  During 2016-17 the Juno and Cassini spacecraft will both perform close
eccentric orbits of Jupiter and Saturn, respectively, obtaining high-precision
gravity measurements for these planets. This data will be used to estimate the
depth of the observed surface flows on these planets. All models to date,
relating the winds to the gravity field, have been in the forward direction,
thus allowing only calculation of the gravity field from given wind models.
However, there is a need to do the inverse problem since the new observations
will be of the gravity field. Here, an inverse dynamical model, is developed to
relate the expected measurable gravity field, to perturbations of the density
and wind fields, and therefore to the observed cloud-level winds. In order to
invert the gravity field into the 3D circulation, an adjoint model is
constructed for the dynamical model, thus allowing backward integration. This
tool is used for examination of various scenarios, simulating cases in which
the depth of the wind depends on latitude. We show that it is possible to use
the gravity measurements to derive the depth of the winds, both on Jupiter and
Saturn, taking into account also measurement errors. Calculating the solution
uncertainties, we show that the wind depth can be determined more precisely in
the low-to-midlatitudes. In addition, the gravitational moments are found to be
particularly sensitive to flows at the equatorial intermediate depths,
therefore we expect that if deep winds exist on these planets they will have a
measurable signature by Juno and Cassini.
