Unsupervised Estimation of Urban Floodwater Depth Using Post-Event Aerial Imagery and Digital Terrain Models

1. Introduction

Globally, the compounding effects of climate change, rapid urbanization, and land-use dynamics are increasing the frequency and severity of urban flooding [1]. In urban environments, inadequate drainage infrastructure and expanding impervious surfaces limit rainfall infiltration, accelerating surface runoff and leading to destructive flash floods. Consequently, assessing flood risk and exposure is critical. Recent projections estimate that by 2100, nearly half of all built assets will be at risk of severe flooding [2].

While flood extent mapping using global aerial and satellite imagery data is a well established practice [3,4,5,6], with a broad set of datasets for semantic segmentation available [7,8,9,10], the estimation of floodwater depth in urban areas remains a significant challenge. Depth is a vital metric for damage assessment, emergency response, and risk analysis. Traditionally, hydrodynamic models, such as LISFLOOD [11,12] or MIKE FLOOD [13,14,15], have been used to simulate flood depth, but these require extensive computational resources and rely heavily on high quality, dense ground-based gauge data for calibration. In urban settings characterized by fragmented waterlogging and complex infrastructure, such continuous monitoring is often cost prohibitive and practically infeasible.

To overcome the data limitations of traditional hydrodynamic modeling, recent research has pivoted toward remote sensing and machine learning frameworks [16]. For instance, recent studies have leveraged high resolution aerial imagery and Digital Terrain Models (DTMs) alongside supervised deep learning (DL) architectures to predict flood depth by relying on the hydrostatic equilibrium principle [17,18,19]. However, a major bottleneck of these supervised methods, including state of the art Convolutional Neural Networks (CNNs), is their reliance on meticulously labeled, manually corrected ground truth masks for training. In rapid disaster response scenarios, generating high quality training data is often impossible, which limits the scalability and generalization of supervised models to unseen, data-scarce catchments. Furthermore, model generalization remains intrinsically limited because floodwater depth is highly subjective to the specific topological characteristics of each geographic region, preventing models from easily transferring learned spatial priors across different landscapes.

To address these gaps, this paper introduces a fully unsupervised framework for estimating urban floodwater depth. Utilizing high resolution post flood aerial imagery and LiDAR-derived DTMs from 12 urban and peri-urban sites in the Carolinas impacted by Hurricanes Matthew and Florence, we bypass the need for labeled training data. First, the flood extent map is automatically delineated using an unsupervised approach based on UFS-HT-REM [6]. Subsequently, flood depth is estimated by integrating these derived extents with the underlying DTM leveraging sample points on the extent border and the principle of hydrostatic equilibrium. By removing the dependency on annotated data, this approach offers a scalable, rapid response alternative for flood depth estimation in heterogeneous urban landscapes.

The main contributions of our work can be summarized as follows:

• To the best of our knowledge, this work presents the first fully unsupervised, calibration-free framework for flood depth estimation in urban/peri-urban environments, relying exclusively on post-event RGB imagery and DTMs.
• We introduce a training free framework that tightly couples unsupervised horizontal flood extent delineation with hydrostatic equilibrium principles on LiDAR-derived DTMs, completely bypassing the bottleneck of meticulously labeled ground truth masks.
• We address the spatial heterogeneity of flood dynamics by leveraging localized sample points along the automatically extracted boundaries, adapting the depth estimations dynamically to the unique topographic characteristics of each specific catchment landscape.
• We provide an extensive validation of our framework across a number of highly heterogeneous urban and peri-urban sites yielding spatially coherent and physically plausible depth estimates.
• We present a highly scalable and computationally efficient alternative to data-heavy supervised approaches.

These contributions highlight the novelty and effectiveness of our method, particularly its suitability for rapid and efficient deployment in disaster response scenarios.

The remainder of this manuscript is structured as follows. Section 2 provides a review of pertinent literature and existing methodologies. Section 3 introduces the primary dataset utilized in this study, namely the post event remote sensing imagery and DTMs. Section 4 delineates the proposed unsupervised framework. Implementation details and evaluation metrics are presented in Section 5. Section 6 details the experimental evaluation and discusses the implications of the findings. Finally, Section 7 concludes the study and identifies potential avenues for future research.

2. Related Work

The increasing frequency and severity of global flood events have made accurate and timely floodwater depth estimation critical for disaster response and mitigation. Traditionally, depth estimation relied heavily on costly field measurements or computationally intensive hydrodynamic models. However, recent years have witnessed a significant paradigm shift toward data driven machine learning approaches and Earth Observation (EO) technologies [20]. Machine learning models have proven highly effective at handling large scale, complex datasets, thereby significantly improving the computational efficiency and accuracy of flood depth predictions. Concurrently, the integration of 2D satellite imagery, such as Synthetic Aperture Radar (SAR) and optical data, with 3D geospatial datasets like Digital Terrain Models (DTMs), has empowered operational emergency services to move beyond simple surface inundation mapping to rapid volumetric depth estimation. Together, these advancements provide the foundational context for the evolution of modern flood modeling techniques [16].

The growing availability of remote sensing and crowd-sourced imagery has spurred the development of data-driven methods for flood depth estimation. Several studies have utilized deep learning-based object detection to infer depth from partially submerged reference objects in urban imagery. For example, Mask R-CNN has been applied using traffic sign poles as scale references [21,22], while other frameworks have leveraged YOLOv4 in a similar context to infer street level flood depth from submerged traffic signs [23].

Furthermore, especially architectures like YOLOv4 and YOLOv8 have been utilized to estimate depth based on the standardized dimensions of submerged vehicles [24,25,26]. Other researchers have employed attention enhanced CNNs, such as CA-ResNet and ResNet50, to extract depth levels from social media and Internet of Things (IoT) imagery [27,28] using vehicles as reference. Building upon these individual approaches, hybrid frameworks have also emerged that integrate both architectures. For instance, leveraging YOLO to first detect and localize the submerged vehicles, followed by a ResNet50 module to precisely estimate the flood depth from the localized bounding boxes [29]. Despite their novelty, this entire class of object-reference methods remains fundamentally constrained by the physical availability and standardized dimensions of the target features (e.g., traffic signs or vehicles), heavily limiting their spatial scalability across uncalibrated, diverse geographic regions.

In addition to vehicles and traffic infrastructure, researchers have expanded the scope of reference objects in deep computer vision models to include humans and buildings. For instance, object detection algorithms have been applied to estimate flood depths by analyzing the varying submersion levels of human bodies captured in web and social media imagery with Mask R-CNN and YOLO based models [30,31]. Similarly, buildings and their associated architectural features, obtained using machine learning methods, have been leveraged as reference structures to automatically calculate actual flood depth from pre and post flood images [32]. While these approaches capitalize on the high availability and real time nature of online visual data, their accuracy can be significantly influenced by variable factors such as image quality, lighting conditions, and shooting angles.

Recent advancements at the intersection of computer vision and natural language processing have driven a paradigm shift from task-specific architectures toward Large Multimodal Models (LMMs) and Vision Language Models (VLMs) capable of zero-shot, continuous flood depth estimation from ground level imagery. Unlike conventional deep learning pipelines that require training distinct models for isolated reference features, foundation visual linguistic systems can jointly interpret textual metadata prompts and visual cues to compute depth measurements [33,34,35]. FloodDepth-GPT [33] leverages proprietary engines like GPT-4 Vision alongside structured object height prompts, encompassing pedestrians, vehicles, and traffic signs, to calculate localized waterlines without explicit model training. In [35] the usability of prominent LMMs, such as GPT-4 and Gemini, for estimating urban floodwater depth from ground level imagery have been systematically evaluated, examining performance degradation based on prompt syntax. Benchmarked against a supervised YOLOv5 model on static and time sequenced surveillance datasets, this framework demonstrated that GPT-4 successfully captures flood peak dynamics ($\rho =0.88$) without prior training.

Despite their adaptability and strong synthetic to real transfer capabilities, VLM powered depth estimators are fundamentally constrained by severe structural and spatial limitations. Zero-shot prompting of closed source models yields significant performance variability and inconsistent outputs under complex environmental circumstances. The accuracy of these models remains highly dependent on the alignment between prompt complexity and visual scene data.

To achieve spatially continuous depth mapping, researchers have increasingly integrated remote sensing imagery with topographic data. Supervised deep learning models, such as U-Net or FCN-8, have been trained on hyetographs and topographic variables to predict two dimensional flood depth maps [36,37], though this approach necessitates large volumes of accurately annotated data. Similarly, conditional Generative Adversarial Networks (cGANs) incorporating rainfall, land use, and topography have been shown to generate predictions significantly faster than traditional HEC-RAS hydrodynamic models [38], yet they frequently encounter spatial transferability challenges when applied to unseen domains. Alongside these machine learning advancements, direct integration of multispectral UAV data has also proven effective for near real time response, allowing researchers to compute water depth directly by subtracting terrain elevation from the water surface elevation [39], but its broader application is ultimately limited by UAV operational and deployment constraints. Operating on a broader satellite scale, Geographic Information System (GIS) approaches, like the Floodwater Depth Estimation Tool (FwDET v2.0), bypass hydrodynamic simulations entirely by extracting elevation values at the boundaries of 2D inundation polygons and allocating them to interior pixels [40]. While computationally efficient, this deterministic method relies heavily on externally provided inundation masks and is highly sensitive to the spatial resolution of the input Digital Elevation Model (DEM).

Recently, in [18] a terrain-informed deep learning framework utilizing the same dataset analyzed in this study was proposed. Their framework is grounded in the hydrostatic equilibrium principle, which assumes that stationary floodwater forms a relatively smooth surface dictated by underlying terrain elevation. Within this framework, the authors evaluated various ResNet and Swin U-Net architectures. Their enhanced models incorporated terrain descriptors, such as Topographic Wetness Index (TWI), slope, and curvature, alongside binary flood extents and DTMs, to estimate depth. While their ResNet18 model achieved high predictive accuracy, it experienced a marked decline in performance when applied to unseen catchments due to regional overfitting and limited validation data. Furthermore, their methodology required substantial manual intervention to correct omission and commission errors in the segmented "open water" training masks.

A comparative overview of the flood depth estimation methodologies discussed in this section, detailing their underlying models and depth indicators, is presented in Table 1. While supervised models demonstrate high accuracy within their training domains, their reliance on large volumes of accurately annotated data severely restricts their rapid deployment during active flood events. Unsupervised methods provide a critical pathway to bypass this limitation. By automatically deriving flood extents without prior labeling, such as the unsupervised techniques explored in this study, and physically constraining depth estimates using available DTMs, it is possible to achieve scalable, near real time flood depth mapping. This unsupervised paradigm addresses the spatial transferability challenges observed in existing supervised frameworks, providing a highly pragmatic tool for emergency response in data-scarce environments.

3. Materials

The Inundation2Depth dataset [19] utilized in this study comprises 12 flood affected urban and peri-urban locations across the Southeastern United States, specifically within North Carolina and South Carolina. These study areas were severely impacted by two major storm events: Hurricane Matthew in 2016 (which affected nine of the selected sites) and Hurricane Florence in 2018 (which affected the remaining three sites). Collectively, these regions cover approximately 25,059 acres of developed land, providing a diverse array of topographic and hydrological conditions for evaluating flood mapping methodologies. Each of these spatial domains is supported by high resolution post event remote sensing imagery coupled with empirical ground truth flood depth measurements [18].

Table 2 summarizes the topographical profiles, flood inundation metrics, and imagery dimensions for all twelve study sites. Given that the underlying DTMs possess a spatial resolution of 1 m [19], the absolute flood area for each site is directly derived from the total pixel count of the corresponding ground truth flood masks. The analysis of the descriptive statistics across the twelve study sites in Table 2 underscores the topographic and hydrological heterogeneity within the dataset. The maximum recorded flood depth exhibits a wide variation, ranging from 0.96 m in Nichols to 8.74 m in Kinston1, while the mean depth spans from 0.31 m to 1.71 m across the locations. Crucially, severe inundation events occur across distinctly contrasting terrain profiles. For instance, Greenville2 is characterized by a very low mean elevation of 2.70 m yet experiences a substantial maximum water depth of 6.12 m. Conversely, Lumberton represents a high-altitude catchment with a mean elevation of 36.54 m, yet still suffers from a critical maximum flood depth of 6.96 m. In terms of spatial scale, Lumberton presents the largest absolute flooded surface, exceeding 8.7 million m², despite having a relatively low flood cover percentage of 20.74% on the given DTM. In contrast, topographically smaller catchments like Wallace and Chinquaqin exhibit the highest relative inundation density, with 65.57% and 62.01% of their areas submerged, respectively. This broad spectrum of elevational gradients, spatial scales (ranging from 287 × 954 pixels up to 5848 × 7244 pixels), and flood severity metrics establishes a rigorous benchmark for evaluating the generalization and spatial transferability of the proposed unsupervised framework.

The sole inputs for our unsupervised framework, as illustrated in Figure 1, consist exclusively of post event aerial imagery and Digital Terrain Models (DTMs):

• Post Flood Aerial Imagery: High resolution optical imagery (approximately 0.25 m spatial resolution) captured shortly after the storm events. Imagery for Hurricane Matthew was acquired between October 10 and 15, 2016, and for Hurricane Florence on September 18, 2018, originally sourced from the NOAA Storms Archive.
• Digital Terrain Models: High resolution elevation data (approximately 1 m spatial resolution) derived from LiDAR point clouds. These were originally sourced from the North Carolina Emergency Management Spatial Data Portal and the USGS 3D Elevation Program (3DEP).

To demonstrate the rapid response capabilities of the proposed unsupervised approach, the data was ingested and processed using the raw, whole remote sensing images. Unlike previous supervised methodologies [2] applied to this dataset, the images were not subdivided or cropped into standardized tiles (e.g., $256\times 256$ pixel patches). To ensure spatial alignment between the RGB visual data and the topographic data, all aerial images were resized to match the exact spatial dimensions of their corresponding DTMs.

Furthermore, the depth estimation relied solely on the extracted flood extent and the raw DTM. No additional hydrologically relevant terrain derivatives, such as slope, surface curvature, or Topographic Wetness Index (TWI), were computed or utilized in this study. By eliminating the requirement for supervised training data, the proposed framework was directly evaluated across the entirety of all 12 urban and peri-urban study sites.

4. Methodology

The proposed framework operates in two distinct phases to achieve scalable, near real time floodwater depth estimation. In the first phase, a binary flood extent map is automatically delineated from high resolution aerial imagery using a modified unsupervised image segmentation algorithm. In the second phase, the extracted two dimensional flood footprints are integrated with the underlying Digital Terrain Model (DTM) to calculate the volumetric flood depth based on the principle of hydrostatic equilibrium. A detailed schematic overview of the proposed framework and its sequential processing stages is presented in Figure 1.

4.1. Unsupervised Flood Extent Delineation

To automatically extract the flood extent without the need for labeled training data or pre disaster imagery, we adopt the foundational architecture of the Unsupervised Flood Segmentation (UFS-HT-REM) framework proposed in [6]. The core principle of this algorithm is to iteratively exclude image regions that confidently belong to the non flood background using a series of parameter free binary masks. While we maintain the overall probability optimization and region growing pipeline of the original framework, we introduce critical modifications to the initialization masks. These adaptations are essential to account for the distinct spatial resolutions, spectral signatures, and atmospheric variables inherent to satellite remote sensing imagery, transitioning the methodology from its original focus on low altitude UAV captures to broad scale satellite observation.

• The extraction process consists of the following sequential steps:

• Greenery Exclusion via RGB Vegetation Index

In both urban and rural landscapes, dense vegetation and tree canopies are highly unlikely to be completely submerged. To prevent vegetation from being misclassified as floodwater, we compute the RGB Vegetation Index (RGBVI) [41], which is defined as the normalized difference between the squared green reflectance and the product of blue and red reflectance.

To differentiate vegetation from floodwater across diverse urban and rural topologies, an RGBVI threshold was empirically selected in the context of [6]. The threshold value of $0.0$ accounts for the specific lighting conditions and atmospheric noise present in post hurricane RGB satellite imagery. Pixels with $RGBVI>0.0$ are classified as definitive background, yielding the first exclusion mask $M_{RGBVI}$.

• LAB Color Space Masking

The imagery is converted into the perceptually uniform CIELAB color space [42], which decouples luminance (L) from chromaticity (A and B). This transformation provides a wide color gamut and enhances robustness against variable illumination conditions [43]. In low altitude acquisitions, such as UAV imagery, floodwater typically exhibits elevated values across all three LAB components relative to the surrounding terrestrial background [6]. Conversely, satellite imagery reveals a distinct spectral behavior: while the A component (red-green axis) remains elevated, both the luminance (L) and the B component (yellow-blue axis) exhibit depressed values. This spectral shift can be attributed to the strong absorption of visible light by water bodies, yielding a lower overall top of atmosphere luminance compared to highly reflective urban surfaces, coupled with atmospheric path radiance (Rayleigh scattering) that introduces a blue spectral bias (lower B), and the high concentration of optical impurities typical of turbid floodwaters that selectively increases reflectance in the red spectrum (higher A).

To systematically isolate these regions, binary exclusion masks ($M_L$, $M_A$, $M_B$) are computed for each LAB component according to adapted thresholds. Whereas the original framework [6] established classification thresholds by uniformly subtracting the standard deviation from the mean for each LAB component, the aforementioned distinct spectral signatures of satellite derived floodwaters necessitate a adapted approach for components L and B. Consequently, we introduce modified mathematical formulations to define the classification thresholds:

$$\begin{array}{ccc}\hfill {\Delta }_L& =& {\mu }_L+{\sigma }_L,\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill {\Delta }_A& =& {\mu }_A-{\sigma }_A,\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill {\Delta }_B& =& {\mu }_B+{\sigma }_B\hfill \end{array}$$

(3)

$M_A$ is labeled as non flood, when the component’s pixel value is smaller than ${\Delta }_A$, while $M_L$ and $M_B$ are labeled as non flood when the component’s pixel value is higher than ${\Delta }_L$ and ${\Delta }_B$, respectively. Figure 2 shows the average value of (a) L, (b) A, and (c) B color components computed on flood (blue curve) and background (red curve) pixels for each image of the Inundation2Depth dataset, sorted in ascending order. The yellow curves represent the corresponding thresholds ${\Delta }_L$, ${\Delta }_A$, and ${\Delta }_B$ used to create the three masks.

These tailored calculations significantly improve algorithmic robustness against the highly heterogeneous spectral signatures characteristic of urban floodwaters, ensuring that only confident background areas are excluded. Furthermore, an edge detection mask ($M_{Edge}$) is derived from the luminance component using the Canny edge detection algorithm [44] on the blurred L component resulting in dilated edges, as to explicitly exclude object boundaries and structural contours from the final flood classification.

• Dominant Color Estimation and Probability Mapping

The union of the five aforementioned masks defines the absolute non flood areas. Consistent classification across all individual masks synergistically increases the confidence of non flood pixel exclusion. Conversely, in regions where a single mask exhibits low discriminative power, the complementary nature of the remaining masks provides functional redundancy, thereby mitigating misclassification. Subsequently, a morphological closing operation is applied to enforce spatial continuity, effectively bridging narrow discontinuities and consolidating proximate, disjointed regions into unified segments.

The remaining unmasked pixels are designated as Potential Flood Areas (PFAs). Under the assumption that pixels further away from confirmed background areas have a higher likelihood of being actual floodwater, a Euclidean distance transform [45] is applied to assign spatial weights to the PFAs. These weighted pixels are then used to estimate the dominant spectral signature of the floodwater. Based on this estimated dominant color, a Gaussian probability map is computed, assigning every pixel in the image a confidence score [0, 1] representing its likelihood of being flooded. The complete mathematical formulation governing this spatial weighting and probability estimation process is detailed in [6].

• Final Segmentation via Hysteresis Thresholding

To finalize the binary extent map, hysteresis thresholding is applied to the probability map. Pixels with a probability exceeding a high threshold ($T_H=0.75$) are designated as strong flood seeds. A connectivity based region growing algorithm then expands the flood area from these seeds, absorbing neighboring pixels that exceed a lower threshold ($T_L=0.01$), with both thresholds derived from the ablation study in [6]. This dual threshold approach ensures spatial continuity of the water body while rejecting isolated noise, resulting in a highly accurate, unsupervised flood extent map. Following hysteresis thresholding, the final segmentation undergoes morphological dilation for edge refinement [6].

Ultimately, this unsupervised pipeline transitions from broad background exclusion to precise, probability driven region growing, successfully delineating the flood boundaries within complex urban and rural topographies. A schematic overview illustrating the chronological progression of these unsupervised segmentation steps, accompanied by representative visual outputs for each stage, is presented in Figure 3.

4.2. DTM-Based Hydrostatic Flood Depth Estimation

Following the delineation of the two dimensional flood extent map, the volumetric floodwater depth is derived by integrating the spatial flood footprints with the underlying Digital Terrain Model (DTM). This derivation is governed by the physical principle of hydrostatic equilibrium. In the absence of significant momentum, a condition typical of standing or slowly receding post event floodwaters, water distributes itself to conform to the underlying topography, establishing a relatively smooth, horizontal free surface elevation. Under these steady state conditions, the local flood depth at any given spatial coordinate can be analytically calculated as the vertical differential between the established water surface elevation (WSE) and the localized ground elevation provided by the DTM. The principle is expressed as:

$$H(x,y)=W(x,y)-Z(x,y),$$

(4)

where H, W, and Z are the flood depth, the WSE, and the ground elevation derived from the DTM at coordinates $(x,y)$, respectively, thereby linking the water stage/volume with the flood extent [46].

Because inundation patterns in urban and peri-urban environments are frequently fragmented, yielding multiple spatially disjointed flooded zones, a global WSE cannot be assumed. Therefore, we conceptualize the binary flood extent mask as a set of N discrete, non overlapping connected components, denoted as $\mathcal{C}=\{C_1,C_2,\dots ,C_N\}$, utilizing 8-way pixel connectivity. Since the water surface elevation is assumed to be constant within each hydraulically connected flooded region $C_i$, $i\in \{1,\dots ,N\}$, Equation (4) can be reformulated by replacing the spatially varying water surface elevation $W(x,y)$ with a constant reference level $c_i$, $i\in \{1,\dots ,N\}$, representing the estimated water surface elevation for the corresponding flood component $C_i$ (see Figure 4). Consequently, the local flood depth is expressed as:

$$H(x,y)=max\left(0,c_i-Z(x,y)\right)$$

(5)

The $max(0,·)$ operator enforces the physical constraint that flood depth cannot be negative, ensuring that topographic points located above the estimated water surface are appropriately assigned a zero depth. This generic formulation establishes the mathematical foundation for our localized depth estimators. To operationalize this model, the reference constant $c_i$ is systematically parameterized using specific elevation estimates derived from the discrete flood boundaries, namely the maximum and mean topographic values, as explicitly formalized in Equations (8) and (9).

Algorithmically, this depth estimation protocol is executed independently for each distinct flooded region $C_i$. For any given connected component $C_i$, the physical interaction between the floodwater and the terrain occurs at its spatial boundary. The morphological perimeter of the component $\partial C_i$ contains a set of $|\partial C_i|$ pixels. Under ideal hydrostatic conditions, the local water surface elevation is bounded by the highest topographic point intersecting this boundary. Thus, the local topography at the perimeter dictates the potential maximum WSE for that specific flooded depression. The absolute maximum terrain elevation along the perimeter is defined as:

$$Z_{max}\left(i\right)=\underset{(x,y)\in \partial C_i}{max}Z(x,y).$$

(6)

However, relying exclusively on $Z_{max}\left(i\right)$ renders the estimation highly susceptible to localized DTM anomalies, such as unclassified canopy remnants, minor structural artifacts, or boundary misclassifications. To mitigate these outlier effects and establish a more robust surface estimate, the mean topographical elevation along the entire perimeter is concurrently calculated:

$$Z_{mean}\left(i\right)={\displaystyle \frac{1}{|\partial C_i|}}\sum _{(x,y)\in \partial C_i}Z(x,y)$$

(7)

To estimate the local flood depth for each pixel $(x,y)\in C_i$, we introduce a weighted formulation based on two independent local depth estimates. The first estimate is derived from the maximum boundary elevation, while the second is derived from the mean boundary elevation. These quantities are defined as

$$H_{max}(x,y)=\left\{\begin{array}{cc}max\left(0,Z_{max}\left(i\right)-Z(x,y)\right),\hfill & (x,y)\in C,\hfill \\ 0,\hfill & (x,y)\notin C,\hfill \end{array}\right.$$

(8)

$$H_{\mathrm{mean}}(x,y)=\left\{\begin{array}{cc}max\left(0,Z_{mean}\left(i\right)-Z(x,y)\right),\hfill & (x,y)\in C,\hfill \\ 0,\hfill & (x,y)\notin C.\hfill \end{array}\right.$$

(9)

The flood depth is then computed as a weighted linear combination of the two estimates:

$$\widehat{H}(x,y)=(1-w)·H_{max}(x,y)+w·H_{\mathrm{mean}}(x,y),$$

(10)

where $w\in [0,1]$ serves as a weighting coefficient that regulates the influence of the mean elevation against the absolute physical upper bound. In our experiments, we set $w={\displaystyle \frac{1}{2}}$, such that the two depth components contribute equally to the flood depth estimate $\widehat{H}$. Consequently, $\widehat{H}(x,y)$ corresponds to the arithmetic average of $H_{max}(x,y)$ and $H_{\mathrm{mean}}(x,y)$, ensuring that neither the maximum- nor the mean-boundary-based estimate dominates the solution. This choice reflects a neutral weighting scheme that balances the physically conservative nature of the maximum elevation criterion with the more representative characterization provided by the mean boundary elevation. The application of the $max(·,·)$ operator to each individual depth estimate is mathematically critical to ensure that internal topographic features (e.g., elevated terrain or structural islands enclosed within the flood boundary) correctly yield zero local flood depth. This preserves the physical constraints imposed by the terrain and prevents the generation of non-physical negative depth values.

Finally, to eliminate sharp local discontinuities and yield a physically realistic, smooth final depth profile H, the aggregated depth map $\widehat{H}$ undergoes spatial smoothing via convolution with a $5\times 5$ Gaussian kernel $G_{\sigma }$ with $\sigma =1$:

$$H(x,y)=\widehat{H}(x,y)*G_{\sigma }(x,y),$$

(11)

where * denotes the 2D convolution operator.

Figure 4 provides a conceptual overview of the proposed DTM-based flood depth estimation framework. The cross-sectional view in Figure 4(a) illustrates the generic depth formulation, where the flood depth at any point $p(x,y)$ is defined as the positive difference between the reference water surface elevation $c_i$ and the corresponding terrain elevation $Z(x,y)$. The plan-view representation in Figure 4(b) demonstrates the proposed spatial implementation of this principle, where the floodwater boundary (red) defines the flooded region used to estimate the reference water level, and the blue and gray color gradients depict the resulting flood depth distribution and the underlying terrain elevation, respectively. Together, these schematics illustrate the physical basis of the proposed method and its application to DTM-based flood depth estimation.

Figure 5 visually illustrates the sequential outcomes of this depth estimation framework. Specifically, Figure 5(a) and Figure 5(b) display the independent depth profiles generated using the absolute maximum ($H_{max}$) and mean ($H_{\mathrm{mean}}$) perimeter elevations, respectively. The arithmetic average of these two bounds, representing the balanced but weighted depth ($\widehat{H}$), is shown in Figure 5(c). Finally, the post-processed depth map following Gaussian convolution (H) is presented in Figure 5(d), demonstrating the successful mitigation of sharp local discontinuities to yield a continuous, physically plausible volumetric floodwater profile.

5. Experimental Setup

• Implementation Details: The proposed methodology was implemented using MATLAB 2023b. All experimental simulations and evaluations were executed on a standard workstation equipped with an Intel Core i7 CPU processor operating at 2.3 GHz and 40 GB of RAM. A significant architectural advantage of the proposed unsupervised algorithm is its computational efficiency. The method achieves a typical inference time of approximately 3 seconds per image, with an image resolution of 2000 × 1400, using purely CPU-based processing, successfully bypassing the necessity for the power-intensive GPU cores required by heavy DL models. Because of its low computational complexity and minimal hardware resource requirements, the proposed framework is highly suitable for rapid, on-site deployment on standard laptops, enabling immediate execution by emergency response personnel during active disaster scenarios.

To ensure reproducibility and facilitate future research, the source code implementing the proposed method, alongside detailed experimental results, will be made publicly available upon publication at the following repository: https://sites.google.com/site/costaspanagiotakis/research/flood-detection.

• Evaluation metrics: To assess the dual outputs of our framework, distinct evaluation metrics are employed for the 2D flood extent segmentation and the 3D volumetric depth estimation. For the binary flood extent map, the spatial agreement between the predicted flooded areas and the ground truth annotations is quantified using the F₁-score metric, defined as:

$$F1={\displaystyle \frac{2·TP}{2·TP+FP+FN}},$$

(12)

where $TP$, $FP$, and $FN$ represent the number of true positive, false positive, and false negative pixels, respectively.

For the flood depth estimation, the accuracy of the predicted water levels is evaluated using standard regression metrics, specifically the Root Mean Square Error (RMSE) and the Mean Absolute Error (MAE). Let $C_{gt}$ denote the set of spatial coordinates corresponding to the valid flooded pixels within the ground truth mask, such that for all $(x,y)\in C_{gt}$, the ground truth depth is strictly positive ($H_{gt}(x,y)>0$). The total number of these valid pixels is represented by $|C_{gt}|$. To quantitatively evaluate the predictive accuracy of a flood depth estimation model, the Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) are calculated over these localized flooded regions. Measured in meters (m), these absolute evaluation metrics are formally defined as follows:

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{|C_{gt}|}}{\sum }_{(x,y)\in C_{gt}}{\left(H(x,y)-H_{gt}(x,y)\right)}^2}$$

(13)

$$\mathrm{MAE}={\displaystyle \frac{1}{|C_{gt}|}}\sum _{(x,y)\in C_{gt}}\left|H(x,y)-H_{gt}(x,y)\right|$$

(14)

where $H(x,y)$ is the estimated flood depth at pixel $(x,y)$, and $H_{gt}(x,y)$ is the corresponding true depth derived from the ground truth data.

Furthermore, to provide a scale-independent assessment of relative depth accuracy across catchments with differing flood magnitudes, we report the maximum-normalized RMSE (nRMSE) and maximum-normalized MAE (nMAE). Normalizing by the maximum ground truth depth ensures mathematical stability and prevents the artificial error inflation that typically plagues traditional relative metrics at shallow inundation boundaries. These are defined as:

$$\mathrm{nRMSE}={\displaystyle \frac{\mathrm{RMSE}}{max\left(H_{gt}\right)}},\mathrm{and}\mathrm{nMAE}={\displaystyle \frac{\mathrm{MAE}}{max\left(H_{gt}\right)}},$$

(15)

where $max\left(H_{gt}\right)$ represents the maximum recorded ground truth depth within the specific catchment.

6. Experimental Results and Discussion

In this section, we present a comprehensive experimental evaluation of the proposed unsupervised framework. To rigorously assess the dual outputs of our pipeline, we provide both quantitative metrics and qualitative visual analyses for its two core stages: (1) flood region segmentation, and (2) flood depth estimation. Furthermore, a parameter sensitivity analysis is conducted to systematically evaluate the weighted contributions of the maximum and mean boundary elevation estimates, as defined in Equations (8) and (9) respectively, to the final predictive accuracy. Finally, to contextualize the algorithmic efficiency and physical validity of the proposed framework, we conclude with a comparative evaluation against iterative, optimization-based baselines.

6.1. Flood Region Segmentation Performance

The evaluation of the flood extent delineation stage, summarized under the semantic segmentation metrics in Table 3, demonstrates that the proposed unsupervised method achieves robust and reliable segmentation performance across highly heterogeneous environments. Because our depth estimation stage relies entirely on hydrostatic equilibrium constraints sampled from extracted boundaries, establishing a high-fidelity flood extent mask is vital. The experimental results show that our framework successfully captures intricate flood distributions without requiring any manual supervision or pre-training.

When categorized by overall segmentation quality, the highest F₁-scores were obtained in catchments HancheysStore (F₁ = 0.9512, IoU = 0.9069, Accuracy = 0.9410), Chinquaqin (F₁ = 0.9199, IoU = 0.8517, Accuracy = 0.9045), and Wallace (F₁ = 0.9084, IoU = 0.8321, Accuracy = 0.8855). These outstanding metrics indicate that the unsupervised boundary delineation module handles continuous, wide-area water layers excellently, producing extremely sharp region outlines that minimize both false positive and false negative pixel assignments. Moreover, the model maintains high structural stability even in more fragmented peri-urban landscapes. Sites like Lumberton and Greenville1 exhibit stable segmentation capabilities with comparable F₁-scores of 0.8030 and 0.7836, respectively, combined with strong pixel-level accuracy metrics exceeding 90%.

Furthermore, an analysis of the relationship between Precision and Recall reveals the balanced nature of our unsupervised delineation module. High Precision scores, such as those seen in Chinquaqin (0.9576) and Wallace (0.9556), demonstrate that the method is highly resilient against false alarms caused by non flooded urban features with dark spectral responses. Conversely, the lowest segmentation performance was recorded at the Kinston1 and Goldsboro2 catchments, yielding F₁-scores of 0.6955 and 0.6407, respectively. This localized degradation is primarily driven by extreme visual fragmentation and structural occlusions inherent to the site’s layout, shadow and reflectance issues.

Overall, despite the slight performance variations the framework maintains a pixel accuracy above 84% across 11 of the 12 study catchments. This consistent performance underscores the generalizability and effectiveness of our zero-shot unsupervised approach, validating its suitability as a robust, automated mask generation module for downstream hydrological calculations.

6.2. Flood Depth Estimation Performance

From the flood depth estimation metrics (columns 7-10 of Table 3), the maximum-normalized RMSE (nRMSE) serves as our primary performance metric, as it provides a scale-independent assessment of relative depth accuracy across catchments with differing flood magnitudes. As demonstrated in Table 3, our unsupervised framework achieved strong relative precision (nRMSE < 0.17) in sites such as Princeville (nRMSE = 0.1386), Goldsboro1 (RMSE = 0.1541), and Nichols (RMSE = 0.1645). Hydrologically, these low nRMSE values are tied to substantial, continuous inundation scenarios. In these well-defined flooding events (see Figure 6), minor spatial misalignments along the delineated edge borders translate into negligible relative depth errors, validating the robustness of our hydrostatic equilibrium assumptions across wide-area topographies.

Moving beyond the top performers, catchments such as Kinston2 and HancheysStore demonstrate intermediate, yet highly stable, depth estimation accuracy, yielding nRMSE values of 0.1852 and 0.1910, respectively. In these mid-performing scenarios, the absolute discrepancies remain well within acceptable operational tolerances for rapid response, recording MAEs of 0.6864 m and 0.8886 m. Analyzing these results in the context of the spatial classification metrics highlights the non-linear relationship between 2D pixel accuracy and 3D hydrostatic fidelity. For instance, HancheysStore achieves an outstanding global segmentation score (F₁ = 0.9512) and a near-perfect Recall (0.9637), indicating exceptional overall flood coverage. However, because the terrain-to-depth mapping phase extracts its vertical references strictly from the delineated margins, even mathematically negligible boundary variations, which barely impact the high global F₁-score, can occasionally sample unrepresentative topographic elevations, gently warping the localized hydrostatic plane and elevating the nRMSE to 0.1910 in this case. Conversely, Kinston2 records a slightly lower relative depth error (nRMSE = 0.1852) despite a significant drop in horizontal segmentation quality (F₁ = 0.7401). This 2D degradation in Kinston2 is characterized by a lower Recall (0.6916), reflecting boundary omission errors that shift the sampling margin inward. The fact that Kinston2 maintains comparable depth errors to HancheysStore despite these 2D omissions emphasizes that the severity of the 3D depth error is ultimately dictated by the specific topographic gradients present at the boundary pixels, rather than global segmentation metrics alone.

The most pronounced degradation in depth estimation occurs in catchments where the quantitative relationship between horizontal segmentation and vertical elevation sampling breaks down, specifically Goldsboro2 (nRMSE = 0.2564) and Greenville1 (nRMSE =0.2609). In Goldsboro2, the severe absolute depth discrepancy (RMSE = 1.7196 m, MAE = 1.3089 m) is a direct algorithmic consequence of a deficient horizontal segmentation stage (F₁ = 0.6407). Notably, this catchment recorded an exceptionally low Recall of 0.5274, indicating severe omission errors in the 2D mask. Because the terrain-to-depth mapping phase extracts its vertical references strictly from the detected water margins, missing nearly half of the true flood boundary forces the algorithm to sample elevations from erroneous, interior topographic pixels, propagating artificially skewed baselines throughout the hydrostatic-based depth estimation module. In contrast, Greenville1 maintains a substantially higher horizontal segmentation accuracy (F₁ = 0.7836, Recall = 0.8733) yet still records the highest relative depth error (nRMSE = 0.2609). This quantitative divergence demonstrates that even when 2D boundaries are reasonably well-captured, the algorithm remains highly sensitive to localized topological subjectivity in the underlying DTM. If the accurately segmented boundary pixels coincidentally align with steep elevation gradients or uncorrected micro-topographic artifacts, the sampled baseline will inherently misrepresent the true hydrostatic plane. Nevertheless, bounding the maximum relative error at 0.26 across the most computationally challenging scenarios highlights the practical resilience of this zero-shot, training-free deployment strategy.

Notably, the only competing method on the examined dataset is the one by Blay et al. [18]. This work utilized the same dataset to evaluate terrain-informed deep learning architectures for flood depth estimation. While both studies (ours and [18]) report RMSE and MAE, a direct quantitative comparison between their absolute error values and ours is structurally invalid due to three fundamental methodological divergences. First, their framework relies on a supervised learning paradigm, dividing the dataset to train their models on thousands of localized patches, giving the network prior spatial knowledge of the regional topography. In contrast, our approach is entirely unsupervised and training-free, operating without prior geographic exposure. Second, to establish their reference data, the authors applied systematic manual corrections to their initial flood extent masks to mitigate omission and commission errors, whereas our pipeline derives horizontal boundaries completely automatically. Finally, and most critically for metric interpretation, their reported site-level depth errors are calculated exclusively at sparse High Water Mark (HWM) sample points, often relying on as few as two to four reference points for entire catchments like Wallace, Chinquaqin, and HancheysStore. Conversely, our performance metrics are computed continuously across the entire dense 2D flood extent. Consequently, while both studies provide valuable insights into depth estimation, their point-based supervised errors and our pixel-wise unsupervised metrics evaluate fundamentally different spatial scales and operational paradigms.

6.3. Qualitative Analysis and Depth Error Mapping

To evaluate the spatial robustness of the proposed unsupervised depth estimation framework, Figure 6 illustrates a side-by-side qualitative comparison across six representative catchments. The rows are arranged hierarchically by performance (maximum-normalized RMSE), presenting the best-performing cases in the first two rows (Princeville and Goldsboro1), mid-performing catchments in the third and fourth rows (Kinston2 and HancheysStore), and the worst-performing cases (Goldsboro2, Greenville1) in the last two rows.

The first two rows demonstrate that for our best-performing cases, Princeville and Goldsboro1, the unsupervised depth maps closely mirror the structural topology of the ground truth. The absolute error maps ($|H_{gt}-H|$) for these regions, depicted in Figure 6 (d) and (h), are primarily characterized by a highly uniform, low-magnitude error profile dominated by the pale cyan and white spectrum, which corresponds to near-zero vertical discrepancies. Because these catchments generally feature widespread, continuous flood coverage, the boundary extraction module successfully isolates clean land-water interfaces. This allows the localized hydrostatic equilibrium sampler to anchor its vertical height references accurately, minimizing relative depth fluctuations across the vast majority of the water surface. Notably, while the absolute error in Princeville is exceptionally uniform, Goldsboro1 reveals a concentrated pocket of higher discrepancy (visible as the dark blue patch in the central-right corridor in Figure 6 (h)). This indicates that even within highly accurate global estimations, isolated DTM anomalies, undetected structural barriers, or localized boundary occlusions can still perturb the hydrostatic plane on a microscale.

For the mid-performing catchments in the third and fourth rows (Kinston2 and HancheysStore), an inspection of the absolute error maps (Figure 6 (l), (p)) reveals that their moderate depth discrepancies originate from two distinctly different visual challenges. In Kinston2, the visual layout is characterized by multiple small, geometrically complex flooded regions (Figure 6 (j)). Here, the moderate depth error is tied to a drop in horizontal segmentation quality (F₁ = 0.7401). When the unsupervised module occasionally misses true boundaries, the hydrostatic-based depth estimation module samples from shifted topographic pixels. Conversely, HancheysStore achieves an exceptional global segmentation (F₁ = 0.9512) but is visually characterized by high flood coverage interspersed with dense tree canopies and scattered housing (Figure 6 (n)). The complex textural transitions around these green spaces introduce minor, localized boundary segmentation errors. While mathematically negligible to the 2D mask, sampling from these structural microedges, such as the elevated terrain pad of a scattered house, gently warps the local hydrostatic plane (Figure 6 (o)).

The final rows represent the highly complex catchments of Goldsboro2 and Greenville1, which exhibit the highest vertical discrepancies. Visually, Goldsboro2 suffers from a heavily degraded segmentation (F₁ = 0.6407), as can be observed by comparing Figure 6 (r) and (s), where large flooded areas failed to be segmented. A visual inspection reveals a flooded airport facility and adjacent basins where dark, turbid water blends seamlessly with dense greenery and prominent cloud shadows. This lack of spectral contrast leads the unsupervised method to miss a massive subregion of water. Because our terrain-to-depth mapping phase extracts its vertical references directly from these water margins, this massive omission error forces the hydrostatic-based depth estimation module to sample incorrect interior elevations, propagating severe errors throughout the pipeline (nRMSE = 0.2564).

Conversely, Greenville1 achieves a higher segmentation score (F₁ = 0.7836), yet struggles with depth estimation (nRMSE = 0.2609). The raw imagery depicts a densely forested residential area heavily obscured by a massive optical shadow (Figure 6 (u)). The algorithm successfully segments around the scattered houses and canopies, but this creates a highly porous, fragmented mask. Consequently, the boundary pixels physically align with elevated rooftops and dense root systems rather than the true bare-earth waterline (Figure 6 (w)). Sampling from these artificial DTM elevations induces the extreme localized depth fluctuations seen in Figure 6 (x). Furthermore, this highlights a fundamental limitation: the accuracy of any terrain-to-depth mapping is bounded by the quality of the underlying DTM. In environments like Greenville1, when uncorrected micro-topographic variations or vegetation biases coincide with the extracted flood margins, they introduce erroneous vertical references into the hydrostatic-based depth estimation module, driving severe depth estimation errors despite a highly accurate 2D segmentation.

6.4. Parameter Sensitivity Study

To systematically evaluate the impact of the weighting coefficient w introduced in Equation (10), a parameter sensitivity analysis was conducted across all 12 catchments of the dataset. The primary purpose of the weight w is to regulate the operational balance between the physically conservative maximum boundary elevation estimate ($H_{max}$) and the statistically representative mean boundary elevation estimate ($H_{mean}$). Because unsupervised horizontal edge delineation can be susceptible to localized topographic anomalies or minor boundary misclassifications, tuning this parameter controls the model’s sensitivity to outlier elevation values along the extracted perimeters.

In our core framework, we empirically selected a balanced baseline value of $w=0.5$. This setting dictates that both depth components contribute equally to the final volumetric floodwater profile, serving as a neutral weighting scheme that balances the absolute physical upper bound with the average terrain elevation of the boundary, thereby maintaining the strictly unsupervised nature of our method.

As illustrated by the sensitivity curves in Figure 7, varying w from 0 to 1 directly alters the global performance across the regression metrics (RMSE and MAE). Rather than converging at a single intersection, the optimal performance metrics exhibit distinct minima. Despite this spread, the optimal performance range remains highly proximate to our selected default value of $w=0.5$. The relatively shallow gradients of the error curves around this baseline demonstrate that our framework is highly stable and resilient to minor parameter perturbations. This validates the robustness of the neutral weighting scheme for zero-shot deployment in unmapped environments without requiring training data or site-specific hyperparameter tuning. If we were to optimize the framework by finding the mathematically ideal value for the weight w, the optimal configuration emerges at the intersection of the two evaluation curves at approximately $w=0.55$. At this optimal vertex, the model slightly favors the mean boundary elevation ($H_{mean}$), which further absorbs localized DTM outliers. Crucially, this optimal value of $0.55$ remains highly proximate to our selected default value of $0.5$. The minimal performance variance between $0.5$ and $0.55$ demonstrates that our framework is highly stable and resilient to minor parameter perturbations, validating the robustness of the neutral weighting scheme for zero-shot deployment in unmapped environments without requiring training data or site-specific hyperparameter tuning.

6.5. Comparative Evaluation

To evaluate the predictive performance of our proposed framework on the given dataset and determine how close the derived depth estimation errors are to the theoretical minimums using the hydrostatic model formalized in Equation (10), we benchmark our approach against two optimization-based baseline methods. These baselines operate on the same multimodal input data but leverage ground truth depth profiles to calibrate and optimize the model parameters using Particle Swarm Optimization (PSO) [47,48,49]. PSO is a population-based metaheuristic algorithm that solves complex optimization problems by iteratively improving a swarm of candidate solutions. In the context of this study, PSO serves as a robust numerical solver to optimize the localized water surface elevation.

• First, to assess the practical applicability and comparative performance of the proposed hydrostatic model formalized in Equation (10), we establish a baseline, called the calibrated Auto-Extent Particle Swarm Optimization (AE-PSO), that utilizes our automatically derived flood extent map. For each discrete connected flood component (flood blob), the swarm explores the continuous parameter space of potential elevation constants ($c_i$), dynamically converging on the value that minimizes the RMSE between the simulated hydrostatic depth and the actual ground truth measurements based on Equation (5).
• Second, to isolate the error inherited from flood segmentation inaccuracies and the proposed hydrostatic model, we evaluate a ground truth baseline, called calibrated True-Extent Particle Swarm Optimization (TE-PSO). This secondary approach implements the previous PSO-driven parameter estimation scheme to find the optimal constant $c_i$ per i-blob, but substitutes the estimated flood extend mask of the proposed method with the ground truth flood extent mask.

More specifically, under AE-PSO and TE-PSO, for each isolated flooded region, the swarm explores the continuous parameter space of potential elevation constants ($c_i$), dynamically converging on the precise value that minimizes the RMSE between the simulated hydrostatic depth and the actual ground truth measurements. This ensures that both the AE-PSO and TE-PSO baselines represent the absolute theoretical best-case performance of the depth estimation model under their respective boundary conditions.

To quantify how closely the proposed framework to these optimized theoretical models is, relative performance is reported as the percentage change in RMSE against each respective baseline, formulated as

$${\Delta }_{baseline}=100\times {\displaystyle \frac{{\mathrm{RMSE}}_{baseline}-{\mathrm{RMSE}}_{proposed}}{{\mathrm{RMSE}}_{baseline}}}$$

(16)

Under this convention, negative values indicate a performance deficit (error increase). Such negative values are expected, as the proposed unsupervised framework is being compared against baselines that explicitly utilize actual ground truth depth data to optimally calibrate their parameters. However, positive values of ${\Delta }_{baseline}$ (which denote a performance gain achieved by our method), or small negative values close to zero, highlight the robustness of the proposed framework compared to optimized theoretical models.

Table 4 contextualizes the depth estimation accuracy of our pixel-wise framework against the two theoretical baseline approaches AE-PSO and TE-PSO. An analysis of the comparative results demonstrates that the proposed single-pass, pixel-wise framework remains highly competitive against the optimized theoretical methods. When evaluated on the same unsupervised segmentation masks (${\Delta }_{\mathrm{AE}\text{-}\mathrm{PSO}}$), our framework successfully outperforms the optimization heuristic in the Nichols (+4.70%) and Wallace (+4.41%) catchments. Furthermore, across half of the remaining study sites (e.g., Kinston2, Greenville2, Goldsboro2, Goldsboro1, Chinquaqin, and Princeville), the proposed method operates within a tight 10% error margin of the first AE-PSO baseline. In catchments where the performance deficit is more pronounced (e.g., Lumberton at -23.39% and Kinston1 at -21.64%), the ability of the theoretical optimized AE-PSO method to use the ground truth flood depth data, allows it to mathematically smooth out the micro-topographical DTM anomalies. By explicitly minimizing the volumetric discrepancy against the true flood depth, the AE-PSO effectively bypasses erroneous elevation spikes caused by vegetation or structural artifacts at the water’s edge, whereas our approach remains constrained by these local topographic features.

When compared against the theoretical upper bound TE-PSO method (${\Delta }_{\mathrm{TE}\text{-}\mathrm{PSO}}$), the data fundamentally reinforces our earlier qualitative findings: the majority of the 3D depth error budget is inherited directly from 2D boundary omissions. The massive performance gaps in highly obscured catchments like Goldsboro2 (-90.68%) and Kinston1 (-61.15%) confirm that supplying perfect ground truth flood extent masks to a hydrostatic-based depth estimation module drastically reduces depth discrepancies. However, a notable exception in Table 4 occurs in the Greenville2 catchment, where our proposed unsupervised method actually outperforms even the theoretical ground truth TE-PSO baseline, yielding a performance gain of +3.31%. This finding validates a critical hydrological premise of our methodology. By calculating depth on a localized, pixel-wise basis, our framework based on the proposed hydrostatic model formalized in Equation (10) better captures the nuanced hydraulic gradients and topological compartmentalization present in Greenville2.

7. Conclusions and Future Work

This study introduced a fully unsupervised, training-free framework for rapid, floodwater depth estimation in urban and peri-urban environments by integrating post event aerial imagery with Digital Terrain Models (DTMs). Operating without the bottleneck of manual annotations or task-specific training data, the proposed method addresses the complex spatial nature of flood dynamics by leveraging localized terrain constraints and the physical principle of hydrostatic equilibrium. Experimental validation across 12 heterogeneous catchments from the Inundation2Depth dataset [19] demonstrated robust generalization capabilities. By automatically extracting continuous, high-fidelity flood boundaries, eliminating the need for manual mask corrections, the framework yields spatially coherent and physically plausible depth estimates. Furthermore, our continuous, pixel-wise evaluation demonstrates that this approach offers a highly scalable and computationally efficient alternative to data-heavy supervised baselines, which often rely on sparse, point-based validation.

However, the extensive experimental evaluation also establishes a critical architectural insight regarding multimodal sensor dependency: the performance of our training-free pipeline is structurally coupled with the baseline quality of the input satellite data. Because the geometric depth module relies on sampling topographic points exactly where the horizontal land-water interface is detected, any visual ambiguities in the optical imagery, such as structural building shadows, dense cloud cover, or a lack of spectral contrast in highly turbid waters, directly dictate the positioning of the extraction boundary. When these horizontal variances align with steep elevation gradients in the underlying DTM, they introduce artificial vertical references into the hydrostatic-based depth estimation module. Rather than exposing a flaw in the physical modeling phase, this dependency emphasizes that the optimization of unsupervised flood mapping is inherently bounded by the fidelity of the raw inputs, highlighting the necessity for high-resolution, cloud-cleared data feeds during rapid disaster response mapping.

While the current unsupervised paradigm delivers highly stable depth mapping across diverse topographic terrain profiles, several avenues remain open for future investigation. A primary objective for subsequent research is transitioning this framework into an end-to-end unsupervised or self-supervised deep learning pipeline [50]. Instead of relying on a decoupled two-stage approach, neural networks could be designed to directly predict volumetric flood depth from raw multimodal inputs (RGB satellite imagery and DTMs) without requiring labeled ground truth data. One promising direction involves self-supervised contrastive learning or masked autoencoders [51] optimized for multi-source remote sensing data, where the model learns robust structural representations by masking and reconstructing paired RGB-DTM patches. Another crucial advancement would be the development of unsupervised physics-guided neural networks (PGNNs) [52]. By embedding hydrostatic equilibrium constraints and terrain elevation boundaries directly into the network’s loss function, the deep model could optimize its depth predictions autonomously. This end-to-end integration would enable the framework to learn complex spatial priors, gracefully handling micro-topographical architectural noise and dense canopy occlusions that traditionally challenge static geometric solvers.

Beyond algorithmic advancements, future work must also address the inherent data scarcity constraints associated with acquiring synchronous, paired aerial imagery and high-fidelity DTMs for newly impacted geographic regions. To mitigate this bottleneck, research will focus on generative synthetic data augmentation [53]. By synthetically co-generating diverse, simulated post-flood aerial textures aligned with varying artificial or real DTM structures, it becomes possible to simulate rare and extreme multi-meter inundation scenarios. Crucially, these rich synthetic environments can be leveraged to train multimodal deep neural architectures that ingest both RGB imagery and topographic streams simultaneously. This synthetic training pipeline will facilitate extensive domain generalization studies, ultimately helping to robustly close the gap for downstream data driven flood management models without the need for human annotated real world records.