The first hydrodynamic model is developed in Delft3D-FM using an unstructured finite-volume grid that consists of triangular cells with a spatially varying size. Unstructured grids help capture geomorphological and urban features with greater detail than conventional nested, structured grids (Kumar et al., 2009; Muñoz et al., 2022a). These features include the G-Bay entrance, artificial channels in Houston, intracoastal waterways, lateral floodplains, wetland regions, and bottleneck-like connections between G-Bay and both the Buffalo Bayou and San Jacinto rivers (Fig. 1c). Triangular cell sizes are set to increase from 3 km at the open-ocean boundary in the Gulf of Mexico up to 5 m in Harris County. This ensures a detailed simulation of CF dynamics in natural and urban settings. Similarly, the second hydrodynamic model is developed in 2D HEC-RAS using an unstructured finite-volume grid. The mesh consists of polygons of varying cell size and the numerical scheme to solve the shallow water equations is set to the Eulerian–Lagrangian (SWL-ELM) method. This, in turn, ensures that the model solves all terms in Eqs. (1)–(3), except for atmospheric pressure due to the current model capabilities of 2D HEC-RAS. In addition, we force the mesh generation with a cell size and spatial distribution similar to that of the Delft3D-FM model. Although there is no a straightforward procedure to transfer the mesh properties and/or spatial characteristics between the two hydrodynamic models, we ensure that geomorphological and urban features are correctly delineated by conducting an extensive mesh refinement in critical locations, as suggested in similar studies (Muñoz et al., 2021; Shustikova et al., 2019). The time step is controlled by the Courant–Friedrichs–Lewy condition, with a maximum value of 0.7 for both models. Also, model outputs are generated with an hourly interval for calibration and validation purposes.

After the mesh generation process, we consider multiple USGS river discharge stations in the G-Bay model as upstream BCs, including Whiteoak Bayou (USGS 08074500), Buffalo Bayou (USGS 08073700), Brays Bayou (USGS 08075000), Sims Bayou (USGS 08075500), Berry Bayou (USGS 08075605), Greens Bayou (USGS 08076000), Hunting Bayou (USGS 08075770), Vince Bayou (USGS 08075730), Clear Creek (USGS 08077600), Goose Creek (USGS 08067525), Cedar Creek (USGS 08067500), and Trinity River (USGS 08067252). Also, due to the lack of available river-gauge stations located immediately downstream of Lake Houston dam, river flow from Lake Houston dam to the San Jacinto River is estimated as the sum of upstream freshwater input to the lake (Fig. 1). Such an estimation is realistic because the dam is not operating as a flood control structure anymore and was overflowed by extreme river discharge as a result of Hurricane Harvey (Sebastian et al., 2021; Valle-Levinson et al., 2020). Regarding the downstream BCs, we force the G-Bay model with storm tides using WL records from two tide-gauge stations, namely, Freeport Harbor (NOAA 8772471) and Galveston Bay Entrance (NOAA 8771341). These stations are located offshore and are a good proxy for coastal WL propagating from the open-ocean boundary. To account for WL variability arising from atmospheric variables, we include reanalysis data of 10 m wind velocity and atmospheric (sea level) pressure in the model simulations. Note that the latest versions of 2D HEC-RAS allow the user to simulate wind speeds even though the atmospheric pressure component is yet to be implemented in Eqs. (2) and (3).

Lastly, we retrieve rainfall data from a relative dense rain-gauge network of the Harris County Flood Warning System. These data have been validated and further used to estimate flood damages in G-Bay (Sebastian et al., 2021). In addition, we complement these data with “total precipitation” reanalysis datasets from ERA5 in order to estimate rainfall patterns in coastal areas beyond Harris County and over the Gulf of Mexico (Fig. 1). Specifically, we use the “inverse distance weight” as the interpolation method in ArcGIS with an output cell size of 1 km (e.g., shortest Euclidean distance between existing rain gauges), a search radius of 5 points, and a power function of 2. The interpolation method as well as the aforementioned values follow those suggested in other studies and are validated through sensitivity analysis (Sebastian et al., 2021; Ahrens, 2006; Otieno et al., 2014).

G-Bay is influenced by multiple flood drivers, including local rainfall, river discharge, and storm tides. At low river flow rates, tides propagate from the ocean boundary in the landward direction where they attenuate and eventually vanish due to bottom friction (Hoitink and Jay, 2016; Bolla Pittaluga et al., 2015). Therefore, we ensure that the model setup (Sect. 2.3.1) is adequate to simulate tidal propagation across the model domain. Specifically, we simulate tidal dynamics in Delft3D-FM by setting barotropic tides from the TPXO 8.0 global inverse tide model as forcing data at the open-ocean boundary (Egbert and Erofeeva, 2002). We then run 100 ensemble model realizations for a 1-year simulation window without incorporating any additional forcing (e.g., only tides) and considering a range of plausible Manning's roughness values (n) derived from pertinent literature and technical reports (Liu et al., 2018; Arcement and Schneider, 1989). A combination of plausible n values are generated with the Latin hypercube sampling (LHS) technique (Helton and Davis, 2003) and evaluated via ensemble model simulations in a high-performance computing (HPC) system (Table 1). We next identify an optimal (calibrated) value for the “open water” category that achieves the lowest root-mean-square error (RMSE) and mean absolute error (MAE) as well as the highest Kling–Gupta efficiency (KGE) and Nash–Sutcliffe efficiency (NSE). These metrics and underlying equations have been presented in another study on G-Bay (Muñoz et al., 2022a).

Furthermore, we conduct harmonic analyses and retrieve tidal amplitudes and phases for the main tidal constituents in G-Bay including K2, S2, M2, N2, K1, S1, P1, O1, and Q1 (NOAA, 2000). We then compare observed and simulated tidal characteristics at each tide-gauge station using a 1:1 line assessment as well as concentric circles. In general, the evaluation metrics suggest that the model setup is adequate to simulate tidal propagation in G-Bay. Also, the optimal n value for open water helps minimize both the RMSE and MAE and achieve high NSE and KGE scores (Fig. 2a). The 1:1 line shows a satisfactory agreement between observed and simulated tidal amplitudes, with a maximum MAE of 1.53 cm (Fig. 2b). Likewise, observed and simulated tidal phases are in good agreement for the majority of tidal constituents and stations in G-Bay. The closer the colored (simulated) and gray (observed) markers in each concentric circle, the more accurate the simulated phase. Note that we only evaluate tidal propagation using the Delft3D-FM model because this software is installed in our HPC system, whereas 2D HEC-RAS is run on a desktop computer. Moreover, the former model can run in parallel, whereas recent 2D HEC-RAS versions in Linux run in series within the HPC system. Thus, the latter would require significantly more computational resources and time to accomplish 100 ensemble model realizations for a 1-year simulation window. Nevertheless, we expect similar score metrics regarding tidal propagation for a 1-year simulation window, as the model setup of 2D HEC-RAS is similar to that of Delft3D-FM (i.e., mesh extent, refinement regions, and cell size grid. Also, we support this claim based on calibration and validation results of CF events as explained later on, i.e., Hurricane Ike and Harvey.

Next, we calibrate n values of the navigation channel, riverine water, coastal wetlands, and urban areas for Hurricane Ike and Hurricane Harvey (Table 1). We follow an identical process to identify optimal n values for both the Delft3D-FM and 2D HEC-RAS models, i.e., LHS technique with 100 ensemble members, but this time we keep the calibrated n value for the “open water” category invariant to ensure accurate tidal propagation. It is worth noting that both models are independently calibrated in order to ensure a reliable assessment of model parameters and model structure errors, especially in riverine areas in Houston. The roughness coefficient of open water is common for both models, as it simulates tidal propagation from the ocean boundary. Also, we set a 1-month warm-up period for the model to reach equilibrium and consider a 1-week simulation window (centered on the peak WL) to assess the accuracy of model simulations (Fig. 3 and Figs. S1 and S2 in the Supplement). In general, both hydrodynamic models perform satisfactorily, especially at downstream tide-gauge stations where the RMSE is below 0.30 cm. However, the RMSE progressively increases at upstream tide-gauge stations due to a relatively large riverine influence and underlying nonlinear interactions with storm tides (Fig. 3d, h). The latter is more evident in model simulations of Hurricane Harvey, as the peak river discharge at the Buffalo Bayou station (USGS 08073700), located upstream of the Manchester tide-gauge station (NOAA 8770777), is 3.5 times higher than that of Hurricane Ike. Also, inaccuracies in the topobathy data along the Buffalo Bayou river and its tributaries might have reduced the model's performance in the upstream part of G-Bay, as reported in flood hazard and damage studies (Wing et al., 2019; Jafarzadegan et al., 2021a; Valle-Levinson et al., 2020).

To validate the hydrodynamic models, we generate composite maps representing maximum WLs within the simulation period of Hurricane Ike and Hurricane Harvey and compare those maps against USGS's high-water marks collected in the aftermath of both hurricanes (Fig. 4a). We evaluate the accuracy of the composite maps by comparing observed and simulated maximum WLs (Fig. 4b). Data points that fall along the 1:1 (diagonal) line represent a perfect match between those maximum WLs. The validation process indicates that Delft3D-FM and 2D HEC-RAS produce maximum WLs that are in close agreement with available high-water marks in G-Bay; moreover, both the RMSE and MAE agree well with the corresponding metrics reported in similar CF studies (Huang et al., 2021; Lee et al., 2024; Sebastian et al., 2021; Wing et al., 2019). In addition, we validate peak WLs at selected tide-gauge stations (Figs. 3, S1, S2) with respect to CF hazard maps generated in Delft3D-FM for Hurricane Ike and Hurricane Harvey (Fig. 4c and d, respectively). Note that we mask out values below 0.10 m in order to enhance visualization of water depths above ground. As expected, the peak WL of Hurricane Ike (∼3 m) triggers a relatively large flood extent and water depth in the lower part of G-Bay, whereas the peak WL of Hurricane Harvey (∼1.25 m) produces moderate coastal flooding over the same area. However, flood extent and water depth in Harris County are relatively large compared with those triggered by Hurricane Ike due to the compounding effects of heavy rainfall, extreme river discharge, and storm tides. Such compounding effects are evident at Manchester tide-gauge station where freshwater input controls WL dynamics starting on 27 August (Fig. 3h). Also, note that Delft3D-FM constantly outperforms 2D HEC-RAS based on the evaluation metrics and 1:1 line assessment (Figs. 3, 4). Following this, we hereinafter consider Delft3D-FM as the best hydrodynamic model to analyze cascading uncertainty in G-Bay with respect to 2D HEC-RAS.

We propose five scenarios to analyze the effects of isolated and total uncertainty on CF hazard assessment for Hurricane Harvey (Table 2). The first scenario focuses on the initial condition of the system, including topographic and bathymetric data in coastal DEMs. Recently, Holmquist and Windham-Myers (2022) produced a DEM of relative tidal marsh elevation for the conterminous US using land cover classes derived from the 2010 Coastal Change Analysis Program (C-CAP). As this DEM accounts for elevation errors within coastal wetlands, we conveniently evaluate uncertainty from the initial condition of the system using the aforementioned DEM and NOAA's CUDEM in hydrodynamic simulations (see Sect. 2.2). The second scenario represents uncertainty derived from forcing conditions that are often neglected in real-time hurricane-induced flood forecasts and advisories (CERA, 2023). Specifically, such flood forecasts assume that riverine flow and local rainfall contributions to flooding are relatively small compared with those driven by storm surge. Following this reasoning, we will analyze the impact of local rainfall and riverine flow in CF dynamics by turning off those two forcing conditions in hydrodynamic simulations.

The third scenario analyzes uncertainty stemming from model parameters, including bed channel and floodplain roughness coefficients associated with land cover classes from the NLCD (Fig. 1a, Table 1). Here, we simulate compound coastal flooding triggered by Hurricane Harvey using two sets of optimal (calibrated) n values. The first set consists of n values calibrated for Hurricane Ike (September 2008), whereas the second set comes from the actual model calibration of Hurricane Harvey (August 2017). The reasoning is that a realistic estimation (or proxy) of calibrated n values for future flood scenarios consists of those already calibrated for past flood events (Domeneghetti et al., 2013; Meresa et al., 2021). In this regard, Hurricane Ike is the closest and most relevant CF event that impacted G-Bay prior to Hurricane Harvey. The fourth scenario represents uncertainty derived from model structure, including the model setup and configuration necessary to simulate flood extent and inundation depth. Therefore, we analyze this source of uncertainty by comparing model outputs of Delft3D-FM and 2D HEC-RAS (see Sect. 2.3). Lastly, the fifth scenario is named “total uncertainty” and involves all sources of uncertainty including their cascading effect on CF hazard assessment.

WL residuals obtained from physically based model simulations are used as input features for multiple linear regression (MLR) and nonlinear models (Fig. 5). First, we fit the input features using an MLR model and considered it as a benchmark model to further evaluate the benefits of nonlinear models including physics-informed machine learning (PB–ML). The goal of the MLR model is to estimate total uncertainty as the target variable in terms of WL residuals. As we are dealing with WL residuals in meters that have a comparable order of magnitude, we do not scale or normalize the input features prior to the fitting process. This is also convenient for the evaluation purposes of the fitted model and the relative importance quantification of each source of uncertainty based on fitted regression weights. We do, however, identify and remove outliers in the input features, especially those arising at the edges of the mesh and those around the BCs. We use the “statsmodels API” package in Python to conduct a robust fitting of input features (https://www.statsmodels.org/stable/api.html, last access: 6 June 2024) and report regression coefficients with the underlying statistical significance and confidence intervals (see Sect. 3.2).

We conducted a preliminary analysis to identify the best nonlinear ML model that predicts total uncertainty (not shown here for brevity). Among those models, we notice that a random forest regressor model outperforms support vector machine and artificial neural networks (e.g., multiple-layer perceptron model), and the latter agrees with results of multiple flood studies (Chen et al., 2020; Mosavi et al., 2018; Schoppa et al., 2020). Random forest (RF) is a nonparametric ensemble algorithm that builds multiple decision trees based on random bootstrapped samples through replacement (Breiman, 2001). The advantage of RF over other nonlinear regression models lies in its simplicity and easy implementation for efficient regression and classification tasks. Also, RF connects input and target features with complex and nonlinear associations and provides estimates of feature importance to predict the target variable (Alipour et al., 2020; Wang et al., 2015). We develop an RF regressor model and conduct a thorough model evaluation in Python using the “scikit-learn” package. WL residuals from each scenario are set as input (S1–S4) and target (S5) features, and our analysis is focused on both time-evolving and maximum residuals across the model domain. We split input features into training (80 %) and validation (20 %) datasets using the total number of data points after outlier removal. In the context of hydrodynamic modeling, outliers are unrealistic WLs emerging around upstream and downstream BC lines as well as the edges of the model domain. Such values are extreme values, either positive or negative, that do not reflect WL dynamics within the model domain. Therefore, we masked out such values using a buffer polygon in ArcGIS and proceeded with the training and validation dataset using realistic WLs (e.g., 1 093 501 data points). Those data points represent the number of grid nodes generated in Delft3D-FM (see Sect. 2.3.1). Next, we conduct hyperparameter tuning to build decision trees and estimate optimal (calibrated) values for each parameter using an HPC system (Table 3).

We use the scikit-learn package to find optimal values and also account for overfitting issues through a cross-validation (CV) process. For this, we generate an initial grid of parameter values within a specified range using the LHS technique (Helton and Davis, 2003). Then, we use a “k-fold” approach to conduct CV using the training dataset. The number of k-folds is set to five, and the parameter grid is generated with a fixed seed value to ensure reproducibility. We then use the grid in the “RandomizedSearchCV” function to randomly sample a set of hyperparameter values and conduct a k-fold CV for each combination of values. The number of combinations to be tested is defined by the “n_iter” parameter that is set to 100 here. We repeat the splitting and sampling process for time-evolving WL residuals and conduct hyperparameter tuning for the resulting 40 datasets using the HPC system. Nevertheless, we notice that the optimal values associated with each dataset and those reported in Table 3 lead to similar score metrics in the validation dataset (e.g., R2, R, KGE, and RMSE). Also, the main differences in the score metrics are observed in the third or fourth decimal place (see Sect. 3.2).

We fit and train MLR and ML models using maximum WL residuals as an indicator of CF hazard in G-Bay (Table 4). Regression weights are statistically significant for all four scenarios (p value < 0.05), including S3 in spite of the negative correlation. Also, the Pearson and Kendall correlation coefficients reveal a statistically significant linear correlation between total uncertainty and isolated uncertainty (Fig. 7). Note that the rank of such correlations, in descending order, agrees well with the maximum WL residuals and underlying patterns resulting from the model scenarios (Fig. 6). Among them, uncertainty stemming from model parameters has a negative linear correlation with total uncertainty that partially explains the discrepancy of residual patterns already mentioned in the previous section. Particularly, a negative correlation suggests that an increase in WL residuals from scenario S3 results in a decrease in those from scenario S5. Note that the MLR model estimates total uncertainty with a relatively high accuracy, as corroborated by the score metrics, i.e., R2=0.76, R=0.87, RMSE = 0.42 m, and KGE = 0.82 (Fig. 7c).

Confidence intervals are obtained from the statsmodels API package available in Python.

Overall, the absolute magnitude of regression weights agrees well with the rank resulting from either Pearson's or Kendall's correlation coefficients. This suggests that uncertainty stemming from the forcing conditions and model structure are crucial for estimating total uncertainty in CF hazard assessment (Fig. 7g). Other flood studies have shown similar results and demonstrated that uncertainty stemming from forcing conditions is even more important than that of the remaining sources, especially for flood prediction and inundation mapping in riverine systems (Alipour et al., 2020; Jafarzadegan et al., 2023; Pappenberger et al., 2008; Savage et al., 2016). The initial condition of the system and model parameters are also relevant sources of uncertainty, but they show a relatively low regression weight. Note that if WL residuals of scenarios S1, S3, and S4 are kept invariant, any perturbations of scenario S2 will result in a nearly identical response of scenario S5 as well as a negative offset of 12 cm. Although MLR models help analyze the influence of each input feature to estimating total uncertainty, they do not capture hidden associations and/or interactions among the input features. This, in turn, reduces the effectiveness of MLR models to characterize cascading effects on total uncertainty.

To overcome this limitation, we determine whether or not the proposed PB–ML model (e.g., RF regressor) outperforms the MLR model based on the aforementioned evaluation metrics. In this regard, the score metrics evince a substantial improvement by the PB–ML model: the RMSE decreases to 0.28 m, whereas R2, R, and the KGE increase up to 0.90, 0.95, and 0.92, respectively (Fig. 7f). Next, we conduct a rank analysis focused on the contribution of each input feature to the model's variance reduction and overall performance for total uncertainty estimation (Fig. 7h and i, respectively). In the proposed RF regressor model, feature importance measures the mean decrease in variance within each selected decision tree, whereas permutation importance measures the decrease in a predefined score metric when individual input features are randomly shuffled (Breiman, 2001). Specifically, the four input features are shuffled multiple times and the RF regressor model is refitted to estimate their importance in the model's performance. Here, we shuffle the input features 100 times and set the RMSE as the evaluation metric to be consistent with the ensemble members and score metrics used in the calibration and validation process (see Sect. 2.3.2). Notably, the rank of feature importance agrees well with that of the regression weights, indicating that the forcing conditions are not only the main contributor to total uncertainty in CF hazard assessment (56 %) but also a key feature for variance reduction (Fig. 7h). It is noted that the model structure, the input conditions, and the model parameters contribute 20 %, 18 %, and 6 % to the variance reduction, respectively.

The advantage of permutation over feature importance analysis is that the former method circumvents any overfitting issues by focusing the analysis on validation data (∼219000 data points). Also, permutation importance is not biased towards input features with a large number of unique values, compared with feature importance, i.e., high cardinality. Nevertheless, note that we address potential issues of overfitting and high cardinality through a 5-fold CV process in the training process (see Sect. 2.5.2). Similarly, the rank of permutation importance agrees well with both the ranks of regression weights (or slope terms) and those derived from feature importance (Fig. 7i). Nevertheless, note that the permutation importance of forcing conditions (61 %) is higher than that of feature importance; this can be partially explained by the number of data points used in the computation of the corresponding importance as well as the objective function, i.e., training vs. test datasets and mean variance decrease vs. RMSE. Note that model structure, input, and parameters contribute 24 %, 9 %, and 6 % to the overall model performance, respectively. Following these results, we recommend that any efforts to improve CF modeling and hazard assessment should focus on accounting for all relevant forcing (boundary) conditions and implementing a suitable hydrodynamic model to simulate complex compound coastal flood dynamics. For example, recent studies have shown that estimating water deficits between upstream and downstream flows and distributing such deficits using lateral flows and vertical fluxes (as additional forcing conditions) improve the performance of flood modeling during extreme events (Jafarzadegan et al., 2021b; Oruc Baci et al., 2024).

We track the trajectory of the four sources of uncertainty during the onset, peak, and dissipation of Hurricane Harvey using a 6 h interval for the entire simulation period (Fig. 8a). For this, we fit MLR and train PB–ML models for each of the 40 datasets containing WL residuals and report relative and cumulative contributions as well as models' performance in terms of the R2, R, KGE, and RMSE. The model structure and initial condition of the system are the main sources of total uncertainty at the onset of Hurricane Harvey with a contribution of 60 % and 20 %, respectively (Fig. 8b). However, their relative contribution drops around the peak WL because forcing conditions become a more relevant source of uncertainty until the dissipation of WLs, i.e., drastic increase from 10 % to 50 %. The contribution of model parameters is estimated at 10 % and remains almost invariant during the onset, peak, and dissipation of Hurricane Harvey. Furthermore, the cumulative contribution of each source of uncertainty helps visualize their overall importance for total uncertainty estimation in the simulation period (Fig. 8d). In contrast to the rank of contributions established for maximum WL residuals (see Sect. 3.2.1), note that the model scenario and/or input feature derived from model structure is the main contributor to variance reduction in the RF regressor model (49 %) followed by that of the forcing conditions (23 %), initial condition (20 %), and model parameters (8 %).

These results are somehow similar to other studies that analyzed how the influence of forcing (boundary) conditions and model parameters change during flood events (Alipour et al., 2022; Jafarzadegan et al., 2021b; Savage et al., 2016). Although those studies identify forcing conditions as the most influential factor for flood inundation mapping, uncertainty stemming from model structure is not explicitly analyzed; however, it is recognized as a determinant factor of the results. Lastly, the evaluation metrics computed for the 40 datasets indicate that RF regressor models (dashed line) outperform the benchmark MLR models (solid line) in the simulation period (Fig. 8c). Note that the R2, R, and KGE display a sudden drop around the peak WL, suggesting that MLR models cannot fully characterize total uncertainty, whereas the same evaluation metrics of RF regressor models remain almost invariant due to the ability of nonlinear models to capture any complex interactions as well as cascading effects arising from the four sources of uncertainty. Likewise, note that the RMSE drops to ∼11 cm when characterizing cascading uncertainty with RF regressor models. Also, there is a rather constant RMSE of ∼25 cm when estimating total uncertainty in terms of WL residuals in the simulation period.