Seismic event relocation techniques
- 1 Seismic event relocation overview
- 2 When is relocation completed?
- 3 How is it done?
- 4 Successful application of event relocation in mining applications
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 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:
where rik = (tobs - tcal)ik, tobs and tcal are the observed and theoretical travel time, respectively, and Δmi = (Δxi, Δyi, Δzi, Δ τ i). Equation (2) is appropriate for use with measured arrival times. However, cross-correlation methods measure travel-time differences between events, (tki - tkj)obs, and as a consequence, equation (2) can not be used directly. Frechet (1985) obtained an equation for the relative hypocentral parameters between two events i and j by taking the difference between equation (2) for a pair of events,
where Δmij_(Δdxij, Δdyij, Δdzij, Δdsij) is the change in the relative hypo central parameters between the two events, and the partial derivatives of t with respect to m are the components of the slowness vector of the ray connecting the source and receiver measured at the source (e.g., Aki and Richards, 1980). Note that in equation (3) the source is actually the centroid of the two hypocenters, assuming a constant slowness vector for the two events. drikj in equation (3)m L. Ellsworth is the residual between observed and calculated differential travel time between the two events defined as
We define equation (4) as the double-difference. Note that equation (4) may use either phases with measured arrival times where the observables are absolute travel times, t, or cross-correlation relative travel-time differences. The assumption of a constant slowness vector is valid for events that are sufficiently close together, but breaks down in the case where the events are farther apart. A generally valid equation for the change in hypo central distance between two events i and j is obtained by taking the difference between equation (2) and using the appropriate slowness vector and origin time term for each event
or written out in full
The partial derivatives of the travel times, t, for events i and j, with respect to their locations (x, y, z) and origin times (s), respectively, are calculated for the current hypocenters and the location of the station where the kth phase was recorded. Δx, Δy, Δz, and Δs are the changes required in the hypo central parameters to make the model better fit the data. We combine equation (6) from all hypocentral pairs for a station, and for all stations to form a system of linear equations of the form
where G defines a matrix of size M _ 4N (M, number of double-difference observations; N, number of events) containing the partial derivatives, d is the data vector containing the double-differences (4), m is a vector of length 4N, [Δx, Δy, Δz, ΔT]T, containing the changes in hypocentral parameters we wish to determine, and W is a diagonal matrix to weight each equation. We may constrain the mean shift of all earthquakes during relocation to zero by extending (7) by four equations so that
for each coordinate direction and origin time, respectively. Note that this is a crude way to apply a constraint, but appropriate for a solution constructed by conjugate gradients (see Lawson and Hanson  for more exact solutions of constrained least squares). As shown later, the double difference algorithm is also sensitive to errors in the absolute location of a cluster. Thus, equation (8) is usually down weighted during inversion to allow the cluster centroid to move slightly and correct for possible errors in the initial absolute locations.
Master event (ME) method
In contrast to the double-difference method, all relocated events are linked back to single, master event. This is a major hindrance to the method, as any location error in the master event will propagate throughout all relocated events.
The procedure for executing the double difference algorithm is adapted from (Ito, 1985). Other examples of the Master Event Approach include (Scherbaum and Wendler, 1986; Fremont and Malone, 1987; Van Decar and Crosson, 1990; Deichmann and Garcia-Fernandez, 1992; Lees, 1998). This example was chosen for it’s relative simplicity and straight forward execution.
Master event (ME) event workflow
Differences of P-wave onset times are used to determine relative hyocenters of two events. P-wave onsets i- and j-th events at k-th station, Pik and Pjk, are written as
Respectively, where Oi is the origin of the i-th event, Tik the travel time of the P-wave from i-th event to k-th station, dk the total instrumental delay at k-th station, and so on. Since seismic waves from two events are recorded at a station using exactly the same observation system, all the instruemtnal time delays, dk, can be canceled out in arrival time differences. Therefore, difference of P-wave onse times between i- and j-th events observed at k-th station τijk is written as
The difference in τij between k- and l-th stations Δτkl is
Therefore Δτkl would be due totally to a change in hypocenters, between i- and j-th events
In determination of relative hypocenters, we assume that the medium is uniform with the P and S wave velocities. This simplification is considered because we need structure parameters only for the source region. Relative hypocenters will be determined from station-to-station differences in arrival time differences between two events, referring to the location of a master event. This is done by using 4 station-to-station time differences, in order to determine three coordinates of a relative hypocenter (X,Y,Z). The limitation of 4 time-differences out of 6 expected from 4 independent signals results from the fact that the signal of S02 can not be matched to those of S-3 and S-4, because of the difference in the time delay unit.
Make some comment about how many measurements you need to solve for three parameters, for the system to be uniquely determined.
Simultaneous event method
Descriptive intro text goes here
Simultaneous event relocation workflow
Successful application of event relocation in mining applications
Application of extended double difference relocation