  Radially inhomogeneous gamma-ray burst (GRB) jets release variable
photospheric emission and can have internal shocks occurring above the
photosphere. We generically formulate a photospheric emission model of GRBs
including Compton up-scattered photospheric (UP) emission off the electrons
(and positrons) in the internal shocks, and find that the photospheric emission
may correspond to the traditional (Band) component at <~1 MeV and the UP
emission to the high-energy emission observed by Fermi/LAT for some GRBs at >~
10 MeV. The two components can be separate in the spectrum in some cases or can
mimic a smooth broad Band spectrum in other cases. We apply our formulation to
the well-studied long and short LAT GRBs, GRB 080916C, GRB 090902B, and GRB
090510, and typically find reasonable parameters for fitting the time-binned
spectra, although fine tuning of several parameters is required. The observed
delays of the high-energy emission with respect to the MeV emission which are
large compared to the variability times are unlikely to be due to simple
kinematic effects of a non-evolving jet. These delays may be attributed to the
temporal evolution of the physical parameters of the jet, and thus the delay
timescales could provide a potential tool for investigating the structures of
GRB jets themselves and their progenitors. The difference of the delay
timescales of long and short GRBs inferred from the Fermi data might be due to
the differences in the progenitors of long and short GRBs. Some other
properties and consequences of this model are discussed, including temporal
correlations among the prompt optical, the soft X-ray, and the distinct
high-energy component as well as the Band component.
