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 measurements or an unfortunate starting guess may result in the inversion converging to a physically impossible "solution". Even given perfect data, multiple solutions may exist that predict the observed traveltimes equally well. We desire the inversion algorithm to converge not just to a physically possible solution, but to the "best" (i.e. most physically likely) solution of all those allowed. We present a new parameterization that attempts to solve these difficulties. The search space is limited to physically realizable media by making use of the Kelvin eigenstiffness-eigentensor representation of the 6x6 elastic stiffness matrix. Instead of 21 stiffnesses, there are 6 "eigenstiffness parameters" and 15 "rotational parameters". The rotational parameters are defined using a Lie-algebra representation that avoids the artificial degeneracies and coordinate-system bias that can occur with standard polar representations. For any choice of these 21 real parameters, the corresponding stiffness matrix is guaranteed to be physically realizable. Furthermore, all physically realizable matrices can be represented in this way. This new parameterization still leaves considerable latitude as to which linear combinations of the Kelvin parameters to use, and how they should be ordered. We demonstrate that by careful choice and ordering of the parameters, the inversion can be "relaxed' from higher to lower symmetry simply by adding a few more parameters at 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 isotropy of all those that fit the data.