Geiger's Method
Geiger's method [1] is an iterative procedure using Gauss-Newton
optimization to determine the location of an earthquake, or seismic event.
Originally his method was developed to obtain the origin time and Epicentre
but it is easily extended to include the Focal Depth
for Hypocentre
determination.
Given a set of
arrival times
find the origin time
and
the hypocentre in cartesian coordinatios
which minimize
the objective function
 |
(1) |
Here,
is the difference between observed and calculated arrival times
 |
(2) |
and the unknown parameter
vector
is
 |
(3) |
In matrix
form (1) becomes
 |
(4) |
The Gauss-Newton procedure requires an initial guess of the
sought parameters, denoted here as
 |
(5) |
which are then used to calculate the adjustment vector
 |
(6) |
in
 |
(7) |
The Jacobian matrix
is defined as
 |
(8) |
The partial derivatives are evaluated at the initial guess, or trial vector,
. Equation (7) can be rewritten as
 |
(9) |
Using (9) and an initial guess
an adjustment
vector can be calculated. The initial guess can then be updated
and used as the inital guess in the next
run of the algorithm. In this manner the sought parameters
can
be determined to some tolerance.
-
- 1
- Geiger, L., ``Probability method for the determination of
earthquake epicenters from the arrival time only.'' Bull. St. Louis
Univ. vol. 8, pp. 60-71.
- 2
- Lee, W. H. K. and Stewart, S. W. Principles and Applications
of Microearthquake Networks, Academic Press, New York. 1981
- 3
- Gibowicz, S. J. and Kijko, A. An Introduction to Mining
Seismology, Academic Press, New York. 1994.
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