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 but it is easily extended to include the Focal Depth for 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 function $$ F(\mathbf{X})=\sum_{i=1}^{M}r_i^2. $$ Here, $$r_i$$ is the difference between observed and calculated arrival times $$ r_i=t_i-t_0-T_i, $$ and the unknown parameter vector is $$ \mathbf{X}=(t_0,x_0,y_0,z_0)^{\mathrm{T}} $$ In matrix form (1) becomes $$ F(\mathbf{X})=\mathbf{r}^{\mathrm{T}}\mathbf{r} $$ The Gauss--Newton procedure requires an initial guess of the sought parameters, denoted here as $$ \mathbf{X}^*=(t_0^*,x_0^*,y_0^*,z_0^*)^{\mathbf{T}}, $$ which are then used to calculate the adjustment vector $$ \delta\mathbf{X}=(\delta t_0,\delta x_0,\delta y_0,\delta z_0)^{\mathrm{T}} $$ in $$ (1) \mathbf{A}^{\mathrm{T}}\mathbf{A}\delta\mathbf{X}=-\mathbf{A}^{\mathrm{T}}\mathbf{r}. $$ The Jacobian matrix $$\mathbf{A}$$ is defined as $$ \mathbf{A}=\left( \begin{matrix} \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 \\

\vdots & \vdots & \vdots & \vdots \\

\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{matrix}\right). $$ The partial derivatives are evaluated at the initial guess, or trial vector, $$\mathbf{X}^*$$. Equation (45) can be rewritten as $$ (2) \mathbf{G}\delta\mathbf{X}=\mathbf{g}. $$ Using (46) and an initial guess $$\mathbf{X}^*$$ an adjustment vector can be calculated. The initial guess can then be updated $$\mathbf{X}^*+\delta \mathbf{X}$$ and used as the inital guess in the next run of the algorithm. In this manner the sought parameters $$\mathbf{X}$$ can be determined to some tolerance.