A seminumerical technique developed for solving the transport equation in slab geometry is adopted for a point isotropic source of gamma rays in spherical geometry. Only Compton scattering is dealt with currently. Two quantities, namely surface flux and surface source, are introduced to circumvent singularities at the origin. A collision-by-collision iterative approach is followed to solve the coupled form of integral transport equations separating the spatial and energy transmission kernels. The spatial transmission kernel for obtaining surface flux from surface source is derived. The energy angle transmission kernels are evaluated by taking recourse to Legendre polynomial expansions. The uncollided and first collision surface fluxes are obtained analytically. An appropriate functional form is chosen for the spatial interpolation of flux and source facilitating large spatial mesh widths. The computer program ASFIT-Sphere is written on the basis of these formulations. Energy flux spectra and angular distributions obtained by the current method of scattered photons 2 and 3 mean-free-paths away from a 137Cs source in water are compared with the data of ANS-6 shielding benchmark experiments. Comparison with calculations by other methods is also included.