A Very Rapid Scan for Hypocentre Foci Driven by a Conformal Transformation

Abstract

This paper aims to suggest and demonstrate a method whereby very rapid scans for the Hypocentre foci can take place in real-time or near real-time. These scans are executed by point-to-point (P2P) ray-tracers, which, for each depth-point in the scan construct rays from the depth-point in question to each of the active seismographs which are in possession of a P-wave or S-wave onset timing. The P2P tracers may employ any of a set of 1-D (radial) Earth velocity models. Two methods are available to construct the set of segments that are to form the rays: Varying the radius, or varying the subtended angle, in the construction. The crux of the method is to use a conformal complex mapping from a disc on the z-plane to a rectangular area on the w-plane. The required ray-traces can take place in the w-plane. The geometry developed for any ray can be transformed back into discoid form in the z-plane and ray-path timings and path-lengths can be extracted. The indicators chosen here to define the possible depth coordinates of the Hypocentre foci, during and after scanning are the Fisher F-statistic and Fisher z-transform It can be suggested that a conformal mapping from the w-plane rectangular strip can proceed to an ellipse, (and hence encompass spheroidal Earth geometry), on a u-plane via the discoid on the z-plane. Otherwise, this suggested scanning system encompasses a spherical Earth model only.