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Polarized correlation between three-component seismograms
PHYSICS OFTHE EARTH ANDPLANETARY INTERIORS Physics of the Earth and Planetary Interiors 97 (1996) 197-204
ELSEVIER
Polarized correlation between three-component seismograms Chiou-Fen Shieh Institute of Seismology and Applied Geophysics, National Chung Cheng University, Chia-yi. Taiwan. ROC
Received 28 March 1995;revised 7 November 1995; accepted 13 November 1995
Abstract
A comparison of all polarization characteristics is considered to be the best approach for discriminating signal similarity. This paper presents an alternative polarized correlation method to measure signal similarity from different stations, or the same station but from different earthquakes. The polarization characteristics used in this paper include phase difference between the vertical and horizontal components as well as the correlation of rectilinearity, linear polarization, phase difference and inner products of two vector wavefields. These characteristics are combined into a total polarized correlation function, which may be used as a polarization filter as well. This function can be used effectively to discriminate different wave types appearing in different time windows. In this study, synthetic data are used to test the accuracy of this method, and some real cases are used to outline further potential applications.
1. Introduction
Polarization analysis has become an important method for signal processing. This includes the real covariance matrix method in the time domain by Flinn (1965) and Jurkevics (1988), the coherency matrix method in the frequency domain by Samson and Olson (1980) and the revised coherency matrix method composed of analytic signals in the time domain by Vidale (1986). However, except for the method of Jurkevics (1988), who averaged the covariance matrix over several stations, these methods are only used on single station three-component data. Recently, Rutty and Greenhalgh (1993), using Vidale's method at two stations, completed a 6 x 6 coherency matrix to analyze the relationship among signals. From their simulation results, it was found that both signals and coherent noise could be discemed. The purpose of this paper is to introduce a method
that can be used to study the relationship among signals from different testing stations, or from the same station but from different earthquakes. Two individual 3 x 3 coherency matrices instead of one 6 x 6 coherency matrix are used to analyze the polarization characteristics as well as to define the total polarized correlation coefficients (Tc). The value of Tc equals unity only under one condition: when the signals are exactly the same. Hence, any unwanted wave types are suppressed or completely eliminated by multiplying this coefficient. Polarization parameters are usually found from the characteristics of eigenvalues and eigenvectors, such as in the studies conducted by Flinn (1965), Samson (1983), and Vidale (1986). The last used the eigenvector corresponding to the dominant eigenvalue to obtain the strike and dip of the maximum polarization, and designed a search scheme for rotational angles to maximize the real part of the eigenvector and to obtain ellipticity. For a better under-
0031-9201/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. SSDI 0031-9201(95)03131-6
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C.-F. Mlieh / Phy,~ics ~f'the Earth and Planetar 3 Interiors 97 (1996) 197 -204
standing of this study, the characteristics of the eigenvectors of complete polarization (including all types of waves) are investigated first, as some of them are not well interpreted in the literature. The second step is to find and define new polarization parameters from such characteristics of the eigenvector.
2. Eigenvector o f c o m p l e t e polarization
The station coordinate (z, x,y) is taken to represent the traditional directions in which the instrument is usually set, where z, x, and y stand for vertical, east-west and south-north, respectively. The Hilbert transformation yields the analytic triaxial signal:
z(t) =azexp[j4'z(t)],
x(t) =axexp[j4'h(t)],
y ( t ) = ayexp[ J4'h( t)]
or the complete lbrm of
U=
ahcos 0cos 4' - JahCOS0sin4' ah sin 0cos 4' -- fih sin 0 sin 4'
where a,, a h, a x and a,, are the normalized directional projections. The form of Eq. (3) is used below for the dominant eigenvector. It is clear that strike (0) can be easily calculated from the real or imaginary parts of Eq. (2), tan 0 =
Yr X r
--
Yi
Xi
which was used by Vidale (1986), even though he only adopted the real part. The horizontal projection is calculated by
+ x )°5
(t)
The horizontal component is assumed to be h(t)= ahexp[j4'h(t)] and a x = ahCOS0, ay ahsin0, where 0 is the strike of polarization. In this study, only through the three-dimensional receiver could the signals be recorded completely, because as two-dimensional plane waves they might be incident in an arbitrary direction. However, the two horizontal components, x and y, still have the same phase (qbh(t)) as expressed in Eq. (1). The coherency matrix is defined as (Mott, 1986)
(3)
ah =
( 4)
COS0
=
and the phase difference (+) between the vertical and horizontal components can be calculated by 4' = arctan - -
xr
U=
< axaze-J6 > < a2x > < axas > < ayaze-J4~ > < axay > < ay2 >
where ~ = q b : - qbh represents the phase difference between the vertical and horizontal components. From the eigensolution of the C matrix, it is determined that the sum of the three eigenvalues equals the sum of the trace elements, namely, A = < a ~ > -t- < a2x > + < a y 2 > However, for complete polarization (without any noise), only one eigenvalue exists, and the corresponding eigenvector is found to take on the form of
U=
{Zr ( x~+jxi
Yr +jYi
=
~cos4'-fi,.sin4'
/ ayCOS 4' -- j~ySin4'
(2)
(y)
(5)
Once a z, a h and + are found, the three-dimensional case may be simplified into the two-dimensional one, and Eq. (3) can be rewritten as
< a 2. > < azaxeJ6 > < azayeJ4~ > C=
= arctan
(
~fihsin ,
)
(6)
In this condition, the polarization of seismic waves is confined to a plane, and the ellipticity (e) can be calculated from the eigenvector described by Samson and Olson (1980) or Shieh and Herrmann (1990). In this study, however, the three-dimensional search scheme for the rotational angles method proposed by Vidale (1986) is used.
3. P o l a r i z e d correlations b e t w e e n data sets
With one signal selected as a reference wave, the technique of analyzing the polarization characteristics has proven to be very helpful in identifying the one which is the most similar from another set of data, such as in comparing the P waves of two
C.-F. Shieh / Physics of the Earth and Planetary" Interiors 97 (1996) 197-204
different sets of data. In this case, the polarization characteristics should be analyzed well first. The signal selected at the window range of T seconds of reference seismograms is processed through polarization analysis. Then certain variables are obtained: the eigenvector Uref, phase difference ~b~f, strength of polarization Pr~f (Eq. (8) of Vidale (1986)) and rectilinearity Rr~f. For the readers' convenience, Pr~ and RF~f are rewritten here as A 2 -I- A 3 P~e~. = I At Rre f =
1- e
where AI, A2 and A3 are the eigenvalues of the reference wave, in order from maximum to minimum, and e is the ellipticity. The next step is to use the same time (T seconds) as the moving window and to perform polarization analysis for the compared seismograms. It is to be noted that, theoretically, the time window for the compared seismograms has to be the same as that in the reference seismograms. The time-dependent polarization parameters for the compared seismograms are the eigenvector U(t), phase difference +(t), strength of polarization P ( t ) and rectilinearity R(t). Now several polarized correlation terms are defined: chc(t) = I&ref -- ~b(t)l
(7)
Rc(t ) = 1 -IRre f - R(t)l
(8)
Pc(t) =
1 -Ieref-P(t)l
(9)
c~(t) = cos2[q~c(/)]
(10)
I,~(t) = IU~dS/J * (t)l
(11)
Eq. (7) and Eq. (8) represent the correlation of the phase difference and rectilinearity between the correlated signals. The following examples can be used to explain these equations: (1) when the correlated signals are exactly the same, + c ( t ) = 0 and Re(t)= 1; (2) when the correlated signals are P and S waves, qbc(t) = 180 and R c ( t ) = 1; (3) when the correlated signals are P waves and circular polarization, +c(t) = 90 ° and Rc(t) = 0. Eq. (9) shows the relationships between signals under two conditions: (1) if the correlated signals show complete polarization, such as P and S waves, Pr~f= P(t) = 1, and then Pc(t) =
199
1; (2) if the correlated signals are P waves and random noise, P~ef= 1, P(t) = 0, and then Pc(t) = O. Therefore, it may be expected that noise and polarized signals are completely uncorrelated. Also, it should be noted that if Pref is lOW (owing to a poor choice of reference wave) and if the correlated signal is highly polarized (high P(t)), then Pc(t) is low. This can indicate that these two signals are not similar in terms of the strength of polarization. Eq. (10) is used to suppress different wave types, such as correlating P wave and circular polarization results in + c ( t ) = 90°; consequently, Co(t)= 0. When the correlated signals are P waves and other nonlinear waves, C c ( t ) < 1. Eq. (11) represents the absolute value of the inner products of two eigenvectors, where (3 and * represent inner products and complex conjugate, respectively. For signals with parallel polarization, It(t)= 1, whereas for those with normal polarization, l c ( t ) = 0. Therefore, It(t) is designed to remove particle motions perpendicular to each other. For instance, when the correlated signals, P and S waves, have the same angle of incidence, cbc(t) = 180 and Co(t) = 1; under this condition, it is impossible to suppress the S waves. However, in the case of l c ( t ) = 0 , the S wave can be removed. Finally, the total polarized correlation coefficients are defined as
re(t) = Re(t) Pc(t) c o ( t ) i (t)
(12)
This equation equals unity only when the correlated signals are exactly the same. Although it may simpler to use Eq. (9) and Eq. (11) to compare the signals, doing so would not yield such good results as by using Eq. (12). The reason for this is the same as that in the explanation of Eq. (7) and Eq. (8). Hence, Eq. (12) is very adequate in measuring the similarity between signals. Referring to synthetic and real data, the following sections will support this method.
4. Synthetic tests In Fig. 1, five different Ricker wavelets with a central frequency of 30Hz and a strike of 30° are simulated. Fig. l(a) represents the data without noise; the square windows from the first to the fifth represent the following phase differences: 0 °, 45 °, 90 °,
C.-F. Shieh / Phys'icv q[ the Earth and Planetary Interiors 97 (1996) 197 204
200
135 °, 180 ° (the same as in the study by Shieh and Herrmann (1990)). The first and the fifth are P and S waves; the second and the fourth are 45 ° and 135 ° elliptic polarization; the third is 90 ° circular polarization. Fig. 3(b) shows the same data as in Fig. l(a) but with 20% random noise added. Comparing each selected window in Fig. l(a) with the seismograms in Fig. 3(b), five results are obtained, and these are listed in the following section. For the first two cases, detailed discussions are provided, but for the other three, only the final results are given.
4.1. Case 1 : 0 ° linear polarization as the reference wave The 0 ° linear wave in the first window of Fig. 3(a) is extracted as the reference P wave; this has polarization characteristics with qbref= O, Pref = 1, e r e f = 3. We let Fig. l(b) be the compared data and perform the polarized correlation as mentioned above. In Fig. 2(a), the calculated phase difference (qb(t)) is discussed in five time ranges. The phase difference in the first time range, 0 . 8 1 - 3 . 2 1 s (P wave), is + ( t ) --- 0°; in the second time range, 1.56-3.96 s (45 ° elliptic polarization), it is + ( t ) = 45°; in the third time range, 2.31-2.71 s (circular polarization), it is + ( t ) = 9 0 ° ; in the fourth time range, 3 . 0 6 - 3 . 4 6 s (135 ° elliptic polarization), it is qb(t)= 135°; in the last time range, 3.81-4.23 s (S wave), it is qb(t)--_+ 380 °.
x
[b]
Z
(a)
X Z
- - -
I
0.0
2 .I5
i
5.0
SECOND Fig. 1. Five Ricker wavelets of different polarizations are displayed: (a) noise free; (b) with 20% random noise added. The five signals windowed in (a) correspond to 0 °, 45 °, 90 °, 135 ° and 180 ° phase difference. Each windowed signal is taken as an independent reference wave used later.
[g] o. [-f}
MAX 1MUM VALUE
- - - - ~
o.8~
o , ~ ~ .
le] o
.
0.94
~
~.oo
[d] o.~/'4~
~
~'J'J~
~.oo
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~-'L~'~"~
"~J~
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~
~_
~so.oo
. IO.O
t 215 SECOND
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Fig. 2. Case 1:0 ° polarization of the first windowed signal in Fig.
l(a) as the reference wave. The results of total polarized correlations: (a) phase difference of correlated seismograms (~b(t)); (b) correlation of phase difference (Oc(t)) between signals as calculated by Eq. (7); (c) cos2[Oc(t)]; (d) correlation of rectilinearity; (e) inner products of two eigenvectors; (f) correlation of the strength of polarization; (g) total polarized correlation coefficients (To(t)) as calculated by Eq. (12).
In Fig. 2(b), the correlations of the phase difference, ~bc(t), corresponding to the above five time ranges are: (3) ~b(t) --- 0 ° (both of them are P waves); (2) qb(t)= 45°; (3) qb(t)= 90°; (4) + ( t ) = 135°; (5) + ( t ) = 180 °. The calculated Co(t) from Eq. (10) is shown in Fig. 2(c), where two maximum values almost equal 1.0 in the first and the last time ranges, as d?c(t) is near 0 ° and 180 °, respectively; the minim u m value in the third time range almost equals zero because ~b~(t)-= 90 °. The intemediate values in the second and the fourth time ranges lie between zero and one. In the correlation of phase differences, P and S waves are completely correlated, P waves and circular polarization are completely uncorrelated, and P waves and other elliptic polarizations are partially correlated. In terms of correlation of the rectilinearities calculated from Eq. (8), Fig. 2(d) shows that P and S waves are completely correlated because their particle motions are all linear, P waves and circular polarizations are completely uncorrelated, and other elliptic polarizations are partially correlated. In Fig. 2(e), the inner products (l~(t)) of the two vector
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C.-F. Shieh / Physics of the Earth and Planetary Interiors 97 (1996) 197-204
wavefields calculated from Eq. (11) indicate that P and S waves are completely uncorrelated, yet other nonlinear waves are partially correlated with P waves. In this synthetic case, the particle motion of P and S waves are perpendicular to each other, causing their inner products to equal zero. This allows for the removal of unwanted S waves. It should be noted that in Fig. 2(e), the It(t) has a higher value for 45 ° elliptic polarization than for 135 °, as the former has a right inclination (the major axis of ellipse lies in the first and third quadrants) and contains more P-wave component, whereas the latter has a left inclination (the major axis of ellipse lies in the second and fourth quadrants) and contains more S-wave components. In contrast, for the reference S wave, the lc(t) has a higher value for 135 ° elliptic polarization than for 45 °. In Fig. 2(f), the correlation of the strength of polarization calculated from Eq. (9) shows that all five wave types are almost completely correlated. This is not surprising, as all of them exhibit nearly complete polarization and, thus the same strength of polarization. Nevertheless, it is used to suppress random noise. Combining all the conditions mentioned above, the author defines the total polarized correlation function as in Eq. (12) in such a way that only P and P waves are highly correlated, as shown in Fig. 2(g). In this figure, circular polarization and S waves are completely uncorrelated, whereas 45 ° and 135 ° elliptic polarizations are partially correlated with the reference P wave. It is apparent that when the reference wave type is P, the highly correlated P wave may be singled out from the compared seismograms. Certainly, the existing noise disturbs the analysis in a finite length window. However, the synthetic test indicates that this method works when the noise level is not very high.
_
MAXIMUM VALUE
~
{41 O.
~
0.91
lel 0. ~
0.gtt
Idl o.
].0o
It) 0
.
~
I
-
I
0.0
1.00
E
1
2.5 SECOND
5.0
Fig. 3. Case 2:45 ° elliptic polarization of the second windowed signal in Fig. 2(a) as the reference wave. The results of total polarized correlation: (a) correlation of phase difference (4~c(t)); (b) cos214>c(/)];(c) correlation of rectilinearity; (d) inner products of two eigenvectors; (e) correlation of the strength of polarization; (f) total polarized correlation coefficients.
from that in Fig. 2. In other words, the arrangement of Fig. 3(a)-(f) is the same as that of Fig. 2(b)-(g). Obviously, in Fig. 3(f), the highest correlation coefficients occur in the time range of 45 ° elliptic polarization. As the other relevant characteristics are easy to understand, it is not necessary to repeat explanation here. Because the particle motions of 45 ° and 135 ° elliptic polarizations are perpendicular to each other, based on Eq. (10) they are completely uncorrelated in phase difference.
M A X 1M U M
VALUE (c] 0
~
(b} o. ~
....
~,~-J~"q,--~'~
0.93 o.9]
4.2. Case 2." 45 ° elliptic polarization as the reference waue
The 45 ° elliptic wave of the second window in Fig. l(a) is taken as the reference wave and then compared with Fig. l(b) through polarized correlation analysis. The results are shown in Fig. 3. The phase difference (~b(t)) displayed in Fig. 2(a) is the same in this case so it is not shown in Fig. 3. Therefore, the order in Fig. 3 is slightly different
(a] O.~ r 0.0
~
~
~ 2.S SECOND
0.80
-
- - 1 S.O
Fig. 4. The results of total polarized correlation coefficients for (a)
90° circular polarization of the third windowed signal, (b) 135° elliptic polarization of the fourth windowed signal, and (c) 180°
polarization of the fifth windowed signal in Fig. l(a) as reference waves.
202
C.-F. Shieh / Physics tff'the Earth aml Planetary Interiors 97 (1996) 197-204 MAX] MUM VRLUE
O. -
,~.,.,_.,A,_,,.,_/llk_,..,.,._J1~x,~.,.._~n.._._,.,J~lk_.,A.,~_Jl~
I
0.00
I
2.56
P
S
El.i
.......
Ew t
O. 7 8 6 3 6 q 0
1
5.12
SECOND
f"
0,0
Fig. 5. The correlation coefficient of a simple one-dimensional correlation method.
In the remaining three cases, with 90 ° circular polarization, 135 ° elliptic polarization and 180 ° linear polarization as reference waves, the total polarized correlation results are displayed in Fig. 4(a), Fig. 4(b), and Fig. 4(c), respectively. The results again show that the high correlation coefficients only occur in the time range of the same wave type. One may at first suspect that a simple one-dimensional correlation method can provide the same results for such synthetic cases. However, the result of a simple correlation between the reference P wave and the seismogram (the same situation as in Case 1) is shown in Fig. 5, and it obvious that to distinguish wave types by such a simple method is very difficult indeed. From the test results, it may be concluded that only when the polarized characteristics are very similar can high total polarized correlation coefficients be obtained. Consequently, this can be considered to isolate a particular wave type, as demonstrated in the following two real cases.
•
i0.2
SECOND
Fig. 6. Reference earthquake data collected by Station ALS. The first window contains the first P arrival, and the second contains the first S arrival.
in the bottom three traces of Fig. 7. In Fig. 6, two square windows indicate the P and S first arrivals. We let the P wave in the first window be the reference wave, then carry out the polarized correlation analysis. The result is shown in Fig. 7. It is found that, in Fig. 7(g), high correlation coefficients (Tc(t)) appear only in the range of the first P arrival. The results of the same processing for the first S MAX lM U M VALUE O. 99
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o.
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o
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l. O0
(c)
o.
l, O0
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5. Applications lal
0.
180.00
The analyses above can be used effectively in seismology. Two examples adequately explain the first application.
5.1. Example 1 Testing Station ALS located in the Chia-Nan area of Taiwan is selected, and data from two different earthquakes are collected. The first earthquake occurred at 19:42h UT, 15 February 1992, with a focal depth of 20.48 km, and the second occurred at 05:43 h LIT, 16 February 1992, with a focal depth of 16.42 km. Three-component seismograms of the first and the second earthquakes are shown in Fig. 6 and
7q5,00
580. O0
~ uJ
Z
O.
~0.0
qq2. 013
51 1 SECOND
lO.2
Fig. 7. "llle bottom three traces are correlated seismograms collected by Station ALS. Results of polarized correlation with the first P arrival in Fig. 6: (a) phase difference; (b) correlation of phase difference (4%0)); (c) (d) correlation of rectilinearity; (e) inner products of two eigenvectors; (f) correlation of the strength of polarization; (g) total polarized correlation coefficients.
COS2[~bc(t)];
C.-F. Shieh / Physics of the Earth anzl Planetary Interiors 97 (1996) 197-204 P
1 II (g}
0
(4)
o
S
I *
NOISE
@-
-I
I
. ~
0. J"J
~
(d)
o. J ~ ] ' ~ l ~ _ , / ~
(:/
o
(a)
o 0.0
~
/
EN
0,92
Z
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1.00
VV~ ~ ~/-~//L
l.oo
]
.
N5
VALUE
~
le)
.
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1 I
~
.
~
i
~
~ 1
203
NS
,.
EN
[a)
Z
0.0
.I SECOND
10.2
1.00
Fig. 9. The filtered results of the second earthquake: (a) the first P arrival; (b) the first S arrival.
18o.oo
arrivals, respectively, of the first earthquake. This analysis requires that the two compared signals have nearly the same frequency content, thereby making it more valuable for local earthquake data or two teleseismic stations that are not too far apart. In addition, it is not limited to the first arrivals; any converted phases can also be used.
10.2
SECOND Fig. 8. Results of total polarized correlation for the first S arrival: (a) phase difference; (b) correlation of phase difference; (c) cos2[(bc(t)]; (d) correlation of rectilinearity; (e) inner products of two eigenvectors; (f) correlation of the strength of polarization; (g) total polarized correlation coefficients.
arrival are illustrated in Fig. 8, where high correlation coefficients also appear in the range of the first S arrival. However, the variation of To(t) in Fig. 8(g) is much higher than that in Fig. 7(g), which complicates the correlation relationships for the first S wave. In the three portions of Fig. 8(g), the first portion before the first P arrival consists of background noise, and the high value of Tc(t) indicates that the noise contains significant S-wave components; in the second portion, between the first P and S arrivals, the high values of T~(t) indicate that there are many converted S waves (P to S); the last portion, after the first S arrival, is composed of surface waves, and the high value of T~(t) indicates that the surface waves contain significant S-wave energy. Although the interpretation is not new, it had been proven by the polarized correlation method. It should be pointed out that even though the reference S wave contains noise, the method works well. To see how similar signals can be extracted, the compared seismograms are multiplied by the function of T.(t). The results for P and S waves are displayed in Fig. 9(a) and Fig. 9(b). The results show that the first P and S arrivals of the second earthquake are highly correlated with the first P and S
5.2. Example 2 Surface waves may well be of interest too. Fig. 10(a) and Fig. 10(b) list two sets of three-component exploration data for geophone distance of 30.4m. The low-frequency dispersed surface wave in the box in Fig. 10(a) is taken as the reference wave, and then the polarized correlation analysis is performed. The filtered results of Fig. 10(b) are displayed in Fig.
[
O. f}
•
]. [', SECOND
q
~:,.I]
Fig. 10. (a) Reference dispersed surface is selected in the window; (b) correlated data recorded from another geophone at a distance of 30.4m; (c) filtered dispersed surface waves.
204
C.-F. Shieh / Physics of the Earth and Planetary Interiors 97 (1996) 197 204
10(c), and it is found that the dispersed surface wave has been enhanced. From the examples above, it is seen that polarized correlation analysis does offer tremendous gains in conducting advanced studies on the similarities among wave types.
6. Concluding remarks and discussion In this study, the characteristics of the eigenvector for complete polarization is investigated. In addition, information on each element in the eigenvector is completely discovered for the first time. The total polarized correlations of two different seismograms are compared from four viewpoints: phase difference, rectilinearity, strength of polarization and inner products. The phase difference calculated from the eigenvector is a very useful polarization parameter which has not been used before. The inner products provide a superior way to compare the direction of particle motion such that two orthogonal signals can be well identified. Only when signals have exactly the same polarization characteristics can the total polarized correlation coefficient equal unity. Under other situations, the coefficient is smaller than unity, including the case of two P waves with different angles of incidence (owing to the inner products being less than unity). The method used in this paper is very effective in discriminating P and S waves, reducing unwanted wave types and in comparing any signals that include surface waves and converted phases. The main purpose of this study is to compare the similarities between two signals. It can be illustrated in the following example. For the first P arrival of two different stations, if their polarized correlation is
weak, one may consider such causes as path effect, site effect, etc., with the help of other obtained sources. However, the traditional polarization analysis does not offer this kind of information. As the window length is limited to that which is chosen, the compared signals must have nearly the same frequency content, which limits this analysis to local earthquake data or teleseismic data of two different stations with a very short distance between them.
Acknowledgements The author would like to thank two anonymous reviewers for their constructive suggestions and comments. Gratitude is also extended to the editor, David Gubbins, for the time he spent processing and checking the manuscript.
References Flinn, E.A., 1965. Signal analysis using rectilinearity and direction of particle motion. Proc. IEEE, 53: 1874-1876. Jurkevics, A., 1988. Polarization analysis of three-component array data. Bull. Seismol. Soc. Am., 78: 1725-1743. Mott, H., 1986. Polarization in Antennas and Radar. Wiley, New York, 297 pp. Rutty, H.J. and Greenhalgh, S.A., 1993. The correlation of seismic events on multicomponent data in the presence of coherent noise. Geophys. J. Int., 113: 343-358. Samson, J.C., 1983. Pure states, polarized waves, and principal components in the spectra of multiple, geophysical time-series. Geophys. J. R. Astron. Soc., 61: 115-129. Samson, J.C. and Olson, J.V., 1980. Some comments on the description of the polarization states of waves. Geophys. J. R. Astron. Soc., 61:115-129. Shieh, C.F. and Herrmann, R.B., 1990. Ground roll: rejection using polarization filters. Geophysics, 55:1216-1222. Vidale, J.E., 1986. Complex polarization analysis of panicle motion. Bull. Seismol. Soc. Am., 76: 1393-1405.
ELSEVIER
Polarized correlation between three-component seismograms Chiou-Fen Shieh Institute of Seismology and Applied Geophysics, National Chung Cheng University, Chia-yi. Taiwan. ROC
Received 28 March 1995;revised 7 November 1995; accepted 13 November 1995
Abstract
A comparison of all polarization characteristics is considered to be the best approach for discriminating signal similarity. This paper presents an alternative polarized correlation method to measure signal similarity from different stations, or the same station but from different earthquakes. The polarization characteristics used in this paper include phase difference between the vertical and horizontal components as well as the correlation of rectilinearity, linear polarization, phase difference and inner products of two vector wavefields. These characteristics are combined into a total polarized correlation function, which may be used as a polarization filter as well. This function can be used effectively to discriminate different wave types appearing in different time windows. In this study, synthetic data are used to test the accuracy of this method, and some real cases are used to outline further potential applications.
1. Introduction
Polarization analysis has become an important method for signal processing. This includes the real covariance matrix method in the time domain by Flinn (1965) and Jurkevics (1988), the coherency matrix method in the frequency domain by Samson and Olson (1980) and the revised coherency matrix method composed of analytic signals in the time domain by Vidale (1986). However, except for the method of Jurkevics (1988), who averaged the covariance matrix over several stations, these methods are only used on single station three-component data. Recently, Rutty and Greenhalgh (1993), using Vidale's method at two stations, completed a 6 x 6 coherency matrix to analyze the relationship among signals. From their simulation results, it was found that both signals and coherent noise could be discemed. The purpose of this paper is to introduce a method
that can be used to study the relationship among signals from different testing stations, or from the same station but from different earthquakes. Two individual 3 x 3 coherency matrices instead of one 6 x 6 coherency matrix are used to analyze the polarization characteristics as well as to define the total polarized correlation coefficients (Tc). The value of Tc equals unity only under one condition: when the signals are exactly the same. Hence, any unwanted wave types are suppressed or completely eliminated by multiplying this coefficient. Polarization parameters are usually found from the characteristics of eigenvalues and eigenvectors, such as in the studies conducted by Flinn (1965), Samson (1983), and Vidale (1986). The last used the eigenvector corresponding to the dominant eigenvalue to obtain the strike and dip of the maximum polarization, and designed a search scheme for rotational angles to maximize the real part of the eigenvector and to obtain ellipticity. For a better under-
0031-9201/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. SSDI 0031-9201(95)03131-6
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C.-F. Mlieh / Phy,~ics ~f'the Earth and Planetar 3 Interiors 97 (1996) 197 -204
standing of this study, the characteristics of the eigenvectors of complete polarization (including all types of waves) are investigated first, as some of them are not well interpreted in the literature. The second step is to find and define new polarization parameters from such characteristics of the eigenvector.
2. Eigenvector o f c o m p l e t e polarization
The station coordinate (z, x,y) is taken to represent the traditional directions in which the instrument is usually set, where z, x, and y stand for vertical, east-west and south-north, respectively. The Hilbert transformation yields the analytic triaxial signal:
z(t) =azexp[j4'z(t)],
x(t) =axexp[j4'h(t)],
y ( t ) = ayexp[ J4'h( t)]
or the complete lbrm of
U=
ahcos 0cos 4' - JahCOS0sin4' ah sin 0cos 4' -- fih sin 0 sin 4'
where a,, a h, a x and a,, are the normalized directional projections. The form of Eq. (3) is used below for the dominant eigenvector. It is clear that strike (0) can be easily calculated from the real or imaginary parts of Eq. (2), tan 0 =
Yr X r
--
Yi
Xi
which was used by Vidale (1986), even though he only adopted the real part. The horizontal projection is calculated by
+ x )°5
(t)
The horizontal component is assumed to be h(t)= ahexp[j4'h(t)] and a x = ahCOS0, ay ahsin0, where 0 is the strike of polarization. In this study, only through the three-dimensional receiver could the signals be recorded completely, because as two-dimensional plane waves they might be incident in an arbitrary direction. However, the two horizontal components, x and y, still have the same phase (qbh(t)) as expressed in Eq. (1). The coherency matrix is defined as (Mott, 1986)
(3)
ah =
( 4)
COS0
=
and the phase difference (+) between the vertical and horizontal components can be calculated by 4' = arctan - -
xr
U=
< axaze-J6 > < a2x > < axas > < ayaze-J4~ > < axay > < ay2 >
where ~ = q b : - qbh represents the phase difference between the vertical and horizontal components. From the eigensolution of the C matrix, it is determined that the sum of the three eigenvalues equals the sum of the trace elements, namely, A = < a ~ > -t- < a2x > + < a y 2 > However, for complete polarization (without any noise), only one eigenvalue exists, and the corresponding eigenvector is found to take on the form of
U=
{Zr ( x~+jxi
Yr +jYi
=
~cos4'-fi,.sin4'
/ ayCOS 4' -- j~ySin4'
(2)
(y)
(5)
Once a z, a h and + are found, the three-dimensional case may be simplified into the two-dimensional one, and Eq. (3) can be rewritten as
< a 2. > < azaxeJ6 > < azayeJ4~ > C=
= arctan
(
~fihsin ,
)
(6)
In this condition, the polarization of seismic waves is confined to a plane, and the ellipticity (e) can be calculated from the eigenvector described by Samson and Olson (1980) or Shieh and Herrmann (1990). In this study, however, the three-dimensional search scheme for the rotational angles method proposed by Vidale (1986) is used.
3. P o l a r i z e d correlations b e t w e e n data sets
With one signal selected as a reference wave, the technique of analyzing the polarization characteristics has proven to be very helpful in identifying the one which is the most similar from another set of data, such as in comparing the P waves of two
C.-F. Shieh / Physics of the Earth and Planetary" Interiors 97 (1996) 197-204
different sets of data. In this case, the polarization characteristics should be analyzed well first. The signal selected at the window range of T seconds of reference seismograms is processed through polarization analysis. Then certain variables are obtained: the eigenvector Uref, phase difference ~b~f, strength of polarization Pr~f (Eq. (8) of Vidale (1986)) and rectilinearity Rr~f. For the readers' convenience, Pr~ and RF~f are rewritten here as A 2 -I- A 3 P~e~. = I At Rre f =
1- e
where AI, A2 and A3 are the eigenvalues of the reference wave, in order from maximum to minimum, and e is the ellipticity. The next step is to use the same time (T seconds) as the moving window and to perform polarization analysis for the compared seismograms. It is to be noted that, theoretically, the time window for the compared seismograms has to be the same as that in the reference seismograms. The time-dependent polarization parameters for the compared seismograms are the eigenvector U(t), phase difference +(t), strength of polarization P ( t ) and rectilinearity R(t). Now several polarized correlation terms are defined: chc(t) = I&ref -- ~b(t)l
(7)
Rc(t ) = 1 -IRre f - R(t)l
(8)
Pc(t) =
1 -Ieref-P(t)l
(9)
c~(t) = cos2[q~c(/)]
(10)
I,~(t) = IU~dS/J * (t)l
(11)
Eq. (7) and Eq. (8) represent the correlation of the phase difference and rectilinearity between the correlated signals. The following examples can be used to explain these equations: (1) when the correlated signals are exactly the same, + c ( t ) = 0 and Re(t)= 1; (2) when the correlated signals are P and S waves, qbc(t) = 180 and R c ( t ) = 1; (3) when the correlated signals are P waves and circular polarization, +c(t) = 90 ° and Rc(t) = 0. Eq. (9) shows the relationships between signals under two conditions: (1) if the correlated signals show complete polarization, such as P and S waves, Pr~f= P(t) = 1, and then Pc(t) =
199
1; (2) if the correlated signals are P waves and random noise, P~ef= 1, P(t) = 0, and then Pc(t) = O. Therefore, it may be expected that noise and polarized signals are completely uncorrelated. Also, it should be noted that if Pref is lOW (owing to a poor choice of reference wave) and if the correlated signal is highly polarized (high P(t)), then Pc(t) is low. This can indicate that these two signals are not similar in terms of the strength of polarization. Eq. (10) is used to suppress different wave types, such as correlating P wave and circular polarization results in + c ( t ) = 90°; consequently, Co(t)= 0. When the correlated signals are P waves and other nonlinear waves, C c ( t ) < 1. Eq. (11) represents the absolute value of the inner products of two eigenvectors, where (3 and * represent inner products and complex conjugate, respectively. For signals with parallel polarization, It(t)= 1, whereas for those with normal polarization, l c ( t ) = 0. Therefore, It(t) is designed to remove particle motions perpendicular to each other. For instance, when the correlated signals, P and S waves, have the same angle of incidence, cbc(t) = 180 and Co(t) = 1; under this condition, it is impossible to suppress the S waves. However, in the case of l c ( t ) = 0 , the S wave can be removed. Finally, the total polarized correlation coefficients are defined as
re(t) = Re(t) Pc(t) c o ( t ) i (t)
(12)
This equation equals unity only when the correlated signals are exactly the same. Although it may simpler to use Eq. (9) and Eq. (11) to compare the signals, doing so would not yield such good results as by using Eq. (12). The reason for this is the same as that in the explanation of Eq. (7) and Eq. (8). Hence, Eq. (12) is very adequate in measuring the similarity between signals. Referring to synthetic and real data, the following sections will support this method.
4. Synthetic tests In Fig. 1, five different Ricker wavelets with a central frequency of 30Hz and a strike of 30° are simulated. Fig. l(a) represents the data without noise; the square windows from the first to the fifth represent the following phase differences: 0 °, 45 °, 90 °,
C.-F. Shieh / Phys'icv q[ the Earth and Planetary Interiors 97 (1996) 197 204
200
135 °, 180 ° (the same as in the study by Shieh and Herrmann (1990)). The first and the fifth are P and S waves; the second and the fourth are 45 ° and 135 ° elliptic polarization; the third is 90 ° circular polarization. Fig. 3(b) shows the same data as in Fig. l(a) but with 20% random noise added. Comparing each selected window in Fig. l(a) with the seismograms in Fig. 3(b), five results are obtained, and these are listed in the following section. For the first two cases, detailed discussions are provided, but for the other three, only the final results are given.
4.1. Case 1 : 0 ° linear polarization as the reference wave The 0 ° linear wave in the first window of Fig. 3(a) is extracted as the reference P wave; this has polarization characteristics with qbref= O, Pref = 1, e r e f = 3. We let Fig. l(b) be the compared data and perform the polarized correlation as mentioned above. In Fig. 2(a), the calculated phase difference (qb(t)) is discussed in five time ranges. The phase difference in the first time range, 0 . 8 1 - 3 . 2 1 s (P wave), is + ( t ) --- 0°; in the second time range, 1.56-3.96 s (45 ° elliptic polarization), it is + ( t ) = 45°; in the third time range, 2.31-2.71 s (circular polarization), it is + ( t ) = 9 0 ° ; in the fourth time range, 3 . 0 6 - 3 . 4 6 s (135 ° elliptic polarization), it is qb(t)= 135°; in the last time range, 3.81-4.23 s (S wave), it is qb(t)--_+ 380 °.
x
[b]
Z
(a)
X Z
- - -
I
0.0
2 .I5
i
5.0
SECOND Fig. 1. Five Ricker wavelets of different polarizations are displayed: (a) noise free; (b) with 20% random noise added. The five signals windowed in (a) correspond to 0 °, 45 °, 90 °, 135 ° and 180 ° phase difference. Each windowed signal is taken as an independent reference wave used later.
[g] o. [-f}
MAX 1MUM VALUE
- - - - ~
o.8~
o , ~ ~ .
le] o
.
0.94
~
~.oo
[d] o.~/'4~
~
~'J'J~
~.oo
{c} o.3k'~J
~-'L~'~"~
"~J~
J.oo
[al o
~
~_
~so.oo
. IO.O
t 215 SECOND
5.0 j
Fig. 2. Case 1:0 ° polarization of the first windowed signal in Fig.
l(a) as the reference wave. The results of total polarized correlations: (a) phase difference of correlated seismograms (~b(t)); (b) correlation of phase difference (Oc(t)) between signals as calculated by Eq. (7); (c) cos2[Oc(t)]; (d) correlation of rectilinearity; (e) inner products of two eigenvectors; (f) correlation of the strength of polarization; (g) total polarized correlation coefficients (To(t)) as calculated by Eq. (12).
In Fig. 2(b), the correlations of the phase difference, ~bc(t), corresponding to the above five time ranges are: (3) ~b(t) --- 0 ° (both of them are P waves); (2) qb(t)= 45°; (3) qb(t)= 90°; (4) + ( t ) = 135°; (5) + ( t ) = 180 °. The calculated Co(t) from Eq. (10) is shown in Fig. 2(c), where two maximum values almost equal 1.0 in the first and the last time ranges, as d?c(t) is near 0 ° and 180 °, respectively; the minim u m value in the third time range almost equals zero because ~b~(t)-= 90 °. The intemediate values in the second and the fourth time ranges lie between zero and one. In the correlation of phase differences, P and S waves are completely correlated, P waves and circular polarization are completely uncorrelated, and P waves and other elliptic polarizations are partially correlated. In terms of correlation of the rectilinearities calculated from Eq. (8), Fig. 2(d) shows that P and S waves are completely correlated because their particle motions are all linear, P waves and circular polarizations are completely uncorrelated, and other elliptic polarizations are partially correlated. In Fig. 2(e), the inner products (l~(t)) of the two vector
201
C.-F. Shieh / Physics of the Earth and Planetary Interiors 97 (1996) 197-204
wavefields calculated from Eq. (11) indicate that P and S waves are completely uncorrelated, yet other nonlinear waves are partially correlated with P waves. In this synthetic case, the particle motion of P and S waves are perpendicular to each other, causing their inner products to equal zero. This allows for the removal of unwanted S waves. It should be noted that in Fig. 2(e), the It(t) has a higher value for 45 ° elliptic polarization than for 135 °, as the former has a right inclination (the major axis of ellipse lies in the first and third quadrants) and contains more P-wave component, whereas the latter has a left inclination (the major axis of ellipse lies in the second and fourth quadrants) and contains more S-wave components. In contrast, for the reference S wave, the lc(t) has a higher value for 135 ° elliptic polarization than for 45 °. In Fig. 2(f), the correlation of the strength of polarization calculated from Eq. (9) shows that all five wave types are almost completely correlated. This is not surprising, as all of them exhibit nearly complete polarization and, thus the same strength of polarization. Nevertheless, it is used to suppress random noise. Combining all the conditions mentioned above, the author defines the total polarized correlation function as in Eq. (12) in such a way that only P and P waves are highly correlated, as shown in Fig. 2(g). In this figure, circular polarization and S waves are completely uncorrelated, whereas 45 ° and 135 ° elliptic polarizations are partially correlated with the reference P wave. It is apparent that when the reference wave type is P, the highly correlated P wave may be singled out from the compared seismograms. Certainly, the existing noise disturbs the analysis in a finite length window. However, the synthetic test indicates that this method works when the noise level is not very high.
_
MAXIMUM VALUE
~
{41 O.
~
0.91
lel 0. ~
0.gtt
Idl o.
].0o
It) 0
.
~
I
-
I
0.0
1.00
E
1
2.5 SECOND
5.0
Fig. 3. Case 2:45 ° elliptic polarization of the second windowed signal in Fig. 2(a) as the reference wave. The results of total polarized correlation: (a) correlation of phase difference (4~c(t)); (b) cos214>c(/)];(c) correlation of rectilinearity; (d) inner products of two eigenvectors; (e) correlation of the strength of polarization; (f) total polarized correlation coefficients.
from that in Fig. 2. In other words, the arrangement of Fig. 3(a)-(f) is the same as that of Fig. 2(b)-(g). Obviously, in Fig. 3(f), the highest correlation coefficients occur in the time range of 45 ° elliptic polarization. As the other relevant characteristics are easy to understand, it is not necessary to repeat explanation here. Because the particle motions of 45 ° and 135 ° elliptic polarizations are perpendicular to each other, based on Eq. (10) they are completely uncorrelated in phase difference.
M A X 1M U M
VALUE (c] 0
~
(b} o. ~
....
~,~-J~"q,--~'~
0.93 o.9]
4.2. Case 2." 45 ° elliptic polarization as the reference waue
The 45 ° elliptic wave of the second window in Fig. l(a) is taken as the reference wave and then compared with Fig. l(b) through polarized correlation analysis. The results are shown in Fig. 3. The phase difference (~b(t)) displayed in Fig. 2(a) is the same in this case so it is not shown in Fig. 3. Therefore, the order in Fig. 3 is slightly different
(a] O.~ r 0.0
~
~
~ 2.S SECOND
0.80
-
- - 1 S.O
Fig. 4. The results of total polarized correlation coefficients for (a)
90° circular polarization of the third windowed signal, (b) 135° elliptic polarization of the fourth windowed signal, and (c) 180°
polarization of the fifth windowed signal in Fig. l(a) as reference waves.
202
C.-F. Shieh / Physics tff'the Earth aml Planetary Interiors 97 (1996) 197-204 MAX] MUM VRLUE
O. -
,~.,.,_.,A,_,,.,_/llk_,..,.,._J1~x,~.,.._~n.._._,.,J~lk_.,A.,~_Jl~
I
0.00
I
2.56
P
S
El.i
.......
Ew t
O. 7 8 6 3 6 q 0
1
5.12
SECOND
f"
0,0
Fig. 5. The correlation coefficient of a simple one-dimensional correlation method.
In the remaining three cases, with 90 ° circular polarization, 135 ° elliptic polarization and 180 ° linear polarization as reference waves, the total polarized correlation results are displayed in Fig. 4(a), Fig. 4(b), and Fig. 4(c), respectively. The results again show that the high correlation coefficients only occur in the time range of the same wave type. One may at first suspect that a simple one-dimensional correlation method can provide the same results for such synthetic cases. However, the result of a simple correlation between the reference P wave and the seismogram (the same situation as in Case 1) is shown in Fig. 5, and it obvious that to distinguish wave types by such a simple method is very difficult indeed. From the test results, it may be concluded that only when the polarized characteristics are very similar can high total polarized correlation coefficients be obtained. Consequently, this can be considered to isolate a particular wave type, as demonstrated in the following two real cases.
•
i0.2
SECOND
Fig. 6. Reference earthquake data collected by Station ALS. The first window contains the first P arrival, and the second contains the first S arrival.
in the bottom three traces of Fig. 7. In Fig. 6, two square windows indicate the P and S first arrivals. We let the P wave in the first window be the reference wave, then carry out the polarized correlation analysis. The result is shown in Fig. 7. It is found that, in Fig. 7(g), high correlation coefficients (Tc(t)) appear only in the range of the first P arrival. The results of the same processing for the first S MAX lM U M VALUE O. 99
(g}
o.
(4)
o
.
~
l.OO
(el
o
.
~
l. O0
(d)
o.~ ~ # ~ ~ i ~ ~
l. O0
(c)
o.
l, O0
222.2q
5. Applications lal
0.
180.00
The analyses above can be used effectively in seismology. Two examples adequately explain the first application.
5.1. Example 1 Testing Station ALS located in the Chia-Nan area of Taiwan is selected, and data from two different earthquakes are collected. The first earthquake occurred at 19:42h UT, 15 February 1992, with a focal depth of 20.48 km, and the second occurred at 05:43 h LIT, 16 February 1992, with a focal depth of 16.42 km. Three-component seismograms of the first and the second earthquakes are shown in Fig. 6 and
7q5,00
580. O0
~ uJ
Z
O.
~0.0
qq2. 013
51 1 SECOND
lO.2
Fig. 7. "llle bottom three traces are correlated seismograms collected by Station ALS. Results of polarized correlation with the first P arrival in Fig. 6: (a) phase difference; (b) correlation of phase difference (4%0)); (c) (d) correlation of rectilinearity; (e) inner products of two eigenvectors; (f) correlation of the strength of polarization; (g) total polarized correlation coefficients.
COS2[~bc(t)];
C.-F. Shieh / Physics of the Earth anzl Planetary Interiors 97 (1996) 197-204 P
1 II (g}
0
(4)
o
S
I *
NOISE
@-
-I
I
. ~
0. J"J
~
(d)
o. J ~ ] ' ~ l ~ _ , / ~
(:/
o
(a)
o 0.0
~
/
EN
0,92
Z
,.oo "-"~ V-''~7
1.00
VV~ ~ ~/-~//L
l.oo
]
.
N5
VALUE
~
le)
.
MRXIMUM
1 I
~
.
~
i
~
~ 1
203
NS
,.
EN
[a)
Z
0.0
.I SECOND
10.2
1.00
Fig. 9. The filtered results of the second earthquake: (a) the first P arrival; (b) the first S arrival.
18o.oo
arrivals, respectively, of the first earthquake. This analysis requires that the two compared signals have nearly the same frequency content, thereby making it more valuable for local earthquake data or two teleseismic stations that are not too far apart. In addition, it is not limited to the first arrivals; any converted phases can also be used.
10.2
SECOND Fig. 8. Results of total polarized correlation for the first S arrival: (a) phase difference; (b) correlation of phase difference; (c) cos2[(bc(t)]; (d) correlation of rectilinearity; (e) inner products of two eigenvectors; (f) correlation of the strength of polarization; (g) total polarized correlation coefficients.
arrival are illustrated in Fig. 8, where high correlation coefficients also appear in the range of the first S arrival. However, the variation of To(t) in Fig. 8(g) is much higher than that in Fig. 7(g), which complicates the correlation relationships for the first S wave. In the three portions of Fig. 8(g), the first portion before the first P arrival consists of background noise, and the high value of Tc(t) indicates that the noise contains significant S-wave components; in the second portion, between the first P and S arrivals, the high values of T~(t) indicate that there are many converted S waves (P to S); the last portion, after the first S arrival, is composed of surface waves, and the high value of T~(t) indicates that the surface waves contain significant S-wave energy. Although the interpretation is not new, it had been proven by the polarized correlation method. It should be pointed out that even though the reference S wave contains noise, the method works well. To see how similar signals can be extracted, the compared seismograms are multiplied by the function of T.(t). The results for P and S waves are displayed in Fig. 9(a) and Fig. 9(b). The results show that the first P and S arrivals of the second earthquake are highly correlated with the first P and S
5.2. Example 2 Surface waves may well be of interest too. Fig. 10(a) and Fig. 10(b) list two sets of three-component exploration data for geophone distance of 30.4m. The low-frequency dispersed surface wave in the box in Fig. 10(a) is taken as the reference wave, and then the polarized correlation analysis is performed. The filtered results of Fig. 10(b) are displayed in Fig.
[
O. f}
•
]. [', SECOND
q
~:,.I]
Fig. 10. (a) Reference dispersed surface is selected in the window; (b) correlated data recorded from another geophone at a distance of 30.4m; (c) filtered dispersed surface waves.
204
C.-F. Shieh / Physics of the Earth and Planetary Interiors 97 (1996) 197 204
10(c), and it is found that the dispersed surface wave has been enhanced. From the examples above, it is seen that polarized correlation analysis does offer tremendous gains in conducting advanced studies on the similarities among wave types.
6. Concluding remarks and discussion In this study, the characteristics of the eigenvector for complete polarization is investigated. In addition, information on each element in the eigenvector is completely discovered for the first time. The total polarized correlations of two different seismograms are compared from four viewpoints: phase difference, rectilinearity, strength of polarization and inner products. The phase difference calculated from the eigenvector is a very useful polarization parameter which has not been used before. The inner products provide a superior way to compare the direction of particle motion such that two orthogonal signals can be well identified. Only when signals have exactly the same polarization characteristics can the total polarized correlation coefficient equal unity. Under other situations, the coefficient is smaller than unity, including the case of two P waves with different angles of incidence (owing to the inner products being less than unity). The method used in this paper is very effective in discriminating P and S waves, reducing unwanted wave types and in comparing any signals that include surface waves and converted phases. The main purpose of this study is to compare the similarities between two signals. It can be illustrated in the following example. For the first P arrival of two different stations, if their polarized correlation is
weak, one may consider such causes as path effect, site effect, etc., with the help of other obtained sources. However, the traditional polarization analysis does not offer this kind of information. As the window length is limited to that which is chosen, the compared signals must have nearly the same frequency content, which limits this analysis to local earthquake data or teleseismic data of two different stations with a very short distance between them.
Acknowledgements The author would like to thank two anonymous reviewers for their constructive suggestions and comments. Gratitude is also extended to the editor, David Gubbins, for the time he spent processing and checking the manuscript.
References Flinn, E.A., 1965. Signal analysis using rectilinearity and direction of particle motion. Proc. IEEE, 53: 1874-1876. Jurkevics, A., 1988. Polarization analysis of three-component array data. Bull. Seismol. Soc. Am., 78: 1725-1743. Mott, H., 1986. Polarization in Antennas and Radar. Wiley, New York, 297 pp. Rutty, H.J. and Greenhalgh, S.A., 1993. The correlation of seismic events on multicomponent data in the presence of coherent noise. Geophys. J. Int., 113: 343-358. Samson, J.C., 1983. Pure states, polarized waves, and principal components in the spectra of multiple, geophysical time-series. Geophys. J. R. Astron. Soc., 61: 115-129. Samson, J.C. and Olson, J.V., 1980. Some comments on the description of the polarization states of waves. Geophys. J. R. Astron. Soc., 61:115-129. Shieh, C.F. and Herrmann, R.B., 1990. Ground roll: rejection using polarization filters. Geophysics, 55:1216-1222. Vidale, J.E., 1986. Complex polarization analysis of panicle motion. Bull. Seismol. Soc. Am., 76: 1393-1405.