So, this approach requires one more layer of effort: converting the height map (or at least its visible tiles) into a "normal map" which is smooth and natural enough to be subjected to lighting effects and then displayed on screen.
The first filter that is applied to the height map is a simple Gaussian blur, which the GPU handles with ease, but which -- if left to its own devices -- would simply smooth out all the dramatic extremes in the height map and leave us with a bunch of bland rolling hills.