Calibrating the Priestley–Taylor model for evapotranspiration across different substrate depths in green roofs Open Access

Urbanization has led to the widespread environmental issue of the urban heat island effect, which urgently requires attention. Green roofs (GRs), as an innovative urban greening strategy, have garnered widespread attention for their significant effects in mitigating the heat island effect, improving urban microclimate, and promoting biodiversity (Zhang & He 2021; Alim et al. 2022). Evapotranspiration (ET) serves as the primary means by which GRs regulate the energy and water balance of buildings, mitigating urban heat island and stormwater runoff issues. By releasing moisture from the substrate, GRs can restore their water-holding capacity, making room for the next rainfall. The cooling effect of water vaporization (e.g. via ET) is a key factor in regulating the thermal environment of buildings, improving the microclimate, and mitigating the urban heat island effect (Ouedraogo et al. 2023).

Currently, the calculation of ET for GRs primarily employs an integrated conversion method, which involves first estimating potential ET (PET) and then converting it into actual evapotranspiration (AET). PET is a theoretical value that represents the maximum amount of evaporation that the surface could achieve under certain climatic conditions if there is a sufficient water supply (Wadzuk et al. 2013). AET is the actual amount of water evaporated and transpired from the surface (including soil and vegetation). It is influenced by surface, subsurface, and meteorological conditions. AET is a water-limited ET, meaning it is constrained by the availability of soil water. As soil moisture (SM) decreases from field capacity, AET also decreases (Marasco et al. 2015). In the current study, a summary table of PET calculation methods is presented in Table 1.

It is worth noting that the Penman–Monteith (PM) equation (Monteith 1965) supersedes the Penman equation (Penman 1948). The Food and Agriculture Organization (FAO) of the United Nations recommends a simplified version of the American Society of Civil Engineers (ASCE)-PM equation for a short crop reference surface (similar to grass) with a vegetation height, surface resistance, and albedo of 0.12 m, 70 s/m, and 0.23, respectively, which is widely known as the FAO-56 PM equation (Allan et al. 1998). Jensen et al. (1990) presented a PM-based ET equation through the ASCE Manual 70, which is known as the ASCE-PM equation. The ASCE recommends two versions of the ASCE-PM equation for short and tall crop reference surfaces, which are known as ASCE standardized PM equations (ASCE, Technical Committee on Standardization of Reference Evapotranspiration 2005).

The Priestley–Taylor model, with the benefits of simplified calculations, easy implementation, and wide application, has been extensively used in the fields of hydrology, agricultural meteorology, and environmental engineering (Kim et al. 2024; Yang et al. 2024) and has also received considerable attention in the estimation of ET for GRs (Table 2).

After comparing Blaney–Criddle, Hargreaves, Slatyer–McIlroy, Priestley–Taylor, and ASCE-PM for predicting ET against measured ET, Wadzuk et al. (2013) found that the Priestley–Taylor method tends to slightly underestimate the measured ET but is generally close to the observed data, indicating its high predictive accuracy.

In the study by Marasco et al. (2015), the Priestley–Taylor model outperformed other evaluated ET models (Hargreaves, Penman, and ASCE-PM) in predicting ET for two large GRs in New York City. The Priestley–Taylor model showed the highest correlation (r²) with dynamic chamber measurements, indicating that this model could more accurately predict the AET conditions. Additionally, the Priestley–Taylor model was able to capture the seasonal variations of ET well, especially under non-water-limited conditions, with its predictions closely aligning with the trends of actual measured data. However, despite the Priestley–Taylor model's excellent performance in predicting ET, Marasco et al. (2015) also identified systematic errors in the model under extreme winter and summer conditions, where the model might overestimate ET in winter and underestimate it in summer. This indicates that under specific climatic conditions, further model improvements or the integration of other methods may be necessary to enhance the accuracy of predictions.

It is particularly noteworthy that, although there were two GRs with different SDs in the study by Marasco et al. (2015), their construction types, construction years, geographical locations, roof areas, saturated water-holding capacities, field capacities, and saturated hydraulic conductivities were all different. The study did not provide specific values or adjustment methods for the Priestley–Taylor parameter α under different scenarios. Therefore, the study can indeed be considered as the repeated application of the Priestley–Taylor model in two different GR scenarios, without specifically discussing the differences in parameter α under different circumstances.

Mobilia et al. (2017) utilized the Priestley–Taylor model to calculate PET, which was subsequently employed in two AET estimation methods: antecedent precipitation index (API) and advection-aridity (AA). The effectiveness of the Priestley–Taylor model in estimating PET for GRs was validated through a comparison with actual observational data, demonstrating the model's efficacy in this context.

Previous studies (e.g. Table 2) affirmed the reliability of the Priestley–Taylor model in estimating AET for GRs, but there are still gaps in these studies. First, previous research does not clarify the significance of calibration and validation procedures for the Priestley–Taylor estimating AET for GRs, often employing the default value of the Priestley–Taylor coefficient α for estimating or not specifically discussing the differences in parameter α under various scenarios.

The empirical coefficient α in the Priestley–Taylor model is provided to correct for the impact of advection on ET. This parameter reflects the influence of advection to a certain extent. Ideally, when the value of α equals 1, it implies that the ground is neither wet nor dry, and advection is close to zero; a value of 1.26 signifies a saturated surface or a moist condition (Priestley & Taylor 1972; ASCE, Technical Committee on Standardization of Reference Evapotranspiration 2005). However, numerous studies have indicated that the value of α varies with the geographical characteristics of the study area and exhibits interannual and seasonal temporal variations (Wang et al. 2024). Moreover, the value of α has been found to exhibit considerable variation on sub-daily, daily, and seasonal bases (Aschonitis et al. 2015), tending to decrease with increasing vapor pressure deficit (Szilágyi et al. 2014). Specifically, Nikolaou et al. (2023) adjusted α to accommodate greenhouse environments under Mediterranean climates. They found that in greenhouse compartments equipped with different cooling systems, the estimated values of α were 0.86 and 0.72, respectively, which differ from the commonly recommended values. Therefore, for the Priestley–Taylor model, the empirical coefficient α requires necessary adjustments based on the surrounding environment.

The substrate depth (SD) is a key factor affecting GR water dynamics and plant growth and significantly influences ET estimation (Nektarios et al. 2021; Schrieke et al. 2023). Not only that, Xing et al. (2021) have targeted the ET across the entirety of China by incorporating control of SM at various depths into soil evaporation and canopy transpiration, thereby updating the original Priestley Taylor–Jet Propulsion Laboratory algorithm to the Priestley Taylor–soil moisture evapotranspiration (PT-SM ET) algorithm. After assessment by 17 eddy covariance towers across different biomes in China, it was discovered that the inclusion of SM at different depths significantly enhanced the accuracy of ET estimation, particularly in areas where water is a limiting factor.

However, previous studies (e.g. Table 2) focus on the GR of a single SD, employing the default value of the Priestley–Taylor coefficient α for ET estimating, failing to fully consider the impact of different SDs on ET and how this impact is reflected in the Priestley–Taylor parameter α.

In practical applications, the design of GRs often needs to consider various SDs to accommodate different building structures and environmental conditions (Leite & Antunes 2023; Monteiro et al. 2023). Therefore, the gaps in existing research may lead to estimation accuracy uncertainty: The absence of α parameter calibration for specific GRs may result in reduced precision in ET estimation. This is crucial for predicting the hydrological benefits of GRs and designing optimal irrigation systems. Inaccurate ET estimations could lead to suboptimal water management strategies, potentially affecting the overall performance and economic efficiency of GRs.

This study employs a one-way analysis of variance (ANOVA) based on IBM SPSS Statistics (Version 24) (IBM Corporation Released 2016) to quantify and assess the impact of SD on ET from GRs. Should SD significantly influence AET, it implies that the Priestley–Taylor model may adjust the α-value rather than use the default value to improve its predictive accuracy. By comparing the Priestley–Taylor model's ET estimates before and after calibration and validation procedures with the AET from GRs of varying SDs, the following questions can be addressed:

• (i) What discrepancies exist between the AET estimated by the uncalibrated Priestley–Taylor model and the measured AET from GRs with different SDs?

• (ii) How well does the ET estimated by the Priestley–Taylor model fit the measured values after calibration and validation?

• (iii) How do the values of the calibrated Priestley–Taylor coefficient α change for AET estimation with different SDs?

The GR experimental design was structured using a parallel approach, where SD was the sole variable manipulated in this investigation. For extensive vegetated roofs, the SD typically varies from 50 to 150 mm (Kosareo & Ries 2007; Berndtsson 2010). In contrast, intensive GRs, which are characterized by a deeper substrate (exceeding 15 cm in thickness), are reserved for the most robust structures due to their capacity to support vegetation with deeper root systems (Ladani et al. 2019). The SD for intensive vegetated roofs is defined to be between 150 and 300 mm, according to Razzaghmanesh & Beecham (2014). In alignment with the construction criteria of GRs and the referenced literature, the SD treatments were standardized at 50, 150, and 300 mm for the purposes of this study, GR experimental parameters were shown in Table S2-1 (Supplementary Material).

The methodology consisted of three main steps:

• (i) Utilization of ANOVA to evaluate the effect of SD on AET and determine the need for detailed analysis at different SDs.

• (ii) Comparison of measured AET with the uncalibrated Priestley–Taylor model estimates to expose the limitations of the default parameter α across SDs.

• (iii) Calibration of the Priestley–Taylor model using the NSGA-II algorithm, with the calibrated model's predictions compared to measured AET to assess the impact of calibration.

The Priestley–Taylor model is the relationship between heat flux and evaporation, which is suitable when detailed underlying surface and aerodynamic measurements are not available. The Priestley–Taylor approach for estimating PET from a wet vegetated surface with minimal advection (Priestley & Taylor 1972). The parameters of the Priestley–Taylor model are listed in Table 3.

Table 3

The parameter of the Priestley–Taylor model

Parameter .	Description .	Unit .
Rn	Net radiation	MJ m−2 day−1
G0	Soil heat flux	MJ m−2 day−1
	The slope of the saturation vapor pressure–temperature relationship	kPa °C−1
γ	Psychrometric constant	kPa °C−1
	Priestley–Taylor empirical coefficient	–

Parameter .	Description .	Unit .
Rn	Net radiation	MJ m−2 day−1
G0	Soil heat flux	MJ m−2 day−1
	The slope of the saturation vapor pressure–temperature relationship	kPa °C−1
γ	Psychrometric constant	kPa °C−1
	Priestley–Taylor empirical coefficient	–

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(mm/day) is the equilibrium evaporation and is independent from windspeed and only based upon temperature and radiant energy (Stewart & Rouse 1976).

is the Priestley–Taylor empirical coefficient. Under humid conditions, the value of α may be less than 1, while under arid conditions, it may approach 2. The default value is 1.26, indicating a saturated surface or a wet scenario (Priestley & Taylor 1972; ASCE, Technical Committee on Standardization of Reference Evapotranspiration 2005).

The field capacity of the GR substrate is obtained by concurrent measurements of substrate moisture and discharge in the lysimeter. After each rainfall or irrigation-causing discharge, when discharge ceased, the remaining substrate moisture of the GR was assumed to be equal to the substrate's (Jahanfar et al. 2018). Transpiration is a biological process, and thus when substrate moisture drops below ⁠, we have estimated AET as zero (Jahanfar et al. 2018).

The NSGA-II model (Deb et al. 2002) was employed to identify the optimal Priestley–Taylor coefficient ⁠, which is the one that maximizes the Nash–Sutcliffe efficiency (NSE) and minimizes the root mean square error (RMSE) between the simulated and observed AET values.

In this study, equal weight was given to the F1 and F2 objective functions, with a weight of 0.5 assigned to both. To enhance the model's capability in selecting the optimal solution, a value of 2 was assigned to parameter p in Equation (10) (Goorani & Shabanlou 2021).

In this study, we conducted parallel experiments to collect empirical data and employed ANOVA to quantify the impact of SD on AET and its statistical significance, Table S2-1(Supplementary Material). This method enables the comparison of AET across varying SDs, ascertaining the criticality of SD in AET dynamics. The analysis informs the role of SD in GR design and management, aiding in the optimization of water resource management strategies (Morrison 2005; Warne 2014).

SD is a key characteristic that distinguishes GRs as an artificial greening facility from other terrestrial vegetation systems, such as farmlands and forests. Since GRs are constructed on top of buildings, they typically include one or multiple layers of substrate, whereas other terrestrial vegetation systems are directly situated on the ground, rendering the concept of SD inapplicable in these systems (Leite & Antunes 2023; Monteiro et al. 2023).

If SD significantly affects AET, this suggests that when simulating the ET of GRs using the Priestley–Taylor model, it may be necessary to adjust the α-value within the model, rather than simply using the default value.

From April 2021 to May 2023 the average temperature was 17.94 °C, with the highest temperature reaching 40.32 °C and the lowest temperature dropping to −6.87 °C, and the median temperature was 18.26 °C. The average humidity was 75.13%, with the highest humidity at 100.00% and the lowest at 18.40%, and the median humidity was 78.72%. The average precipitation was 0.18 mm/h, peaking at 30.54 mm/h for the highest precipitation and reaching 0.00 mm/h for the lowest. The average wind speed was 2.38 m/s, with the highest wind speed at 9.98 m/s and the lowest at 0.01 m/s, and the median wind speed was 2.13 m/s.

From 25 April 2021 to 26 April 2023, the average daily AET for SD of 300, 150, and 50 mm were 2.38, 2.27, and 1.96 mm, respectively. The maximum daily AET peaked at 7.11, 6.45, and 7.03 mm, while the minimum AET was 0.14, 0.01, and 0.01 mm, respectively. The results of measured AET are not in conflict with current research findings (Cascone et al. 2019; Ebrahimian et al. 2019).

ANOVA was employed to assess the statistical significance of differences in AET among experimental groups with varying SDs. Given the unequal variances in AET data across different SD groups (Table S3-1, Supplementary Material, significance = 0.00 < 0.05), a variance non-homogeneity method (Games–Howell) was selected for the ANOVA. The results are presented in Table S3-2 (Supplementary Material). The findings indicate that SD significantly influences the AET process on GR and this effect is statistically significant (F-value = 18.524, significance = 0.000 < 0.05). This result concurs with studies that agree that it is the SD that has a major influence on the AET of GRs (O'Carroll et al. 2023; Schrieke et al. 2023). Therefore, this implies that when using the Priestley–Taylor model to simulate the ET of GRs, employing default values may not yield equivalent simulation efficacy for all SDs. Further discussion is needed on whether adjustments to the α-value within the model are required to enhance the model's predictive accuracy.

The was calculated using Equation (7), and the results are presented in Table S3-3 (Supplementary Material). The table displays the seasonal and annual statistical data for aiming to provide a basis for model parameter calibration.

Under the condition of SD = 300 mm, the annual mean of is 1.97, with a maximum of 3.52 and a minimum of 0.07. The data for spring show a slightly higher mean than the annual average, at 2.04, while the means for summer and autumn decrease to 1.43 and 1.35, respectively, indicating seasonal variation. The mean for winter is close to the annual average, at 1.20. For SD = 150 mm, the annual mean, maximum, and minimum values of are 1.53, 4.99, and 0.00, respectively. The highest mean occurs in spring, reaching 1.75. Regarding SD = 50 mm, the annual mean of is 1.92, with a maximum of 3.40 and a minimum of 0.10. The highest mean is observed in spring, at 2.13, while the means for summer and autumn are 2.04 and 1.66, respectively, and winter drops to 1.79. From Table S3-3, the calibration range for α is determined to be 0–4.99.

In studies by Wadzuk et al. (2013), Marasco et al. (2015), and Mobilia et al. (2017), the Priestley–Taylor coefficient (α), which is essential for calculating ET in GRs, was utilized without the standard calibration and validation processes. The default value of 1.26 (Priestley & Taylor 1972; ASCE, Technical Committee on Standardization of Reference Evapotranspiration 2005) was adopted for the coefficient α.

In order to investigate the deviations between the Priestley–Taylor model's estimations of and the from GRs with different SDs, this section did not incorporate the calibration and validation processes; the default value of 1.26 was used for the Priestley–Taylor coefficient (α) in calculation. The results are presented in Table S3-4 (Supplementary Material).

Based on Table S3-4, the uncalibrated Priestley–Taylor model exhibits relatively good simulation performance for GR with an SD value of 150 mm, achieving an NSE value of 0.49, and an RMSE value of 0.91. This suggests that at a moderate SD, the model can capture the dynamics of ET with a reasonable degree of accuracy. These findings are consistent with the results reported by Marasco et al. (2015) and Mobilia et al. (2017).

However, for SD = 300 mm, there is a significant degradation in the model's performance, with an NSE value of 0.13 and an RMSE value of 1.32. This indicates that the uncalibrated model is unable to accurately predict ET under conditions of deeper substrate, possibly due to the neglect of the impact of SD on water retention capacity and plant transpiration. The discrepancy in performance between the two SDs highlights the necessity for model parameterization that accounts for the specific characteristics of the GRs, including SD, to improve the accuracy of ET predictions.

For SD = 50 mm, the Priestley–Taylor model yields an NSE value of 0.35 and an RMSE value of 1.02. Although these results are superior to those obtained for SD = 300 mm, they cannot be considered a good representation of what occurs in reality. This outcome indicates that while the Priestley–Taylor model may provide reasonable estimates of ET in certain scenarios, the variability in performance across different SDs suggests that the model parameter α requires calibration to suit specific conditions.

In summary, the uncalibrated Priestley–Taylor model has limitations when it comes to simulating ET in GRs, particularly when dealing with varying SDs. Therefore, it is recommended that future research should focus on calibrating the model parameters to better accommodate different GR designs and climatic conditions. Additionally, this sets the stage for a comparative assessment of the model's performance after calibration in subsequent sections, providing a more comprehensive understanding of the impact of parameter calibration on model performance.

In this section, calibration of the model parameters was conducted using daily AET over 348 days from 25 April 2021 to 25April 2022. Model validation was then performed on data over 349 days from 26 April 2022 to 26 April 2023. Results are presented in Table 4.

Table 4

The performance of the calibrated Priestley–Taylor model in simulating AET for SDs with NSE, RMSE

.	.	SD = 300 mm .	SD = 150 mm .	SD = 50 mm .
		1.86	1.54	1.71
Calibration	NSE	0.66	0.73	0.64
	RMSE	0.82	0.71	0.73
Validation	NSE	0.54	0.81	0.69
	RMSE	0.96	0.67	0.77

.	.	SD = 300 mm .	SD = 150 mm .	SD = 50 mm .
		1.86	1.54	1.71
Calibration	NSE	0.66	0.73	0.64
	RMSE	0.82	0.71	0.73
Validation	NSE	0.54	0.81	0.69
	RMSE	0.96	0.67	0.77

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From Table 4, the best simulation was observed for SD = 150 mm, with NSE values of 0.73 and 0.81, and RMSE values of 0.71 and 0.67 during calibration and validation.

Comparison between Table 3 and 4 reveal that calibration improved the simulation performance of the Priestley–Taylor model for all SDs. The calibrated Priestley–Taylor model provided improvement to the RMSE by 28 and 27% (1.02–0.73, 1.02–0.77), and NSE by 45 and 49% (0.64–0.35, 0.69–0.35), respectively, under SD = 50 mm. For SD = 150 mm, RMSE improved by 22 and 26% (0.91–0.71, 0.91–0.67); NSE improved by 33 and 40% (0.73–0.49, 0.81–0.49). For SD = 300 mm, errors were significantly reduced. RMSE improved by 38 and 27% (1.32–0.82, 1.32–0.96); NSE improved by 80 and 76% (0.66–0.13, 0.54–0.13).

For SD = 50 mm, the data points are primarily concentrated below the diagonal line, suggesting that the model tends to underestimate the AET. This may be due to the model's conservative estimation of certain factors affecting AET, such as SD, or a lack of sensitivity in the model's response to these factors.

Incorporating the analytical outcomes presented in Table 4 and Figure 4, we can deduce that although the calibrated Priestley–Taylor model demonstrates enhanced predictive capabilities for ET, variations in its performance across different SDs persist. These insights emphasize the critical need for tailored calibration of the model parameters, thereby charting a course for subsequent investigations aimed at model refinement and the augmentation of forecasting precision. Furthermore, this revelation prompts contemplation of the SD's influence when employing the model, suggesting that additional research is warranted to determine the most effective parameterization techniques under varying circumstances.

While this study has made significant contributions to calibrating the Priestley–Taylor model for estimating ET across different SDs in GRs, it is not without its limitations. First, the research was confined to a specific geographic location, which may restrict the generalizability of the findings to regions with varying meteorological conditions. Second, the study primarily focused on the impact of SD on ET, neglecting other factors such as vegetation type and soil properties that could significantly influence ET rates.

In light of these identified limitations, future research should expand its scope to include a variety of vegetation types and soil properties. This will aid in gaining a more comprehensive understanding of how different GR configurations interact with the performance of the Priestley–Taylor model. Here are some specific areas for future exploration:

• (i) Impact of vegetation types: Future studies should investigate the effects of different vegetation types on ET rates. The selection of plant species can alter water use efficiency and transpiration rates, which in turn affect the overall ET process. Comparative analysis of the ET contributions of different vegetation types can provide valuable insights for optimizing GR design.

• (ii) Soil properties: The physical and chemical properties of soil, such as texture, structure, and organic matter content, play a crucial role in water retention and release. Subsequent research should assess how these properties, especially in relation to different SDs, influence the accuracy of the Priestley–Taylor model and how the model can be adjusted accordingly.

• (iii) Long-term monitoring and model validation: Long-term monitoring of GRs is essential for validating and refining the Priestley–Taylor model under real-world conditions. This will help capture seasonal variations and long-term trends that may not be evident in shorter studies.

In conclusion, although this study has made progress in enhancing the applicability of the Priestley–Taylor model for GRs across different SDs, further research is needed to address the complexity of GR hydrology. By broadening the scope to include vegetation and soil dynamics and considering a wider environmental context, future research can make greater contributions to the development of more resilient and efficient GR systems.

Through an in-depth analysis of this study, the following key conclusions have been drawn:

• (i) For GRs with SDs of 300, 150, and 50 mm, the corresponding average daily AET values were 2.38, 2.27, and 1.96 mm, respectively. These findings suggest that SD has a significant impact on the AET rate, with a larger SD (300 mm) leading to the highest average AET under the study's conditions.

• (ii) The uncalibrated Priestley–Taylor model can provide relatively accurate ET estimation on GRs with a medium SD (150 mm). However, for deeper (300 mm) and shallower (50 mm) SDs, the model's predictive performance significantly declines, indicating that the model parameters are sensitive to SD.

• (iii) After calibrating the Priestley–Taylor model parameter α using the NSGA-II optimization algorithm and CP method, the model showed significantly improved performance across all tested SDs. Particularly on the 300 mm depth substrate, the performance enhancement after calibration was the most notable, underscoring the importance of parameter calibration in enhancing the accuracy of ET estimation.

• (iv) This study provides a new perspective for future research on ET in GRs, namely, by improving model parameter calibration methods to adapt to different environmental conditions. Future research could further explore the impact of various factors, including climate change, vegetation types, and urban microclimate, on model parameter calibration, and how to integrate these factors into the model for more precise and reliable ET prediction.

The authors express their gratitude for the financial support provided by the Nanxu Scholars Program for Young Scholars of ZJWEU (RC2023021227).

Data cannot be made publicly available; readers should contact the corresponding author for details.

The authors declare there is no conflict.