Remove tripod shadows
Many panographers like to present a pure spherical image, with no sign of the photographer or his/her apparatus. It is widely known how to use an additional "nadir shot" to replace the image of a tripod, but somehow it seems more difficult to eliminate the shadows of the tripod, pole, and photographer from a pano taken at a strongly lighted spot. However, this can be done by applying the same principle: combine several shots in which the shadows are in different places.
If all images are taken from a fixed lens position, the shadow of the camera itself must be at the same place in all of them. But we can move the shadow of the support, simply by moving the support. This is easiest if the camera is mounted on a telescoping monopod or pole, but can be done with a conventional tripod as well. And when the camera shadow is sufficiently far away, it is possible to get a completely clean 'patch shot' by shifting the camera slightly to one side of the panocenter.
As in all other aspects of photography, the key to success is visualizing problems before shooting, and planning ways around them. First, you must be aware of the direction of the light that is making the shadows. You will need to move things at right angles to that direction in order to to shift the shadows. Second, choose a spot where the shadow of the camera itself is either far away or in a dark place, perhaps inside some natural shadow that will be part of the picture. Make sure the chosen spot gives you enough room to maneuver yourself and your equipment. If using a tripod, try to place it so the shadows of two legs coincide.
When taking the fixed shots from panocenter, take care that your own shadow is not in any of them; or when that is not possible -- for example when shooting from a hand-supported pole or using a cable release -- take each shot in which your shadow appears twice, once standing to the left of the camera and once standing to the right.
To take your 'shadow patch' shots, slant the support off to the side (at right angles to the light) while keeping the lens as nearly as possible at panocenter and pointed at the shadow area. This is easiest with a pole, especially one that rests on the ground. In that case you can get a pretty stable shooting position by extending the pole enough to compensate for the slant. for example:
slant 15 degrees: no compensation needed; slant 30 degrees: 1/6 of the distance you move the foot; slant 45 degrees: 2/5 of the distance you move the foot.
If using a tripod, extend its center column as much as possible for the fixed shooting. Then fold the legs and use it like a pole. You will need more slant than with a pole, to let the camera see past the tripod's shoulders. It may be feasible to extend the legs, but I usually just hold it up in the air.
Before moving the support, sight on a couple of reference points to help you judge lens position, and tighten the lock screw on your rotator if using one.
Sometimes the same shot can serve as both nadir patch and shadow patch, but it is best to take them separately, optimizing each for its particular function. In fact it is a good plan to take several shadow patches, for two reasons. It is much easier to align a small patch well than a big one. But more important, in a single shot you can easily have part of your shadow where part of the tripod shadow was -- it isn't there anymore, so it's hard to tell. If you take two, standing on opposite sides of the shadow area, with the pole slanted in opposite directions, it is much more likely you will cover everything.
When stitching, handle your shadow patches like a nadir patch. Enable viewpoint optimization. Make sure all control points are near the shadow region, and more or less on one plane. That does not have to be a horizontal plane, or even a flat surface, in fact just picking points roughly equidistant from the camera also works well. Mask out all shadows in the fixed images, mask the patch images completely, then erase the masks just where needed to fill the gaps.
In difficult cases you can try advanced patching tricks such as using the same patch twice with different control points and mask. But if you plan and execute your shooting carefully that should not be necessary.