  Previously, we modelled the X-ray spectra of the narrow-line Seyfert 1 galaxy
IRAS 13224$-$3809 using a disc reflection model with a fixed electron density
of $10^{15}$ cm$^{-3}$. An additional blackbody component was required to fit
the soft X-ray excess below 2 keV. In this work, we analyse simultaneously five
flux-resolved XMM-Newton spectra of this source comprising data collected over
2 Ms. A disc reflection model with an electron density of $n_{\rm
e}\approx10^{20}$ cm$^{-3}$ and an iron abundance of $Z_{\rm
Fe}=3.2\pm0.5Z_{\odot}$ is used to fit the broad-band spectra of this source.
No additional component is required to fit the soft excess. Our best-fit model
provides consistent measurements of black hole spin and disc inclination angle
as in previous models where a low disc density was assumed. In the end, we
calculate the average illumination distance between the corona and the
reflection region in the disc of IRAS 13224$-$3809 based on best-fit density
and ionisation parameters, which changes from 0.43$\sqrt{f_{\rm AD}/f_{\rm
INF}}$ $r_{\rm g}$ in the lowest flux state to 1.71$\sqrt{f_{\rm AD}/f_{\rm
INF}}$ $r_{\rm g}$ in the highest flux state assuming a black hole mass of
$2\times10^{6}M_{\odot}$. $f_{\rm AD}/f_{\rm INF}$ is the ratio between the
flux of the coronal emission that reaches the accretion disc and infinity. This
ratio depends on the geometry of the coronal region in IRAS 13224$-$3809. So we
only discuss its value based on the simple `lamp-post' model, although detailed
modelling of the disc emissivity profile of IRAS 13224$-$3809 is required in
future to reveal the exact geometry of the corona.
