Locating microseismic events and traveltime mapping using locally spherical wavefronts
John C. Bancroft, Xiang Du
A method is presented for identifying the source of a locally circular or spherical wavefront given the traveltimes at arbitrary locations. For 2D data, the center of the circular wavefront is computed from three traveltimes recorded at three arbitrary locations. Application to 3D data requires four traveltimes recorded at four arbitrary locations.
This method is suited for a number of applications such as mapping traveltimes that are computed along sparse raypaths to gridded traveltimes, the monitoring of microseismic events caused by fraccing, or to the possible prediction of landslides in geologically unstable areas.
The analytic solution for the 2D problem is found using the tangency of circles, a problem that was originally solved about 200 BC by Apollonius, a Greek mathematician. The 3D solution involves the tangency of spheres and was obtained by using a parallel development to the 2D solution. Both the 2D and 3D problems produce two solutions from which one must be chosen.