Model Parameter Estimation Experiment Research Articles

Budyko-based past and future disaggregation of climate and catchment effects on streamflow changes

Budyko-based approaches are widely used to separate climate and catchment effects on streamflow changes. Considering the potential changes in streamflow, such separation is essential for future water resources management. In this study, we propose a method that allows for continuous disaggregation of climate and catchment impacts from the past to the future. We utilize a Generalized Linear Model to represent catchment parameters based on climate and catchment features. Subsequently, we determine the future positions of the catchment in the Budyko-state space. This framework is applied to four Model Parameter Estimation Experiment (MOPEX) catchments with varying aridity indices in the USA. The results indicate that, in the recent historical period (1971-2011), climate change has contributed to increased streamflow in three catchments compared to the baseline (1948-1970) period. However, future projections indicate a mixed pattern of contributions in two of these catchments due to increased potential evapotranspiration, resulting in decreased streamflow.

A Differentiable Hydrology Approach for Modeling With Time‐Varying Parameters

AbstractIn the analysis of observational data with mathematical models, it is often desirable to let one or more model parameters vary with time to account for changing environmental conditions or to offer more flexibility. A standard approach for dynamic models is to allow for the number of parameters to grow as a function of time, leading to a high‐dimensional inference problem for long data records. We propose using Hamiltonian Monte Carlo, a gradient‐based Markov chain Monte Carlo (MCMC) method, to perform Bayesian inference for time‐varying parameters (TVP) in hydrology models. As derivatives of model error functions with regard to parameters are not available in closed form, we implemented the GR4J rainfall‐runoff models in Theano/PyMC3, and Jax/NumPyro, allowing for the application of automatic differentiation techniques. A simulation experiment assessing the viability of these methods for recovering underlying temporal variation in parameters indicates that a discrete Gaussian random walk‐based prior appears to be best suited across a range of scenarios while the continuous‐time Gaussian process compares relatively poorly. In an analysis of 20 years of daily real‐world streamflow records from the Model Parameter Estimation Experiment (MOPEX) hydrology data set with increasing data sparsity and at multiple temporal resolutions, we find that TVP inference with gradient‐based MCMC is a flexible and reliable approach for analyzing hydrology models with dynamic parameter sets.

Towards an Extension of the Model Conditional Processor: Predictive Uncertainty Quantification of Monthly Streamflow via Gaussian Mixture Models and Clusters

This research develops an extension of the Model Conditional Processor (MCP), which merges clusters with Gaussian mixture models to offer an alternative solution to manage heteroscedastic errors. The new method is called the Gaussian mixture clustering post-processor (GMCP). The results of the proposed post-processor were compared to the traditional MCP and MCP using a truncated Normal distribution (MCPt) by applying multiple deterministic and probabilistic verification indices. This research also assesses the GMCP’s capacity to estimate the predictive uncertainty of the monthly streamflow under different climate conditions in the “Second Workshop on Model Parameter Estimation Experiment” (MOPEX) catchments distributed in the SE part of the USA. The results indicate that all three post-processors showed promising results. However, the GMCP post-processor has shown significant potential in generating more reliable, sharp, and accurate monthly streamflow predictions than the MCP and MCPt methods, especially in dry catchments. Moreover, the MCP and MCPt provided similar performances for monthly streamflow and better performances in wet catchments than in dry catchments. The GMCP constitutes a promising solution to handle heteroscedastic errors in monthly streamflow, therefore moving towards a more realistic monthly hydrological prediction to support effective decision-making in planning and managing water resources.

Probabilistic Budyko-based Separation of Climate and Catchment Effects on Streamflow

Streamflow variability is affected by both climate change and direct human impacts within the catchment. Quantification of the role of these two factors have significant implications for water resources. In recent years, Budyko-based analytical methods have been widely used for such quantification due to their simplicity and efficacy. This paper presents a comparison of such methods to attribute changes in streamflow in four Model Parameter Estimation Experiment (MOPEX) catchments in the contiguous United States. Further, a new, improved Two-stage Decomposition method to separate the role of climate and catchment changes on streamflow is proposed, considering the uncertainty in the catchment parameter that further leads to a range of evaporative index values within a catchment. We find that climate and catchment effects are sensitive to the choice of Budyko equation, disaggregation algorithm and approximation technique. Further, the catchment parameter can be expressed probabilistically, to account for the uncertainty because of limited hydro-meteorological records. The proposed approach employs a bootstrap analysis to obtain the stochastic range of the catchment parameter, considering different choices of Budyko equations. We argue that rather than relying on a single Budyko-based approach for separation of climate and catchment effects on streamflow, the proposed method offers a more robust way by considering the entire range of uncertainty of the catchment parameter.

Automatic procedure for selecting flood events and identifying flood characteristics from daily streamflow data

The selection of flood events and determination of flood characteristics (e.g. start and end dates, peak discharge, volume, and duration) are the first critical steps for flood analyses. To obtain the key flood information accurately, we used the automatic peak over threshold (POT) model for flood sampling and proposed an automatic approach to determine flood characteristics using the master recession curve analysis (MRC) method. We further developed a graphical user interface (GUI) and toolbox for this procedure in MATLAB. Model parameter estimation experiment (MOPEX) data from 423 stations were used to evaluate the proposed method. Our results suggest that the proposed procedure performs well for watersheds with diverse characteristics. The developed toolbox can be conveniently applied to other watersheds for flood sampling and the characterisation of flood events, thus helping reduce the uncertainty in subsequent flood analyses, such as multivariable flood frequency and trend analyses.

Role of hydrological model structure in the assimilation of soil moisture for streamflow prediction

Soil moisture data assimilation (SMDA) has found growing application in the hydrological research community because of readily available satellite-based soil moisture (SM) data. Several past studies have explored the capability to assimilate observed SM in hydrological model to enhance streamflow prediction. However, the impact of the conceptual hydrological model (CHM) structure on the SMDA in streamflow prediction has not been investigated yet. In this study, to understand the CHM structure's role, we used three different CHMs for the SMDA: dynamic Budyko (DB), Génie Rural à 4 paramètres Journalier (GR4J), and Probability Distributed Model (PDM) model. The SM obtained from Global Land Data Assimilation System (GLDAS) was assimilated using Ensemble Kalman Filter (EnKF) for 43 Model Parameter Estimation Experiment (MOPEX) basins. The GR4J model was found to be performed best from calibration and validation results, and the DB model was found to be performed least. The results show that the performance of the DB model improved during assimilation compared to its open-loop version for all the 43 basins. However, deterioration in model performance was observed for the GR4J and PDM model during assimilation for all basins except a few. The assimilated model performance was evaluated in respect of Assimilation Efficiency (AE) and ranged from 2.56 to 81.24%, −70.9 to 17.65%, and −71.38 to 24.86%, respectively, for the DB, GR4J, and PDM model. Further, we hypothesized that if the GLDAS SM is better than the model-simulated SM, then improvement in the model performance is observed due to the SMDA. The coefficient of determination (r2) between the SM simulated by all the three models without assimilation and the GLDAS SM was found for all the basins. Results indicated that the GR4J and PDM model captured SM better than the DB model. Moreover, to strengthen this hypothesis, we run the GR4J and PDM model deterministically, considering the daily GLDAS SM as the initial condition for streamflow simulation instead of the SM simulated by these models for the basins where deterioration in model performance was observed. For all the basins, it is found that the GR4J and PDM model performance deteriorated. Even though the GLDAS SM is used as every day's initial condition to simulate the streamflow, no improvement of model performance is observed. In general, our study underlines the significance of the CHM structure in the SMDA exercise.

Simulating runoff under changing climatic conditions: A comparison of the long short-term memory network with two conceptual hydrologic models

Hydrologic models are commonly used to assess climate change impact on water resources. Several studies have reported that hydrologic models often experience severe performance degradation under climatic conditions different from calibration periods. With the advancement of artificial intelligence technology, the long short-term memory (LSTM) network has recently shown great potentials in rainfall-runoff modeling. However, little is known about the robustness of the LSTM network when used in changing climatic conditions. In this study, we compare the robustness of the LSTM network and two conceptual hydrologic models in runoff prediction in changing climatic conditions in 278 Model Parameter Estimation Experiment (MOPEX) basins. For calibration periods, the two hydrologic models have better performance in wet periods than in dry periods, while the LSTM network shows little performance difference under different climatic conditions. For validation periods, the three models suffer the largest performance loss when calibrated in a wet period and validated in a dry period. The performance losses of the LSTM network are primarily affected by the climate contrast between calibration and validation periods, while the performance losses of the two hydrologic models are mainly dependent on the climatic condition of validation periods. We also find that the length of the calibration period is an important factor affecting the relative performance of the models. Increasing the length of the calibration period has little effect on the validation performance of the two hydrologic models but enhances the LSTM network's performance. If sufficient calibration data is available, the LSTM network is a preferred tool for runoff simulation. On the other hand, the hydrologic models could have more advantages over the LSTM network in case of limited calibration data available.

Predictability of Monthly Streamflow Time Series and its Relationship with Basin Characteristics: an Empirical Study Based on the MOPEX Basins

Machine learning (ML) models have been applied to monthly streamflow forecasting in recent decades. In this study, forecasting skills of eight ML models are evaluated based on the Model Parameter Estimation Experiment (MOPEX) dataset. We consider two skill scores, i.e., the Nash–Sutcliffe efficiency (NSE) and the adjusted NSE (ANSE), and the latter is the skill score based on the interannual mean monthly value (MMV) as the reference (benchmark) model. Furthermore, NSE of the MMV model (NSEmmv) is used as a measure of the seasonality of monthly streamflow, as it is the ratio of variance explained by the MMV process. An important result is that forecasting skills of ML models for monthly streamflow are largely controlled by NSEmmv. Moreover, based on comparisons of different ML models, we have found that the selection of models is not a dominating factor impacting the final skill. Three key factors influencing NSE, i.e., NSEmmv, the base flow index (BFI) and the aridity index (AI), are explored in this paper. Specifically, NSEmmv impacts NSE directly and is the predominant factor; BFI influences the memory of the monthly streamflow and therefore influences NSE. The relationship between AI and NSE is much complex and indirect. Firstly, basins with higher AI tend to have lower NSEmmv, and this will lead to lower NSE; secondly, basins with higher AI tend to have lower BFI, which will also lead to lower NSE; thirdly, for a given BFI level, basins with higher AI tend to have higher memory and higher NSE. For ANSE, basins with AI between 1 and 2 show higher ANSE, which corresponds to higher autocorrelation coefficients.

Climate and Landscape Controls of Regional Patterns of Flow Duration Curves Across the Continental United States: Statistical Approach

AbstractThe flow duration curve (FDC) is a hydrologically meaningful representation of the statistical distribution of daily streamflows. The complexity of processes contributing to the FDC introduces challenges for the direct exploration of physical controls on FDC. In this paper, the controls of climate and catchment characteristics on FDC are explored using a stochastic framework that enables construction of the FDC from three components of streamflow: fast and slow flow (during wet days) and slow flow during dry days. The FDC during wet days (FDCw) is computed as the statistical sum of the fast flow duration curve (FFDC) and the slow flow duration curve (SFDCw), considering their dependency. FDC is modeled as the mixture distribution of FDCw and the slow flow duration curve during dry days (SFDCd), by considering the fraction of wet days (δ) for perennial streams and both δ and the fraction of days of zero streamflow for ephemeral streams. The Kappa distribution is employed to fit the FFDC, SFDCw, and SFDCd for 300 catchments from Model Parameter Estimation Experiment (MOPEX) across the United States. Results show that the 0–20th percentile of FDC is controlled by FFDC and SFDCw, the 90–100th percentile of FDC is controlled by SFDCd, and the 20–90th percentile of FDC is controlled by three components. The relationships between estimated Kappa distribution parameters and climate and catchment characteristics reveal that the aridity index, the coefficient of variation of daily precipitation, timing of precipitation, time interval between storms, snow, topographic slope, and slope of recession slope curve are dominant controlling factors.

Derivation of Interannual Climate Elasticity of Streamflow

AbstractBesides climate change (i.e., change on an average condition), streamflow change is also affected by interannual variability of climate (i.e., change from year to year). To quantify the impact of interannual climatic variability, in this study, we first clarify the content and definition of interannual climate elasticity that represents the sensitivity of streamflow to climatic variability from year to year and equals to the ratio of proportional annual change in streamflow and a climatic variable. Then, we analytically derive the interannual climate (i.e., potential evapotranspiration and precipitation) elasticities of streamflow by integrating the annual water balance with a conceptual hydrologic model (i.e., extended Budyko framework). At interannual scale, the terrestrial water storage change (i.e., ∆S) is a substantial component of water balance equation. Consequently, the analytical solutions of the interannual climate elasticities are functions of precipitation, potential evapotranspiration, and ∆S. The newly derived interannual climate elasticities are applied to 335 Model Parameter Estimation Experiment watersheds as a case study to evaluate the hypotheses in this study and construct the regional map of the streamflow sensitivity to interannual variability of climate. Results show that the interannual precipitation elasticity is generally larger than the absolute interannual potential evapotranspiration elasticity across study watersheds. Spatially, streamflow in the arid region tends to be more sensitive to interannual variability of climate than that in humid region. This study may be useful for understanding the extreme events and annual streamflow change caused by the ∆S (e.g., groundwater abstraction) across watersheds or regions.

Searching for an Optimized Single-objective Function Matching Multiple Objectives with Automatic Calibration of Hydrological Models

In the calibration of hydrological models, evaluation criteria are explicitly and quantitatively defined as single- or multi-objective functions when utilizing automatic calibration approaches. In most previous studies, there is a general opinion that no single-objective function can represent all important characteristics of even one specific hydrological variable (e.g., streamflow). Thus hydrologists must turn to multi-objective calibration. In this study, we demonstrated that an optimized single-objective function can compromise multi-response modes (i.e., multi-objective functions) of the hydrograph, which is defined as summation of a power function of the absolute error between observed and simulated streamflow with the exponent of power function optimized for specific watersheds. The new objective function was applied to 196 model parameter estimation experiment (MOPEX) watersheds across the eastern United States using the semi-distributed Xinanjiang hydrological model. The optimized exponent value for each watershed was obtained by targeting four popular objective functions focusing on peak flows, low flows, water balance, and flashiness, respectively. Results showed that the optimized single-objective function can achieve a better hydrograph simulation compared to the traditional single-objective function Nash-Sutcliffe efficiency coefficient for most watersheds, and balance high flow part and low flow part of the hydrograph without substantial differences compared to multi-objective calibration. The proposed optimal single-objective function can be practically adopted in the hydrological modeling if the optimal exponent value could be determined a priori according to hydrological/climatic/landscape characteristics in a specific watershed.

Heteroscedastic and symmetric efficiency for hydrological model evaluation criteria

Abstract Evaluation criteria play a key role in assessing the performances of hydrological models. Most previous criteria are based on the standard least square method, which assumes model residuals to be homoscedastic and is, therefore, not suitable for assessing cases with heteroscedastic residuals. Here, we compared a heteroscedastic and symmetric efficiency (HSE) criterion with the Nash–Sutcliffe efficiency (NSE) and the heteroscedastic maximum-likelihood estimator (HMLE) by running a monthly water balance model with four parameters (i.e., the abcd model) in 138 basins located in the continental United States derived from the Model Parameter Estimation Experiment dataset. The results show that compared to the NSE, the HSE and HMLE are both more effective for stabilizing variance and producing more uniform performances with flow magnitude, and the latter is slightly more effective than the former on stabilizing the residual heteroscedasticity, with the aid of an additional parameter.

Improving probabilistic hydroclimatic projections through high-resolution convection-permitting climate modeling and Markov chain Monte Carlo simulations

Understanding future changes in hydroclimatic variables plays a crucial role in improving resilience and adaptation to extreme weather events such as floods and droughts. In this study, we develop high-resolution climate projections over Texas by using the convection-permitting Weather Research and Forecasting (WRF) model with 4 km horizontal grid spacing, and then produce the Markov chain Monte Carlo (MCMC)-based hydrologic forecasts in the Guadalupe River basin which is the primary concern of the Texas Water Development Board and the Guadalupe-Blanco River Authority. The Parameter-elevation Regressions on Independent Slopes Model (PRISM) dataset is used to verify the WRF climate simulations. The Model Parameter Estimation Experiment (MOPEX) dataset is used to validate probabilistic hydrologic predictions. Projected changes in precipitation, potential evapotranspiration (PET) and streamflow at different temporal scales are examined by dynamically downscaling climate projections derived from 15 Coupled Model Intercomparison Project Phase 5 (CMIP5) general circulation models (GCMs). Our findings reveal that the Upper Coast Climate Division of Texas is projected to experience the most remarkable wetting caused by precipitation and PET changes, whereas the most significant drying is expected to occur for the North Central Texas Climate Division. The dry Guadalupe River basin is projected to become drier with a substantial increase in future drought risks, especially for the summer season. And the extreme precipitation events are projected to increase in frequency and intensity with a reduction in overall precipitation frequency, which may result in more frequent occurrences of flash floods and drought episodes in the Guadalupe River basin.

Hydrological Interpretation of a Statistical Measure of Basin Complexity

AbstractThis paper studies how streamflow predictability varies with basin characteristics. We introduce an index of basin complexity that is based on a model of least statistical complexity that is needed to reliably predict daily streamflow of the basin. We then relate it with climate, vegetation and soil characteristics of the basin. Daily streamflow is modeled using k nearest neighbor model of lagged streamflow that predicts next time step streamflow based on the occurrences of similar streamflow events from the past. In order to calculate basin complexity, we identify difficult streamflow events of the basin and then use Vapnik‐Chervonenkis generalization theory, which trades off model performance with Vapnik‐Chervonenkis dimension (i.e., a measure of model complexity), to find a k nearest neighbor model of appropriate complexity for predicting a difficult streamflow event of the basin. The average of selected model complexities corresponding to difficult events is then defined as the basin's complexity. Basin complexity of 412 Model Parameter Estimation Experiment basins from continental United States are then related with its six basin characteristics. All the characteristics have been derived from the Model Parameter Estimation Experiment database to represent climate, vegetation and soil characteristics of the basins in a concise manner. Results find that more complex basins that are drier have less seasonal rainfall, vegetation with more storage capacity (i.e., smaller 5‐week Normalized Difference Vegetation Index gradient), and faster responsive soils. The results reaffirm prior observations that minimum complexity that is required to model a basin depends on its climate and landscape characteristics (e.g., complex models do not perform well in dry basins).

The CAMELS data set: catchment attributes and meteorology for large-sample studies

Abstract. We present a new data set of attributes for 671 catchments in the contiguous United States (CONUS) minimally impacted by human activities. This complements the daily time series of meteorological forcing and streamflow provided by Newman et al. (2015b). To produce this extension, we synthesized diverse and complementary data sets to describe six main classes of attributes at the catchment scale: topography, climate, streamflow, land cover, soil, and geology. The spatial variations among basins over the CONUS are discussed and compared using a series of maps. The large number of catchments, combined with the diversity of the attributes we extracted, makes this new data set well suited for large-sample studies and comparative hydrology. In comparison to the similar Model Parameter Estimation Experiment (MOPEX) data set, this data set relies on more recent data, it covers a wider range of attributes, and its catchments are more evenly distributed across the CONUS. This study also involves assessments of the limitations of the source data sets used to compute catchment attributes, as well as detailed descriptions of how the attributes were computed. The hydrometeorological time series provided by Newman et al. (2015b, https://doi.org/10.5065/D6MW2F4D) together with the catchment attributes introduced in this paper (https://doi.org/10.5065/D6G73C3Q) constitute the freely available CAMELS data set, which stands for Catchment Attributes and MEteorology for Large-sample Studies.

A Framework for Validation of Remotely Sensed Precipitation and Evapotranspiration Based on the Budyko Hypothesis

AbstractDespite offering spatially and temporally continuous measurements, the use of remotely sensed P and E in data‐scarce catchments is hindered by the lack of ground‐based measurements that enable comprehensive validation. This study proposes a novel validation framework that characterizes the combined error in the long‐term average estimates of remotely sensed P and E by making use of the Budyko hypothesis, specifically Fu's equation. A Root Mean Square Error (RMSE)‐based error metric that is capable of translating individual biases in P and E estimates onto the Budyko space is developed. A controlled sensitivity experiment using data from Model Parameter Estimation Experiment (MOPEX) catchments in the United States showed that the developed error metric is more sensitive to biases in P compared to biases in estimates of E. Validating the framework using combinations of different satellite‐based estimates of P and E revealed that the framework succeeds in arriving at the same conclusions as a traditional validation method with regards to the quality of P and E data sets. The framework offers a physically consistent, parametrically efficient basis for the selection of remotely sensed P and E data sets for hydrologic studies. Due to lack of consideration for catchment storage in the formulation of Fu's equation, the developed error metric is limited to long temporal time scales. As a result, the error metric is capable of characterizing the bias in P and E data sets and not the variance.

A comparison of the discrete cosine and wavelet transforms for hydrologic model input data reduction

Abstract. The treatment of input data uncertainty in hydrologic models is of crucial importance in the analysis, diagnosis and detection of model structural errors. Data reduction techniques decrease the dimensionality of input data, thus allowing modern parameter estimation algorithms to more efficiently estimate errors associated with input uncertainty and model structure. The discrete cosine transform (DCT) and discrete wavelet transform (DWT) are used to reduce the dimensionality of observed rainfall time series for the 438 catchments in the Model Parameter Estimation Experiment (MOPEX) data set. The rainfall time signals are then reconstructed and compared to the observed hyetographs using standard simulation performance summary metrics and descriptive statistics. The results convincingly demonstrate that the DWT is superior to the DCT in preserving and characterizing the observed rainfall data records. It is recommended that the DWT be used for model input data reduction in hydrology in preference over the DCT.

Climate change impact assessment on hydropower generation using multi-model climate ensemble

Hydropower is the primary renewable source of energy that harnesses the power of the naturally flowing water streams and its potential is strongly impacted by the hydrological regime. The objective of the present research is to carry out a hydrological model based study to assess the impacts of climate change on hydropower generation by using the regional climate model (RCM) data available through coordinated regional downscaling experiment (CORDEX), and subsequently to analyse the effect of using model ensemble in projecting the future hydrology and energy generation scenario. C.H.Corn hydroelectric project located on River Ochlockonee near Tallahassee in Florida, USA, has been considered as a case study. A hydrologic model of the basin, draining into the dam, is developed using a conceptual model HYMOD with the historical climate and flow data extracted from the model parameter estimation experiment (MOPEX) dataset. The future projected climate scenario (2091–2100) is generated following the Representative Concentrated Pathways (RCP) 4.5 with the ensembles of six climate models. The impact of the future climate on water availability indicates a significant seasonal variability among the model ensemble results with an overall annual increase of statistics for the climate variables, corresponding inflows and a marginal increase in the energy generation.

Regionalization of runoff models derived by genetic programming

The aim of this study is to assess the potential of hydrological models derived by genetic programming (GP) to estimate runoff at ungauged catchments by regionalization. A set of 176 catchments from the MOPEX (Model Parameter Estimation Experiment) project was used for our analysis. Runoff models for each catchment were derived by genetic programming (hereafter GP models). A comparison of efficiency was made between GP models and three conceptual models (SAC-SMA, BTOPMC, GR4J). The efficiency of the GP models was in general comparable with that of the SAC-SMA and BTOPMC models but slightly lower (up to 10% for calibration and 15% in validation) than for the GR4J model. The relationship between the efficiency of the GP models and catchment descriptors (CDs) was investigated. From 13 available CDs the aridity index and mean catchment elevation explained most of the variation in the efficiency of the GP models. The runoff for each catchment was then estimated considering GP models from single or multiple physically similar catchments (donors). Better results were obtained with multiple donor catchments. Increasing the number of CDs used for quantification of physical similarity improves the efficiency of the GP models in runoff simulation. The best regionalization results were obtained with 6 CDs together with 6 donors. Our results show that transfer of the GP models is possible and leads to satisfactory results when applied at physically similar catchments. The GP models can be therefore used as an alternative for runoff modelling at ungauged catchments if similar gauged catchments can be identified and successfully simulated.

Infective endocarditis involving multiple sites

Streamflow variability is affected by both climate change and direct human impacts within the catchment. Quantification of the role of these two factors have significant implications for water resources. In recent years, Budyko-based analytical methods have been widely used for such quantification due to their simplicity and efficacy. This paper presents a comparison of such methods to attribute changes in streamflow in four Model Parameter Estimation Experiment (MOPEX) catchments in the contiguous United States. Further, a new, improved Two-stage Decomposition method to separate the role of climate and catchment changes on streamflow is proposed, considering the uncertainty in the catchment parameter that further leads to a range of evaporative index values within a catchment. We find that climate and catchment effects are sensitive to the choice of Budyko equation, disaggregation algorithm and approximation technique. Further, the catchment parameter can be expressed probabilistically, to account for the uncertainty because of limited hydro-meteorological records. The proposed approach employs a bootstrap analysis to obtain the stochastic range of the catchment parameter, considering different choices of Budyko equations. We argue that rather than relying on a single Budyko-based approach for separation of climate and catchment effects on streamflow, the proposed method offers a more robust way by considering the entire range of uncertainty of the catchment parameter.