# Case Study: A Real-Time Flood Forecasting System with Predictive Uncertainty Estimation for the Godavari River, India

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Predictive Uncertainty Assessment: The Model Conditional Processor in the Multi-Temporal Approach

- First, the observations, $y$, and the forecasts, ${\widehat{y}}_{k}$ (k = 1, …, M, M = number of available forecasts provided by M different forecasting models), are converted into the normal space using the NQT (Figure 1, blue box on the left side). The original variables, $y$ and ${\widehat{y}}_{k}$, whose empirical cumulative distribution functions are computed using the Weibull plotting position (see Equations (1) and (2)), are converted to their transformed values $\eta $ and ${\widehat{\eta}}_{k}$, respectively, which are normally distributed with zero mean and unit variance. According to the NQT definition, the probability of each element of $\eta $ and ${\widehat{\eta}}_{k}$ is the same as its original corresponding value in $y$ and ${\widehat{y}}_{k}$. Thus, the relation between the original variables and their transformed values is:$$P(y<{y}_{i})=\frac{i}{n+1}=P(\eta <{\eta}_{i})$$$$P({\widehat{y}}_{k}<{\widehat{y}}_{ki})=\frac{i}{n+1}=P({\widehat{\eta}}_{k}<{\widehat{\eta}}_{ki})$$
- In the normal space, the joint probability distribution of observed and predicted variables, $f(\eta ,{\widehat{\eta}}_{k})$, is assumed to be a Normal Multivariate Distribution in the first formulation [19] (Figure 1, red box on the left side) or composed by two Truncated Normal Multivariate Distributions as proposed by Coccia and Todini [16].
- The predictive density is obtained applying the Bayes Theorem (Figure 1, green box on the right side):$$f(\eta |{\widehat{\eta}}_{k})=\frac{f(\eta ,{\widehat{\eta}}_{1},.......,{\widehat{\eta}}_{M})}{f({\widehat{\eta}}_{1},.......,{\widehat{\eta}}_{M})}$$It is normally distributed with mean, $\mu (\eta |{\widehat{\eta}}_{k})$, and variance, ${\sigma}^{2}(\eta |{\widehat{\eta}}_{k})$, defined as:$$\mu (\eta |{\widehat{\eta}}_{k})={\Sigma}_{\eta \widehat{\eta}}\cdot {{\Sigma}_{\eta \widehat{\eta}}}^{-1}\cdot \left[\begin{array}{l}{\widehat{\eta}}_{1}\\ .\\ .\\ {\widehat{\eta}}_{M}\end{array}\right]$$$${\sigma}^{2}(\eta |\widehat{\eta})=1-{\Sigma}_{\eta \widehat{\eta}}\cdot {{\Sigma}_{\eta \widehat{\eta}}}^{-1}\cdot {{\Sigma}_{\eta \widehat{\eta}}}^{T}$$
- The PU in the normal space is finally reconverted to the real space by applying the Inverse NQT (Figure 1, grey box on the right side).

## 3. Forecasting Model: STAFOM-RCM

**STAFOM**provides a first estimate of the forecast stage (preliminary forecast) at the downstream end, ${h}_{d}^{\prime}$, computed as:$${h}_{d}^{\prime}\left({t}_{f}+\Delta {t}^{*}\right)={\left\{\frac{1}{\lambda}\left[{C}_{1}^{*}\left({Q}_{u}\left({t}_{f}\right)+{q}_{for}\left({t}_{f}\right)L\right)+{C}_{2}^{*}{\left({h}_{d}\left({t}_{f}\right)\right)}^{\delta}\right]\right\}}^{1/\delta}$$_{L}, of the reach), ${Q}_{u}\left({t}_{f}\right)$ is the observed upstream discharge at ${t}_{f}$, L is the river reach length, and $\lambda $ and $\delta $ are parameters of the downstream rating curve (${Q}_{d}=\lambda {h}_{d}^{\delta}$). ${C}_{1}^{*}$ and ${C}_{2}^{*}$ refer to the Muskingum parameters K and θ respecting the constraint $\Delta {t}^{*}=2K\theta $:$${C}_{1}^{*}=\frac{K\theta +0.5\Delta {t}^{*}}{K-K\theta +0.5\Delta {t}^{*}};\text{\hspace{1em}}{C}_{2}^{*}=\frac{K-K\theta -0.5\Delta {t}^{*}}{K-K\theta +0.5\Delta {t}^{*}}$$_{for}is the lateral flow contribution for unit channel length estimated as [27]:$${q}_{for}\left({t}_{f}\right)=\frac{{A}_{d}\left({t}_{f}\right)-{A}_{u}\left({t}_{f}-{T}_{L}\right)}{{T}_{L}}$$_{L}is the flood wave travel time assumed equal to the lead-time, $\Delta {t}^{*}$, for forecasting purposes. The lateral flow is assumed uniformly distributed along the branch and, hence, the total lateral discharge entering in the reach in the time interval $({t}_{f};{t}_{f}+\Delta {t}^{*})$, ${Q}_{l}$, is equal to ${q}_{for}({t}_{f})L$.**RCM**, improves the preliminary forecast stage from STAFOM by exploiting the following relationship between the upstream and downstream discharge, ${Q}_{d}$, [27,28]:$${Q}_{d}\left(t+{T}_{L}\right)=\alpha \frac{{A}_{d}\left(t+{T}_{L}\right)}{{A}_{u}\left(t\right)}{Q}_{u}\left(t\right)+\beta $$

## 4. Study Area, Model Setting and Dataset

^{2}that is about 10% of India’s total geographical area. The Godavari River rises at an elevation of 1067 m a.s.l. in the Western Ghats, near Thriambak Hills in the Nasik district of Maharashrta, and after flowing for about 1465 km, in a generally southeast direction, it falls into the Bay of Bengal (see Figure 2). The mean annual rainfall varies from 1000 to 3000 mm. The Godavari basin receives its maximum rainfall during the Southwest monsoon; specifically, 84% of the annual rainfall falls during the period starting in mid June and ending by mid October. The monsoon currents strike the West Coast of the peninsula from west and southwest, meet the Western Ghats or Sahyadri Range which present almost an uninterrupted barrier ranging from 600 to 2100 m a.s.l. Rainfall is governed largely by the orography of the area, which leads to variation in the amount of precipitation.

^{2}) and Bhadrachalam (drainage area = 280,505 km

^{2}), and downstream by the hydrometric site of Polavaram (drainage area = 307,800 km

^{2}) are selected for the application of the STAFOM-RCM model.

^{2}, which represents 13% of the entire catchment. It has a mean observed wave travel time between 20 and 24 h. The shorter investigated reach, Bhadrachalam-Polavaram, is 73 km long and is characterized by an intermediate drainage area that is about 9% of the downstream total one. The mean wave travel time of the reach is found equal to about 10–12 h. The mean section width of the Godavari River between the gauged site of Perur and Polavaram is about 1300 m. The main properties of the investigated river reaches are summarized in Table 1, while the geometry of the Polavaram gauged section, where the stage forecast and the PU estimate are provided, is represented in Figure 3.

## 5. Results and Discussion

#### 5.1. Performance Evaluation Measures

_{obsi}is the ith ordinate of the observed stage-hydrograph; ${\overline{h}}_{obs}$ is the mean of the observed stage-hydrograph ordinates; h

_{fori}is the ith ordinate of the forecast stage-hydrograph; and N is the total number of stage-hydrograph ordinates to be forecasted.

#### 5.2. Forecasting Model

#### 5.3. Predictive Uncertainty Estimate Using MCP-MT

_{tr}, is equal to 1.79 when STAFOM-RCM is applied to reach 1 (Figure 5a), while a different value of 0.5 is identified for reach 2 (Figure 5b). Specifically, the threshold identified for reach 2 divides the data into two samples, the first corresponding to low flows that include 68% of the data and the second one referring to high flows and containing 32% of the entire sample.

_{obs_i}is the number of occurrences below the ith percentile and n is the sample size. The line y = x identifies the perfect behavior (red diagonal), while the deviation from the bisector suggests if the PU estimated percentiles are underestimated or overestimated.

#### 5.3.1. Calibration and Validation

_{i}, z

_{i}= P(x

_{i}), versus their corresponding empirical distribution function, R

_{i}/n, with R

_{i}= ranks and n = sample size. The PPR indicates if the uniformity test is passed or not and, also, the shape of the resulting curve gives information on the possible causes behind deviations from uniformity, i.e., placement of the points along the 1:1 line [10,31]. Moreover, the Kolmogorov confidence band can be displayed in the PPR indicating if the uniformity test is passed (the curve is inside the band) or not [31]. The PPR is here developed for the case study of MCP-MT results for the longer branch, reach 2, lead-time 24 h and focusing on the 2001 severe monsoon season. Following Laio and Tamea [31], we use 24 sub-series obtaining the results shown in Figure 9. Based on the indications provided by Laio and Tamea [31] to evaluate the results, it is clear that the forecasts provided by the probabilistic method are reliable (most of the forecasts remains inside the Kolmogorov band with 5% significance), even if the shape of the curves indicates that the predictions are large around the central value. The probability plot shows a large steepness of the curves, i.e., more z

_{i}points concentration, in the vicinity of 0.4–0.5 points.

#### 5.3.2. Probability of Hydrometric Thresholds Exceedance: Flooding Probability within a Time Horizon and Contingency Table

_{att}), “warning” (th

_{war}) and “alarm” (th

_{alar}) threshold. It is worth noting that these critical levels are not based on operational values defined by the authority in charge of decision in case of flood, but they are set by the authors. Specifically, the warning and alarm threshold are identified on the basis of the section geometry (see Figure 3), while the lowest one is assumed equal to the 95th percentile of the historical observed river level data with the aim of investigating a critical level exceeded several times during the available dataset.

_{att}, we see 14 hits and one false alarm for 12 h lead-time and 15 hits and two false alarms for 24 h lead-time. By inspecting the results for MCP-MT calibrated by using the whole available dataset, a better performance can be seen both in terms of mean and 95th percentile. For example, the two false alarms for th

_{war}and 12 h lead-time are no more observed for the expected value. Moreover, the two false alarms for reach 2 (24 h) of the 95th percentile are characterized by a maximum value of the exceedance probability provided by MCP-MT lower than 40% and 10%. It is also worth noting that the maximum water level observed during the only miss for th

_{att}provided by the expected value of MCP-MT is found only two centimeters above the critical level.

_{war}. As concerns th

_{att}, the theshold is reached five and nine times during the calibration and the validation period, respectively, when the dataset of reach 1 is considered. If reach 2 is investigated, it is seen that th

_{att}is exceeded six times during the calibration period and nine times in the validation time series. The results of reach 1 (lead-time = 12 h) show for the calibration period five hits with 0 false alarms and misses (for both the deteministic model and the MCP-MT outcomes), and for the validation period, nine. Threrefore, the th

_{att}overcoming is always correctly predicted when it actually occurs, however one false alarm is also observed in the validation period for MCP-MT as well as for the deterministic model. When the 24 h lead-time case study is analyzed, it is seen that in the calibration period six hits are obtained and only one false alarm for the 95th percentile that, however, is characterized by a maximum probability threshold exceedance equal to 22%. Finally, for the validation dataset, eight and nine hits are found for the expected value and the 95th percentile, respectively, with one miss for the mean of MCP-MT and one false alarm for both that is characterized by a maximum exceedance probability of about 50%.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**Godavari River: geometry of the Polavaram section where the stage forecast and the PU estimate are provided. The zero gauge level is shown along with the assumed hydrometric thresholds.

**Figure 4.**Godavari River, reach 1 (Bhadrachalam-Polavaram): performance measures for the deterministic forecast and the MCP-MT application for lead-times from 1 to 12 h: (

**a**) mean of the absolute error on stage forecast; (

**b**) standard deviation of the absolute error on stage forecast; (

**c**) root mean square error, RMSE; and (

**d**) Nash–Sutcliffe coefficient, NS. The measures are computed for the entire database.

**Figure 5.**Division of the joint distribution in the transferred normal space (i.e., space fulfilling assumptions of normality) into two bivariate truncated normal distributions for: (

**a**) Bhadrachalam-Polavaram reach (lead-time = 12 h); and (

**b**) Perur-Polavaram reach (lead-time = 24 h). The red line represents the mean value, while the light blue lines represent the 5% and the 95% quantiles. The black dashed line represents the threshold used in order to identify the two TNDs.

**Figure 6.**Comparison between the MCP-MT estimated percentiles and the corresponding observed occurrences (n

_{obs_i}= number of occurrences below the ith percentile, n = sample size) for: (

**a**,

**b**) all calibration dataset ((

**a**) reach 1; and (

**b**) reach 2); (

**c**,

**d**) calibration period 2001–2004 ((

**c**) reach 1; and (

**d**) reach 2); and (

**e**,

**f**) validation period 2005, 2007 and 2010 ((

**e**) reach 1; and (

**f**) reach 2). The red line represents the perfect behavior, and the blue shows the MCP-MT response.

**Figure 7.**Polavaram section (lead-time = 12 h): comparison between observed and forecast stages provided STAFOM-RCM applied to the shorter reach for the flood event occurred on the period: (

**a**) 19–31 July 2003; and (

**b**) 10–27 August 2004. The 90% uncertainty band along with the expected value assessed through the multi-temporal approach of MCP-MT are also shown.

**Figure 8.**As for Figure 7, but for STAFOM-RCM applied to the longer reach (lead-time = 24 h) and for the events occurred on the period: (

**a**) 24 July–26 August 2005; and (

**b**) 4–12 August 2010.

**Figure 9.**Godavari River, monsoon season 2001 (reach 2, lead-time = 24 h): probability plot representation of the probabilistic forecast. The Kolmogorov 5% significance band is shown as dashed lines.

**Figure 10.**Polavaram section (warning threshold = 14.27 m): comparison between the observed exceedance threshold probability and the one computed by the MCP-MT within: (

**a**) the next 12 h (reach 1); and (

**b**) the next 24 h (reach 2) for all the available dataset.

**Figure 11.**Polavaram section (lead-time 24 h): overtopping warning threshold exceedance probability within the following 24 h estimated by MCP-MT (the threshold exceedance probability at time t refers to the interval t–t + 24 h) for the flood occurred on: (

**a**) 21–22 August 2001; and (

**b**) 6–10 August 2010. The comparison between deterministic and probabilistic stage forecasts is also shown along with the observed occurrences.

**Table 1.**Godavari River: main properties of the selected river reaches (L, length; Aup and Adown, upstream and downstream drainage area; Aint, intermediate drainage area; S

_{0}, mean bed slope; B, mean section width; T

_{L}, mean wave travel time).

River Reach | L (km) | Aup (km^{2}) | Adown (km^{2}) | Aint (km^{2}) | S_{0} | B (m) | T_{L} (h) | |
---|---|---|---|---|---|---|---|---|

reach 1 | Bhadrachalam-Polavaram | 73 | 280,505 | 307,800 | 27,295 (9%) | 0.00025 | 1300 | 10–12 |

reach 2 | Perur-Polavaram | 206 | 268,200 | 39,600 (13%) | 0.0003 | 20–24 |

**Table 2.**Polavaram section (MCP-MT calibrated considering all the available dataset): mean (m) and standard deviation (σ) of the absolute error on stage forecast (er_h), Nash–Sutcliffe coefficient (NS), root mean square error (RMSE), and coefficient of persistence (PC), for the deterministic forecasting model, STAFOM-RCM, and the expected value estimated by the MCP-MT for the two investigated reaches.

River Reach | Lead-Time (h) | STAFOM-RCM | MCP-MT Expected Value | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

er_h (m) | NS | RMSE (m) | PC | er_h (m) | NS | RMSE (m) | PC | ||||

m | σ | m | σ | ||||||||

reach 1 | 10 | 0.196 | 0.223 | 0.989 | 0.297 | 0.56 | 0.117 | 0.159 | 0.995 | 0.198 | 0.81 |

12 | 0.202 | 0.220 | 0.989 | 0.298 | 0.67 | 0.136 | 0.184 | 0.993 | 0.228 | 0.81 | |

reach 2 | 20 | 0.344 | 0.403 | 0.960 | 0.530 | 0.34 | 0.283 | 0.328 | 0.973 | 0.433 | 0.56 |

24 | 0.353 | 0.413 | 0.958 | 0.543 | 0.48 | 0.269 | 0.317 | 0.975 | 0.416 | 0.70 |

**Table 3.**Polavaram section (MCP-MT calibrated considering all the available dataset): percentage of observed data that fall inside the 90% uncertainty band (Perc90%), and mean and standard deviation of the uncertainty band width.

River Reach | Lead-Time (h) | Perc90% | Width of the 90% Uncertainty Band | |
---|---|---|---|---|

Mean (m) | Standard Deviation (m) | |||

reach 1 | 10 | 93.7 | 0.73 | 0.20 |

12 | 93.6 | 0.84 | 0.23 | |

reach 2 | 20 | 93.1 | 1.42 | 0.32 |

24 | 93.2 | 1.40 | 0.30 |

**Table 4.**As for Table 2, but considering a separated calibration and validation dataset for MCP-MT.

River Reach | Lead-Time (h) | STAFOM-RCM | MCP-MT Expected Value | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

er_h (m) | NS | RMSE (m) | PC | er_h (m) | NS | RMSE (m) | PC | ||||

m | σ | m | σ | ||||||||

Calibration (2001–2004) | |||||||||||

reach 1 | 12 | 0.184 | 0.194 | 0.99 | 0.267 | 0.73 | 0.115 | 0.137 | 0.996 | 0.178 | 0.88 |

reach 2 | 24 | 0.343 | 0.400 | 0.958 | 0.527 | 0.4 | 0.263 | 0.306 | 0.975 | 0.403 | 0.74 |

Validation (2005, 2007, 2010) | |||||||||||

reach 1 | 12 | 0.221 | 0.243 | 0.987 | 0.328 | 0.58 | 0.142 | 0.212 | 0.992 | 0.255 | 0.75 |

reach 2 | 24 | 0.362 | 0.426 | 0.958 | 0.56 | 0.55 | 0.292 | 0.348 | 0.973 | 0.455 | 0.71 |

**Table 5.**As for Table 3, but considering a separated calibration and validation dataset for MCP-MT.

River Reach | Lead-Time (h) | Perc90% | Width of the 90% Uncertainty Band | |
---|---|---|---|---|

Mean (m) | Standard Deviation (m) | |||

Calibration (2001–2004) | ||||

reach 1 | 12 | 92.3 | 0.68 | 0.22 |

reach 2 | 24 | 92.3 | 1.33 | 0.33 |

Validation (2005, 2007, 2010) | ||||

reach 1 | 12 | 92.0 | 0.74 | 0.20 |

reach 2 | 24 | 92.2 | 1.4 | 0.29 |

**Table 6.**Contingency tables showing the capability of STAFOM-RCM and of the expected value and the 95th percentile provided by MCP-MT in hydrometric thresholds exceedance/non-exceedance prediction.

River Reach | Warning Threshold (14.27 m) | Attention Threshold (11.80 m) | |||||
---|---|---|---|---|---|---|---|

Hits | False Alarms | Misses | Hits | False Alarms | Misses | ||

All Dataset (2001–2010) | |||||||

reach 1 (12 h) | STAFOM-RCM | 3 | 2 | 0 | 14 | 1 | 0 |

Exp. value | 3 | 0 | 0 | 14 | 1 | 0 | |

95th perc. | 3 | 2 | 0 | 14 | 1 | 0 | |

reach 2 (24 h) | STAFOM-RCM | 2 | 1 | 1 | 15 | 2 | 0 |

Exp. value | 2 | 1 | 1 | 14 | 1 | 1 | |

95th perc. | 3 | 2 | 0 | 15 | 2 | 0 | |

Calibration (2001–2004) | |||||||

reach 1 (12 h) | STAFOM-RCM | 0 | 1 | 0 | 5 | 0 | 0 |

Exp. value | 0 | 0 | 0 | 5 | 0 | 0 | |

95th perc. | 0 | 1 | 0 | 5 | 0 | 0 | |

reach 2 (24 h) | STAFOM-RCM | 0 | 1 | 0 | 6 | 1 | 0 |

Exp. value | 0 | 0 | 0 | 6 | 0 | 0 | |

95th perc. | 0 | 0 | 0 | 6 | 1 | 0 | |

Validation (2005, 2007, 2010) | |||||||

reach 1 (12 h) | STAFOM-RCM | 3 | 1 | 0 | 9 | 1 | 0 |

Exp. value | 3 | 0 | 0 | 9 | 1 | 0 | |

95th perc. | 3 | 1 | 0 | 9 | 1 | 0 | |

reach 2 (24 h) | STAFOM-RCM | 2 | 0 | 1 | 9 | 1 | 0 |

Exp. value | 2 | 0 | 1 | 8 | 1 | 1 | |

95th perc. | 2 | 0 | 1 | 9 | 1 | 0 |

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**MDPI and ACS Style**

Barbetta, S.; Coccia, G.; Moramarco, T.; Todini, E. Case Study: A Real-Time Flood Forecasting System with Predictive Uncertainty Estimation for the Godavari River, India. *Water* **2016**, *8*, 463.
https://doi.org/10.3390/w8100463

**AMA Style**

Barbetta S, Coccia G, Moramarco T, Todini E. Case Study: A Real-Time Flood Forecasting System with Predictive Uncertainty Estimation for the Godavari River, India. *Water*. 2016; 8(10):463.
https://doi.org/10.3390/w8100463

**Chicago/Turabian Style**

Barbetta, Silvia, Gabriele Coccia, Tommaso Moramarco, and Ezio Todini. 2016. "Case Study: A Real-Time Flood Forecasting System with Predictive Uncertainty Estimation for the Godavari River, India" *Water* 8, no. 10: 463.
https://doi.org/10.3390/w8100463