  The ramp up signals of gravitational waves appearing in the numerical
simulations could be important signals to estimate parameters of the
protoneutron star (PNS) at supernova explosions. To identify the signals with
PNS oscillations, we make a linear perturbation analysis and compare the
resultant eigenfrequencies with the ramp up signals obtained via the
two-dimensional numerical simulations. Then, we find that the ramp up signals
correspond well to the $g_1$-mode in the early phase and to the $f$-mode, to
which the $g_1$-mode is exchanged via the avoided crossing. We also confirm
that the $f$- and $g_1$-modes are almost independent of the selection of the
PNS surface density in the later phase after core bounce. In addition, we
successfully find that the fitting formula of $g_1$- and $f$-modes, which
correspond to the ramp up signals in the numerical simulation, as a function of
the PNS average density. That is, via the direct observation of the
gravitational waves after supernova explosion, one could extract the time
evolution of the PNS average density by using our fitting formula.
