Using transient tracers: the regularization problem

The generic problem of deducing fluid properties, velocity, mixing, and boundary conditions from realistic tracer observations is formulated and attacked. The focus is on oceanic transient tracers, but the same problem elements appear in many other fields including meteorology,medicine and chemical engineering. It becomes clear that use of tracers, transient or otherwise, requires at some stage the solution of the advection/diffusion equation either upstream, or backwards in time, or both, a situation notorious for its instability because diffusive processes progressively destroy information in the “forward” direction. We consider an example in simple finite difference form for one space dimension and time. The problem is run “backwards” but regularized by solving it through linear programming. Linear programming is shown also to have several potential advantages for stable and unstable forward problems, and the inverse problem: knowledge of the form of the answer is easily used; data uncertainty is accommodated; the positivity constraints on concentrations are implicit; sparse system software is available; questions of best observational strategy are addressed by the dualsolution; and the strong instabilities of the inverse and backwards problems are suppressed.