  (abridged) In the last decade, the advent of enormous galaxy surveys has
motivated the development of automated morphological classification schemes to
deal with large data volumes. Existing automated schemes can successfully
distinguish between early and late type galaxies and identify merger
candidates, but are inadequate for studying detailed morphologies of red
sequence galaxies. To fill this need, we present a new automated classification
scheme that focuses on making finer distinctions between early types roughly
corresponding to Hubble types E, S0, and Sa. We visually classify a sample of
984 non-starforming SDSS galaxies with apparent sizes >14". We then develop an
automated method to closely reproduce the visual classifications, which both
provides a check on the visual results and makes it possible to extend
morphological analysis to much larger samples. We visually classify the
galaxies into three bulge classes (BC) by the shape of the light profile in the
outer regions: discs have sharp edges and bulges do not, while some galaxies
are intermediate. We separately identify galaxies with features: spiral arms,
bars, clumps, rings, and dust. We find general agreement between BC and the
bulge fraction B/T measured by the galaxy modeling package GIM2D, but many
visual discs have B/T>0.5. Three additional automated parameters -- smoothness,
axis ratio, and concentration -- can identify many of these high-B/T discs to
yield automated classifications that agree ~70% with the visual classifications
(>90% within one BC). Both methods are used to study the bulge vs. disc
frequency as a function of four measures of galaxy 'size': luminosity, stellar
mass, velocity dispersion, and radius. All size indicators show a fall in disc
fraction and a rise in bulge fraction among larger galaxies.
