# Difference between revisions of "Seismic event relocation techniques"

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where τ is the origin time of event i, u is the slowness field, and ds is an element of path length. Due to the nonlinear relationship between travel time and event location, a truncated Taylor series expansion (Geiger, 1910) is generally used to linearize equation (1). The resulting problem then is one in which the travel-time residuals, r, for an event i are linearly related to perturbations, Δm, to the four current hypocentral parameters for each observation k: |
where τ is the origin time of event i, u is the slowness field, and ds is an element of path length. Due to the nonlinear relationship between travel time and event location, a truncated Taylor series expansion (Geiger, 1910) is generally used to linearize equation (1). The resulting problem then is one in which the travel-time residuals, r, for an event i are linearly related to perturbations, Δm, to the four current hypocentral parameters for each observation k: |
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− | <math> \partial t^i_k \over \partial m </math> |
+ | <math> \partial t^i_k \over \partial m \delata m^i </math> |

=== Master event (ME) method=== |
=== Master event (ME) method=== |

## Revision as of 17:29, 24 February 2017

Intro discussion

## Contents

## Seismic event relocation overview

In the study of mining induced seismicity, the accuracy of event location and magnitudes has improved through the technological advancement of hardware and processing algorithms. Seismic events are typically located using P and S wave arrival times at different receivers paired to a velocity model. One of the most influential developments in micro seismic data processing is event relocation. Research in the field of crustal seismology has developed a wide array of techniques to increase the accuracy of event location. These techniques fall into the broad category of “event relocation” or “relative location methods”. Adding event relocation to a micro seismic data processing flow aids interpretation by concentrating previously diffuse event clouds by reducing positional uncertainty. This increased confidence in event location can add context to the geometry of the source mechanism; adding value to the seismic data set.

Initial (absolute) event location accuracy is generally hampered by the following

- Errors in P and S wave arrival time picks either from auto-picking or human error
- Limited understanding or over-simplification of regional velocity model
- Limited spatial coverage of geophones in mining applications

The techniques outlined below are deemed “relocation” techniques because they are done after an initial (absolute) location solution has been calculated. These types of process can also be considered “post-processing” or “reprocessing” in the seismic data processing workflow. The case examples presented are related specifically to mining induced seismicity, though the development and use of the techniques transcend the field, with applications in hydraulic fracture monitoring and crustal seismology.

## When is relocation completed?

In the seismic data processing workflow, event relocation can be performed immediately after an initial (absolute) solution for an event’s location has been calculated. Often a relocation algorithm will be applied to heritage data as a relatively inexpensive way to add value to pre-existing datasets. Due to the advance in computational power in seismic monitoring solutions relocation algorithms can be completed “on the fly” as part of the standard event location processing flow to reduce uncertainty in calculated event locations.

## How are is it done?

Seismic event relocation algorithms describe a style of process, with many variations on a handful of themes. The most popular and arguably most effective theme being the Double-difference technique which differs from other relocation algorithms due to its assumptions and simultaneous relocation of large numbers of events over large distances. Other themes include “Master Event Relocation”, where events are moved relative to single “master” event which is computationally straight forward, but propagates location errors from the initial placement of the master event through to the relocated “relative events”. Alternatively, there are “Simultaneous Relocation” approaches. First implemented by Got et al, they determined cross-correlation time delays for all possible event pairs in question and combined them into a system of linear equation which was solved via least squares approximation and converted to positional corrections. For the remainder of this article reference is made to the three aforementioned classifications; Double-difference, Master Event, and Simultaneous Event relocation algorithms to summarize the current state of event relocation techniques and enable direct comparison between them.

### Double-difference (DD) method

The double-difference approach, first introduced by Waldhauser in 2000 has been applied successfully as an event relocation algorithm option in mining induced seismicity applications.

#### Double-difference (DD) workflow

The following procedure for executing the double difference algorithm is adapted from (Waldhauser, Ellsworth 2000).

First, both P and S-wave differential travel times are derived from cross-spectral (cross correlation) methods with travel-time differences formed from catalog data and minimize residual differences (or double differences) for pairs of earthquakes by adjusting the vector difference between their hypocenters. From this calculated “double difference” we are able to determine interevent distances between correlated events that form a single multiplet (group of similar events) to the accuracy of the cross-correlation data while simultaneously determining the relative locations of other multiplets and uncorrelated events to the accuracy of the absolute travel-time data, without the use of station corrections.

The arrival time, T, for an earthquake, i, to a seismic station, k, is expressed using ray theory as a path integral along the ray,

where τ is the origin time of event i, u is the slowness field, and ds is an element of path length. Due to the nonlinear relationship between travel time and event location, a truncated Taylor series expansion (Geiger, 1910) is generally used to linearize equation (1). The resulting problem then is one in which the travel-time residuals, r, for an event i are linearly related to perturbations, Δm, to the four current hypocentral parameters for each observation k:

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