The minimum angular difference; an objective criterion for selection of the best option in the direct linear inversion method

The direct linear inversion is a rapid and robust method for paleostress estimation from fault-slip observations. The method uses 6- and 9-Dimensional spaces and requires judgement of the best option from the four possible solutions, 6D, 6Df, 9D and 9Df. We propose an objective criterion for selection of the best option. The criterion identifies the best solution as the tensor that gives the minimum average angular difference between the observed and theoretically predicted slip vectors. Success of the criterion is demonstrated in a variety of tectonic settings represented by 180 synthetic and 6 natural examples of fault-slip observations. Except for the one example of natural fault-slip observations, there is no vorticity component in any of the test examples. In both, the inclined-triaxial and the Andersonian stress states, a pair made of 6D/6Df and 9D/9Df, provides the correct solution. In the inclined-uniaxial stress states, only 6D/6Df is the correct solution and the use of 9-Dimensional space fails to invert the observations. In the natural example, that contains a vorticity component, only 9D gave the correct result. The best option, in all the tests, is invariably the tensor that yields the minimum angular difference between the observed and theoretically predicted slip vectors.