The complete moment tensors of seismic sources in homogeneous or vertically inhomogeneous isotropic structures cannot be retrieved using receivers deployed in one vertical borehole. The complete moment tensors can be retrieved from amplitudes of P waves, provided receivers are deployed in three boreholes at least. Using amplitudes of P and S waves, two boreholes are, in principle, sufficient. The similar rules apply also to transversely isotropic media with a vertical axis of symmetry.
In the case of limited observations, the inversion can be stabilized by imposing the zero-trace constraint on the moment tensors. However, this constraint is legitimate only if applied to observations of shear faulting on planar faults in isotropic media, which produces double-couple (DC) mechanisms. For shear faulting on non-planar faults, for tensile faulting, and for shear faulting in anisotropic media, the zero-trace constraint is no longer valid and can distort the retrieved moment tensor and bias the fault plane solution.
Numerical modelling simulating the inversion of the DC mechanism from real data reveals that the errors in the DC and non-DC percentages of the moment tensors rapidly decrease with increasing the number of boreholes used. For noisy P and S amplitudes with noise of 15% of the top amplitude at each channel and for a velocity model biased by 10%, the errors in the DC percentage attain 25, 13 and 6% when inverting for the DC mechanism from one, two and three boreholes.
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