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GEOPHYSICS, VOL. 74, NO. 2 �MARCH-APRIL 2009�; P. WA113–WA122, 20 FIGS.10.1190/1.3068426

easuring velocity dispersion and attenuationn the exploration seismic frequency band

angqiu F. Sun1, Bernd Milkereit1, and Douglas R. Schmitt2

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ABSTRACT

No perfectly elastic medium exists in the earth. In an ane-lastic medium, seismic waves are distorted by attenuationand velocity dispersion. Velocity dispersion depends on thepetrophysical properties of reservoir rocks, such as porosity,fractures, fluid mobility, and the scale of heterogeneities.However, velocity dispersion usually is neglected in seismicdata processing partly because of the insufficiency of obser-vations in the exploration seismic frequency band ��5through 200 Hz�. The feasibility of determining velocity dis-persion in this band is investigated. Four methods are used inmeasuring velocity dispersion from uncorrelated vibratorvertical seismic profile �VSP� data: the moving windowcrosscorrelation �MWCC� method, instantaneous phasemethod, time-frequency spectral decomposition method, andcross-spectrum method. The MWCC method is a new meth-od that is satisfactorily robust, accurate, and efficient in mea-suring the frequency-dependent traveltime in uncorrelatedvibrator records. The MWCC method is applied to the uncor-related vibrator VSP data acquired in the Mallik gas hydrateresearch well. For the first time, continuous velocity disper-sion is observed in the exploration seismic frequency bandusing uncorrelated vibrator VSP data. The observed velocitydispersion is fitted to a straight line with respect to log fre-quency to calculate Q. This provides an alternative methodfor Q measurement.

INTRODUCTION

The energy of seismic waves propagating in an anelastic mediums attenuated by various dissipation mechanisms. This phenomenonas been observed in experiments on different solids, liquids, andeismograms �e.g., Gutenberg, 1958�. Early attempts to account forhis phenomenon include various modifications of Hooke’s law to

Manuscript received by the Editor 20 January 2008; revised manuscript rec1University of Toronto, Department of Physics, Toronto, Ontario, Canada.2University ofAlberta, Department of Physics, Institute for Geophysical R2009 Society of Exploration Geophysicists.All rights reserved.

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lter perfect elasticity. For example, Zener �1948� presented thetandard linear solid model, in which the internal friction Q�1 �where

is quality factor; Knopoff and MacDonald, 1958� shows a peak atcertain frequency. This type of attenuation model is referred to as

he Debye peak model.By investigating some attenuation models, Knopoff and Mac-

onald �1958� concluded that Q and wave velocity are frequencyependent in a linear medium, i.e., one in which stress is related lin-arly to strain. The strain induced by passing seismic waves is small,nd the earth materials can be reasonably assumed to be linear. How-ver, attenuation observations �e.g., Knopoff, 1964� support aonstant-Q model; that is, Q is constant in a broad frequency bandfL through fH�. Liu et al. �1976� explained the contradiction betweenbservation and theory by superposing a series of Debye peaks toroduce a nearly constant-Q model �Figure 1�.

The causality of seismic waves requires seismic velocity to be fre-uency dependent �i.e., velocity dispersion exists� in a medium withttenuation. Furthermore, the linkage between velocity dispersionnd attenuation is the Kramers-Krönig relation �Futterman, 1962�.elocity dispersion and Q can be written explicitly as �Bourbié et al.,987�

V�� 2�M��2

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nd

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here � is density, � � 2� f is angular frequency, and M is the com-lex elastic modulus defined by the ratio of stress to strain. The realart of M, MR, and the imaginary part of M, MI, constitute a Kram-rs-Krönig pair:

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5 September 2008; published online 11 March [email protected]; [email protected]., Edmonton,Alberta, Canada. E-mail: [email protected].

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Thus, in theory, Q can be calculated from velocity dispersion.his alternative method of determining Q was deemed impracticalecause velocity dispersion is very difficult to measure in reflectioneismograms, especially with noise �Jannsen et al., 1985�.

In a constant-Q model, seismic velocity increases with frequencyKjartansson, 1979�:

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here f1 and f2 are in the fL through fH frequency band. For example,f Q � 20, and the seismic velocity is 2.6 km/s at 20 Hz, then fromquation 1, the velocity is 2.7 km/s at 200 Hz, increased by 4%.

When �� · ln�f2/f1��1 and tan�1�1/Q��1/Q, using Taylor’s ex-ansion, the above equation becomes

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n the exploration frequency band, �log�f2/f1��4, hence equation 1s valid when Q�10. This type of velocity dispersion is referred tos linear velocity dispersion �see Figure 1�. Equation 1 is identical tohe velocity dispersion in Liu et al. �1976� for a nearly constant-Q

odel.The later research on attenuation and velocity dispersion includesodeling studies on various petrophysical conditions, e.g., partial

as saturation �White, 1975�, squirt flow �Mavko et al., 1998�,atchy-saturated porous media �Johnson, 2001�, double-porosityual-permeability materials �Pride and Berryman, 2003�, and frac-ured porous media with fluid flow �Brajanovski et al., 2006�. These

odeling studies indicate that attenuation and velocity dispersion inaturated porous rocks depend on petrophysical properties, e.g., po-osity, fractures, and fluid fill.

Velocity dispersion in the sonic and ultrasonic frequency band isbserved in rock experiments �e.g., Spencer, 1981; Winkler, 1986;ones, 1986; Adam et al., 2006; Batzle et al., 2006�. Velocity disper-ion in the seismic and sonic frequency band is observed in field sur-eys �e.g., Brown and Seifert, 1997; Sams et al., 1997�. These obser-

igure 1. The constant-Q model resulted from superposed Debyeeaks and the corresponding velocity dispersion. Here fL and fH de-ote the frequency range in which Q is constant. After Liu et al.1976�.


ations demonstrate that velocity dispersion could be a strong indi-ator of petrophysical properties.

However, most velocity dispersion data are in the sonic and ultra-onic frequency band, and are sparse in the seismic frequency band10 through 200 Hz�. Therefore, the existing velocity dispersionbservations are insufficient because they do not provide enough in-ormation in the seismic frequency band, which is of greater interestn exploration seismology because the range of seismic wavelengths a few meters through a few hundred meters, making it possible tossess the bulk rock properties in a rock volume.

Velocity dispersion usually is neglected in conventional seismicata processing because the effect is small and difficult to measure inmedium with Q�30 �Futterman, 1962�. However, in high-attenu-tion media with Q30, velocity dispersion is not negligibleMolyneux and Schmitt, 1999�. A frequency-dependent phase shiftill be introduced into the seismic data because the traveltime variesith frequency. Figure 2 shows how attenuation and velocity disper-

ion distort the conventional crosscorrelation process for vibratorata. In the absence of velocity dispersion, the correlation wavelet isero phase. Assuming a constant Q and linear velocity dispersion inhe frequency band of the pilot vibrator sweep, as the sweep propa-ates, the high-frequency component is attenuated quickly as a re-ult of the constant-Q attenuation, and the correlation wavelet be-omes mixed phased because of velocity dispersion.

In summary, attenuation and velocity dispersion can provide newnsight into physical rock properties. However, present observationsn the exploration seismic frequency band are insufficient to makeny valuable deductions. In addition, seismic data processing can be

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igure 2. Distortion of correlation wavelets of vibrator sweeps in aedium with attenuation and velocity dispersion. The input vibrator

weep is linear, 12 s long, 8–180 Hz, with 0.15 s tapering. �a� The Qnd corresponding velocity dispersion model in the 8–180-Hz band.b� Correlation wavelets of the vibrator sweeps changing with theource-receiver distance.


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ffected severely if attenuation and velocity dispersion are intense.ence, it is necessary to develop a robust method to measure veloci-

y dispersion in field seismic data.

ethodology

uitable data type: Uncorrelated vibrator data

Attenuation alters the shape of a signal’s amplitude spectrum,hereas phase velocity dispersion changes the phase spectrum.herefore, uncorrelated vibrator data are appropriate to measure themall velocity dispersion in the seismic frequency band. The advan-ages of using uncorrelated vibrator sweeps include �1� the ability toontrol or to measure both the amplitude and phase spectra of the pi-ot sweep, and �2� the retention of both the amplitude and phase spec-ra of the received sweep, which is no longer possible once the signalas been correlated.

In a vibrator survey, seismic energy of varying frequencies isaunched from the source and arrives at the receiver at differentimes, forming a time-frequency �t-f� relation. This t-f relation of ailot sweep �pilot t-f relation� is predetermined in a vibrator survey.igure 3 shows the predetermined t-f relation of a pilot sweep, the pi-

ot sweep, received sweep, and their t-f spectra. Different events,uch as the direct wave, reflections, harmonics, and noise, are sepa-ated satisfactorily in the t-f spectra, making it possible to analyzehe t-f relation of a particular event. The most visible event in a re-eived t-f spectrum usually is the direct wave. Only the direct waves discussed in this paper; “received t-f relation” refers to the t-f rela-ion of the direct wave in a received sweep.

When velocity dispersion is negligible, the received t-f relationill be parallel to the pilot t-f relation. Otherwise, the received t-f re-

ation will deviate. In a constant-Q attenuation model, seismic ve-ocity is larger at higher frequencies; hence the higher-frequencyomponent always arrives earlier than expected. Although the phys-cs of the problems differs, in many ways this frequency dispersionould be compared with the Doppler effect in marine vibrator sur-eys using a moving boat �Dragoset, 1988�. Thus, if a vibrator sweeps distorted by velocity dispersion, the distortion can be correctedith the similar method presented by Dragoset.The difference between the traveltimes at a high frequency and at

low frequency t can be used to determine velocity dispersion. Thet usually is very small in the seismic band. For example, if f1

20 Hz, f2 � 200 Hz, V�f1� � 2.6 km/s, V�f2� � 2.7 km/s, andhe source-to-receiver distance is 1 km, then t is only 14 ms in thef1 through f2 band.

From equation 1, t increases as f1/f2 increases. In addition, tncreases if the source-receiver distance increases. Therefore, to op-imize observability of the small velocity dispersion in uncorrelatedibrator sweeps, broadband, long-baseline data are desirable. As forata acquisition geometry, vertical seismic profile �VSP� data arereferred because transmission seismograms are easier to analyzehan reflection seismograms. Moreover, the method to measure ve-ocity dispersion in the seismic frequency band needs to be accuratend robust for field data.

ethods of measuring velocity dispersionn uncorrelated vibrator data

There are various methods of measuring velocity dispersion inncorrelated vibrator sweeps. A desirable method must be �1� accu-


ate because velocity dispersion is small in the exploration frequen-y band, �2� robust in the presence of noise, and �3� capable of sepa-ating different events because a field vibrator record usually is con-aminated by noise and contains multiple events �e.g., reflections,ube waves, and harmonics�.

The cross-spectrum �CS� method �Donald and Butt, 2004�, in-tantaneous phase �IPH� method �Taner and Sheriff, 1977�, t-f spec-ral decomposition �TFSD� method �Castagna and Sun, 2006�, and

oving window crosscorrelation �MWCC� method have been testedsing synthetic data. The MWCC method is presented below. Thether methods and synthetic tests are discussed in the appendix.

The MWCC method measures the t-f relation of an uncorrelatedibrator sweep in the time domain. The difference between the pilotnd received t-f relations provides the frequency-dependent travel-ime, T�f�. Dispersion of this raypath-average velocity can be calcu-ated by dividing the source-receiver distance by T�f�. The proce-ure consists of six steps:

tep 1� Take a part of the pilot sweep with a tapering window. De-note f i as the central frequency of this part of the pilot sweep.The window must be longer than a few wavelengths so thatthe correlation in the next step is valid.

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igure 3. An example of uncorrelated vibrator sweeps. �a� The de-igned t-f relation of a pilot sweep; �b� the pilot sweep; �c� t-f spec-rum of the pilot sweep; �d� a received sweep; �e� t-f spectrum of theeceived sweep; �f� a schematic illustration of the events in �e�. Therayscale in �c� and �e� is amplitude. The spectral decompositionethod for �c� and �e� is discrete Fourier transform using a 1-s-long

osine window. The data are from the 3L-38 Mallik gas hydrate re-earch well, MacKenzie Delta, Northwest Territories, Canada. Theeceiver is in the borehole at a depth of 1085 m. The vibrator-bore-ole offset is 22 m. The sampling interval is 2 ms.


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tep 2� Crosscorrelate this part of the pilot sweep with the entire re-ceived sweep to produce a wavelet of correlation coeffi-cient.

tep 3� Calculate the envelope of the correlation wavelet so that apeak is produced at the arrival time of an event. This is thearrival time of the frequency component centered at f i. Themost significant peak usually is the direct wave.

tep 4� Move the tapering window along the pilot sweep for anotherf i, and repeat steps 1 through 4 so that a pseudo t-f spectrumis produced for an uncorrelated vibrator sweep. Similarly toa t-f spectrum from spectral decomposition, different eventscan be separated satisfactorily in a pseudo t-f spectrum, andthe most significant event usually is the direct wave.

tep 5� Pick the arrival time of the direct wave in the pseudo t-fspectrum for different f i to obtain the received t-f relation.For high-quality data, the arrival time can be picked contin-uously at each frequency component using an automatic ar-rival picking algorithm. Otherwise, the arrival time must bepicked manually at a few control frequencies.

tep 6� Calculate the difference between the pilot and received t-frelations to obtain the T�f�.

Steps 1 through 3 are shown in Figure 4, and steps 4 through 6 arehown in Figure 5. The uncorrelated vibrator data in Figures 3–5ere acquired in the Mallik gas hydrate research well �Dallimore et

l., 2005� at a depth of 1085 m and offset 22 m.It must be noted that the MWCC method by itself cannot deter-ine the central frequency f i of any part of a pilot sweep. This meth-

d determines only the arrival time of a wave pattern from the pilotweep. The f i must be determined by other methods, for example, thePH method.

Although a pseudo t-f spectrum from the MWCC method resem-les a t-f spectrum from spectral decomposition, they are different.he MWCC method is a time-domain method, and the values in aseudo t-f spectrum are envelopes of correlation coefficients.Apeakn a pseudo t-f spectrum implies the maximum likelihood of the ar-ival of an event, whereas a peak in a t-f spectrum represents highmplitude at a certain frequency.

According to the results of synthetic tests �see Appendix A�, theS method is fast, but it fails when noise or multiple events exist in

eceived vibrator sweeps. The IPH method is fast and suitable foreasuring the t-f relation of pilot sweeps, but it is not appropriate for

eceived sweeps. The TFSD method is the most expensive and can-ot separate multiple events as satisfactorily as the MWCC method.he MWCC method is less expensive than the TFSD method and

igure 4. Steps to calculate the arrival time of part of a pilot sweepith central frequency f i using the MWCC method. The dot in step 1

tands for multiplication of the tapering window and pilot sweep.he asterisk in step 2 stands for crosscorrelation of this part of the pi-

ot sweep and the entire received sweep.


rovides satisfactory results in the presence of different types ofoise and multiple events, but it requires the pilot t-f relation to bebtained from another method.

Combining the IPH and MWCC methods provides the best reso-ution because the IPH method can measure the pilot t-f relation, andhe MWCC can be used to measure the received t-f relation. The CSnd TFSD methods can be applied for comparison and quality con-rol.

ield data examplesThe Mallik gas hydrate field, located in the MacKenzie Delta,

orthwest Territories, Canada, is of historic interest to gas hydratenvestigators and a site of ongoing research since 1971 �Dallimore etl., 2005�. There has been abundant research on the petrophysicalnd geophysical properties of this area. For example, Lee �2002�odels the P- and S-wave velocities in gas hydrate-bearing sedi-ents using the Biot-Gassmann theory. Winters et al. �2005� pre-

ents profiles of different petrophysical properties. In general, theone of interest within the upper kilometer of sediment consists ofermafrost, water-saturated sediments, and gas hydrate-saturatedediments.

igure 5. The pseudo t-f spectrum of the uncorrelated receivedweep shown in Figure 3d from the MWCC method using a 2-s-longosine window. The pseudo t-f spectrum is obtained by repeating theteps shown in Figure 4 for different f i. �a� Displayed in grayscale,hich represents the envelope of the correlation coefficients. �b�oomed in on part of the pseudo t-f spectrum and displayed in wig-les. The dashed line shows the t-f relation of the pilot sweep. Therosses show the arrival time picks of the direct wave at different fre-uencies.


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Athree-component multioffset vibrator VSP survey was conduct-d in the 3L-38 Mallik gas hydrate research well in 2002. The uncor-elated vibrator data from this survey are analyzed to determine theeismic attenuation and velocity dispersion in this area. For the vi-rator data set discussed below, the source-borehole offset is 22 m.he receivers are at 15-m intervals in a 560–1145-m depth range.he sampling interval is 2 ms. The pilot sweep is linear, 14 s long,–120 Hz above 935 m, and 8–180 Hz at, and deeper than, 935 m.igure 6 shows the vertical component of the correlated VSP data,nd the raypath-average P-wave velocity derived from well logging.

The uncorrelated pilot and received sweeps are analyzed with thePH and MWCC methods, respectively. Arrival time of the directave is picked continuously at each frequency in the pseudo t-f

pectrum and compared with the pilot t-f relation to obtain T�f�. Asn example, T�f� measured in the vibrator sweep recorded at theepth of 1085 m is shown in Figure 7. The seismograms and pseudo-f spectrum for Figure 7 are shown in Figures 3 and 5, respectively.

Figure 8 shows the dispersion of the raypath-average velocity, in-icating that, in general, seismic velocity increases with frequency.his trend can be explained with the linear velocity dispersion in aonstant-Q model. By obtaining a best-fit straight line with respecto log frequency, the linear velocity dispersion is determined to be% in the 8–180-Hz band. Using equation 1, a raypath average Q of5 is obtained.

a) b)

Figure 6. �a� The 0.1–0.5 s of the correlated vibrator VSP sectionfrom the 3L-38 Mallik gas hydrate research well, vertical compo-nent. The source-borehole offset is 22 m. �b� The raypath-average P-wave velocity obtained from well logging, compared with the geo-logic setting.

igure 7. The T�f� of the direct wave in the vibrator sweep shown inigure 3d, obtained by comparing the pilot and received t-f relations.he t-f relation of the pilot sweep, which is shown in Figure 3b, isalculated with the IPH method. The received t-f relation is mea-ured with the MWCC method, with the pseudo t-f spectrum shownn Figure 5.


The above procedure is applied to all uncorrelated sweeps fromhe 3L-38 well at the 22-m source-borehole offset. Figure 9 showshe T�f� profile for all depth levels. For convenience, only the 8–20-Hz band is shown. The trend of the high-frequency componenteing faster is clear. Superposed on this trend are narrow bands witharlier or later arrivals, a phenomenon that is systematic in this dataet and inexplicable with constant-Q attenuation. Figure 10 showshe linear velocity dispersion calculated from the T�f� profile, com-

igure 8. Solid line: dispersion of the raypath-average P-wave ve-ocity in the vibrator sweep shown in Figure 3d, calculated from the�f� shown in Figure 7. Dashed line: linear fitting of the solid line.he linear velocity dispersion is 6% in the 8–180-Hz band, whichives a Q estimate of 15. Note that frequency is in log scale.

igure 9. The T�f� profile for the direct wave in the vibrator sweepseceived at different depth levels from the 3L-38 Mallik gas hydrateesearch well. The source-borehole offset is 22 m.

igure 10. The profile of linear velocity dispersion for the Mallikata, calculated from the T�f� profile in Figure 9, compared with theeologic setting. The contours are raypath-average P-wave velocity.ote that frequency is in log scale.


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ared with the geologic setting. Although the velocity values are theaypath-average velocities, the relationship between the velocityispersion and geologic setting can be observed.

Araypath-average Q profile is calculated from linear velocity dis-ersion using equation 1, and is shown in Figure 11. This Q profile isompared with the Q profile obtained from the spectral ratio �Tonn,991� of the first arrivals in the correlated vibrator traces. The two Qrofiles are comparable and consistent with the results of Guerin etl. �2005�, Gauer et al. �2005�, and Pratt et al. �2005�. The Q values ofP-wave can be as small as 15 in gas hydrate-bearing sediments andre higher in gas hydrate-free sediments.

Uncorrelated vibrator VSP data from other areas with differenteologic settings have been analyzed �Sun et al., 2007a, 2007b�. Ve-ocity dispersion as large as 10% in the 30–150-Hz band and ex-remely low Q have been observed in the data from McArthur River

ines, Athabasca Basin, Canada, where the metamorphic sediments highly fractured and saturated with fluids. Velocity dispersion lesshan 1% in the 30–160-Hz band and moderate Q have been observedn the uncorrelated vibrator data acquired in fractured crystallineocks in Outokumpu, Finland.

CONCLUSIONS

A new method, the moving window crosscorrelation method, isresented for determining velocity dispersion with satisfactory ac-uracy and affordable cost in the exploration seismic frequencyand, using uncorrelated vibrator VSP data. This determination ofelocity dispersion provides an alternative method for measuring Q.

Using the MWCC method, continuous velocity dispersion haseen observed in the uncorrelated vibrator VSP data acquired in theallik gas hydrate research well. By fitting a straight line of the ve-

ocity dispersion data with respect to log frequency, a Q profile is ob-ained for the Mallik data. This Q profile is consistent with the onealculated using the spectral ratio method.

Attenuation and velocity dispersion can provide better insightnto physical rock properties. Therefore, future research will focusn the analysis of uncorrelated vibrator data from other regions withifferent geologic and petrophysical settings. Well logs and core

igure 11. The Q profiles for the Mallik data compared with the geo-ogic setting. The QKK: calculated from the linear velocity dispersionhown in Figure 10. The QSR: calculated from the spectral ratio ofrst arrivals in the correlated vibrator data, which are shown in Fig-re 6.


amples will be studied to determine the relationship between thehysical rock properties and observed attenuation and velocity dis-ersion. Various petrophysical models will be tested to interpret thebserved attenuation and velocity dispersion in terms of fracture dis-ribution, fluid fill, porosity, and other rock properties.

ACKNOWLEDGMENTS

The Mallik 2002 Gas Hydrate Production Well Research Programas supported by the International Continental Scientific Drillingrogram. Funding for the VSP data acquisition was provided by aniversity of Toronto start-up grant to Bernd Milkereit. We ac-nowledge the excellent field operations conducted by Schlum-erger Ltd. for the offset VSP survey.

APPENDIX A

METHODS TO MEASUREVELOCITY DISPERSION IN

UNCORRELATED VIBRATOR SWEEPS

ross-spectrum (CS) method

Donald and Butt �2004� measured velocity dispersion in theirock experiments from the cross-spectrum of the pilot and receivedignals

Ssr � Sr · Ss*,

here Sr is Fourier transform of the received signal, and Ss* is the

onjugate of Fourier transform of the pilot signal. The frequency-de-endent traveltime T�f� then is

T�f� �1

2�

d��f�df

, �A-1�

here ��f� is the unwrapped phase of Ssr.The CS method directly calculates the difference between the pi-

ot and received t-f relations, other than individually. This method ispplicable to seismic data of any source type. For example, theource signal in rock experiments by Donald and Butt �2004� was anmpulse-like wavelet, whereas in this case it consists of uncorrelatedibrator sweeps.

This method relies on the assumption that the wave group propa-ates as a single mode and therefore is not applicable if multiplevents exist in a seismic record, as shown in Figure A-1.

nstantaneous phase (IPH) method

The instantaneous phase was introduced by Taner and Sheriff1977�. The instantaneous frequency at a time t is

f�t� �1

2�

d��t�dt

, �A-2�

here ��t� is the unwrapped phase of the Hilbert transform of an un-orrelated vibrator sweep. Similarly to the CS method, this methodelies on the assumption that the wave group propagates as a singleode.This method measures the frequency at a certain time but not the

rrival time as a function of frequency. To obtain a received t-f rela-


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ion, f�t� must be converted to obtain the frequency-dependent arriv-l time. Thus, this method is convenient for measuring the t-f rela-ion of a pilot sweep, but not for a received sweep.

The derivatives in equations A-1 and A-2 can be calculated withifference quotient formulas.

ime-frequency spectral decomposition (TFSD) method

The t-f relation of an uncorrelated vibrator sweep can be detectedy picking the arrival time of an event at different frequencies in the-f spectrum. A t-f spectrum is calculated using a time-frequencypectral decomposition algorithm. The clearest event in a received-f spectrum usually is the direct wave. The T�f� is the difference be-ween the pilot and received t-f relations.

The TFSD method can be used to calculate the t-f relation of bothilot sweeps and received sweeps. The key procedure of this methods an appropriate TFSD to generate a t-f spectrum in which differentvents are separated satisfactorily. There are different approaches toonduct TFSD, as discussed by Castagna and Sun �2006�. The spec-ral decomposition algorithm used in this appendix is the discreteourier transform �DFT�.

omputing cost of the four methods

The CS, IPH, TFSD, and MWCC methods are implemented inATLAB 7.0 on a PC with a dual-core CPU and 2 GB memory.The computing time to obtain a t-f spectrum using the TFSDethod, or a pseudo t-f spectrum using the MWCC method, increas-

s with window length and decreases with the time step at which theindow moves along a vibrator sweep:

Computing time �Window length��

�Time step�� , �A-3�

here � �1, � �2 for the TFSD method, and � �0.7, � �0.8 forhe MWCC method.

Comparing the computing time of the four methods, and settinghe CS method to be 1, the relative computing time of the other meth-ds are as follows: the IPH method is 2; the TFSD method is 7�104

hen the window length is 1000 sampling intervals and the time steps 1 sampling interval; and the MWCC method is 102 when the win-ow length and time step are 2000 and 50 sampling intervals, respec-ively.

easuring the t-f relation of a pilot sweep

In a vibrator survey, the pilot sweep often is not exactly the sames predetermined because of site response, harmonics, and so forth.ence the recorded pilot sweep should be used. The pilot t-f relation

s obtained by measuring the instantaneous frequency of the pilotweep using the IPH or TFSD method.

Synthetic vibrator data were used in the tests discussed in this ap-endix. The sampling interval is 1 ms. The pilot sweep is linear, 12 song, 8–180 Hz, and with 0.15 s taper, as shown in Figure A-1.

Figure A-2 shows the errors in the pilot t-f relation measured us-ng the IPH method without noise. The errors, small in the centralart but large at both ends, are caused mainly by improper taperingf the pilot sweep. This is because the Hilbert transform is imple-ented using Fourier transform, in which a short taper results in aavy spectrum. These errors can be reduced using a mean filter, as

hown in Figure A-2.


Figure A-3 shows errors in the pilot t-f relation measured usinghe TFSD method without noise. The t-f spectrum is calculated withFT using a 1-s cosine window. The errors in the central part are

aused by the reduced frequency resolution caused by the short win-ow. The errors always are greater than 0.1 Hz and are large at bothnds in the time range marked by Tunst. The large errors at both endsesult because a partial sweep being extracted from an end of a pilotweep contains less effective data points.

Magnitude of errors in the central part and the Tunst depend on theindow length:

�Error in f�t�� 0.5/Window length,

nd

Tunst � 0.5 � Window length.

n Figure A-3, the window length is 1 s, so that the error amplitude is.5 Hz, and the Tunst is about 0.5 s. Therefore, increasing the windowength reduces errors in the central part, but increases the Tunst.

The TFSD method is remarkably more expensive but not neces-arily more accurate than the IPH method. Therefore, the IPH meth-d is preferable when determining pilot t-f relations.

On the pilot sweep in Figure A-1, white noise is added to test thePH method. The errors can be controlled by applying a mean filtern ��t�. Increasing filter length reduces errors in the central part butxpands the unstable range at both ends.

)

)

)

igure A-1. The pilot sweep used in the synthetic tests, withoutoise. This sweep is linear, 12 s long, 8–180 Hz, sampling intervalms. �a� The first 2 s of the pilot sweep; �b� the designed t-f relation

f the pilot sweep; �c� amplitude spectrum in the 0–200-Hz band ofhe pilot sweep.

igure A-2. Errors in the measured f�t� of the pilot sweep �Figure-1� using the IPH method. No noise is added. Black line: no

moothing applied. Gray line: a 10-point mean filter has been ap-lied on ��t�.


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If stationary noise exists at a frequency, the result of the IPH meth-d is not stable in the time range in which the instantaneous frequen-y of the signal is close to the noise frequency. The large errors cane reduced by applying a median filter on ��t�.

In theory, the IPH method cannot handle harmonics because of theingle mode assumption. Fortunately, synthetic tests show that whenhe harmonics are weak, the result of the IPH method still is accept-ble, as long as ��t� is smoothed properly. Figure A-4 is an example.mplitude of the harmonic is about 20% of the signal. Errors in the.74–11.92-s range are less than 0.1 Hz. However, if the harmonic istrong, e.g., 70% of the signal, the IPH method fails. A solution is topply a pure phase-shift filter �Li et al., 1995� on the pilot sweep be-ore measuring f�t� using the IPH method.

To conclude, the IPH method is efficient and applicable for pilotweeps of ordinary data quality. The TFSD method is expensive andoes not provide better results than the IPH method. Therefore, thePH method is preferable.

easuring the frequency-dependent traveltime

The CS method measures T�f� directly, whereas the other meth-ds compare the pilot and received t-f relations to calculate T�f�.easuring the pilot t-f relation was discussed in the previous sec-

ion. This section is a discussion of measuring the received t-f rela-ion, i.e., frequency-dependent arrival time.

In the tests below, the noise-free pilot sweep �Figure A-1� is usedo better assess the influence of noise in the received sweep. The re-eived sweep, shown in Figure A-5, is calculated by assuming thathe pilot sweep has propagated for 1 km in a medium with Q � 20nd linear velocity dispersion �Figure 2b�. The four methods are test-d in the presence of noise and multiple events.

)

) c)

igure A-3. Errors in the measured f�t� of the pilot sweep, using theFT method with a 1-s cosine window. �a�The full length; �b� zoom-

ng in for the central 1 s; �c� zooming for the last 1 s, with the arrownd Tunst marking the time range in which the DFT method could notrovide stable results.

igure A-4. Errors in the measured f�t� of the pilot sweep using thePH method in the presence of a harmonic with an amplitude of 20%f the signal. Two 100-point mean filters were applied on the ��t�.


Figure A-6 compares results from the CS, TFSD, and MWCCethods when the received sweep is free of noise. No smoothing is

pplied for the CS method. Errors in the CS method are negligible.A-s cosine window is used for the TFSD and MWCC method. Re-ults of the TFSD and MWCC methods are not stable at both ends.

For the TFSD method �Figure A-6b�, when there is no noise, theagnitude of the errors in the central part depends on the time step athich the window moves along the received sweep:

�Error in T�f��DFT � 0.5 � Time step.

he errors are 0.5 ms in the central part because the window movesample by sample, i.e., 1 ms. Therefore, to obtain T�f� measure-ents of satisfactory accuracy, the time step must be small enough,hich makes the TFSD method extremely time-consuming �equa-

ion A-3�.For the MWCC method �Figure A-6c�, when there is no noise,agnitude error in the central part is determined by the sampling in-

erval:

�Error in T�f��MWCC � 0.5 � Sampling interval.

he errors are within 0.5 ms in the central part because the samplingnterval is 1 ms.Accuracy of the MWCC method does not depend onhe time step at which the window moves along the pilot sweep;ence, the time step can be increased to save computing time �equa-ion A-3�. This results in fewer f i readings.

If white noise is added to the received sweep, none of the methodsre stable when the signal-to-noise ratio �S/N� is smaller than 1.

hen S/N�1, mean filters need to be applied to ��f� for the CSethod. For the MWCC method, a longer window length should be

sed to better tackle the noise. However, increasing the windowength in the MWCC method smears the T�f�. After proper smooth-ng, the results of the CS, TFSD, and MWCC methods are compara-le.

For stationary noise at a frequency fnoise, results of the TFSD andWCC methods are not affected significantly because stationary

oise can be separated satisfactorily in a t-f spectrum or a pseudo t-f

)

)

)

igure A-5. The received sweep used in the synthetic tests withoutoise. This sweep was obtained by propagating the pilot sweep inigure A-1 for 1 km in a medium with the Q and velocity dispersionhown in Figure 2a. �a� The first 4 s of the received sweep. �b� The�f� calculated from the velocity model. �c� Amplitude spectrum of

he received sweep compared with that of the pilot sweep in the–180-Hz band.


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pectrum. The CS method failed around fnoise, but the large errors cane eliminated using a medium filter.

In field data, there always are multiple events in one receivedweep, whereas the T�f� of the direct event is desired. The CS meth-d cannot separate different events. Notches appear in the measured�f�, resembling the “ghosts” in marine seismic surveys as a result of

nterference. The TFSD and MWCC methods can provide satisfac-ory results if the events are well separated in time; otherwise, inter-erence occurs.

As an example, a secondary event is added with a time delay T tohe received sweep in Figure A-2. As shown in Figure A-7, the CS

ethod cannot separate multiple events even if the T is large andhe amplitude of the second event is significantly smaller than the

ain event.Capability of the TFSD method in separating multiple events de-

ends on the window length; a shorter window better separates dif-erent events because the events cannot be distinguished in an ampli-ude spectrum. Figure A-8 is an example. In Figure A-8, T�f� mea-urements of the main event are satisfactory when the windowength is 0.8 s but have large errors when the window length is 2 s.

For the MWCC method, the ability to separate multiple events in-reases with window length. Figure A-9 shows that the errors whenhe window length is 4 s are smaller than those when the windowength is 3 s. However, a long window will smear the result.

Conclusions of synthetic tests on determining T�f� of a receivedweep are as follows:

� The CS method is efficient but not suitable for field data be-cause it cannot provide accurate results when multiple eventsexist.

)

)

)

igure A-6. Errors in the measured T�f� from �a� the CS method, �b�he DFT method, and �c� the MWCC method. The pilot sweep ishown in Figure A-1, and the received sweep is shown in Figure A-5.o noise is added. No smoothing is applied for the CS method.A1-s

osine window is used for the DFT and MWCC methods.


)

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igure A-7. Measuring T�f� of a main event using the CS methodhen another event exists in the received sweep. �a� The correlated

eismogram of the received sweep. �b� The measured T�f� compared

a)

b)

c)

igure A-8. Measuring the T�f� of a main event in the presence of aecond event using the TFSD method. �a� The correlated seismo-ram of the received sweep. �b� Errors in the T�f� measurementshen a 0.8-s cosine window is used. �c� Errors in the T�f� measure-ents when a 2-s cosine window is used.

)

b)

igure A-9. Measuring the T�f� of a main event in the presence of aecond event using the MWCC method. �a� The correlated wave-orm of the received sweep. �b� Errors in the measured T�f� using a 3s �black line� and a 4-s �gray line� cosine window.


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� The IPH method is suitable for measuring the pilot t-f relationbecause it is efficient and provides satisfactory results for thepilot sweeps of ordinary data quality, e.g., when S/N of whitenoise is larger than 1 and harmonics are weak.

� The TFSD method is stable with white noise and stationarynoise, but the capability of separating different events is not assatisfactory as that of the MWCC method. This method is themost expensive.

� The MWCC method is appropriate for measuring T�f� in a re-ceived vibrator sweep because it is stable with noise and cansatisfactorily separate different events in the pseudo t-f spec-trum. However, the window length must be selected carefully,as a longer window better tackles the noise and better separatesdifferent events, but more severely smears the variation in theT�f�.

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measuring velocity dispersion and attenuation

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