Seismic event relocation techniques
- 1 Seismic event relocation overview 
- 2 When is relocation completed?
- 3 How is it done? - Styles of relocations   
- 4 Successful application of event relocation in mining applications  
- 5 References
Seismic event relocation overview 
Initial (absolute) event location accuracy is generally hampered by the following (also known as Single Event (SE) location)
- 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
Relative (relocation) event location accuracy is generally hampered by the following 
- Accuracy of subsurface velocity model
- Type of seismicity (induced vs natural)
- Geometry of the recording seismic network
- Size of the controlled area
- 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 . 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.
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.
Figure 1: In signal processing, cross-correlation is a measure of similarity of two series as a function of displacement of one relative to another. This is also known as a sliding dot product or sliding inner product. This technique is used in most relocation algorithms to identify similar events, also known as multiplets. If a group of events share many of the same characteristics they may be grouped together with the goal of "collapsing" the hypocenters to improve their positional certainty 
Figure 2: Representative seismic traces ready for cross-correlation. All traces appear to have a high degree of coherency (similarity). From the cross-correlation algorithm time delays will be derived between each receiver which can be converted to positional adjustments through a velocity model. 
- Figure 3:
CLICK IMAGE ABOVE FOR VIDEOTwo synthetic time series are cross-correlated to determine their level of coherency (Rho), as the two function overlap they are identical, and a Rho value of one is derived. This Rho value can be used to classify events, eliminate outliers and improve the accuracy of relocation algorithms 
Examples of the different workflows provided below are suggested from (Waldhauser & Ellsworth, 2000) . The following papers present a more rigorous explanation of the techniques with case studies.
- 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 
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.
|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|
Figure 4:Above is an illustration of the double-difference relocation algorithm. Solid and open circles represent trial hypocenters that are linked to neighboring events by cross-correlation (solid lines) or catalog (dashed lines) data. For two events, i and j, the initial locations (open circles) and corresponding slowness vectors, s, with respect to two stations, k and l, are shown. Ray paths from the sources to the stations are indicated. Thick arrows (Δx) indicate the relocation vector for events i and j obtained from the full set of equations (5) and dt is the travel-time difference between the events i and j observed at station k and l, respectively. 
Figure 5: Relocation of 28 highly correlated events with three independent data sets (a) NCSN locations (b)Double-difference locations computed using the difference of NCSN catalog times for each event pair (c)P-wave cross correlation travel-time differences (d)S-wave cross-correlation travel-time differences and (e) all three data types.
Double-difference (DD) workflow 
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,
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:
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. 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
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 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
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 
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.
|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.
Figure 6: All shapes (circles, triangles and rectangles) are events that have been relocated with respect to single master event. Error bars show 1 standard deviation of the epicenters. 
Figure 7: Cross-correlation example for the P-waves on the vertical component of a single station (top) and a sing S-wave component. The top trace of each signal pair is the master event. The tick marks correspond to 0.1s. Sampling rate is 100 Hz. For the P-wave correlation a time window of 0.2s was shifted between the maximum lags of ±0.1s. For the S-waves the time window was 0.2s and the maximum lag 0.2s. The vertical bar on the traces of the master event shows the arrival time picked by eye, while on the lower trace it shows the the corresponding arrival determined from the correlation. The small diamonds in the right side of each diagram denote the discrete values of the cross-correlation for each shift of one sampling interval. The continuous curve is the parabola fitted by least squares through the peak of the discrete values. The resulting time differences between the arrivals of the master event and slave event are given by the value delta. 
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).
Simultaneous event method 
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.
|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)|
Figure 8: Plot of all coherency measurements for all possible pairs of initial location (1095 events). The top (open circles) represent the coherency for the relocated multiplet (252 events). The bottom dots represent the coherency for the non-relocated part of the initial selection (843 events). Each dot or open circle indicates a coherence greater than 90% for the corresponding pair of events. Each coordinate is the index of the each event event in the chronological list of earthquakes selected. Notice the higher number of coherent pairs (doublets) in the initial selection,, and the strong proportion of doublets from the initial section finally relocation (top). 
Figure 9:Projection of the relocated cluster on a N-S vertical plane shwoing absolute locations (stars) and relative relocation with their 90% confidence ellipses (open ellipses). 
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 
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.
Figure 10: 2-D a posteriori PDF for the origin time T and depth Z coordinates obtained by a single event relocation algorithm. Note the strong and nonlinear correlation between these two parameters which is not visibible in the DD solution (Figure 11) 
Figure 11: 2-D a posteriori PDF for origin time and depth (T,Z) for one of the events from the cluster obtained using the DD algorithm. There is almost no correlation between Z and T, which means that the depth is resolved independently from T. This is quite different from he results of the single-event relocation techniques. 
Relocation of mining-induced seismic events in the upper silesian coal basin by use of double difference method 
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.
Figure 12: Systematic shift of the maximum likelihood solutions with respect to the true one as a function of the velocity for three relocation algorithms (Single event, Double-difference, and Extended Double-difference". 
Figure 13:Dependence of the location errors estimated from the a posteriori covariance matrix on the velocity for the considered relocation algorithms (Single event, Double difference, and Extended double difference) 
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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