1. Introduction
Floods are among the major causes of yearly infrastructure damage and loss of human lives across the globe (Tanoue et al. 2016; Wallemacq and House 2018). These events are especially impactful in India, where floods driven by summer monsoon (June–September) rainfall events—which contribute to more than 80% of total annual rainfall—are frequently observed and are caused by synoptic-scale cyclonic depressions (Hunt et al. 2016; Hunt and Fletcher 2019). In addition, an increase in frequency and intensity of extreme precipitation events is projected due to anthropogenic climate change (Wasko and Sharma 2017; Ali and Mishra 2018; Papalexiou and Montanari 2019). Thus, skillful daily streamflow forecasts on river networks are crucial for flood mitigation, especially in rainfall-driven river basins, as in the case of India during the monsoon season. However, efforts to develop early flood warning systems in India are limited (CWC 2012).
India’s current operational flood forecasting system is based on statistical models with gauge-to-gauge discharge correlations and coaxial correlations that use discharge and rainfall for most locations (CWC 2015). While gauge-to-gauge correlation-based streamflow forecasting frameworks may be effective for large river basins, they may also yield lower skill in basins with short times of concentration (Ghimire and Krajewski 2020; Krajewski et al. 2020). In view of this, the implementation of physically based hydrological models is an attractive alternative since they can be configured in a spatially distributed fashion, providing detailed information on the spatiotemporal evolution of states and fluxes (e.g., Ercolani and Castelli 2017; Jay-Allemand et al. 2020). Nevertheless, physically based models suffer from structural, parameter, and data errors, yielding biases in hydrologic model output (Liu and Gupta 2007; Clark et al. 2017). Thus, the development of postprocessing techniques (Li et al. 2017b) that allow reducing these errors is needed, although they are rarely implemented in an automated way (Pagano et al. 2014).
Postprocessing methods were originally developed in atmospheric sciences to correct numerical weather prediction (NWP) model output (e.g., Bremnes 2004; Glahn and Lowry 1972; Mendoza et al. 2015b). These methods have evolved for hydrological applications, either to improve the raw skill and accuracy of meteorological variables that force hydrology models (e.g., Clark et al. 2004; Verkade et al. 2013; Grillakis et al. 2018) or for direct application to hydrological models’ output (i.e., streamflow) for operational flood forecasting (e.g., Zalachori et al. 2012; Demargne et al. 2014; Muhammad et al. 2018). In principle, postprocessing techniques seek to estimate the conditional probability of a variable given “raw” (single value or ensemble) simulations or forecasts. In the case of hydrological variables such as streamflow, these methods have to deal with additional difficulties of heteroscedastic errors, poor representation of extreme events due to limited samples (conditional bias), and reduced skill in forecasts of streamflow peak timing, which are important for operational flood forecasting (Li et al. 2017b).
Several postprocessing approaches have been implemented for hydrological simulations or forecasts produced with physically based models, including nonparametric methods, regression-based techniques, ensemble dressing methods, and conditional distribution-based approaches (for a detailed review, see Li et al. 2017b). Nonparametric approaches attempt to map the target variable to historical distributions in order to reduce biases, and examples include quantile mapping (QM; Hashino et al. 2007; Lucatero et al. 2018; Tiwari et al. 2022) and conditional-bias-penalized indicator cokriging (CBP-ICK; Brown and Seo 2010, 2013). Regression-based methods used for hydrological forecasts include the General Linear Model postprocessor (GLMPP; Zhao et al. 2011; Ye et al. 2014, 2015), autoregressive models (AR; Bogner and Pappenberger 2011; McInerney et al. 2017, 2018; Woldemeskel et al. 2018), restricted autoregressive models (restricted AR; Li et al. 2015, 2016, 2017a; Bennett et al. 2021), and quantile regression (QR; Weerts et al. 2011; Tyralis et al. 2019). Conditional distribution-based methods applied for hydrological forecasts include the hydrological uncertainty processor (HUP; Krzysztofowicz and Kelly 2000; Reggiani et al. 2009), model conditional processor (MCP; Todini 2008; Anele et al. 2018), and Bayesian joint probability (BJP; Wang et al. 2009; Pokhrel et al. 2013; Zhao et al. 2015, 2016).
Since physically based hydrological models contain deterministic conservation equations that simulate hydrological processes and their interactions, ensemble simulations or forecasts can be produced when multiple initial conditions, parameterization options, parameter sets, or meteorological traces are provided. An example of the latter is the ensemble streamflow prediction (ESP) system (Day 1985)—used for operational purposes—which initializes a hydrologic model with current land surface hydrologic conditions and drives it with an ensemble of historical meteorological forcings. However, many agencies (e.g., National Weather Service) provide only a few quantiles, instead of the forecast’s distribution. Alternatively, some postprocessing approaches can provide forecast ensembles given deterministic raw forecasts. Among these, AR, restricted AR, BJP, and MCP can deal with the heteroscedasticity of the residuals through the use of variance stabilizing transformation and bivariate normal distributions (Zhao et al. 2015; McInerney et al. 2018; Li et al. 2015) or multivariate truncated normal distributions (Todini 2008; Anele et al. 2018). However, most of these approaches rely on the autocorrelation of residuals (streamflow time persistence), and thus a continuous record of observed streamflow is required for their implementation in real time, which can be an obstacle in basins where recent observations are not readily available.
Given the existing techniques, their limitation for real-time application, and the need for risk-based flood hazard mitigation, we build upon the Bayesian hierarchical network model developed by Ossandón et al. (2021) to propose a Bayesian hierarchical model (BHM) for postprocessing daily streamflow simulations across a river network. Namely, our BHM formulation aims to reduce residual errors arising from historical forcings, model structure, and parametric uncertainties, providing a starting point for future applications to real-time forecasting. Specifically, we seek to test if the proposed BHM framework (i) enhances the skill from a deterministic, physically based hydrological model; (ii) deals with the heteroscedasticity of the residuals; and (iii) is capable of correcting systematic and nonsystematic biases. To this end, we demonstrate the potential of the BHM to postprocess daily monsoon (July–August) streamflow simulations produced with the Variable Infiltration Capacity (VIC; Liang et al. 1994, 1996) model at five gauges in the Narmada River basin network in central India. For the remainder of this paper, we use the term “daily monsoon streamflow” to refer to daily streamflows over the period July–August.
2. Study domain and observational datasets
a. Study region
We used the Narmada River basin in India as a testbed to develop and evaluate the Bayesian hierarchical framework proposed here. The Narmada River basin (97 882 km2) is a narrow and elongated catchment (Fig. 1) that stretches from the headwaters in the Amarkantak hills of central India toward the Arabian Sea. The basin-averaged mean annual rainfall is 1120 mm (period 1951–2018), with most of it arriving during the summer monsoon season (June–September), and the upper portion of the basin receives larger precipitation amounts than the lower basin (Banerjee 2009). Because this basin is an essential water source for the populous states of Madhya Pradesh and Gujarat, several reservoirs have been built within the basin for irrigation during the nonmonsoon season, while others have been designed for flood control. Flood events in the basin occur mainly during July–August, the target season of our application.
Map of the Narmada River basin in India, showing 200-m elevation bands, the locations of five subbasin outlets (Sandiya, Hoshangabad, Handia, Mandleshwar, and Garudeshwar), and some of the major dams in the basin: Bargi, Tawa, Indirasagar, Jobat, and Sardar Sarovar (from upstream to downstream direction). The gray circle represents an artificial gauge created for the postprocessing model.
Citation: Journal of Hydrometeorology 23, 6; 10.1175/JHM-D-21-0167.1
b. Historical datasets
We used a 0.25° gridded daily precipitation dataset for the period 1951–2018, developed by the India Meteorology Department (IMD; Pai et al. 2014) through an inverse distance weighting scheme that interpolates observations from 6995 meteorological stations across India (Pai et al. 2014). Gridded daily maximum and minimum temperature fields were obtained by spatially interpolating observations from 395 stations from India to the same 0.25° horizontal resolution grid, using the synergraphic mapping approach (SYMAP; Maurer et al. 2002). These daily gridded precipitation and maximum and minimum temperature fields from IMD have been widely used in previous hydrometeorological studies (e.g., Ali et al. 2019; Shah and Mishra 2016). Daily wind speed was obtained from the National Centers for Environmental Prediction (NCEP)–National Centers for Atmospheric Research (NCAR) reanalysis dataset (Kalnay et al. 1996; Kistler et al. 2001), and regridded to the same 0.25° spatial resolution grid to keep consistency with the other variables.
Observed daily streamflow at five gauges in the Narmada River basin—Sandiya, Hoshangabad, Handia, Mandleshwar, and Garudeshwar (Fig. 1)—were obtained from the India Water Resource Information System (IWRIS) for the period 1970–2014.
3. Approach
Our proposed framework has two components: (i) deterministic streamflow simulations from a physically based hydrological model and (ii) a Bayesian hierarchical model to postprocess the deterministic output to provide posterior distribution of streamflow. These two components are described below.
a. Hydrological modeling
1) Model description and configuration
We used the VIC macroscale hydrological model with three soil layers (Liang et al. 1994, 1996; Liang and Xie 2001; Cherkauer et al. 2003), implemented at a 0.25° horizontal resolution. VIC is a semidistributed model that independently solves water and energy balance at each grid cell. The top soil layer is the dynamic layer representing the saturation excess and infiltration excess runoff, while the bottom layers simulated delayed baseflow responses. In the third soil layer, the slow runoff component is estimated nonlinearly using the ARNO method (Franchini and Pacciani 1991). A stand-alone routing model developed by Lohmann et al. (1996) is used for routing the gridded runoff values generated by the VIC model at specific locations along the main river reach. The routing model solves linearized St. Venant equations for channel routing, with a unit hydrograph model for estimating overland flow (Lohmann et al. 1996). The VIC model has been widely used for simulating streamflow, soil moisture, and other water balance components in the United States, India, and other parts of the world (Schumann et al. 2013; Zhou and Guo 2013; Wu et al. 2014; Bohn and Vivoni 2016; Shah and Mishra 2016; Vásquez et al. 2021). Although the construction of multiple reservoirs in our study domain regulates the river flow, we did not consider their influence for simulating streamflow.
In addition to gridded meteorological forcings (i.e., daily precipitation, maximum and minimum temperatures at 2 m, and wind speed at 10-m height from the surface), the VIC model requires land surface characteristics such as soil, vegetation, and topographic parameters as inputs. We obtained the vegetation parameters from Advanced Very High-Resolution Radiometers (AVHRR; Hansen et al. 2000; Sheffield and Wood 2007) and vegetation library from the Land Data Assimilation System (LDAS; Rodell et al. 2004). Soil parameters were obtained from the Harmonized World Soil Database (HWSD; FAO et al. 2012).
2) Calibration and validation
We calibrated and evaluated the VIC model at three locations in the Narmada River basin (Sandiya, Handia, and Garudeshwar) using observed daily streamflow. The model parameters were calibrated manually using data for the period (1980–94) based on the Pearson’s correlation coefficient (R) and the Nash–Sutcliffe efficiency (NSE; Nash and Sutcliffe 1970), using the first five years of the record (1951–55) as spinup. Due to the lack of a common period with streamflow records at all three stations, the model evaluation was conducted during the 1998–2002 period at the Sandiya and Handia stations, and during the 1996–2000 period at Garudeshwar station. Values of the calibrated parameters can be found in Table S2. Scatterplots for the calibration and validation periods; performance metrics R, NSE, and the Kling–Gupta efficiency (KGE; Gupta et al. 2009); and times series with observed and simulated streamflow for the validation period can be found in Fig. S1 in the online supplemental material. We obtained R > 0.8 between observed and simulated daily streamflow for the calibration period and all subbasins, while NSE and KGE values were higher than 0.6 and 0.7, respectively. For the validation period, we obtained R, NSE, and KGE values higher than 0.8, 0.5, and 0.65, respectively. Additionally, a negative bias for high flows is observed at the gauges (see time series in Fig. S1). After calibrating the model, we simulated streamflow at the five locations in the Narmada basin (Fig. 1) for the 1951–2018 period.
b. Bayesian hierarchical model for streamflow postprocessing
Streamflow probability distributions—typically estimated from ensembles—are required to provide risk estimates for operational and management decisions. Although ensembles can be produced with complex, process-based models, their lack of agility (Mendoza et al. 2015a)—in particular, the computational cost associated with high space-time resolution—makes it difficult to implement in real-time forecasting and decision-making applications. To overcome this issue, we propose a Bayesian hierarchical model to postprocess simulated daily streamflow from the VIC model at five gauges of the Narmada River basin: 1) Garudeshwar, 2) Mandleshwar, 3) Handia, 4) Hoshangabad, and 5) Sandiya. The numbers in brackets are used for model formulation, which is based on the Bayesian hierarchical network model proposed by Ossandón et al. (2021). Although we postprocess streamflow simulations in this paper, this framework is envisioned for implementation in real-time forecasting.
1) General model structure
2) Potential covariates
We considered three types of potential covariates for the monsoon season: (i) daily streamflow simulated with the VIC model (hereafter referred to as “VIC streamflow”), (ii) spatially averaged precipitation for the area between two consecutive gauges, and (iii) a nonlinear covariate obtained from the product between simulated streamflow and precipitation. We did not consider the lagged observed streamflow as a potential covariate because such measurements are discontinuous of it since 2019 at most of the gauges, precluding the implementation of the BHM framework in real time.
For the first type of covariate (VIC streamflow for station i), we considered both daily VIC streamflow at gauge i (
Even after parameter calibration, conditional biases in VIC streamflow and time delays in simulated peak flows were detected at the four stations. Figure 2 displays the scatterplot of VIC streamflow against observations at the Handia gauge during the monsoon seasons for the period 1978–2014, and their time series for 2013 (wet year). The results show that, although observed streamflow is highly correlated with VIC streamflow for the entire period (R = 0.76, Fig. 2a), there is a conditional bias, i.e., a positive (negative) bias for low (high) flows (see Fig. 3b and Fig. S2 for results in other gauges). Further, a time delay of 1 day is observed for VIC streamflow peaks compared to observed streamflow peaks. To address this issue, we included a nonlinear covariate for gauge i (
(a) Scatterplot with VIC streamflow vs observations at the Handia gauge during monsoon season for the period 1978–2014 and (b) time series for 2013 (high flow year). Light blue ellipses in (b) show examples of conditional bias and time delay of VIC streamflow simulations. The R value is statistically significant (p value < 0.05).
Citation: Journal of Hydrometeorology 23, 6; 10.1175/JHM-D-21-0167.1
Conceptual diagram of the network Bayesian model for the Narmada River basin;
Citation: Journal of Hydrometeorology 23, 6; 10.1175/JHM-D-21-0167.1
We explored the strength of the relationship between the covariates and daily streamflow at each gauge by computing the Spearman’s rank correlation coefficient (see Table S3), finding statistically significant correlations.
3) Implementation and model fitting
We fitted various candidate BHMs using different combinations of covariates and alternative unconditional distributions of daily streamflow (i.e., gamma and lognormal). For the gamma distribution, rate and shape parameters were formulated as a function of the mean and variance (Wilks 2011). For the lognormal distribution, we applied a log transformation to the predictands and covariates, and the standard deviation was set stationary because of its positive restriction (σ > 0). Since the Garudeshwar gauge has missing periods (summers of 1988, 1989, and 1995), we fitted the BHM including only years with complete records (34 years for Garudeshwar, and 37 years for the other gauges) within the period 1978–2014. To select the best BHM, we used the leave-one-out cross-validation information criteria (LOOIC; Vehtari et al. 2017), which enables the estimation of pointwise out-of-sample prediction accuracy using the log-likelihood evaluated at the posterior simulations of the parameter values. Luo and Al-Harbi (2017) showed that LOOIC performs better than other traditional model selection metrics such as the Akaike information criterion (Akaike 1974) and the deviance information criterion (DIC; Spiegelhalter et al. 2002). For each candidate BHM, we computed the LOOIC using the R package loo (Vehtari et al. 2020) and selected the model with the lowest LOOIC value.
Candidate BHMs were implemented in R (R Core Team 2017) using the program STAN (Stan Development Team 2014) and the R package RStan (Stan Development Team 2020), which provides an interface from R to the STAN library for Bayesian data analysis. Posterior parameter distributions and predictive posterior streamflow distributions (ensembles) were estimated using the No-U-Turn Sampler (NUTS; Hoffman and Gelman 2014) for the Markov chain Monte Carlo method (Gelman and Hill 2006; Robert and Casella 2011), using the priors described in section 3b(1). We ran three parallel chains—each with a length of 4000 simulations (iterations) and a burn-in size of 2000 to ensure convergence—with different initial values. To reduce the sample dependence (autocorrelation), we chose a thinning factor of 2, yielding 3000 samples (1000 samples from each chain). The scale reduction factor
4) Cross-validation and verification methods
To test the out-of-sample predictability of the model, we performed leave-one-year-out cross validation by dropping one year (validation year) at a time from 37 (34) years with complete records in the period 1978–2014, and fitting the best BHM using the remaining (training) years for all the gauges (except Garudeshwar). The fitted model is used to predict the dropped year.
To assess the performance of the proposed BHM, we considered deterministic metrics—i.e., the coefficient of efficiency (CE; Briffa et al. 1988), the Pearson correlation coefficient (R), and the relative bias (BIAS)—and probabilistic verification measures, including the continuous ranked probability skill score (CRPSS), rank histogram, reliability diagram, and discrimination diagram. A detailed description of the verification methods is provided in the supplementary material.
In this work, we compute accuracy metrics and rank histograms for different types of predicted events via sample stratification (Bellier et al. 2017). Therefore, we defined the event categories as (strata) “low flow,” “midrange flow,” and “high flow” simulated events, as days when the ensemble mean simulated streamflow is below the observed median (Q50th), between 50th and 80th observed quantiles (Q50th and Q80th) and above the 80th observed quantile (Q80th), respectively. With these categories, we seek to answer the question: how did our postprocessing method perform, given that we simulated a low-/high-flow event?
Reliability and discrimination diagrams were generated for three thresholds obtained from quantiles Q30th, Q50th, and Q80th of streamflow observations at each gauge (see Table S4). Time series of observed daily streamflow at five stations showed that most high-flow events during 1978–2014 are above the upper threshold (Q80th), while low-flow events are between 0 and the lower threshold (Q30th, see Figs. S3 and S4). This is because heavy rainfalls yield significant flows during the monsoon season, but the rest of the days are characterized by low or midrange flows.
4. Results
a. Selection of the best BHM
LOOIC values from candidate BHMs, for which the significance of the regression coefficients (β and ϕ) was checked (i.e., that posterior PDFs do not contain zero in the 95% credible interval), can be found in Table S5. The best BHM (model 11 in Table S5), considered a gamma model for the marginal distribution, daily VIC streamflow (
Since the nonlinear covariate,
Figure 4 shows times series of observations, VIC streamflow, and calibrated ensembles of daily streamflow produced with four candidate BHMs for monsoon season 2013, along with scatterplots of BHM ensemble median versus observations for the entire period (1978–2014) at the Handia gauge (i = 3). The BHM model 5 (with
Calibrated daily streamflow ensembles for monsoon season 2013 at the Handia gauge (i = 3), obtained from BHMs calibrated for the full period considering as covariates (a)
Citation: Journal of Hydrometeorology 23, 6; 10.1175/JHM-D-21-0167.1
b. Cross validation
Figure 5 shows cross-validated streamflow ensembles from the best BHM model at the terminal (Garudeshwar), middle (Handia), and headwater (Sandiya) gauges. At Garudeshwar, the BHM can fix the time delay and attenuate the positive bias for low and midrange flows in the ensemble mean (Figs. 5a,b). Although a high correlation is obtained for the entire period (R = 0.78 compared to 0.7 for VIC streamflow) and the ensemble spread is able to capture observations, an increase in the negative bias is observed for high flows compared to VIC streamflow. At Handia (Figs. 5c,d) and Sandiya (Figs. 5e,f), the best BHM model is able to fix the time delay and attenuate the conditional bias in terms of the ensemble mean (blue line and scatterplots for the entire period). Further, R values for Handia and Sandiya are 0.86, and observed values are captured by the ensemble spread at both gauges (Figs. 5d,f). Similar performances are observed for Mandleshwar and Hoshangabad gauges (Fig. S5).
Scatterplots of cross-validated BHM ensemble mean (best model) vs observed streamflow for 1978–2014, and times series with observed, VIC, and cross-validated BHM ensembles of daily monsoon streamflow for 2013 at (a),(b) Garudeshwar, (c),(d) Handia, and (e),(f) Sandiya gauges. Time series of ensembles are presented as boxplots, and outliers are not displayed. Red lines correspond to observed streamflow, dark yellow to VIC streamflow, and blue lines to BHM ensemble streamflow mean. All R values presented here are statistically significant (p value < 0.05).
Citation: Journal of Hydrometeorology 23, 6; 10.1175/JHM-D-21-0167.1
1) Homoscedasticity of the residuals
Homoscedasticity of the residuals—i.e., constant residual (or error) variance for all magnitude of simulated flows is desired. However, heteroscedasticity is not uncommon for higher flow values. To assess the degree of heteroscedasticity, we examine the relationships between simulated streamflow and the corresponding simulated error (historical flow minus simulated flow from the ensembles, Fig. 6) for calibration and cross-validation periods from the BHM at the five gauges. The ensemble spread of the errors is represented by 50% (gray bands) and 95% (light gray bands) confidence intervals, while the thick blue line is the ensemble mean and the red circles the corresponding values from VIC simulated streamflow. For both calibration and cross validation, the BHM simulated flows exhibit very low heteroscedasticity in terms of the ensemble mean (blue lines oscillating around the null error line) compared to VIC simulated streamflow (red points), for all gauges excepting Garudeshwar, where the BHM ensembles overestimate high flows (Figs. 6a,f). In terms of the ensemble spread, the 50% confidence intervals (gray bands) also exhibit very low heteroscedasticity at all five gauges; however, the 95% confidence intervals (light gray bands) tend to increase with streamflow magnitude. This feature can be explained by the nonstationary variance assumption of the BHM model at all gauges, being also present in other approaches that handle the heteroscedasticity of the residuals (BJP; Zhao et al. 2015).
Relationships between simulated streamflow and simulated error. Rows are associated with error uncertainty obtained from BHM calibration and cross validation for the entire period (1978–2014), and columns are associated with different gauges. Blue lines correspond to the ensemble mean; gray bands to 50% confidence interval; light gray bands to 95% confidence interval; and the red dots to errors in VIC simulated streamflow.
Citation: Journal of Hydrometeorology 23, 6; 10.1175/JHM-D-21-0167.1
2) Deterministic accuracy metrics
Figure 7 displays boxplots with performance metrics (CE, R, and BIAS) of VIC and BHM models for the entire period (1978–2014) and the three strata (low, midrange, and high simulated events) defined in section 3b(4). For CE (left panels), VIC streamflow was considered as the reference model. For the entire period, BHM outperforms VIC significantly based on all accuracy metrics. Indeed, the distribution of CE (95% interval) and medians are above 0.38 and 0.48, respectively, for all gauges (Fig. 7a). The increase of median R relative to VIC forecast is roughly 0.1 for each gauge (Fig. 7b). Although negative biases are obtained at all gauges, their distributions are close to zero and absolute reductions range from 48% to 78% for the median BIAS, and their distributions do not overlap (Fig. 7c). For R, a nonparametric test (Gibbons and Chakraborti 1992) showed that differences between BHM and VIC medians are statistically significant at a 95% confidence level.
Boxplots of (left) CE, (center) R, and (right) BIAS at the five stations of the Narmada River basin for (a)–(c) the entire period, (d)–(f) low [
Citation: Journal of Hydrometeorology 23, 6; 10.1175/JHM-D-21-0167.1
For the strata associated with simulated low-flow events, considerable improvements over VIC streamflow are observed at all station gauges for CE and BIAS, since the entire CE distribution is above 0.36 (Fig. 7d). The absolute median reduction in BIAS is larger than 50% (Fig. 7f). For all stations but Mandleshwar (Fig. 7e), we obtained a higher R for VIC streamflow, being Garudeshwar the only site where BHM yielded a high reduction in R. These results indicate that BHM could reduce the positive bias in VIC streamflow (Fig. 2) when a low-flow event is simulated. Figures 7g–I show that BHM outperforms VIC in all metrics (except R for Garudeshwar) when a midrange-flow event is simulated, with bias close to zero and CE distributions above 0.6. Finally, BHM outperforms VIC at all gauges for high-flow simulated events strata (Figs. 7j–l): the CE distributions are above 0, median R improvements range 0.22–0.3, and absolute median BIAS reductions reach ∼20% with distributions close to zero, except for Garudeshwar (median BIAS of ∼−20%). These results indicate that when a high-flow event is simulated at each gauge, the postprocessing model provides improvements over raw VIC output in terms of time delay (higher R values) and bias reduction (lower absolute BIAS values).
3) Probabilistic accuracy metrics
Figure 8 shows CRPSS results from BHM model output—using (i) VIC streamflow and (ii) climatology as the reference—for the entire period (1978–2014) and the three strata (low, midrange, and high-flow simulated events strata). Compared to VIC streamflow (Fig. 8a), the proposed BHM framework yields a substantial increase in skill (95% of the distributions above 0.56) at all gauges considering the entire period, low- and midrange-flow events strata, excepting Mandleshwar for low-flow simulated events (95% of the distribution above 0.48). The smallest skill improvements are achieved for the high-flow simulated events strata (95% of the distributions are between 0.35 and 0.65). On the other hand, BHM yields a significant increase in skill (95% of the distribution above 0.35) at all gauges compared to observed climatology (Fig. 8b) considering the entire period, low- and high-flow events strata; even for the low-flow events strata, 95% of the CRPSS distribution at all gauges is above 0.72. The lowest increases in skill are obtained for the midrange-flow events strata (95% of the distributions are between 0.22 and 0.36).
Boxplots of CRPSS considering (a) VIC simulated streamflow and (b) observed climatology as the reference forecast at the five gauges for the entire period, low [
Citation: Journal of Hydrometeorology 23, 6; 10.1175/JHM-D-21-0167.1
4) Reliability
Figure 9 displays rank histograms for cross-validated BHM streamflow ensembles along with discrepancy indices (DI) for each station (columns), the entire period and the three strata (displayed in different rows). For Garudeshwar (first column of Fig. 9), the shape of rank histograms for the entire period, midrange and high-flow simulated events strata reveals a combination of over dispersive (underconfident) and systematic positive bias shapes. Rank histograms of the low-flow simulated events strata shows a slight under dispersive (overconfident) shape. DI values range 33%–41% for the entire period, low- and midrange-flow simulated strata, and increases up to 66% for the high-flow simulated strata. At the remaining sites, most rank histograms show a close to uniform shape for the entire period, and low and midrange-flow simulated events strata, with DI values generally below 18%. An exception is the high-flow simulated strata at all gauges, where rank histograms exhibit a slight undersimulation shape, with DI values below 33%. Additionally, a slight positive bias is observed at Handia and Hoshangabad gauges for the low-flow simulated strata. Similar features are observed in reliability diagrams at the other four stations (Fig. S6).
Rank histograms for cross-validated BHM daily streamflow ensembles during July–August for the entire period, low [
Citation: Journal of Hydrometeorology 23, 6; 10.1175/JHM-D-21-0167.1
5) Discrimination
Figure 10 displays discrimination diagrams with BHM results for each station (rows) and three streamflow thresholds (columns). The metrics included in each panel are discrimination distance (d), overlapping area (A), and standard deviations (σ0 and σ1) from the two conditional likelihoods. A good discrimination is given by sharp (i.e., small σ0 and σ1), well separated (large d), and no overlapping (small A) likelihoods. In this case, we omit confidence limits in discrimination diagrams because there were only minor differences in the resampled PDFs. At Garudeshwar (the first column of Fig. 10), conditional likelihoods are well separated (high d), with little overlap and increased PDF sharpness for nonoccurrence events (i.e., decrease of σ0) as the threshold increases. No clear relationships with thresholds are observed for the standard deviation of event occurrence (σ1). For Q30th the shape of nonoccurrence PDF (o = 0) is not skewed to the left, which implies a failure of the BHM system to detect nonoccurrence of low-flow events. The opposite is observed during midrange and high-flow events, for which PDFs shape of occurrence (o = 1, red lines) for Q50th and Q80th are not skewed to the right, showing difficulties to detect midrange and high-flow events.
Discrimination diagrams for cross-validated BHM streamflows computed over July–August days during the period 1978–2014. Rows are associated with different daily streamflow thresholds, and columns are associated to different gauges. The PDF for cases where o = 0 (no occurrence) is represented by the dark line (diamonds), and the PDF for cases with o = 1 (occurrence) is represented by the red line (circles). The metrics displayed in each panel are discrimination distance (d), overlapping area (A), and standard deviations (σ0 and σ1) from the two conditional likelihoods.
Citation: Journal of Hydrometeorology 23, 6; 10.1175/JHM-D-21-0167.1
At the remaining four stations (columns 2–5 of Fig. 10), the proposed BHM framework provides good discrimination, with a decrease (increase) in PDF sharpness of event occurrence (nonoccurrence) for larger thresholds—i.e., increase of σ1 and decrease of σ0 for larger thresholds. The PDFs shapes of event occurrence (o = 1, red lines) and nonoccurrence (o = 0, black lines) for Q30th and Q50th thresholds are skewed to the right and left, respectively, reflecting success in detecting low- and midrange-flow events. For the Q80th threshold, the PDF of occurrence at three gauges (Mandleshwar, Hoshangabad, and Sandiya; Figs. 10l,n,o) is not skewed to the right, reflecting a failure to assign high probabilities for high-streamflow events.
5. Discussion
The BHM postprocessing framework presented here has several appealing features. First, it can be easily adapted for different types of hydrometeorological variables by selecting suitable marginal distributions (e.g., extreme value, Weibull, binomial, Poisson) at the data layer. Nonlinearities can be incorporated into the process layer by adding nonlinear terms in the regression equations. The approach also enables handling the heteroscedasticity of the residuals, similar to MCP (Todini 2008) and BJP (Zhao et al. 2015). The BHM approach can be implemented with single trace or ensemble forecasts from multiple sources (i.e., models) as potential covariates, enabling multimodel combinations (e.g., Mendoza et al. 2014). Qualitative information about the flows can also be incorporated in the process layers as covariates or conditions of the steps functions. For example, information about reservoirs can be incorporated into the BHM as conditions for the steps functions or covariates.
Initial implementations of the BHM considered other VIC outputs as potential covariates, such as the soil moisture from the three layers. Nevertheless, this variable was finally discarded because complex nonlinear transformations were needed to achieve acceptable skill. Additionally, soil moisture is a local storage within the watershed, while streamflow results from the combination of different hydrological processes at various spatiotemporal scales.
Although previous studies showed that LOOIC performs better than other traditional model selection metrics such as the AIC and DIC (Luo and Al-Harbi 2017), we noticed that the LOOIC metric may not necessarily help to avoid an unnecessary increase of model complexity. For example, a previous implementation of this framework considered to increase the model structure complexity using three-step functions for the mean and variance, with the hypothesis that including a third step for high-flow events in the step functions would handle the sampling error. Then, we selected a BHM model with three-step functions as the best model according to the LOOIC criteria. However, a comprehensive cross-validation performance assessment of both models (BHM implemented here and one with three-step functions) revealed that the improvement in performance obtained by adding a step was marginal to justify an increase in model complexity (6 additional coefficients). Thus, we recommend for future applications of the framework to carefully examine tradeoffs between model complexity and gain in performance—and not be constrained with a single metric.
In this work, we did not compare BHM results against generalized linear models (GLMs) as a reference since they yield lower performance than BHM in terms of probabilistic accuracy metrics (Ossandón et al. 2021). In comparison with postprocessing methods that rely on streamflow persistence (Li et al. 2015, 2016, 2017a; McInerney et al. 2018; Bennett et al. 2021), BHM can reach similar skill improvements without incorporating observed streamflow as a covariate, which is helpful for basins (e.g., Narmada) where discontinuous streamflow records are a problem for model development. A detailed comparison of BHM against other postprocessing methods (Wood and Schaake 2008; Weerts et al. 2011) is out of the scope of this paper, being left for future work.
A key limitation of this study is the exclusion of reservoir operations in the Narmada River basin from hydrologic model simulations, since we did not have reliable information about operation policies. We speculate that such limitation could affect some of the results presented here. For example, although BHM provides a bias reduction and increased skill and correlation relative to VIC at Garudeshwar, it also yields a negative bias for high-flow simulated events strata, poor reliability (combination of over dispersive and positive bias shapes) for the entire period, midrange and high-flow simulated events strata, and poor discrimination. At this gauge, a poor performance of VIC simulated streamflow was obtained during the monsoon season, with large biases likely attributed to reservoir effects that are not captured by VIC. This problem is also observed during July 2013 (Fig. 5b)—when VIC overestimates observed low flows—and other water years (not shown). Future work could explore, besides the incorporation of reservoir operation rules, other parameter estimation strategies, including alternative calibration algorithms (e.g., Gharari et al. 2013; Fenicia et al. 2018) and objective functions (e.g., Shafii and Tolson 2015; Fowler et al. 2018). We expect that improved hydrologic model performance should translate into enhanced skill in postprocessed ensembles.
The proposed framework could be easily implemented in other major flood-affected basins worldwide, with different river network topologies, and using other conceptual or physically based hydrological models, suitable parametric distributions, and covariates. Also, in the case that hydrologic model simulations cannot capture well the spatiotemporal dependence of the observed data, copulas can be included in the BHM for simultaneous multisite postprocessing.
6. Summary and conclusions
We developed a Bayesian hierarchical model (BHM) for the simultaneous postprocessing of daily summer (July–August) streamflow obtained from a physically based model at multiple sites. The framework was tested at five streamflow gauges in central India’s Narmada River Basin network, using outputs from the Variable Infiltration Capacity (VIC) model for the summer season 1978–2014. Daily streamflow at each location was modeled using a Gamma distribution with time-varying parameters, which were modeled as a linear function of VIC streamflow and a nonlinear covariate—the product of VIC streamflow and 2-day spatial average precipitation from the area between two consecutive gauges. This nonlinear covariate could fix VIC streamflow peaks’ conditional bias and time delay. The best model (i.e., gamma distribution and the covariates) was selected using LOOIC—a cross-validation criterion. Finally, the inclusion of suitable priors for the regression coefficients helped to correctly simulate posterior parameter distributions and predictive posterior distribution of streamflows for each day and location.
Deterministic accuracy metrics showed enhanced performance of BHM ensembles compared to raw VIC simulated streamflow—i.e., absolute bias reductions of at least 83% and skill improvements of at least 38% and 46% for CE and CRPSS, respectively, across the basin for the entire period. The BHM skill results were consistently better than VIC streamflow for the entire period and the three strata (low-, midrange-, and high-flow events strata) at all gauges. Further, rank histograms and discrimination diagrams showed that the BHM frameworks provide streamflow ensembles with a good spread and discrimination at most sites, except for the terminal gauge (Garudeshwar). Overall, the results presented here illustrate the enormous potential of this framework for short- to medium-range streamflow forecasting in the Narmada River basin, complementing and enhancing the current operational flood forecasting system (CWC 2015). Additionally, the proposed approach could be implemented and tested in other river basin networks across the globe, contributing to improved flood hazard mitigation.
Acknowledgments.
This research was funded by the Monsoon Mission project of the Ministry of Earth Sciences, India. We also acknowledge the support from the Fulbright Foreign Student Program and the National Agency for Research and Development (ANID) Scholarship Program/DOCTORADO BECAS CHILE/2015-56150013 to the first author. P. A. Mendoza acknowledges support from Fondecyt projects 11200142 and CONICYT/PIA Project AFB180004.
Data availability statement.
The dataset used in this study, which consists of time series of potential covariates and daily monsoon period (July–August) streamflow for four station gauges in the Narmada River basin, can be downloaded from HydroShare https://www.hydroshare.org/resource/e2e4924a3ed2416b831b0e6a0cb0bdaf/.
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