Statistical blending of global-gridded climatological products: an approach to inverse hydrological model

The calculation of water balance components and their quantification are two challenges in water resources management. Previously, the main sources for calculating and estimating the components were ground measurements and statistical estimates (the experimental formula). Recently, most researchers tend to use large-scale or remotely sensed datasets or both for this purpose, and different research projects have been conducted to investigate using such data in hydrological models (Moreira et al. 2019; Guo et al. 2022). Most of these research projects have focused on evaluating the accuracy of the products and comparing them with observed and in situ values (Shayeghi et al. 2020). In addition, a few studies have also tested the performance of these datasets as input or state variables in hydrological models (Duan et al. 2019).

These products are practical tools to estimate the components of hydrological models, especially in areas with gapped, missed, or scattered ground information (Huang et al. 2020). However, this information sometimes has considerable uncertainty and error because of the low resolution of spatiotemporal ground information, which can be due to the remotely sensed information and the algorithms used in their estimations (Nguyen et al. 2021). Therefore, extensive efforts have been made to solve this problem lately. Following this issue, one of the most useful actions is to blend different remote sensing or large-scale products to achieve higher accuracy, lower error, and improved spatiotemporal estimates (Aires 2014; Munier et al. 2014; Schoups & Nasseri 2021). The studies conducted in the field of blending remotely sensed information are divided into three categories:

• Blending ground and satellite data to obtain improved, consolidated information.

• Comparing different statistical and probabilistic methods to blend the initial datasets.

• Evaluation of the blended information's effectiveness in the secondary environment, like hydrological models.

Ground data and one or more large-scale datasets have been merged in the first category. They are used to improve the reliability of information coverage using various statistical and probabilistic methods, such as Xie and Arkin's maximum likelihood estimation method, nonparametric kernel smoothing, the Random Forest-based MErging Procedure (RF-MEP), and the Bayesian approach (Huffman et al. 1997; Xie & Arkin 1997; Li & Shao 2010; Rozante et al. 2010; Xie & Xiong 2011; Manz et al. 2016; Yang et al. 2017; Kimani et al. 2018; Baez-Villanueva et al. 2020; Lu et al. 2021; Ma et al. 2021; Nguyen et al. 2021; Zandi et al. 2022). In the second category, several titles can be mentioned compared to the first one. For instance, simple averaging, statistical collocation, and the Bayesian approach are the most implemented methods for blending large-scale datasets without ground information (Jongjin et al. 2016; Baik et al. 2018; Yumnam et al. 2022).

The third category included studies in which the second model (the water balance model in the current research) was used to validate (or calibrate) the blended products. This type of blending approach is the focus of the current research. Sahoo et al. (2011) and Pan et al. (2012) used various satellites and observed precipitation and EvapoTranspiration (ET) products to blend multiple sources of water cycle variables, using large-scale water balance and the assimilation approach. Aires (2014) proposed four different statistical methods to blend different datasets of multiple water balance components. Schoups & Nasseri (2021) suggested a new probabilistic method to combine different climatological variables to close a large-scale water balance model.

The noble point of the current research is to blend various ET and precipitation products using inverse conceptual hydrological modeling. In the previous studies (Pan et al. 2012; Aires 2014; Schoups & Nasseri 2021), a large-scale water balance model (a macro-scale terrestrial water balance model) has been used, but in the current research, the terminology has been developed for the conceptual model in a mid-size watershed scale. The main idea of inverse hydrological modeling (or backward hydrology) has been introduced and developed by Vrugt et al. (2008), Kirchner (2009), and Herrnegger et al. (2015). It means going backward from streamflow as the output variable to define (or tune) precipitation, ET, or both as the input variables via statistical and probabilistic methods. In this research, different precipitation products (Climate Hazards Group Infrared Precipitation with Stations (CHIRPS), ERA5-Land of the European Center for Medium-Range Weather Forecasts (ECMWF) (ERA), Integrated Multi-Satellite Retrievals for GPM (IMERG), Tropical Rainfall Measuring Mission (TRMM), and Terra) and Actual EvapoTranspiration (AET) datasets (Global Land Evaporation Amsterdam Model (GLEAM), SSEBop, Moderate Resolution Imaging Spectroradiometer (MODIS), Terra, and ERA) are used to combine linearly through an inverse modeling approach. The following section presents and discusses the used information, the study area, and the modeling subject.

The mountainous basin of Gheshlagh is one of the sub-basins of the border river watershed in the west of Iran and has an area of 1,062.12 km². The geographic coordinates of this basin are located between 46° 46′ and 47° 20′ east longitude and 35° 24′ and 35° 43′ north latitude (Figure 1). Due to its mountainous nature, the watershed is snow-covered during December to March. In addition, the case study has no alluvial aquifer (based on the official report of Iran Water Resources Management Company-IWRMC). The average annual rainfall of the basin is 454.4 mm, and the average annual temperature is 13.3 °C.

Since the conceptual and monthly water balance models are used, it is necessary to aggregate the daily in situ information of precipitation, streamflow, temperature, and pan evaporation datasets collected by the IWRMC climatological and hydrometric stations. The quality of the datasets has been checked using run-test and double-mass curve methods. Then, climatological information (including precipitation, temperature, and pan evaporation datasets) was spatially estimated over the watershed with a 5 km resolution using the Moving Least Square (MLS) method on the computational grids (Lancaster & Salkauskas 1981; Amini & Nasseri 2021). To do so, geographical characteristics of altitude, longitude, and latitude values of the stations (climatological and synoptic stations) are considered as input variables to calibrate the MLS model and the input variables have been estimated over the computational grids. In addition, due to the full coverage of information during the computational period (from 2003 to 2015), gap-filling methods are not applicable. The input variables of the selected monthly water balance model have been prepared by aggregating the spatially estimated precipitation, temperature, and pan evaporation datasets. It is worthwhile to mention that the watershed has been the subject of a few previous research projects by Sadeghi et al. (2013), Taheri et al. (2022), and Mousavi et al. (2022). The outlet of the selected basin is Gheshlagh Dam and the input streamflow to the reservoir is the target of the water balance modeling, as well.

The specifications of the recommended large-scale and gridded climatological products of precipitation and AET used in the current research (based on Javadian et al. 2019; Moshir Panahi et al. 2021; Schoups & Nasseri 2021; Ghomlaghi et al. 2022; Nasseri et al. 2022) are briefly discussed as follows.

In the current research, five global-gridded and large-scale precipitation datasets (including CHIRPS, IMERG, ERA5-Land of the ECMWF, TRMM, and Terra Climate) with monthly temporal resolution have been used.

Table 1 shows their spatiotemporal specifications and descriptions. The CHIRPS, which combines satellite images and ground station data for generating rain time series to analyze precipitation trends and monitor the drought, was developed by the US Geological Survey (USGS) and the Climate Hazards Group at the University of California, Santa Barbara (UCSB) (Funk et al. 2014).

A combination of multiple satellites' microwave precipitation products, microwave-calibrated infrared (IR) satellite estimates, values of ground gauges, and other precipitation estimators on a suitable time and space scale forms the IMERG algorithm which is the GPM satellite's product for estimating surface precipitation (Huffman et al. 2019). The re-analyzed dataset named ERA5-Land (ERA), which combines historical observations of global estimates with meteorological and data-driven models, was developed by the ECMWF and has been available monthly since 1950 (Hersbach et al. 2019). The Terra product, available monthly since 1958, combines CRU Ts 4.0 and 55-year-old Japanese re-evaluated products (Abatzoglou et al. 2018). The TRMM integrates microwave and IR satellite information and is a joint space mission between NASA and the Japan Aerospace Exploration Agency (JAXA) (Huffman et al. 2010).

It should be noted that only the CHIRPS dataset has the same spatial resolution with the selected computational grid (5 km), and the other products have been resampled on the spatial grids.

Table 1 summarizes the specifications of the large-scale ET information used in this study. The product of MODIS (Moderate Resolution Imaging Spectroradiometer) with the name MOD16 has been produced on a global scale with a spatial accuracy of 500 m based on the Penman–Monteith equation logic since 2001 (Running et al. 2017). The GLEAM is a set of large-scale climatic results that separately estimate the various components of ET. The product is based on the Priestley–Taylor equation (Martens et al. 2017).

The Simplified Surface Energy Balance (SSEB) approach has been used to develop the SSEBop AET product since 2003. It is accomplished by combining the ET produced by MODIS 10-day remote sensing thermal images with the reference ET. The main structure used for estimating the ET product is based on the use of applied hot or cold pixels in SEBAL (Bastiaanssen et al. 1998) and METRIC (Allen et al. 2007) methods (Senay et al. 2020).

Since 1958, Terra Climate has been a collection of monthly climate datasets on global-scale terrestrial surfaces. With a monthly spatiotemporal resolution of 4 km, this dataset provides critical inputs for global-scale environmental and hydrological studies. Terra Climate also generates monthly surface water balance components using a water balance model (Abatzoglou et al. 2018).

ERA5-Land of ECMWF (ERA), the last used re-analyzed dataset, has been modified using observations from around the world into a complete and consistent global dataset via a physically based model since 1981. The spatial accuracy of the product is about 9 km (Hersbach et al. 2019).

In this section, the proposed method of blending different climatological information using water balance models, the selected uncertainty assessment approach, and evaluation metrics have been introduced.

The water balance model used in the current research is based on Wang's water balance model (Wang et al. 2011). The model focuses on monthly hydro-climatic structure and surface processes to simulate streamflow and its water balance components. The usual structure of Wang's water balance model was implemented in this research (in the Supplementary material, Figure S1). However, its snow hydrology because of snow-covered and mountainous nature of the area has been modified using the exponential degree-day relationship to separate rainfall and snow from the monthly precipitation values used by Guo et al. (2005). However, the four parameters in the original model and to better consideration of snow storage and initial storage values (soil and snow storages), the modified water balance model has seven (Mousavi et al. 2022; Taheri et al. 2022). A comprehensive simulation to detect the snow hydrology of the watershed with the same water balance model has been reported by Taheri et al. (2022). The final water balance model has a single-layer soil storage to play role in generating subsurface flow in the model, without considering groundwater and aquifer storage. Therefore, the final streamflow in the model is the sum of surface, subsurface, and snowmelt flows. The flow diagram of the used water balance model has been depicted in the supplementary materials (Supplementary material, Figure S1).

Blending the large-scale climate products provides more reliable and compatible inputs with the recorded streamflow of a river network (Aires 2014; Schoups & Nasseri 2021). For example, precipitation is calculated using recorded information from rain gauge stations, and its results have been used in lump, semi-distributed, and distributed hydrologic models. Because of the low density of stations or recorded values tied to different uncertainty sources (especially in high-altitude areas), these uncertain estimations cause inconsistency in the hydrological modeling response. Therefore, merging or combining raw precipitation datasets (without any correction with ground information) creates a chance to use the advantages of each precipitation product (even recorded values) in simulating the hydro-climatological behavior of the considered areas. On the other hand, the ET values are environmental variables that can be calculated or measured in different ways. Its appropriate determination is one of the most important challenges in hydrological and water resource modeling. Combining different ET products and determining their acceptable values are the goals of this research.

Against the usual hydrological modeling that goes from the main drivers to streamflow, the purpose of blending is fulfilled in the current study via inverse modeling, which means moving backward from observed streamflow to the main hydrologic drivers such as precipitation and ET. This hydrological modeling approach has formerly been used by other researchers, such as Herrnegger et al. (2015) and Vrugt et al. (2008).

To achieve the blending input variables for water balance modeling and their evaluations, three different scenarios have been introduced. The first scenario, known as the reference scenario, used the observed precipitation values and an experimental method to estimate ET and recorded streamflow at the basin outlet to calibrate the parameters of the selected water balance model. The calibrated parameters and AET time series have been used for the evaluation of the proposed scenarios in the next steps. It should be noted that the spatially averaged monthly precipitation in the basin of interest has been calculated using the MLS method.

Most of the previous studies with the aim of modifying and combining large-scale remote sensing products are based on validation, analysis, comparison, and evaluation with in situ information. The effects of using these products (or their modified ones) on the uncertainty behavior of hydrological models and applications have not been studied. This type of evaluation is one of the other fundamental aspects of the current research. Generally, uncertainty originates from three sources, including model structure, datasets (including input and output), and model parameters (Chen et al. 2016). Various methods have been proposed in the literature to evaluate the uncertainty caused by the mentioned causes. The Generalized Likelihood Uncertainty Estimation (GLUE) method, proposed by Beven & Binley (1992), has been employed to assess the model parameters' uncertainty in this research. The basis of the method is the sequential sampling-simulation approach, considering their performance. GLUE has been widely used in evaluating the uncertainty of hydrological models individually or as a benchmark method (Ahmadi & Nasseri 2020). GLUE has no limit for the number of parameters and ultimately provides the overall uncertainty of the model structure and its parameters (Beven & Binley 1992). At this step, to evaluate the effectiveness of using the investigated scenarios in calibrating the balance model (the reference and modified models with the combination of remote sensing information), their parametric uncertainties were measured using the GLUE method. To maintain the effect of the generated random parameters, Nash-Sutcliffe Efficiency (NSE) has been considered to have an acceptance level threshold of 0.75. With this desirability limit, 10,000 acceptable random parameter sets (with an NSE of greater than 0.75) were included to form the statistical distribution of the parameters.

A decimal-based and single-objective genetic algorithm has been used to calibrate the water balance model. The computational period was from 2003 to 2015 due to the availability and coverage of ground and satellite data. This time range (149 months) was randomly divided into calibration (104 months) and validation (45 months) periods, which are considered the same throughout the evaluation of different scenarios of this model. The fitness function is set to maximize the NSE metric between the observed and calculated streamflow values of the outlet. The statistical metrics and parameters of the basic model and the second scenario models, which include 101 models in the study area, are presented in Supplementary material, Table S1. In addition, five statistics of the water balance models (Mean Squared Error (MSE), Normal Mean Squared Error (NMSE), Root Mean Square Error (RMSE), coefficient of determination (R²), and Taylor skill's score (TS)) have been reported in Supplementary material, Table S1. The seven significant parameters of the model, the initial range, and its optimized value in the reference model, which is called the first scenario, are reported in Table 3. The initial limits of the parameter range are on loan from Wang et al. (2013).

According to Table 3, the acceptability limits of the models for simulating the streamflow surpass 0.5 with the values of the NSE and KGE indices in the monthly time step at the basin scale (Kouchi et al. 2017). In other words, models with more significant index values (more than 0.5) are acceptable. Therefore, these values were selected as adequate limits in the statistical evaluation to validate the models. Considering the combination of five precipitation and five ET products in the second scenario, 100 models were calibrated. According to the results achieved from the comparison of the statistical indicators (details in Supplementary material, Table S1), the first and second scenarios have almost the same level of performance. About 60% of the calibrated models have met the presented statistical requirements (see Table 2). The next two conditions have also been used to filter the accepted models.

The second condition is a KGE index greater than the reference model in the calibration and validation periods (Figure 2(b)). The KGE metrics for 35 models were greater than 0.75 and 0.77 for the calibration and validation periods (for the reference model), respectively.

In the next step, the models that met the aforementioned conditions were checked using other criteria. In this investigation, two parameters, alpha (α) and beta (β), which are respectively optimized as the blending coefficients for the precipitation and ET products of the models, should be calibrated in the range of (0, 1) (not exactly equal to zero or one). This means that the two products are combined, and neither is used alone in the model. By applying these two conditions to the previous models, 12 models were selected out of calibrated ones (Figure 2(c)).

The diameter of the drawn circles in this figure is proportional to the multiplication root value of two parameters, alpha and beta. The smaller the diameter of these circles (circles with a darker color), the smaller the optimized parameter. The details of the indicators and optimized parameters of the 12 models are mentioned in the supplementary materials (Supplementary material, Table S2). The important point in these selected models is that the ERA product has participated in all 12 models, which shows the product's greater contributions in this way of modeling than the other products. In choosing the models, superiority over the reference model has been considered. Therefore, it was concluded that the performance of the model will be in a good range (considering Table 2) if it has these three conditions simultaneously,

• An NSE value over 0.73 and 0.54 in the calibration and validation periods.

• A KGE value over 0.75 and 0.77 in the calibration and validation periods.

• The statistical superiority over the reference model.

Although the figures have a similar trend in the case of combined and calculated ET values in the reference model, different optimal combinations in different models have greater change ranges compared to the precipitation graphs. The highest amount of ET in May relates to the P ERA & TRMM-E Gleam & ERA model with a value of 121.84 mm, and the lowest value of 81.99 mm in this month relates to the P ERA & TRMM-E Terra & ERA model. Although the same procedure is followed, the ET peaked in the combination results this month. Notably, the depicted ET is achieved after applying the ET parameter's coefficient in the model.

It appears that based on the blending coefficients and their combinations, blended AET results have more diversity in their responses than blended precipitation results. This convergence in blended precipitation and variation in merged ET values represent the behavioral similarity and dissimilarity of precipitation and ET products. The values of the calibration and uncertainty evaluation metrics of the selected models (12 models) and the reference model are presented in Table 4.

Their KGE and NSE values are presented for two calibration and validation periods of the proposed scenarios (blended precipitation and ET products). Based on the table, the values of both metrics in the calibration and validation periods are well adjusted, and there is no significant difference between the models' performances in the simulation periods. The combinations of precipitation and ET values in all developed models have led to improved modeling results. Based on the results in Table 4, the P ERA and TRMM-E Gleam and ERA models have acceptable values in almost all metrics.

The uncertainties of the model parameters and simulated streamflow are among the other aspects that can be considered for the evaluation of hydrological models using remotely sensed and large-scale datasets. Determining the aforementioned uncertainties is the aim of this part of the paper, which is the second phase of comprehensive evaluation and behavior assessment. In the first step, the reference model based on the information from the calibration period has been employed to calibrate its parameters considering the observed streamflow at the basin outlet. The reference model has also been used as a benchmark scenario to compare the performance of other models from the perspective of evaluating the uncertainty of the calibrated model's parameters.

As mentioned before, parametric uncertainty has been evaluated using the GLUE method. Table 5 presents the values of the uncertainty evaluation metrics of the reference models. Interpreting the indicators' mean values, a lower ARIL and a higher P_level result in a lower effect of uncertainty sources on the model's behavior. Therefore, a more stable model causes this fact. The NUE metric provides a general representation of uncertain simulation performance using a combination of P_level and ARIL metrics. The higher value of this metric causes more efficiency and reliability in uncertainty simulation.

According to the figure, it is necessary to explain that the ERA precipitation product contributed to the combined precipitation with a more significant coefficient. Moreover, the behavior of CHIRPS and ERA in the first and second categories is similar. The second parameter used to combine the ET products is the beta parameter (Figure 4(b)). Three different patterns can be inferred from the various combinations of the products. In the first one, that is the combination of MODIS & ERA and SSEBop & ERA products (regardless of the used precipitation products), MODIS and SSEBop datasets had less contributions than ERA product which played a vital role in the dual combination of the products. The second pattern is related to the category of blended information that GLEAM contributed to it. In these combinations, despite the wholly defeated combination with more participation of ERA, the GLEAM product has equal or almost equal participation. In the third pattern, which is the combination of TERRA & ERA, the contribution of ERA is more than that of TERRA, but its amount is less than in the first pattern.

In Table 6, the uncertainty metrics resulting from the developed scenarios related to blended precipitation, ET, and streamflow have been reported. According to the table, the ET uncertainty metrics have the highest NUE and lowest ARIL values when compared to the P CHIRPS and ERA MODIS and ERA models, which are 3.88 and 9.01, respectively.

The highest P_level value of 45.83 was related to the P ERA & IMERG model as well. Regarding the precipitation variable, the maximum value of NUE and minimum value of ARIL related to P ERA & TRMM E MODIS & ERA and P ERA & TRMM E SSEBop & ERA models have been obtained at 28.85 and 0.76, respectively. Since, there is no reference model for these two variables; these metrics have only been compared with each other. Generally, ERA & TRMM and ERA & Terra precipitation models had less uncertainty.

Based on the statistics reported in Table 6, the ARIL values of streamflow uncertainties vary in the range of [2.12, 3.82] percent compared to the base model. P ERA & IMERG – E Gleam & ERA and P CHIRPS & ERA – E MODIS & ERA have the highest and lowest metric values, respectively. Furthermore, regarding the percentage of the observed values grouped with the upper and lower bounds of the simulated uncertainty (P_level), only the P CHIRPS & ERA-E MODIS & ERA combination is less than the reference model, and the rest models have greater values of the metric compared to the reference model. In the rest of the models, their NUE values are higher than the metrics of the reference model, except for one case of P ERA & IMERG E Gleam & ERA, which indicates the better performance of the models compared to the reference model. Table 6 shows the uncertainty of the simulated streamflow using blended P ERA & Terra – E GLEAM & ERA which is the most efficient model considering its NUE values (equal to 12.51).

On the other hand, the lowest efficiency (with an NUE value of 10.48) was related to the calibrated model using blended P ERA & IMERG E Gleam & ERA datasets. According to the contribution of ET products in these two models, the results show that the type of precipitation product combined with ERA has influenced the efficiency of the modeling uncertainty.

According to Table 6 and Figure 5, the developed blended precipitation datasets have significantly less uncertainty than the blended ET values. Comparing the results, the most important differences between the new precipitation datasets and ground information are their monthly peak values (in the months of February, March, and April). These scenarios are in good agreement in the middle and low monthly precipitation. In addition, the calculated uncertain peaks are randomly lower and higher than the precipitation amount of the reference model. Considering the values of the NUE reported in Table 6, uncertainty of the blended precipitation for the models P ERA & TRMM-E GLEAM & ERA and P ERA & TRMM-E SSEBop & ERA show the best performance and are innately the best inverse and uncertain simulated precipitation datasets. Based on the NUE metric, the PCHIRPS and ERA-E combination models produced the most disproportionate blended certain and uncertain precipitation dataset.

However, in the ET model, the simulated uncertain blended values are greater than the ET values of the reference model except in the model calibrated using P CHIRPS & ERA-E GLEAM & Terra input variables. Not only is the uncertain range of ET significant in the months with the most extensive range of changes (April, May, and June), but also in cold months, when ET has lower values, it is even remarkable in an uncertain way. Likewise, according to Figure 6, the combined uncertain ET in dry months shows a higher value than the reference model. Based on the NUE values presented in Table 6, the best uncertain blended dataset compared to the calculated ET values of the model would be the P CHIRPS & ERA – E MODIS & ERA, which is a scenario of product combinations.

The comparison of Table 6 and Figures 5 and 6 shows that due to the objective nature of precipitation and the conceptual nature of ET (without observed values), they have less uncertainty, and the correction resulting from their combination only includes cold months with high local amounts of precipitation. However, due to the non-objective and conceptual nature of ET, it has an estimated uncertain range in the combined model even in cold months, and naturally, this range will increase significantly in hot months. The addition of ET in the blended scenarios compared to the values of the reference model indicates the tendency of the model to remove moisture in cold months to simulate the streamflow behavior in these months with low values in contrast to the dry and wet months.

The current study's innovation is the creation of new precipitation and ET datasets based on linear combinations to be available in two certain and uncertain types for many large-scale products at the basin level.

The proposed blending method has been used to evaluate the conceptual structure of the water balance model and the inverse modeling procedure. In addition to evaluating the effectiveness of hydrologic modeling using the blended values of large-scale precipitation (such as CHIRPS, ERA, TRMM, Terra, and IMERG) and ET products (such as GLEAM, MODIS, SSEBop, Terra, and ERA), a new criterion is presented in this article.

This criterion was introduced to measure the suitability of this large-scale information not with ground information but in a complex process such as the hydrological model. Although some products have been combined and modified in some research projects in the form of Terrestrial Water Storage (TWS) macro-water balance modeling (Schoups & Nasseri 2021), doing this process in the form of a model with many details and on a small spatial scale is the novel focus point of the present research.

In this procedure, the large-scale information is dually and linearly (uncertainly and certainly) blended to reach the best precipitation and ET signals to generate streamflow with an inverse approach. The results show the conformity of the pattern and behavior of the blended precipitation obtained using the MLS method in the study area. This type of blending precipitation led to slight differences in the cold months of February, March, April, and (roughly) May. Due to the mountainous nature of the study area, these differences can be caused by the insufficient expansion of the precipitation-monitoring network, especially in the highlands. This proposed method provides a clear suggestion for investigating and analyzing precipitation patterns in areas without rainfall recording stations or with variable topography, where the uncertainty of the rainfall amount on the ground is notable.

This combination process was carried out for the large-scale ET product data (in the form of two combination scenarios with precipitation). To evaluate the changes caused by this combination, the results of the corresponding scenarios were compared with the amount of ET calculated in the calibrated reference model, which was created using recorded environmental information (precipitation, temperature, pan evaporation, and streamflow). In addition to following the general procedure of ET in the base scenario, the combined values have mainly high uncertainty, and compared to the used reference balance model, they bring more ET in cold months. Using this method for measuring the optimal pattern of large-scale precipitation and ET products' combination results is a new improvement in the production of calibrated information of hydrological models at the basin scale. It also provides a controlled approximation in areas without input information or an information gap. This method can use large-scale snow-water equivalent products in its evaluation format, and by entering the temperature combination, snow-water equivalent supplies the main components of hydrological modeling at the watershed scale.

Examining these components as a leading research step plays a role in establishing a process model like a water balance model in areas without stations or calibrating large-scale information from a process path. The use of data mining methods to gradually verify the ratio of information combinations, especially in online hydrology modeling, can be considered the next step in modifying and calibrating large-scale information with a process purpose.

All relevant data are available from an online repository or repositories http://dx.doi.org/10.5281/zenodo.8062625.

The authors declare there is no conflict.