I'm trying to use Surface Volume tool in ArcMap to calculate 2D and 3D area of a catchment.
What would be the best projection to use and how do i check the maximum error of the estimation?
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The components of this question have been addressed in other threads:
For delineating the catchment itself as well as computing the slopes, use a conformal projection adapted to the study area. (New Zealand is small enough that such a conformal projection will introduce little error in the slopes--and even that can be corrected if you like.)
Reproject the results to any equal-area projection.
Then apply a slope correction to compute the "3D" surface area.
Unless you are using extremely accurate and precise data (such as LIDAR), the measurement errors in your dataset will be greater than inaccuracies introduced by these calculations.