Pc. Mcintosh et G. Veronis, SOLVING UNDERDETERMINED TRACER INVERSE PROBLEMS BY SPATIAL SMOOTHING AND CROSS VALIDATION, Journal of physical oceanography, 23(4), 1993, pp. 716-730
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.