Signal and traveltime parameter estimation using singular

value decomposition

Bjorn Ursin

The Norwegian University of Science and Technology (NTNU),

Department of Petroleum Engineering and Applied Geophysics.

S.P. Andersensvei 15A, NO - 7491 Trondheim, Norway.

Michelangelo G. Silva

Centro de Pesquisa em Geofısica e Geologia (CPPG/UFBA)

Instituto de Geociencias, Universidade Federal da Bahia

Campus Universitario da Federacao, Salvador, Bahia, Brazil.

Milton J. Porsani

Centro de Pesquisa em Geofısica e Geologia (CPPG/UFBA) and National

Institute of Science and Technology of Petroleum Geophysics (INCT-GP/CNPQ).

Instituto de Geociencias, Universidade Federal da Bahia

Campus Universitario da Federacao, Salvador, Bahia, Brazil.

March 18, 2014

1

ABSTRACT

Signal detection and traveltime parameter estimation can be performed by comput-

ing a coherence function in a data window centered around a traveltime function

defined by its parameters. We use SVD of the data matrix, not eigen decomposition

of a covariance matrix, to review the most commonly used coherence measures. This

results in a new reduced semblance coefficient defined from the first eigenimage, as-

suming that the signal amplitude is the same on all data channels (as in classical

semblance). In a second signal model the time signal is constant on each channel,

but the amplitude changes. Then semblance coefficient is the square of the first

singular value divided by the energy of the data. Two normalized crosscorrelation

coefficients derived from the first eigenimage can also be used as coherence measure.

The normalized crosscorrelation of the spatial singular vector with a vector with all

elements equal to one, and the normalized crosscorrelation of the temporal singular

vector and the average time signal (the stacked trace). We define a multiple signal

classification (MUSIC) measure as the inverse of one minus any of the normalized

coherence measures described above. In order to reduce the numerical range we

prefer to use log10MUSIC. Numerical examples with different coherence measures

applied to seismic velocity analysis of synthetic and real data show that the nor-

malized crosscorrelation coefficients perform poorly, and that log MUSIC give no

resolution enhancement on real data. The normalized eigenimage-energy coherence

measure perform poorly on synthetic data, but give the best result for a simulated

reflection with a polarity reversal. It also give good time resolution on the real

data. The classical semblance coefficient and the reduced semblance coefficient give

similar results with the reduced semblance coefficient having better resolution.

2

INTRODUCTION

The detection and estimation of traveltime parameters in multi-channel recorded

data is a classical problem in signal processing. Velocity analysis in seismic data

processing (Taner and Koehler, 1969) and the detection of multiple arriving plane

waves (Schmidt, 1986) are typical examples. In Appendix A a longer list of travel-

time functions is given.

In order to estimate the traveltime parameters one may use a window in the data,

centered around an assumed traveltime function, and compute a coherence measure

for the data in this window (Taner and Koehler, 1969). This is done for many pa-

rameter values, and the maxima in the coherence measure indicate arriving signals.

This is a standard approach in seismic velocity estimation (Yilmaz, 1987), and we

shall use this in our numerical examples. Semblance (Taner and Koehler, 1969) is

the most common coherence measure used in seismic velocity analysis. It is defined

as the ratio between the energy of the estimated signal and the energy of the data

in the analysis window, and it is normalized to be between zero and one.

A crosscorrelation coherence measure is obtained by summing the crosscorrelation

coefficient between different traces in the data window. By using the normal-

ized crosscorrelation coefficient this gives a normalized coherence measure which

is a scaled and shifted version of semblance, as presented by Neidell and Taner

(1971), who reported that semblance gave superior results in seismic velocity anal-

ysis. Larner and Celis (2007) proposed a crosscorrelation coherence measure which,

in the summation, they only included traces with relative differential moveout of

reflections exceeding a chosen threshold value. Sguazzero and Vesnaver (1987) used

the complex seismic trace (Taner et al, 1979) to define a normalized complex cross-

3

correlation coherence measure.

Semblance is an optimal coherence measure when the signal on each trace has the

same shape and the same amplitude (Ursin, 1979). For a constant time value Fomel

(2009) showed that semblance is a normalized crosscorrelation coefficient between

the data on all channels and a vector with all elements equal to one. He then pro-

posed a coherence measure for an amplitude which is equal to a constant plus a

known function with unknown amplitude. Ursin and Dahl (1990) proposed to esti-

mate the amplitude for constant time as a sum of polynomials. This was extended

by Johansen et al (1995) to an amplitude function which is a sum of orthogonal

polynomials. The advantage of this procedure is that a new polynomial coefficient

can be computed independent of the previous ones. Ursin and Ekren (1995) ex-

tended the approach by Ursin and Dahl (1990) to define semblance for data in a

window with polynomial amplitude variation. Spagnolini (1994) investigated the

ambiguity in the estimation of traveltime parameters and amplitude variations with

offset (AVO) effects simultaneously.

Schmidt (1986) proposed the multiple signal classification (MUSIC) coherence mea-

sure. This is defined as the inverse of the data projected onto the noise subspace.

It can also be expressed as the inverse of one minus the projection of the data onto

the signal subspace. Barros et al (2012) define MUSIC in terms of normalized cross-

correation measures. A general expression for MUSIC is the inverse of one minus a

normalized coherence measure which is normalized between zero and one, and the

MUSIC measure is unbounded and greater than one.

Douze and Laster (1979) proposed a coherence measure based on an approximation

of the F -statistics that is a scaled version of the signal-to-noise ratio (SNR). With

4

the definition of MUSIC above, the SNR is equal to MUSIC minus one or semblance

divided by one minus semblance (Gulunay, 1991). The SNR is unbounded and

greater than zero. Gulunay (1991) also proposed to use the logarithm of MUSIC as

a coherence measure. This is bounded below by zero, and it has less variation in

numerical values than MUSIC itself.

In Appendix B we show that the optimal estimate of the traveltime parameters is

the same for related coherence measure and MUSIC. For high SNR (coherence close

to one) the peak in MUSIC is much sharper than the peak in semblance, indicating

better resolution. The resolution power of log MUSIC is better than for semblance,

but not as good as for MUSIC. For low SNR (small values of coherence) the three

coherence measures have similar performance.

The relationship between semblance and MUSIC coherence measures has been anal-

ysed using the eigenstructure of the spatial covariance matrix (Biondi and Kostov,

1989; Kirlin, 1992; Key and Smithson, 1990; Sacchi, 1998). Singular value decom-

position (SVD, Golub and van Loan, 1996) has been used in seismic data processing

for noise removal (Freire and Ulrych, 1988; Porsani et al, 2013), but it has rarely

been used in analysing coherence measures. An exception is Spagnolini et al, (1993)

who used SVD to define a matched filter for seismic velocity analysis.

SVD separates the data into a sum of eigenimages sorted according to relative en-

ergy, with the first eigenimage having the largest energy. We shall use this first

eigenimage to define two types of generalized semblance coefficients. First, we as-

sume a common time signal with constant amplitude in each data channel and use

the first eigenimage to compute an average signal amplitude. This results in a

new reduced semblance coefficient which takes into account only signal in the first

5

eigenimage.

A second type of semblance coefficient is obtained by assuming a signal model which

has the same time signal on each trace multiplied by a spatially variant amplitude

function. The optimal signal estimate is then the first eigenimage of the data,

and the semblance is the first singular value squared divided by the energy of the

data. This was proposed by Gersztenkorn and Marfurt (1999), and it is related to

a coherence measure proposed by Key and Smithson (1990) and Sacchi (1998).

Barros et al (2012) proposed two versions of the MUSIC coherence measure based

on the first eigenvectors (corresponding to the largest eigenvalue) of the covariance

matrices. They use the normalized crosscorrelation coefficient between the first

eigenvector of the spatial covariance matrix and a vector which has all elements equal

to one. In the second coherence measure they use the normalized crosscorrelation

coefficient between the first eigenvector of the temporal covariance matrix and a

signal estimate which is the average time signal on the different channels (the stacked

trace). These crosscorrelation coefficients can be obtained directly from the first

eigenimage of the data.

In the numerical examples we compare the classical semblance coefficient with four

coherence measures derived from the first eigenimage. These are the reduced sem-

blance coefficient, the normalized energy of the first eigenimage and the two normal-

ized correlation coefficients derived from the first eigenimage. We compute the four

coherence measures and the corresponding logarithm of the MUSIC measure, in two

synthetic data examples with various levels of noise. In the first case there are two

signals with unit amplitude arriving with the same zero-offset traveltime and with

different normal moveout (NMO) velocities. In the second, there are two signals

6

arriving with the same NMO velocity and slightly different zero-offset traveltime.

There is also included a separate arriving signal with a polarity reversal, indicating

strong AVO behaviour.

We apply the four coherence measures and one log MUSIC coherence measure to

marine seismic data from offshore north-east Brazil.

NORMALIZED COHERENCE MEASURES

We assume that the data d(t, x), function of time t and direction x, can be repre-

sented by a sum of arriving pulses, sk(t), with different arrival times T (θk, x) which

have the same form, but different values of the traveltime parameters vector θ. That

is

d(t, x) =K∑k=1

sk(t− T (θk, x)) + n(t, x) (1)

where n(t, x) is a noise term. Some of the most common traveltime functions are

listed in Appendix A. In standard seismic velocity analysis the hyperbolic approxi-

mation (Mayne, 1962; Taner and Koehler, 1969)

T (θ, x) =

[T (0)2 +

x2

v2NMO

]1/2, θ = [T (0), vNMO], (2)

is used. Here T (0) is the zero-offset traveltime and vNMO is the normal moveout

velocity, often approximated by the stacking velocity, vs. We shall use this traveltime

function in our numerical examples on seismic velocity analysis.

With a trial value for the traveltime parameter vector θ, the data within a window

7

T (θ, x)± ((Nt − 1)/2)∆t are aligned with T (θ, x) to give

d(tj, xn) = Djn

j = 1, . . . , Nt

n = 1, . . . , Nx

(3)

with Nt ≤ Nx. Nt and Nx represent the time and spacial length of the data analysis

window. This procedure is illustrated in Figure 1, and we use linear interpolation

to obtain the data values between sampling grid points. The data matrix D is the

sum of signal plus noise

D = W + N (4)

where the signal matrix W and the noise matrix N, both of dimension Nt×Nx, are

assumed to be uncorrelated.

The classical coherence measure used in seismic velocity analysis is semblance (Taner

and Koehler, 1969), defined as the ratio between the signal energy and the energy

of the data. That is

S =||W||2F||D||2F

(5)

where the Frobenius matrix norm is

||D||F =

[Nt∑i=1

Nx∑j=1

D2ij

]1/2(6)

Schmidt (1986) introduced the multiple signal classification (MUSIC) algorithm.

The MUSIC coherence measure is defined as the inverse of the data projected onto

the noise subspace. With the assumption of uncorrelated signal and noise, MUSIC

can also be defined as the inverse of one minus the projection of the data onto the

signal subspace. In this way, a generalized MUSIC coherence measure is

P =1

1− S(7)

8

where S can be any coherence measure normalized so that 0 ≤ S ≤ 1. Equation (7)

is equivalent to equation 4 for MUSIC in Barros et al (2012).

Since the number of parameters is small (often only two), one computes and plots

the coherence measure for a number of traveltime parameters, as for example in

seismic velocity analysis. When plotting the results, semblance is most convenient

because it is bounded between zero and one. MUSIC has a lower bound of one, but

it may take on very large values for high SNR (S close to one). Therefore one may

use the logarithm of MUSIC (Gulunay, 1991)

log10 P = − log10(1− S) (8)

The log MUSIC coherence measure is positive and unbounded, and its numerical

values have much less variation than MUSIC itself.

In Appendix B it is shown that the coherence measures S, MUSIC and log MUSIC

give the same optimal estimate for the traveltime parameters. Using the sharpness

in the peak in the coherence measure (that is, its second-order derivatives with

respect to the traveltime parameters) as an indication of resolution power, for high

SNR (S close to one) the resolution power of log MUSIC is larger than for semblance

but less than for MUSIC. For poor SNR S is small, and MUSIC becomes P ≈ 1+S,

and

log10 P ≈ S log10 e ≈ 0.43S . (9)

In this case there is no resolution enhancement, and the coherence measures S,

MUSIC and log MUSIC behave similarly.

9

SVD SUBSPACE SEMBLANCE

The classical semblance coefficient (Taner and Koehler, 1969) is based on the signal

model

W = seT (10)

where eT = [1, 1, . . . , 1], e is an Nx × 1 stacking vector of only one’s and the Nt × 1

vector s is the time signal which is assumed to be the same on all channels. An

optimal estimate of the signal is (Ursin, 1979):

s =1

Nx

De (11)

This is the unweighted stack over all channels, or the average signal on the traces.

Then the semblance is, according to equation 5,

S =Nx||s||2

||D||2F=||De||2

Nx||D||2F(12)

where ||s||2 =∑Nt

j=1 s(tj)2.

In the following we shall use the reduced SVD (Golub and van Loan, 1996) of the

data matrix

D = UΣVT =Nt∑k=1

ukσkvTk (13)

where the Nt × 1 vectors uk are orthonormal, and the Nx × 1 vectors vk are also

orthonormal. The singular values σk are sorted such that σ1 ≥ σ2 ≥ . . . ≥ σNt ≥ 0,

assuming Nt ≤ Nx. With equation 13, semblance in equation 12 becomes

S =

∑Nt

k=1 σ2kv

2k

Nx

∑Nt

k=1 σ2k

(14)

Now we shall use only the first eigenimage D1 = u1σ1vT1 , and the signal estimate

s1 =1

Nx

D1e =σ1Nx

u1vT1 e (15)

10

The corresponding semblance is, using equation 12

SR =Nx||s1||2

||D||2F=||D1e||2

Nx||D2||F=

σ21|vT

1 e|2

Nx

∑Nt

k=1 σ2k

(16)

From equation 14 and 16 we have:

SR = S −∑Nt

2 σ2k|vT

k e|2

Nx

∑Nt

k=1 σ2k

(17)

so that we call this the reduced semblance coefficient. It uses information only from

the first eigenimage. We also note that

SR =|uT

1 De|2

Nx||D||2F=|uT

1 s|2

||D||2F(18)

where s is the signal estimate in equation 11.

NORMALIZED EIGENIMAGE ENERGY

The signal model in equation 10 corresponds to a common signal with unit amplitude

on each trace. An extension to this is to assume that the signal shape, s(t), is the

same, but that there is an amplitude variation a(x). Then the signal model is

W = saT (19)

where aT = (a1, . . . , aNx), a is a vector with the amplitudes on each channel. A

least-squares estimate of the signal is the first eigenimage (Golub and van Loan,

1996):

W = u1σ1vT1 (20)

Semblance is computed from equation 5 which gives

SE =σ21

||D||2F=

σ21∑Nt

k=1 σ2k

(21)

This was proposed by Gersztenkorn and Marfurt (1999).

11

NORMALIZED EIGENVECTOR CROSSCORRELATION

COEFFICIENTS

Barros et al (2012) defined MUSIC as in equation 7 with two different normalized

crosscorrelation coherence measures. They use the first eigenvector (correspond-

ing to the largest eigenvalue) of the spatial covariance matrix correlated with the

stacking vector e. That is, using the SVD of the data matrix in equation 13, the

coherence measure

SM =|vT

1 e|2

||e||2||v1||2=

1

Nx

|vT1 e|2 (22)

We note that

SM =SR

SE

(23)

where the semblance coefficients SR and SE are defined in equations 16 and 21

respectively.

Barros et al (2012) also used a coherence measure defined as the normalized cross-

correlation between the first eigenvector of the temporal covariance matrix and the

signal estimate in equation 11. That is, using SVD of the data matrix

ST =|uT

1 s|2

||u1||2||s||2=|uT

1 De|2

||De||2=σ21|vT

1 e|2

||De||2(24)

We note that

ST =SR

S(25)

where SR is defined in equation 16, and S is the standard semblance coefficient

defined in equation 12.

Both crosscorrelation coherence measures SM and ST are normalized between zero

and one, and they can be expressed as ratios between two semblance coefficients.

12

They will have larger numerical values than the reduced semblance coefficient and

therefore perform better in a MUSIC algorithm. They are also more sensitive to

noise, as they become ratios between two small semblance coefficients when SNR is

poor.

NUMERICAL ALGORITHM

We want to compute the first eigenimage of the data matrix D. The temporal

covariance matrix is

R = DDT = UΣ2UT (26)

We use the power method (Golub and van Loan, 1996, Barros et al, 2012) to compute

u1. With u(0) = e we iterate

u(n+1) =Ru(n)

||Ru(n)||(27)

untill ||u(n+1) − u(n)|| < ε , a prescribed threshold which controls the accuracy of

the algorithm. Then u1 = u(n+1), and next

DTu1 = v = σ1v1 (28)

so that σ1 = ||v|| and v1 = v/||v||. This completes the computation of the first

eigenimage.

The energy of the data matrix is the sum of all the autocorrelation coefficients in

the covariance matrix, that is

||D||2F =Nt∑i=1

Nx∑j=1

D2ij = tr{DDT} (29)

We note that all information related to the first eigenimage can be obtained by

iteration with the temporal covariance matrix. Barros et al (2012) also used the

13

spatial covariance matrix which normally has a higher dimension than the temporal

one.

DATA EXAMPLES

Two synthetic data examples were analysed. All data sets consist of 60 traces

from x = 20 m to x = 1200 m with a sampling distance ∆x = 20 m. The data

were computed from t = 0.8 s to t = 1.4 s with a sampling interval ∆t = 1 ms. All

coherent signals were computed with a Ricker wavelet with centre frequency f0 = 30

Hz. In each example three data sets were analysed with little noise, medium noise

and much noise, respectively. The noise was generated by white noise filtered with

the same Ricker wavelet as used for the signals. In all computations of coherence

measures we used a time window of 41 samples or 41 ms. This corresponds to

the main duration of the Ricker wavelet. In the data examples we compare the

normalized coherence measures S, SR, SE and SM and their respective log MUSIC

coherence measures. The coherence measure ST is large and very sensitive to noise,

so it was not useful. Instead we computed the corresponding log MUSIC measure,

log PT , for the two data examples.

In the first data example we want to test the velocity resolution of the different

coherence measures. Synthetic data were generated with two reflections with the

same amplitude on all channels and with the same T (0) = 1 s and vNMO = 2500 m/s

and 3000 m/s, respectively. Filtered white noise was added to these data with three

different amounts, as shown in Fig. 2. We computed the four normalized coherence

measures and their respective log MUSIC values as shown in Fig. 3 for T (0) = 1 s

and different velocities. The correct velocity values are indicated by vertical dashed

14

lines. Fig. 4-6 show standard velocity analysis for these data with different levels

of noise. The correct parameter values are marked by crosses with dashed lines.

Analysing Fig. 3 we see that SE does not work for this data set. It produces a

false event between the two real reflections for low and medium noise level. SM

has broader peaks than S and SR, log MUSIC gives slightly sharper peaks than the

respective normalized coherence measures for low and medium noise levels. For the

much-noise data set log MUSIC is a scaled version of the corresponding normalized

coherence measures, as predicted by equation 9. For this data set S and SR give

very similar results, and they are superior to SM and SE.

In the second data example we want to investigate time resolution and AVO effects.

Data were generated with three reflections with the same vNMO = 3000 m/s. The

two first reflections have the same amplitude on all channels and T (0) = 1 s and

T (0) = 1.017 s, respectively. The separation in arrival time is 17 ms which cor-

responds to half the period of the dominant frequency f0 = 30 Hz of the Ricker

wavelet. The last reflection has T (0) = 1.2 s, and the amplitude varies from 1 at

x = 20 m to −0.5 at x = 1200 m, with zero amplitude at about 950 m. The same

levels of filtered noise as in the previous example were added to these data, as shown

in Fig. 7.

In Fig. 8 we show the normalized coherence measures S, SR, SE and SM in the left

panels and the respective log MUSIC functions in the right panels for vNMO = 3000

m/s and different times. The vertical dashed lines indicate the correct values of

T (0) for the three reflections. Fig. 9-11 show standard velocity analysis plots for

these data with different noise levels. The correct parameter values are marked by

crosses with dashed lines. From Fig. 8 we see that SM performs poorly, and that

S and SR behave similarly. But for high noise level SR has better time resolution

15

than S. SE detects the reflection with severe AVO effects best for all noise levels, as

expected. From Fig. 9-11 it is seen that log MUSIC has better resolution than the

normalized coherence measures for low and medium noise levels, but for high noise

level there is no improvement.

The coherence measure ST behaved very poorly for both data sets with many er-

roneous peaks in the coherence panels. However, the corresponding log MUSIC

function, log PT , enhanced the largest peaks of ST and performed well. This is

shown in Fig. 12 for the two data sets where again the correct parameter values are

marked with crosses. Comparing the plots on the left side for the first dataset with

Fig. 4-6, it is seen that log PT performs very well for low and medium noise levels,

but it breaks down for high level of noise. When we compare the data on the right

side of Fig. 12 for the second data example with Fig. 9-11, it is seen that for low

and medium noise level log PT performs very well in separating the two reflections

with close arrival times. For the reflection with AVO effects log PE is still clearly

the best.

Figure 13, top right, shows a CMP gather from a marine seismic data set from

north-east Brazil. It is recorded from t = 0 s to t = 4 s with a sampling interval

of ∆t = 4 ms. There are 60 traces with a minimum offset equal to 150 m and a

distance between data channels ∆x = 25 m. Figure 13 also displays S, SR, SM , SE

and log10 PT . These coherence measures were computed with a time window of 17

samples or 68 ms. Details of the velocity analysis in Figure 13 are shown in Figure

14 where log10 PE also has been included. It is seen that SR and SE have better

resolution than S, and that SE has the best time resolution.

16

CONCLUSIONS

We use SVD on the data window with traveltime corrected data to review most

published coherence measures, and to define a new reduced semblance coefficient

from the first eigenimage. The signal model for the reduced semblance coefficient is

that the signal is equal to the temporal singular vector of the first eigenimage with

the same amplitude on each channel equal to the average value of the first spatial

singular vector. The signal model for the normalized eigenimage-energy coherence

measure is that the signal shape on each channel is the same, but that the amplitudes

are different. We also derive two normalized crosscorrelation coefficients using the

singular vectors in the first eigenimage.

A MUSIC coherence measure is given by the inverse of one minus any coherence

measure normalized between zero and one. To reduce its numerical range we choose

to use log MUSIC or -log10(1 − S) where S is any of the normalized coherence

measures described above. We compare the different coherent measures in seis-

mic velocity analysis, both for synthetic and real data. In the data examples the

normalized crosscorrelation coefficients give poor results. The log MUSIC for the

normalized crosscorrelation between the stacked trace and the temporal eigenvector

of the first eigenimage give good results for synthetic data with high and medium

SNR. However, for low SNR and for real data the performance is not good. The

reduced semblance coefficient, SR, and the standard semblance coefficient give com-

parable results, but SR has better resolution. The normalized eigenimage-energy

coherence measure, SE, has poor performance on the synthetic data, except for

the reflection with strong AVO effects. On this latter it is clearly superior to the

other coherence measures. Also on the real data it performs well, with good time

17

resolution.

18

ACKNOWLEDGEMENTS

We thank the associate editor and two reviewers for comments that helped improve

the paper. We also thank INCT-GP/CNPq/MCT, PETROBRAS, ANP, FINEP,

FAPESB Brazil for financial support and LANDMARK for the licenses granted to

CPGG-UFBA. Bjorn Ursin has received financial support from Statoil and from the

Norwegian Research Council through the ROSE project.

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for multichannel data: Geophysics 36, no. 3, 482-297.

Porsani, M. J., B. Ursin, M.G. Silva, and P.E.M. Melo, 2013, Dip-adaptive

singular-value decomposition filtering for seismic reflection enhancement: Geo-

physical Prospecting, 61 no. 1, 42-52, doi: 10.1111/j.1365-2478.2012.01059.x

Sacchi, M. D., 1998, A bootstrap procedure for high-resolution velocity analysis:

Geophysics, 63, 1716-1725.

21

Schmidt, R., 1986, Multiple emitter location and signal parameter estimation:

IEEE Trans. Antennas Propagat., 34, 276-280.

Sguazzero, P., and A. Vesnaver, 1987, A comparative analysis of algorithms for

stacking velocity estimation, in Bernabini, M., Carrion P., Jacovitti, G. Rocca,

F., Treitel, S., Eds., Deconvolution and inversion: Blackwell Scientific publ.,

267-286.

Shah, P. M., 1973, Use of wavefront curvature to stack seismic data with sub-

surface parameters: Geophysics, 38, 812-825.

Spagnolini, U., 1994, Compound events decomposition and the interaction be-

tween AVO and velocity information: Geophysical Prospecting, 42, 241-259.

Spagnolini, U., L. Macciotta, and A. Manni, 1993, Velocity analysis by truncated

singular value decomposition: Ann. Internat. Mtg., Soc. Expl. Geophys.,

Expanded Abstracts, 677-680.

Taner, M. T., and F. Koehler, 1969, Velocity spectra: digital computer derivation

and application of velocity functions: Geophysics, 34, 859-881, doi: 10.1190/1.1440058.

Taner, M. T.,F. Koehler and R. E. Sheriff, 1979, Complex trace analysis: Geo-

physics, 44, 1041-1063.

Ursin, B., 1979, Seismic signal detection and parameter estimation: Geophysical

Prospecting, 27, 1-15.

Ursin, B., 1982, Quadratic wavefront and traveltime approximations in inhomo-

geneous layered media with curved interfaces: Geophysics, 47, 1012-1021.

22

Ursin, B., and T. Dahl, 1990, Least-squares estimation of reflectivity polynomials:

60th Ann. Internat. Mtg. Soc. Expl. Geophys., Expanded Abstract, 1069-

1071.

Ursin, B., and B. O. Ekren, 1995, Robust AVO analysis: Geophysics, 60, 317-326.

Ursin, B., and A. Stovas, 2005, Traveltime approximations for a layered trans-

versely isotropic medium: Geophysics, 71, no. 2, D23-D33.

Yilmaz, O., 1987, Seismic data processing: SEG, Tulsa.

23

APPENDIX A - TRAVELTIME FUNCTIONS

The traveltime or arrival time function in equation 1 may have different forms. For

a plane wave (Schmidt, 1986; Biondi and Kostov, 1989)

T (θ, x) = T (0) + px, θ = [T (0), p], (A-1)

where T (0) = zero-offset traveltime and p = slowness.

For a parabolic wavefront (Hampson, 1986; Abbad et al, 2011)

T (θ, x) = T (0) + qx2, θ = [T (0), q], (A-2)

where the new parameter is q = curvature parameter.

The hyperbolic traveltime approximation used in standard seismic velocity analysis

(Mayne, 1962; Taner and Koehler, 1969; Yilmaz, 1987) is

T (θ, x) =

[T (0)2 +

x2

v2NMO

]1/2, θ = [T (0), vNMO], (A-3)

where vNMO = the normal moveout velocity is often approximated by vs, the stack-

ing velocity. We have used this traveltime function in our numerical examples.

A long offset or anisotropy traveltime approximation (Alkhalifah and Tsvankin,

1995, Ursin and Stovas, 2005; Abbad et al, 2009, 2010) is:

T (θ, x) =

{T (0)2 +

x2

v2NMO

− 2ηx4

v2NMO[T (0)2 v2NMO + (1 + 2η)x2]

}1/2

θ = [T (0), vNMO, η, ] ,

(A-4)

where the new parameter is η = effective anellipticity parameter.

A multi-CMP traveltime approximation (Shah, 1973), also referred to as the CRS

24

approximation (Jager et al, 2001; Asgedom et al, 2011) is:

T (θ, x) =

[[T (0) + p(m−m0)]

2 + q(m−m0)2 +

h2

v2NMO

]1/2θ = [T (0), p, q, vNMO]

(A-5)

Here the coordinate h is offset, m is CMP or source-receiver midpoint position and

m0 is the reference CMP position of the output of the stacking process. The new

parameters are p = linear term in CMP coordinate and q = quadratic term in CMP

coordinate. A three-dimensional multi-CMP traveltime approximation correspond-

ing to equation A-5 has been given by Ursin (1982).

A double-square-root approximation for time-migration moveout (Claerbout, 1985;

Iversen et al, 2012) is:

T (θ, x) =

[T (0)2

4+

(s−m0)2

v2M

]1/2+

[T (0)2

4+

(r −m0)2

v2M

]1/2,

θ = [T (0), vM ]

(A-6)

Here s and r are the source and receiver coordinates, m0 is the output coordinate,

and vM = time migration velocity.

Alkhalifah (2000) has given a very accurate double-square-root traveltime approxi-

mation for VTI media with parameters θ = (τ, v, η) which are similarly defined as

in equation A-4.

25

APPENDIX B

RESOLUTION POWER OF NORMALIZED COHERENCE MEASURES

A normalized coherence measure S(θ) is maximized with respect to the traveltime

parameter vector θ. There are normally many arriving signals, so there will be many

local maxima. At a maximum we have

∂S

∂θk= 0 (B-1)

for the optimal estimate θ = θ. The sharpness of a maximum peak is determined

by

∂2S

∂θi∂θk(B-2)

at this point in parameter space. For the corresponding

MUSIC-type coherence measure P in equation 7 the maximum occurs at

∂P

∂θk=

1

(1− S)2∂S

∂θk= 0 (B-3)

which is at the same point as for semblance. The curvature of the P (θ)-surface near

the maximum is determined by

∂2P

∂θi∂θk=

1

(1− S)2∂2S

∂θi∂θk(B-4)

where we have used equation (B-1).

For log10 MUSIC we use the relation

log10P = log10e logeP (B-5)

and the optimal parameter estimate satisfies

∂log10P

∂θk=log10e

1− S∂S

∂θk(B-6)

26

which again is at the same point as for semblance. The curvature of the log MUSIC

surface at the optimal parameter values is

∂log10P

∂θi∂θk=log10e

1− S∂2S

∂θi∂θk(B-7)

where we again have used equation B-1. In

all cases the optimal parameters θk occur at ∂S∂θk

= 0. But for large values of S

the curvature for P is much larger than for S, so that the parameters are better

determined using P . The separation of closely spaced arriving signals will be better,

and in this case the maxima of semblance may not coincide with the maxima of P.

The curvature of the log10P (θ)-surface is larger than for S(θ) but less than for P (θ).

Thus the resolution power of log MUSIC is better than for semblance but not as

good as for MUSIC.

27

LIST OF FIGURES

1 Data analysis window.

2 Synthetic data for velocity resolution: Little noise (a), Medium noise (b),

Much noise (c).

3 Coherence measure (left panels) and log MUSIC (right panels) for velocity-

resolution data for T (0) = 1 s and different velocities. Little-noise data (top panels),

medium-noise data (middle panels) and much-noise data (bottom panels). The ver-

tical dashed lines indicate the correct NMO velocities.

4 Little-noise velocity-resolution data. Coherence functions (top panels), and

the respective log MUSIC functions (bottom panels). The crosses with dashed lines

indicate the correct parameter values.

5 Medium noise velocity-resolution data. Coherence functions (top panels),

and the respective log MUSIC functions (bottom panels). The crosses with dashed

lines indicate the correct parameter values.

6 Much-noise velocity-resolution data. Coherence functions (top panels), and

the respective log MUSIC functions (bottom panels). The crosses with dashed lines

indicate the correct parameter values.

7 Synthetic data for time resolution and AVO effects: Little noise (a), Medium

noise (b), Much noise (c).

8 Coherence measure (left panels) and log MUSIC (right panels) for time-

resolution and AVO data for vNMO= 3000 m/s and different times. Little-noise data

(top panels), medium-noise data (middle panels) and much-noise data (bottom pan-

els). The vertical dashed lines indicate the correct values of T (0).

9 Little-noise time-resolution and AVO data. Coherence functions (top pan-

els), and the respective log MUSIC functions (bottom panels). The crosses with

28

dashed lines indicate the correct parameter values.

10 Medium-noise time-resolution and AVO data. Coherence functions (top

panels), and the respective log MUSIC functions (bottom panels). The crosses with

dashed lines indicate the correct parameter values.

11 Much-noise time-resolution and AVO data. Coherence functions (top pan-

els), and the respective log MUSIC functions (bottom panels). The crosses with

dashed lines indicate the correct parameter values.

12 Log MUSIC corresponding to the coherence measure ST . Velocity-resolution

data in the top panels, and time-resolution and AVO data in the bottom panels.

Low-noise data (left panels), middle-noise data (middle panels), and much-noise

data (right panels). The crosses with dashed lines indicate the correct parameter

values.

13 Different coherence measures in velocity analysis of marine seismic data.

14 Details from the velocity analyses in Fig. 13. We have also included log10 PE

at the bottom right.

29

Figure 1: Data analysis window.

30

0.8

1.0

1.2

1.4

Time(s)

2040

60T

race

s

(a)

0.8

1.0

1.2

1.4

2040

60T

race

s

(b)

0.8

1.0

1.2

1.4

2040

60T

race

s

(c)

Fig

ure

2:Synth

etic

dat

afo

rve

loci

tyre

solu

tion

:L

ittl

enoi

se(a

),M

ediu

mnoi

se(b

),M

uch

noi

se(c

).

31

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2500 3000 1000 5000

Val

ue C

oher

ence

Velocity(m/s)

SSMSRSE

0

0.2

0.4

0.6

0.8

1

1.2

1.4

2500 3000 1000 5000

Val

ue C

oher

ence

Velocity(m/s)

log Plog PMlog PRlog PE

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

2500 3000 1000 5000

Val

ue C

oher

ence

Velocity(m/s)

SSMSRSE

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

2500 3000 1000 5000

Val

ue C

oher

ence

Velocity(m/s)

log Plog PMlog PRlog PE

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

2500 3000 1000 5000

Val

ue C

oher

ence

Velocity(m/s)

SSMSRSE

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

2500 3000 1000 5000

Val

ue C

oher

ence

Velocity(m/s)

log Plog PMlog PRlog PE

Figure 3: Coherence measure (left panels) and log MUSIC (right panels) for velocity-

resolution data for T (0) = 1 s and different velocities. Little-noise data (top panels),

medium-noise data (middle panels) and much-noise data (bottom panels). The

vertical dashed lines indicate the correct NMO velocities.

32

0.8

0.9

1.0

1.1

1.2

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

S

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.8

0.9

1.0

1.1

1.2

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

SM

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.8

0.9

1.0

1.1

1.2

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

SE

0.6

5

0.7

0

0.7

5

0.8

0

0.8

5

0.9

0

0.9

5

0.8

0.9

1.0

1.1

1.2

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

SR

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.8

0.9

1.0

1.1

1.2

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

log

P

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.8

0.9

1.0

1.1

1.2

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

log

PM

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.8

0.9

1.0

1.1

1.2

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

log

PE

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

0.8

0.9

1.0

1.1

1.2

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

log

PR

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Fig

ure

4:L

ittl

e-noi

seve

loci

ty-r

esol

uti

ondat

a.C

oher

ence

funct

ions

(top

pan

els)

,an

dth

ere

spec

tive

log

MU

SIC

funct

ions

(bot

tom

pan

els)

.T

he

cros

ses

wit

hdas

hed

lines

indic

ate

the

corr

ect

par

amet

erva

lues

.

33

0.8

0.9

1.0

1.1

1.2

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

S

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.8

0.9

1.0

1.1

1.2

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

SM

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.8

0.9

1.0

1.1

1.2

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

SE

0.5

0

0.5

5

0.6

0

0.6

5

0.7

0

0.7

5

0.8

0

0.8

5

0.9

0

0.9

5

0.8

0.9

1.0

1.1

1.2

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

SR

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.8

0.9

1.0

1.1

1.2

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

log

P

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

log

PM

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

log

PE

0.3

0.4

0.5

0.6

0.7

0.8

0.8

0.9

1.0

1.1

1.2

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

log

PR

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Fig

ure

5:M

ediu

mnoi

seve

loci

ty-r

esol

uti

ondat

a.C

oher

ence

funct

ions

(top

pan

els)

,an

dth

ere

spec

tive

log

MU

SIC

funct

ions

(bot

tom

pan

els)

.T

he

cros

ses

wit

hdas

hed

lines

indic

ate

the

corr

ect

par

amet

erva

lues

.

34

0.8

0.9

1.0

1.1

1.2

Time(s)

1000

2000

3000

4000

5000

Velo

city(m

/s)

S

0

0.0

5

0.1

0

0.1

5

0.2

0

0.2

5

0.3

0

0.3

5

0.4

0

0.8

0.9

1.0

1.1

1.2

Time(s)

1000

2000

3000

4000

5000

Velo

city(m

/s)

SM

0

0.0

5

0.1

0

0.1

5

0.2

0

0.2

5

0.3

0

0.3

5

0.4

0

0.8

0.9

1.0

1.1

1.2

Time(s)

1000

2000

3000

4000

5000

Velo

city(m

/s)

SE

0.4

5

0.5

0

0.5

5

0.6

0

0.6

5

0.8

0.9

1.0

1.1

1.2

Time(s)

1000

2000

3000

4000

5000

Velo

city(m

/s)

SR

0

0.0

5

0.1

0

0.1

5

0.2

0

0.2

5

0.8

0.9

1.0

1.1

1.2

Time(s)

1000

2000

3000

4000

5000

Velo

city(m

/s)

log

P

0

0.0

5

0.1

0

0.1

5

0.2

0

0.8

0.9

1.0

1.1

1.2

Time(s)

1000

2000

3000

4000

5000

Velo

city(m

/s)

log

PM

0

0.0

5

0.1

0

0.1

5

0.2

0

0.2

5

0.8

0.9

1.0

1.1

1.2

Time(s)

1000

2000

3000

4000

5000

Velo

city(m

/s)

log

PE

0.2

5

0.3

0

0.3

5

0.4

0

0.4

5

0.8

0.9

1.0

1.1

1.2

Time(s)

1000

2000

3000

4000

5000

Velo

city(m

/s)

log

PR

0

0.0

5

0.1

0

0.1

5

0.2

0

0.2

5

Fig

ure

6:M

uch

-noi

seve

loci

ty-r

esol

uti

ondat

a.C

oher

ence

funct

ions

(top

pan

els)

,an

dth

ere

spec

tive

log

MU

SIC

funct

ions

(bot

tom

pan

els)

.T

he

cros

ses

wit

hdas

hed

lines

indic

ate

the

corr

ect

par

amet

erva

lues

.

35

0.8

1.0

1.2

1.4

Time(s)

2040

60T

race

s

(a)

0.8

1.0

1.2

1.4

2040

60T

race

s

(b)

0.8

1.0

1.2

1.4

2040

60T

race

s

(c)

Fig

ure

7:Synth

etic

dat

afo

rti

me

reso

luti

onan

dA

VO

effec

ts:

Lit

tle

noi

se(a

),M

ediu

mnoi

se(b

),M

uch

noi

se(c

).

36

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 1.2 0.9 1.3

Val

ue C

oher

ence

Time(s)

SSMSRSE

0

0.5

1

1.5

2

2.5

1 1.2 0.9 1.3

Val

ue C

oher

ence

Time(s)

log Plog PMlog PRlog PE

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 1.2 0.9 1.3

Val

ue C

oher

ence

Time(s)

SSMSRSE

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 1.2 0.9 1.3

Val

ue C

oher

ence

Time(s)

log Plog PMlog PRlog PE

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 1.2 0.9 1.3

Val

ue C

oher

ence

Time(s)

SSMSRSE

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

1 1.2 0.9 1.3

Val

ue C

oher

ence

Time(s)

log Plog PMlog PRlog PE

Figure 8: Coherence measure (left panels) and log MUSIC (right panels) for time-

resolution and AVO data for vNMO= 3000 m/s and different times. Little-noise

data (top panels), medium-noise data (middle panels) and much-noise data (bottom

panels). The vertical dashed lines indicate the correct values of T (0).

37

0.8

1.0

1.2

1.4

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

S

0

0.2

0.4

0.6

0.8

1.0

0.8

1.0

1.2

1.4

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

SM

0

0.2

0.4

0.6

0.8

1.0

0.8

1.0

1.2

1.4

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

SE

0.4

0.5

0.6

0.7

0.8

0.9

0.8

1.0

1.2

1.4

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

SR

0

0.2

0.4

0.6

0.8

1.0

0.8

1.0

1.2

1.4

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

log

P

0

0.5

1.0

1.5

2.0

0.8

1.0

1.2

1.4

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

log

PM

0

0.5

1.0

1.5

2.0

0.8

1.0

1.2

1.4

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

log

PE

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.8

1.0

1.2

1.4

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

log

PR

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Fig

ure

9:L

ittl

e-noi

seti

me-

reso

luti

onan

dA

VO

dat

a.C

oher

ence

funct

ions

(top

pan

els)

,an

dth

ere

spec

tive

log

MU

SIC

funct

ions

(bot

tom

pan

els)

.T

he

cros

ses

wit

hdas

hed

lines

indic

ate

the

corr

ect

par

amet

erva

lues

.

38

0.8

1.0

1.2

1.4

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

S

0

0.2

0.4

0.6

0.8

1.0

0.8

1.0

1.2

1.4

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

SM

0

0.2

0.4

0.6

0.8

0.8

1.0

1.2

1.4

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

SE

0.4

0.5

0.6

0.7

0.8

0.9

0.8

1.0

1.2

1.4

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

SR

0.4

0.5

0.6

0.7

0.8

0.9

0.8

1.0

1.2

1.4

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

log

P

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1.0

1.2

1.4

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

log

PM

0

0.2

0.4

0.6

0.8

1.0

1.2

0.8

1.0

1.2

1.4

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

log

PE

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.8

1.0

1.2

1.4

Time(s)

10

00

20

00

30

00

40

00

50

00

Ve

locity(m

/s)

log

PR

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Fig

ure

10:

Med

ium

-noi

seti

me-

reso

luti

onan

dA

VO

dat

a.C

oher

ence

funct

ions

(top

pan

els)

,an

dth

ere

spec

tive

log

MU

SIC

funct

ions

(bot

tom

pan

els)

.T

he

cros

ses

wit

hdas

hed

lines

indic

ate

the

corr

ect

par

amet

erva

lues

.

39

0.8

1.0

1.2

1.4

Time(s)

1000

2000

3000

4000

5000

Velo

city(m

/s)

S

0

0.0

5

0.1

0

0.1

5

0.2

0

0.2

5

0.3

0

0.3

5

0.4

0

0.8

1.0

1.2

1.4

Time(s)

1000

2000

3000

4000

5000

Velo

city(m

/s)

SM

0

0.1

0.2

0.3

0.4

0.5

0.6

0.8

1.0

1.2

1.4

Time(s)

1000

2000

3000

4000

5000

Velo

city(m

/s)

SE

0.5

0

0.5

2

0.5

4

0.5

6

0.5

8

0.6

0

0.6

2

0.6

4

0.6

6

0.6

8

0.7

0

0.8

1.0

1.2

1.4

Time(s)

1000

2000

3000

4000

5000

Velo

city(m

/s)

SR

0

0.0

5

0.1

0

0.1

5

0.2

0

0.2

5

0.3

0

0.3

5

0.4

0

0.8

1.0

1.2

1.4

Time(s)

1000

2000

3000

4000

5000

Velo

city(m

/s)

log

P

0

0.0

5

0.1

0

0.1

5

0.2

0

0.2

5

0.8

1.0

1.2

1.4

Time(s)

1000

2000

3000

4000

5000

Velo

city(m

/s)

log

PM

0

0.0

5

0.1

0

0.1

5

0.2

0

0.2

5

0.3

0

0.3

5

0.4

0

0.8

1.0

1.2

1.4

Time(s)

1000

2000

3000

4000

5000

Velo

city(m

/s)

log

PE

0.3

2

0.3

4

0.3

6

0.3

8

0.4

0

0.4

2

0.4

4

0.4

6

0.4

8

0.8

1.0

1.2

1.4

Time(s)

1000

2000

3000

4000

5000

Velo

city(m

/s)

log

PR

0

0.0

5

0.1

0

0.1

5

0.2

0

Fig

ure

11:

Much

-noi

seti

me-

reso

luti

onan

dA

VO

dat

a.C

oher

ence

funct

ions

(top

pan

els)

,an

dth

ere

spec

tive

log

MU

SIC

funct

ions

(bot

tom

pan

els)

.T

he

cros

ses

wit

hdas

hed

lines

indic

ate

the

corr

ect

par

amet

erva

lues

.

40

0.8

0.9

1.0

1.1

1.2

Time(s)

1000

2000

3000

4000

5000

Velo

city(m

/s)

log

PT

1.0

1.5

2.0

2.5

3.0

3.5

0.8

0.9

1.0

1.1

1.2

Time(s)

1000

2000

3000

4000

5000

Velo

city(m

/s)

log

PT

1.0

1.5

2.0

2.5

3.0

3.5

0.8

0.9

1.0

1.1

1.2

Time(s)

1000

2000

3000

4000

5000

Velo

city(m

/s)

log

PT

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.8

1.0

1.2

1.4

Time(s)

1000

2000

3000

4000

5000

Velo

city(m

/s)

log

PT

123456

0.8

1.0

1.2

1.4

Time(s)

1000

2000

3000

4000

5000

Velo

city(m

/s)

log

PT

1.0

1.5

2.0

2.5

3.0

3.5

0.8

1.0

1.2

1.4

Time(s)

1000

2000

3000

4000

5000

Velo

city(m

/s)

log

PT

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

Fig

ure

12:

Log

MU

SIC

corr

esp

ondin

gto

the

coher

ence

mea

sureST

.V

eloci

ty-r

esol

uti

ondat

ain

the

top

pan

els,

and

tim

e-

reso

luti

onan

dA

VO

dat

ain

the

bot

tom

pan

els.

Low

-noi

sedat

a(l

eft

pan

els)

,m

iddle

-noi

sedat

a(m

iddle

pan

els)

,an

dm

uch

-noi

se

dat

a(r

ight

pan

els)

.T

he

cros

ses

wit

hdas

hed

lines

indic

ate

the

corr

ect

par

amet

erva

lues

.

41

0

1

2

3

4

Tim

e(s)

1000 2000 3000 4000Velocity(m/s)

S

0.2

0.4

0.6

0

1

2

3

4T

ime(

s)

20 40 60Traces

Data

0

1

2

3

4

Tim

e(s)

1000 2000 3000 4000Velocity(m/s)

SM

0.2

0.4

0.6

0.8

1.0

0

1

2

3

4

Tim

e(s)

1000 2000 3000 4000Velocity(m/s)

SR

0.2

0.4

0.6

0

1

2

3

4

Tim

e(s)

1000 2000 3000 4000Velocity(m/s)

log PT

1.5

2.0

2.5

3.0

3.5

0

1

2

3

4

Tim

e(s)

1000 2000 3000 4000Velocity(m/s)

SE

0.6

0.7

0.8

0.9

Figure 13: Different coherence measures in velocity analysis of marine seismic data.

42

2.1

2.2

2.3

2.4

Time(s)

1000

2000

3000

Vel

ocity

(m/s

)

S0

0.2

0.4

0.6

2.1

2.2

2.3

2.4

Time(s)

1000

2000

3000

Vel

ocity

(m/s

)

SM

0

0.2

0.4

0.6

0.8

1.0

2.1

2.2

2.3

2.4

Time(s)

1000

2000

3000

Vel

ocity

(m/s

)

SR

0

0.2

0.4

0.6

2.1

2.2

2.3

2.4

Time(s)

1000

2000

3000

Vel

ocity

(m/s

)

SE

0.6

0.7

0.8

0.9

2.1

2.2

2.3

2.4

Time(s)

1000

2000

3000

Vel

ocity

(m/s

)

log

PT

1.0

1.5

2.0

2.5

3.0

3.5

2.1

2.2

2.3

2.4

Time(s)

1000

2000

3000

Vel

ocity

(m/s

)

log

PE

0.4

0.6

0.8

1.0

1.2

Fig

ure

14:

Det

ails

from

the

velo

city

anal

yse

sin

Fig

.13

.W

ehav

eal

soin

cluded

log10PE

atth

eb

otto

mri

ght.

43