  We study the surface Andreev bound state (ABS) of superconducting
Sr$_2$RuO$_4$, which is a candidate material for the realization of the chiral
$p$-wave superconducting state. In order to clarify the role of chiral edge
modes as ABSs, the surface density of states and the tunneling conductance is
calculated in the normal metal/Sr$_2$RuO$_4$ junction within the framework of
recursive Green's function method, while taking into account the orbital
degrees of freedom (including Spin-Orbit interactions) with realistic material
parameters. In Sr$_2$RuO$_4$, there are two bands $\alpha$ and $\beta$
originating from quasi-one-dimensional orbitals $d_{yz}$ and $d_{zx}$ and a
two-dimensional band $\gamma$ originating from $d_{xy}$ orbital. We discuss
about the contributions of various electronic bands to LDOS and the influence
of atomic spin-orbit interaction (SOI). In the light of our calculations,
quasi-one-dimensional model with dominant pair potentials in $\alpha$ and
$\beta$ bands is consistent with conductance measurements in
Au/Sr$_{2}$RuO$_{4}$ junctions.
