## Sunday, May 14, 2023

### Determining the cell size of a Digital Elevation Model from point density

We regulary refer to high resolution topography (HRT) as having Digital Elevation Model (DEM) pixel sizes with their edges less than 1 m. Many raster DEMs derived from airborne laser swath mapping available for example from OpenTopography are delivered at 1m/pix. In many cases, the point density might be high enough to support even higher resolution or smaller pixels. This seems like a fairly simple question: what is the recommended cell size of a DEM for a given point density?

The Langridge, et al., 2013 paper cites Hu, 2003 and suggests the following:

where S is the estimated cell size (typically in m), n is the number of sample points and A is the area containing the sample points.

Here is a little spreadsheet to solve this equation given A and n: link.

In the trivial case, that equation can be rearranged and show that 1 point per sq. m is consistent with a 1 m DEM. That seems ok in the sense that then on average there is one point for each pixel which would be the logic of this equation. It assumes then that the points are well distributed and that whatever average for point density that we might use to estimate a cell size represents the data well. A complication is the method of DEM computation: local function or linear interpolation. An example of the local implementation is the Points2Grid tool (see prototype here and here). Tinning or the use of triangular irregular networks linearly interpolates between (selected) points to estimate DEM pixel elevations (see blast2dem for example).

Here are some illustrations from the Jemez River Basin dataset cited below. These data have a stated point density of 9.68 points/sq. m and were computed within the OpenTopography portal using the TIN approach.
Here is an example from a site called Sulphur Creek (1 m/pix for this Digital Surface Model, DSM):

Here is a zoom to that site with the 1m/pix DSM hillshade and 14% of the points displayed from ArcMap. We can see the 1 m pixels and that there are a decent number (about 9.7) of points for each one. This has all of the points (all classes): Here is the same view but with a 0.32 m/pix (recommended resolution from equation above), but applied to the Digital Terrain Model (DTM). Note this is not really correct resolution estimate because the number of points classified as ground is only about 1/5 of the total. And, this shows the ground returns only. Zooming in even more, we can see the big triangular facets where the TINNING algorithm spanned the data gaps in the ground classified points for this 0.32 m/pix DTM: Finally, zooming back out with the 1 m/pix DTM displayed, we can see that at this scale, the DTM landscape is well represented in general. We do need the interpolation by tinning across the gaps in the ground points:

References Hu, Y., 2003. Automated Extraction of Digital Terrain Models, Roads and Buildings Using Airborne LiDAR Data (PhD thesis). Department of Geomatics Engineering, The University of Calgary, Calgary, Alberta, Canada.

Jemez River Basin Snow-off LiDAR Survey. Distributed by OpenTopography. https://doi.org/10.5069/G9RB72JV . Accessed: 2023-05-14

Langridge, R.M., et al., Developing sub 5-m LiDAR DEMs for forested sections of the Alpine and Hope faults, South Island, New Zealand: Implications for structural interpretations, Journal of Structural Geology (2013), http://dx.doi.org/10.1016/j.jsg.2013.11.007