Appendix C: FINITE DIFFERENCE FORM OF THE THIRD TYPE BOUNDARY CONDITIONAs shown in equations (3.15) and (3.16) in the Section 3.2.1, the third type boundary condition at the top of the soil column is expressed as:(C.1) (C.2) If we assume D is constant over the top pixels involved here, the finite difference form of the first term of the left hand of the above equations can be written (omitting the subscript w) as: (C.3) _{} If we assume q_{w} is constant over top two pixels involved here, the second term of the equations (C.1) and (C.2) becomes (C.4) _{ } By substituting the equations (C.3) and (C.4) into (C.1) and (C.2), (C.5) _{} where C_{0} becomes zero when t>t_{0}. When i = 1 (i.e., at the first cell), the equation (C.5) can be written as: (C.6) _{} Note that the C_{0}^{k+1} and C_{0}^{k} terms represent outside the column boundary. The same terms also can be found in the governing equation (3.36) of Section 3.3.1 by applying i = 1. When i = 1, the equation (3.36) becomes as follows. (C.7) _{} Rewriting equation (C.7) and rearranging the C^{k+1} terms into the right hand side and C^{k} terms into the left side of equation, the following can be derived. (C.8) where
Because we assumed D and q_{w} are constant, M_{1}^{'} and N_{1}^{'} become 0. Then
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