Computational Engine
Bivariate Taylor Polynomial Fitting and Multi-Scale Analysis
Rather than relying on discrete finite-difference approximations between adjacent grid cells, a continuous, differentiable mathematical surface is fitted across each local neighborhood via ordinary least squares. The partial derivatives of elevation are then analytically derived from this surface to compute precise local morphometric variables.
Minimum 5×5 Neighborhood — Moving Beyond 3×3
Deriving a 10-parameter polynomial surface model using only 9 grid nodes from a conventional 3×3 window creates an underdetermined, mathematically ill-posed system. By establishing a minimum 5×5 window, the calculation encompasses 25 spatial nodes. This provides 15 degrees of freedom, creating an overdetermined system that enables Ordinary Least Squares (OLS) optimization to robustly fit the surface and effectively suppress high-frequency topographic noise.
Numerical Stabilization in Extended Neighborhoods
Prior to constructing the design matrix, geodesic distances are normalized relative to the maximum neighborhood extent. This critical geometric scaling mitigates matrix ill-conditioning, ensuring robust inversion and numerical stability across extensive analytical windows.
Explicit Scale Definition — Beyond Spatial Filtering
The moving window dimension explicitly defines the physical scale of the terrain analysis. For instance, on a 30-meter SRTM DEM, a 5×5 neighborhood encompasses a 150×150 meter spatial extent; on a 1-meter LiDAR grid, the identical analytical window captures a highly localized 5×5 meter footprint.
Adaptive Boundary Reconstruction
At raster boundaries or adjacent to NoData regions, if a minimum of 15 valid grid nodes remain within the local neighborhood, the design matrix is dynamically reconfigured and inverted for that specific asymmetric spatial mask. This mathematically rigorous treatment minimizes edge artifacts and ensures maximum data retention.
Rather than acting as a conventional spatial filter, the window dimension explicitly dictates the physical scale of the morphometric analysis. To preserve geometric symmetry around the central grid node, its value is strictly constrained to odd integers, establishing a minimum 5×5 baseline to ensure mathematical stability.
Seamless ArcGIS Pro Integration
The Geomorphometry toolbox is distributed as a standard ArcGIS Pro toolbox (.atbx). It eliminates the need for complex standalone installations, external dependencies, or Python environment configurations.
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Plug & Play: Directly accessible from the Catalog pane. No complex setups required.
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Intuitive UI: Standard geoprocessing parameters including Input DEM, Window Size, and Target Attribute selection.
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Optimized Backend: The simple interface elegantly hides a powerful Python core that handles complex spheroidal matrix operations.
* Zero mandatory external dependencies. Core mathematical and topological operations run natively using the standard ArcGIS Pro Python environment (arcpy, numpy, scipy). numba and gdal are automatically engaged if present to deliver extreme multi-threaded performance and BigTIFF chunking.
Geomorphometry.atbx inside ArcGIS Pro Catalog and Geoprocessing Panes
Spheroidal Hydrology Engine
Rather than relying on an assumed planar constant, the true geodetic area of each grid cell is analytically computed as a spheroidal trapezoid. This engine integrates three fundamental flow-routing algorithms (D8, MFD, and D-Infinity) with a rigorous topological treatment of flat surfaces via the Garbrecht–Martz (1997) double-gradient method, ensuring mathematically exact hydrological simulations.
D8 steepest descent
The total flow accumulation of a grid cell is routed to the single downslope neighbor exhibiting the maximum gradient. This deterministic approach is highly optimal for extracting primary linear drainage networks and channelized flow.
MFD / FD8 Quinn 1991
Flow is distributed proportionally among all downslope neighbors based on the local gradient and a latitude-dependent effective contour width. Incorporating a convergence exponent of p=1.1 (Freeman 1991), this method realistically models dispersive flow across convex or divergent topographies.
D-Infinity Tarboton 1997
Flow direction is parameterized as a continuous angle across eight local triangular facets. The conventional algorithm is rigorously generalized to accommodate the irregular, latitude-dependent dimensions (dx≠dy) of spheroidal grid cells, providing the most mathematically robust approximation of specific catchment area.
