In the analysis of astrophysical data assumed to lie on the celestial sphere, new directional data techniques are necessary when the monitoring station is on the earth's surface. Weighted spherical random variables are introduced and investigated for the cases when the true underlying distributions are either uniform, or of the Fisher or Dimroth-Watson forms. The uniform distribution on a rotating cap is examined and numerical examples are given with data for the arrival directions of ultra-high energy cosmic rays.