Introduction
Flooding is among the most devastating natural disasters in the world, causing massive economic loss and loss of life. The rise of climate change to exacerbate harsh climatic conditions makes it imperative now more than ever to develop accurate flood forecasting and flood management systems. Geographic Information Systems (GIS) and remote sensing technologies have revolutionized how we assess flood hazards and design mitigation strategies. At the heart of these procedures are Digital Elevation Models (DEMs), which provide the fundamental topographic data used to understand how water flows through environments.
This blog discusses how DEMs help in assessing and managing flood risk and reviewing their development, uses, constraints, and the mathematical concepts underpinning their application in hydrology and flood modeling.
Understanding Digital Elevation Models
What is a DEM?
A Digital Elevation Model (DEM) is a 3D model of the surface of a terrain generated from height data. DEMs give continuous values of elevation over an area, normally stored as a raster grid whose cells hold an elevation value. DEMs are the basis for many spatial analyses, especially hydrological analyses and flood modeling.
Types of DEMs
DEM Type Resolution Source Best Applications
SRTM (Shuttle Radar Topography Mission) 30m globally, 1 arc-second Space shuttle radar Regional flood modeling, medium-scale watershed analysis
ASTER GDEM 30m globally Stereoscopic satellite imagery Regional terrain analysis, medium-scale hydrological modeling
LiDAR-derived DEMs 0.5-2m Airborne laser scanning High-precision urban flood modeling, detailed floodplain mapping
Photogrammetry-derived DEMs 1-5m Aerial imagery Local flood studies, infrastructure planning
Bathymetric DEMs Variable Sonar, satellite altimetry Coastal flooding, combined riverine and coastal flood modeling

DEM Acquisition Methods
DEMs can be acquired through various remote sensing techniques:

1. LiDAR (Light Detection and Ranging): Returns high-resolution elevation information by measuring the time it requires emitted laser pulses to travel back after reflecting off the Earth's surface.
2. Radar Interferometry: This method is used in missions like SRTM to find out how high something is by measuring the phase differences between radar signals.
3. Photogrammetry: This method finds elevation by comparing overlapping aerial or satellite photos and finding points that match.
4. Field Surveying: A traditional method that uses GPS and total stations to get very accurate but limited elevation data.

Figure 1: Visualization of a digital elevation model with color-coded elevation ranges
(Source:html/images/remotesensing-14-02776-g001-550.jpg)

DEM Processing for Hydrological Applications
Prior to the utilization of DEMs for flood modeling, they generally need to undergo preprocessing so that they are hydrologically correct.
DEM Preprocessing Steps

1. Filling depressions/sinks: Removing artificial depressions that would improperly catch water flow in the model
2. Stream burning: Adding known stream networks so that there is correct flow direction
3. Resampling: Changing resolution to reconcile required analysis scale
4. Edge matching: Providing smooth transitions between adjacent DEM tiles Deriving Hydrological Parameters From a hydrologically corrected DEM, a number of important parameters may be obtained: Flow Direction Flow direction determines the direction in which water will move from each cell to its steepest downslope neighbor. The D8 method gives each cell a value depending on steepest descent direction:

Figure 2: D8 flow direction coding (RASTER and VECTOR)
(Source:https://pro.arcgis.com/en/pro-app/latest/tool-reference/spatial-analyst)

Flow Accumulation
Flow accumulation computes the amount of cells upstream draining into every cell, essentially outlining stream networks and contributing areas.
Watershed Delineation
Watersheds (catchments) are defined as the area of land from which all precipitation flows to a single outlet. DEM-based watershed delineation is basic for flood modeling.

Figure 3: Watershed delineation from a DEM
(Source: https://pro.arcgis.com/en/pro-app/latest/tool-reference/spatial-analyst/GUID-E473F9D8-D2F7-475B-BEC8-DC7065CD4605-web.png)

Mathematical Principles of DEM-Based Flood Modeling
Slope and Aspect Calculation
Slope is the rate of elevation change and is important for the understanding of water speed and infiltration rates. For a DEM with cell dimension a, the slope is estimated as follows:
Slope (degrees) = arctan(√[(dz/dx)² + (dz/dy)²]) × (180/π)
Where:
•\tdz/dx is the change rate in the x direction
•\tdz/dy is the change rate in the y direction
With a 3×3 moving window, dz/dx and dz/dy can be derived using the following equations:
dz/dx = ((c + 2f + i) - (a + 2d + g)) / (8 × cell size)
dz/dy = ((g + 2h + i) - (a + 2b + c)) / (8 × cell size)
Where a-i are the elevation values in the 3×3 window:
a b c
d e f
g h i
Example Calculation
Take a 3×3 window from a DEM of 10m cell size and these elevation values (in meters):
78 72 69
74 67 56
69 53 44
Step 1: Find dz/dx
dz/dx = (((69 + 2*56 + 44) - (78 + 2*74 + 69)) / (8 * 10))
= ((69 + 112 + 44 - 78 - 148 - 69) / 80)
= (225 - 295) / 80
= -70 / 80
= -0.875
Step 2: Find dz/dy
dz/dy = ((69 + 2(53) + 44) - (78 + 2(72) + 69)) / (8 × 10)
= (69 + 106 + 44 - 78 - 144 - 69) / 80
= (219 - 291) / 80
= -72 / 80
= -0.9
Step 3: Calculate slope
Slope = arctan(√[(-0.875)² + (-0.9)²]) × (180/π)
= arctan(√[0.766 + 0.81]) × (180/π)
= arctan(√1.576) × (180/π)
= arctan(1.255) × (180/π)
= 51.4° × (180/π)
= 51.4° × 57.3
= 10.19°
This steep slope (10.19°) indicates rapid water flow, suggesting higher flood velocities in this area.
Flow Accumulation Algorithms
The most basic flow accumulation algorithm operates by:

1. Initializing each cell with a starting value of 1 (for its own share)
2. Following directions of flow and adding values downstream
3. The resulting accumulation value is the sum of all the cells draining through each point Recursive algorithms or queue-based algorithms are typically employed for large DEMs to handle the computational demands in an efficient manner. Flood simulation using DEMs Basic Flood Inundation Modeling Height Above Nearest Drainage (HAND) Model The HAND approach makes use of relative height difference of each DEM cell to the closest drainage point in the creation of flood susceptibility maps. The computationally efficient approach gives rapid determinations of the possible flood areas. Each cell's HAND value is calculated as: HAND = Cell Elevation - Nearest Drainage Point Elevation Cells with lower HAND values than predicted flood levels are identified as potentially flooded.

Figure 4: Comparison of DEM (a) and HAND model (b) showing relative heights above drainage network
(Source:https://www.researchgate.net/profile/Luz-Cuartas-2/Procedure-to-ge)

Hydraulic Modeling Integration
Advanced flood models combine DEMs with hydrodynamic modeling software like HEC-RAS, MIKE FLOOD, or TUFLOW. The models solve complex equations (Saint-Venant equations) numerically for the water displacement to simulate a flood.
Manning's Equation
Among the most important equations used in flood modeling is one that relates water velocity to channel characteristics:
V = (1/n) × (R)^(2/3) × (S)^(1/2)Where:
• V is velocity (m/s)
• n is Manning's roughness coefficient
• R is hydraulic radius (m)
• S is water surface slope (derived from DEM)

Case Study: 2D Flood Modeling with HEC-RAS
HEC-RAS (Hydrologic Engineering Center's River Analysis System) is widely used in flood modeling. All of its 2D applications highly rely on good-quality DEMs.
The procedure involves

1. Exporting a DEM to HEC-RAS
2. Defining a 2D flow area mesh
3. Boundary condition definition (inflow hydrographs, etc.)

4. Simulation runs to produce extent, depth, and velocity maps of floods

5. Model Parameter 6. Common Value Range 7. Sensitivity 8. Data Source
   DEM Resolution 1-30m High LiDAR/SRTM/Local survey
   Manning's n 0.01-0.10 Moderate Land cover maps, field measurements
   Inflow Hydrograph Site-specific High Gauge data, hydrologic modeling
   Boundary Condition Type Normal depth, rating curve Moderate Field measurements
   Computational Time Step 1-60 seconds Low-moderate Based on grid size and stability

Figure 5: HEC-RAS 2D flood inundation model output showing flood depths in an urban area
(Source: https://enc_8aWBtXwbWuYI5ADYphOiQ&s)

Real-World Applications
Urban Flood Risk Assessment
Urban landscapes present unique challenges to flood modeling due to sophisticated drainage systems, impervious surfaces, and built infrastructure. It requires high-resolution DEMs from LiDAR (typically 1m or higher) to adequately model urban flood hazard.
Applications are:
• Stormwater system planning
• Flood vulnerability assessment building
• Evacuation route planning
• Insurance risk mapping
Climate Change Adaptation Planning
DEM-based flood models help communities prepare for upcoming climate conditions:
• Sea level rise impact assessment
• More intense rainfall events
• Infrastructure adaptation planning
• Economic appraisal of flood defense measure
Early Warning Systems
DEMs augmented by near-real-time meteorological data and hydrological modeling allow the generation of early warning flash flood systems:
• Flash flood potential mapping
• Warning time estimation
• Evacuation planning
• Critical infrastructure protection

Challenges and Limitations
DEM Accuracy Considerations
The quality of flood modeling is directly dependent on DEM accuracy:
Error Source Potential Impact Mitigation Strategy
Vertical Accuracy Significant over-/under-estimation of flood extent Use highest resolution data available; field verification
Feature Representation Missing flood barriers or channels Manual editing; incorporation of engineered features
Temporal Currency Outdated terrain information Regular updates, especially post-development
Vegetation Effects False elevation readings Use Digital Terrain Models (bare earth)
Urban Features Buildings appearing as "terrain" Building removal algorithms; manual editing

Case Study: Impact of DEM Resolution on Flood Extent
A study by Savage et al. (2016) examined how DEM resolution affects flood model outcomes:
DEM Resolution Flooded Area (km²) Computational Time Notable Differences
50m 12.4 4 minutes Missed narrow floodplains
10m 10.8 18 minutes Moderate detail, practical compromise
5m 9.7 45 minutes Detected flood barriers
1m 8.9 240 minutes Captured microtopography, roads, levees

The increased resolution always produced a more accurate but reduced flood extent by identifying terrain features that constrict water flow.
Future Directions
Emerging Technologies
Several technical developments improve DEM-based flood modeling:

1. Drone-derived DEMs: Low-cost, on-demand high-resolution elevation data
2. Structure-from-Motion (SfM): High-end photogrammetry methods for fast DEM production
3. Global Navigation Satellite System (GNSS): Enhanced ground control vertical accuracy
4. Near real-time DEM updates: Satellite data for updating DEMs after high-magnitude landscape change

Integration with Machine Learning
Machine learning techniques are enhancing flood prediction by:
• DEM feature extraction automation
• Enhancing flood-risk area classification
• Developing hybrid models that combine the physical and statistical approaches
• Computational requirements reduction through surrogate modeling
Conclusion
Digital Elevation Models are essential tools for modern flood risk analysis and management. They are applied from simple watershed delineation to advanced 2D hydraulic modeling. With the advancement of technology, DEM resolution, accuracy, and availability improve step by step, enabling more precise flood prediction and better decision-making.
Their mathematical underpinning remains the same, but their implementation has been transformed with the advancement in computational capabilities. DEMs provide the basic topographic foundation upon which effective flood management strategies are formulated for urban planning, for climatic resilience, or for emergency response.
For climatologists, emergency managers, urban planners, or climate adaptation specialists, proficiency in being able to effectively use DEMs to simulate floods is a skill that they need. With climate change generating more intense weather events, these technologies will increasingly become essential to building resilient communities.
References

1. Bates, P.D. (2022). "Flood inundation modelling: Current capabilities and future challenges." Journal of Flood Risk Management, 15(1), e12784.
2. Grimaldi, S., Li, Y., Pauwels, V.R., & Walker, J.P. (2018). "Remote sensing-derived water extent and level to constrain hydraulic flood forecasting models: Opportunities and challenges." Surveys in Geophysics, 39(4), 809-840.
3. Hawker, L., Neal, J., & Bates, P. (2020). "Accuracy assessment of the TanDEM-X 90 Digital Elevation Model for selected floodplain sites." Remote Sensing of Environment, 232, 111319.
4. Jarihani, A.A., Callow, J.N., McVicar, T.R., Van Niel, T.G., & Larsen, J.R. (2015). "Satellite-derived Digital Elevation Model (DEM) selection, preparation and correction for hydrodynamic modelling in large, low-gradient and data-sparse catchments." Journal of Hydrology, 524, 489-506.
5. Nobre, A.D., Cuartas, L.A., Hodnett, M., Rennó, C.D., Rodrigues, G., Silveira, A., Waterloo, M., & Saleska, S. (2011). "Height Above the Nearest Drainage–a hydrologically relevant new terrain model." Journal of Hydrology, 404(1-2), 13-29.
6. Savage, J.T.S., Bates, P., Freer, J., Neal, J., & Aronica, G. (2016). "When does spatial resolution become spurious in probabilistic flood inundation predictions?" Hydrological Processes, 30(13), 2014-2032.
7. Schumann, G.J.P., & Bates, P.D. (2018). "The need for a high-accuracy, open-access global DEM." Frontiers in Earth Science, 6, 225.
8. Yamazaki, D., Ikeshima, D., Tawatari, R., Yamaguchi, T., O'Loughlin, F., Neal, J.C., Sampson, C.C., Kanae, S., & Bates, P.D. (2017). "A high-accuracy map of global terrain elevations." Geophysical Research Letters, 44(11), 5844-5853.