Real-Time Earthquake Localisation and the Elliptic Correction

Abstract

We have proposed fast algorithms for the concurrent localisation of Earthquake Epi- and Hypocentres in real-time. In tests, a “reduced simulation” regime is followed in which these algorithms absorb P-wave 1^st arrival timings sequentially into the object data set, in the order in which the expanding wave-field impinged on the sensors, (seismographs), in the network considered. Most localisation algorithms at some point apply a correction for the Earth’s ellipticity. However, the bulk of the calculation is done assuming a spherical Earth – with the correction added on as an after-thought. In this paper, we will broach a proposal for implanting the Earth’s spheroidal geometry into localization algorithms directly. This implantation can act either within a tabular scan delivering a concurrent localization of Epi- and Hypocentres, or within a purely Hypocentral scan (given knowledge of the Epicentre). This would obviate the need for a “tag-on” correction. Output from both these spheroidal options is included with comment. The consequence for the computational power that would be needed to support such a set of calculations is also treated.

G. R. Daglish—Consultant (Seismology).

I. P. Sizov—Retired Consultant (Electro Magnetics).

Appendix A

Amplification of Derivation of the Calculation for the Surface Geodesic Curve Constant

The square of the distance between two neighbouring points is:

$$ ds^{2 } = dx^{2} + dy^{2} + dz^{2} = \mathop \sum \limits_{u,v} h_{u,v}^{2} .dudv $$

We have:

$$ dx^{2} = \left( {\frac{\partial x}{\partial u}} \right)^{2} \left( {du} \right)^{2} + 2.\frac{\partial x}{\partial u}\frac{\partial x}{\partial v}dudv + \left( {\frac{\partial x}{\partial v}} \right)^{2} \left( {dv} \right)^{2} $$

And, similarly, for $dy$ and $dz$.

There is an array of Scale Factors (the metric):

$$ \begin{array}{*{20}c} {h_{u,u}^{2} } & {h_{u,v}^{2} } \\ {h_{v,u}^{2} } & {h_{v,v}^{2} } \\ \end{array} $$

Equating like with like and assuming Orthogonality:

$$ h_{u,u}^{2} = \left( {\frac{\partial x}{\partial u}} \right)^{2} + \left( {\frac{\partial y}{\partial u}} \right)^{2} + \left( {\frac{\partial z}{\partial u}} \right)^{2} \left( { \triangleq h_{u}^{2} } \right) $$

$$ h_{v,v}^{2} = \left( {\frac{\partial x}{\partial v}} \right)^{2} + \left( {\frac{\partial y}{\partial v}} \right)^{2} + \left( {\frac{\partial z}{\partial v}} \right)^{2} \left( { \triangleq h_{v}^{2} } \right) $$

This leads to [15]:

$$ \begin{aligned} h_{u}^{2} & = r_{e}^{2} sin^{2} \left( u \right) + r_{p}^{2} cos^{2} \left( u \right) \\ & \quad h_{v}^{2} = r_{e}^{2} cos^{2} \left( u \right) \\ \end{aligned} $$

We also have:

$$ ds^{2} = h_{u}^{2} .du^{2} + h_{v}^{2} .dv^{2} $$

$$ \frac{{ds^{2} }}{{du^{2} }} = \left( {\frac{ds}{du}} \right)^{2} = h_{u}^{2} + h_{v}^{2} .\left( {\frac{dv}{du}} \right)^{2} $$

$$ \sqrt {\left( {\frac{ds}{du}} \right)^{2} } .du = \sqrt {\left( {r_{e}^{2} sin^{2} \left( u \right) + r_{p}^{2} cos^{2} \left( u \right) + r_{e}^{2} cos^{2} \left( u \right).V^{2} } \right)} .du $$

$$ = F\left( {u,V} \right).du; \quad \left\{ {V = \frac{dv}{du}} \right\} $$

Paraphrasing: “The integral,$\int_{{\lambda_{1} }}^{{\lambda_{2} }} {F(u,\,V).\,du}$, [which gives arc-length in Cartesian 3-space as a function of displacements in Latitude and Longitude, $(u,\,v)$, and whose end points are fixed], is stationary for weak variations, if v satisfies the condition in the differential equation below” [16].

So, by Theorem 3, at [16], and setting:

$$ \frac{\partial F}{\partial V} = k $$

we can generate required geodesic paths on integration if we can find a $v$ that can satisfy this differential equation. However, a constant $k$ can be found which will correspond to the requirements of the function V, in the above, as:

$$ \frac{dF}{dV} = \frac{{r_{e}^{2} .cos^{2} \left( u \right).V}}{{\sqrt {\left( {r_{e}^{2} sin^{2} \left( u \right) + r_{p}^{2} cos^{2} \left( u \right) + r_{e}^{2} cos^{2} \left( u \right).V^{2} } \right)} }} = k $$

$$ r_{e}^{4} .cos^{4} \left( u \right).V^{2} = k^{2} .\left( {r_{e}^{2} sin^{2} \left( u \right) + r_{p}^{2} cos^{2} \left( u \right) + r_{e}^{2} cos^{2} \left( u \right).V^{2} } \right) $$

$$ V^{2} = \left( {\frac{dv}{du}} \right)^{2} = \frac{{k^{2} \left( {r_{e}^{2} sin^{2} \left( u \right) + r_{p}^{2} cos^{2} \left( u \right)} \right)}}{{r_{e}^{2} cos^{2} \left( u \right).\left( {r_{e}^{2} cos^{2} \left( u \right) - k^{2} } \right)}} $$

If $k$ is to remain constant, then this integral equation must hold:

$$ v = \frac{k}{{r_{e} }}.\smallint \frac{1}{cos\left( u \right)}\sqrt {\frac{{r_{e}^{2} sin^{2} \left( u \right) + r_{p}^{2} cos^{2} \left( u \right)}}{{\left( {r_{e}^{2} cos^{2} \left( u \right) - k^{2} } \right)}}.} du $$

The constant $k$ can be found by numerical means for a specific geodesic trajectory, defined by its end-points as (Latitude, Longitude) pairs. (For example, by scanning in $k$, searching for a crossing and then performing a linear interpolation).

Finally, the process:

$$ \phi_{i} = \frac{k}{{r_{e} }}\mathop \smallint \limits_{{\lambda_{1} }}^{{\lambda_{i} }} \frac{1}{cos\left( u \right)}\sqrt {\frac{{r_{e}^{2} sin^{2} \left( u \right) + r_{p}^{2} cos^{2} \left( u \right)}}{{\left( {r_{e}^{2} cos^{2} \left( u \right) - k^{2} } \right)}}.} du + \phi_{1} $$

can be evaluated numerically to give the trajectory of the geodesic associated with the corresponding value of $k:$ $\left\{ {(\lambda ,\phi )_{i} } \right\}$. This will occur on the surface of the Spheroid, between, and including, the end-points: $(\lambda ,\phi )_{1}$ and $(\lambda ,\phi )_{2}$, as an explicit function of $u$, the Latitude.