Kulesh M Holschneider M Ohrnb er ger E uck Institute for Mathematics Univ ersit of otsdam Am Neuen alais 10 14469 otsdam German Institute for Geoscience Univ ersit of otsdam KarlLiebknec tStrasse 2425 14414 otsdam German SUMMAR In this pap er sho ho ID: 26905 Download Pdf

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Kulesh M Holschneider M Ohrnb er ger E uck Institute for Mathematics Univ ersit of otsdam Am Neuen alais 10 14469 otsdam German Institute for Geoscience Univ ersit of otsdam KarlLiebknec tStrasse 2425 14414 otsdam German SUMMAR In this pap er sho ho

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Mo deling of Disp ersion Using Con tin uous elet ransforms I: elet Based requency-V elo cit Analysis M. Kulesh M. Holschneider M. Ohrnb er ger E. uck Institute for Mathematics, Univ ersit of otsdam, Am Neuen alais 10, 14469 otsdam, German Institute for Geoscience, Univ ersit of otsdam, Karl-Liebknec t-Strasse 24-25, 14414 otsdam, German SUMMAR In this pap er, sho ho to estimate the phase elo cities of ultimo de signals as presen in 2-D shallo seismic surv eys along seismic line with the help of metho that is based on the deformation of the elet sp ectra of the seismic traces.

In analogy with frequency-w en um er analysis, erform "frequency-v elo cit y" analysis using the correlations et een phases of the elet sp ectra. Our metho has tuning parameters the parameter of an analyzing elet and the parameter of threshold op eration. Numerical and exp erimen tal examples are presen ted to illustrate ho the metho accurately extracts the phase elo cit from single- and ultimo de signals. Key ords Con tin uous elet transform, ultimo de signal, disp ersion, phase deformation. INTR ODUCTION In this pap er, con tin ue the series of studies dedicated to the mo delling of disp

ersed and atten uated propagation with the help of the con tin uous elet transform. consider 2-D shallo seismic surv ey (stations along line) and assume single source and geometrical conguration where the source and the receiv er stations are aligned. In this situation ha sho wn previously in Kulesh et al. (2005) ho the elet transform of the source and the propagated signals are related through transformation op erator (w elet propagator) that explicitly incorp orates the phase and group elo cities as ell as the atten uation factor of the medium. In (Holsc hneider et al., 2005a),

discuss ho minimazation of cost functional based on this transformation op erator allo ws the

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estimation of the disp ersion prop erties in the case of single-mo de signals. This metho es its robustness to the fact that the minimization pro cess in olv es not only the mo dulus but also the phase of the elet transform, th us making it generally ossible to reconstruct the disp ersed signal from the manipulated elet co ecien ts. Recen tly Holsc hneider et al. (2005b) expanded this op erator using complex alued Cauc elet in the case when the relationship et een the phase

elo cit and the atten uation is satised causalit constrain t. or more complex signals where the individual mo des can not easily separated in the time-frequency represen tation, pre-pro cessing step is required for the metho prop osed in Holsc hneider et al. (2005a). Within this step, it is necessary to dene the um er of propagating mo des con tained in the signals observ ed at aligned stations. or eac mo de, as is required for the optimization metho d, need to also then sp ecify frequency-dep enden phase elo cit curv as an initial condition. In this pap er, prop ose solution

for nding suc initial guesses for the optimization problem. The elet propagator in tro duced Kulesh et al. (2005) sho ws that the group elo cit is function, whic "deforms" the image of the absolute alue of the source sig- nal's elet sp ectrum. The phase elo cit "deforms" the image of the elet sp ectrum phase, and the atten uation function determines the frequency-dep enden real co ecien whic the sp ectrum is ultiplied. It allo ws us to use the correlations et een phases of the signals' elet sp ectra to erform "frequency-v elo cit y" analysis in analogy to frequency- en um er

analysis. requency-w en um er tec hniques ha een widely used for estimating apparen phase elo cities of transien seismic phases (for review see e.g. Rost and Thomas, (2002)). They are also suited for deriving phase elo cit curv es. The follo wing metho ds are the most widely utilized tec hniques applied in seismological applications: con en tional frequency- en um er metho (b eamformer) in tro duced (Burg, 1964), the high resolution analysis (Cap on, 1969) and the MU ltiple SI gnal lassication (MUSIC) metho (Sc hmidt, 1986). All of these metho ds usually require substan tial um er of

sim ultaneous

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recordings to allo for stable and reliable phase elo cit estimates. In this pap er, sug- gest new mathematical implemen tation of analysis based on the con tin uous elet transform and its correlations. In general, cross elet analysis of time series is not new metho d. or example, the elet cross sp ectra and elet coherency as discussed Maraum, Kurths Holsc hnei- der (2007). Gurley et al. (2003) analyzed the time-frequency elet-based coherence map in ligh of the inheren noise in estimates. In comparison with the ab e-men tioned pap ers, the ma jor no elt in our

suggested metho is the application of correlations et een phases of elet sp ectra, whic do not con tain an amplitude information. The pap er is organized as follo ws. In the rst section, after the in tro duction and short erview of the con tin uous elet transforms metho d, presen our concept of propagation mo deling in the elet domain with the use of elet propagator. In the next section, presen the extension of elet propagator to the "frequency-v elo cit y" analysis of ulti-mo de signals and demonstrate this tec hnique with syn thetic examples. Finally using exp erimen tal data,

compared our metho with the results from indep enden metho ds suc as MUSIC and Cap on's sp ectral analysis. The last section is dev oted to discussions and concluding remarks, where note some merits of the presen ted approac h. ASYMPTOTIC PR OP GA TOR IN VELET SP CE Let us assume that and represen signals observ ed at stations, distance apart. If disp ersiv and dissipativ haracteristics of the medium are represen ted the frequency-dep enden en um er and atten uation co ecien ), the relation et een the ourier transforms of these signals reads iN (1) where is an in teger um er and is the

complex en um er whic can dened real functions and as i ).

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Using the metho fully describ ed in (Kulesh et al., 2005), express the sp ectral propa- gator (1) in terms of the elet transform of the source signal, t; and propagated signal, t; ). The elet transform of signal with resp ect to mother elet is set of {scalar pro ducts of all dilated and translated elets with an arbitrary signal to analyzed (Holsc hneider, 1995): t; t;a 1 d where t;a (( =a is generated from elet through dilation and translation The sym ol denotes the complex conjugate. The elet is

assumed to function whic is ell lo calized in time and frequency and ob eys the oscillation condition 1 dt 0. The elet transform can expressed in terms of the ourier transform of as t; 1 a it d (2) and the in erse of the scale =a ma asso ciated with frequency measured in units of the cen tral frequency of ). If the cen tral frequency of the elet is assumed to eac scale can related to the ph ysical frequency =f Therefore, if select elet with unit cen tral frequency it is ossible to obtain the ph ysical frequency directly taking the in erse of the scale. An example of the elet

with unit cen tral frequency is the complex Morlet elet (Holsc hneider, 1995). This elet is con enien for the analysis of seismic signals and can written with its ourier transform as it where is the circular frequency and parameter describ es the ariance of the elet. Characteristic represen tation of Morlet elet in time and frequency domain is sho wn in Figure 1.

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rom equations (1) and (2), the elet transform of expressed in terms of the ourier transform of yields t; 1 =f it iN d Let us assume that frequency-dep ended en um er and atten uation are slo wly arying with

regard to the frequency range of the mother elet. or mo derate disp ersion, the complex en um er can appro ximated the rst terms of its ylor series around the frequency ). Up on inserting this appro ximation in to the in tegral ab obtain t; )] iN 1 =f (2 )] d (3) There are dieren ys to calculate the in tegral in equation (3). ollo wing the rst sp ecial elet lik Cauc elet can used, whic can absorb the imaginary part of time shift concerned with the atten uation. As result, obtain an asymptotic propagator in elet space in the case where the relationship et een the

phase elo cit and the atten uation is satised causalit constrain (Holsc hneider et al., 2005b). Instead of the use of sp ecial elets, assume that the atten uation sho ws nearly linear frequency dep endence. In suc case, and the asymptotic propagator in the elet space has the form (Kulesh et al., 2005): t; )] iN (4) In the sp ecial case, with the assumption that the analyzing elet has linear phase (with time-deriv ativ appro ximately equal to as it is the case for the Morlet elet), the appro ximation can written in terms of the phase and group elo cities as: t; exp arg (5) where (6)

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The relationship (5) has the follo wing in terpretation. The group elo cit is function that "deforms" the image of the absolute alue of the source signal's elet sp ectrum, the phase elo cit "deforms" the image of the elet sp ectrum phase, and the atten uation function determines the frequency-dep enden real co ecien whic the sp ectrum is ultiplied. This eha viour is demonstrated in Figure 2, where consider syn thetic signal ). In this example, use propagation phase and group elo cities whic are not based on ph ysical mo del. These frequency-dep enden elo cities are sho

wn in Figures 2c,d. erform the propagation of signal using the equation (1) and obtain propagated coun terpart ). The gra y-scaled images in Figure sho the absolute alues and phases of elet sp ectra t; and t; ). see that the deformations of images lab eled as jW t; and arg t; agree in general with elo cities curv es and accordingly but ha small distinctions that demonstrate the asymptotic prop erties of the equation (5). VELET BASED FREQUENCY-VELOCITY ANAL YSIS The equation (5) and Figure allo us to form ulate an idea as to ho the frequency- dep enden phase elo cit can obtained using the elet

sp ectra' phases of source and propagated signals. Using the correlations et een sp ectra, can erform "frequency- elo cit y" analysis on the analogy of frequency-w en um er metho d. The main part of this analysis consists in the calculation of correlation sp ectrum as follo ws: max min n;m n;m d max min n;m exp )) exp n;m d (7) where min max indicate the total time range for whic the elet sp ectrum as calcu- lated, min max is an un ound ariable corresp onding to the phase elo cit is complex-v alued elet phase, is real-v alued elet phase and n;m is the distance et een stations indexed and j;j

0.

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Correlation sp ectrum is calculated using only the correlations et een phases of elet sp ectra; these phases do not con tain an amplitude information. Using this fact, can use an alternativ denition of ultitrace correlation expression: max min =1 ;n d (8) The elet phases ( and ( in equations (7) and (8) are dened as t; jW t; arg (9) decrease the inuence of lo w-amplitude noise in source signal on the correlation sp ectrum ), can consider threshold op eration based on the amplitude of elet co ecien ts instead of total phases

dened in equation (9). or example, suc threshold op eration for real-v alued elet phase can in tro duced as ( if jW otherwise (10) obtain the phase elo cities, plot as surface on "frequency" and "v elo cit y" axes. The lo cal maxima in this surface corresp ond to phase elo cities of the propagating mo des within the medium and observ ed in the seismograms. impro the con trast of suc scalograms for the purp ose of visualization, instead of can plot er function of the normalized correlation co ecien ts as max min ;C max (11) where consider the maxim um of correlation sp

ectrum er elo cit min max for eac frequency separately demonstrate this concept, syn thetic seismogram ha ving mo des with dieren um ers but with the same frequency con ten is analyzed. consider the situation without atten uation: 0. ik ik (12) Giv en dieren phase elo cities, from whic the en um ers are computed, prop- agation mo deling as erformed using equation (1); the syn thetic traces are formed then

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adding the pulses obtained from the in erse ourier transform of equation (1). In this manner, sev en seismic traces ere generated to sim ulate the observ

ation at sev en successiv stations. These traces are presen ted in Figure 3a. The phase elo cities used for this seis- mogram generation are plotted in Figures 3b,c as solid curv es; fundamen tally this situation describ es rst symmetric and asymmetric mo des of Lam e. erform the "frequency-v elo cit y" analysis of this syn thetic seismogram using oth equations (7) and (8) with threshold op eration giv en in equation (10) where 1% of maxim um mo dulus of elet transform. Then plot the er function of normalized correlation co ecien ts in tro duced equation (11) with 1. The

greyscale bac kground image in Figure 3b sho ws the result of correlation et een real-v alued elet phases us- ing equation (8), and Figure 3c sho ws the correlation result when using complex-v alued phases and equation (7). The agreemen of the correlation co ecien ts' maxim um lines with theoretical phase elo cities is ery go in oth metho ds. our extra "pseudomo des" presen ted in Figure 3b,c are justied on the basis of -cycle skips et een the stations in tro duced in equation (1) and remaining in the elet propaga- tor (5) as =f item. These "pseudomo des" can ltered

analysing the group elo cities obtained from the maxim um lines of the correlation image with the help of equations (6). In order to demonstrate the noise stabilit of this metho d, erform the "frequency- elo cit y" for the same example but add random noise to the source signal. These noisy seismograms are plotted in Figure 4a; Figures 4b,c sho the correlation images using meth- ds (8) and (7) accordingly rom this example, can conclude that the noise stabilit of the presen ted approac is sucien tly high for the successful analysis of signals disturb ed additiv uncorrelated noise.

APPLICA TION TO EXPERIMENT AL In this section, use the eld data to erify the abilit of the prop osed metho to estimate phase elo cities. preciously analyzed this data in (Holsc hneider et al., 2005a) where successfully estimated the phase and group elo cities as ell as atten uation using

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minimization pro cedure for the in ersion of equation (5). Here use this equation again in order to in tro duce the correlation computation giv en in equation (8). The exp erimen tal data sho ed in Figure consist of 2-D shallo seismic surv ey (sta- tions along line) at Kerp en,

particular site in the Lo er Rhine em ba ymen where the buried scarp of historically activ fault is presumed. Sev eral proles of 48 hannels with in ter-receiv er spacing ere collected using hammer blo ws as seismic source. selected seismogram prole with prominen lo frequency high amplitude arriv als that corresp ond to the surface arriv als in tend to haracterize. selected subsections from the seismograms for our analysis. These subsections are lab eled "subsection A" and "subsection B" in Figure 5. or eac subsections, erform the "frequency-v elo cit y" analysis with the use of

correlation et een real-v alued elet phases using equation (8) with threshold op eration giv en in equation (10) where 1% of maxim um mo dulus of elet transform. sho the results of this analysis for subsections in Figures 6b and 7b as greyscale bac kground image, where plot the slo wness =C ). hec the qualit of our correlation metho d, ev aluated the slo wness using alternativ metho ds as describ ed in Holsc hneider et al. (2005a): CAPON high resolution metho (Cap on, 1969). An estimate of the phase elo cit of plane fron recorded sensors at osition ectors Cap on's algo- rithm is obtained

maximizing the expression (13) where are the steering ectors accoun ting for the tra el time dela ys of the plane fron with comp onen ts The ector denotes the horizon tal en um er ector with =C and the direction of arriv al is giv en arctan( =k ). is cross sp ectral matrix formed from the pairwise cross sp ectra of

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10 the signals as or the linear receiv er conguration of the hammer shot dataset, erform the maximiza- tion of equation (13) in grid searc manner along the one-dimensional en um er grid pro jected along the source-receiv er direction. MUSIC high resolution

metho (Sc hmidt, 1986). This metho elongs to the class of subspace-metho ds. Those metho ds rely on the orthogonalit et een signal and noise subspaces spanned the eigen ectors of the cross sp ectral matrix to deduce signal propagation haracteristics for ultiple signal con tributions. The phase elo cities of ultiple sources (arriv als) can deriv ed searc hing for the ro ots of the signal steering ectors pro jected on to the noise subspace or equiv alen tly maximizing the function +1 (14) where are the normalized eigen ectors of ), denotes the um er of acting sources and are the steering ectors

as ab e. The maximization of equation (14) is erformed again grid searc approac and the um er of sources is estimated an information theoretical approac (W ax Kailath, 1985) based on Ak aik e's criterion (Ak aik e, 1973). selected the complete eform windo ws of sub-arra ys corresp onding to subsections and along the shot prole to compute in equations (13) and (14) for horizon tal en um ers sampled equidistan tly in one dimension, and for set of 200 discrete fre- quencies spaced equidistan tly on logarithmic frequency scale from 10 to 50 Hz. In Figures 6c and 7c, sho the results of this

analysis for sub-arra ys. Eac sub-gure displa ys the results of the MUSIC approac as greyscale bac kground image. or etter visibilit normalized the maxima to one for eac individual analysis frequency The greyscale bac k- ground depicting the result from the MUSIC approac sho ws decreased resolution of the

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11 elo cit in the frequency range where the surface is highly atten uated, indicated the eak broadening along the ertical scale. The sup erimp osed con tour lines giv the distribution of from Cap on's analysis metho d. It is in teresting to note that the con tour

plot of Cap on's analysis results do not reac the frequency range where the surface is highly atten uated. The region where the phase elo cit estimates from all metho ds come quite close corre- sp onds to those where the energy of the surface arriv al are signican tly high (around 25 Hz and 35 Hz). The relativ ely large mist et een the elo cit estimates from the presen ted metho and those from MUSIC and Cap on's analysis is probably due to the sensitivit of the metho ds to the reduced signal to noise ratio in this frequency range. second ossible cause for this mist ma

the appro ximation base of the elet propagator (5). In spite of this mist, can establish go agreemen et een all considered metho ds. CONCLUSIONS AND DISCUSSION In this study extended the elet propagator prop osed in our previous ork to the metho of "frequency-v elo cit y" analysis in analogy to the classical frequency-w en um er analysis metho ds. The elet propagator is mathematical mo del for establishing link et een the con tin uous elet transform of signal and its propagated coun terpart in disp ersiv and atten uating medium. rom this prop ert erform the "frequency- elo cit y"

analysis using correlation calculations et een phases of signals' elet sp ectra. The "frequency-v elo cit y" analysis is an appro ximate approac and therefore there is some mist et een the elo cit estimates from the presen ted metho on one hand and those from another high-resolution metho ds on the other. demonstrate it the analysis of exp erimen tal eld data. Note that the elet propagator (5) con tains oth phase and group elo cities as pa- rameters. rst sigh t, the correlation calculations et een absolute alues of the signals' elet sp ectra seem useful for obtaining

the group elo cities. Ho ev er, the umerical

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12 sim ulations sho that the correlation of absolute alues is ery sensible to noise and therefore is not eectiv when applied to real data. rom analysis of syn thetic and real data, can note some merits of our approac as compared with other metho ds: Our metho has tuning parameters, suc as the parameter of the elet and threshold of phase sensivit These parameters allo us to ary the sensitivit of metho in compliance with considered signals. The metho do es not ha an constrain ts as to um er of analyzed traces in seis- mogram;

it is ossible to use only or three traces. The determination of sev eral mo de branc hes is feasible. CKNO WLEDGMENTS This pro ject is supp orted gran from the Deutsc he orsc ungsgemeinsc haft (DF G) within the framew ork of the priorit program SPP 1114 "Mathematical metho ds for time series analysis and digital image pro cessing". REFERENCES Ak aik e, H., 1973. Information theory and an extension of the maxim um lik eliho principle, Pr c. 2nd Int. Symp. Inform. The ory pp. 267{281. Burg, J. ., 1964. Three-dimensional ltering with an arra of seismometers, Ge ophysics 29 (5), 693{713.

Cap on, J., 1969. High-resolution frequency-w en um er sp ectrum analysis, Pr c. IEEE 57 1408{ 1418. Gurley K., Kijewski, T., Kareem, A., 2003. First- and higher-order correlation detection using elet transforms, J. Engr g. Me ch. 129 (2), 188{201. Holsc hneider, M., 1995. Wavelets: an nalysis ol Clarendon Press, Oxford. Holsc hneider, M., Diallo, M. S., Ohrn erger, M. K. M., uc k, E., Sc herbaum, F., 2005. Char- acterization of disp ersiv surface es using con tin uous elet transforms, Ge ophys. J. Int. 163 (2), 463{478. Holsc hneider, M., Kulesh, M., Diallo, M. S., Kurenna a, K., Sc herbaum,

F., 2005. Mo deling of

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13 disp ersion using con tin uous elet transforms: incorp orating causalit constrain with non-linear frequency-dep enden atten uation, Eos ans. GU, 86(52), al Me et. Suppl., b- str act S33A-0289 Kulesh, M., Holsc hneider, M., Diallo, M. S., Xie, Q., Sc herbaum, F., 2005. Mo deling of disp ersion using con tin uous elet transforms, Pur and Applie Ge ophysics 162 (5), 843{855. Maraun, D., Kurths, J., Holsc hneider, M., 2007. Nonstationary gaussian pro cesses in elet domain: syn thesis, estimation and signicance testing, Phys. ev. (in pr ess) Rost,

S. Thomas, C., 2002. Arra seismology: Metho ds and applications, eviews of Ge ophysics 40 (3), 2{1 2{27. Sc hmidt, R. O., 1986. Multiple source df signal pro cessing: an exp erimen tal system, IEEE ans. nt. Pr op 34 (3), 281{290. ax, M. Kailath, T., 1985. Detection of signals information theoretic criteria, IEEE ans- actions on ASSP 33 (2), 387{392.

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14 −3 −2 −1 −1 −0.5 0.5 Time (a) real part imaginary part 5 Frequency (b) Figure 1. Represen tation of Morlet elet in (a) time and (b) frequency domain, 2. elet is progressiv and has the unit cen tral

frequency

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15 (t) Frequency (Hz) (a) |W S (t,f)| 10 20 30 40 50 60 70 80 90 Time (s) Frequency (Hz) (b) arg W S (t,f) 0.5 1.5 10 20 30 40 50 60 70 80 90 (t) (c) |W S (t,f)| (f) 10 20 30 40 50 60 70 80 90 Time (s) (d) arg W S (t,f) (f) 0.5 1.5 10 20 30 40 50 60 70 80 90 Figure 2. Propagated syn thetic signal and its elet transform: (a),(c) are the er (absolute alue squared) of the elet co ecien ts and (b),(d) are the corresp onding phase images. The lines in (c) and (d) sho frequency-dep enden group and phase elo cities used in propagation mo del.

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16

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time (s) Trace Number (a) Frequency (Hz) Phase velocity (m/s) (b) 50 100 150 200 250 1200 1250 1300 1350 1400 1450 Frequency (Hz) (c) 50 100 150 200 250 Figure 3. requency-v elo cit analysis of seismic arriv als that consist of in terfering pulses of dieren disp ersion haracteristics: (a) the syn thetic seismogram, correlation sp ectrum using (b) real-v alued elet phases and (c) complex-v alued phases.

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17 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time (s) Trace Number (a) Frequency (Hz) Phase velocity (m/s) (b) 50 100 150 200 250 1200

1250 1300 1350 1400 1450 Frequency (Hz) (c) 50 100 150 200 250 Figure 4. requency-v elo cit analysis of noised seismic arriv als with in terfering pulses. Noise lev el is ab out %.

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18 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 Time (s) Subsec. A Subsec. B Receiver station Figure 5. Observ ed seismograms obtained from shallo seismic exp erimen using sledgeham- mer as source. The distance et een consecutiv stations is m.

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19 0.05 0.1 0.15 0.2

0.25 0.3 0.35 0.4 0.45 0.5 Time (s) (a) Subsection A 15 20 25 30 35 40 45 3.5 4.5 5.5 6.5 7.5 8.5 x 10 3 Frequency (Hz) Slowness (s/km) (b) 15 20 25 30 35 40 45 Frequency (Hz) (c) Figure 6. The "frequency-v elo cit y" analysis of subsection A.

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20 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time (s) (a) Subsection B 15 20 25 30 35 40 45 3.5 4.5 5.5 6.5 7.5 8.5 x 10 3 Frequency (Hz) Slowness (s/km) (b) 15 20 25 30 35 40 45 Frequency (Hz) (c) Figure 7. The "frequency-v elo cit y" analysis of subsection B.

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