Determining flow fields and mixing rates from chemical tracer distributions is a challenging and important oceanographic problem. Thus the conclusion, that solutions obtained for underdetermined systems were “devoid of physical content”, drawn by Fiadeiro and Veronis after attempting to “invert” a simple tracer distribution in a known advective-diffusive field, is particularly disturbing. The problem they formulated is reexamined here. The procedures used differ from theirs in making use of the full machinery of inverse methods; even in the grossly underdetermined case, it is possible to (i) obtain useful information about the underlying flow field, (ii) to deduce the structure of the parts that are not determinable, (iii) to find formal error bars arising both from data noise and from indeterminate components, (iv) to make use of a priori statistical information and a posteriori tests, and, in general, (v) to extract much useful information about the field.