Development of temperature and mass transfer-based empirical models for estimating reference evapotranspiration in Nigeria

Water is a precious resource, and efficient water management in agriculture is crucial for sustainable development in agricultural planning (García-Tejero et al. 2011; Zaman et al. 2017). Reference evapotranspiration (ET₀) is critical to understanding crops' water needs and planning effective irrigation strategies (Bai et al. 2017; Roy et al. 2021; Dong et al. 2024). ET₀ models serve as fundamental tools in irrigation and water resource management and represent the water requirement of well-watered reference crops, forming the basis for calculating crop water requirements and climate change. Accurate ET₀ estimation enables water resource managers to optimise irrigation scheduling, conserve water, and enhance agricultural productivity. In regions like Nigeria, where water management is a recurring challenge, precise ET₀ models are essential for sustainable water use.

Despite the global development of various empirical ET₀ models, none have been specifically tailored to the unique climatic conditions in Nigeria. This gap in the literature underscores the need for region-specific models that can account for the intricacies of Nigeria's diverse climate. Empirical models, which rely on observed data and are often more adaptable to local conditions, become paramount in addressing this gap and providing accurate ET₀ estimates for specific needs in Nigeria.

Many empirical models have been developed worldwide to estimate ET₀, often using readily available meteorological data, e.g., temperature, wind speed, and relative humidity (Trabert 1896; Meyer 1926; Linacre 1977; Ahooghalandari et al. 2016; Muhammad et al. 2022). While these models offer valuable tools, their accuracy can be significantly limited when applied in regions with unique climatic characteristics. Nigeria has a diverse landscape, ranging from humid coastal regions to arid northern savannas, each experiencing distinct rainfall patterns, wind speed, temperature regimes, and solar radiation profiles (Oguntunde et al. 2011; Shiru et al. 2020; Oloyede et al. 2022; Animashaun et al. 2023). This climatic heterogeneity challenges existing global ET₀ models, often leading to inaccurate estimates within the country.

The consequences of inaccurate ET₀ estimation can be substantial for irrigation, water resource management, and food security (Bashir et al. 2023; Awhari et al. 2024). Overestimating ET₀ can lead to excessive irrigation, wasting precious water resources, and potentially damaging soil through salinisation (Ankindawa & Awhari 2010; Al-Bakri et al. 2022). Conversely, underestimating ET₀ can cause water deficits, endanger crop yields, and contribute to food insecurity (Nikolaou et al. 2020). Therefore, developing dependable and region-specific ET₀ models for the Nigerian climate is crucial for optimising water management strategies and ensuring sustainable agricultural practices.

Numerous ET₀ models have been developed using temperature, solar radiation, wind speed, and various combinations to accurately predict evapotranspiration rates from surfaces such as soil, water, and vegetation canopies (Huang et al. 2021). The FAO-PM equation is widely recognised as the most effective approach for estimating ET₀ across various climatic conditions (Hernández-Bedolla et al. 2023; Wu et al. 2023). FAO-PM estimation is resource-intensive, and the necessary data are not readily accessible in underdeveloped countries like Nigeria. An inherent constraint in the widespread use of the universally recognised FAO-PM approach is the lack of access to the extensive data it requires at multiple meteorological stations, compounded by concerns over the quality of these data points.

This study aimed not to develop a novel approach that substitutes FAO-PM but to devise a straightforward method based on temperature and mass transfer-based models. This method relies solely on maximum temperature, wind speed, and relative humidity, making it suitable for areas where other climatic parameters are unavailable at the desired level of detail. The FAO-PM was used as the reference for formulating these novel methodologies. This research evaluated the model's effectiveness in accurately representing the spatial distribution of the performance metrics and compared it with FAO-PM as reference ET₀.

This study first assessed 31 empirical models, as shown in Table 1, consisting of 20 models based on temperature and 11 models based on mass transfer. Kling–Gupta efficiency (KGE) (Gupta & Kling 2011) and percentage bias (Pbias) were used for this purpose. KGE measures correlation, variability ratio, and mean bias between two datasets and is considered a better metric for assessing model performance. Pbias provides the mean bias in ET₀ estimation and thus serves as a simple measure of model deviation from the observation.

Table 1

Reference evapotranspiration model methods

No. .	Model .	Inputs .	Equation .	KGE .	Pbias .	References .
Temperature-based models
1	Linacre	T		0.092	61.0	Linacre (1977)
2	Hamon	T		−47.00	−89.4	Hamon (1963)
3	FAO-Blaney	T		−5,199.18	−11,031.0	Dooreng & Pruitt (1977)
4	Kharrufa	T		−0.48	86.6	Kharrufa (1985)
5	Ahooghalandari-1	T, RH		0.001	61.5	Ahooghalandari et al. (2016)
6	Ahooghalandari-2	T, RH		−0.10	68.1	Ahooghalandari et al. (2016)
7	Ivanov	T, RH		0.33	54.2	Romanenko (1961)
8	Papadakis	T, RH		0.19	73.8	Papadakis (1965)
9	Hargreaves –Samani	T, Tmax,Tmin,Ra		−4.30 × 106	−9.20 × 106	Hargreaves & Samani (1985)
10	Baier-Robertson	T, Tmax,Tmin,Ra		−537.1	−1,072.7	Baier & Robert (1965)
11	Modified Baier	T, Tmax, Tmin, Ra, RH		8.77 × 105	1.93 × 106	Baier (1971)
12	Modified H-S1	T, Tmax, Tmin, Ra		1.14 × 106	−2.45 × 106	Droogers & Allen (2002)
13	Modified H-S2	T, Tmax, Tmin, Ra		−4.67 × 105	1 × 108	Droogers & Allen (2002)
14	Trajkovic	T, Tmax, Tmin, Ra		−371.43	−813.5	Trajkovic (2007)
15	Ravazzani	T, Tmax, Tmin, Ra		−9.39 × 107	−2.01 × 107	Ravazzani et al. (2012)
16	Schendel	T, RH		−1.64	163.7	Schendel (1967)
17	Dorji	T, Tmax, Tmin, Ra		−332.22	−695.8	Dorji et al. (2016)
18	Berti	T, Tmax, Tmin, Ra		−1,031.3	−2,259.8	Berti et al. (2014))
19	Talee and Tabari	T, Tmax, Tmin, R		−1,817.3	−2,706.3	Tabari & Talaee (2011)
20	Drogers and Allen	T, Tmax, Tmin, R		−1,336.9	−2,903.2	Droogers & Allen (2002)
Mass transfer-based models
21	Mahringer	T, RH, u		0.74	18.96	Mahringer (1970)
22	Penman	T, RH, u		0.72	−10.17	Penman (1948)
23	Rohwer-2	T, RH, u		0.20	59.62	Rohwer (1931)
24	Szasz	T, RH, u		0.37	55.16	Szasz (1973)
25	Trabert	T, RH, u		0.62	27.92	Trabert (1896)
26	WMO	RH, u		0.89	6.63	WMO et al. (1966)
27	Albrecht	T, RH, u		−0.33	90.10	Albrecht (1950)
28	Meyer	T, RH, u		0.28	54.67	Meyer (1926)
29	Brockamp–Wenner	T, RH, u		−0.69	121.70	Brockamp & Wenner (1963)
30	Fitzgerald	T, RH, u		−3.62 × 105	7.69 × 105	FitzGerald (1886)
31	Saif	RH, u		−6.38 × 105	1.34 × 106	Islam et al. (2020)

No. .	Model .	Inputs .	Equation .	KGE .	Pbias .	References .
Temperature-based models
1	Linacre	T		0.092	61.0	Linacre (1977)
2	Hamon	T		−47.00	−89.4	Hamon (1963)
3	FAO-Blaney	T		−5,199.18	−11,031.0	Dooreng & Pruitt (1977)
4	Kharrufa	T		−0.48	86.6	Kharrufa (1985)
5	Ahooghalandari-1	T, RH		0.001	61.5	Ahooghalandari et al. (2016)
6	Ahooghalandari-2	T, RH		−0.10	68.1	Ahooghalandari et al. (2016)
7	Ivanov	T, RH		0.33	54.2	Romanenko (1961)
8	Papadakis	T, RH		0.19	73.8	Papadakis (1965)
9	Hargreaves –Samani	T, Tmax,Tmin,Ra		−4.30 × 106	−9.20 × 106	Hargreaves & Samani (1985)
10	Baier-Robertson	T, Tmax,Tmin,Ra		−537.1	−1,072.7	Baier & Robert (1965)
11	Modified Baier	T, Tmax, Tmin, Ra, RH		8.77 × 105	1.93 × 106	Baier (1971)
12	Modified H-S1	T, Tmax, Tmin, Ra		1.14 × 106	−2.45 × 106	Droogers & Allen (2002)
13	Modified H-S2	T, Tmax, Tmin, Ra		−4.67 × 105	1 × 108	Droogers & Allen (2002)
14	Trajkovic	T, Tmax, Tmin, Ra		−371.43	−813.5	Trajkovic (2007)
15	Ravazzani	T, Tmax, Tmin, Ra		−9.39 × 107	−2.01 × 107	Ravazzani et al. (2012)
16	Schendel	T, RH		−1.64	163.7	Schendel (1967)
17	Dorji	T, Tmax, Tmin, Ra		−332.22	−695.8	Dorji et al. (2016)
18	Berti	T, Tmax, Tmin, Ra		−1,031.3	−2,259.8	Berti et al. (2014))
19	Talee and Tabari	T, Tmax, Tmin, R		−1,817.3	−2,706.3	Tabari & Talaee (2011)
20	Drogers and Allen	T, Tmax, Tmin, R		−1,336.9	−2,903.2	Droogers & Allen (2002)
Mass transfer-based models
21	Mahringer	T, RH, u		0.74	18.96	Mahringer (1970)
22	Penman	T, RH, u		0.72	−10.17	Penman (1948)
23	Rohwer-2	T, RH, u		0.20	59.62	Rohwer (1931)
24	Szasz	T, RH, u		0.37	55.16	Szasz (1973)
25	Trabert	T, RH, u		0.62	27.92	Trabert (1896)
26	WMO	RH, u		0.89	6.63	WMO et al. (1966)
27	Albrecht	T, RH, u		−0.33	90.10	Albrecht (1950)
28	Meyer	T, RH, u		0.28	54.67	Meyer (1926)
29	Brockamp–Wenner	T, RH, u		−0.69	121.70	Brockamp & Wenner (1963)
30	Fitzgerald	T, RH, u		−3.62 × 105	7.69 × 105	FitzGerald (1886)
31	Saif	RH, u		−6.38 × 105	1.34 × 106	Islam et al. (2020)

Note: ET₀ is the reference evapotranspiration (mm/day), ND is the length of time interval (day), and f is a correction factor that varies with the average annual temperature (T_a < 23.5 °C, F = 0.01). R_a is the extra-terrestrial radiation (MJ/m²/day), u is the wind speed at 2 m (m/s), RH is the relative humidity (%), T, T_max, T_min are average, maximum, and minimum air temperature (°C), T_d is the daily dewpoint air temperature in (°C), TD is the difference between the T_max and T_min (°C), e_ma is the saturation vapour pressure at the monthly mean daily maximum temperature (kPa), and e_a and e_s represent actual and saturation vapour pressure (kPa), except for Penman, Papadakis, and Rohwer (kPa), RHOSAT is the saturated vapour density (g/m³), KPEC is a calibration coefficient (1.2), p is a constant having two values (0.27 and 0.28), f(u) denotes a function of wind speed, L is the latitude at each point (degree), z is the elevation (m), λ represents latent heat of evaporation (MJ/kg), Δ is the slope of saturation vapour pressure–temperature curve (kPa/°C), and γ is the psychometric constant (kPa/°C).

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Ivanov and World Meteorological Organization (WMO) performed better among these models based on their KGE values, as shown in Table 1. These models were the main focus of our investigation. A novel temperature technique that utilises maximum temperature and relative humidity to determine ET₀ was derived. Ivanov generally performs accurately across the specified region among the temperature-based models. There is a need to build a straightforward temperature approach that can effectively work at all locations throughout the nation. As a result, a novel empirical temperature equation using maximum temperature and relative humidity was developed. In developing a new technique, the aerodynamic term was calculated based on the mean vapour pressure deficit (Dav). It is suggested that Dav may be estimated as 1 − RH/100, where RH represents the relative humidity. ET₀ is closely associated with extra-terrestrial radiation (Ahooghalandari et al. 2016). The new equations were derived by integrating the expression (1 − RH/100) with the maximum temperature.

The two models, Awhari1 (temperature-based) and Awhari2 (mass transfer-based), were developed through curve-fitting techniques and particle swarm optimisation (PSO). These equations evaluate the fitness function for ET₀ estimation, incorporating varying ‘a’ and ‘b’ values and discrepancies between simulated and observed ET₀ values. To optimise model performance, we implemented PSO, a technique that iteratively refines the objective function for each particle. This involves the concurrent execution of both the ET₀ model and the simulated evapotranspiration model, with dynamic adjustments to each particle's velocity and position based on historical best positions and those discovered within the swarm. The optimised models, identified through personal and global best positions, underwent rigorous evaluation by comparing simulated and observed ET₀. The two constants, a and b, were set with options and bounds for coefficients [a, b], lb = lower bounds for coefficients [a, b], ub = upper bounds for coefficients, and max iterations of 100. The two constant values for the temperature and mass transfer-based models, a and b, were calculated using PSO over the years 1950–2022.

The probability density function (PDF) of the estimated ET₀ using Awhari1, Ivanov, and PM models for the whole study region is shown in Figure 6(b). PM and Awhari1 showed peak densities, ET₀ of 0.18 and 2.3 mm/day, and median ET₀ values of 4.6 mm/day, respectively. Compared with the PM ET₀, Ivanov exhibits a lower peak density of 0.14 and a slightly lower median ET₀ of 4.4 mm/day, suggesting a tendency to underestimate ET₀. The tail behaviour of PM ET₀ towards tails gradually fades towards zero, reflecting the decreasing probability of extreme ET₀ values and the Awharil1 model. The shorter left tail implies a lower probability of underestimating extreme ET₀ events. The Ivanov model's longer right tail suggests a higher chance of overestimating extreme ET₀ events.

The PDF of the ET₀ estimated using PM, WMO, and Awhari2 models for the whole study region is shown in Figure 9(b). PM and Awhari2 peaked at 0.17 and 0.16 and ET₀ of 2.6 mm/day and median values of 4.6 mm/day for both the PM and the models. WMO exhibits a higher peak density at 0.2. The tail behaviour of the Awharil2 model's longer left tail implies a higher probability of overestimating extreme ET₀ events. The WMO model's shorter right tail suggests a higher chance of underestimating extreme ET₀ events.

The present study developed empirical PSO models for reliable ET₀ estimation using the ERA5 reanalysis temperature, wind speed, and relative humidity data. Several studies have endorsed the use of ERA5 reanalysis data for investigating the climate of West Africa (Quagraine et al. 2020; Steinkopf & Engelbrecht 2022; Bodjrenou et al. 2023; Gbode et al. 2023). Compared with its predecessor, ERA-Interim, ERA5 notably improves in representing various climatic aspects. Quagraine et al. (2020) demonstrated that ERA5 provides a more accurate depiction of precipitation patterns across West Africa, particularly by capturing crucial interannual variability and regional variations in summer monsoon rainfall. ERA5 also shows reduced biases in the annual cycle, resulting in a more precise representation of seasonal temperature fluctuations (Gbode et al. 2023). In addition, ERA5 exhibits improved capabilities in capturing interannual temperature variations (Bodjrenou et al. 2023). ERA5 data have also been found suitable for agricultural study (Araghi et al. 2022). Therefore, the results obtained in this study using ERA5 reanalysis climate data can be considered reliable.

The comparison of the newly developed models showed higher performance in terms of the statistical indices KGE and Pbias compared with the existing widely used models. The performance of the models showed considerable and promising improvement. For example, the newly developed temperature-based Awhari1 shows a mean KGE of 075 and Pbias of 6.49% compared with the Ivanov with KGE values of 0.33 and Pbias of 54.20%. In contrast, the WMO shows KGE values of 0.89 and Pbias of −6.63, and Awhari2 exhibits even better performance with KGE values of 0.92 and Pbias of 5.67%. This indicates a significant improvement in both the KGE and Pbias and the capability of newly developed models to estimate ET₀ in Nigeria with higher accuracy and less uncertainty. The spatial variability of a model's parameter development is significant because it provides an inside look at all the grid points where a model can be calibrated and validated to know it, thereby addressing the daily percentage bias varies in design irrigation and water resource management for crop production. The striking comparison that can be made on the performance of an equation can be seen from the spatial point of the observed and computed daily evaporation. This study brings an insight into the spatial performance of models that can easily be identified where the Pbias of the models developed can be addressed between the observed values across the grid cell and adjusted to the required values. This means that an acceptance level of confidence can be achieved when transferring the model's applicability from one region to another, even within a large region.

PSO is a potent and adaptable optimisation technique because of its global search powers, resilience, and ease of implementation (Juneja & Nagar 2016; Wang et al. 2018). It is a viable solution for many optimisation issues because of its unique properties, despite its strengths and shortcomings. It efficiently navigates the search space, avoiding being trapped in local optima (Gad 2022; Zhang & Kong 2023). This is accomplished by the swarm intelligence process, in which particles exchange information and draw insights from one another's achievements, thus avoiding premature settling on less-than-ideal solutions. PSO tolerates noise and complicated search landscapes, making it appropriate for situations with non-smooth objective functions or noisy data. It may be easily adjusted for different optimisation issues by parameter modifications or the inclusion of other processes. PSO is more computationally efficient than other population-based algorithms (Xu et al. 2023; Zhang & Kong 2023). Furthermore, PSO promotes information exchange among particles, accelerating algorithmic convergence. Population-based approaches are effective for smooth objective functions with well-defined gradients but may be susceptible to local optima and struggle with noisy functions. In contrast, PSO is more resilient (Gad 2022). This advantage has contributed to the development of superior ET₀ models.

Developing innovative evapotranspiration models in Nigeria might significantly impact agriculture, water resource management, environmental sustainability, and other areas. Accurate estimation of ET₀ is a challenge in Nigeria due to the unavailability of reliable data on the variables required for its estimation (Adesogan & Sasanya 2023). Enhanced ET₀ data using readily accessible information may guide irrigation strategies, improve water efficiency and minimise waste. This may result in higher crop yields, enhanced food security, reduced irrigation expenses for farmers, and, ultimately, greater economic sustainability and profitability. Irrigation water management is a major issue in agricultural development in Nigeria (Oluniyi & Bala 2021). Localised ET₀ models include area climate and soil variables to help farmers make irrigation choices tailored to their crops and localities. Drought significantly affects agriculture in some parts of Nigeria (Shiru et al. 2020). ET₀ models may pinpoint regions with elevated ET₀ levels, aiding in creating drought readiness plans, reducing crop losses, and promoting agricultural resilience.

The new models contribute to sustainable water management in Nigeria by improving the accuracy of ET₀ estimates and enabling better irrigation planning and water resource allocation. Precise evapotranspiration data may assist water managers in distributing water resources more effectively across different sectors, such as agricultural, residential consumption, and industrial requirements. Comprehending ET₀ patterns may help develop reservoir operating plans to optimise water storage and release to fulfil needs while preserving ecological balance. Integrating ET₀ into hydrological models may enhance the precision of flood predictions, facilitating prompt preparation and mitigation measures. Optimising water use in agriculture reduces water footprints, decreases environmental effects, and conserves valuable resources.

Standard statistical metrics often focus solely on overall ET₀ magnitude agreement, overlooking crucial aspects such as timing, frequency, and variability of ET₀ events. These characteristics are vital for comprehending drought dynamics, where short-term deficits and episodic droughts matter more than long-term averages. While accurate ET₀ estimation based on statistical metrics is valuable, it does not guarantee optimal performance in various hydrological applications. A model might accurately estimate observed ET₀, yet its simulations can yield different drought occurrence estimations using standard indices or depict distinct aridity patterns. This discrepancy arises from neglecting crucial ET₀ characteristics relevant to drought and aridity assessments, which were not evaluated during model development. In addition, models might perform differently when used in conjunction with other variables. Further study can be conducted to assess the performance of the models in accurately estimating drought, aridity, or other hydro-climatic events.

This study has pioneered a novel methodology for estimating ET₀ tailored for Nigeria. Its focus on Nigeria may restrict its immediate generalizability. However, developing new models that effectively estimate ET₀ using a limited set of meteorological variables significantly contributed to Nigeria's irrigation and water resources management. Moreover, the research contributes significantly to developing a methodology for constructing ET₀ models, setting a precedent for future endeavours to formulate ET₀ equations for diverse geographical regions beyond Nigeria.

In conclusion, this study has successfully advanced the ET₀ estimation by developing empirical models using PSO with temperature, wind speed, and relative humidity data. The comparison of these models against widely used existing ones revealed a substantial and promising improvement in performance, as evidenced by higher statistical indices, specifically KGE and Pbias. The temperature-based model, Awhari1, emerged as a standout, showing a mean KGE of 0.75 and a Pbias of 6.49%, showing a remarkable enhancement compared with established models like Ivanov. Awhari2 demonstrated an even more impressive performance, with KGE values of 0.92 and a Pbias of 5.67. These results demonstrate that the new models can estimate ET₀ in diverse climatic conditions with Nigeria with heightened accuracy and reduced uncertainty. The spatial distribution analysis of model parameters, allowing for comprehensive calibration and validation at different grid points, provides a notable novelty and addresses the insight of daily percentage bias, offering valuable implications for designing irrigation and water resource management strategies in agricultural production. Moreover, a spatial performance comparison of observed and computed daily evaporation across grid cells provides a striking visualisation of model effectiveness. The ability to address Pbias variations between observed and model values at the grid cell level demonstrates the adaptability of the developed models, fostering confidence in transferring parameter values from one region to another, even within a diverse environmental context. These findings significantly contribute to ET₀ estimation, particularly in regions with limited data availability and diverse climatic conditions.

The authors received no financial support for this article's research, authorship, and publication.

The codes used for data processing can be provided by the corresponding author at request.

All the authors consented to publish the paper.

A.P.D.: Conceptualisation, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Data Curation, Writing – Original Draft, Writing – Review & Editing, Visualisation; M.H.J.: Formal analysis, Investigation, Data Curation, Writing – Original Draft, Writing – Review & Editing, Visualisation, Supervision; M.K.I.M.: Formal analysis, Investigation, Data Curation, Writing – Original Draft, Writing – Review & Editing, Visualisation, Supervision; K.B.M.: Methodology, Formal analysis; Z.M.Y.: Methodology, Software, Formal analysis, Writing – Review & Editing, Visualisation, Supervision; S.S.: Methodology, Software, Validation, Formal analysis, Investigation, Data Curation, Writing – Review & Editing, Visualisation, Supervision.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.