  In this work we explore accurate new ways to derive the ionization parameter
($U$) and the equivalent effective temperature ($T_*$) in HII regions using
emission-line intensities from the ionized gas. The so-called softness
parameter ($\eta$), based on [OII], [OIII], [SII], and [SIII] has been proposed
to estimate the hardening of the ionizing incident field of radiation, but the
simplest relation of this parameter with $T_*$ also depends on $U$ and
metallicity ($Z$). Here we provide a Bayesian-like code (HCm-Teff) that
compares the observed emission lines of $\eta$ with the predictions of a large
grid of photoionization models giving precise estimations of both $U$ and $T_*$
when $Z$ is known. We also study the radial variation of these parameters in
well-studied disc galaxies observed by the CHAOS collaboration. Our results
indicate that the observed radial decreasing of $\eta$ can be attributed to a
radial hardening of $T_*$, across galactic discs as in NGC~628 and NGC~5457. On
the other hand NGC~5194, which presents a positive slope of the fitting of the
softness parameter, has a flat slope in $T_*$. On the contrary the three
galaxies do not seem to present large radial variations of the ionization
parameter. When we inspect a larger sample of galaxies we observe steeper
radial variations of $T_*$ in less bright and later-type galaxies, mimicking a
similar trend observed for $Z$ but the studied sample should be enlarged to
obtain more statistically significant conclusions.
