Matlab software for the analysis of seismic waves recorded by threeelement arrays*
A. Pignatelli¹, A. Giuntini2 and R.Console3
1
Istituto Nazionale di Geofisica e Vulcanologia, via di Vigna Murata 605, 00143 Roma, Italy (email: pignatelli@ingv.it, tel: +39 06 36915605, fax: +39 06 36915617).
2
Istituto Nazionale di Geofisica e Vulcanologia, via di Vigna Murata 605, 00143 Roma, Italy (email: giuntini@ingv.it, tel: +39 06 36915628, fax: +39 06 36915617).
3
Italian National Data Center viale Pinturicchio 23/E 00196 Roma, Italy.(e- mail: console@ingv.it,
tel: +39 06 36915657, fax: +39 06 36915617).
* Code available from server at http://www.iamg.org/CGEditor/index.htm
Abstract
We develop and implement an algorithm for inverting three-element array data on a Matlab
platform. The algorithm allows reliable estimation of back-azimuth and apparent velocity from
seismic records under low signal-to-noise conditions. We start with a cubic-spline interpolation of
the waveforms and determine the differences between arrival times at pairs of array elements. The
time differences are directly computed from cross-correlation functions. The advantages of this
technique are: (a) manual picking of the onset of each arrival is not necessary at each array element;
(b) interpolation makes it possible to estimate time differences at a higher resolution than the
sampling rate of the digital waveforms; (c) consistency among three independent determinations
provides a reliability check; and (d) the value of apparent velocity indicates the nature of the
recorded wavelet and physically checks the results. The algorithm was tested on data collected by a
tri-partite array (with an aperture of ~250 m) deployed in 1998 by the National Data Center of
Israel, during a field experiment in southern Israel, 20 km southwest of the Dead Sea. The data
include shallow explosions and natural earthquakes under both high and low signal-to-noise
conditions. The procedure developed in this study is considered suitable for searching of small
aftershocks subsequent to an underground explosion, in the context of On-Site Inspections
according to the Comprehensive Nuclear-Test-Ban Treaty (CTBT).
1
Keywords: cross-correlation, location, automation, array processing, CTBT
1. Introduction
Seismic event location is crucial for studying seismogenic processes, but it is also very useful
for verifying a nuclear test ban treaty. The CTBT, opened for signature in 1996, contains provisions
for the use of passive seismic techniques in carrying out On-Site Inspections in areas of suspected
clandestine underground nuclear tests. Seismological surveys are thought to be capable of detecting
and locating very weak aftershocks triggered by nuclear explosions (Richards and Ekstrom, 1995;
Adushkin and Spivak, 1995). Small aperture tri-partite arrays are proposed as flexible and
convenient tools for this purpose.
The goal of this work is to develop and test an algorithm to obtain reliable back azimuth and
apparent velocity estimates from the seismic waveforms of very low magnitude events recorded by
a tri-partite array. The algorithm is entirely original computer code on a Matlab platform, and has a
very simple Graphical User Interface (GUI) intended to facilitate analysis. The interactive Matlab
GUI allows results to be rapidly processed and immediately visualized.
In order to run the program, a version of Matlab 6.5 or higher with the mapping toolbox is
required. So far the program is designed only for Windows, but because it is written in an
interpreter (Matlab) rather than a compiler, it is portable to other platforms.
2. Methodology
The algorithm has three basic tools:
a) Waveform interpolation
Seismic data collected from the field is typically digitized at discrete intervals. The user may
wish to interpolate the seismograms to form a function that fits the raw data with shorter sampling
intervals than the original. This improves the precision of the time intervals, assuming that the
dominant frequency of signal is significantly lower than the sampling frequency. Many
interpolation algorithms can be applied (e.g.: power, rational, Gaussian, Weibull, Fourier, and so
on; see Matlab user manual); this work uses a cubic spline interpolation that fits the discrete points
2
with a 3rd degree polynomial. The user can choose a different method by changing just two lines of
code, but other interpolation techniques do not change the results significantly. The algorithm also
allows the choice of the outcome sampling rate.
b) Time difference estimation
As stated earlier, location accuracy can be improved by estimating the difference between
arrival times at two array elements more precisely. In order to accomplish this, we use crosscorrelation. Cross-correlation relies on the similarity of seismograms at spatially close sensors for a
given event; in fact, we expect waveforms to be similar because the source-receiver paths cross the
same heterogeneities. This assumption is valid up to some frequency threshold.
The time difference between the two arrivals is taken to be the time shift of one record with
respect to the other that yields the maximum of the correlation function in the time domain.
Correlation between two digital waveform segments ({ X 1 , X 2 ,...., X N },{ Y1 , Y2 ,...., YN }) is defined as
(Taylor, 1982):
N
R
N
N
i 1
i 1
N  X iYi   X i  Yi
i 1
2
 N 2  N
   N 2  N  
 N  X i    X i   *  N  Yi    Yi  
 i 1    i 1
 i 1  
 i 1
(1)
2
Where N is the number of samples of the correlation window.
When the two seismograms are identical, R assumes the maximum value of 1 at zero time
shift, while two different waveforms yield low cross-correlation values.
The correlation function is obtained by shifting one segment with respect to the other by
different delays.
c) Azimuth and apparent velocity
If analyzed events are local we can neglect the curvature of the Earth. Furthermore, if the
elements of a tri-partite array are very close to each other with respect to the distance from the
events, we can consider the wave fronts to be planar.
Under this hypothesis, consider an event E recorded by three different sensors S1 , S 2 and S 3 of
the tri-partite array (Fig. 1).
3
Consider, for the moment, the case where sensor S1 is the reference sensor of the tri-partite
array, so there are two possible pairs: ( S1 , S 2 ) and ( S1 , S 3 ). The same reasoning applies to any
other reference sensor.
The time difference between the arrival times of the pair of sensors yields two equations with
two unknowns: azimuth and apparent velocity. Assuming an homogeneous elastic medium, the
wave front takes the same time to cover the distance between S1 and S 2 as between the projection
of the two sensors on the ray: S 1' and S 2' .
So the time difference time between the arrivals can be calculated by:
T12  T2  T1 
S1' S2'
va
,
(2)
where va is the apparent velocity of the wave front.
By simple plane geometry, equation (2) can be written as a function of the sensor coordinates
and the event’s azimuth:
( y 2  y1 ) cos   ( x2  x1 ) sin 
va
T12 
(3)
T13 can be expressed in the same way. T12 and T13 are observed quantities, so the ratio
between the two equations defines the azimuth of the event:
tan  
T12 ( y 3  y1 )  T13 ( y 2  y1 )
T12 ( x3  x1 )  T13 ( x2  x1 )
(4)
Once the azimuth has been calculated, it is possible to define the apparent velocity of the
wave through equation(3):
va 
( y 2  y1 ) cos   ( x2  x1 ) sin 
T12
(5)
Similar equations can be written for each reference sensor. For tri-partite arrays there will be
three azimuths and apparent velocity estimates. These three pairs of values are independent, and if
time differences are estimated accurately, these values must be more or less similar.
By simple synthetic tests we have proved that the planar wave front model provides an
acceptable approximation even if the distance between the center of the array and the seismic
source is only few times larger than the distance between the sensors. For instance, with an array
4
250 m wide and the seismic source at 0.5 km distance, the maximum difference between the true
azimuth and that computed by the planar wave front approximation is smaller than 6°.
A more sophisticated geometry would be necessary if the distance between the elements of
the array were comparable to the distance between the source and the array elements (Almendros et
al., 1999). In this case a standard location algorithm based on absolute arrival times at single
sensors could be used instead of the tri-partite array algorithm.
3. Program operation
To start the program, all the provided files must be placed in a folder, Matlab must be
started, and the file WaveAnalyzer.m run.
The Matlab GUI is organized into ten menus, which appear at the top of the window when
the program is launched. The first seven menus are standard, so only the last three menus (Load,
Execute, Save) are detailed below.
The program needs two input files: (I) an ASCII file containing the digital waveforms as a
four column matrix (the first column contains the sample time and the three others contain the
amplitudes of each station) and (II) an ASCII file containing the geographical coordinates of the
stations. File (I) can be loaded using the command button Load Wave forms, and file (II) by means
of the submenu Station Coordinates in menu Load. In order to test the functionality of the program,
example files Waves.txt and Stations.txt are supplied.
The time difference between waveforms must be calculated with respect to a reference
station. After loading the waveforms, the user can visualize them in the GUI. The reference wave
can be chosen using the combo box Station of reference (Fig. 2).
The Matlab standard submenu zoom in in menu View can be used to analyze the reference
wave in detail. When the correct level of zoom is chosen, the user can refresh all the other plots
with the button Update seismograms. Note that the plot coordinates of the reference field shown on
the first axis are not updated in real time when zoom in is selected. For this reason we suggest
selecting this menu only when the user does not have to click on the reference wave plot.
Next, the processing interval needs to be chosen. This interval can be selected by simply
moving the pointer onto the reference wave plot and pressing spacebar. The limits are indicated by
vertical straight lines on the plot (see Fig. 3). The (x,y) values of the pointer position (time,
amplitude) are indicated to facilitate the clicking procedure.
5
If the reference sensor is changed, the window limits disappear but the chosen values are not
deleted. The user can choose to use the stored parameters in three different ways using the
following three buttons:
(I)
Reload previous window. The previous window is loaded and the straight lines
automatically appear on the axis.
(II)
Reload previous origin time. The previous lower limit is loaded and the user only
needs to click on the higher limit.
(III)
Reload previous window size. The previous window size is loaded and the user
must click on the lower limit. The upper limit will be automatically selected.
The last two inputs required to perform the correlation calculation are:
(I)
Step of correlations. This indicates the interpolation step and can be inserted by the
corresponding text input.
(II)
Time shift. This will be requested after clicking on the button Correlation.
It is very important to click on Data Confirmation before computing the correlation.
After calculating the correlation function, the software automatically computes its
maximum. The user can decide if this value is acceptable or if manual picking is preferred (Fig. 4).
In this case, the user can simply set the differences by clicking on the output plots. Finally, the user
can calculate the azimuth and the wave speed with the "Execute" menu; the results are shown in a
dialog box (Fig. 5).
Upon starting the program, an archive session is opened, and every time the "Execute" menu
is chosen, all the parameters and results are stored in this archive. This archive can be deleted or
exported as an ASCII file at any time using the menu Save, as in the example shown in Table I.
4. Application and results
We were able to check the algorithm efficiency and the program functionality using a set of
data collected by the National Data Center of Israel during an experiment from July 1998 to
February 1999 with a tri-partite array. Each element of the array was equipped with a three
component short period (1 Hz) seismometer system with a sampling rate of 62.5 Hz. The array was
installed in southern Israel, near the Dead Sea; the three elements formed a triangle with an aperture
of ~250 m. If the seismic waves travel with an apparent velocity greater than 1 km/s, they must
cross the array in less than 0.25 s (~15 samples). So if the arrival times were correctly calculated,
the time difference should not be larger than this quantity. We considered 11 chemical explosions
6
and 4 earthquakes with local magnitudes ranging from 1.6 to 2.7 (see Table II). Fig. 6 shows their
spatial distribution (from Bartal et al., 2000).
Most of the events were close to the array, with distances from the sensors ranging from 6 km
to 37 km (130 times the aperture of the array). There were only two distant events, ID 7 and 11
(80 km and 100 km away, respectively).
The following criteria were used in our analysis:
-
window length one and half times the dominant period of the signal,
-
identical window beginnings for the three reference sensors,
-
interpolation sampling of 0.002 s,
-
application to P, S and Rg phases according to their clarity, and
-
use of the maximum amplitude wavelet of each phase; when convenient,
the results are reported in Table III.
For it was possible to analyze the S phase for nine events, and for two of them the Rg phase
as well; the results obtained are not always consistent with those of the P phase, with azimuth
differences from 15° to 30°.
Fig. 7 shows the comparison between the azimuths obtained in this study and those computed
by the Israel national seismic network (Bartal et al., 2000). These azimuths are assumed to be
accurate enough to be taken as reference.
The square symbols are the results obtained using S phases, while the triangles are those
obtained with surface phases (Rg), generally recorded as the maximum amplitude of the
seismogram. The straight line shows the theoretical one–to-one ratio between the two solutions.
There are only few observations that fall nearly exactly on this line, but most of the others are fairly
close. Fig. 8 reports the velocity values determined for phases P, S and Rg versus the epicentral
distance. The conventions of Fig. 7 are used for the symbols.
The solutions exhibit an average velocity of 5.7 km/s for the P waves and 3.0 km/s for the S
waves. For events closer than 40 km to the array site, the P average velocity is smaller than 7 km/s
except for event 13, which is characterized by an anomalous velocity of 9 km/s. This value has
proved stable even when changing some of the free parameters in the algorithm. This could be
because this high apparent velocity is related to the arrival of a P wave reflected by the interface
between two crustal layers. The P arrivals of the two more distant events (7 and 11) exhibit
velocities close to or higher than 7 km/s, indicating a refraction under the crustal interface. Last, the
smallest apparent velocities, under 2 km/s, belong to the two surface wave arrivals (Rg).
7
6. Discussion and conclusions
The technique described in this paper and implemented in the software could be easily applied
to a seismological survey of aftershocks triggered by an underground explosion, in an On-Site
Inspection, or in any other situation in which we have a tri-partite array and low signal-to-noise.
This situation makes it difficult to pick up the exact arrival times of the seismic waves, especially P
waves, when using traditional single-station location techniques. In this context our technique may
be useful because it does not need to estimate the exact arrival time, but can instead compare the
corresponding waveform segments at different elements. Furthermore, our tests prove that small
variations of the initial instant of the time windows used for cross-correlation do not significantly
affect the obtained values of azimuth and apparent velocity.
Using cross-correlation between the waveforms provides a higher resolution than the
sampling rate in estimating time difference. In spite of this time resolution, the precision allowed by
a very small aperture tri-partite array in determining event azimuths of the order of 15°-30° is not
very high because of a number of reasons, including systematic bias in wave propagation.
Nevertheless, this precision is high enough when the network of arrays is spaced by few km,
because the plane wave approximation is valid also at these short distances.
Moreover, developing an interactive Matlab GUI allows very fast analysis and estimation of
the speed and back-azimuth. The software is designed to make it easy to change the parameters, the
time-window analysis, the reference wave, the step of the interpolation, and all the other parameters
necessary to perform the computation. Every parameter can be changed independently without
stopping the program execution.
Every test and computation is automatically recorded in an ASCII file containing the numeric
parameters and results in order to allow a complete analysis of all the tests performed.
Finally, the use of a high level interpreter like Matlab allows the use of optimized algorithms
to perform the interpolation and cross-correlation, and makes it easy to set the reference wave and
time windows by a simple click of the mouse.
Acknowledgements
We are grateful to the Italian and Israeli National Data Centers for the support to this study.
We wish in particular to acknowledge the contributions of Massimo Chiappini and Yochai Ben
Horin. Special acknowledgement is due to David Jepsen for his scientific and linguistic suggestions.
8
We also thank the two anonymous reviewers for their useful comments and suggestions.
9
References
Adushkin, V., Spivak, A., 1995. Aftershocks of underground nuclear explosions, In: Console R. and
Nikolaev A. (Ed.) Earthquakes Induced by Underground Nuclear explosions, NATO ASI Series,
Partnership Sub-Series, 2. Environment, Springer-Verlag Berlin Heidelberg, Vol.4, pp. 21-34.
Almendros, J., Ibanez, J. M., Alguacil, G., Del Pezzo, E., 1999. Array analysis using circular-wavefront geometry: an application to locate the nearby seismo-volcanic source. Geophysical Journal.
International, 136, 159-170.
Bartal, Y. ,Villagran, M., Ben Horin, Y., Leonard, G., Joswig, M., 2000. Definition of exclusive
zones using seismic data. Pure and Applied Geophysics 158, 397-419.
Richards, P.G., and Ekstrom, G, 1995. Earthquake activity associated with underground nuclear
explosions, In: Console R. and Nikolaev A. (Ed.) Earthquakes Induced by Underground Nuclear
explosions, NATO ASI Series, Partnership Sub-Series, 2. Environment, Springer-Verlag Berlin
Heidelberg, Vol.4, pp. 35-49.
Taylor, J. R., 1982. An Introduction to Error Analysis, the Study of Uncertainties in Physical
Measurements. University Science Books, Sausalito, CA, 327 pp..
10
Figure captions
Fig. 1 – Sketch of a tri-partite array. S1 , S 2 and S3 are three elements, dashed line shows direction of wave
propagation, and 1 , 2 , 3 and  are azimuths of the three elements and azimuth of event with respect to the origin of
rectangular coordinates.
Fig. 2 Main frame of program for event 980730. First element, called Dim1, has been chosen as reference.
Fig. 3 Frame of program. Reference window has been chosen and is shown with thin lines.
Fig. 4 Frame of program. At this point, the user can choose to manually select maximum of correlation functions or to
accept absolute maximum values automatically computed by program.
Fig. 5 Frame of program. At this point the user can calculate azimuth and apparent velocity of waves with “Execute”
menu. Results are shown by means of a message box and are automatically recorded in the session archive.
Fig. 6 Spatial distribution of events analysed in this study. After Y. Bartal et al., 2000.
Fig. 7 Comparison between azimuth obtained in this study and azimuth of national network, considered as reference.
Open circles refer to results obtained by P phases. Squares indicate azimuths obtained with S phases, and triangles those
obtained with Rg phases. Numbers correspond to events listed in Table I. M symbol accompanying event number
indicates that maximum amplitude wavelet has been used.
Fig. 8 Apparent velocity found with phases P and when possible with later phases. See Fig. 7 for symbols meanings.
11
Table I
An example (event 980730) of obtained results
Reference_ Advance Step Number_
Sensor
(s)
(s) of_Steps
1
2
3
0.317
0.317
0.317
0.002
0.002
0.002
317
317
317
Window_ Window_
Time_
Time_
Azimuth Velocity
Start
Lenght Correlation_1 Shift_1 Correlation_2 Shift_2
(°)
(km/s)
(s)
(s)
(s)
(s)
54.675
0.317
0.979
0.024
0.989
0.052 25.820
4.896
54.675
0.317
0.977
-0.025
0.979
0.026
26.49
4.988
54.675
0.317
0.986
-0.053
0.987
-0.029 24.604
4.882
For each sensor of reference:
- Window_Start is the instant at which reference window starts;
- Advance is the instant at which calculation of cross-correlation begins;
- Step is interpolated sampling interval;
- Correlation_1 and Time_Shift_1 are first maximum of cross-correlation and corresponding time difference
found, respectively; (Correlation_2 and Time_Shift_2 are the second values found)
-Azimuth and Velocity are solutions obtained.
12
Table II
Event Locations from Y. Bartal et al., 2000
ID
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Name of
Event
980724
980726
980730
980803_0
980803_1
980814
980817
980824
980901
980904
980906_0
980906_1
980907
980908
981009
Lat N
(deg)
31.143
30.858
31.094
31.034
31.293
30.724
31.909
30.723
30.869
31.106
31.689
31.077
30.783
31.013
30.876
Long E
(deg)
34.973
34.999
35.169
35.235
35.064
35.329
35.482
35.308
35.109
34.996
35.507
35.198
35.040
35.248
35.107
Dep
(km)
0.0F
0.0F
0.0F
0.0F
0.0F
11.8
14.3
7.2
0.0F
0.0F
16.0
0.0F
0.0F
0.0F
0.0F
Azimuth
(deg)
305.95.8
221.15.5
2.013.8
79.215.6
342.05.9
154.77.4
16.82.2
157.77.4
197.69.3
299.65.6
23.53.1
26.930.9
204.37.2
98.716.3
199.019.7
Dist
(km)
22.72
24.25
7.90
6.71
31.48
36.60
102.72
35.90
18.01
18.60
80.54
6.63
29.18
7.91
17.22
Apparent Velocity
(km/s)
4.361.33
6.452.64
5.312.42
4.131.06
2.310.47
4.571.96
7.815.74
3.250.96
4.131.48
4.371.58
7.815.74
4.621.98
2.250.39
3.610.93
7.043.61
Ml
Ev
2.0
1.7
2.1
2.2
2.0
1.6
2.6
2.0
2.7
2.3
2.1
2.2
1.6
2.1
2.6
E
E
E
E
E
Q
Q
Q
E
E
Q
E
E
E
E
Each row corresponds to one event, and describes: identified number, name, latitude, longitude, depth, azimuth,
distance from site of the array, apparent velocity, and local magnitude. In last column, E or Q identifies the nature of the
event: E is explosion and Q earthquake.
13
Table III
Results obtained
1
Name of
Event
980724
X
2
980726
X
3
980730
X
4
980803_0
X
5
6
7
8
980803_1
980814
980817
980824
X
X
X
9
980901
X
10
980904
X
11
980906_0
X
XM
12
13
14
980906_1
980907
980908
X
X
X
15
981009
X
ID
P
S
R
XM
XM
X
X
X
X
X
X
X
X
X
Width of
Window (s)
0.255
0.290
0.298
0.297
0.301
0.305
0.265
0.265
1.34
0.291
0.22
0.278
0.400
0.308
1.056
0.312
0.525
0.225
0.225
0.225
0.296
0.291
0.264
0.364
0.344
0.47
Azimuth
Average (deg)
311.23
324.65
207.67
176.72
25.19
12.89
91.95
82.93
345.71
147.58
44.20
136.06
139.06
169.93
200.33
298.96
332.69
50.40
52.90
34.38
29.38
193.28
97.83
80.61
186.16
171.49
Error
() (deg)
1.917
14.93
2.3
12.13
0
0.338
0.511
0.794
2.884
1.31
0.286
2.116
0.478
0.174
1.608
3.065
14.84
1.947
1.681
0.355
0.598
3.115
2.152
3.52
2.78
8.3
Apparent Velocity
Average (km/s)
6.35
3.41
6.14
4.43
4.98
4.10
3.80
2.72
1.58
3.79
7.36
5.02
2.39
4.79
1.34
6.69
4.03
6.62
7.52
3.65
5.12
9.00
4.22
2.85
6.05
2.87
Error ()
(km/s)
0.424
0.826
0.055
0.793
0
0.09
0.052
0.082
0.089
0.057
0.058
0.081
0.03
0.099
0.051
0.376
0.529
0.006
0.25
0.053
0.101
0.371
0.40
0.13
0.07
0.35
Each row includes the identified number of the event, its name and signal phase used, the length of the reference
window, and the parameters desired: the azimuth average and relative error and the apparent velocity and relative error.
When M accompanies the phase, the maximum amplitude wavelet was used.
14
Fig 1
15
Fig. 2
16
Fig. 3
17
Fig. 4
18
Fig. 5
19
Fig. 6
20
Fig. 7
21
Fig. 8
22