Okay I've got a quad, see picture. And want to find the height at any point in the quad, is there a way to do this accurately as bilinear interpolation doesn't work in this case.

As you can see the trees trunk is well beneath the terrain and should be at the same height as the trunks around it. I know the altitude of the vertices marked by a red circle and the centre red circle is roughly where the base should be, I've also crudely added a line as that surface is not flat.