The potential of combining MATISSE and ALMA observations: Constraining
  the structure of the innermost region in protoplanetary discs
  In order to study the initial conditions of planet formation, it is crucial
to obtain spatially resolved multi-wavelength observations of the innermost
region of protoplanetary discs. We evaluate the advantage of combining
observations with MATISSE/VLTI and ALMA to constrain the radial and vertical
structure of the dust in the innermost region of circumstellar discs in nearby
star-forming regions. Based on a disc model with a parameterized dust density
distribution, we apply 3D radiative-transfer simulations to obtain ideal
intensity maps. These are used to derive the corresponding wavelength-dependent
visibilities we would obtain with MATISSE as well as ALMA maps simulated with
CASA. Within the considered parameter space, we find that constraining the dust
density structure in the innermost $5\,$au around the central star is
challenging with MATISSE alone, whereas ALMA observations with reasonable
integration times allow us to derive significant constraints on the disc
surface density. However, we find that the estimation of the different disc
parameters can be considerably improved by combining MATISSE and ALMA
observations. For example, combining a 30-minute ALMA observation (at 310 GHz
with an angular resolution of 0.03$^{\prime\prime}$) for MATISSE observations
in the L and M bands (with visibility accuracies of about $3\,\%$) allows the
radial density slope and the dust surface density profile to be constrained to
within $\Delta \alpha=0.3$ and $\Delta (\alpha-\beta)=0.15$, respectively. For
an accuracy of ${\sim 1\%}$ even the disc flaring can be constrained to within
$\Delta \beta=0.1$. To constrain the scale height to within $5\,$au, M band
accuracies of $0.8\,\%$ are required. While ALMA is sensitive to the number of
large dust grains settled to the disc midplane we find that the impact of the
surface density distribution of the large grains on the observed quantities is
small.
