant τ (day) and an equilibrium temperature T e (°C):

T T T T t
water i e water i e exp 1 /W The water-body time constant (τ) is calculated based on the De Bruin (1982) method: where ρ w is the density of water (kg•m -3 ), c w the specific heat capacity of water (0.004185 MJ•kg -1 •K -1 ), and d the water depth (m), T wet the wet-bulb temperature, γ the psychrometric constant, ∆T wet (kPa•°C -1 ) the slope of the saturation water vapour vs temperature relationship at the wet-bulb temperature and f(U) the wind function that is usually derived empirically for a particular location. The f(U) above water is computed using the Harbeck (1962) where f(U 2 ) is the wind function for wind speed measured at a height of 2 m above the surface (MJ• m -2 •kPa -1 ) and A is the surface area of the water storage (m 2 ).
For open water, the net irradiance at the wet bulb, instead of the water-predicted temperature was used to avoid any calculations involving water depth. For daily open water evaporation, Penman

Data collection and processing
Daily measurements of meteorological variables such as R s land , T a , T min , T max , RH min , RH max , U land , U water , T water , DEWP min , DEWP max , R s water , r water R s water , L d water , L u water and R n water were acquired from all 5 sites. The record period and available data from each site are presented in Table 1 All datasets underwent a data quality control routine to identify and remove all erroneous, suspicious and impossible values, following the procedure of Allen et al. (1998). Only good-quality datasets were used for the evaluation of the DPMETHS model. At the Stratus Ocean site where measurements of R n water were missing, a constant r water value of 0.08 was used to estimate r water R s water from measurements of R s water above the ocean surface. The estimates of L u water from the ocean surface were computed from T water using Eq. 10. The DEWP min and DEWP max were used to estimate RH min and RH max , respectively, using the procedure of Allen et al. (1998). Finally, the daily measurements of R n water were computed using Eq. 1, replacing R s land with R s water only at the Stratus Ocean site due to the lack of nearby measurements of R s land .

Data analysis
The root mean square error (RMSE, MJ•m -2 ), mean bias error (MBE, MJ•m -2 ) and index of agreement (d) were used to evaluate the performance of the DPMETHS model estimates against daily measurements of R n water and were calculated following the procedure of Willmott et al. (1985)  where R ne water (MJ•m -2 ) is the estimated net irradiance of open water, R n water is the measured net irradiance of open water, R n water is the mean of R n water and n is the number of observations. Additionally, a linear regression between R ne water and R n water values was calculated as: where m is the slope and c (MJ•m -2 ) is the y-intercept. The coefficient of determination (r 2 ) was used as a measure of precision. Based on these statistics, RMSE, MBE and c values approaching zero whilst d, r 2 and m values approaching 1 indicate the best model performances (Willmott et al., 1985).

Weather conditions during the study period at all 5 sites
Newly developed models to estimate R n water from land-based meteorological data still require evaluation against in-situ measurements collected over a wide range of climatic conditions before they can be used with confidence. The meteorological data used for model evaluation illustrated a wide range of climatic conditions which had implications for the further interpretation of the results ( Table 2).

Evaluation of the DPMETHS model at all 5 sites
To evaluate the performance of the DPMETHS model, comparisons were made between the daily estimates of R n water and measurements of R n water at all 5 sites. The relationships between estimated net irradiance (R ne water ) and measured net irradiance (R n water ) were reasonable at all sites ( Fig. 1; Table 3).

Performance of the DPMETHS model at all 5 sites
The correlation between R n water and R ne water indicated a statistically significant relationship, with r 2 values ranging from 0.  , 2010). Consequently, the DPMETHS model over-estimated R n water due to under-estimations of L u water . These findings suggested that using land-based meteorological data that do not represent weather conditions above open water surfaces could result in significant errors in R ne water predicted from the DPMETHS model. Thus, it is recommended that the land-based meteorological data should be acquired with caution from a nearby weather station that represents the prevailing weather conditions above water storage of interest (Everson, 1999).

Applicability and limitations of the DPMETHS model
The DPMETHS model, a daily model, uses the daily-averaged U land as an input and, therefore, this model does not explicitly account for night-time R n water which is dominated by L u water that is directly governed by T water . For example, Savage et al. (2017) reported that U water was a maximum during night-time and minimal early in the morning at Midmar Dam. Consequently, higher U water at nighttime than expected could result in surface cooling and decreased T water . Consequently, the DPMETHS model is likely to overestimate R n water due to under-simulations of L u water during clear and windy days. Furthermore, some of the discrepancies between R n water and R ne water could be attributed to the poor estimation of L d water within the DPMETHS model, since this model only estimates the cloud fraction with no optical properties. However, whether the presence of clouds will have a net cooling or warming effect at the water surface depends on the cloud's optical properties such as the cloud's altitude, its size, and the make-up of the particles that form the cloud (Key et al., 1996).
The findings of this study indicate that the performance of the DPMETHS model depends on the representativeness of the landbased daily meteorological data to the weather conditions above the open water surface. Therefore, future research on measuring and modelling of R n water for the estimation of open water evaporation purposes should be cautious of the possible contrasts of weather conditions between land and water surfaces. Despite the discrepancies between R n water and R ne water , the findings of this study indicated that the DPMETHS model can be reliably used to estimate R n water for estimating open water evaporation over a wide range of climatic conditions. The DPMETHS model is a promising and user-friendly model for estimating R n water for the estimation of open water evaporation at high resolution with minimal landbased meteorological data that are often readily available from a standard weather station. Furthermore, the DPMETHS model uses universally applicable scientific theories and assumptions to estimate daily R n water accurately. The spreadsheet-based iterative procedure of the DPMETHS model evaluated in this study allows easy data handling and visual inspection.

CONCLUSIONS
The DPMETHS model to estimate daily R n water was evaluated using daily R n water in-situ measurements acquired from 5 sites, representing different climatic conditions.
The DPMETHS model reliably estimates R n water for the estimation of open water evaporation over a wide range of climatic conditions. Major discrepancies between R n water and R ne water were attributed to the use of the land-based meteorological data that do not represent weather conditions over open water surfaces. Therefore, it is recommended that the land-based weather stations should be selected with caution, such that they represent the weather conditions above water storage of interest.
The spreadsheet-based iterative procedure of the DPMETHS model to estimate daily R n water using minimal land-based meteorological data is user-friendly, with minimal computational requirements, and is quick and reliable. It also allows easy data handling and visual inspection. One of the limitations of the DPMETHS model is that the model utilizes the daily meteorological data which might not be a true representation of climatic conditions for the entire day, since most of the weather variables had a wide range of diurnal variability. Therefore, a sub-daily version of the DPMETHS model is recommended for improved estimation of R n water for open water evaporation.

AUTHOR CONTRIBUTIONS
Conceptualization -L Myeni, MJ Savage and AD Clulow; methodology -L Myeni and MJ Savage; data analysis -L Myeni; original draft preparation and writing -L Myeni L; review and editing -L Myeni, MJ Savage and AD Clulow; supervision -MJ Savage and AD Clulow.