Probabilistic Flood Forecasts Research Articles

Uncertainty in three dimensions: the challenges of communicating probabilistic flood forecast maps

Abstract. Real-time operational flood forecasting most often concentrates on issuing streamflow predictions at specific points along the rivers of a watershed. However, we are now witnessing an increasing number of studies aimed at also including flood mapping as part of the forecasting system. While this additional new information (flood extent, depth, velocity, etc.) can potentially be useful for decision-makers, it could also be overwhelming. This is especially true for probabilistic and ensemble forecasting systems. While ensemble streamflow forecasts for a given point in space can be visualized relatively easily, the visualization and communication of probabilistic forecasts for water depth and extent pose additional challenges. Confusion typically arises from too much information, counterintuitive interpretation, or simply too much complexity in the representation of the forecast. The communication and visualization of probabilistic streamflow forecasts has been studied in the past, but this is not the case for the probabilistic flood forecast map, which is still an emerging product. In this paper, we synthesize the results of a large-scale survey (28 government representatives, 52 municipalities, 9 organizations, and 38 citizens and farmers, for a total of 140 people) regarding the users' preferences in terms of visualizing probabilistic flood forecasts over an entire river reach. The survey was performed through interviews, during which the interviewees were asked about their needs in terms of hydrological forecasting. We also presented the interviewees with four prototypes representing alternative visualizations of the same probabilistic forecast in order to understand their preferences in terms of colour maps, wording, and the representation of uncertainty. Our results highlight several issues related to the understanding of probabilities in the specific context of visualizing forecasted flood maps. We propose several suggestions for visualizing probabilistic flood maps and also describe potential adaptations for different categories of end users. This study is the first to investigate the visualization of probabilistic flood maps, which are gaining popularity. Given that the interview questions were not tied to a specific geographical location, our findings are applicable outside of the study area and, therefore, to other operational centres interested in providing probabilistic flood forecast maps to decision-making organizations and citizens.

Fully integrating probabilistic flood forecasts into the decision-making process across southern Quebec, Canada: some factors to consider

Flood forecasts can enable decision-makers to plan for mitigation measures, ensure the safety of people and reduce damages to private and public property. However, while flood forecasting systems are becoming increasingly sophisticated, those technical improvements do not always translate into a reduction of damages. In particular, ensemble and probabilistic streamflow forecasts provide more information regarding the uncertainty, but such forecasts are also more difficult to interpret. The connection between streamflow, water depth and flood extent is also not straightforward, as this connection requires extensive previous experience with a given river reach. Streamflow forecasts can be transformed in predictive flood maps through a hydraulic model, and those maps might have a more intuitive interpretation. However, if the system is designed to also represent the uncertainty of both water depth and extent, the interpretation becomes considerably more complex. Within this context of continuous improvement of flood forecasting systems, a broad consultation (90 participants) of forecast users was conducted across southern Quebec, Canada, with the aim of understanding their perception of flood forecasts and how they use those forecasts for decision-making. Participants responses emphasize the importance of contextualizing the forecasts according to potential consequences, providing an accurate representation of the time evolution of flood events, and facilitating the communication between forecasters and decision-makers. Following this, our main recommendation is to organize workshops to provide training for decision-makers on how to interpret hydrological forecasts correctly. Another recommendation is to implement additional communication channels between forecasters and end-users.

Evaluating storm surge predictability on subseasonal timescales for flood forecasting applications: A case study for Hurricane Isabel and Katrina

Coastal flooding operational forecasting in the US is limited to short-range temporal scales (3–7 days), which limits the response time for emergency preparation and planning. The sub-seasonal prediction project (SubX), which produces weather forecasts with a lead time of up to four weeks, provides an opportunity to assess the potential for creating probabilistic flood forecasts with longer lead times. Using the ADCIRC hydrodynamic model for coastal storm surge, two major hurricanes, Isabel (2003) and Katrina (2005), were used as case studies to test coastal flood predictions induced by wind and pressure fields generated from five global weather models within SubX. The storm surges simulations are forced by Sea Level Pressure (SLP) and 10 m winds fields from SubX models for a lead-time of up to 30 days before storm landfall. The subseasonal surge forecasts are evaluated temporally and spatially at 1–4 weeks lead-time against the NOAA tide gages observations and a verification dataset derived by forcing the storm surge model with wind and pressure fields from the NCEP-Reanalysis. The results are evaluated in terms of lead-time and forecast skill metrics. The storm surge forecast skill is measured using the mean square error skill score (MSESS) relative to the verification dataset and an approximate of the climatology. A skill score greater than 0.55 is considered here useful for flood forecasting. The multi-model ensemble (MME) mean surge forecasts driven by several members of SubX models demonstrate skill greater than 0.55 up to a 4-day and 10-day lead for Katrina and Isabel, respectively. A sharper decrease in MSESS was noted from week 1 to week 3 lead-times for Katrina, in comparison to Isabel. Some ensemble members forecasted hurricanes and storm surges as early as 3–4 weeks lead-time. However, due to the offsets developed in the timing and magnitude of the peak at these lead-times, and based on a sample size of only two events, it is hard to establish the significance of these longer lead-time results. While a follow up study involving flood reforecasts over the entire SubX reforecast period (1999–2015) is needed to support more robust statistics of the forecast skill, our case studies demonstrate the feasibility of probabilistic flood forecasting at subseasonal timescales using the SubX models.

Applying a precipitation error model to numerical weather predictions for probabilistic flood forecasts

This work investigates the use of a stochastic error model (the 2-Dimensional Satellite Rainfall Error Model-SREM2D) to generate an ensemble of rainfall fields, based on the forecasts from the Eta regional weather forecast model. To evaluate the usefulness of this approach against traditional techniques, streamflow probabilistic forecasts from a distributed hydrological model forced with two sources of rainfall data are compared in the Tocantins-Araguaia basin in Brazil. The first dataset is an empirical rainfall ensemble produced by the SREM2D model applied to the Eta model, and the second is a state-of-the-art rainfall ensemble produced by the ECMWF model. Results show the potential of the stochastic error model to generate precipitation ensemble fields from a regional numerical weather forecasting model removing around 60% and 12% of the systematic and random error, respectively. Moreover, SREM2D is proven to be an efficient technique that involves a low computational cost when compared to the more sophisticated ensemble techniques used by the ECMWF model.

Towards an ensemble-based short-term flood forecasting using an event-based flood model- incorporating catchment-average estimates of soil moisture

Extreme floods pose a threat to life and property and accurate and reliable flood forecasts are required to mitigate the consequences of these events. In this paper, we propose a novel Monte-Carlo based framework that utilizes an ensemble-based short-term flood forecasting model. An event-based rainfall-runoff model is selected due to its simplicity and wide use by industry practitioners. However, a challenge with event models is that they are unable to account for the initial condition of the catchment wetness at the commencement of a particular rainfall event. To address this issue, we used independent catchment-average estimates of soil moisture to estimate catchment losses at the commencement and over the duration of the event. The adopted framework enables uncertainty in catchment losses and rainfall depth and patterns to be quantified and incorporated in forecasts. The accuracy and reliability of stochastic components of this framework are evaluated using ensemble verification measures. Later, these measures are used to evaluate four scenarios based on different combinations of deterministic and stochastic conditions. In the stochastic scenarios, the uncertainty of the flood forecasts is reliably quantified (88% reliability on average). The results show that operationally available estimates of catchment wetness can be used to account for uncertainty from event-based loss parameters, and that the uncertainty from average forecast rainfall depth is considerably more important than the uncertainty from forecast losses and the spatio-temporal patterns of the rainfall. This study has shown that event-based models have the potential to be applied within a probabilistic framework to operationally generate probabilistic flood forecasts.

Development of a Hydrological Ensemble Prediction System to Assist with Decision-Making for Floods during Typhoons

Hydrological ensemble prediction systems (HEPSs) can provide decision makers with early warning information, such as peak stage and peak time, with enough lead time to take the necessary measures to mitigate disasters. This study develops a HEPS that integrates meteorological, hydrological, storm surge, and global tidal models. It is established to understand information about the uncertainty of numerical weather predictions and then to provide probabilistic flood forecasts instead of commonly adopted deterministic forecasts. The accuracy of flood forecasting is increased. However, the spatiotemporal uncertainty associated with these numerical models in the HEPS and the difficulty in interpreting the model results hinder effective decision-making during emergency response situations. As a result, the efficiency of decision-making is not always increased. Thus, this study also presents a visualization method to interpret the ensemble results to enhance the understanding of probabilistic runoff forecasts for operational purposes. A small watershed with area of 100 km2 and four historical typhoon events were selected to evaluate the performance of the method. The results showed that the proposed HEPS along with the visualization approach improved the intelligibility of forecasts of the peak stages and peak times compared to that of approaches previously described in the literature. The capture rate is greater than 50%, which is considered practical for decision makers. The proposed HEPS with the visualization method has potential for both decreasing the uncertainty of numerical rainfall forecasts and improving the efficiency of decision-making for flood forecasts.

Real-Time Probabilistic Flood Forecasting Using Multiple Machine Learning Methods

Probabilistic flood forecasting, which provides uncertain information in the forecasting of floods, is practical and informative for implementing flood-mitigation countermeasures. This study adopted various machine learning methods, including support vector regression (SVR), a fuzzy inference model (FIM), and the k-nearest neighbors (k-NN) method, to establish a probabilistic forecasting model. The probabilistic forecasting method is a combination of a deterministic forecast produced using SVR and a probability distribution of forecast errors determined by the FIM and k-NN method. This study proposed an FIM with a modified defuzzification scheme to transform the FIM’s output into a probability distribution, and k-NN was employed to refine the probability distribution. The probabilistic forecasting model was applied to forecast flash floods with lead times of 1–3 hours in Yilan River, Taiwan. Validation results revealed the deterministic forecasting to be accurate, and the probabilistic forecasting was promising in view of a forecasted hydrograph and quantitative assessment concerning the confidence level.

Improving the Reliability of Probabilistic Multi-Step-Ahead Flood Forecasting by Fusing Unscented Kalman Filter with Recurrent Neural Network

It is fundamentally challenging to quantify the uncertainty of data-driven flood forecasting. This study introduces a general framework for probabilistic flood forecasting conditional on point forecasts. We adopt an unscented Kalman filter (UKF) post-processing technique to model the point forecasts made by a recurrent neural network and their corresponding observations. The methodology is tested by using a long-term 6-h timescale inflow series of the Three Gorges Reservoir in China. The main merits of the proposed approach lie in: first, overcoming the under-prediction phenomena in data-driven flood forecasting; second, alleviating the uncertainty encountered in data-driven flood forecasting. Two commonly used artificial neural networks, a recurrent and a static neural network, were used to make the point forecasts. Then the UKF approach driven by the point forecasts demonstrated its competency in increasing the reliability of probabilistic flood forecasts significantly, where predictive distributions encountered in multi-step-ahead flood forecasts were effectively reduced to small ranges. The results demonstrated that the UKF plus recurrent neural network approach could suitably extract the complex non-linear dependence structure between the model’s outputs and observed inflows and overcome the systematic error so that model reliability as well as forecast accuracy for future horizons could be significantly improved.

Probabilistic Flood Forecasting Using Hydrologic Uncertainty Processor with Ensemble Weather Forecasts

Recent advances in the field of flood forecasting have shown increased interests in probabilistic forecasting as it provides not only the point forecast but also the assessment of associated uncertainty. Here, an investigation of a hydrologic uncertainty processor (HUP) as a postprocessor of ensemble forecasts to generate probabilistic flood forecasts is presented. The main purpose is to quantify dominant uncertainties and enhance flood forecast reliability. HUP is based on Bayes’s theorem and designed to capture hydrologic uncertainty. Ensemble forecasts are forced by ensemble weather forecasts from the Global Ensemble Prediction System (GEPS) that are inherently uncertain, and the input uncertainty propagates through the model chain and integrates with hydrologic uncertainty in HUP. The bias of GEPS was removed using multivariate bias correction, and several scenarios were developed by different combinations of GEPS with HUP. The performance of different forecast horizons for these scenarios was compared using multifaceted evaluation metrics. Results show that HUP is able to improve the performance for both short- and medium-range forecasts; the improvement is significant for short lead times and becomes less obvious with increasing lead time. Overall, the performances for short-range forecasts when using HUP are promising, and the most satisfactory result for the short range is obtained by applying bias correction to each ensemble member plus applying the HUP postprocessor.

Ensemble Bayesian forecasting system Part II: Experiments and properties

A typical deficiency of ensemble forecasts is the lack of calibration; e.g., ensemble members are under-dispersed and, consequently, probabilities estimated from them cannot be taken at their face values. Self-calibration is one of the unique theoretic properties of the Bayesian forecasting system, BFS. Its ensemble version, EBFS, inherits this property, provided the ensemble size is large enough. That requirement motivated the version with randomization, EBFSR, presented in part I. Its unique advantage is the operational feasibility of generating a hydrologic ensemble forecast of large size, M (having hundreds or even thousands of members), from a meteorologic input ensemble forecast of small size, MI (having only tens or hundreds of members): MI<M, with M=MIR, where R stands for randomization factor (R>1 and integer). This R-fold enlargement of the ensemble size is achieved through a Monte Carlo generator (sans extra runs of the hydrologic model) and, therefore, requires little extra effort or computing time. But what are the statistical implications?The objective of this part II is to identify experimentally the sampling properties of the EBFSR so that its advantage can be properly harnessed in operational forecasting. The core matter to understand and to quantify is the tradeoff between the computing efficiency and the statistical efficiency of the EBFSR. This is so because the user must specify (i) the largest acceptable sampling error [measured in terms of the average expected maximum absolute difference, E¯(MAD), between the true but unknown predictive distribution function of the forecasted variate and the empirical distribution function estimated from a sample of size m, a fraction of the ensemble size M], and (ii) the tradeoff between MI and R. Critical to deciding this tradeoff are two relationships: E¯MAD=ξ(m) and ME=τ(MP,R), where MP is the size of the input sub-ensemble with positive precipitation amounts, and ME is the effective sample size of the output sub-ensemble of size MPR. In terms of output/input ensemble sizes, the computing efficiency is M/MI=R>1, and the statistical efficiency is ME/MPR<1 because MP<ME<MPR.Based on results of large numerical experiments, general parametric forms of ξ and τ are identified, and the dependence of the parameter values is established on factors such as the forecast type (probabilistic river stage forecast, probabilistic stage transition forecast, probabilistic flood forecast), the forecast lead time, the magnitude of the precipitation input uncertainty, and the informativeness of the deterministic hydrologic model employed by the EBFSR. Finally, two algorithms are formulated to aid the user in specifying the values of M and R, such that E¯(MAD) does not exceed the largest acceptable value and that operational constraints are satisfied (e.g., constraints regarding the input ensemble size MI or the feasible number of runs of the hydrologic model).

An Effective Modelling Approach to Support Probabilistic Flood Forecasting in Coastal Cities—Case Study: Can Tho, Mekong Delta, Vietnam

Probabilistic flood forecasting requires flood models that are simple and fast. Many of the modelling applications in the literature tend to be complex and slow, making them unsuitable for probabilistic applications which require thousands of individual simulations. This article focusses on the development of such a modelling approach to support probabilistic assessment of flood hazards, while accounting for forcing and system uncertainty. Here, we demonstrate the feasibility of using the open-source SWMM (Storm Water Management Model), focussing on Can Tho city, Mekong Delta, Vietnam. SWMM is a dynamic rainfall-runoff simulation model which is generally used for single event or long-term (continuous) simulation of runoff quantity and quality and its application for probabilistic riverflow modelling is atypical. In this study, a detailed SWMM model of the entire Mekong Delta was built based on an existing ISIS model containing 575 nodes and 592 links of the same study area. The detailed SWMM model was then systematically reduced by strategically removing nodes and links to eventually arrive at a level of detail that provides sufficiently accurate predictions of water levels for Can Tho for the purpose of simulating urban flooding, which is the target diagnostic of this study. After a comprehensive assessment (based on trials with the varying levels of complexity), a much reduced SWMM model comprising 37 nodes and 40 links was determined to be able to provide a sufficiently accurate result while being fast enough to support probabilistic future flood forecasting and, further, to support flood risk reduction management.

Performance assessment of hydrologic uncertainty processor through integration of the principal components analysis

Abstract Uncertainty analysis is important and should be always considered when using models for flood forecasting. In this paper, the ‘Principal Components Analysis-Hydrologic Uncertainty Processor’ (PCA-HUP) was developed for probabilistic flood forecasting (PFF) and further evaluated in the middle Yellow River, China. Due to the severe sediment erosion, small and medium floods drain in the main channel (normal floods) while large floods would spill over the bank and drain in river floodplains (overbank floods). Thus, the practical routing methods were used to provide the deterministic flood forecasting (DFF) input for PCA-HUP. PCA-HUP quantifies the forecast uncertainty and provides PFF results. The comparison of performance between the DFF and PFF outputs indicated that PFF could also provide a good accuracy of deterministic hydrograph. In order to explore the performance decay of DFF and PFF with lead time increasing, the lead times n = 1, 6 and 10 hours were chosen for comparison. Results suggested that, with the increasing lead time, the performances of both DFF and PFF decayed accordingly. As a consequence, this study proved the practicability of PCA-HUP in the operational forecasting for both normal and overbank floods in the middle reach of Yellow River.

Flood Forecasting and Decision Making in the new Millennium. Where are We?

This paper reviews the development of real time flood forecasting systems from the early 1970 approaches to the recent probabilistic ones. A preliminary discussion on the motivations for developing real time flood forecasting systems is introduced to explain their evolution in the last four to five decades. It will be shown how recent probabilistic flood forecasts are more robust and effective than the traditional deterministic ones. In particular, when combined with Bayesian decision approaches, probabilistic forecasts are the most appropriate tools for rational decision making in flood warning and flood management. Moreover, they allow taking into account the information from several models to be taken into account by combining into a unique predictive density the deterministic predictions of several hydrological or hydraulic models of a different nature, while in the multi-temporal forecasting extensions, they provide to answers questions such as: Which is the probability of overtopping a dyke in the next 24 h? When will this event be more likely to occur during the next 24 h? The work concludes with a discussion on the still unresolved problems, namely how decisions makers can fully take advantage of the probabilistic forecasts and how these forecasts must be communicated to them in order to meet this objective.

A real-time probabilistic channel flood-forecasting model based on the Bayesian particle filter approach

Reliable real-time probabilistic flood forecasting is critical for effective water management and flood protection all over the world. In this study, we develop a real-time probabilistic channel flood-forecasting model by combining a channel hydraulic model with the Bayesian particle filter approach. The new model is tested in the upstream river reach of Three Gorges Dam (TGD) on the Yangtze River, China. Stage observations at seven hydrological stations are used simultaneously to adjust the Manning's roughness coefficients and to update discharges and stages along the river reach to attain reliable probabilistic flood forecasting. The synthetic experiments are applied to demonstrate the new model's correction and forecasting performances. The real-world experiments show that the new model can make accurate flood forecasting as well as derive reliable intervals for different confidence levels. The new probabilistic flood forecasting model not only outperforms the existing deterministic channel flood-forecasting models in accuracy, but also provides a more robust tool with which to incorporate uncertainty into flood-control efforts.

Willingness-to-pay for a probabilistic flood forecast: a risk-based decision-making game

Abstract. Probabilistic hydro-meteorological forecasts have over the last decades been used more frequently to communicate forecast uncertainty. This uncertainty is twofold, as it constitutes both an added value and a challenge for the forecaster and the user of the forecasts. Many authors have demonstrated the added (economic) value of probabilistic over deterministic forecasts across the water sector (e.g. flood protection, hydroelectric power management and navigation). However, the richness of the information is also a source of challenges for operational uses, due partially to the difficulty in transforming the probability of occurrence of an event into a binary decision. This paper presents the results of a risk-based decision-making game on the topic of flood protection mitigation, called "How much are you prepared to pay for a forecast?". The game was played at several workshops in 2015, which were attended by operational forecasters and academics working in the field of hydro-meteorology. The aim of this game was to better understand the role of probabilistic forecasts in decision-making processes and their perceived value by decision-makers. Based on the participants' willingness-to-pay for a forecast, the results of the game show that the value (or the usefulness) of a forecast depends on several factors, including the way users perceive the quality of their forecasts and link it to the perception of their own performances as decision-makers.

Ensemble Bayesian forecasting system Part I: Theory and algorithms

Summary The ensemble Bayesian forecasting system (EBFS), whose theory was published in 2001, is developed for the purpose of quantifying the total uncertainty about a discrete-time, continuous-state, non-stationary stochastic process such as a time series of stages, discharges, or volumes at a river gauge. The EBFS is built of three components: an input ensemble forecaster (IEF), which simulates the uncertainty associated with random inputs; a deterministic hydrologic model (of any complexity), which simulates physical processes within a river basin; and a hydrologic uncertainty processor (HUP), which simulates the hydrologic uncertainty (an aggregate of all uncertainties except input). It works as a Monte Carlo simulator: an ensemble of time series of inputs (e.g., precipitation amounts) generated by the IEF is transformed deterministically through a hydrologic model into an ensemble of time series of outputs, which is next transformed stochastically by the HUP into an ensemble of time series of predictands (e.g., river stages). Previous research indicated that in order to attain an acceptable sampling error, the ensemble size must be on the order of hundreds (for probabilistic river stage forecasts and probabilistic flood forecasts) or even thousands (for probabilistic stage transition forecasts). The computing time needed to run the hydrologic model this many times renders the straightforward simulations operationally infeasible. This motivates the development of the ensemble Bayesian forecasting system with randomization (EBFSR), which takes full advantage of the analytic meta-Gaussian HUP and generates multiple ensemble members after each run of the hydrologic model; this auxiliary randomization reduces the required size of the meteorological input ensemble and makes it operationally feasible to generate a Bayesian ensemble forecast of large size. Such a forecast quantifies the total uncertainty, is well calibrated against the prior (climatic) distribution of predictand, possesses a Bayesian coherence property, constitutes a random sample of the predictand, and has an acceptable sampling error—which makes it suitable for rational decision making under uncertainty.

Probabilistic forecasting and the reshaping of flood risk management

Abstract Advances in probabilistic forecasting, notably based on ensemble prediction systems, are transforming flood risk management. Four trends shaping the assimilation of probabilistic flood forecasting into flood risk management are longer forecasting lead times, advances in decision-making aids, inclusion of probabilistic forecasting in hazard mitigation and collaboration between researchers and managers. Confronting how to use probabilistic flood forecasts to make binary management decisions for reducing flood losses requires developing institutional capacity while acknowledging flood risk estimation is one component of decision making under uncertainty in an evolving policy landscape.

Improving flood forecasts for better flood preparedness in the UK (and beyond)

This winter (2013/14) coastal storms and an unprecedented amount of rainfall led to significant and widespread flooding across the southern UK. Despite much criticism and blame surrounding the flood events, the Flood Forecasting Centre, a recent development in national‐level flood forecasting capabilities for the government and emergency response communities, has received considerable praise. Here we consider how scientific developments and organisational change have led to improvements in the forecasting and flood preparedness seen in this winter's flooding. Although such improvements are admirable, there are many technical and communication challenges that remain for probabilistic flood forecasts to achieve their full potential.

Probabilistic flood forecast: Exact and approximate predictive distributions

For quantification of predictive uncertainty at the forecast time t0, the future hydrograph is viewed as a discrete-time continuous-state stochastic process {Hn:n=1,…,N}, where Hn is the river stage at time instance tn>t0. The probabilistic flood forecast (PFF) should specify a sequence of exceedance functions {F‾n:n=1,…,N} such that F‾n(h)=P(Zn>h), where P stands for probability, and Zn is the maximum river stage within time interval (t0,tn], practically Zn=max{H1,…,Hn}. This article presents a method for deriving the exact PFF from a probabilistic stage transition forecast (PSTF) produced by the Bayesian forecasting system (BFS). It then recalls (i) the bounds on F‾n, which can be derived cheaply from a probabilistic river stage forecast (PRSF) produced by a simpler version of the BFS, and (ii) an approximation to F‾n, which can be constructed from the bounds via a recursive linear interpolator (RLI) without information about the stochastic dependence in the process {H1,…,Hn}, as this information is not provided by the PRSF. The RLI is substantiated by comparing the approximate PFF against the exact PFF. Being reasonably accurate and very simple, the RLI may be attractive for real-time flood forecasting in systems of lesser complexity. All methods are illustrated with a case study for a 1430km2 headwater basin wherein the PFF is produced for a 72-h interval discretized into 6-h steps.

Addendum to Bayesian ensemble forecast of river stages and ensemble size requirements

Summary Ensemble size requirements were analyzed using Bayesian ensemble forecasts ( Herr and Krzysztofowicz, 2010 ). These forecasts were generated using output from an analytic–numerical Bayesian forecasting system (BFS) within a Monte-Carlo algorithm. Empirical distributions were estimated from the ensemble forecasts and compared with the analytic distributions output by the BFS. Using the maximum absolute difference as a measure of the quality of fit, it was shown that a single graph described the relationship between the ensemble size and the expected maximum absolute difference. Within the scope of the experiments, the graph was insensitive to forecast point, precipitation event (rain or no rain), and lead time. However, the graph only applied to probabilistic river stage forecast and probabilistic flood forecast, but not to probabilistic stage transition forecast, which is specified by a family of conditional predictive one-step transition distributions. Herein, further mathematical analysis of the estimation algorithm shows that, with a small modification, the single graph can be applied to all three types of forecasts.