Obtaining Velocities from Tracer Distributions

Abstract

A simple advective-diffusive system with uniform, horizontal flow in a channel and fixed boundary concentrations of two tracers is analyzed for the tracer concentrations in the interior. The deduced concentrations are then treated as given information to invert the analysis to obtain the velocity field. The methodology of the inverse procedure is studied here in order to determine the types of information and the conditions under which the original velocity field can be recovered in limited portions of the channel. When the inverse problem is underdetermined, e.g., when only a single tracer distribution and no other information is given, the inversion leads to a flow that bears no resemblance to the known velocity. An overdetermined inverse problem with data that are free of error and noise recovers the original velocity. Also studied are systems with more realistic features in the given data, such as truncation error, noise, qualitatively correct (or incorrect) auxiliary velocity information and combinations of the above. When an underdetermined system is made overdetermined by the addition of qualitatively correct information, the results are much improved even if the latter information is given a low weight. When qualitatively incorrect information is given a low weight, it is rejected by the system. An empirical criterion based on the condition number of the matrix and the rms residual of an overdetermined system is offered as a tentative measure of how well the actual velocity field is approximated by the inverse solution.