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 that receivers are deployed in at least three boreholes. Using amplitudes of P- and S-waves, two boreholes are, in principle, sufficient. Similar rules also apply 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 valid only if applied to observations of shear faulting on planar faults in isotropic media, which produces double-couple 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on non-planar faults, for tensile faulting, and for shear faulting in anisotropic media, the zero-trace constraint is no longer valid and can distort the retrieved moment tensor and bias the fault-plane solution.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 double-couple mechanism from real data reveals that the errors in the double-couple and non-double-couple percentages of the moment tensors rapidly decrease with increase in the number of boreholes used. For noisy P- and S-wave amplitudes with noise of 15% of the top amplitude at each channel and for a velocity model biased by 10%, the errors in the double-couple percentage attain 25, 13 and 6% when inverting for the double-couple mechanism from one, two and three boreholes.
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