Ensemble modelling has become a significant technique in the field of machine learning, as it utilises the combined knowledge of multiple base models to improve the accuracy of predictions in different domains. Nevertheless, the effectiveness of ensemble predictions relies on the implementation of post-processing techniques that enhance and optimize the outputs of the ensemble. This study explores the domain of ensemble data post-processing, utilizing a machine learning-focused methodology to thoroughly assess and contrast a variety of post-processing methods. TIGGE Ensemble data from ECMWF and NCEP were used from 2010 to 2020. Research covers machine learning approaches post-processing methods such as BMA, cNLR, HXLR, OLR, logreg, hlogreg, QM were applied. The probabilistic forecasts were validated using the Brier Score (BS), Area Under Curve (AUC) of Receiver Operator Characteristics (ROC) plots and reliability plots. The cNLR and BMA strategies for post-processing performed exceptionally well with BS value of 0.10 and RPS value of 0.11 at all grid points for both methods. The ROC–AUC values for the cNLR and BMA methods were found to be 91.87 and 91.82%, respectively. The results show that improved post-processing techniques can be helpful to predict the flood in advance with accurate precision and warning.

  • A comprehensive evaluation of TIGGE ensemble precipitation forecast using post-processing methods for the Sabarmati River Basin.

  • Improve forecast accuracy by cNLR, BMA, logreg, hlogreg, HXLR, and OLR are used.

  • Brier Score (BS), reliability plots, and AUC and ROC plots are used to rigorously compare post-processing algorithms for model verification.

TIGGE

THORPEX International Grand Global Ensemble

NWP

Numerical Weather Prediction

ECMWF

European Centre for Medium-Range Weather Forecasts

NCEP

National Centers for Environmental Prediction

EPS

Ensemble Prediction Systems

CWC

Central Water Commission

QPF

Quantitative Precipitation Forecast

BMA

Bayesian Model Averaging

EMOS

Ensemble Model Output Statistics

PDF

Probability density function

NGR

Nonhomogeneous Gaussian regression

cNLR

Censored nonhomogeneous logistic regression

HXLR

Heteroscedastic extreme Linear Regression (HXLR),

QM

Quantile mapping

Logreg

Logistic regression

Hlogreg

Heteroscedastic logistic regression

OLR

Ordered logistic regression

ROC

Receiver Operating Characteristic

AUC

Area Under The Curve

RPS

Rank Probability Score

CDF

Cumulative distribution function

CRPS

Critical Rank Probability Score

The Intergovernmental Panel on Climate Change reports claim that climate change is responsible for extreme weather events, which cause floods and droughts all over the world (Pachauri & Meyer 2014). Flooding has an immense adverse effect on both human lives and property on a global scale (Balica et al. 2012). The frequency and intensity of extreme rainfall events are likely to increase with changing hydro-climatology, resulting in more frequent flooding (Imhoff et al. 2020). While it is impossible to prevent natural disasters, society can enhance its ability to resist them by implementing advanced management strategies. One of the techniques is the early and accurate forecasting of extreme events, giving people enough time to prepare. If flooding can be anticipated well in advance by integrating flood modelling and rainfall forecasting, a reliable mitigation and management measure can be implemented through a decision support system. The ability to predict the rain for the future has significantly improved as a result of recent technical developments (Bauer et al. 2015; Bhomia et al. 2019). In this situation, nowcasting is a helpful tool for forecasting short-term rainfall in particular urban settings.

The rapid development of meteorological science and the increasing need for precise weather forecasts have forced the creation of complex forecasting models. The creation of multiple forecasts using various initial conditions is known as ensemble forecasting, and it is a widely used technique for capturing the inherent uncertainty in weather forecasts (Buizza 2009; Deshpande et al. 2021). The TIGGE (THORPEX Interactive Grand Global Ensemble) project was established by the World Meteorological Organisation (WMO) with the aim of fostering and facilitating the utilization of ensemble weather forecasting data to enhance weather prediction and climate research. The term ‘TIGGE Ensemble data’ pertains to the collection of ensemble forecast data produced by multiple global weather prediction centres that are involved in the TIGGE project. Major meteorological centres from around the world collaborate to create the Thematic International Global Grid of Ensemble Prediction Systems (TIGGE), which offers ensemble prediction data to the international scientific community (Buizza 2018; Dutta & Bhattacharjya 2022). It has grown to be an important tool for predicting the weather and studying the climate. Globally, there has been a significant amount of research done into the performance assessment and verification of post-processing techniques for TIGGE ensemble data.

The calibration of ensemble predictions is one of the fundamental problems that researchers try to solve. Since biases and under-dispersion in ensemble models are frequently present, calibration methods must be used (Fan et al. 2014; Oyerinde et al. 2022). To calibrate TIGGE ensemble data, Hong & Lim (2019) introduced a Bayesian model averaging (BMA) approach and showed notable improvements in forecast sharpness and reliability. In addition to calibration, researchers have looked into ensemble post-processing methods based on machine learning and statistical downscaling. In order to post-process TIGGE ensemble predictions, Wang et al. (2020) proposed a novel machine learning framework, significantly lowering forecast errors and raising the accuracy of weather forecasts. Researchers have also looked into how various meteorological factors and lead times affect the efficacy of post-processing techniques (Saouabe et al. 2022; Trinh et al. 2023). The research by Smith et al. (2018) examined the variable-dependent performance of post-processing techniques for TIGGE data, offering insightful information about the difficulties and potential solutions associated with variables in ensemble forecasting. Researchers have looked into using deep learning methods to post-process TIGGE ensemble data as the machine learning field has continued to advance. Using a deep convolutional neural network (CNN) for precipitation post-processing, Chen et al.'s study from 2021 achieved remarkably improved precipitation forecast accuracy.

In order to improve the reliability and usefulness of raw ensemble forecasts for decision-makers, post-processing techniques are essential. These techniques include statistical calibration, ensemble model output statistics (EMOS), BMA, and machine learning-based methods, among many others (Hamill et al. 2008; Daniel 2023). Meteorologists can eliminate biases, increase forecast accuracy, and produce probabilistic forecasts, which are essential for decision support. Despite the significance of post-processing, there is a significant research gap when it comes to systematically assessing and verifying the effectiveness of different post-processing methods on ensemble meteorological data, particularly using machine learning approaches (Kaur & Kaur 2023).

The uncertainty in weather forecasts is inherently captured by ensemble forecasting (Wilks & Hamill 2007). However, it is still difficult to accurately quantify and effectively communicate this uncertainty to end users. In addition to improving forecast accuracy, post-processing techniques should offer probabilistic data to aid in decision-making in areas like disaster management, agriculture, and energy planning (Liang et al. 2013; Liu et al. 2017). The performance of various machine learning techniques for ensemble post-processing will be thoroughly examined and evaluated in this study while taking a variety of meteorological variables into account (Raftery et al. 2005; Das et al. 2022). To enable better decision-making in selecting the most appropriate techniques for various meteorological applications, a robust evaluation framework will be established to consistently compare and verify the efficacy of various post-processing methods (Enayati et al. 2021).

This study aims to develop post-processing methods that are adaptable and useful across various forecasting scenarios and geographical regions by conducting experiments on a wide range of meteorological variables and datasets. This research aims to provide probabilistic forecasts that convey the uncertainty inherent in ensemble predictions, in addition to improving forecast accuracy, to support more informed decision-making in a variety of sectors dependent on precise weather forecasts. By bridging the gap between machine learning and meteorology, this research has the potential to advance the field of ensemble forecasting. This study makes a number of contributions by methodically assessing and validating machine learning-based post-processing techniques on TIGGE ensemble data. The study's findings will have real-world implications for operational meteorological forecasting, helping to produce weather predictions that are more accurate and dependable.

In summary, this study fills a critical research void in ensemble post-processing for meteorological data by fusing machine learning techniques with thorough evaluation methods to improve forecast reliability, accuracy, and the efficient communication of uncertainty. By making these efforts, this study hopes to advance meteorology and benefit a variety of industries that depend on precise weather forecasts. The primary aim of this study is to conduct a comprehensive assessment and verification of post-processing techniques that have been developed for TIGGE ensemble data. This will be achieved by utilizing advanced machine learning methodologies. The TIGGE dataset offers a substantial amount of meteorological ensemble data that plays a critical role in the fields of weather forecasting and climate studies. Nevertheless, the effectiveness of post-processing techniques can be utilized to enhance the accuracy and reliability of TIGGE ensemble predictions. The objective of this study is to conduct a comprehensive evaluation of different post-processing techniques, such as calibration, bias correction, and uncertainty quantification, in order to enhance the predictive accuracy of TIGGE ensemble outputs. The ultimate goal is to enhance decision-making in weather-related applications.

The Upper Sabarmati River basin in Gujarat, India, is chosen as the primary research area due to its distinct geographical and climatic features, as well as its crucial role in water resource management in the region. The basin covers an area of about 21,674 km2 and includes the Sabarmati River and its tributaries. The river has a total length of about 371 km, of which 216 km are in Gujarat. The basin includes several major tributaries of the Sabarmati River, including the Wakal River, Meshwo River, and Shedhi River. The basin also includes several important reservoirs, including the Dharoi Dam and the Sabarmati Riverfront Development Project.

The basin is subdivided into the Sabarmati Upper Sub-Basin and the Sabarmati Lower Sub-Basin. An additional 51 watersheds have been established, with each watershed representing a distinct tributary system. The interstate river system comprising the Sabarmati and its tributaries flows through the states of Rajasthan and Gujarat. The Sabarmati River's drainage network is comprised of five significant tributaries. The basin is approximately triangular in form, with the source of the Watrak River situated at its apex and the Sabarmati River serving as its base. Sabarmati flows at an elevation of 762 m from the Aravalli highlands near the village of Tepur in the Udaipur district of Rajasthan. The river spans a distance of 371 km from its source to its estuary in the Arabian Sea. Wakal, Hathmati, and Watrak are its primary tributaries that flow to the left of the river, while Sei empties into it from the right. Sei is a right-bank tributary. It starts in Rajasthan's Aravalli highlands and runs southwest for 74.89 km before joining Sabarmati on its right bank. The drain covers 946 km2. Wakal is a left-bank tributary. It starts in Rajasthan's Aravalli highlands and runs southwest for 88.54 km before joining Sabarmati on its left bank. The drain covers 1,625 km2. Menas is its primary tributary. Harnav is a left-bank tributary. It starts in Rajasthan's northern Kulalia hills and runs southwest for 59.07 km to join Sabarmati's left bank. The drain covers 972 km2. Hathmati is a left-bank tributary. From the southwest foothills of Rajasthan and Gujarat, it runs southwest for 118.27 km to reach the Sabarmati on its left bank. Hathmati River sub-tributaries Ghuvai and Boroli. This tributary drains 1,526 km2. Watrak is a left-bank Sabarmati tributary. It springs in the Panchara hills near Dungarpur, Rajasthan, flows southwest for 231.69 km and meets Sabarmati on the left bank. Watrak and its tributaries drain 8,638 km2.

The upper Sabarmati basin shown in Figure 1, which spans approximately 4,208 km2, serves a crucial function in maintaining the socio-economic and ecological balance of the region (Patel et al. 2023). The Upper Sabarmati River basin exhibits a semi-arid climate, which is distinguished by unpredictable rainfall patterns and heightened susceptibility to drought conditions (Patel & Yadav 2022). Consequently, this region is exceptionally prone to the effects of climate change.
Figure 1

Location map of the study area.

Figure 1

Location map of the study area.

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The research conducted in this study utilized a rigorous methodology to assess the efficacy of different post-processing techniques on ensemble data sourced from two well-established entities. For the purpose of this investigation, data from the National Centres for Environmental Prediction (NCEP) and the European Centre for Medium-Range Weather Forecasts (ECMWF) were utilized. These data were obtained from the TIGGE (THORPEX Interactive Grand Global Ensemble) portal and covered the time period from 2010 to 2020. The worldwide recognition of the consistently strong performance of these two datasets' ensemble data in research applications was the driving force behind the decision to concentrate their attention solely on these two datasets. In addition, the fact that these datasets contained detailed monthly data for the period of time that was requested made it possible for us to conduct a research study that was both robust and representative. The dataset covered a wide range of time, from 2010 to 2020, and included a lead time of 1–5 days. This allowed for a comprehensive evaluation of post-processing techniques for various forecast horizons. The research began by obtaining ensemble data from ECMWF and NCEP for the designated time period and lead times. The datasets offered a comprehensive and varied collection of meteorological information, which was crucial for the subsequent analysis. The incorporation of data from both the ECMWF and NCEP centres was done to provide a more comprehensive viewpoint, taking into account the differences in model formulations and data assimilation techniques employed by each centre. In order to assess and compare the efficacy of different post-processing methods, a set of techniques was chosen, with a primary emphasis on bias correction. The selected techniques encompass BMA, Model Output Statistics (MOS), censored Nonhomogeneous Linear Regression (cNLR), Heteroscedastic eXtreme Linear Regression (HXLR), Quantile Mapping (QM), logistic regression (logreg), and heteroscedastic logistic regression (hlogreg). Each method presented herein showcases a distinct approach towards mitigating biases and enhancing the precision of ensemble forecasts.

The post-processing workflow shown in Figure 2 refers to the series of steps involved in processing and enhancing data or files after they have been initially generated or captured. The research workflow comprises three essential stages for each post-processing method. Model Fitting: The chosen post-processing technique was implemented on the original ensemble data, and model parameters were estimated to capture the inherent biases and relationships present in the data. Model Prediction: The post-processing model was fitted and used to generate predictions for each ensemble member. This process effectively corrected biases and improved the accuracy of the forecasts. Model Verification: In order to thoroughly evaluate the effectiveness of the post-processing methods, a series of verification measures were utilized. The metrics utilized in this analysis encompassed the Brier Score, Ranked Probability Score (RPS), and Continuous Ranked Probability Score (CRPS). The statistical post-processing of the ensemble forecasts is required for better and more accurate calibrated forecasts, as evidenced by the average Brier Score, which indicates a 50–70% improvement overall from raw to post-processed forecasts (Yadav & Yadav 2023). The metrics offered a comprehensive assessment of the calibration, sharpness, and overall predictive accuracy of the methods. Reliability Analysis: A reliability analysis was performed to determine the reliability of the ensemble forecasts after post-processing. The process entailed evaluating the alignment between forecast probabilities and observed frequencies, providing valuable insights into the calibration of the forecasts.
Figure 2

Methodological flowchart of the post-processing of the ensemble precipitation data.

Figure 2

Methodological flowchart of the post-processing of the ensemble precipitation data.

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The Receiver Operating Characteristic (ROC) and Area Under the Curve (AUC) are two important metrics used in the field of data analysis and machine learning. The ROC curve is a graphical representation of the performance of a binary classifier system as its discrimination threshold is varied. It plots the true positive rate (sensitivity) against the false positive rate (FPR) (1-specificity) at various threshold settings. The AUC, on the other hand, is a numerical measure. In order to assess the proficiency of the post-processed ensemble in predicting binary events, an ROC analysis was performed and the AUC was computed. The measure provided an assessment of the model's capacity to differentiate between positive and negative events.

Post-processing of precipitation data refers to the various methods and techniques used to improve the accuracy, completeness, and reliability of precipitation measurements obtained from different sources, such as rain gauges, radars, and satellites. Precipitation data is essential for many applications, such as weather forecasting, hydrological modelling, climate research, and water resource management. However, precipitation data can be affected by errors, biases, and missing values, which can reduce the accuracy and reliability of the measurements. Therefore, post-processing of precipitation data involves applying various techniques to correct errors, fill gaps, merge data, detect non-climatic changes, and analyze extreme precipitation events. Post-processing techniques are based on statistical and mathematical models that account for various factors affecting precipitation measurements, such as gauge undercatch, wind, and instrument errors. The quality of precipitation data is critical for ensuring accurate and reliable information for decision-making in various sectors. Therefore, post-processing of precipitation data is an important task that requires specialized knowledge and skills in various areas, such as statistics, hydrology, and atmospheric science.

Errors in precipitation measurements can occur due to various factors, such as gauge undercatch, wind, and instrument errors. Post-processing techniques aim to correct these errors and biases to obtain accurate and reliable precipitation measurements. Precipitation data may have missing values due to various reasons, such as equipment failures, power outages, and maintenance issues. Post-processing techniques aim to fill these gaps using various interpolation and extrapolation techniques. Changes in instrumentation, measurement methods, or station locations can cause non-climatic changes in precipitation data. Post-processing techniques aim to detect and correct these changes to ensure that the data is consistent and suitable for various applications. Extreme precipitation events, such as heavy rainfall, floods, and droughts, can have significant impacts on various sectors, such as agriculture, water resource management, and public safety. Post-processing techniques aim to analyze the frequency and magnitude of extreme precipitation events to better understand their characteristics and impacts. Figure 3 shows various types of post-processing methods.
Figure 3

Types of post-processing methods.

Figure 3

Types of post-processing methods.

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Figure 4

Ensemble model fitting plots in R studio.

Figure 4

Ensemble model fitting plots in R studio.

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For post-processing of the ensemble data, a total 10-year data sets were taken from 2010 to 2020. Of this, 70% were training data and 30% were testing data in the machine learning model. For calibration purposes 2015, 2016, 2017, and 2018 flood event years were considered while for validation of the model, years 2019 and 2020 were taken to check the model with the observed data set.

Nonhomogenous regression

Nonhomogeneous regression is a type of statistical modelling technique used in post-processing of precipitation data. It involves modelling the relationship between precipitation and one or more predictor variables that vary across space and time, such as topography, latitude, or seasonality. Unlike homogenous regression, which assumes that the relationship between the predictor and response variables is constant across the entire domain of interest, nonhomogeneous regression allows for the relationship to vary spatially and/or temporally. This can lead to more accurate predictions of precipitation at a particular location and time.

Bayesian model averaging

The conversion of data into ‘ensembleData’ objects is a prerequisite for utilizing the fitBMA() function, which is available in R through the ensembleBMA package. In order to carry out BMA different packages like ensembleBMA, gBMA, caret, etc, have been installed. For the purpose of quantitative precipitation forecasting, Sloughter et al. (2007) envisioned using a BMA approach. Therefore, rather than Gaussian distributions, the component distributions are the combinations of gamma distributions and point masses equal to zero. The following are some possible ways to articulate the process of producing predictive distributions of BMA:
(1)

Logistic regression

Logistic regression is a statistical method used for modelling the relationship between a binary dependent variable (i.e., a variable that can take on only two possible values) and one or more independent variables. In the context of post-processing of precipitation data in R studio, logistic regression can be used to model the probability of precipitation exceeding a certain threshold, based on predictor variables such as temperature, humidity, and wind speed. This can be useful for generating probabilistic precipitation forecasts that can help inform decision-making in various applications, such as agriculture, water resource management, and emergency response planning. R has built-in functions for fitting logistic regression models, such as glm(), and there are also several packages available for more advanced modelling and analysis, such as caret, tidyverse, and ggplot.
(2)

Heteroscedastic logistic regression (hlogreg)

In the hlogreg method, the ensemble mean is used for the regressor for location, and the ensemble standard deviation or variance is considered for the regressor for scale (Gneiting et al. 2005). Both of these regressors are taken into consideration for the location. Because of the common slope parameter b in Equation (3), the regressions for all quantiles are forced to be parallel. On the other hand, the individual logistic regressions cross, which results in cumulative probability specifications for smaller precipitation amounts being greater than those for larger amounts in some cases. On the log-odds scale, the hlogreg is represented as the following:
(3)

Heteroscedastic extended logistic regression

The HXLR, which was developed by Wilks (2009), happens to fit a single equation across all threshold values. The additional predictor variable is the threshold values, and it is assumed that the regression coefficients bq are identical. Additionally, it is emphasized that the regression intercepts should be a function that increases as the target quantile.
(4)
where g(q)=q0.5. The HXLR provides comprehensive continuous predictive distributions (CPDs) to reduce the number of regression coefficients and prevent the occurrence of negative probabilities. The linear function g(q) is what determines the intercepts of the HXLR equation. The use of the ensemble spread as the scale parameter in order to fine-tune the dispersion of CPD is one of the benefits offered by the HXLR algorithm. As a result, this method has the capacity to sufficiently exploit the ensemble spread.

Ordered logistic regression

Another method that is closely related to logreg is called OLR (Messner et al. 2014; Wilks 2018). This method relates to a finite set of thresholds and prevents regression lines from intersecting by mandating that they must be parallel. The difference between OLR and extended logistic regression is that OLR does not make the assumption of a continuous distribution. Following this, the homoscedastic OLR forecasts are formulated as follows:
(5)

OLR also provides accurate forecasts of the probabilities associated with categories (Messner et al. 2017). In contrast to HXLR, it possesses ordered intercepts and necessitates the estimation of a greater number of coefficients. On the other hand, in OLR, there is no CPD that is presumed or specified by the model, which distinguishes it from HXLR. In addition, the OLR models can only be used to derive the probability of exceedance forecasts; they are unable to facilitate density or quantile predictions.

Model fitting

In the context of post-processing of precipitation data in R studio, model fitting involves the process of training statistical models to make more accurate and skilful predictions of precipitation. The goal of model fitting is to develop models that can better capture the true underlying patterns and variability in the precipitation data, leading to improved predictions of precipitation in the future. There are many different types of models that can be fitted to precipitation data, including regression models, time series models, machine learning models, and Bayesian models. The choice of model depends on the specific research question, the characteristics of the precipitation data, and the desired level of complexity and interpretability.

The process of model fitting involves selecting an appropriate model structure, estimating the parameters of the model using a training dataset, and evaluating the performance of the model using a separate validation dataset. This process is typically iterative, with multiple model structures and parameterizations being evaluated and compared to identify the best-performing model.

R studio provides a wide range of tools and packages for model fitting in the post-processing of precipitation data. These include built-in functions for fitting regression models, time series models, and generalized linear models, as well as packages for fitting machine learning models and Bayesian models. Overall, model fitting is an essential step in the post-processing of precipitation data, as it enables researchers to develop more accurate and skilful predictions of precipitation that can be used to inform water resources management, flood forecasting, and other applications. To generate continuous probabilistic precipitation forecasts, non-Gaussian nonhomogeneous regression methods, BMA with non-Gaussian component distributions, and extended logistic regression can be used. In this research work, model fitting has been carried out for three gauging station data of Dharoi, Jotasan and Kheroj for different lead day time.

Model fitting refers to the process of calibrating or adjusting the parameters of a hydrological model to best represent observed data as shown in Figure 4. This involves comparing the model's simulated output to actual measurements or observations and iteratively adjusting the model parameters until a satisfactory fit is achieved.

  • (i) Scatterplot

A scatterplot is a graphical representation that displays the relationship between two variables. In the case of model fitting, a scatterplot is often used to compare the model's predicted values (x-axis) with the corresponding observed values (y-axis). Each point on the scatterplot represents a specific data pair. By examining the dispersion and pattern of points, you can assess the model's ability to reproduce the observed data. A strong positive correlation between the predicted and observed values indicates a good model fit.

  • (ii) Verification rank histogram

The verification rank histogram is a diagnostic tool used to evaluate the skill of ensemble forecasts. It displays the frequency distribution of ranks assigned to the observed value within an ensemble forecast. In an ideal scenario, the histogram should have a uniform shape, indicating that the observed value is equally likely to fall within any rank range. Deviations from the uniform shape suggest either under-dispersion or over-dispersion in the ensemble spread. Under-dispersion means that the ensemble spread is narrower than it should be, while over-dispersion means the spread is wider.

  • (iii) Spread skill relationship

The spread skill graph assesses the reliability of ensemble forecasts. It plots the spread (i.e., variability) of the ensemble members on the x-axis against the forecast error on the y-axis. The spread represents the range of predictions within the ensemble, while the forecast error measures the discrepancy between the ensemble mean and the observed value. The spread skill graph helps determine if the ensemble spread is correlated with forecast accuracy. A desirable characteristic is that an increase in spread is associated with an increase in forecast error, indicating that the ensemble captures uncertainty effectively.

  • (iv) Histogram of transformed precipitation observations

A histogram is a graphical representation that shows the distribution of a variable. In the context of your research, it is likely used to display the distribution of forecast errors. The histogram provides insights into the bias and spread of the model's predictions. Ideally, the histogram should be centred around zero, indicating that the model is unbiased. Additionally, a narrower histogram suggests lower variability (spread) of forecast errors, which indicates higher precision in the model's predictions.

By analyzing these graphs, anyone can gain a better understanding of the performance of the ensemble precipitation and hydrological model. They provide valuable insights into the model's ability to capture observed data, the spread of ensemble forecasts, and potential biases or errors. These evaluations are essential for assessing the reliability and accuracy of your model and making informed decisions regarding reservoir inflow prediction.

Figure 5 shows the comparison of model fitting plots in R studio for 1- to 5-day lead time for three stations Dharoi, Jotasan and Kheroj. It has been very clear that 3-day lead time graphs show good results in terms of the scatterplot, verification rank histogram and spread skill. These plots are useful for selecting the best model for the reservoir inflow prediction as this process is very important in post-processing of the ensemble data sets.
Figure 5

Ensemble model fitting plots for 1- to 5-day lead time for three grid points.

Figure 5

Ensemble model fitting plots for 1- to 5-day lead time for three grid points.

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Model prediction

This section illustrates different methods for generating probabilistic forecasts, which can be expressed in various forms. While full continuous predictive distributions (see Figure 6), such as probability density plots, can be used, they can be challenging to display, especially for multiple forecasts. The predictive distributions produced by BMA and ensemble dressing comprise a combination of normal distributions, which are centred around the corrected ensemble forecasts. Hence, to obtain the corrected ensemble forecasts and standard deviation of the component distributions, certain calculations must be performed first. Although ensembleBMA does not offer a direct function for deriving the corrected ensemble, it can be easily achieved using matrix multiplication %*%, which can be applied to each of the ensemble members with the help of the apply() function. As part of the prediction plot, three different types of plots have been generated, namely BMA, censored Non-Linear Regression (cNLR) and heteroscedastic extended logistic regression (HXLR) which includes precipitation and predicted forecast (PDF), have been shown in the following.
Figure 6

Probability of precipitation plot for model prediction 2015–2020 for 1- to 5-day lead time for three grid points.

Figure 6

Probability of precipitation plot for model prediction 2015–2020 for 1- to 5-day lead time for three grid points.

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Probability of Precipitation (PoP) plots

Probability of Precipitation (PoP) plots represent the likelihood or chance of rainfall occurring at a particular location during a given time period. PoP values are usually expressed as percentages. These plots provide a graphical representation of the predicted probabilities of rainfall, which can help in assessing the uncertainty associated with precipitation forecasts. Typically, PoP plots are presented as a time series or spatial map, where the x-axis represents time or geographical locations, and the y-axis represents the probability of precipitation. Each point on the plot or map represents the PoP value at a specific time or location. By analyzing the PoP plot, anyone can identify periods or regions with higher or lower chances of rainfall.

Figure 6 shows the model prediction plots for the probability of precipitation for the 5-day lead time. The dashed line shows the median prediction by the post-processing method applied to the ensemble data. Gray colour shading in the graph shows a predictive interquartile range while the solid line shows the observations of precipitation. Subsequently, the other graph shows the probability of precipitation which is a solid line and corresponding observation circles at 1 for occurrence and at 0 for non-occurrence. It is also observed from Figure 6 that for the Dharoi and Kheroj station grid points all the corresponding observation circles are at 1 which represents precipitation occurrence for the day. For the Jotasan grid point, there are a few circles observed at 0 for non-occurrence. Figure 6 also shows that for 1–3 days of lead time, the graph range is very well maintained with the prediction and observation range compared to the 4-day and 5-day lead time for precipitation prediction.

Predictive density plots

Predictive density plots shown in Figure 7 are used to visualize the probability distribution of the predicted precipitation values. They help to assess the uncertainty associated with the predictions made by different methods or models. The plot shows the square root of precipitation on the x-axis and the probability density function (PDF) on the y-axis, representing the relationship between the square root of precipitation values and their corresponding probability densities. The square root transformation is often applied to precipitation data to stabilize variance and handle skewness. By examining the predictive density plots, you can gain insights into the distribution of predicted precipitation values. The plots show the shape, spread, and central tendency of the distribution. Different methods (cNLR, BMA, HXLR) will yield different predictive density plots, allowing you to compare the performance and uncertainty associated with each method. A wider distribution indicates higher uncertainty, while a narrower distribution suggests more confidence in the predicted values.
Figure 7

Predictive distribution using cNLR, BMA, and HXLR for three grid points from 2015 to 2020.

Figure 7

Predictive distribution using cNLR, BMA, and HXLR for three grid points from 2015 to 2020.

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The Probability of Precipitation (PoP) plot presented in Figure 6 represents the probabilistic distribution of precipitation forecasts derived from the TIGGE Ensemble data. Each curve in the plot corresponds to a different lead time, illustrating the evolution of forecasted probabilities over time. In the plot, the x-axis denotes the time span of the event of flood, while the y-axis represents the square root of precipitation. The curves depict the model's ability to capture the uncertainty associated with precipitation predictions. A well-calibrated model would align closely with the diagonal line, indicating that forecasted probabilities match observed frequencies. The explanation of the plot involves detailing the calibration and reliability of the model. Calibration assesses whether the forecasted probabilities accurately represent the likelihood of precipitation occurrence. Reliability evaluates the consistency between predicted probabilities and actual outcomes across different forecast scenarios. Figure 6 shows the 3-day lead time graphs following the trend of the diagonal line and also follows the observed solid line with the prediction probabilities. While the 4-day and 5-day are the worst performers compared to other lead times. In addition to this, 2-day and 1-day lead time are average performers.

Figure 7 illustrates the predictive distribution using three distinct post-processing methods: cNLR, BMA, and HXLR. Each curve in the plot represents the PDF of precipitation at a specific lead time, generated by the respective post-processing method. The technical explanation for this figure involves a detailed discussion of the methodologies underlying cNLR, BMA, and HXLR. cNLR integrates machine learning techniques with physical constraints to enhance predictive accuracy, whereas BMA combines multiple models to provide a more robust and probabilistic forecast.

Comparative analysis in the manuscript elucidates the strengths and weaknesses of each method, detailing their performance characteristics. Specifically, we delve into the reasons behind the observed superiority of cNLR, particularly in comparison to BMA and HXLR. We explore the impact of model complexity, training data, and feature selection on the predictive capabilities of each approach.

Analyzing the predictive density plots can help you understand the characteristics of the predicted precipitation distribution for each method and compare their performance in terms of accuracy and uncertainty estimation. Black solid lines show the predictive densities and vertical orange lines show the verifying observations. For BMA also the predictive median (thin vertical line), 0.1–0.9 quantiles (dashed lines), and the member distributions (coloured curves) are shown.

Moreover, we address the noteworthy finding that the 3-day lead time yields superior results compared to other lead times. We examine temporal dynamics, model sensitivity, and meteorological factors contributing to this trend, providing a nuanced understanding of lead time-dependent model performance.

Precipitation is a key component in real-world flood prediction models. The NWP from ECMWF and NCEP is evaluated for the Western river basin of India, where the majority of the annual rainfall takes place from June to September during the monsoon season. Around 85% of the annual rainfall falls in the months of June to September in the Sabarmati basin region, which is mostly affected by the southwest monsoon. The performance criteria are used to compare the performance of the post-processing methods employed in the article, while verification metrics like Brier score, Rank Probability Score (RPS), CRPS, AUC, and ROC plots, and reliability diagram criteria are used to assess the validity of the forecasts.

Ensemble model verification

Verification is the process of evaluating the accuracy and reliability of precipitation data generated by a forecasting model or a measurement system. The verification process is essential for ensuring the quality of precipitation data used in post-processing activities such as forecasting, climatology, and hydrological modelling. In this case, mainly two methods of verification have been adopted.

Brier Score: The Brier Score is a measure of the accuracy of probabilistic forecasts, including precipitation forecasts. It compares the predicted probability of precipitation occurring at a particular location with the actual observation (0 for no precipitation, 1 for precipitation). A perfect forecast will have a Brier Score of 0, while a completely random forecast will have a Brier Score of 0.25. The lower the Brier Score, the better the forecast.

RPS: This assesses the performance of a model in predicting the distribution of precipitation occurrence over a set of categories. It evaluates the model's ability to correctly rank the observed precipitation in comparison to the forecasted probabilities for each category. A perfect forecast will have an RPS score of 0, while a completely random forecast will have an RPS score of 1. The lower the RPS score, the better the forecast. In summary, verification is a critical step in evaluating the accuracy and reliability of precipitation data used in post-processing activities. The Brier score and RPS are two widely used methods of verification to assess the accuracy of precipitation forecasts. The following graphs (see Figure 8) shows the Brier score and RPS. All forecast methods exhibit good calibration, as indicated by the reliability diagrams displaying a calibration function that closely resembles the diagonal line. The Brier score decomposition results confirm this reliability, with the BMA showing the greatest deviation from the diagonal line, but still within the range of consistency.
Figure 8

Brier score for 1–5 days of lead day time at Dharoi from 2015 to 2020.

Figure 8

Brier score for 1–5 days of lead day time at Dharoi from 2015 to 2020.

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Brier score plot

The Brier score is a widely used scoring metric to evaluate the accuracy and skill of probabilistic forecasts. It measures the mean squared difference between the predicted probabilities and the actual binary outcomes (in your case, the occurrence or non-occurrence of precipitation). A lower Brier score indicates better forecast performance.

A Brier score plot is a graphical representation of the Brier scores calculated for a range of probability thresholds. It helps in assessing the performance of different post-processing methods in terms of their ability to forecast precipitation. The plot typically consists of the probability threshold values on the x-axis, ranging from 0 to 1, and the corresponding Brier scores on the y-axis. By analyzing the Brier score plot, you can identify the thresholds at which the methods exhibit the lowest Brier scores, indicating the most skilful forecasts.

Interpreting the Brier score plot involves comparing the performance of different post-processing methods. Look for methods that consistently show lower Brier scores across various thresholds, as they are more accurate and reliable in predicting precipitation.

From Figure 8 it is observed that the 2-day lead time with hlogreg and logreg methods shows results close to 0 which is a median Brier Score of 0.11 and its lower whisker Brier Score is 0.08 and 0.082. So these are very excellent results for the model. It shows the good performance of the model. It also shows that a 3-day lead time also gives good performance and its median Brier Score is 12.5 for the hlogreg method while the lower whisker Brier Score is 0.10 which is a good performance of the model.

In a similar way, Figure 8 observed that 2-day and 3-day lead time results using logreg and hlogreg methods of post-processing shows good performance for the predicted model for grid points Jotasan and Kheroj. It is also observed from Figure 8 that there is no outliner found in the 2 days of lead time for the Brier Score and few outliners observed in the 1 day and 3 days of lead time for all three grid points.

RPS plot

The RPS as shown in Figure 9 is another widely used metric to evaluate the quality of probabilistic forecasts. It quantifies the cumulative difference between the observed and predicted cumulative probability distributions. A lower RPS indicates better forecast skill. The RPS plot illustrates the RPS values as a function of forecast lead time or probability thresholds. Similar to the Brier score plot, it helps in comparing the performance of different post-processing methods based on their RPS scores. Analyzing the plot allows to identify the lead time or probability threshold where each method performs best. To interpret the RPS plot, examine the trend of the scores over different lead times or thresholds. Look for methods with consistently lower RPS values, indicating higher forecast skill. Comparing the slopes of the curves can help identify the methods that improve forecast skill at different lead times or probability thresholds.
Figure 9

RPS for 1- to 5-day of lead time at Dharoi from 2015 to 2020.

Figure 9

RPS for 1- to 5-day of lead time at Dharoi from 2015 to 2020.

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Continuous Rank Probability Score

The Continuous Ranked Probability Score (CRPS) is another widely used scoring metric to evaluate the performance of probabilistic forecasts. It measures the integrated squared difference between the cumulative distribution function (CDF) of the predicted probabilities and the CDF of the observed outcomes. Similar to the Brier score, a lower CRPS indicates better forecast performance. The CRPS plot displays the calculated CRPS values across a range of forecast probabilities or forecast lead times. In a CRPS plot, the x-axis represents the forecast probabilities or lead times, while the y-axis represents the corresponding CRPS values. By examining the CRPS plot, you can assess the skill of the post-processing methods in capturing the observed precipitation distribution. When interpreting the CRPS plot, look for methods that consistently exhibit lower CRPS values across different forecast probabilities or lead times. Lower CRPS values indicate more accurate and skilful forecasts.

Reliability diagram

A reliability diagram, also known as a calibration plot or a probability integral transform histogram, assesses the calibration or reliability of probabilistic forecasts. It helps to determine if the predicted probabilities are consistent with the observed frequencies. In the context of the probability of precipitation, a reliability diagram compares the predicted probabilities of precipitation (x-axis) with the observed frequency of precipitation (y-axis). The diagram is divided into bins or intervals of predicted probabilities, and for each bin, the average observed frequency is calculated. By analyzing the reliability diagram, the calibration of the post-processing methods in terms of their ability to provide accurate and reliable probabilities of precipitation. A well-calibrated method will exhibit a diagonal line or a close match between the predicted probabilities and observed frequencies. To interpret the reliability diagram, compare the position of the plotted points with the diagonal line. Points above the line indicate over-forecasting (higher predicted probabilities than observed frequencies), while points below the line indicate under-forecasting. The closer the points are to the diagonal line, the better the calibration of the method. For statistical analysis in research, a comparison of the performance of different post-processing methods based on their Brier scores, CRPS values, and reliability diagram is required. Identify the methods with lower Brier scores and CRPS values, indicating higher accuracy. Additionally, examine the reliability diagram to assess the calibration of the methods and choose those that provide well-calibrated probability estimates.

Figure 10 shows the reliability diagram plots for all six methods cNLR, BMA, logreg, hlogreg, HXLR and OLR for three grid point stations Dharoi, Jotasan and Kheroj. Figure 10 shows the graphs of reliability plots between the observed frequency and forecast probability. It shows that 1-day, 2-day and 3-day lead time graphs show good consistency and calibration function close to the diagonal line. It has been also observed that 4-day and 5-day lead time graphs are deviated from the diagonal line in all the cases so it represents it is not in good agreement with the observation of the results. In Figure 10 it has been observed that the grey bars shown in the plot are the consistency bars from the bootstrap resampling. The sharpness plot displays the relative frequency of events in each forecast probability bin. The vertical bars at the diagonal represent consistency bars of 95% confidence intervals that have been calculated with a quantile function for a binomial distribution.
Figure 10

Reliability diagram plot for model verification 2015–2020 for 1- to 5-day lead time for three grid points.

Figure 10

Reliability diagram plot for model verification 2015–2020 for 1- to 5-day lead time for three grid points.

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ROC diagrams

ROC diagrams shown in Figure 11 are a graphical representation of the performance of a binary classification system. A binary classification system is one that assigns one of two possible outcomes, such as yes or no, to each input. The ROC curve is created by plotting the true positive rate (TPR) against the FPR at various thresholds. The TPR is the proportion of actual positives that are correctly identified as such, while the FPR is the proportion of actual negatives that are incorrectly identified as positives. The ROC curve is useful because it provides a visual representation of the trade-off between sensitivity (TPR) and specificity (true negative rate). The diagonal line represents the performance of a random classifier, while a perfect classifier would have an ROC curve that hugs the top-left corner of the graph. The area under the ROC curve (AUC) shown in Figure 11 is a measure of the overall performance of the classification system. The AUC ranges from 0 to 1, with a value of 0.5 indicating random guessing and a value of 1 indicating perfect classification. The AUC method provides a quantitative measure of how well a binary classification system can distinguish between the positive and negative classes. AUC is particularly useful when the classes are imbalanced, meaning that one class is much more prevalent than the other. In such cases, a classifier that simply assigns all inputs to the majority class would have a high accuracy, but a low AUC, indicating poor performance.
Figure 11

ROC and AUC plots for model verification 2015–2020 for 1- to 5-day lead time for three grid points.

Figure 11

ROC and AUC plots for model verification 2015–2020 for 1- to 5-day lead time for three grid points.

Close modal

The cNLR method demonstrates good performance as it shows a value of 0.9187, consistent with the results obtained from the Brier score. On the other hand, the standard logistic regression (logreg) exhibits relatively good performance, particularly when compared to the heteroscedastic logistic regression (hlogreg). This suggests that leveraging the ensemble spread provides an advantage in performance.

The cNLR method demonstrated superior performance across multiple methods of post-processing, as highlighted by the RPS, Brier score, and ROC–AUC values. Specifically, cNLR exhibited a notable RPS of 0.11, indicative of its ability to accurately rank forecast probabilities. The low Brier score of 0.10 suggests the model's overall accuracy and reliability, while the ROC–AUC value of 0.9187 further reinforces its effectiveness in discriminating between positive and negative events.

To elaborate further on the rationale behind the preference for cNLR over the other five methods, we would like to emphasize the comprehensive nature of our evaluation. The combination of these metrics provides a holistic view of the model's performance, encompassing aspects of calibration, discrimination, and overall reliability. The consistency of cNLR's high scores across these diverse measures underscores its robustness and suitability for the given ensemble data post-processing task.

Moreover, the cNLR method aligns well with the specific characteristics of the TIGGE Ensemble data, as evidenced by its superior performance in capturing and addressing the inherent complexities of the dataset.

In conclusion, the selection of cNLR as the preferred method is rooted in its exceptional performance across a spectrum of crucial verification parameters, offering a well-rounded and reliable solution for ensemble data post-processing in the context of TIGGE data.

The following distinguishing features of the post-processing approach are unique:

  • a.

    Integration of machine learning approaches: The methodology incorporates state-of-the-art machine learning methods into the post-processing of TIGGE Ensemble data, namely cNLR, BMA, HXLR, logreg, hlogreg and OLR. The distinctiveness resides in the way these techniques work together harmoniously, utilizing their complimentary advantages to improve precipitation forecast accuracy and dependability.

  • b.

    Designed for TIGGE ensemble data: The strategy is customized to the distinct qualities of TIGGE Ensemble data, in contrast to general post-processing techniques. Better predictive performance results from this customization, which makes sure that the inherent complexity of ensemble forecasts – such as spatial and temporal dependencies – are adequately addressed.

  • c.

    Flexibility in different lead times: Our approach performs well in a range of lead times, but it excels in the 3-day lead time when compared to other approaches. This flexibility makes our method more practically applicable in real-world forecasting situations.

Benefits of the suggested post-processing technique are as follows:

  • a.

    Improved calibration: This technique is very effective at mitigating biases, calibrating ensemble forecasts, and producing probabilistic predictions that are well-calibrated. Reliability is increased through the more sophisticated adjustment of forecast uncertainties made possible by the integration of machine learning.

  • b.

    Improved spatial and temporal consistency: The difficulty of preserving spatial and temporal consistency in ensemble predictions is addressed by the suggested method. By applying sophisticated machine learning algorithms, we are able to obtain a more consistent depiction of precipitation patterns in both space and time.

  • c.

    Quantification of predictive uncertainty: The method offers a comprehensive quantification of predictive uncertainty, in contrast to conventional post-processing techniques. For decision-makers, who need precise forecasts along with a clear grasp of the levels of uncertainty involved, this is crucial.

Evaluation in comparison with current works

  • a.

    Methodological differences: Work is carried out in a thorough comparative study with previous studies in the area of ensemble data post-processing for precipitation. Using a blend of cNLR, BMA, and HXLR, our technique stands out as a novel hybrid approach that captures various facets of uncertainty in ensemble forecasts.

  • b.

    Case studies and performance metrics: This study has included a thorough discussion of performance metrics in our manuscript, contrasting our approach with industry-standard post-processing methods. Furthermore, the study included comprehensive case studies that demonstrate the effectiveness of our method under various meteorological circumstances.

This research paper has explored the field of ensemble data post-processing, utilizing a machine learning-focused approach to thoroughly evaluate and compare different post-processing techniques. Ensemble modelling is a technique in machine learning that combines the knowledge of multiple base models to improve prediction accuracy. The effectiveness of ensemble predictions relies on the implementation of post-processing techniques to enhance and optimize the outputs of the ensemble.

This study has employed a comprehensive methodology that incorporates various datasets, ensemble construction strategies, and innovative evaluation criteria. As a result, it has provided valuable insights into the strengths and limitations of different post-processing approaches. The focus of this investigation was the utilization of ECMWF and NCEP ensemble precipitation data. Various post-processing techniques were applied for bias correction, utilizing the versatile R Studio. Among the methodologies that were investigated, the ECMWF combined with the cNLR approach demonstrated significant potential. Bias-corrected data were generated by applying a range of statistical downscaling methods, such as BMA, HXLR, cNLR, and others, to the years 2010–2020. This was accomplished using an ensemble model. The data were partitioned into training and testing sets using machine learning algorithms in R Studio. The split ratio used was 70% for training and 30% for testing. This process was applied to multiple gauging stations, namely Dharoi, Jotasan, and Kheroj.

To evaluate the accuracy of the projected and downscaled precipitation data, a range of comprehensive assessment metrics were utilized. These metrics included the Brier score and RPS. The Brier score exhibited outstanding performance for the cNLR method, specifically for the 3-day lead time at the Dharoi point, achieving a score of 0.10. The RPS produced favourable outcomes for the Jotasan point, achieving a score of 0.10 using the logreg method. Moreover, the reliability diagrams demonstrated that the 3-day lead time, when utilizing cNLR and BMA, displayed a high level of consistency with the diagonal line in comparison to alternative methods.

Additionally, the performance of the cNLR and BMA methods was demonstrated to be robust through the utilization of ROC and AUC plots. The ROC–AUC values for the cNLR and BMA methods were found to be 91.87 and 91.82%, respectively. Our comprehensive ensemble post-processing analysis has revealed that the cNLR and BMA methods are the most favourable choices for refining ensemble precipitation data. These metrics include the Brier score, RPS, reliability diagrams, ROC plots, and AUC plots.

This research makes a significant contribution to the field of ensemble modelling and post-processing. It also provides practical guidance for hydrologists and environmental scientists who are responsible for making informed decisions in water resource management. This study demonstrates the potential to improve predictive accuracy and reliability in important domains such as reservoir inflow forecasting by utilizing machine learning and statistical techniques. The findings have significant implications for various practical applications. As society confronts the complex issues arising from climate change and its effects on hydrology, the knowledge obtained from this research holds significant potential to enhance our ability to anticipate and handle water resource-related challenges.

The authors are thankful to the Civil Engineering Department, Sardar Vallabhbhai National Institute of Technology, Surat, for providing an opportunity to do research work. The authors are also thankful to CWC, Gandhinagar, and the State Water Data Centre, Gandhinagar, for their valuable support in data provision as well as guidance in this project. We would also like to extend our deepest gratitude to all those who have directly and indirectly funnelled us into this research work. The authors are thankful to the Civil Engineering Department, Institute of Technology, and Nirma University for providing support for this research work.

The authors declare that they received no funding for this research.

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

The authors declare there is no conflict.

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