Derivatives

Derivatives are the measures you compute directly from the shape of the surface around each cell: how steep it is, which way it faces, and how it bends.

Slope and aspect are the two everyone uses. The twenty one curvature tools that follow describe the bending of the surface in different ways, and they matter for erosion, deposition, water flow and landform analysis.

Start here

Slope

Steepness, computed with the Zevenbergen and Thorne method. The single most used terrain derivative, and an input to almost everything else.

Parameter	What to set
Input	Your elevation raster.
Units	Degrees, from 0 flat to 90 vertical, the most readable. Percent, the rise over run used in engineering and road design. Radians, for feeding another computation.
Z factor	Vertical exaggeration. Leave at 1 when vertical and horizontal units match.
Output	The slope raster.

Slope depends on the resolution of your DEM. The same hillside measured on a 1 metre LiDAR model and on a 30 metre global model will not give the same steepness, because each cell averages a different amount of ground.

Aspect

The compass direction the slope faces, in degrees: 0 is north, 90 east, 180 south, 270 west. It drives how much sun a slope receives, and with it snow melt, soil moisture and vegetation.

Aspect is circular, which has a practical consequence: 359 and 1 are neighbours, not opposites. Do not average it as an ordinary number, and style it with a cyclic colour scheme so north does not appear twice at both ends of the ramp.

Flat cells have no meaningful aspect.

Plan curvature and Profile curvature

The two curvatures worth knowing if you never use the others.

Plan curvature measures bending across the slope, at right angles to the flow direction. Positive values mark places where flow converges, which is where water and material concentrate. Negative values mark places where it spreads out.

Profile curvature measures bending along the slope. It tells you where flow accelerates, which is where erosion happens, and where it slows down, which is where material is deposited.

Together they describe the local shape of the ground better than slope alone: a uniform 10 degree hillside and a 10 degree hollow behave very differently.

All the tools

Tool	What it does
Slope	Steepness in degrees, radians or percent, by the Zevenbergen and Thorne method.
Aspect	The compass direction of the steepest descent, in degrees from north.
Relative Aspect	Aspect expressed relative to an azimuth you choose, from 0 to 180 degrees.
Plan Curvature	Bending across the slope. Positive means converging flow, negative means diverging.
Profile Curvature	Bending along the slope. Positive means accelerating flow, negative means decelerating.
Tangential Curvature	Lateral curvature perpendicular to the flow line, close to plan curvature but directional.
Total Curvature	Overall curvature independent of direction. High at peaks and pits, low on planes.
Mean Curvature	Average of the two principal curvatures. Emphasises how smooth or abrupt the surface is.
Gaussian Curvature	Product of the principal curvatures. Positive is a bowl or a dome, negative is a saddle, zero is a cylinder.
Maximal Curvature	The larger of the two principal curvatures.
Minimal Curvature	The smaller of the two principal curvatures.
Principal Curvature Direction	The direction, in degrees, in which the surface bends most.
Accumulation Curvature	The product of profile and tangential curvature, marking zones of flow accumulation.
Casorati Curvature	A single measure of overall bending derived from the principal curvatures.
Curvedness	How strongly the surface is curved, regardless of shape.
Shape Index	The kind of shape, from cup through saddle to cap, independent of how strongly it curves.
Difference Curvature	Half the difference between vertical and horizontal curvature.
Generating Function	A combined curvature descriptor used in surface classification.
Horizontal Excess Curvature	How much the horizontal curvature exceeds the minimal curvature.
Vertical Excess Curvature	How much the vertical curvature exceeds the minimal curvature.
Ring Curvature	The squared twisting of flow lines.
Rotor	The twisting of flow lines, signed.
Unsphericity	Half the difference between the principal curvatures, marking departure from a sphere.

Notes

Curvature values are small. They are expressed per unit of distance, so the raw numbers look like tiny fractions. Style by percentile or with a diverging ramp centred on zero rather than by the raw range.

Smooth first. Curvature amplifies noise, so a LiDAR derived model is usually run through Feature Preserving Smoothing in General before any curvature is computed.

Naming. Whitebox follows the geomorphometric convention where positive plan curvature is convergent. Other software sometimes uses the opposite sign, which is worth remembering when comparing results.