SOLVING UNDERDETERMINED TRACER INVERSE PROBLEMS BY SPATIAL SMOOTHING AND CROSS VALIDATION

Tracer conservation equations may be inverted to determine the flow fi eld and macroscopic diffusion coefficients from known tracer distribut ions. An underdetermined system leads to an infinite number of possibl e solutions. The solution that is selected is the one that is as smoot h as possible while still reproducing the tracer observations. The pro cedure suggested here is to define a penalty function that balances so lution smoothness, based on spatial derivatives of the solution, again st residuals in the conservation equations. The ratio of detail in the solution to equation error is controlled by one or more smoothing par ameters, which will not usually be known prior to the inversion. A par ameter estimation technique known as generalized cross-validation is u sed to determine the degree of smoothing based on optimizing the predi ction of withheld information. The method is tested for the case of st eady flow containing a range of spatial scales in a two-dimensional ch annel with a spatially varying diffusion coefficient. It is shown that the correct flow field and diffusivity may be reproduced relatively a ccurately from a knowledge of the distribution of two tracers for a va riety of flow configurations. The impact on the solution of errors in the equations and errors in the tracer data is studied. It is found th at relatively large (correlated) errors in the equations due to numeri cal truncation error have the same effect as relatively small random e rrors in the data. A useful qualitative diagnostic measure of the valu e of an inverse solution is introduced. It is a measure of the loss of independent information due to smoothing the solution and is related to the data resolution matrix of classical discrete inverse theory.