Hello,
as initial condition you can provide the initial Free Surface Elevation or the initial Water Depth. Sorry, but not quite understanding the meanings of the your equations and the "total depth".
If you provide the initial Free Surface elevation, Telemac-2D will compute in the first step / initialize the Water Depth = initial Free Surface Elevation - Bottom Elevation, since Telemac-2D solves for the unknown velocities U and V and water depth H.
Best regards,
Clemens