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absolute value ACOUSTIC EMISSION SOURCE ANALYSIS By Francis analytical control appendix approximation array geometry arrival assumptions Av/v AxP+1 Ayaay Bureau of Mines calculated constant velocity Contour of error converges coordinate data inaccuracies dimensional directional derivative emission source location equa equation 15 equation 26 equation D-17 equation D-ll equations A-2 error coefficients error estimates figures first-order formulation Fortran fractional velocity change geometrical control geophones higher order terms i i i i i i'th phone increment independent variables initial solution least-squares equations least-squares solution Leighton and Duvall location error location solution location techniques microseism source minR minRj Newton's method nonlinear equation nonlinear location equations P-wave parameter partial derivatives redundancy respect to fractional singularity solu solution estimate solving a nonlinear source location equations squares standard deviation statistical error systematic error Taylor series tion transducer channel traveltime vector velocity errors zeroth phone