Scintillometer measurements allow for estimations of the refractive index structure parameter $C_n^2$ over large areas in the atmospheric surface layer. Turbulent fluxes of heat and momentum are inferred through coupled sets of equations derived from the Monin--Obukhov similarity hypothesis. One-dimensional sensitivity functions have been produced that relate the sensitivity of heat fluxes to uncertainties in single values of beam height over flat terrain. However, real field sites include variable topography. We develop here, using functional derivatives, the first analysis of the sensitivity of scintillometer-derived sensible heat fluxes to uncertainties in spatially distributed topographic measurements. Sensitivity is shown to be concentrated in areas near the center of the beam path and where the underlying topography is closest to the beam height. Relative uncertainty contributions to the sensible heat flux from uncertainties in topography can reach 20\,\% of the heat flux in some cases. Uncertainty may be greatly reduced by focusing accurate topographic measurements in these specific areas. A~new two-dimensional variable terrain sensitivity function is developed for quantitative error analysis. This function is compared with the previous one-dimensional sensitivity function for the same measurement strategy over flat terrain. Additionally, a~new method of solution to the set of coupled equations is produced that eliminates computational error. ; Comment: 11 pages, 11 figures. Published in Atmospheric Measurement Techniques. http://www.atmos-meas-tech.net/7/2361/2014/amt-7-2361-2014.html