A technique has been developed to estimate the directions of principal stresses from focal mechanism solutions, under the assumption that the stress is homogeneous throughout the seismic zone. That method is called the Multiple Solution per Earthquake Technique (MSET), and utilizes each member of multiple focal mechanism solution set as a possible solution. The MSET is useful when applied to small data sets, and differs from existing techniques in that (1) the use of multiple focal mechanisms for individual earthquakes allows for a range in the possible orientation of the fault geometry, while preserving fit to the original polarity and amplitude ratio data, and (2) the differences between the observed and theoretical fault slip is used as a weighting scheme for the results of the tensor estimation. Other methods, which rotate the single observed focal mechanism solution from its original configuration to estimate misfit, do not take into consideration the fit of that final solution to the original input data but assume that a minimization of errors between the theoretical stress model and the focal mechanism solution indicates a reasonable fit. For large data sets that assumption is likely to be met.

The MSET was applied to a set of 32 earthquakes to estimate the principal stress orientations for seismically active zones in the Southeastern United States, using focal mechanism solution sets derived by the program FOCMEC. Eight events were studied for the Giles County Seismic Zone, 13 for the Central Virginia Seismic Zone, and 11 for the Eastern Tennessee Seismic Zone. After testing against approximately 25000 theoretical solutions, an average of sixteen focal mechanism solutions fit the input polarity and (SV/P)₂ amplitude ratio data for each earthquake. The P-axes of the multiple focal mechanism solutions were averaged to determine a provisional single P-axis direction for each earthquake. P-axes for the Giles County and Eastern Tennessee Seismic Zones trended generally NE-SW, while those of the Central Virginia Zone varied with depth, with the P-axes of events above approximately 8 km trending NE-SW, and those below trending NW-SE.

Application of the MSET resulted in consistent principal stress orientations for the Giles County and Eastern Tennessee Zones, with the horizontal component of the maximum compressive stress direction (σ₁) trending about N40°E and N50°E, respectively. Results for the Central Virginia Zone also suggested differently oriented stress regimes above and below a depth of 8 km. The direction of the σ₁ axis above that boundary was N70°E, while below it was east-west, with a shallow plunge to the west. While those results were not as pronounced as suggested initially by the P-axis data alone, the hypothesis of two stress tensors produced better MSET results than for a single stress tensor for the combined data set.

The technique developed for this study produces comparable results to other methods when applied to identical data sets. Estimation of error is based on subjective criteria, and includes the lit of the original seismic polarity and amplitude ratio data with the focal mechanism solutions. The error associated with each step in the process (e.g. distribution and reliability of the polarity and ratio data, calculation of focal mechanism solutions and estimation for the stress field) is very difficult to parameterize, and thus, no formal statistical analyses were undertaken.

After the estimate of the homogeneous stress field was made for each zone, a single best focal mechanism solution for each earthquake could be objectively chosen by constraining the slip associated with each mechanism to be aligned with the resolved stress derived from the principal stress directions. In that manner, focal mechanism solutions could be identified which fit the sparse input polarity and amplitude ratio data, but which were not compatible with the calculated stresses. Also, in that same procedure, the fault plane was chosen from the set of two nodal planes for each focal mechanism solution by examination of the theoretical slip on each of those planes. The faults within the Giles County Seismic Zone matched the direction found in previous seismic reflection surveys, with an average strike of N25°E. In the Eastern Tennessee Seismic Zone, faulting also occurred on planes oriented predominantly NE-SW. For the five shallow Central Virginia events, faults trended NW-SE, while for the deeper events there was no consistent trend.

A comparison was made between the P-axis and σ₁ axis derived for each earthquake. Although in 81% of the cases σ₁ was within 35° of at least one P-axis of the focal mechanism solution set, no further empirical relationship was found.

The MSET has proven itself useful in two ways when applied to sparse data sets. First of all, the primary seismic data (polarities and amplitude ratios) are not overlooked when deriving the orientation of a stress tensor associated with local faulting. Secondly, the MSET is an objective method for defining the best fitting solution among a family of focal mechanism solutions by requiring compatibility with the regional stresses. In the future, after integration with a program such as FOCMEC, regional stress tensors may be derived by the MSET which incorporate reasonable statistical parameters based on the fit of that primary data.