Determining Relative Source Locations using
Coda Waves: Applying the theory to data
Mark Vrij landt
Center for Wave Phenomena, Department of Geophysics
Colorado School of Mines, Golden, Colorado 80401
April 14, 2004
Abstract
The location of seismic events can be determined from waveforms recorded at different stations using a velocity model. Errors in this velocity model introduce an error in the computed locations of the events. Determining the relative location between two events is less dependent on errors in the velocity model than determining the abso-lute location. Snieder constructed a theory [1] to use the coda of the waveforms, instead of first arrivals, to compute relative source loca-tions. In this study we set out to test this theory using synthetic and real data. If the distance between the sources is small and both events are recorded at the same station, the theory can be applied. The rel-ative distances computed from the coda of the synthetic waveforms agree with the actual distance between the sources. We compare the relative distances computed from earthquake data with relative dis-tances resulting from the double difference algorithm by Waldhauser and Ellsworth [2]. The distances computed from the coda agree with the distances from the double difference algorithm. A next step could be to automize the process and to incorporate it as an extra constraint in an earthquake location algorithm.
1 Introduction
Determining source locations is useful in earthquake studies and reservoir characterization. When the relative positions of events closely spaced
to-Figure 1: Two events occuring close together and their direct wavepaths.
gether are known, the location and orientation of the fault zone can be mapped [3, 4]. In earthquake research i t is important to know the orien-tation and position of the fault plane to understand the tectonic setting. A moving pressure front in a producing reservoir can produce seismicity [5 along pre-existing faults. Faults in a reservoir influence the availability of any hydrocarbons that are present.
Relative source locations can be obtained by first computing the absolute locations of both events and subsequently determining the relative location. When a source is located absolutely, the velocity along the whole path from source to receiver is used. Errors in the velocity model along the whole path map erroneously into the source location. I n relative source location is it assumed that waves from both sources travel along the same path outside the source region (see Figure 1). This approach is advantageous because i t only uses the velocity in the region of the sources [2, 6, 7]. The sources have to occur close together for this assumption to hold.
To compute the relative distance between two sources using the coda of the waveforms we also assume that the sources occur close together. Subse-quently we assume that the path from the first scatterer to the receiver is the same for both source locations and that the effect of geometrical spreading is subdominant to the effect of the travel time difference. Finally we assume that the sources in the two locations ai'e similar, meaning that they have the same radiation pattern. The difference in travel time between the waves due to a change in the soure position occurs between the source and the first scatterer (see Figure 2). The main difference between using direct arrivals and coda waves is that energy radiated in all directions from the source is present in the coda, whereas the take-off direction of the direct arrivals can be calculated. I n the coda not a certain time difference is present but the measured time difference is an average of many scattered waves that travelled
Figure 2: Two events occuring close togetiier and two scattered wavepaths. The main difference between the wavepaths occurs between the source and the first scatterer.
along different paths.
In this paper we first give an overview of the theory of Snieder [1]. This is followed by an application of the theory to synthetic data which are con-structed using finite differences. Finally, we use VSP data acquired by the Reservoir Characterization Project at Colorado School of Mines and earth-quake data from the Northern California Seismic Network to test the theory on real data.
2 Theory
In this section we present the theory of Snieder [1] which was used to de-termine the relative location between seismic events. For a more thourough presentation of the theory we refer to this paper.
Assume that the displacement at the receiver resulting from the source can be written as a superposition of waves with different wavepaths
u^-\t)=Y:AT{t). (1)
TThe summation over the different trajectories includes all possible wavepaths and wave modes (P-wave or S-wave) from source to receiver. The superscript (u) tells us that this is the wavefield resulting from the original (unperturbed)
source position. Subsequently we assume that changes in geometrical spread-ing and radiation are subdominant to the effect of the travel time difierence. Then, the wavepaths resulting from the perturbed source position only differ from the unperturbed wavepaths by a time difference due to the difference in pathlength. W i t h these assumptions the perturbed wavefield can be written as
U^\t) = Y:AT{t-TT). (2)
T
We use the cross correlation to investigate the travel time perturbation (the lag) and the similarity between the two waveforms, We compute the cross correlation for different time windows along the seismograms. The normalised cross correlation for a time-window with center t and duration 2ty, is given by
r'^u^''\t') u^\t'+ ts)dt'
^ ' • " ^ ( ^ ^ ) = ^ . . . r " : . . . . . . . .xv^
where ts is the lag time for the correlation. Applying a Taylor expansion for small values of {tg - TT) Snieder [8] shows that E^*'*"") reaches a maximum when ts = {r)(t,t^, where {T){tu '^^ ^'^^ average lag in a certain time-window. Subsequently, this maximum equals
^ ^ 1 ^ ^ = 1 - (4) where is the average of the square of the dominant frequency within the
time-window given by
_ / U^^'^t') ü^''\t')dt' / ü^^^\t')dt'
/ u^^^\t!)dt! \ u^^^^{t!)dt!
J i j — J i — i j y j
and (Tr is the variance of the travel time perturbation of the arrival times given by
where (r) is the average lag weighted by the energy within the time-window. The maximum of the cross correlation and can be computed from the unperturbed and perturbed waveforms. So with equation (4) we can deter-mine the variance of the travel time perturbations for two waveforms. Now, how does ar relate to the distance between the sources which caused the perturbations?
When the distance between the two sources is small with respect to the distance between the source and the first scatterer, it can be assumed that waves from both sources have a similar angle of approach to the first scatterer. As a result, the unperturbed and perturbed paths are similar- from the first scatterer on to the source (see Figure 2), W i t h this assumption the relative time diflference between the two paths occurs between the source and the first scatterer. The time difference corresponds to the distance between the sources projected onto the direction in which the wave takes off from the source
r = - i ( r • Ö) (7) where 5 is the distance between the sources, v is the velocity in the source
region and f is the direction i n which the wave takes off from the sources. The time perturbation computed using the time-windowed cross corre-lation is not the time difference for a certain wavepath but an average for all the waves arriving i n the time-window, All these waves interfere result-ing in the measured waveforms. We use a Green's function to describe the wavefield between the source and the first scatterer. W i t h equation (7) i t can then be shown that the average travel time perturbation (r) is zero [1]. This can also be shown intuitively. Waves leaving the sources in all direc-tions can reach the source by scattering through the medium. The travel time perturbation depends on the take off direction and orientation of the sources. For some takeoff directions the perturbation wül be positive, for other directions negative. The travel time perturbation can even be zero if the talce off direction is perpendicular to the distance between the sources. I f the scatterers are distributed homogeneously around the sources the average travel time perturbation will be zero (see Figure 3). This does not hold for large earthquakes, in this case the energy is not radiated homogeneously in aU directions. As the average travel time perturbation is zero: = (r^). Using equation (7) and the decomposition into the Green's function and the source spectrum, cr^ is expressed as a function of the distance between the two sources. This expression depends on the Green's function, which in turn.
Figure 3: Two events and direct wavepaths to several scatters.
depends on the type of medium and source.
Consider a simple case: an isotropic source in an acoustic medium, the setting for the synthetic experiment. Snieder [1] investigated the 3D case, however the synthetic experiment is carried out in 2 dimensions, in which case the wavefield can be written as follows [9
= G'^{T)S{UJ) = - ^ ^ « ( f c r ) S ( a ; ) (8)
where G'^^{r) is the Green's function for the Helmholtz equation in two dimensions, HQ^\kr) is the first Hankel function of degree zero and S{u}) is the source spectrum. Together with equation (7) this wiU result in the expression
[ f ^ { r - Sf I - '-Hi'\kr)S{uj)\''dndw
( ^ 2 ^
= LUi ..
4 (9)J l \-lH^'\kr)S{uj)\''dÜdu
where v is the velocity around the sources and ƒ ƒ ' ' ' dÜdu} represents an integration over all outgoing directions and frequencies. I f the scatterers are
Figure 4: Two double couple sources.
randomly distributed around the sources, r is on average the same for all directions. Then expression (9) simplifies to
Using polar coordinates
f cosO \
r= oi./) (11) and aligning Ö with the y-axis gives ( r • öf = ö'^sinPO and therefore
For the earthquake data consider a double couple source in an elastic medium. A n analogous derivation as for an isotropic source in an acoustic medium (however more complicated) is carried out by Snieder [1]. The re-sulting relationship between the variance of the perturbation of the travel time and the distance between the two sources is given by
(4 +
i )
^//ja y . , ^ s y ^//fanU +
^[il +
i)Ö±fault- 7/ 2 , 3 ^ • ^ ^ ^ ^
For aftershocks that occur on the same fault i t is generaUy true that both sources have the same fault plane and that the displacement between the two sources is in this fault plane (see figure 4), I n this case 5±fault = 0 and then equation (13) yields a relationship between cr-r and the distance between the two sources. The assumption of a Poisson medium {vp = V^Vs) simplifies expression (13) into
This equation (14) is used to test the theory. However, if only S-waves are considered (the P-wave velocity is set to zero), equation (13) simplifies to
\ | 3t;2 ~ V3 (15)
which is the same result as for an isotropic source in an acoustic medium in 3D. The similarity between equations (14) and (15) also confirms the idea that S-waves are dominant over P-waves in the coda [10],
3 Applying the theory to data
We compute the distance between two events using the coda of the waveforms in the following way. Firstly, we plot waveforms of both events recorded at the same station. Comparing the first arrivals allows checking for repeatability of the source mechanism. Then a time window of certain length is slid along the waveforms. We choose the length of the time window in such a way that it contains a multitude of wavelengths (at least 10 wavelengths). We use a timestep between two subsequent windows of one tenth of the length of the time window. This means that adjacent time windows overlap. We apply a taper within each time window to counter edge effects.
Subsequently we compute the maximum of the normalized cross correla-tion for each time window. We convert the maximum of the cross correlacorrela-tion into the variance of the travel time perturbation, which we subsequently convert into distance using expressions (4) and (12). To obtain the distance using this single receiver, we average the distances that were computed in a time window within the coda.
4 Synthetic Data
We construct a synthetic experiment to test the theory. A finite differences code from CREWES (Consortium for Research i n Elastic Wave Exploration Seismology, University of Calgary) computes the synthetic waveforms. This code uses absorbing boundary conditions. We compute the synthetic wave-forms for a 2D grid of 20 km by 20 km in size. The gridspacing in both the vertical and horizontal direction is 20 m. The setup of the experiment is depicted in Figure 5. The R's represent the positions of the receivers. The
(x = 0,z = 0) r ( x = 20 km,z = 0) R R R R R R X I X2 R (x = 0,z = 20 km) (x = 20 km,z = 20km)
Figure 5: Setup o f t h e 2D synthetic experiment with a gridsize of 20 m. The R's represent the positions ofthe receivers. We conduct the experiment twice, once with the source in position X I and once with the source in position X2, the distance between X I and X2 is 160 m.
receivers are positioned all around the sources, some receivers will not record a time dilference in the first arrivals because the distance to both sources is approximately the same. We conduct the experiment twice, once with the source in position X I and once with the source in position X2. A l l the other parameters are the same for both experiments. The distance between the two source positions is 160 m, which means 8 gridpoints. The 2D velocity model is given in Figure 6. I t is an Gaussian field which results in multiple scattering. The average velocity is 6000 m/s, the standard deviation of the velocities is 1500 m/s and the correlation distance is 1000 m. I n the theory we assume that the distance between the sources is much smaller than the distance from the source to the first scatterer. Therefore, we smooth the region around the sources. By smoothing the region around the sources, we also avoid placing the sources directly on an anomaly. As a source we use a Richer wavelet with a dominant frequency around 8 Hz.
Having computed the synthetic waveforms we consider two waveforms that ai-e recorded at the same receiver location. Such a pair of waveforms is plotted in Figure 7. The first part of the waveforms is similar with a small lag between the waveforms. This corresponds to the direct arrivals. Less
Figure 6: Gaussian velocity model used in the synthetic experiment. The area around the sources is smoothed.
similarity exists in the coda.
For the time windowed normalised cross correlation a time window of length 1.6 s is used. We compute different values for the cross correlation by sliding the time-window along the waveforms. The maximum for each time-window for the normalised cross correlation is plotted in Figure 8 for a pair of seismograms recorded at the same station. The top panel shows the two seismograms. The horizontal bar indicates the size of the time-window in which we compute the cross correlation. The maximum of the cross correlation for each time-window is plotted in the middle panel. The time at which the cross correlation maximum for a certain time-window is plotted, is the time of the center of the time-window. The high values of
Umax in early times correspond to the high degree of similarities in the first arrivals. For later times Rmax decreases as the time-window moves into the coda. The computed distance is plotted in the bottom graph of Figure 8. The dashed line indicates the actual source displacement of 160 m. The computed distance varies but is close to the actual value. To obtain an estimate for the distance using this single receiver, the average of the distances after t = 5.6 s is taken. This is done to make sure that all distances that are included
t i m e [s]
Figure 7: Synthetic seismograms for two source locations recorded at the same station. The top and bottom panel show close-ups of the direct arrivals and a section of the coda respectively.
time [s]
Figure 8: The cross-correlation and calculated distance for a synthetic pair of seismograms. The top panel shows the two seismograms and the horizontal bar indicates the size of the time-window in which we compute the cross correlation. The maximum of the cross correlation for each time-window is plotted i n the middle panel. The time at which the cross correlation maximum for a certain time-window is plotted, is the time of the center of the time-window. The computed distance is plotted in the bottom graph.
in the average were computed in time-windows that were in the coda and not in the first arrivals. We compute 9 independent distances since there are 9 receivers. The average of these 9 distances is 174 m and the standard deviation is 15 m.
5 VSP data
This section deals with testing the theory using VSP data acquired by the Reservoir Charaterization Project at Colorado School of Mines. I n the the-ory we assume that the scatteres are distributed homogeneously around the sources. For the VSP data this assumption may not be valid because the region is known to consist of approximately horizontal layering. Another problem can arise because of the receivers in the borehole. The interaction between the borehole and the receiver can induce a high level of noise.
Receivers in the borehole record the waveforms with a vibroseis source on the surface. A total of 96 traces were recorded and 8 shots were used to record the traces. In one of every 12 receiver locations two traces were recorded which allows testing for repeatability of the source. All 96 traces are plotted in Figure 9.
The theory was designed to determine relative source locations but be-cause of reciprocity i t can also be used to determine relative receiver locations. The reciprocity holds because the source is a point source (the vibroseis) and the receivers are displacement sensors. The computed distances can be ver-ified as the receiver spacing is known to be 15 m. We computed the lag between the two signals and the lag of the first arrivals converts into a dis-tance using a I D velocity model. This disdis-tance is close to the receiver spacing. Subsequently the lag changes sign corresponding to primary reflections from a horizontal interface below the two receivers. Unfortunately the coda can-not be distinguished from the noise level. As the signal-to-noise level is too low for our purposes, these VSP data are not suitable to test the theory.
6 Earthquake Data
The earthquake data were recorded by the Northern California Seismic Net-work (see Figure 10). We use a cluster of 28 earthquakes that occurred close together (the Berkely Cluster [2]) to test the theory. Waldhauser and
20 40 60 80
Figure 9: The VSP data: 96 traces were recorded. The horizontal axis represents the receiver number, the receiver spacing is 15 m and one in every 12 traces is recorded in the same location. The vertical axis represents time in seconds.
45' 40-i m " ^ 40-i ? ; \ ''\, \ \ \ )j / % l i * J IS % ^ " 6 -104" -122" -120" -118"
NCSN Stations
"Analog - short peiio d <#Digitfll - b orehole 1 DigLtal - bro a db and « Digital - str ong-motlon Figure 10: Map of the station of the NCSN (Northern California Seismic
Net-Ellsworth constructed an algorithm (Double Difference algorithm) that in-verts for hypocenter locations using absolute and relative travel time mea-surements [2]. The Double-Difference algorithm minimizes residuals between observed and theoretical travel times (double differences) for a large dataset. We compare our results to the results of this inversion.
We filtered the waveforms with a band-pass Butterworth filter, filtering out the DC component and frequencies greater than 5 Hz. This decreases the dominant frequency to an average of 3 Hz. We align the seismograms in time so that the first arrivals coincide. This will affect the lag of the cross correlation but not its maximum Rmaa:- Figure 11 shows two events that oc-cured close together and were recorded at the same station. I n Figure 11 one of the two seismograms is scaled because of a difference in magnitude between the two events. As for the synthetic data, the first arrivals are very similar. This imphes that the source mechanism for both events is similar. There is a magnitude difference because the eaiThquakes are of different magnitude, however this does not affect the value of the cross correlation.
We choose the time-window for the cross-correlation to be 10 s. A pair of seismograms recorded at the same station is plotted in the top panel of Figure 12. The middle panel shows the cross correlation maximum for different positions of the time-window. We apply a noise correction to the cross correlation and the cross correlation is only computed when the signal-to-noise ratio reaches a certain value (see Appendix A ) . On average, the cross correlation decreases i n time due to the decreasing amplitude of the waveforms which lowers the signal-to-noise ratio. A higher level of noise decreases the similarity between the two waveforms.
In the bottom panel of Figure 12 the computed distance between the two sources is plotted. The dashed lines indicate the double difference distance from Waldhauser and Ellsworth plus and minus one standard deviation. The distance calculated from coda waves agrees with the double difference dis-tance, calculated from direct arrivals. When the signal becomes weaker the distance from the coda waves starts to deviate.
The distance between two events can be determined from many different receiver locations. Figure 13 shows three different event pairs (each i n a different panel) and the distance for each event pair is computed for four different stations. This results in four indepently computed distances for each event pair. Again the dashed lines give an indication of the double difference distances, whose average values are also displayed as text. The computed distances agree with the results from Waldshauser and Ellsworth, When the
t i m e [s
Figure 11: Seismograms from the earthquake data for two different events recorded at the same station. The top and bottom panel show close-ups of the direct arrivals and a section of the coda respectively. One of the two seismograms is scaled because of a difference i n magnitude between the events
^0.8S
-X
0 „ 3 0 0E
0 2001
100 V) b 0 1 1 20 1 30 40 1 1 50-
1 1 1 -- \-- ^
1 1 1 1 -20 30 40 50 t i m e [ s ]Figure 12: The cross-correlation and calculated distance for a pair of seismo-grams from the earthquake data: event ID's 238295 and 242003 recorded at station CAL The top panel shows the two seismograms and the horizontal bar indicates the size of the time-window in which we compute the cross correlation. The maximum of the cross correlation for each time-window is plotted in the middle panel. A noise correction is applied to the cross cor-relation and the cross corcor-relation is only computed when the signal-to-noise ratio reaches a certain value (see Appendix A ) . The time at which the cross correlation maximum for a certain time-window is plotted, is the time of the center of the time-window. The computed distance is plotted in the bot-tom graph and the dashed lines indicate the double difference distance from Waldhauser and Ellsworth plus and minus one standard deviation.
station CAI station CBW station CMC station CSP 0 300 200 iS 100 c (0 b - / 1 ^ 1 ' 1 DD-distance; 106.85 m T T T A / -1 1 , 1
1
20 30 40 50 300E
o
200 Ü c CO 100 CO b - T DD-distance; 76.53 m ' I ' l 1 -v : - 1 ^ ^ • , ' - ^ 1 , 20 30 time [s] 40 50Figure 13: Calculated distances for 3 different event pairs, each panel corre-sponds to a different event pair: pair number 1, in the top panel, consists of event ID's 238295 and 242003, pair number 2, in the middle panel, consist of event ID's 242003 and 242020, and pair number 3, in the bottom panel, con-sists of event ID's 402093 and 402094. A l l three event pairs were recorded at the stations C A I , CBW, CMC and CSP. The dashed lines give an indication of the double difference distances, whose average values are also displayed as text.
pair number Average Computed Distance DD Distance
1 124 m + / - 19 m 128 m + / - 29 m 2 63 m + / - 11 m 107 m + / - 29 m 3 68 m + / - 14 m 77 m + / - 29 m Table 1: Average computed distance and standard deviation for 3 different event pairs. The double difference distances and standard deviations are also given.signal-to-noise ratio becomes too low the distances start to deviate. The magnitude of the event pair in the top panel is higher relative to the event pairs in the middle and bottom panel. This means that the signal-to-noise ratio is higher and therefore the distances can be computed for longer times. Using these four station the averages and standard deviations are computed (see Table 1). For the averages only the distances are considered before the distance starts deviating. Waldhauser and Ellsworth obtained their standard deviations by statistically resampling. The computed distances for pairs 1 and 3 are close to the DD distances. The computed distance for the second pair differs by just over a standard deviation from the DD distance. The distances computed for different stations agree well for all three event pairs.
7 Discussion
A n approximation is made for Rmax in the theory (see equation (4)), this approximation valid for small values of {ts — TT). Since ts = {T){t,t^) the approximation is valid for small values of a^. To determine the accuracy of the approximation for different values of a^, the actual cross correlation and the approximation are plotted in Figure 14 for a representative time window of the cross correlation of a pair of earthquake waveforms. I n the top panel, the thin line represents the actual cross correlation, the thick line is the approximation used in the theory. To relate the relative error in Rmax
to an error in the computed distance consider expressions (4) and (15). From these two equations it can be concluded that
d5 d<Jr dRfnax Rmax dRmax (16) 6 C T r Rmax 2(1 — Rmax) 2(1 — Rmax)
Using formula (16) we computed the relative error in the distance and plotted the relative error in the bottom panel of Figure 14. When r is 0.06 s the
^ I — 1 — I
—
I
—
I
—
\
—
I i I I I
-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 OJI
[s]
Figure 14: Evaluation of the approximation of the cross correlation. I n the top panel, the thin line represents the actual cross correlation and the thick line (the parabola) is the approximation used in the theory. The bottom panel shows the relative error in distance as a result of the approximation.
relative error in the distance is about 35 percent. The S-wave velocity at the depth ofthe Berkeley cluster is 3300 m/s [11] so this value of r corresponds to 198 m. Since the average dominant frequency content of the signal is about 3 Hz, the dominant wavelength is 1100 m. Therefore, if the distance between the two sources exceeds a fifth of a dominant wavelength, the maximum error in Rjnax will be over 35 percent. This constrains the distance between the sources for this theory to be applied. However, if the distance is larger than a fifth of a wavelength the time differeiice will be clear in the direct arrivals. R-om equation (4) it is clear that influences the approximation. By lowering the frequency content of the data, the average dominant frequency will decrease, thereby increasing the cr^ for which the approximation doesn't deviate too much from the actual value. This is the reason we applied a band pass filter to the data.
The distances computed using coda waves are inaccurate when the dis-tance between sources becomes very small because of the Taylor expansion used in the approximation for Rmax- When the sources are very close together the two waveforms will be very similar and therefore the cross correlation will have a value close to one. Consider the second term in the righthandside of equation (16), if Rmax becomes close to one this term will go to infinity. This means that the relative error in Rmax translates into a large relative error in the computed distance. This effect is countered to a certain extent by a low value of The relative error in Rmax will be low because i f the sources occur close together, the approximation for Rmax will be very accurate.
A measure of the accuracy of the computed distances using coda waves is given by the standard deviations of the indepently determined distances using different stations. For the earthquake data these standard deviations are smaller than for the double difference algorithm. These standard devia-tions were computed from only four indepent measurements while the double difference locations were computed from a large dataset of travel times and waveforms. The application of this theory is limited by the distance be-tween two events. Distances computed from coda waves could be used as an extra constraint for locating earthquakes in algorithms such as the double difference algorithm.
An important difference exists between the resulting relative distances using coda waves and direct arrivals. The time difference of the first arrivals corresponds to the difference in travel time along the direct wavepath. The resulting distance will be the distance between the two sources projected onto the takeoff direction from the source. The distance determined from
coda waves is an absolute distance. These two distances combined can give information the relative orientation between the two sources.
8 Conclusion
We tested the theoiy of determining relative source locations using coda waves on synthetic and earthquake data. This approach is applicable i f the two events occured close together. The actual and computed distance for the synthetic data are within one standard deviation. We compai'ed the results for the eai'thquake data with results from the double difference algorithm by Waldhauser and Ellsworth [2], For two o f t h e three event pairs the distances agree within one standard deviation and for all three of the event pairs the independently determined distances agree amongst each other. The theory seems to work on both synthetic and real data. The next step would be to automize the process to be able to add it as an extra constraint for seismic source localization,
Acknowledgments
Scholarships from Stichting Molengraaff Fonds, Karel Frederik Stichting and Utrecht University enabled me to perform this research at Colorado School of Mines. Special thanks to Alexandre Grêt for useful comments and dis-cussions, Felix Waldhauser for supplying the event ID's for a useful cluster of earthquakes recorded by the Northern California Seismic Network and for supplying me with the results frora the double difference algorithm, Carlos Pacheco for helping me with setting up the synthetic experiment, Ludmila Adam for helping me with the VSP data, Huub Douma, John Stockwell and Greg Wimpey. Also I would like to thank my advisor at Utrecht University, Hanneke Paulssen. M y stay in Colorado was great thanks to many people that I met at Colorado School of Mines, I would like to thank them all, es-pecially Kjetil, Ivan, David, Lotta and Eleanor. Most importantly I would like to thank Roel Snieder for his enthousiastic and patient guidance.
References
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2] F. Waldhauser and W . L . Ellsworth. A double-difference earthquake location algorithm: Method and application to the Northern Hay ward fault, California. Bull. Seism. Soc. Am., 90:1353-1368, 2000.
3] G.S. Fuis, T. Ryberg, W.J. Lutter, and P.L. Ehlig. Seismic mapping of shallow fault zones in the San Gabriel mountains from the Los An-geles region seismic experiment, southern California. J. Geophys. Res., 106:6549-6568, 2001.
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6] P.M. Shearer. Improving local earthquake locations using the Li-norm and waveform cross-correlation: Application to the Whittier Narrows California aftershock sequence. J. Geophys. Res., 102:8269-8283,1997. 7] J.L. Got, J, Frèchet, and F.W. Klein. Deep fault plane geometry inferred
from multiplet relative location beneath the south flank of Kilauea. J. Geophys. Res., 99:15375-15386, 1994.
8] R. Snieder. Coda wave interferometry and the equilibration of energy in elastic media, Phys. Rev. E., 66(046615):l-8, 2002.
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11] F. Waldhauser and W.L. Ellsworth. Fault structure and mechanics ofthe Haywai'd Fault, California, from double-difference earthquake locations. J. Geopphys. Res., 107(B3):10.1029/2000JB000084, 2002.
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A Noise Correction
The noise correction applied to the cross correlation of the earthquake data is given by
1 _
{nl)
uï)\
This equation is derived in Snieder, 2003 [12]. I n expression (17) ( « ^ 2 ) is the average energy of the data with the noise and (n? 2 ) is the average energy of the noise. For the derivation of the noise correction it is assumed that the noise is uncorrelated w i t h the signal. For the earthquake data the noise energy (nf 2) is estimated from the waveforms before the first aiiivals, while (uf 2) is computed for each time window separately. The uncorrected and corrected cross correlation for a pair of earthquake waveforms is plotted in the middle panel of Figure 15. The bold line corresponds to the corrected cross correlation. The uncorrected cross correlation is smaller because the noise decreases the similarity between the two waveforms. I n the bottom panel the cross correlation is converted into distance between the two sources. Again, the bold line corresponds to the distance computed from the corrected cross correlation. The distances have almost the same value for early times because the signal strength is high and consequently the noise level is low. As we move further in the coda the signal-to-noise level becomes lower and the two distances deviate. The corrected distance maintains a more or less constant value further into the coda.
The correction for the cross correlation is only calculated if ( ^ 1 , 2 ) > (ni^i).
If this condition does not hold, the noise level is considered to be too high with respect to the signal level and the value of the cross correlation is discarded.
0.8 0.6 1 1 • • • , 20 1 30 1 40 - Cross Correlation 1 1
-20 1 ' 30 1 40 1 1 30 t i m e [sFigure 15: Uncorrected and corrected cross correlation for event ID's 242003 and 242020 recorded at station CSP. The top panel shows the two waveforms and the horizontal bar indicates the size of the time-window in which we compute the cross correlation. The bold line i n the middle panel corresponds to the corrected cross correlation and the uncorrected cross correlation is given by the thin line. I n the bottom panel the cross correlation is converted into distance between the two sources.
