A
likelihood function that is much more robust in the presence of
outliers than the usual, root-mean-square, L2
norm.

NLL constructs an estimate of
the location *pdf* within the framework of the probabilistic
earthquake location methods of Tarantola and Valette (1982), Moser,
van Eck and Nolet (1992) and Wittlinger et al. (1993). NLL makes
available two different likelihood functions to build the *pdf*.
The first function, LS-L2, incorporates the familiar least-squares, L2
norm (LS-L2), constructed following the formulation of Tarantola and
Valette (1982). The second function, EDT, is based on a generalization
by Font
et al.
(2004)
of the
Equal-Differential-Time (EDT) formulation of Zhou(1994); all of these are extensions of the "method of
hyperbolas" cited by Milne (1886).
The EDT likelihood function is much more robust in the presence of
outliers in the data than are the LS-L2 or other L1 and L2 norms. (An
outlier observation has a residual greater than its nominal
error.) With both the EDT and LS-L2 likelihood functions, the
errors in the
observations (seismic wave arrival times) and in the forward problem
(travel-time calculation) are assumed to be Gaussian. This
assumption allows the direct, analytic calculation of a maximum
likelihood origin time for the LS-L2 likelihood function, while the
EDT determination is inherently independent of any origin time
estimate. Thus the 4D problem of hypocenter location reduces to a 3D
search over latitude, longitude and depth. In this work, this 3D
search is performed with a very efficient, cascading grid-search,
importance-sampling method called Oct-tree.

Most earthquake location
algorithms are based on an L1 or L2 norm of the misfit between
observed and calculated travel times for each observation, given a
nominal error for each observations. Implicitly (in most location
algorithms) or explicitly (in probabilistic location algorithms),
these norms are incorporated into a *pdf*. For the LS-L2 norm,
the *pdf* has the form:

, (1)

where **x** is a point in 3D
space, *t*_{0} is an estimate of the origin time, *k*
is a normalization factor, *Tobs _{i}* and

The LS-L2 *pdf* for most
location problems has a compact form, which may be irregular. Many
global sampling algorithms can produce a fair to good representation
of this *pdf*.

An alternative to the LS-L2
likelihood function that is very robust in the presence of outliers
is given by the Equal Differential Time (EDT) formulation. For the
EDT case, the NLL *pdf* has the form:

, (2)

where *Tobs _{a}*
and

Because it is the intersection
of many EDT surfaces, the EDT *pdf* for location problems with
outlier observations may have a topology that is highly complicated
and irregular. Most global sampling algorithms cannot produce a good
representation of this form of *pdf*, but the Oct-tree method
used here is remarkably stable in almost all cases.