**1. Introduction**

Reliable and accurate flood forecasting is one of the most important tasks of operational hydrology, while it is also very challenge due to the inordinately non-linear hydro-geological features and dynamic nature of climate conditions. High uncertainty encountered in the occurrence and magnitudes of future flood event stimulates the demands for probabilistic flood forecasting. The goal of probabilistic forecasting is to provide information about the uncertainty of the forecast [1]. Most hydrological forecast models produce deterministic forecasts, which provide the best point-value estimates rather than quantify the predictive uncertainty [2]. Nevertheless, when a deterministic forecast turns out to be far from what has taken place, the consequences will probably be worse than a situation where no forecast is available [3]. Probabilistic hydro-meteorological forecasts have been used frequently to communicate forecast uncertainty over the last few decades [4–6]. The transformation from a deterministic approach to a probabilistic approach is a development trend of flood forecasting around the world [6,7]. The optimal mega-reservoir operation for live-saving and resources utilization creates

outreach demands for probabilistic flood forecasting; consequently, scientific research should focus on quantifying and mitigating the uncertainty of probabilistic flood forecasts [8,9]. The reliability of hydrologic forecasts can be affected by input uncertainty, meteorological uncertainty, and hydrologic uncertainty of model structure and parameters. One of the primary techniques to reflect different uncertainties in hydrologic forecasts is to create a probabilistic forecast [10,11]. Probabilistic forecasts can be made using three approaches: a probabilistic pre-processing approach plus a deterministic forecast model; a probabilistic forecast model; and a deterministic forecast model plus a probabilistic post-processing approach [12–14]. The first two approaches quantify uncertainties in inputs and model structure while the third quantifies the overall uncertainty in model structure and parameters. Our study would concentrate on improving hydrologic forecasts using deterministic models plus probabilistic post-processing technique.

Probabilistic post-processing techniques are commonly introduced to complement point-value estimations offered by the deterministic forecast model [15,16]. The Kalman filter (KF) proposed by Kalman [5] provides a theoretical post-processing framework based on model point estimation for reducing forecast uncertainty through recursively calculating a statistically optimal estimate of the prediction. The KF post-processing is a component of the probabilistic post-processing techniques, and is a recursive state estimator for a process that is assumed to be affected by stochastic interference and by stochastic noise [5]. The KF family consists of the linear KF (LKF) and non-linear KF (NKF) [17]. The LKF approach can only identify the linear error estimation whereas the NKF approach can quantify the non-linear error estimation. As is known, the NKF is widely used for extracting non-linear dependence of forecast errors and conquering the white noise with systematic over/under-predicting characteristics [17]. Furthermore, the extended KF (EKF) and the unscented KF (UKF) [17] are two common usages of NKF. Most importantly, the UKF approach has not yet been employed to lessen the uncertainty of multi-step-ahead flood forecasting driven by a recurrent neural network (RNN) according to a review of literature [18–21]. Despite there are several researches associated with the combination of UKF/EKF and hydrological models [22–24] on hydrological domain, all of them concentrate on quantifying the uncertainty of hydrological forecast driven by static (i.e., non-recurrent) artificial neural networks (ANNs), e.g., feed-forward neural network and local linear models as well as the hydraulic model. Bearing this in mind as motivation, for the first time, the UKF is introduced to quantify the uncertainty of multi-step-ahead flood forecasts driven by the RNN (i.e., RNN is more complicated than the static ANNs). Therefore, it is interesting to explore UKF for modeling and lowering the uncertainty appeared in RNN-driven flood forecasts.

Machine-learning techniques have developed fast during the last few decades, and they have been adopted as data-driven methods to model hydrological systems [11,25,26]. For instance, the back-propagation neural network (BPNN), the radial basis function (RBF), the support vector machine, the quantile regression neural network (QRNN), the recurrent neural network (RNN), the long-short term memory (LSTM) and the non-linear auto-regressive with exogenous inputs neural network (NARX) have been widely applied to modeling hydrologic and meteorological time series [27–38]. A number of recent studies indicate that ensemble artificial neural network can improve the probabilistic forecast skill for hydrological events [39–41]. The main advantage of ANN is owing to its ability to discern linear or non-linear relationships even with very limited data inputs and being able to recognize even complex patterns in a data set without a priori understating of the underlying mechanism. The major drawbacks, on the other hand, are that they are prone to under-predict flood series for extreme flood events. Therefore, it is essential to conduct in-depth research on machine-learning models for enhancing model accuracy and reliability through converting deterministic flood forecasts into probabilistic ones using a stochastic post-processing technique.

This study proposes a probabilistic forecasting approach to reduce the prediction intervals of multi-step-ahead flood forecasts, which consists of two parts: the deterministic forecast model and the probabilistic post-processing technique. First, the recurrent neural network (NARX) is introduced to make multi-step-ahead point forecasts. Then, the UKF technique driven by point forecasts is employed to create the prediction intervals of flood forecasts. We concentrate on hydrological uncertainty only (i.e., the uncertainty resulting from imperfect rainfall-runoff modeling), considering "perfect" rainfall as inputs. A static BPNN and a recurrent NARX are used to construct flood forecast models, and the model that produces more accurate point estimations will be employed to carry out probabilistic forecasting. The UKF approach is implemented separately to transform point flood forecasts into probabilistic flood forecasts. The rainfall and inflow datasets of the Three Gorges Reservoir (TGR) in China are used to demonstrate the reliability and applicability of the approach.
