 Crustal mean temperature from conductive heat flow

For a crustal layer of thickness $h_c$ with constant thermal conductivity $k$, no internal heating and a linear conductive profile, the upward heat flux magnitude $q$ obeys Fourier's law, $T(z)=T_s+qz/k$ with depth positive downward. Depth averaging givesThe basal excess temperature is $q{h}_c/k$, twice the mean excess. The formula requires a thickness assumption and a conductive interpretation of the measured flux. Hydrothermal advection, sedimentary thermal effects, internal production and nonlinear conductivity can invalidate a linear extrapolation.