Spectra of late-type stars are usually analyzed with static model atmospheres in local thermodynamic equilibrium (LTE) and a homogeneous plane-parallel or spherically symmetric geometry. The energy balance requires particular attention, as two elements that are particularly difficult to model play an important role: line blanketing and convection. Inversion techniques are able to bypass the difficulties of a detailed description of the energy balance. Assuming that the atmosphere is in hydrostatic equilibrium and LTE, it is possible to constrain its structure from spectroscopic observations. Among the most serious approximations still implicit in the method is a static and homogeneous geometry. In this paper, we take advantage of a realistic three-dimensional radiative hydrodynamical simulation of the solar surface to check the systematic errors incurred by an inversion assuming a plane-parallel horizontally-homogeneous atmosphere. The thermal structure recovered resembles the spatial and time average of the three-dimensional atmosphere. Furthermore, the abundances retrieved are typically within 10% (0.04 dex) of the abundances used to construct the simulation. The application to a fairly complete data set from the solar spectrum provides further confidence in previous analyses of the solar composition. There is only a narrow range of one-dimensional thermal structures able to fit the absorption lines in the spectrum of the Sun. With our carefully selected data set, random errors are about a factor of 2 smaller than systematic errors. A small number of strong metal lines can provide very reliable results. We foresee no major difficulties in applying the technique to other similar stars, and obtaining similar accuracies, using spectra with λ/δλ~5×104 and a signal-to-noise ratio as low as 30.