A Monte Carlo calculation of the atmospheric neutrino fluxes [Barr et al., Phys. Rev. D 39, 3532 (1989); Agrawal et al., ibid. 53, 1314 (1996)] has been extended to take account of the three-dimensional (3D) nature of the problem, including the bending of secondary particles in the geomagnetic field. Emphasis has been placed on minimizing the approximations when introducing the 3D considerations. In this paper, we describe the techniques used and quantify the effects of the small approximations which remain. We compare 3D and 1D calculations using the same physics input in order to evaluate the conditions under which the 3D calculation is required and when the considerably simpler 1D calculation is adequate. We find that the 1D and 3D results are essentially identical for $E_{\nu }>5\mathrm{}\mathrm{GeV}$ except for small effects in the azimuthal distributions due to bending of the secondary muon by the geomagnetic field during their propagation in the atmosphere.

• Received 30 March 2004

DOI:https://doi.org/10.1103/PhysRevD.70.023006