Degree of Compatibility function

This page is subdivided into the following sections:

Introduction

The determine if a candidate hypocenter is falsified by observations or compatible with these observations, the Degree of Compatibility Function DOC(x,y,z,t) is defined.

First, we have to ask how to falsify a single 4D point h(x,y,z,t) as hypocenter:

In the following, I will focus on the treatment of arrival times. Directional data are much easier to handle and I think the concept is easy to transfer.

Data Model

Observation data is usually represented by a single value and an uncertainty measure, inspired by the two parameters needed to define a normal (gaussian) distribution. The mechanism of least squares optimization does indeed imply normal distributions.

Considering arrival times of seismic phases, it is often unclear how to determine, estimate or guess the standard deviation, since phase picking is still a very subjective process. In the case of having only few seismograms of possibly poor quality, the usual data model therefore seems to be ill suited.

Much easier than determining the exact onset time of a seismic phase is to define a time interval which only contains the onset. It is in fact always possible to define a finite length interval which contains, say, the P arrival with 100% certainty. In some cases, this interval would be very long (up to several minutes), but with more than one such interval, it can still be used to constrain the hypocenter.

LOCSMITH therefore defines all data, arrival times as well as angles and other direction data, as intervals: a scalar datum is given by the closed interval [xmin, xmax], such that the true value is between these bounds with 100% certainty. Computer scientists have developed a whole algebra for the so-called interval arithmetic based on this data model.

A strict implementation of interval arithmetic, however, would make the program extremely slow. LOCSMITH only uses the data model, but carries out its computations with normal airthmetics.

Definition of the DOC function

The Degree of Compatibility function DOC(x,y,z,t) simply gives the number of observations with which a candidate solution h(x,y,z,t) is compatible. It is therefore an integer valued function of the hypocenter coordinates.

To determine the DOC for arrival times, follow the following list of steps:

1. Observation at station s:
Choose a time interval [ts,min, ts,max] such that you are sure that the phase arrives within that interval.
2. Choose a candidate 3D hypocenter (x,y,z)
3. Compute theoretical travel times ts,theo from this point to all stations s
4. Subtract ts,theo from ts,min and ts,max to get the origin time intervals [ts,Omin,ts,Omax]=[ts,min-ts,theo, ts,max-ts,theo]
5. Determine origin time intervals [ts,Omin,ts,Omax] of all phases and all stations.
6. Find out where most intervals overlap
7. The degree of compatibility is the largest number of data to which the candidate hypocenter is compatible (see figure: P ans S arrival times at two stations S1 and S2 deinfe overlapping intervals of origin times, only the intersection of all intervals can contain the true origin time.)
8. A point (x,y,z) is part of the solution set if and only if it's degree of compatibility equals the number of valid observations. The t coordinate is the interval defined by the non-falsified origin times, but the spatial coordinates remain scalar, since they are chosen by the user (If the solution set turns out to be empty, you have to check for outlier data or erroneous phase identifications).

Note that the degree of compatibility function DOC(x,y,z,t) is single-valued, integer-valued, positive and non-continuous (stair-like).


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