  Inverting an evolving diffusive scalar field to reconstruct the underlying
velocity field is an underdetermined problem. Here we show, however, that for
two-dimensional incompressible flows, this inverse problem can still be
uniquely solved if high-resolution tracer measurements, as well as velocity
measurements along a curve transverse to the instantaneous scalar contours, are
available. Such measurements enable solving a system of partial differential
equations for the velocity components by the method of characteristics. If the
value of the scalar diffusivity is known, then knowledge of just one velocity
component along a transverse initial curve is sufficient. These conclusions
extend to the shallow-water equations and to flows with spatially dependent
diffusivity. We illustrate our results on velocity reconstruction from tracer
fields for planar Navier- Stokes flows and for a barotropic ocean circulation
model. We also discuss the use of the proposed velocity reconstruction in
oceanographic applications to extend localised velocity measurements to larger
spatial domains with the help of remotely sensed scalar fields.
