Can the number of relativistic solar proton 1 AU crossings be determined from neutron monitor data?

Energetic protons released during solar eruptive events experience scattering during their interplanetary propagation and may cross the spherical surface of radius 1 AU multiple times. Knowledge of N_cross, the average number of 1 AU crossings per particle, is therefore important for deducing the total number of protons in interplanetary space during solar energetic particle events, for example for comparison with the number of interacting protons at the Sun during gamma-ray flares. It has been proposed that for relativistic protons N_cross can be obtained by comparing the relative fluences measured in the sunward and anti-sunward directions by the worldwide network of neutron monitors during ground level enhancements (GLEs). For five recent GLE events, we use neutron monitor data to derive N_cross applying the latter approach and we compare the results with those of full-orbit test particle simulations of relativistic protons in a Parker spiral magnetic field, including the effects of scattering and drifts. We show that the approach based on neutron monitor data significantly underestimates N_cross during highly-anisotropic SEP events. This is due to the data sampling only a very small portion of the 1 AU sphere.