Plate geometry and kinematics generally reflect the mechanical properties of the solid lithosphere rather than those of the fluid mantle underneath, and plate formation and subduction account for most of the heat transport from the Earth's interior. Correspondingly, mantle convection models must incorporate a stiff but mobile boundary layer, like the lithosphere, before they can reproduce the main features of mantle convection. A relatively easy way to accomplish this in numerical models is to combine a temperature-dependent viscosity with an imposed, piecewise constant surface velocity boundary condition. It is shown how surface heat flux, topography, gravity, and geoid (but not plate velocities or stresses) can then be derived. Numerical models confirm that a lithosphere has a first-order effect on the underlying flow structure. For internally heated models, approx-imating the case of whole mantle convection, the calculated surface signatures are in first-order agreement with observations, a level of empirical success which hitherto has not been approached by models of mantle convection. Companion papers exploit the observations more fully to constrain the main features of mantle convection.