This question builds off of the one with the subject line "Calculating Flow Direction and Delineating Basins from Projected vs. Unprojected Data.": Calculating Flow Direction and Delineating Basins from Projected vs. Unprojected DEM Data
This is an entirely separate question, however, as the aforementioned question has established that there are problems with using algorithms (e.g., ArcGIS Flow Direction) that assume Euclidean distance on data in a spherical/unprpojected geographic coordinate system.
We know that map projections are sort of like taking an orange peel and attempting to flatten it out on a desk - you will have some error inherently introduced by the map projection. But, it seems that the benefits of projecting offset any error introduced, particularly when you're running calculations that assume a Cartesian/projected planar surface. In this case, the algorithm I'm interested in is the ArcGIS Flow Direction algorithm which does assume that your data are projected (and this is the assumption taken by most applications based on my research) as it uses a Euclidean approach for calculating distance.
My question is: how could one quantify the error that might be introduced with calculating flow direction in a given study area using unprojected DEM data (DEM data in a geographic coordinate system) vs. projected data (DEM data in an appropriate projection such as a UTM or something conformal)?
Granted, you can derive a flow direction raster using unprojected and then the same DEM data projected. But what then? Since our goal is to model the earth's surface as accurately as we can (and we're not addressing any errors that may be introduced in the process of creating the original DEM etc. - those are a constant as far as I'm concerned)....do we just assume the flow direction data derived from the projected DEM is better, and then compare the individual cell values of the two rasters to identify which cells have different directional values (in the context of the normal D-8 model)? I guess to do this then you'd have to take the flow direction raster derived from unprojected data, and then apply the same projection used with the projected flow direction raster.
What would make the most sense, and what should the unprojected DEM be compared to as a benchmark of accuracy?
Getting into the nitty details of the mathematical equations might, to those that understand it, give you proof at the ground level and be enough for some, but that as well as something that could convey the error to someone that does not have an in-depth understanding of the math but may just know enough geography/GIS to be dangerous would be great (ideally both levels would be good which would resonate with the hardcore geography geeks and the average GIS dabbler). For the higher level folks, saying that the proof is in the math possibly leaves it somewhat open for argument - I'm looking for something more tangible (e.g., akin to attaching a dollar figure to some sort of inefficiency in government).
Any thoughts or ideas on how one might quantify this would be greatly appreciated.