## Determination of moment tensors from borehole data:
a numerical analysis

**Vaclav Vavrycuk**
### Summary

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media, the zero-trace constraint is no longer valid and can distort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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In: Seismic Waves in Complex 3-D Structures, Report 16,
pp. 203-224, Dep. Geophys., Charles Univ., Prague, 2006.

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