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\mbox{if} \, z < z_f  \Rightarrow T_b(t,z) = T_f + (T_G(z) - T_f) \cdot \fracmbox{2erf}{\sqrt{\pi}} \int_0^{( \left( \frac{z_f -z)/}{\sqrt{4at}} e^{-\xi^2} d\xi right)

\mbox{if} \, z_f \leq z \leq z_f + h_f  \Rightarrow T_b(t,z) = T_f

\mbox{if} \, z > z_f + h_f  \Rightarrow T_b(t,z) = T_f + (T_G(z) - T_f) \cdot \fracmbox{2erf}{\sqrt{\pi}} \int_0^{( \left( \frac{z-z_f-h_f)/}{\sqrt{4at}} e^{-\xi^2} d\xi \right)