  Observations of the solar surface reveal the presence of flows with length
scales of around $35$ Mm, commonly referred to as supergranules. Inferring the
sub-surface flow profile of supergranules from measurements of the surface and
photospheric wavefield is an important challenge faced by helioseismology.
Traditionally, the inverse problem has been approached by studying the linear
response of seismic waves in a horizontally translationally invariant
background to the presence of the supergranule; following an iterative approach
that does not depend on horizontal translational invariance might perform
better, since the misfit can be analyzed post iterations. In this work, we
construct synthetic observations using a reference supergranule, and invert for
the flow profile using surface measurements of travel-times of waves belonging
to modal ridges $f$ (surface-gravity) and $p_{1}$ through $p_{7}$ (acoustic).
We study the extent to which individual modes and their combinations contribute
to infer the flow. We show that this method of non-linear iterative inversion
tends to underestimate the flow velocities as well as inferring a shallower
flow profile, with significant deviations from the reference supergranule near
the surface. We carry out a similar analysis for a sound-speed perturbation and
find that analogous near-surface deviations persist, although the iterations
converge faster and more accurately. We conclude that a better approach to
inversion would be to expand the supergranule profile in an appropriate basis,
thereby reducing the number of parameters being inverted for and appropriately
regularizing them.
