Uncertainty in Evapotranspiration Inputs Impacts Hydrological Modeling Open Access

Precipitation and evapotranspiration (ET) serve as pivotal inputs for conceptual hydrological models used in runoff (RO) modeling. The growing demand for precise predictions from hydrological models emphasizes the importance of obtaining more accurate estimates for these variables, ensuring they closely reflect the ‘true’ values (Rajput et al. 2023). While existing literature predominantly addresses the uncertainty in rainfall measurements, it tends to overlook errors in ET (Hong et al. 2006; Kavetski et al. 2006a; Vrugt et al. 2008; Villarini & Krajewski 2010). A prevailing belief in many studies is that hydrological model simulations are more responsive to variations in precipitation input and less sensitive to changes in ET inputs (Paturel et al. 1995; Bai et al. 2016; Essou et al. 2016; Laiti et al. 2018; Massmann 2020). However, contrasting findings exist, indicating a clear dependance of hydrological model performance on variations in ET data (Parmele 1972; Meyer et al. 1989; Nandakumar & Mein 1997; Andréassian et al. 2004). The debate surrounding the significance of addressing errors in ET and whether they should be prioritized or overlooked in the assessment of hydrological model performance remains a subject of discussion. Despite the prevailing trend of neglecting ET errors in the majority of studies, this research endeavors to fill the gap by conducting a comprehensive review of uncertainty quantification specifically related to ET errors in hydrological model simulations. This focus extends to the exclusion of other types of uncertainties, for which a substantial body of literature is already available.

Measuring actual ET in the field proves challenging, even with methods like lysimeters or flux towers, due to the substantial installation and maintenance costs associated with these equipment, as well as the time-consuming nature of the process. Furthermore, because lysimeters only provide point estimates, their large-scale application (in areas covered by different types of plants) has limitations. Therefore, rainfall-RO models make use of potential evapotranspiration (PET) instead of actual ET while simulating discharge from them. PET refers to the ET (sum total of evaporation from the field and transpiration from plants) when there is no limitation of water availability. On the other hand, reference ET is the rate of ET from a hypothetical reference crop (typically a well-watered, uniform grass cover) under specific standard meteorological conditions (e.g., temperature, humidity, wind speed, and solar radiation). Since both these concepts are closely related, reference ET estimated by various methods serves as input in the modeling process. The FAO–Penman–Monteith equation stands out as the most widely employed method for estimating reference ET (FAO-56 PM) (Allen et al. 1989; Rajput et al. 2023; Saranya & Vinish 2023). However, the dependency of this equation on a large number of parameters and the variety of units used for these parameters are cited as limitations of this equation (Nourani et al. 2019; Chen et al. 2020; Talebmorad et al. 2020). Researchers have endeavored to streamline reference ET estimation by relying on fewer meteorological parameters, such as temperature and radiation, in methodologies such as those of Hamon (1961); Jensen & Haise (1963); Hargreaves & Samani (1985); Oudin et al. (2006). Hence, addressing uncertainty in ET inputs involves either considering uncertainty in the ET input data series directly or accounting for uncertainty in the parameters or weather variables from which ET is derived. The Penman–Monteith FAO-56 equation utilizes variables such as relative humidity, temperature, wind speed, and solar radiation to calculate ET. Certain studies have taken into consideration the uncertainty associated with individual parameters such as temperature, relative humidity, wind speed, and solar radiation in the computation of ET using the Penman method (Meyer et al. 1989; Raleigh et al. 2015; Günther et al. 2019). The findings of their experiments suggest that ET estimates were more dependent on relative humidity (RH) and solar radiations than on wind and temperature data. Additionally, investigations employing temperature-based ET equations consider errors in temperature, assessing their impact on the performance of lumped hydrological models (Weerts & El Serafy 2006; Steinschneider et al. 2012; Lü et al. 2016; Tajiki et al. 2020). In the majority of the research addressing input uncertainties, the uncertainties associated with ET data have been overlooked. Among the 290 papers reviewed, which primarily concentrate on input uncertainties, only 3% briefly mentioned ET uncertainties, and even those mentions were not comprehensive. Furthermore, the interaction between ET errors and predicted RO is significantly affected by climatic conditions, especially the rainfall time series. In wet climates, the soil water deficit (SWD) preceding rainfall is closely linked to potential ET. Positive ET errors (underestimated actual ET) increase SWD, reducing RO, while negative ET errors (overestimated actual ET) decrease SWD, increasing RO. This highlights the importance of accurate ET estimations in these regions. Conversely, in dry climates, time between rainfall events often allows SWD to reach maximum capacity, limiting the effect of ET errors on RO predictions. When SWD is maximized, variations in ET have minimal impact on RO generation. Overall, while ET errors significantly influence RO in wet climates, their impact is diminished in dry climates. This underscores the need for studying errors in ET particularly, in case of extreme climatic events, enhancing prediction accuracy, and informing effective water management practices.

Tables 1 and 2 detail the studies that have delved into ET uncertainties. Hence, there is a clear necessity for a comprehensive examination of the uncertainties inherent in ET data. In this work, we will provide a review of ET input uncertainty and its impact on hydrological modeling, an aspect that has received relatively limited attention in existing literature. The main objectives of this study are to (a) conduct a thorough analysis of the literature's existing ET error models; (b) determine if ET errors actually affect RO from hydrological models; (c) determine the effects of systematic and random errors on model performance and outputs; and (d) determine the necessity of incorporating ET errors into future research endeavors. The goal is to guide readers in selecting an appropriate error model when addressing ET uncertainties in hydrological models as well as to assess its potential impacts on a modeling process.

Table 1

Random error models for PET estimation employed in literature

S.No. .	ET error model .	Error distribution .	Reference .	Reason/Result .
1.			Parmele (1972)	(a) Multiplicative form of error model helps to maintain the heteroscedastic nature of the error (higher deviation in higher observed values) (b) The impact of the random error component was overshadowed by the presence of the fixed error in the estimation of PET
2.			Weerts & El Serafy (2006)	The choice of a multiplicative model over an additive one was motivated by the derivation of hourly evaporation values from the long-term monthly mean values of evaporation
3.			Fan et al. (2015)	Constant variance makes computations easy
4.			Salamon & Feyen (2010)	Multiplicative error models provide a straightforward method for integrating error sources. Subsequently, the posterior distributions of the multipliers can be utilized to assess the extent of error or identify any bias in the error source
5.			Oudin et al. (2006)	Exponential error model was incorporated for taking PET errors into account
6.			Weerts & El Serafy (2006)	Higher variance was chosen because temperature Errors have relatively small impact on RO compared with errors in rainfall
7.			Tajiki et al. (2020)
8.			Lü et al. (2016); Steinschneider et al. (2012)
9.			Droogers & Allen (2002)	Errors were introduced into the parameters employed for calculating PET rather than being directly applied to PET itself
10.			Meyer et al. (1989)	Solar radiation and relative humidity exert a greater influence on PET estimates compared to wind speed and temperature

S.No. .	ET error model .	Error distribution .	Reference .	Reason/Result .
1.			Parmele (1972)	(a) Multiplicative form of error model helps to maintain the heteroscedastic nature of the error (higher deviation in higher observed values) (b) The impact of the random error component was overshadowed by the presence of the fixed error in the estimation of PET
2.			Weerts & El Serafy (2006)	The choice of a multiplicative model over an additive one was motivated by the derivation of hourly evaporation values from the long-term monthly mean values of evaporation
3.			Fan et al. (2015)	Constant variance makes computations easy
4.			Salamon & Feyen (2010)	Multiplicative error models provide a straightforward method for integrating error sources. Subsequently, the posterior distributions of the multipliers can be utilized to assess the extent of error or identify any bias in the error source
5.			Oudin et al. (2006)	Exponential error model was incorporated for taking PET errors into account
6.			Weerts & El Serafy (2006)	Higher variance was chosen because temperature Errors have relatively small impact on RO compared with errors in rainfall
7.			Tajiki et al. (2020)
8.			Lü et al. (2016); Steinschneider et al. (2012)
9.			Droogers & Allen (2002)	Errors were introduced into the parameters employed for calculating PET rather than being directly applied to PET itself
10.			Meyer et al. (1989)	Solar radiation and relative humidity exert a greater influence on PET estimates compared to wind speed and temperature

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Table 2

Systematic error models for PET estimation employed in literature

S.No. .	ET error model .	Error distribution .	Reference .	Results .
1.			Parmele (1972)	Negative bias causes more impact on streamflow than positive bias. Random errors are overshadowed by systematic errors in PET
2.			Oudin et al. (2006)	Hydrological models are insensitive to random errors in PET series The performance of the models was typically more impacted when PET was overestimated compared to when it was underestimated
3.			Nandakumar & Mein (1997)	10% bias in PET resulted in 6–10% bias in RO simulations
4.			Jayathilake & Smith (2022)	The model's performance exhibited greater sensitivity to negative biases in PET than to positive biases in mean monthly PET-based datasets. The sensitivity of the model's performance became more pronounced, with notable changes (e.g., a shift in Kling-Gupta efficiency (KGE) of ≥ 15%), starting from a 10% PET error
5.	for precipitation for ET		Wang et al. (2023)	The model's performance deteriorates when precipitation (P) and PET exhibit opposing biases. The overestimation of inputs has a more pronounced and severe impact on the streamflow (Q) outputs

S.No. .	ET error model .	Error distribution .	Reference .	Results .
1.			Parmele (1972)	Negative bias causes more impact on streamflow than positive bias. Random errors are overshadowed by systematic errors in PET
2.			Oudin et al. (2006)	Hydrological models are insensitive to random errors in PET series The performance of the models was typically more impacted when PET was overestimated compared to when it was underestimated
3.			Nandakumar & Mein (1997)	10% bias in PET resulted in 6–10% bias in RO simulations
4.			Jayathilake & Smith (2022)	The model's performance exhibited greater sensitivity to negative biases in PET than to positive biases in mean monthly PET-based datasets. The sensitivity of the model's performance became more pronounced, with notable changes (e.g., a shift in Kling-Gupta efficiency (KGE) of ≥ 15%), starting from a 10% PET error
5.	for precipitation for ET		Wang et al. (2023)	The model's performance deteriorates when precipitation (P) and PET exhibit opposing biases. The overestimation of inputs has a more pronounced and severe impact on the streamflow (Q) outputs

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Input uncertainty, specifically in variables such as rainfall, ET, and discharge, is a significant factor. Rainfall uncertainty arises from sampling and measurement errors (Sokolovskaya et al. 2023). Typically, rainfall is represented by measurements at specific spatial points within a watershed, which may not accurately reflect the annual average rainfall for the entire watershed. Moreover, temporal errors in rainfall measurements contribute to uncertainty. Many hydrological RO models rely on PET instead of actual ET because data on actual ET are often limited across numerous locations. Furthermore, while RO is primarily driven by precipitation, literature has primarily focused on errors in rainfall data, neglecting uncertainties associated with ET. However, due to limitations in estimation methods, lack of real ET data from field measurements, measurement inaccuracies, and spatial variations in ET based on soil and crop types, errors in ET cannot be overlooked. However, errors in ET can significantly impact the RO process. These inaccuracies in the estimation of ET can have a significant effect on the RO predictions due to their direct influence on the water balance within a hydrological system. Overestimating ET reduces the water available for surface RO, resulting in lower RO predictions, while underestimating ET leads to higher predictions due to inflated water availability. Thus, precise ET measurements are essential for reliable RO forecasting, as errors in ET can adversely affect the accuracy of hydrological models. Therefore, it is crucial to consider errors in ET alongside those in rainfall when modeling RO.

Streamflow or RO, usually measured at a single point at the watershed outlet, introduces errors stemming from stage measurement, streamflow measurement, rating-curve inaccuracies, and errors associated with extrapolation to regions with sparse stage measurements (Montanari & Di Baldassarre 2013; McMillan et al. 2018).

Parametric uncertainty emerges from the difficulty in independently measuring or precisely specifying model parameters, requiring their estimation through calibration against observed data (Kuczera 1983). The parameters in hydrological models characterize physical properties or features of the watershed, and their accurate estimation is essential for the model's performance. Factors contributing to parameter uncertainty include challenges in measuring or estimating certain the physical properties, temporal variations in parameters, and the scale-dependent nature of some parameters. Inadequate calibration procedures can also be a source of parameter uncertainty, and the presence of observation errors further compounds this uncertainty within a modeling system.

Structural uncertainty pertains to uncertainties associated with the structure or formulation of the hydrological model itself (Thyer et al. 2009). This uncertainty arises because the actual processes governing hydrological phenomena may not be fully understood or accurately represented by the chosen model structure. The lack of knowledge about underlying processes, simplifications in the model structure, and the inherent complexity of hydrological systems all contribute to structural uncertainty. Different model structures may produce varying results, posing a challenge in selecting an appropriate model structure. Various types of uncertainties in the hydrological modeling of RO, along with the sources of these errors, have been depicted in Figure 1.

In our study, we chose a multiplicative homoscedastic (constant variance) error model to represent errors in ET, with a known mean and variance for the study catchment, as depicted in Figure 2. Using this model, we generated a fictitious daily timescale ET series for 12 years. The generated series was used to calibrate different hydrological models, resulting in different parameter sets during calibration. For each parameter set for each model, we simulated discharges during the validation period, yielding a series of daily simulated discharges. From these simulations, the mean predicted discharge, as well as the 2.5th and 97.5th percentile discharges for each day, can be estimated. This process allows to establish a 95% prediction interval for discharges considering ET errors. Finally, the 95% confidence intervals plotted from ET errors illustrate the uncertainty in RO due to errors in ET only.

Hydrologists and modelers need to acknowledge and address these uncertainties to enhance the reliability of hydrological predictions and better inform water resource management decisions. While the hydrological community widely acknowledges the substantial uncertainties introduced by hydrological models, a significant number of studies in the hydrological literature tend to overlook the quantification of these uncertainties. Both practical and purely scientific modeling endeavors require a quantitative recognition of uncertainty. In practical applications, conducting uncertainty analysis becomes crucial for making well-informed decisions and enhancing predictions from hydrological models. The quantification of uncertainty is essential for understanding the amount of information needed to characterize system behavior, especially in water management applications. This approach can lead to robust risk analysis, cost reduction, and the generation of transparent results that enhance public trust and the confidence of water managers. Below are the main advantages of uncertainty analysis:

• 1. Uncertainty quantification aids in the better identification of catchment processes. Additionally, it has been observed that neglecting model structural uncertainty may result in the non-identifiability of catchment model parameters if not appropriately considered. The interpretation of catchment dynamics is affected by uncertainties, as emphasized by Wagener & Wheater (2006).

• 2. It facilitates improved flow predictions from hydrological models, enabling decision-makers to make appropriate releases.

• 3. Incorporating data uncertainty in water management has proven to lower costs and lead to better decision-making (Montgomery & Sanders 1986; McMillan et al. 2017, 2018).

Quantifying uncertainties in conceptual rainfall-RO models remains an intricate challenge in hydrological modeling. The literature has seen significant strides in addressing errors, with notable contributions from researchers (Kavetski et al. 2002, 2006a, 2006b; Kuczera et al. 2006; Vrugt et al. 2009; Villarini & Krajewski 2010). However, these studies have tended to overlook certain types of uncertainties. Several studies have adopted a multiplicative form of error models to address uncertainty in inputs such as rainfall and ET depths, as demonstrated by works such as Kuczera (1983); Kavetski et al. (2002); Kuczera et al. (2006), (2007); Weerts & El Serafy (2006); Vrugt et al. (2008); Thyer et al. (2009); Salamon & Feyen (2010), as expressed in Equation (1). The true input rainfall/ET depth is thus represented through multiplicative terms. Multiplicative terms are latent variables or multipliers associated with a specific storm index (Kavetski et al. 2006a; Kuczera et al. 2006). Multipliers are generated using the Monte Carlo approach which is a stochastic simulation method used for quantifying input uncertainty in hydrological modeling and involves generating multiple sets of input parameters by randomly sampling from their probability distributions. The application of a multiplicative error model offers the advantage of avoiding the introduction of new dimensions to uncertainty estimations. The multipliers are typically sampled from known probability distributions such as uniform, normal, and log-normal. Table 1 outlines various error distributions used for PET errors, primarily focusing on additive and multiplicative error models. In an additive model, errors are constant across prediction levels, represented as Y = f(X) + ɛ, where ɛ is the error term. A key drawback of the additive model is its inability to account for varying error magnitudes across different prediction levels, which can lead to underestimating uncertainty, especially in rainfall data with heteroscedasticity. In contrast, the multiplicative model expresses errors as a proportional factor of the predicted value, represented as Y = f(X)⋅(1 + ɛ), resulting in larger errors for greater predicted values. However, multiplicative models do not effectively address zero-depth rainfall, which is significant for non-rainy days. The choice between these models should be informed by the data characteristics and the specific analytical context.

A commonly used assumption for quantifying model parameter errors is that these parameters follow a uniform probability distribution. This assumption serves as the basis for setting initial values for adjusting model parameters to achieve the desired performance, as suggested by Beven and Binley in 1992. One widely employed method for parameter uncertainty analysis is the generalized likelihood uncertainty estimation (GLUE), which considers the equifinality hypothesis (Beven & Binley 1992; Beven 2006). The equifinality hypothesis suggests that there are multiple parameter sets that can equally describe hydrological processes or produce the same final result. GLUE combines Monte Carlo simulations with a behavioral threshold measure that distinguishes between hydrologically acceptable and unacceptable parameters and structures. Despite its widespread use, GLUE has faced criticism for its subjectivity in selecting a behavioral threshold and its lack of a formal statistical foundation. Following the GLUE framework, several contributions have sought to formally quantify parameter uncertainty, predominantly relying on Bayesian statistics (Kuczera & Parent 1998; Bates & Campbell 2001; Thiemann et al. 2001; Vrugt et al. 2009). The primary advantage of formal Bayesian statistics is not only the quantification of parameter uncertainty but also the potential reduction of uncertainty through the inclusion of prior knowledge. The latest addition to formal approaches is differential evolution adaptive multi-objective algorithm (DREAM) (Vrugt 2016), widely applied due to its unique combination of the differential evolution algorithm with the adaptive Markov chain Monte Carlo (MCMC) approach (Hastings & Hastings 1970; Storn & Price 1997; Haario et al. 1999). The differential evolution enables efficient exploration of non-linear and discontinuous parameter spaces, while the adaptive MCMC component keeps it within the parameter posterior space.

In contrast to other MCMC schemes, DREAM utilizes multiple chains to exchange information about the sampling space instead of verifying the achievement of the stationary state (Viglione et al. 2007). Nevertheless, a notable difficulty encountered by DREAM and similar Bayesian methods lies in identifying a likelihood function that produces homoscedastic residuals, necessitating the reliance on a robust assumption for a formal likelihood function. This challenge becomes particularly pronounced in cases where observations of streamflow exhibit a skewed distribution, with a predominance of low-flow events coupled with occasional high-flow events.

Quantifying model structural error poses a significant challenge, given the recognition that a singular model structure is insufficient to comprehensively represent all hydrological processes within a watershed (Carrera & Neuman 1986; Gupta & Govindaraju 2023). Different models exhibit varying degrees of proficiency in simulating specific physical processes of the catchment. Consequently, a comparative analysis involving multiple models becomes essential to evaluate uncertainties arising from diverse model structures (Tian et al. 2014). Multi-model averaging has been applied to quantify and reduce structural uncertainty. One extensively employed method for weighted multi-model averaging is Bayesian model averaging (BMA) (Raftery et al. 1997, 2005; Hoeting et al. 1999). BMA and similar Bayesian-based averaging techniques offer advantages due to their interpretable weights, derived from the model posterior performance that combines the model's fitting ability to observations with experts' prior knowledge. Despite its widespread application, BMA faces limitations in addressing structural uncertainty. For instance, despite variations in model performance across hydrograph segments (Duan et al. 2007), RO seasons (Son & Sivapalan 2007), or catchment circumstances (Marshall et al. 2007), BMA assigns constant weights to component models. Additionally, enhancing BMA's performance has been observed when assigning different weights to component models for various quantiles of a hydrograph. Structural uncertainty is also considered in the framework for understanding structural errors (FUSE) (Clark et al. 2008), which investigates this uncertainty by rearranging and combining components of various hydrological models representing alternative formulations of processes. FUSE shares similarities with Monte Carlo (or Bootstrap) methods, but in FUSE, the samples consist of components of a model representing diverse conceptualizations of alternative processes. Additionally, FUSE is advantageous for addressing structural uncertainty due to its ease of parallelization.

A primary criticism common to all multi-model uncertainty analyses is the challenge of identifying and incorporating the entire spectrum of feasible model structures. While the aforementioned methods can accommodate numerous structures, the practical difficulty of identifying and exploring the entire structural space limits the scope of the investigation. Consequently, a comprehensive examination of structural uncertainty is constrained.

Figure 4 demonstrates that the estimation of ET is complicated by errors that can be categorized into systematic and random errors. Systematic errors are biases that consistently affect measurements in one direction, often arising from miscalibrated instruments, incorrect model assumptions, or unaccounted environmental conditions. In contrast, random errors are unpredictable variations caused by factors like measurement noise or environmental fluctuations. To characterize these errors, researchers commonly use additive and multiplicative error models. The additive error model expresses observed ET as the true ET plus a random error term, while the multiplicative error model represents observed ET as the true ET multiplied by a random error factor. The ET estimates, along with their associated errors, are utilized for both model calibration and validation in hydrological modeling to assess the impact of ET errors on RO predictions. By incorporating robust error models into hydrological research, researchers can enhance model calibration, conduct uncertainty analyses, and ultimately improve predictions of water availability, which is essential for sustainable resource management.

There exists considerable ambiguity regarding the choice of an error model when addressing uncertainties in ET data. Surprisingly, the literature has limited studies that account for errors specifically related to ET. In our review of studies on uncertainty analysis in hydrological models, only a handful have considered errors associated with ET as detailed in Tables 1 and 2. There seems to be a lack of exploration in research regarding the sensitivity of watershed models to errors in ET.

Some studies in the literature have asserted that the influence of systematic errors in PET is more pronounced than that of random errors on the responses of hydrological models, as mentioned in Figure 5. Unlike random errors, which tend to cancel each other out over time, systematic errors consistently skew the data in one direction. When systematic errors are overestimated, meaning they are larger than they actually are, the model becomes biased toward these erroneous data. This bias can lead to inaccurate representations of the actual hydrological processes, resulting in poor model performance. Consequently, certain investigations have concentrated solely on addressing systematic errors, as they tend to dominate the impact over random errors when both types of errors are considered, as mentioned in Table 2. The first study was presented by Parmele (1972) who played a pioneering role in investigating the effects of both random and systematic biases in PET input data on streamflow simulated by hydrological models. Parmele (1972) discovered that a consistent 20% bias in PET inputs led to a notable error (2–7%) in RO simulations. Additionally, it was demonstrated that the influence of random PET errors on model performance was less significant compared to systematic PET errors. Furthermore, the study highlighted that the impact of underestimated PET was considerably more substantial than overestimated PET. The influence of random and systematic errors on both performance and parameter values of rainfall-RO models was studied and it was found that systematic errors in PET series have a greater impact than random error (Nandakumar & Mein 1997; Oudin et al. 2006). Moreover, the findings also verified that the efficiency of hydrological models appeared to be more adversely affected by the underestimation of PET than by its overestimation (Nandakumar & Mein 1997). In a recent study conducted in 2023, researchers explored the combined effects of systematic errors in both precipitation (P) and PET on hydrological model simulations (Wang et al. 2023). The findings indicated that the simulations were more significantly influenced by the overestimation of P and PET, as opposed to the impact observed with underestimation substantiating previous findings in the literature. Table 1 outlines the incorporation of random errors, while Table 2 delves into the inclusion of systematic errors. Based on our review of the literature, we recommend addressing systematic errors in future simulation studies, considering their greater impact on RO simulations.

is parameter such as radiation and humidity, windspeed, maximum temperature, and minimum temperature that are used to estimate PET. is a Gaussian random variate, is observed variable standard deviation, and stands for change/error in the variable under study. The superscript ‘true’ represents the corrupted series of the variable, which serves as the true series during model calibration to assess its impact on RO predictions. N represents the normal probability distribution, from which samples are drawn using a specified mean (μ), and standard deviation (σ). Conversely, U denotes the uniform probability distribution, where samples are extracted from a defined interval [a, b], ensuring that each value within this range has an equal probability of selection.

This study delves into an analysis conducted on a densely gauged watershed in the United States, managed by the US Geological Survey (USGS). We investigated the effects of both systematic and random errors on the watershed, corroborating previous findings that random errors may be negligible in uncertainty analyses of hydrological simulation models. Additionally, while a considerable amount of literature exists on quantifying various sources of uncertainty in watershed modeling, this research focuses specifically on the uncertainty surrounding ET.

The NSE values shown in Figure 7 are plotted for the validation period (daily data) from 01 January 2018, to 31 December 2022, reflecting the model's performance when subjected to ET errors. The ‘hydromad’ package in R software was utilized to calibrate the hydrological models GR4J, HYMOD, and BUCKET. The ‘hydromad package’ requires input data on a daily timestep.

Figure 8 demonstrates a close alignment between the predicted and observed RO values, indicating that the model effectively predicts RO. Notably, the peaks of the hydrographs are also accurately represented. Among the three hydrological models assessed, HYMOD performed the best, which is why the observed versus predicted RO plot is specifically presented for this model. From Figure 8, we can conclude that lumped models exhibit a strong capability for forecasting future RO in catchments. While examining the impact of ET errors on RO predictions from different models was not the focus of our study, it presents an opportunity for future research. Additionally, understanding the influence of ET errors on extreme climate events would provide valuable insights for water resource managers aiming for sustainable strategies.

The study underscores the importance of considering errors in ET alongside rainfall uncertainties in hydrological modeling. Numerous investigations have demonstrated the substantial impact of ET errors on both model predictions and parameter identification in hydrological studies. Consequently, overlooking errors arising from ET in flow simulations is not advisable. Furthermore, research has shown that systematic errors in ET data have a more pronounced effect than random errors, leading to a decline in model performance when errors are overestimated. Hence, it is recommended to incorporate systematic errors up to 30% in catchment studies. Adopting a multiplicative error model for PET with errors following a normal distribution in the case of random errors yields favorable results. In addition, the paper provides valuable insights into various strategies for addressing uncertainties related to PET inputs. This includes direct uncertainties stemming from PET data series as well as those associated with model parameters and structure. By exploring a range of methodologies and existing research findings, the study contributes to a deeper understanding of uncertainty quantification in hydrological modeling. Overall, the research underscores the importance of taking a nuanced approach to tackle the diverse sources of uncertainty present in this field. By acknowledging and integrating both systematic and random errors, researchers can significantly enhance the robustness of hydrological models, ultimately leading to more reliable predictions and informed decision-making in water resource management and environmental conservation.

The authors express their sincere appreciation to the Institute Authorities overall, and specifically to the Civil Engineering Department, for their indispensable support in facilitating the execution of this research.

M.A. and M.A.N. conceptualized the process; M.A. wrote the original draft; M.A., M.A.N., and M.A.A. wrote the review and edited the article; M.N. and M.A. visualized the study; M.A.N. supervised the whole work. All authors have read and agreed to the published version of the manuscript.

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

The authors declare there is no conflict.