The coupled modeling system consists of a hydrologic model coupled with a hydraulic model. The hydrologic model is used to compute the runoff at the upstream boundaries of the hydraulic model. The hydrologic model used in this study is SUPERFLEX, which is a framework for conceptual hydrologic modeling introduced by . The model structure is a combination of generic components: reservoirs, connection elements, and lag functions. In this study, a lumped conceptual model and its structure as a combination of three reservoirs are used: an unsaturated soil reservoir with storage SUR, a fast-reacting reservoir with storage SFR, and a slow-reacting reservoir with storage SSR. A lag function has been added at the outlet of the slow- and fast-reacting reservoirs.

The hydraulic model is based on LISFLOOD-FP and simulates flood extent, water levels, and streamflows along the river and on the floodplain. A sub-grid 1D kinematic solver is used for the channel flow. When the storage capacity of the river is exceeded, the water spills into the floodplain and a 2D diffusion wave scheme neglecting the convective acceleration is used for the floodplain flow simulation.

The true meteorological data (i.e., temperature and rainfall) are used as input for the hydrologic–hydraulic model to simulate streamflow and water level time series and to provide binary flood maps, where each pixel is classified as flooded (with value 1) or non-flooded (with value 0) at each assimilation time step. These computational results represent the synthetic truth that will be used to evaluate the performance of the proposed assimilation framework. The true binary flood maps are also used to generate the synthetic SAR observations as described in the next section.

In the proposed synthetic experiment, we generate synthetic SAR images at each assimilation time step. The SAR images are generated with the same spatial resolution of the LISFLOOD-FP maps. Similarly to the study, we make use of a real SAR image, acquired during a flood event in the past, and of the LISFLOOD-FP model to generate true binary flood maps. The histogram of the SAR image backscatter values can be approximated with two Gaussian curves relative to the flooded and non-flooded pixel classes. Generally, the class of flooded pixels is often represented just by a fraction of the SAR image scenes. Therefore, to identify and characterize areas where the flooded and non-flooded classes are more balanced, the hierarchical split-based approach (HSBA; ) is applied to the selected SAR image. The parameters of the Gaussian PDFs are determined by fitting the histogram values of the HSBA selected areas.

Then, random backscatter values, derived from the calibrated Gaussian PDFs, are associated with the pixels of the true binary flood map indicating the presence of water and no-water areas. Once the synthetic SAR images are generated, the procedure is applied and synthetic PFMs are derived. The probability of being flooded, given the recorded backscatter values for each pixel of a SAR image p(F|σ0), is obtained via the Bayes' theorem: 1pF|σ0=pσ0|Fp(F)pσ0=pσ0|Fp(F)pσ0|Fp(F)+pσ0|F‾p(F‾). In Eq. (), p(σ0|F) and p(σ0|F‾) represent, respectively, the probability of the backscatter values of the flooded and non-flooded pixels, p(F) is the prior probability of a pixel being flooded, and p(F‾) is the prior probability of a pixel being non-flooded before any backscatter information is taken into account. The conditional probabilities are derived from the histogram of the backscatter values estimated from the synthetically generated SAR image. The prior probabilities can be estimated from the flood extent model output or through visual interpretation of aerial photography in real cases. However, in general such information is not always available and the prior probabilities are unknown. Consequently, set the prior probability of Eq. () to 0.5 so that both flooded and non-flooded pixels are equally likely. While this study is based on a synthetic experiment, true binary flood extent maps are available. Therefore, the assimilation is realized using both the estimated prior probability (as the ratio between the flooded area and the total area) and the prior probability equal to 0.5. Given the similarity of the results for both cases, in the following sections we only discuss the experiment using the estimated prior probability.

The method proposed by aims to characterize the speckle-induced uncertainty. However, it does not consider any other phenomena leading to a wrong classification in SAR-based flood maps. Particular atmospheric conditions (e.g., wind, snow, and precipitation), water-lookalike areas (e.g., asphalt, sand, and shadow), or obstructing objects (e.g., dense canopy and buildings), as mentioned in , can lead to a wrong classification in the flood maps. Therefore, the areas where such errors could occur should be masked out from the SAR-based flood maps in order to provide reliable flood detection.

In the first part of this study, SAR observations are considered without errors. In the second part, these kinds of errors are integrated in the synthetic SAR observations to evaluate their effect on the DA. Specifically, the pixels along the flood edge of each particle are selected. From this set, a given number of those pixels effectively flooded in the true binary flood map are artificially corrupted so that they belong to dry pixels. The number of corrupted pixels depends on the magnitude of the error that we want to introduce in the SAR observations.

In a PF, the prior and posterior PDFs are approximated by a set of particles. Here, we hypothesize that rainfall is the only source of uncertainty affecting the model-based flood extent simulations. For this reason, an ensemble of rainfall time series is used as input for the coupled hydrologic–hydraulic model. Each rainfall time series is obtained by perturbing, with a multiplicative random noise from a log-normal error distribution, the true rainfall time series following the approach proposed in . Via the hydrologic model, 128 rainfall time series are obtained and forwarded in time.

It is important to note that the same hydrologic–hydraulic model in terms of structure, initial conditions, and parameters is used for all model runs. The reliability of the rainfall ensemble is verified with the statistical metrics proposed by . According to the verification measurement VM1 in Eq. (): 2VM1=<ensk><ensp>≈1. The ensemble spread in Eq. () (where xk,n represents the value of the variable x at time k for each pixel n) 3enspk=1N∑n=1Nxk,n-x‾k2 has to be close to the ensemble skill (Eq. ) 4enskk=x‾k-yk, which is the difference between the mean x‾k over the N particles of the ensemble and the observation yk at time k. VM2 (Eq. ) verifies that the truth is statistically indistinguishable from the random samples of the ensemble. 5VM2=<ensk><mse>≈(N+1)2N with “mse” estimated as: 6msek=1N∑n=1Nxk,n-yk2. VM1 and VM2 are used to assess the quality of the discharge ensemble at the output of the hydrologic model.

The DA framework consists of two main steps: prediction, i.e., model simulations, and analysis, i.e., update of particle probabilities when an observation is available. The prior probability of the model state x at a given time k is represented by a set of N independent random particles xn sampled from the prior probability distribution p(x) as 7p(x)=1N∑n=1Nδx-xn, where δ is the Dirac delta function. In this study, the prior probability distribution is assumed to be uniform. The observations of flooded or non-flooded pixels y are related to the true state xt according to the following equation: 8y=Hxt+ϵ, where H is the observation operator that maps the state vector into the observation space and ϵ represents the observation errors. According to the Bayes' theorem, the observations are assimilated by multiplying the prior PDF p(x) and the likelihood p(y|x), which is the probability density of the observation given the model state, and dividing by the total probability p(y), resulting in 9p(x|y)=p(y|x)p(y)p(x), which is the posterior probability p(x|y), i.e., the probability density function of the model state given the observations. By inserting Eq. () into Eq. () we obtain the following formula: 10p(x|y)≈∑n=1NWnδx-xn, where Wn represents the relative importance in the probability density function (i.e., global weight) given by 11Wn=py|xnN⋅p(y)=py|xnN⋅∫p(y|x)p(x)δx≈py|xn∑py|xj.

In this study, the likelihood (global weight, Wn) is represented by the product of the pixel-based likelihood (local weight, wi), assuming the L pixel observation errors are independent from each other. At time k of the observation, local weights wi,n are defined for each particle n and for each pixel i according to : 12wi,n=piF|σ0Mi,n+1-piF|σ01-Mi,n. wi,n is equal to the probability of a pixel being flooded as derived from the synthetically generated SAR image. Mi,n is equal to 1 if the model predicts the pixel as flooded, whereas Mi,n is equal to 0 if the model predicts the pixels as non-flooded. We convert the model-based water depth maps into binary flood extent maps by considering that a pixel is flooded if its water level is above 10 cm. pi(F|σ0) equals the probability of a pixel being flooded according to the observations, on conversely 1-pi(F|σ0) equals the probability of not being flooded. By applying Eq. () we assign higher probabilities to those pixels where model predictions and observations agree. Next, Wn is estimated for each particle by the normalization of the product of the local weights ensuring that the sum of the global weights is equal to 1 (Eq. ; standard method). 13Wn=∏i=1Lwi,n∑n=1N∏i=1Lwi,n The expectation of the OL is equivalent to the mean of the ensemble because the relative importance of each particle is the same. The global weights are used to compute the expectation of the streamflows (Q) and water levels (h) at time (k) and per pixel (i) after the assimilation (see Eqs.  and ). 14h‾i=∑n=1NWn⋅hi,n15Q‾i=∑n=1NWn⋅Qi,n The particles keep these global weights until the next assimilation time. The particles are then set to the same equal weight before a new analysis step is performed.

Unless the number of particles increases exponentially with the dimension of the system state, the particle filter is likely to degenerate because high probability is assigned to a single particle while all other members will result in small weights . PFs are often subject to degeneracy issues when, due to computational reasons, the number of particles is not sufficiently high . After the application of the standard PF, the variance of the weights tend to increase and only a few particles of the ensemble have a non-negligible weight. To mitigate this problem, in , the global weight defined in Eq. () has been adapted using a tempering coefficient (α, as described by the following Eq. ). 16Wn(α)=∏i=1Lwi,nα∑n=1N∏i=1Lwi,nα Since α and weights values are lower than 1, adding the power of α in the weights formula allows for shifting all weight values closer to 1. This therefore decreases the variance of the weights and inflates the posterior probability. After the assimilation, the number of particles with significant weight depends on the α value. The smaller the α, the higher the variance of the posterior PDF. Consequently, as argued in , when the α coefficient is small enough, this adaptation of the PF helps reduce the degeneracy of the ensemble. While in the previous study by the α value was defined so that the worst model solution would have had a non-zero global weight, in this study we propose to define α based on the desired effective ensemble size (EES). The coefficient α in is site-dependent as it relies on the number of flooded pixels, whereas in this study α is a function of the EES, which is a measure of degeneracy based on the global weights : 17EES(α)=1∑n=1NWn(α)2⋅1N⋅100. The EES is lower than N and its value indicates the level of degeneracy. α is equal to 1 when the standard method is used. Decreasing the α coefficient leads to an increase of the EES.

To carry out the evaluation of the PFM statistics we have used reliability plots. The results of the different assimilation scenarios are evaluated on a spatiotemporal scale with the following performance metrics:

contingency maps and the confusion matrix

The virtual satellite acquisition dates are aligned with the actual Sentinel-1 acquisition frequency. The revisit time over Europe, considering both ascending and descending orbits, is around 3–4 d, meaning that on average two satellite images are available per week. In order to adopt a realistic Sentinel-1-like observation scenario we chose to assimilate four synthetic observations over a period of 10 d.

In Figs.  and , the area corresponds to the hydraulic model domain. The hydrologic model, covering the upstream catchment, is used to compute the input boundary conditions of the hydraulic model. The results are computed and compared within the hydraulic model domain. The synthetic SAR observations are shown in Fig. . The corresponding PFMs are shown in Fig.  and reliability plots are provided in Fig. . In the reliability plots, the points aligned along the 1:1 line indicate a statistically reliable PFM.

The verification measurements VM1 and VM2 (Eqs.  and ) of the ensemble discharge in Bewdley (Fig. ) are equal to 1.047 and 0.527, respectively. These values are close to the ideal values of 1 and 0.5.

The CSI is computed over the entire hydraulic model domain at each assimilation time step.

The general trend of the assimilation effect is positive, as errors tend to decrease at all the assimilation steps with different assimilation methods. Even though the CSI is already high with the OL, the assimilation further improves the results and this becomes particularly clear at the last assimilation time step. From Table  it can be noticed that the CSI, approximately equal to 0.80 with the OL in the worst case (assimilation of the IV image), exceeds 0.96 for the different assimilation types and reaches the maximum value of 0.99 with the standard method. In Fig. , we provide the contingency maps of the OL and of the 5 % EES approach (results of the standard method are similar to those of the 5 % EES approach and therefore not shown). For each pair of images, we show on the left the results of the OL and on the right the results obtained after the assimilation.

In this study, it can be observed that the OL has a tendency to overdetection; the number of red pixels is higher than the number of black pixels and after the assimilation the number of overdetected pixels decreases, confirming the results obtained with the CSI.

The confusion matrix given in Table  provides more details on the fourth assimilation time step. On one hand, the number of pixels wrongly predicted as flooded in the OL is 1196 and more than 90 % of these are correctly classified as non-flooded after the assimilation for both the standard and 5 % EES methods. On the other hand, a few pixels correctly predicted as flooded in the OL are classified as non-flooded after the assimilation. However, it can be argued that the number of 201 wrongly classified pixels after the assimilation is rather low compared to the 1253 px of the OL.

The flood is simulated using an hourly time step. Consequently, it is possible to evaluate the evolution of the performance metrics CSI (Fig. ).

This figure shows that the OL's performance is consistently poor and the standard assimilation performs best compared to the other assimilation runs at all the assimilation time steps.

The assimilation runs with different EES values lie within these two extremes. It can be noted that the more particles that are neglected, which is equivalent to saying the lower the EES, the higher is the performance at the assimilation time step.

Moreover, markedly different CSI time series for the different assimilation experiments are shown in Fig. .

After 27 h from the first assimilation, the performances of the standard and 5 % EES methods, which perform better than the other methods, start decreasing. The lowest values are reached 54 h after the assimilation. One explanation is that the weights assigned to the particles at the first assimilation time are no longer valid when hydraulic conditions change and need to be recomputed.

However, things change after the second assimilation, when the performances of the standard and the 5 % EES assimilation methods remain stable until the end of the simulation time.

The decrease in performance attributed to the standard and 5 % assimilation methods after the first time step is due to a drastic change in the flood extent. The total number of flooded pixels reduces from 8539 to 5494 because the flood started receding.

The spread of the posterior PDF with the standard and 5 % EES methods is small, meaning that only a few particles retain significant importance weight. Consequently, when the flood extent changes and particles evolve in time, it may happen that the uncertainty bounds of the posterior PDF do not encompass the true model state after several time steps. On the contrary, when more particles are considered (higher EES), more particles are used to draw the posterior PDF. This gives more chances for the ensemble to encompass the synthetic truth and increases the overall robustness of the method. This becomes particularly relevant when the hydraulic boundary conditions change and no new observation is available.

The RMSE, reported in Table  decreases by factors larger than 2 and 3 with the standard assimilation and the 5% EES assimilation, respectively. After the first assimilation, carried out close to the flood peak in Saxons Lode, the accuracy of the water level is improved by approximately 20 cm over the entire flood domain. The assimilation of the second and fourth images has a negative effect when the adapted method 50 % EES of the assimilation particle filter is applied; the RMSE increases compared to the OL. As already shown in Table , the standard assimilation and 5 % EES predictions of water levels provide more accurate results (Fig. ). When moving away from the first assimilation, the RMSE of the best performing assimilation methods increases. For instance, after 54 h the RMSE of the standard method is increased by 65 % compared to the RMSE of the OL. When different values of EES are considered, the RMSE values fluctuates significantly in between two assimilations and it becomes difficult to draw any general conclusions. As the number of important particles increases, water levels vary significantly, especially in the area close to the flood edge even though the flood extent does not change too much from one particle to another.

The different assimilation runs are also compared considering the discharges and water levels at different gauge stations along the river Severn. In the right panels of Figs.  and  the different assimilation experiments are compared against the synthetic truth (red line). In the left panels of Figs.  and  the standard method and the 5 % EES assimilation with the important particles and the synthetic truth are shown. The plotted important particles represent 5 % of the ensemble with the largest weight. All 128 particles are equally weighted until the first observation is assimilated. After the first assimilation the number of important particles decreases. At the second assimilation time step, weights are recomputed and the new important particles are selected again and so on. The assimilation of the PFMs improves the predictions of water levels and streamflow at specific points of the river Severn, as in Bewdley and in Saxons Lode (Figs.  and ), for the majority of the assimilation time steps in both underprediction and overprediction cases. The standard method and similarly the 5 % EES assimilation method are the most accurate in forecasting the values of water levels and streamflows. The improvements due to the assimilation persist for a long time: up to 27 h after the first assimilation predictions are still close to the synthetic truth. The local results of water levels suggest that the inaccuracy of the global RMSE values in time is likely due to the evaluation over the entire flood domain.

In the previous section, speckle uncertainty in SAR observations is considered. However, in reality, SAR observations are also susceptible to errors due to the misclassification of wet/dry pixels caused by features on the ground as already mentioned. Therefore, errors are added to the synthetic SAR observations as described in the methodology to investigate the impact on the DA assimilation framework. Figure  shows the RMSE and the CSI obtained at different assimilation time steps. The best performing assimilation methods (i.e., standard and 5 % EES) with no error in the observations are compared with those where error is introduced. With the misclassification of 20 % of the pixels, the assimilation still has beneficial effects: the CSI increases at each assimilation time step with respect to the OL. The RMSE values also tend to be satisfactory after each assimilation. With an increase in the error of 40 %, the performance of the DA framework starts decreasing. The assimilation of the first image still has a positive effect on the predictions. In fact, the CSI and RMSE are improved with respect to the OL even through the improvements are not as significant as in the previous cases. The explanation is arguably to be found in the high number of flooded pixels. It is large enough to counterbalance the misclassified pixels in the SAR image. Performance decreases with the assimilation of the remaining SAR observations when the number of flooded pixels is reduced by half.