# Hypocentroidal Decomposition

mloc is based on the Hypocentroidal Decomposition (HD) method for multiple event relocation introduced by Jordan and Sverdrup (1981). The basic algorithm is completely described in that reference. A PDF of the paper is included in the mloc distribution in the /documents directory. The essence of the HD algorithm is the use of orthogonal projection operators to separate the relocation problem into two parts:

• The cluster vectors, which describe the relative locations in space and time of each event in the cluster. They are defined in kilometers and seconds, relative to the current position of the hypocentroid.
• The hypocentroid, which is defined as the centroid of the current locations of the cluster events. It is defined in geographic coordinates and Coordinated Universal Time (UTC).

Both methods of location calibration implemented in mloc depend fundamentally on this decomposition of the relocation problem. Similar approaches could conceivably be implemented in other multiple event relocation algorithms but it seems likely to be considerably more difficult than it is with hypocentroidal decomposition.

The cluster vectors are defined only in relation to the hypocentroid. The hypocentroid can be thought of as a virtual event with geographic coordinates and origin time in UTC. The orthogonal projection operators act on the data set of arrival times to produce a data set that includes only data that actually bears on the relative location of cluster events, i.e., multiple reports of a given seismic phase at the same station for two or more events in the cluster.

The hypocentroid is located very much as an earthquake would be, except that the data are drawn from all the cluster events. Thus it is typical for the hypocentroid to be determined by many thousands of readings. Nevertheless, the hypocentroid is subject to unknown bias because the theoretical travel times (typically ak135) do not fully account for the three-dimensional velocity structure of the Earth. Geographic locations for the cluster events are found by adding the cluster vectors to the hypocentroid.

The HD method works iteratively. At each iteration, two inversions are performed, first for the cluster vectors relative to the current hypocentroid, then for an improved hypocentroid. The cluster vectors are added to the new hypocentroid to obtain updated absolute coordinates for each event. The convergence criteria are based on the change in relative location of each event (0.5 km) and the change in the hypocentroid (0.005°). The convergence limits for origin time and depth, for cluster vectors and hypocentroid, are 0.1 s and 0.5 km, respectively. Convergence is normally reached in 2 or 3 iterations.

The data sets used for the two problems need not be (and usually are not) the same. Because the inverse problem for changes in cluster vectors is based solely on arrival time differences, baseline errors in the theoretical travel times drop out and it is desirable to use all available phases at all distances outside the immediate source region. For the hypocentroid, baseline errors in theoretical travel times are more important and one may wish to limit the data set to a phase set, e.g., teleseismic P arrivals in the range 30-90°, to achieve a more stable (but uncalibrated) result. The choice of data set for determining the hypocentroid has great importance in the direct calibration method.

Similarly, weighting schemes are different for the two inversions (cluster vectors and hypocentroid) that comprise HD, reflecting the different natures of the two problems. Empirical reading errors for each station-phase pair are used in weighting data for estimating both the hypocentroid and cluster vectors, but the uncertainty of the theoretical travel times, which are estimated empirically for each phase from the residuals of previous runs, is relevant only to the hypocentroid.

The HD algorithm as developed by Jordan and Sverdrup (1981) is used only to obtain improved relative locations for the cluster events, with a geographic location for the cluster as a whole (the hypocentroid) that is biased to an unknown degree by unmodeled Earth structure that has been convolved with the (typically) unbalanced geographic distribution of reporting seismic stations. The capabilities of mloc go far beyond this, including especially the tools implemented for location calibration which attempts to minimize this bias.