KELVIN NOTATION FOR STABILIZING ELASTIC-CONSTANT INVERSION

Inverting a set of core-sample traveltime measurements for a complete set of 21 elastic constants is a difficult problem. If the 21 elastic constants are directly used as the inversion parameters, a few bad mea surements or an unfortunate starting guess may result in the inversion converging to a physically impossible ''solution''. Even given perfec t data, multiple solutions may exist that predict the observed travelt imes equally well. We desire the inversion algorithm to converge not j ust to a physically possible solution, but to the ''best'' (i.e. most physically likely) solution of all those allowed. We present a new par ameterization that attempts to solve these difficulties. The search sp ace is limited to physically realizable media by making use of the Kel vin eigenstiffness-eigentensor representation of the 6x6 elastic stiff ness matrix. Instead of 21 stiffnesses, there are 6 ''eigenstiffness p arameters'' and 15 ''rotational parameters''. The rotational parameter s are defined using a Lie-algebra representation that avoids the artif icial degeneracies and coordinate-system bias that can occur with stan dard polar representations. For any choice of these 21 real parameters , the corresponding stiffness matrix is guaranteed to be physically re alizable. Furthermore, all physically realizable matrices can be repre sented in this way. This new parameterization still leaves considerabl e latitude as to which linear combinations of the Kelvin parameters to use, and how they should be ordered. We demonstrate that by careful c hoice and ordering of the parameters, the inversion can be ''relaxed' from higher to lower symmetry simply by adding a few more parameters a t a time. By starting from isotropy and relaxing to the general result in stages (isotropy, transverse isotropy, orthorhombic, general), we expect that the method should find the solution that is closest to iso tropy of all those that fit the data.