  The Sun is the most studied of all stars. It is a reference for all other
observed stars and a laboratory of fundamental physics helping us understand
processes occuring in conditions irreproducible on Earth. However, our
understanding of the Sun is currently stained by the solar modelling problem
which can stem from various causes, such as the opacities, the equation of
state and the mixing of chemical elements. We combine inversions of sound
speed, an entropy proxy and the Ledoux discriminant with constraints such as
the position of the base of the convective zone and the photospheric helium
abundance. We test various combinations of standard ingredients for solar
modelling such as abundance tables, equation of state, formalism for convection
and diffusion and opacity tables and study the diagnostic potential of the
inversions to constrain ad-hoc modifications of the opacity profile and
additional mixing below the convective envelope. Combining inversions provides
stringent constraints on the modifications on the models, far beyond what is
achievable only from sound speed inversions. We constrain the form and
amplitude of the opacity increase required and show that a 15% increase at log
T=6.35 provides a significant improvement but that a more global increase of
the opacity, within the uncertainties of current tables, coupled with an
additional mixing at the bottom of the convective zone gives the best agreement
for low metallicity models. We show that high metallicity models do not satisfy
all the inversion results. We conclude that the solar problem likely occurs
from various small sources, as ingredients such as the equation of state or the
formalism of convection can induce small but significant changes and that using
phase shift analyses combined with our approach is the next step for a better
understanding of the inaccuracies of solar models just below the convective
envelope.
