  We analyze the asymptotic stability of the quasi-linearly stratified
densities in the 2D inviscid incompressible porous medium equation on $\bbR^2$
with respect to the buoyancy frequency $N$. Our target density of
stratification is the sum of the large background linear profile with its slope
$N$ and the small perturbation that could be both non-linear and non-monotone.
Quantification in $N$ will be performed not only on how large the initial
density disturbance is allowed to be but also on how much the target densities
can deviate from the purely linear density stratification without losing their
stability.
  For the purely linear density stratification, our method robustly applies to
the three fundamental domains $\bbR^2,$ $\bbT^2,$ and $\bbT\times[-1,1]$,
improving both the previous result by Elgindi (On the asymptotic stability of
stationary solutions of the inviscid incompressible porous medium equation,
Archive for Rational Mechanics and Analysis, 225(2), 573-599, 2017) on $\bbR^2$
and $\bbT^2$, and the study by Castro-C\'ordoba-Lear (Global existence of
quasi-stratified solutions for the confined IPM equation. Archive for Rational
Mechanics and Analysis, 232(1), 437-471, 2019) on $\bbT\times[-1,1]$. The
obtained temporal decay rates to the stratified density on $\bbR^2$ and to the
newly found asymptotic density profiles on $\bbT^2$ and $\bbT\times[-1,1]$ are
all sharp, fully realizing the level of the linearized system. We require the
initial disturbance to be small in $H^m$ for any integer $m\geq 4$, which we
even relax to any positive number $m>3$ via a suitable anisotropic commutator
estimate.
