  As they evolve, white dwarfs undergo major changes in surface composition, a
phenomenon known as spectral evolution. In particular, some stars enter the
cooling sequence with helium atmospheres (type DO) but eventually develop
hydrogen atmospheres (type DA), most likely through the upward diffusion of
residual hydrogen. Our empirical knowledge of this process remains scarce: the
fractions of white dwarfs that are born helium-rich and that experience the
DO-to-DA transformation are poorly constrained. We tackle this issue by
performing a detailed model-atmosphere investigation of 1806 hot ($T_{\rm eff}
\ge 30,000$ K) white dwarfs observed spectroscopically by the Sloan Digital Sky
Survey. We first introduce our new generations of model atmospheres and
theoretical cooling tracks, both appropriate for hot white dwarfs. We then
present our spectroscopic analysis, from which we determine the atmospheric and
stellar parameters of our sample objects. We find that $\sim$24% of white
dwarfs begin their degenerate life as DO stars, among which $\sim$2/3 later
become DA stars. We also infer that the DO-to-DA transition occurs at
substantially different temperatures ($75,000 \ {\rm K} > T_{\rm eff} > 30,000$
K) for different objects, implying a broad range of hydrogen content within the
DO population. Furthermore, we identify 127 hybrid white dwarfs, including 31
showing evidence of chemical stratification, and we discuss how these stars fit
in our understanding of the spectral evolution. Finally, we uncover significant
problems in the spectroscopic mass scale of very hot ($T_{\rm eff} > 60,000$ K)
white dwarfs.
