# Seismic event relocation techniques

## Seismic event relocation overview [1]

In the study of mining induced seismicity, the accuracy of event location and magnitudes has improved through the technological advancement of hardware and data 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 contributed greatly to the development of event relocation algorithms. 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 (also known as Single Event (SE) location)

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

Relative (relocation) event location accuracy is generally hampered by the following [2]

1. Accuracy of subsurface velocity model
2. Type of seismicity (induced vs natural)
3. Geometry of the recording seismic network
4. Size of the controlled area
5. Tectonic complexity of the controlled area

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. Though there are different approaches to solve the relocation problem, no single universal relocation algorithm will perform equally well in any situation [2]. The case examples presented are not all related specifically to mining induced seismicity, 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 and increase confidence in pre-existing data sets. 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.

## How is it done? - Styles of relocations [1] [3] [4]

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 “slave” events. Alternatively, there are “Simultaneous Relocation” approaches. Relocations are determined from 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.

Across all approaches there is a need to determine the relative time delay between similar seismic events. Relocations are usually applied to "multiplets" which are groups of similar seismic recordings in both the time and frequency domain. Evaluation of the degree of similarity is often done using signal processing techniques, specifically cross-correlation. Multiplets are often assumed to have originated from the same source mechanism, with the collapsing the hypocenters to a recognizable geologic shape being the goal.

Examples of the different workflows provided below are suggested from (Waldhauser & Ellsworth, 2000) [1]. The following papers present a more rigorous explanation of the techniques with case studies.

Double-difference Method

• Waldhauser and Ellsworth, 2000
• Waldhauser et al, 1999

Master Event Method

• Ito, 1985
• Scherbaum and Wendler, 1986
• Fremont and Malone, 1987
• Van Decar and Crosson, 1990
• Deichmann and Garcia-Fernandez, 1992
• Lees, 1998

Simultaneous Relocation method

• Got et al, 1994
• Dodge et al, 1995

### Double-difference (DD) method [1]

Seismic events origination from mining operations tend to form in clusters, or clouds. This concentration of events lends itself well to relocation algorithms. The Double-difference algorithm is designed for situations, and is less sensitive to velocity perturbations that traditional absolute event relocation. The quality of the result depends on the size of the data set as well, the more events entered into the algorithm, the higher degree of confidence in the calculated result. In mining application the number of high quality seismic recordings can be small, especially when large parts of the subsurface have been excavated, which ultimately hinders the algorithms results. One of the assumptions of the double-difference technique is a requirement that the waves from each source must be recorded at all receivers. This makes implementation of the algorithm difficult in mining applications.

Pro Con
Each event is coupled to its neighbors by direct measurements Hypocentral separation distance must be small compared to event to station distance
Dense network of catalog-based observations constrains the relative locations of multiplets and uncorrelated events Velocity model must be simple (minimal variation over event to station distance)
Relocates large numbers of earthquakes with high resolution

#### Double-difference (DD) workflow [1]

The following procedure for executing the double difference algorithm is sourced 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,

$T^i = \tau^i + \int_i^k uds \qquad (1)$

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 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:

$\frac {\partial t^i_k} {\partial \mathbf{m}} \Delta \mathbf{m}^i = r^i_k \qquad (2)$

where

$r_k^{i} = (t^{obs} - t^{cal})_k^{i}$

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,

$\frac {\partial t_k^{ij}} {\partial \mathbf{m}} \Delta \mathbf{m}^{ij} = dr_k^{ij} \qquad (3)$

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. 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) is the residual between observed and calculated differential travel time between the two events defined as

$dr_k^{ij} = (t_k^i - t_k^j)^{obs} - (t_k^i - t_k^j)^{cal} \qquad (4)$

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

$\frac {\partial t_k^{i}} {\partial \mathbf{m}} \Delta \mathbf{m}^i - \frac {\partial t_k^{i}} {\partial \mathbf{m}} \Delta \mathbf{m}^j = dr_k^{ij} \qquad (5)$

or written out in full

$\frac {\partial t_k^{i}} {\partial {x}} \Delta {x}^i + \frac {\partial t_k^{i}} {\partial {y}} \Delta {y}^i + \frac {\partial t_k^{i}} {\partial {z}} \Delta {z}^i + \Delta \tau^i - \frac {\partial t_k^{j}} {\partial {x^i}} \Delta {x}^j - \frac {\partial t_k^{j}} {\partial {y}} \Delta {y}^j - \frac {\partial t_k^{j}} {\partial {z}} \Delta {z}^j - \Delta \tau^j = dr_k^{ij} \qquad (6)$

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

$\mathbf{WGm} = \mathbf{Wd} \qquad (7)$

where G defines a matrix of size M X 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

$\sum_{i=0}^N \Delta {m}_i = 0 \qquad (8)$

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. 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 [6]

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.

Pro Con
Relatively simple computationally Errors due to the correlation of noise in the master event may propogate through the entire cluster and effect the location of all other events
If geology and acoustic velocity surrounding the event cluster is uniform, consistent results are expected Limited in maximum spatial extension of the cluster that can be relocated, as all events must correlate with a single master event

The procedure for executing the double difference algorithm is sourced from (Ito, 1985) . This example was chosen for it’s relative simplicity and straight forward execution.

#### Master event (ME) event workflow [6]

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

$P_{ik} = O_i + T_{ik} + d_k = 0 \qquad (9)$

And

$P_{jk} = O_j + T_{jk} + d_k = 0 \qquad (10)$

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

$\tau_{ijk} = (O_i + T_{ik}) - (O_j + d_k) \qquad (11)$

The difference in τij between k- and l-th stations Δτkl is

$\Delta \tau_{kl} = \tau_{ijk} - \tau_{ijl} = (T_{ik} - T_{jk}) - (T_{il} - T_{jl}) \qquad (12)$

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).

### Simultaneous event method [8]

For a detailed explanation of the extension from the Master Event relocation to a Simulatenous Relocation algorith see Got et al. (1994) . Got overcame the restrictions of the Master Event algorithm by determining cross-correlation time delays for all possible event pairs and combining them in a system of linear equations that is solved by least-squares methods to determine hypocentroid separations. This method is also completed in Fréchet, 1985. For simplicity, a constant slowness vector was used for each station from all sources. Because only cross-correlation data is considered, this approach cannot relocate uncorrelated clusters relative to each other.

Pro Con
No spatial limitation on relocated events, provided they correlate with neighboring events Because only cross correlation data is considered, cannot relocate uncorrelated clusters relative to each other
Several times correlation computations to be completed
Many “useless” correlation computation done due to low coherency ratings (throw away)

## Successful application of event relocation in mining applications [2] [9]

The following two studies investigated the application of double-difference algorithms to the problem of mining induced seismicty. Results showed some of the fundamental assumptions of the techniques interefered with stable, reliable soultions. Below are the abstracts from the two papers by, Rudziński, L., & Dȩbski, W.

### Application of extended double difference relocation [9]

Abstract: The location of the seismic event hypocenter is the very first task undertaken when studying any seismological problem. The accuracy of the solution can significantly influence consecutive stages of analysis, so there is a continuous demand for new, more efficient and accurate location algorithms. It is important to recognize that there is no single universal location algorithm which will perform equally well in any situation. The type of seismicity, the geometry of the recording seismic network, the size of the controlled area, tectonic complexity, are the most important factors influencing the performance of location algorithms. In this paper, we propose a new location algorithm called the extended double difference (EDD) which combines the insensitivity of the double-difference (DD) algorithm to the velocity structure with the special demands imposed by mining: continuous change of network geometry and a very local recording capability of the network for dominating small induced events. The proposed method provides significantly better estimation of hypocenter depths and origin times compared to the classical and double-difference approaches, the price being greater sensitivity to the velocity structure than the DD approach. The efficiency of both algorithms for the epicentral coordinates is similar.

### Relocation of mining-induced seismic events in the upper silesian coal basin by use of double difference method [2]

Abstract: The application of the double-difference (DD) algorithm to the relocation of induced seismic events from the Upper Silesian Coal Basin is discussed. The method has been enhanced by combining it with the Monte Carlo sampling technique in order to evaluate relocation errors. Results of both synthetic tests and relocation of real events are shown. They are compared with estimates of the classical single-event (SE) approach obtained through the Monte Carlo sampling of the a posteriori probability. On the basis of this comparison we have concluded that the double-difference approach yields better estimates of depth than the classical location technique.

## References

1. 1.0 1.1 1.2 1.3 1.4 1.5 Waldhauser, F. (2000). A Double-Difference Earthquake Location Algorithm: Method and Application to the Northern Hayward Fault, California. Bulletin of the Seismological Society of America, 90(6), 1353-1368. doi:10.1785/0120000006
2. 2.0 2.1 2.2 2.3 2.4 2.5 Rudziński, L., & Dȩbski, W. (2012). Extending the double difference location technique—improving hypocenter depth determination. Journal of Seismology, 17(1), 83-94. doi:10.1007/s10950-012-9322-7
3. Deichmann, N., & Garcia-Fernandez, M. (1992). Rupture geometry from high-precision relative hypocentre locations of microearthquake clusters. Geophysical Journal International, 110(3), 501-517. doi:10.1111/j.1365-246x.1992.tb02088.x
4. Dodge, D. A., Beroza, G. C., & Ellsworth, W. L. (1995). Foreshock sequence of the 1992 Landers, California, earthquake and its implications for earthquake nucleation. Journal of Geophysical Research: Solid Earth, 100(B6), 9865-9880. doi:10.1029/95jb00871
5. The difference between convolution and cross-correlation from a signal-analysis point of view. (n.d.). Retrieved February 26, 2017, from http://dsp.stackexchange.com/questions/27451/the-difference-between-convolution-and-cross-correlation-from-a-signal-analysis
6. 6.0 6.1 6.2 6.3 6.4 Ito, A. (1985). High resolution relative hypocenters of similar earthquakes by cross-spectral analysis method. Journal of Physics of the Earth, 33(4), 279-294. doi:10.4294/jpe1952.33.279
7. S. (2011, May 04). Visualization of Statistical Cross Correlation Output - statistics teaching tool. Retrieved February 28, 2017, from https://www.youtube.com/watch?v=qhFr1ZmWpzs&spfreload=10
8. 8.0 8.1 8.2 Got, J., Fréchet, J., & Klein, F. W. (1994). Deep fault plane geometry inferred from multiplet relative relocation beneath the south flank of Kilauea. Journal of Geophysical Research, 99(B8), 15375. doi:10.1029/94jb00577
9. 9.0 9.1 9.2 9.3 Rudziński, L., & Dȩbski, W. (2005). Relocation of mining-induced seismic events in the upper silesian coal basin, Poland, by a double-difference method. Institute of Geophysics Polish Academy of Sciences, 5(2), 150th ser., 97-104. Retrieved February 23, 2017