  We present a study of the compatibility of some current models of the diffuse
Galactic continuum gamma rays with EGRET data. A set of regions sampling the
whole sky is chosen to provide a comprehensive range of tests. The range of
EGRET data used is extended to 100 GeV. The models are computed with our
GALPROP cosmic-ray propagation and gamma-ray production code. We confirm that
the "conventional model" based on the locally observed electron and nucleon
spectra is inadequate, for all sky regions. A conventional model plus hard
sources in the inner Galaxy is also inadequate, since this cannot explain the
GeV excess away from the Galactic plane. Models with a hard electron injection
spectrum are inconsistent with the local spectrum even considering the expected
fluctuations; they are also inconsistent with the EGRET data above 10 GeV.
  We present a new model which fits the spectrum in all sky regions adequately.
Secondary antiproton data were used to fix the Galactic average proton
spectrum, while the electron spectrum is adjusted using the spectrum of diffuse
emission itself. The derived electron and proton spectra are compatible with
those measured locally considering fluctuations due to energy losses,
propagation, or possibly details of Galactic structure. This model requires a
much less dramatic variation in the electron spectrum than models with a hard
electron injection spectrum, and moreover it fits the gamma-ray spectrum better
and to the highest EGRET energies. It gives a good representation of the
latitude distribution of the gamma-ray emission from the plane to the poles,
and of the longitude distribution. We show that secondary positrons and
electrons make an essential contribution to Galactic diffuse gamma-ray
emission.
