The time integration of 1D thermal conduction equation of ground
surface (A.55)
in appendix A.e is performed by
the CrankNicolson scheme.
The space differencing in (A.55) is evaluated by the second
order centered scheme.
The ground temperature and vertical grid interval are evaluated on the
grid point and the heat flux is evaluated on the half grid point.
The number of vertical grid point is and the suffix
varies from the lowest grid point.
The is assumed to the surface temperature
Where .
When the terms at are moved to left hand side
and the terms at are moved to right hand side, then
Where .
When , this equation can be represented in
matrix form as follows.

(B.61) 
The matrix are J'th order square matrix
and these elements are
Considering the upper boundary condition
(A.56) and
insurate lower boundary,
(B.61) is modified as follows.

(B.62) 
Therefore,

(B.63) 
where the first and th diagonal element
of are represented as follows.
is a column vector whose dimension is
and which elements are represented as follows.
