When conducting analyses with National Elevation Dataset DEMs, projecting the raster correctly is key, otherwise you could end up with slopes greater than 1,000,000% or slope/aspect/hillshade outputs that look like this:
Using the "Project Raster" tool, the first important piece is to output the raster to a projected coordinate system. NED DEMs come from The National Map with a geographic coordinate system, and the linear unit is undefined. If you do an analysis on a raster that requires the use of a z-factor, outputs can look molten or overdone and the resulting cell values can be extreme. Projecting to a projected coordinate system defines a linear unit, avoiding this problem. See this blog post from ESRI for more information.
The next important piece for projecting the DEM is to use a Bilinear interpolation for the "Resampling Technique". ArcMap defaults to a nearest neighbor resampling, but if you do this and subsequently do a slope analysis on this raster, you often see strange grid lines show up in your output (as in the photo above). Selecting BILINEAR instead of NEAREST should avoid the grid-like artifacts that stem from resampling patterns.
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