Solving Underdetermined Tracer Inverse Problems by Spatial Smoothing and Cross Validation

Abstract

Tracer conservation equations may be inverted to determine the flow field and macroscopic diffusion coefficients from known tracer distributions. An underdetermined system leads to an infinite number of possible solutions. The solution that is selected is the one that is as smooth as possible while still reproducing the tracer observations. The procedure suggested here is to define a penalty function that balances solution smoothnness, based on spatial derivatives of the solution, against residuals in the conservation equations. The ratio of detail in the solution to equation error is controlled by one or more smoothing parameters, which will not usually be known prior to the inversion. A parameter estimation technique known as generalized cross-validation is used to determine the degree of smoothing based on optimizing the prediction of withheld information. The method is tested for the case of steady flow containing a range of spatial scales in a two-dimensional channel with a spatially varying diffusion coefficient. It is shown that the correct flow field and diffusivity may be reproduced relatively accurately from a knowledge of the distribution of two tracers for a variety of flow configurations. The impact on the solution of errors in the equations and errors in the tracer data is studied. It is found that relatively large (correlated) errors in the equations due to numerical truncation error have the same effect as relatively small random errors in the data. A useful qualitative diagnostic measure of the value 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.