PlanetPhysics/Geiger's Method

Geiger's method 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/|Epicentre]] but it is easily extended to include the [[../FocalDepth/|Focal Depth]] for [[../Hypocenter/|Hypocentre]] determination.

Given a set of $M$ arrival times $t_i$ find the origin time $t_0$ and the hypocentre in cartesian coordinatios $(x_0,y_0,z_0)$ which minimize

the objective [[../Bijective/|function]]

$F(𝐗)=\sum _{i=1}^M{r}_i^2.$

Here, $r_i$ is the difference between observed and calculated arrival times

$r_i=t_i-t_0-T_i,$

and the unknown [[../Parameter/|parameter]] [[../Vectors/|vector]] is

$𝐗=(t_0,x_0,y_0,z_0{)}^{\mathrm{T}}$

In [[../Matrix/|matrix]] form (1) becomes

$F(𝐗)={𝐫}^{\mathrm{T}}𝐫$

The Gauss--Newton procedure requires an initial guess of the sought parameters, denoted here as

${𝐗}^{*}=(t_0^{*},x_0^{*},y_0^{*},z_0^{*}{)}^{𝐓},$

which are then used to calculate the adjustment vector

$\delta 𝐗=(\delta t_0,\delta x_0,\delta y_0,\delta z_0{)}^{\mathrm{T}}$

in

$(1){𝐀}^{\mathrm{T}}𝐀\delta 𝐗=-{𝐀}^{\mathrm{T}}𝐫.$

The Jacobian matrix $𝐀$ is defined as

$𝐀=\left(\begin{array}{cccc}\partial r_1/\partial t_0& \partial r_1/\partial x_0& \partial r_1/\partial y_0& \partial r_1/\partial z_0\\ \partial r_2/\partial t_0& \partial r_2/\partial x_0& \partial r_2/\partial y_0& \partial r_2/\partial z_0\\ ⋮& ⋮& ⋮& ⋮\\ \partial r_M/\partial t_0& \partial r_M/\partial x_0& \partial r_M/\partial y_0& \partial r_M/\partial z_0\end{array}\right).$

The partial derivatives are evaluated at the initial guess, or trial vector, ${𝐗}^{*}$. Equation (45) can be rewritten as

$(2)𝐆\delta 𝐗=𝐠.$

Using (46) and an initial guess ${𝐗}^{*}$ an adjustment vector can be calculated. The initial guess can then be updated ${𝐗}^{*}+\delta 𝐗$ and used as the inital guess in the next run of the [[../RecursiveFunction/|algorithm]]. In this manner the sought parameters $𝐗$ can be determined to some tolerance.