1D Depth Search Algorithm

The algorithm used to determine the possible depth(s) of the earthquake hypocenter is a one dimensional version of the algorithm used to determine the longitude and latitude coordinates. It is based on an adaptive subdivision of the depth axis, depending on how many data constraints can be satisfied at a given hypocentral depth.

To explain the algorithm, we forget about earthquakes for a while and consider the general problem of one dimensional function optimization:

At what value(s) of the free variable x ∈ R does the function f(x): R → N reach its maximum?

The function to be maximized is
DOC2(R)=max(DOC(lon,lat,R=const,t))
or the maximum of the DOC function on a spherical shell with radius R, and the maximum for a variation of R is searched.

This problem is of course discussed extensively in the literature (e.g. Press et al., 1992) and we would like to consider it as being solved.

Unfortunately, it isn't. Press et al. (1992) describe several methods, and each mathematical handbook gives at least the Newton-method (regula falsi). But all these methods work only with continuous functions f(x): R → R. Some require gradients, and usually it is assumed that a given interval contains only one solution. But as described in the section on the Degree of Compatibility Function, the DOC function has no gradient and is integer valued (and therefore stair-like and non-continuous), and it may have several disconnected maxima. Even Monte Carlo Methods (Simulated Annealing, Genetic Algorithms) have problems with stair functions, since many parameter changes do not change the function value. So we need something different.

LOCSMITH uses a one dimensional version of adaptive grids: an initial R-interval is subdivided into shorter intervals, and the subdivisions occur at the most promising positions, e.g. where the current sampling indicates the highest function values.

It is assumed that the function is known at n ≥ 2 points {R1, R2,... Rn}, defining n-1 intervals. The subdivision of intervals then follows four simple rules:

  1. If DOC2(R) exceeds a predefined fraction ε of the currently highest known function value on at least one end point of a given interval [Rm, Rm+1], insert a new sample in the middle.
  2. Insert new points in the middle of the two neighbour intervals of each interval modified according to rule 1.
  3. If DOC2(R) has the same value at any of the given samples (i.e. function appears to be flat), insert points in the middles of all intervals.
  4. Do not introduce new points into intervals that are shorter than a predefined accuracy δ

This is repeated until rule 4 does not allow to introduce new points any more.

The concept is illustrated in a sequence of figures (which are screenshots from a talk I gave in an internal seminar)

The algorithm is implemented in the MKLSGRIDNEWVERTICAL routine, which drives the insertion of new depth layers into LOCSMITH grids.


eof.