The FCC uses its "HAAT" calculation to figure out how high an antenna should be above the surrounding terrain. http://transition.fcc.gov/mb/audio/bickel/haat_calculator.html
From a mathematical perspective, HAAT appears to be trying to compute the average elevation for circles at 2,4,6,8,10 mile radii from the antenna.
Each of these 5 circles are sampled at 8 or more equal-spaced angles -- e.g., (2*pi*i)/8, i=0..7. The 8 (at least) points from each of these 5 circles are averaged together with equal weights. Suppose you wanted your antenna to cover the maximum _area_ with the _minimum HAAT_. How do you choose the position of your (omnidirectional) antenna? Although the sample points at 2,4,6,8 miles weigh as much in the formula as those at 10 miles, there is more area a 10 miles (and beyond). So, ideally, you position your antenna on a plateau 8 miles in diameter. Alternatively, you position your antenna in the middle of a parabolic peak so that the contribution of the more distant samples scales with the additional area. It isn't clear who gets to choose where the sample points are located: you or the FCC. For example, the terrain could be "crumpled" in radial directions in such a way that the answer could vary widely depending upon the angle of the original baseline relative to the equator. So perhaps the FCC might interpret this as the infimum or supremum over all choices of original baseline. Similarly, the terrain could have thin "walls" at exactly 2,4,6,8,10 miles from the antenna, in which case the average terrain would be lower than the tops of these walls. Of course, in reality, these wall would also create _valleys_ behind them, which would obscure the sight of the antenna. So, taking account of both the FCC and line-of-sight wave propagation, what would be the optimum location? An antenna at the top of a gradually-sloping _conical_ hill would provide the most area covered by line-of-sight of the antenna; the steeper the conical hill, the better (to maximize area and minimize the HAAT number).