The annual variation of daily average soil temperature at different depths is described with the following sinusoidal function ( Hillel, 1982):

where T(z,t) is the soil temperature at time t (d) and depth z (m), T_a, is the average soil temperature (^oC), A₀, is the annual amplitude of the surface soil temperature (^oC), d is the damping depth (m) of annual fluctuation (d = (2D_h/w )^1/2, where D_h is the thermal diffusivity and w = 2p /365 d^-1), and t₀ is the time lag from an arbitrary starting date (taken as January 1 in this software) to the occurrence of the minimum temperature in a year (d).

The sinusoidal temperature model was derived by solving the following partial differential equation ( Hillel, 1982; Marshall and Holmes, 1988):

where T(z,t) is the soil temperature at time t and depth z and D_h is the thermal diffusivity.

The following assumptions are employed in the derivation of the temperature model:

1. A sinusoidal temperature variation at the soil surface z = 0. That is

   

   where T_a is the average soil temperature, A₀, is the amplitude of the annual temperature function, t₀, a time lag from an arbitrary starting date (selected as January 1 in this software) to the occurrence of the minimum temperature in a year.

2. At infinite depth, the soil temperature is constant and is equal to the average soil temperature.

3. The thermal diffusivity is constant throughout the soil profile and throughout the year.