Slope Raster DEM Calculation Focal Calculator
Use a 3 by 3 focal neighborhood from a digital elevation model to estimate slope with the Horn finite difference method. Enter the eight neighbors, center cell elevation, raster cell size, and optional z-factor to compute slope in degrees, percent rise, and gradient components.
dz/dy = ((Z7 + 2Z8 + Z9) – (Z1 + 2Z2 + Z3)) / (8 × cell size)
slope = atan( sqrt( (dz/dx)² + (dz/dy)² ) )
Calculated results
Enter DEM elevations and click Calculate Slope to see the focal slope estimate.
Neighborhood elevation profile
Expert guide to slope raster DEM calculation focal analysis
Slope raster DEM calculation focal analysis is the process of estimating local terrain steepness for each raster cell by looking at a neighborhood around that cell, most commonly a 3 by 3 moving window. In a digital elevation model, every pixel stores elevation. A focal operation treats the center pixel as the location of interest and uses the surrounding neighbors to approximate how rapidly elevation changes in the x and y directions. From those partial derivatives, GIS software computes slope in degrees, percent rise, or sometimes radians. This is one of the foundational terrain metrics in geomorphology, hydrology, hazard screening, road engineering, agriculture, forestry, and site planning.
The calculator above follows a standard finite difference approach commonly associated with the Horn method. In practical GIS workflows, this method is used in many desktop and server tools because it captures local relief from immediate neighbors without requiring a larger moving kernel. The result is a local slope estimate that can be written into a slope raster, classified into terrain steepness categories, or combined with aspect, curvature, flow accumulation, and land cover layers.
What “focal” means in raster terrain analysis
In raster processing, operations are often grouped into local, focal, zonal, and global functions. A local function uses only the value of the current cell. A focal function uses the current cell and its nearby neighbors, usually with a moving window such as 3 by 3, 5 by 5, or circular kernels. Slope belongs to the focal family because the center cell alone does not contain enough information to estimate a gradient. You need nearby elevations to know how the surface tilts.
For a 3 by 3 neighborhood, the cells are typically labeled like this:
- Z1, Z2, Z3 on the top row
- Z4, Z5, Z6 in the middle row
- Z7, Z8, Z9 on the bottom row
The center value Z5 is the target location, but the slope calculation depends strongly on the surrounding eight cells. This is why missing data, voids, edge effects, and inconsistent vertical units can materially change the output slope raster.
How the Horn slope method works
The Horn method approximates the rate of elevation change in two orthogonal directions. One derivative tracks east-west change, and the other tracks north-south change. The neighborhood is weighted so that side neighbors count twice as much as corner neighbors. This weighted stencil stabilizes the derivative estimate and has become a classic choice in GIS terrain analysis. After computing dz/dx and dz/dy, the algorithm combines them into a single gradient magnitude. Taking the arctangent of that magnitude yields slope in radians, which is then converted to degrees if needed.
- dz/dx estimates horizontal change across columns.
- dz/dy estimates vertical change across rows.
- Gradient magnitude equals the square root of the sum of squared derivatives.
- Slope degrees equals arctangent of the gradient, converted from radians.
- Percent rise equals gradient multiplied by 100.
If your DEM elevation units differ from your horizontal map units, a z-factor is required. For example, if elevations are in feet but cell size is in meters, you must convert the vertical component so the math compares like with like. Many GIS users overlook this detail, and it is one of the most common reasons for unrealistic slope values.
Why cell size matters so much
Slope is inherently scale dependent. A 1 meter DEM will detect fine micro-topography such as berms, road cuts, and drainage swales. A 30 meter DEM smooths those features into broader regional terrain. This means the same hillside can yield very different slope rasters depending on source resolution, resampling approach, and whether the DEM was hydrologically conditioned or filtered from lidar. When you compare slope products, always compare products at equivalent spatial resolution and similar preprocessing standards.
| DEM source or product | Typical spatial resolution | Coverage or availability | Practical implication for focal slope |
|---|---|---|---|
| USGS 3DEP lidar-derived DEM | 1 m where available | Large and growing U.S. coverage through 3DEP | Excellent for engineering detail, small drainage features, cut/fill analysis, and local slope break detection. |
| USGS National Elevation Dataset style products / 1 arc-second DEM | About 30 m | Broad U.S. coverage | Well suited to watershed and regional terrain studies, but less sensitive to fine site-scale relief. |
| USGS 1/3 arc-second DEM | About 10 m | Broad U.S. coverage for many applications | A strong middle ground for catchment studies, land suitability mapping, and moderate-scale infrastructure planning. |
| NASA SRTM global DEM | About 30 m globally | Near-global between high latitude limits | Useful for continental and regional analyses, but not ideal for high precision local design. |
The figures above reflect widely cited product resolutions from USGS and NASA programs. For slope mapping, these differences are not just cosmetic. Steep embankments, gullies, levees, and urban grade changes that appear strongly in 1 meter data may be nearly invisible in 30 meter grids.
Degrees versus percent rise
Professionals often switch between slope degrees and percent rise depending on the discipline. Civil designers and road engineers frequently think in percent grade. Geomorphologists, avalanche analysts, and hazard mappers often use degrees because threshold behavior can be tied more directly to angular steepness. Fortunately, these are two ways of expressing the same gradient. Percent rise is simply the tangent of the slope angle multiplied by 100.
| Slope class | Degrees | Approximate percent rise | Typical interpretation |
|---|---|---|---|
| Flat to nearly level | 0° to 2° | 0% to 3.5% | Minimal runoff acceleration, often suitable for many built uses depending on soils and drainage. |
| Gentle | 5° | 8.7% | Common threshold for low-gradient ground and early erosion screening. |
| Moderate | 10° | 17.6% | Noticeable relief, often important for equipment movement and runoff management. |
| Steep | 20° | 36.4% | Higher erosion risk and stronger design constraints for roads, pads, and vegetation management. |
| Very steep | 30° | 57.7% | Substantial grading limitations, frequently used in hazard and hillside development screening. |
| Extreme | 45° | 100% | One-to-one rise over run, often a practical benchmark for severe terrain. |
Best practices for accurate slope raster output
- Use consistent units. Match vertical and horizontal units or apply a correct z-factor.
- Choose the right DEM for the question. Fine-resolution lidar is ideal for local design; 10 meter to 30 meter products can be better for broader regional consistency.
- Check edge cells and nodata. Cells at the edge of a raster or adjacent to voids can produce unstable derivatives.
- Be careful with resampling. Bilinear, cubic, and nearest-neighbor resampling each alter local topography differently before slope is computed.
- Consider smoothing only when justified. Filtering can reduce noise in lidar surfaces, but over-smoothing can erase meaningful breaks in slope.
- Project to an appropriate coordinate system. Geographic coordinates in degrees are not suitable for direct slope calculations without proper handling because x and y spacing vary spatially.
Common use cases for focal slope calculations
Focal slope is one of the most reused terrain derivatives in applied geospatial analysis. Hydrologists use it to understand runoff velocity potential and to parameterize erosion and sediment transport models. Land planners use slope classes to guide zoning, grading constraints, and site suitability. Transportation teams use local slope to review alignments, estimate earthwork difficulty, and evaluate access. Ecologists use topographic steepness as a proxy for moisture gradients, solar exposure interactions, and habitat fragmentation constraints when combined with aspect and curvature.
In a raster workflow, slope can also serve as an input to more advanced models. For example, the Universal Soil Loss Equation family relies on terrain steepness factors. Landslide susceptibility screens often blend slope with geology, land cover, rainfall, and drainage density. Wildfire behavior studies sometimes combine slope with fuels and wind. In each case, the quality of the final model depends heavily on whether the underlying focal slope was calculated on an appropriate DEM using sound preprocessing choices.
Frequent mistakes analysts make
- Using latitude-longitude rasters directly. If the DEM is still in geographic degrees, cell width is not uniform in real distance terms. Reproject first to a suitable projected coordinate system.
- Ignoring unit conversion. Feet-versus-meters mismatches can inflate or suppress slope by a large factor.
- Comparing mixed resolutions. A 1 meter slope raster and a 30 meter slope raster are not directly interchangeable.
- Assuming all GIS tools implement slope identically. Different software may use slightly different kernels, edge handling rules, or geodesic options.
- Overinterpreting noisy surfaces. Bare-earth quality, vegetation artifacts, bridges, and interpolation gaps can create artificial steep cells.
How to interpret the calculator’s output
The calculator returns dz/dx and dz/dy as directional gradients, slope in degrees, percent rise, and the center elevation. The directional derivatives help you diagnose whether the terrain is changing more strongly east-west or north-south. A positive dz/dx means elevations generally increase toward the east side of the kernel relative to the west. A positive dz/dy, given the row layout used here, indicates the southern row is higher than the northern row after weighting. The slope magnitude combines both directional effects into one steepness measure.
If you are checking a GIS output cell by cell, this kind of calculator is very useful for QA. You can plug in the exact 3 by 3 neighborhood from your DEM and verify whether the software’s slope result aligns with the Horn method. This is especially helpful when troubleshooting project-specific questions such as why a retaining wall alignment appears steeper in one dataset than another or why a local slope threshold mask captures more cells than expected.
Recommended authoritative references
- USGS 3D Elevation Program (3DEP) for current information on U.S. elevation data quality and resolution.
- USGS DEM overview for foundational definitions of digital elevation models and elevation products.
- NASA Earthdata SRTM resource for global Shuttle Radar Topography Mission elevation context.
Final takeaway
Slope raster DEM calculation focal analysis is simple in concept but highly sensitive to data quality, resolution, units, and kernel design. A 3 by 3 neighborhood captures local terrain by estimating directional change around each center cell, and the Horn method remains a dependable standard for many GIS applications. When your units are consistent, your DEM is appropriate for the scale of the problem, and your raster has been projected and cleaned properly, focal slope becomes one of the most powerful and interpretable terrain products available. Use the calculator above to validate local neighborhoods, compare degree and percent outputs, and build intuition about how small elevation changes inside a focal window control the final slope raster.