My way of understanding/thinking is:
The function uses the smoothed texture coordinates for this, generated by the last compute terrain prior to the surface layer using the intersect underlying.
These smoothed texture coordinates allow for smoothing a surface layer using the smoothing setting in the last tab. The scale of this smoothing effect is determined by the patch size of the compute terrain before the surface layer.
Actually that's the only part I'm pretty sure about

I think the intersect underlying algorithm tries to look up how the texture coordinates vary over the terrain and measures where the displacements "flatten", being it either a "top" or "valley" of the displacement, by assessing the "flow" of the terrain.
The intersect underlying adds displacement to the parts which cause rises in the terrain or adds displacement on features which cause depressions in the terrain. Displacement intersection does both and smooths out the transition of displacement between rises and depression. The amount and scale of this smoothing is controlled with the smoothing setting.
I always try to imagine this like the terrain is shaped like a sinus and that the tops of the sinus-shape represent the "rises" and the bottoms the "depression". The intersect underlying can emphasize these features.
In case of displacement intersection the settings for shift and minimal shift determine how much displacement you add. A shift value for an overall thickness and a min shift for the minimal value.
Further I tink that the intersection zone determines how fast the intersection "shifts" and how fast the shift and min. shift vary over the terrain.
Smaller values (<0,05 or so) give smoother results whereas larger values more rougher.
Favor rises: adds displacement on raised features
Favor depressions: adds displacement on depressed features of the terrain, kind of "fills" depressions.
Displacement intersection: smoothens out the rises and depression completely when using a big shift value. Or creates an intersecting layer between rises and depressions. So in a sinus-landscape there would be a horizontal line going through the middle of the sinus-shape. You can control the height of the horizontal line using the shift values and the "horizontal-ity" with the intersection zone.
Ghehe, I always feel a bit emberassed when thinking out loud like this. It might be complete rubbish/non-sense

Anyway, with these simple thoughts I can make it work for me, a little though.
Yes the intersect underlying feature is still puzzling and this is not the first time people are wondering what it does and how.
Perhaps asking Matt again for an explanation might help. Tried that before, but I think he missed those.
So Matt, if you find some time could you enlighten us on a description of the feature, it's parameters with a bit of guidance how to use?
Thanks in advance.
Cheers,
Martin