# Slope Calculation Methods For Global Dems: Best Practices

## Definition and Importance of Slope

Slope, defined as the rate of elevation change across a landscape, is a fundamental geographic attribute with widespread utility. Slope calculations from global digital elevation models (DEMs) inform critical environmental research and applications in climatology, hydrology, geomorphology, ecology, natural hazard assessment, and more.

As a first derivative of elevation, slope quantifies the steepness of terrain. Steep slopes are associated with rapid downslope movement of water, sediment, and materials. Gentle slopes promote slower flows and deposition. Thus, accurate slope information underpins sound analysis and decision-making across the environmental sciences.

Global DEMs synthesize massive topographic data sets to model Earth’s complex terrain at consistent spatial resolutions worldwide. From the initial GTOPO30 in 1996 to modern releases like Copernicus DEM and AW3D30, freely available global DEMs support diverse science and conservation goals. As DEM quality and availability continue improving, slope maps derived from these data sets guide research and modeling at global scales.

## Common Slope Calculation Methods

### Finite Difference Method

The finite difference method calculates slope using elevation differences between adjacent DEM grid cells. This technique approximates terrain slope as a discrete first-order derivative based on cell elevations across defined x and y distances. The simple and efficient finite difference algorithm sees widespread use for slope mapping from gridded elevation data sets.

#### First-order Derivatives

First-order derivative functions estimate instantaneous rates of change. For a given grid cell in a DEM with coordinates (x, y) and elevation z, slopes in the x and y directions are approximated by dividing z changes between cells by the cell spacing:

where δx and δy are the DEM x and y resolutions. The steepest downslope gradient for the cell is then computed as the magnitude of the gradient vector:

This efficient finite difference approach sees widespread usage for regional to global scale slope mapping and derivatives for climatology, hydrology, and geomorphology.

#### Example Python Code

A Python script implements the finite difference slope algorithm on a DEM grid (assign elevation array as dem):

import numpy as np dx = dem.dx # x resolution dy = dem.dy # y resolution dzdx = np.gradient(dem, dx, axis=1) # dz/dx dzdy = np.gradient(dem, dy, axis=0) # dz/dy slope = np.sqrt(dzdx**2 + dzdy**2) # Slope magnitude

The script calculates x and y first derivatives using NumPy’s gradient function. Taking the magnitude of the gradient vector slope combines the directional derivatives into a slope surface matching the input DEM grid.

### Regression Analysis Method

In contrast to the discrete cell-by-cell finite difference technique, the regression analysis method fits a continuous statistical model to clusters of DEM points to estimate sloping terrain. Ordinary least squares (OLS) linear regression serves as the predominant approach.

#### Ordinary Least Squares

OLS regression minimizes elevation residuals to fit a planar surface within a moving analysis window. For a 3×3 pixel window, the plane estimation takes the form:

where z represents the center pixel elevation, x and y are coordinate offsets from z, and b terms contain the slope parameters. Solving the OLS minimization through matrix calculations derives the best fit plane. The b1 and b2 values directly provide slope gradients in the x and y directions.

Though more computationally intensive than finite differences, the robust OLS technique allows flexibility in window sizes and fits smooth slope surfaces less susceptible to microtopography and noise.

#### Example R Code

An R script implements OLS slope mapping on a raster DEM (raster layer assigned as dem):

library(raster) slope <- focal(dem, w=matrix(1,3,3), fun=function(z){ df <- as.data.frame(z) mod <- lm(z~x+y, data=df) return(coefficients(mod)[2]) })

The focal function applies a 3x3 moving window with an OLS linear model fitted at each step. Extracting the x coefficient provides the local east-west slope value. Similar processing on a 90 degree rotated layer can extract north-south slope.

## Factors Affecting Slope Calculation Accuracy

While the finite difference and regression techniques show demonstrated utility for mapping global DEM slopes, several factors impact output quality:

### DEM Resolution and Smoothing

Coarser DEM resolutions fail to capture short wavelength topographic variability, underrepresenting steep extreme slopes. Smoothing filters similarly degrade slope accuracy. Finer scale DEMs resolve terrain details more precisely at the cost of noise amplification and artifacts.

### Directional Effects

North-south and east-west slope outputs may diverge due to resampling effects over longer distances. In higher latitudes, increased east-west cell spacing on common geographic projections introduces distortion.

### Handling NoData Values

Grid cells without valid elevation data require special consideration in slope algorithms. Common approaches ignore NoData cells or assign zero slope, but more advanced methods interpolate across small gaps to reduce artifacts.

## Recommendations for Best Practices

Researchers can optimize slope mapping from global DEMs by:

### Assessing DEM Quality

Newer releases generally, but not always, improve over previous versions. Understanding source data limitations and reported accuracy guides appropriate expectations and parameters.

### Addressing Artifacts and Errors

Inspect outputs and filtering noisy surfaces. Leverage slope thresholds to remove unrealistic extremes where useful. Compare directional derivatives and address divergence.

### Selecting Appropriate Methods

Consider tradeoffs between computational efficiency and accuracy needs. account for latency with moving window techniques. Validate outputs against high quality local LiDAR where feasible.

### Reporting Uncertainties

Propagate error estimates into slope uncertainty surfaces to support analytical needs and communicate data quality limitations.