Fixing nadir parallax errors
Here is a set of source photos for a spherical panorama:
They were taken using a one second exposure without a tripod, the camera was stabilised by jamming it against the sides of a palm tree. This means that they don't stitch easily because of the large (approximately 60cm) parallax error:
However the nadir can be stitched by taking advantage of another technique generally used for stitching a linear panorama. The way this works is that if you assume the ground forms a plane just like a mural on a wall; a series of rectilinear Projection images taken from different locations can be assembled by optimising roll, pitch & yaw, Field of View and d & e lens correction parameters separately for each image.
So the first step is to create a series of defished rectilinear images from the fisheye Projection originals. You can do this in any of the GUI front-ends, but you do need to calibrate your lens first:
These can then be stitched together into a single nadir image with minimal errors, in this case I chose a fisheye projection, though a rectilinear or equirectangular projection would work just as well:
This image can then be reinserted into the original panorama project as another source photo with a different lens and stitched as normal. Here is the result as a QTVR, the buildings in the distance are a bit wobbly, but this could have been corrected with vertical control points: