Sequential Data Assimilation for Real-Time Probabilistic 1 Flood Inundation Mapping

19 Real-time probabilistic flood inundation mapping is crucial for flood risk warning and decision 20 making during the emergency of an upcoming flood event. Considering high uncertainties 21 involved in the modeling of a nonlinear and complex flood event, providing a deterministic flood 22 inundation map can be erroneous and misleading for reliable and timely decision making. The 23 conventional flood hazard maps provided for different return periods cannot also represent the 24 actual dynamics of flooding rivers. Therefore, a real-time modeling framework that forecasts the 25 inundation areas before the onset of an upcoming flood is of paramount importance. Sequential 26 Data Assimilation (DA) techniques are well-known for real-time operation of physical models 27 while accounting for existing uncertainties. In this study, we present a Data Assimilation (DA)- 28 hydrodynamic modeling framework where multiple gauge observations are integrated into the 29 LISFLOOD-FP model to improve its performance. This study utilizes the Ensemble Kalman Filter 30 (EnKF) in a multivariate fashion for dual estimation of model state variables and parameters where 31 the correlations among point source observations are taken into account. First, a synthetic 32 experiment is designed to assess the performance of the proposed approach, then the method is 33 used to simulate the Hurricane Harvey flood in 2017. Our results indicate that the multivariate 34 assimilation of point-source observations into hydrodynamic models can improve the accuracy 35 and reliability of probabilistic flood inundation mapping by 5-7% while it also provides the basis 36 for sequential updating and real-time flood inundation mapping. 37 41 42 43 44 45 bathymetry is a more identifiable parameter as it shows the fastest convergence with a minimum degree of uncertainty. However, the channel roughness is less identifiable with the slowest convergence. The scatter plots illustrate the evolution of parameter space at six different time segments. The first day (t=1) includes all 100 ensemble members of parameters and day 30 corresponds to the highest discharge and water stage of flooding when the model parameters reach the highest improvement and get closer to the true value. Figure 3 shows that both model parameters are converging toward the true values as the assimilation proceeds. This indicates the efficacy and usefulness of the proposed DA-hydrodynamic modeling framework developed in this study.

Probabilistic forecasting and uncertainty quantification using DA have been the core of modeling 104 in the atmospheric and oceanic sciences (e.g. Anderson and Anderson, 1999;Courtier et al., 1993). 105 Later, the hydrologic community started to utilize this approach to account for the uncertainties is not adequate for assimilating into these models. To properly estimate the flood inundation 122 extent, a spatial resolution less than river width (e.g. 100 m) is recommended. In addition, dDue 123 to the short duration of floods, satellite data with daily revisit timea sub-daily time scale and spatial 124 resolution less than the river width (e.g. 100 m) is neededrecommended. Since remote sensing 125 products do not provide such high spatiotemporal resolution data for hydrodynamic models, the 126 research on hydrodynamic data assimilation is limited in the literature. Due to the coarse spatial 127 resolution of satellites that provide water surface elevation data, sSome studies have limited their 128 analyses to large rivers with a width of above 1 km (e.g. study of Nile and Amazon) (Brêda et al.,129 2019). However, since the width of the majority of rivers is less than 100 meters, these studies 130 cannot be practically used in many regions. ). This approach can provide high-resolution data that is suitable for the majority of rivers.

139
However, the reliability of this data is concerning because the methods used to convert the flood 140 extent to WSE pose additional errors which that downgrades the quality of the final observed data 141 for assimilation practices. Besides these issues, the major drawback of remote sensing data 142 assimilation pertains to their coarse temporal resolutions. To efficiently monitor the flood 143 dynamics, the assimilation process should be performed at a daily/hourly time scale, however, the 144 revisit frequency of satellites used for capturing the water surface elevationWSE ranges from a 145 week to a month. Therefore, there is a significantly low chance to capture multiple real-time remote 146 sensing images for the majority of inundated catchments during flood events. In the most 147 optimistic scenario, assimilation of satellite data is only limited to one/two updates during the 148 simulation period which may not be sufficient for reliable probabilistic flood inundation mapping.

149
Application of DA in hydrodynamic modeling can be either river monitoring or flood inundation 150 mapping. The goal of hydrodynamic data assimilation for river monitoring is to track variations in 151 the channel roughness and bathymetry in the long run. Therefore, the weekly/monthly satellite 152 data can be well assimilated into the models as the channel characteristics do not change on a daily

159
(2017) performed a Particle Filtering (PF) approach to assimilate the water stage data from six 160 gauges into a hydrodynamic model. In order to calculate the particle weights in the filtering 161 process, they assumed that gauge observations are independent. In this study, however, we 162 consider interconnections among the gauge stations and apply multivariate Ensemble Kalman 163 Filter (EnKF) to a 2D two-dimensional (2D) hydrodynamic model for better characterization and 164 quantification of uncertainty and further improving the accuracy of model simulations.

165
Advancing the probabilistic hydrodynamic modeling with DA techniques is a necessary step to fill 166 the gap between hydrology and hydrodynamics. To address this problem, this study aims to 167 explore the capability of a standard sequential DA technique, namely the EnKF, for real-time  (1) where Q is the flow rate at a given cross-section with the area of A in the main channel, x denotes 230 the location along the channel, t represents time, S0 and Sf are channel bed and friction slopes, and 231 g is the gravitational acceleration.

232
We use the sub-gird channel solver, the most recently developed numerical scheme that considers Using +1 − and forcing data, a model state ensemble and predictions are generated, respectively. and are assumed to be independent.

277
Then we update the parameter ensemble members using the standard Kalman filter equation: where +1 ∈ ℝ × is the Kalman gain matrix for correcting the parameter trajectories and is 280 obtained by: Now using the updated parameter, the new model state trajectories (state forecasts) and prediction 296 trajectories are generated: Model states ensemble is similarly updated as follows: where +1 ∈ ℝ × is the Kalman gain for correcting the state trajectories and is obtained by: where ∑ . +1 ∈ ℝ × is the cross-covariance matrix of states ensemble and prediction ensemble 305 (Eq. 16).
In this study the water depth along the channel is the only state variable (m=1). The channel 309 roughness and bathymetry are two model parameters (l=2) and three point source observations 310 including water discharge at gauge 1 and 2 as well as water stage at gauge 2 (n=3) are assimilated 311 into the LISFLOOD-FP model (Table 1). Therefore, the Kalman gains used to update the model 312 parameters and states (Eqs 5 and 15) are 2 × 3 and 1 × 3 matrices that take advantage of a 313 multivariate point source assimilation while considering the downstream correlation between 314 discharge observations and the correlation between water stage and discharge at gauge 2.  The ensemble of updated water depth (state), bathymetry, and channel roughness (parameters) will 347 beare used within the LISFLOOD-FP to predict an ensemble of water depth for the next time step.

348
The predicted water depth is simply converted to a probabilistic flood inundation map. Using this 349 data assimilation framework, we can generate 1-day forecast of probabilistic flood inundation          In the real experiment, we assimilate the discharge and water stage readings from two internal 492 USGS gauges into the LISFLOOD-FP model. We also run the OL simulation and calculate the 493 ensemble mean to predict the discharge and water stage at these two gauges. Figure 4  Finally, to quantify the performance of EnKF and OL for generating a spatial distribution of water 569 depth over the domain, we illustrate the ROC graphs, the AUC values, and Fit indices in Figure 8.

570
To calculate these measures, we ignore the temporal distributions and only report the maximum 571 inundation maps that represent the union of flooded areas over the entire period of Harvey.

572
Comparing the EnKF and OL in Figure 8.a, the EnKF line (blue) is closer to the northwest of the 573 rfp-rtp space where its AUC is 5% higher than the OL approach. In Figure 8.b, each point 574 represents the Fit indices for the OL and the EnKF approaches corresponding to a given threshold.

575
Using hundred number of100 thresholds that each rangingrange from [0.01,1], the probabilistic  The validation results also demonstrate that the EnKF reduces the underestimation by 7% and 617 outperformed the OL approach by around 5% for probabilistic flood inundation mapping.  Competing interests 698 The authors declare that they have no conflict of interest.