## License Information

Use of this function requires a license for Whitebox Workflows for Python Professional (WbW-Pro). Please visit www.whiteboxgeo.com to purchase a license.

## Description

This tool calculates the hypsometric integral from the elevation distribution contained within the local neighbourhood surrounding each grid cell in an input (`input`) DEM. The hyspometric integral (HI) is the area under the hypsometric curve, which is a plot that relates elevation and area. This plot is a cumulative distribution function with elevation expressed as a proportion of maximum elevation and area expressed as the proportion of the area above. Hypsometry, or area-altitude analysis, is commonly used by geomorphologists and geologists to characterize the erosional history of drainage basins. The HI, ranging between 0 and 1, expresses the volume of land that lies above the lowest point within an area, and thus has not been eroded. Relatively low HI values are indicative of more strongly eroded surfaces.

Some researchers (e.g. Pérez‐Peña et al., 2009) have demonstrated the usefulness of applying hypsometry in a spatially distributed fashion, rather than aggregated by basins, as it is typically applied. While Pérez‐Peña et al. (2009) characterized spatial distributions of HI using coarse grids overlayed overtop a digital elevation model (DEM), this tool uses a filter-based approach instead. Each grid cell in the input DEM (`input`) has an individual HI calculation based on the elevation distribution within a moving kernel. HI values are calculated using the elevation-relief ratio method described by Pike and Wilson (1971).

In actuality, the tool uses a multi-scale approach, much like many of the other tools within the Geomorphometric Analysis toolbox (e.g. max_elevation_deviation, multiscale_std_dev_normals), such that the neighbourhood size is varied according to a range defined by user-specified input parameters. The HI that is reported within each grid cell in the output raster is the minimum HI value measured for each of the tested scales, defined by lower (rL) and upper (rU) ranges.

> HImin=min{HI(r):r=rL...rU},

In this way, it represents a heterogenous, locally scale optimized map of HI distributions. A nonlinear scale sampling interval is used by this tool to ensure that the scale sampling density is higher for short scale ranges, where there is often greater variability in HI values, and coarser at longer tested scales, such that:

> ri = rL + [step × (i - rL)]p

Where ri is the filter radius for step i and p is the nonlinear scaling factor (`step_nonlinearity`) and a step size (`step`) of step.

There are two outputs generated from this tool, including the HImin raster (`out_mag`) and the rmin scale raster (`out_scale`).

## References

Pérez‐Peña, J. V., Azañón, J. M., Booth‐Rea, G., Azor, A., and Delgado, J. (2009). Differentiating geology and tectonics using a spatial autocorrelation technique for the hypsometric integral. Journal of Geophysical Research: Earth Surface, 114(F2).

Pike, R. J., and Wilson, S. E. (1971). Elevation-relief ratio, hypsometric integral, and geomorphic area-altitude analysis. Geological Society of America Bulletin, 82(4), 1079-1084.

## See Also

hypsometric_analysis, max_elevation_deviation, multiscale_std_dev_normals

## Project Links

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