Footprint diameter for a cosmic-ray soil moisture probe: Theory and Monte Carlo simulations

We used a combination of diffusion theory and neutron transport simulations to estimate the lateral footprint for a cosmic-ray soil moisture probe. The footprint is radial and can be described by an exponential function. Our theory assumes, and our simulations confirm that the corresponding exponential folding length is closely related to the moderation length in air, which in this work is defined as the average net displacement experienced by neutrons while traveling from the point of emission from soil to the point of detection in air. These simulations indicate that the effective moderation length is 150 m in dry air at sea level, and that this value is fairly constant over a wide range of detection energies - from 10⁰ to 10⁵ eV. If we define the lateral footprint as the area encompassing two e-fold distances, i.e., the area from which 86% of the recorded neutrons originate, then the footprint diameter is nearly 600 m in dry air. Both theory and simulations indicate that the footprint is inversely proportional to air density and linearly proportional to the height of the sensor above the ground for heights up to 125 m. Furthermore, our simulations indicate that the dependence on soil moisture is small, but the dependence on atmospheric humidity is significant, with a decrease in the footprint diameter of 40 m for every 0.01 kg kg^-1 increase in specific humidity. The good agreement between our theory and transport simulations suggests that the lateral footprint is determined mainly by the properties of air.