velocity analysis using weighted semblance

CWP-652

Velocity analysis using weighted semblance

Simon Luo & Dave HaleCenter for Wave Phenomena, Colorado School of Mines, Golden, CO 80401, USA

(a) (b) (c)

Figure 1. A synthetic common midpoint gather (a), conventional semblance (b) and weighted semblance (c) velocity spectrum.

ABSTRACTIncreasing the resolution of semblance-based velocity spectra, or semblancespectra, can improve the accuracy of normal moveout velocity estimates. Theresolution of semblance spectra depends on the sensitivity of semblance tochanges in velocity. By weighting terms in the semblance calculation that aremore sensitive to changes in velocity, we can increase resolution.Our implementation of weighted semblance is a straightforward extensionof conventional semblance. Somewhat surprisingly, we increase resolution bychoosing a weighting function that minimizes semblance. Compared to conven-tional semblance, weighted semblance better distinguishes semblance peaks forinterfering events.

Key words: semblance resolution velocity analysis

1 INTRODUCTION

Normal moveout (NMO) velocity analysis using sem-blance spectra (Taner & Koehler, 1969) is an importantfirst step toward building a velocity model. The accu-racy of the velocity model depends on one’s ability topick the correct velocity, which in turn depends on theaccuracy and resolution of the semblance spectrum. Incases involving interfering events such as those shownin the common midpoint (CMP) gather in Figure 1a,it may be difficult to distinguish two sets of semblancepeaks in the conventional semblance spectrum shownin Figure 1b. In comparison, it is easier to differenti-ate semblance peaks and pick the correct NMO velocity

in the higher-resolution weighted semblance spectrumshown in Figure 1c.

Semblance is a normalized coherency coefficient. Ithas been shown that emphasizing terms in a coherencycoefficient calculation that are sensitive to changes invelocity can increase the resolution of the correspond-ing velocity spectra. For example, Celis & Larner (2002)introduce a selective-correlation sum that improves theresolution of velocity spectra by discarding crosscorre-lations between traces with relatively small differentialmoveout of events. Selective-correlation is effectively aweighted crosscorrelation sum with weights of eitherzero or unity, depending on the differential moveout be-tween traces.

156 S.Luo & D.Hale

We can likewise increase the resolution of semblancespectra by weighting terms in the conventional sem-blance calculation. Unlike Celis and Larner, however,we do not discard terms in the semblance calculationbut instead weight all terms on the basis of their sen-sitivity to changes in velocity. Our implementation ofweighted semblance is based on work presented in Hale(2009). Hale uses a weighted semblance coefficient toprevent smoothing of seismic images across faults. Wedo something different, i.e. increase resolution, by usinga different weighting scheme.

In this paper we describe a method for computingweighted semblance for the purpose of increasing res-olution of semblance spectra. The method is easy toimplement, and its computational cost is comparable tothat of conventional semblance.

2 SEMBLANCE METHODS

Weighted semblance is a straightforward extension ofconventional semblance. In this section, we will first dis-cuss conventional semblance, and we will introduce ourimplementation of weighted semblance. We will then de-rive the weighting function and show how it is used toincrease resolution.

2.1 Conventional semblance

Conventional semblance is a normalized coherency mea-sure that was first defined by Taner & Koehler (1969). Acomparison of semblance and other coherency measurescan be found in Neidell & Taner (1971). Semblance isroutinely used to estimate NMO velocity as a functionof zero-offset time. Following normal moveout correc-tion of a CMP gather, semblance as defined by Neidelland Taner is computed as

sNT [i] =

i+MXj=i−M

N−1Xk=0

q[j, k]

!2

N

i+MXj=i−M

N−1Xk=0

q[j, k]2, (1)

where i and j are time sample indices, k is a trace num-ber, and q[j, k] is the trace amplitude at time index j andtrace number k of the NMO-corrected gather. The innersums over k correspond to N NMO-corrected traces ina CMP gather, while the outer sums correspond to atime-smoothing window with length 2M +1 centered attime index i. Here, the time-smoothing is performed bya boxcar filter.

In general, we are free to use any time-smoothingfilter, but in practice, it is often a good idea to replacea boxcar filter with one that decays more smoothly. Forthe examples shown in this paper, the boxcar filter isreplaced with a two-sided decaying exponential filter.

We can represent the time-smoothing filter using an ad-ditional weighting function h[j]. The derivations are in-dependent of the choice of h[j], so its exact form is notimportant. We rewrite Neidell and Taner’s conventionalsemblance as

sc[i] =

Xj

h[i− j]

Xk

q[j, k]

!2

NX

j

h[i− j]X

k

q[j, k]2, (2)

where it is assumed that the unspecified summation lim-its include all indices for which the summation terms aredefined.

The semblance value reflects how well the move-out path corresponding to the trial NMO velocity fitsthe moveout of signal in the data. A good fit produces apeak in the semblance spectrum, whereas a poor fit pro-duces semblance values closer to zero. Assuming thereis no noise and no signal amplitude variation with off-set, semblance is maximized when the values of q[j, k]do not vary with index k. That is, s[i] = 1 when theNMO-corrected events are aligned across traces at timeindex i.

The resolution of semblance spectra depends on thesensitivity of NMO times to changes in velocity. If asmall change in trial velocity results in a relatively largechange in NMO time, the semblance value will changerapidly with the mismatch between the NMO times cor-responding to the trial velocity and the correct velocity.The greater the change in NMO time for a change intrial velocity, the higher the resolution of the semblancespectrum.

2.2 Conventional semblance rewritten

Before we consider weighted semblance, let us introducean alternative expression for conventional semblance.We express conventional semblance as a normalized cor-relation coefficient by first defining a reference trace r[j]as a summation over trace number (equivalently, a stackover offset) of the NMO-corrected traces in the CMPgather:

r[j] ≡X

k

q[j, k]. (3)

To simplify notation, we also define

Crq[i] ≡X

j

h[i− j]X

k

r[j]q[j, k],

Crr[i] ≡X

j

h[i− j]X

k

r[j]2,

Cqq[i] ≡X

j

h[i− j]X

k

q[j, k]2.

(4)

Velocity analysis using weighted semblance 157

Conventional semblance sc[i] can then be written as

sc[i] =Crq[i]

2

Crr[i]Cqq[i]. (5)

Equation 5 and equation 2 are equivalent expressionsfor conventional semblance.

2.3 Weighted semblance

To obtain weighted semblance, we modify conventionalsemblance by introducing weights w[j, k] into equations4:

Wrq[i] ≡X

j

h[i− j]X

k

w[j, k]r[j]q[j, k],

Wrr[i] ≡X

j

h[i− j]X

k

w[j, k]r[j]2,

Wqq[i] ≡X

j

h[i− j]X

k

w[j, k]q[j, k]2. (6)

Then, weighted semblance sw[i] is given by

sw[i] =Wrq[i]

2

Wrr[i]Wqq[i]. (7)

Weighted semblance is clearly equal to conventionalsemblance for w[j, k] = 1. Moreover, it can be shown us-ing the Cauchy-Schwarz inequality that weighted sem-blance is bounded between zero and one if the weightsw[j, k] and h[j] are non-negative.

2.4 Weighting function

We use a weighting function w[j, k] to emphasize termsin the semblance calculation that are most sensitive tochanges in velocity.

The form of the weighting function should reflectthe change in NMO time for a given change in veloc-ity; i.e., the weights should vary with both offset andtime. Consider the first-order Taylor series expansionof the hyperbolic moveout equation about the unknowncorrect velocity v:

t[j, k] =p

τ [j]2 + γx[k]2 +x[k]2

2p

τ [j]2 + γx[k]2(γ − γ) ,

(8)where τ [j] is the zero-offset time at time index j, x[k] isthe offset at trace number k, γ ≡ 1/v2, and γ ≡ 1/v2.The correct time is given by t[j, k] =

pτ [j]2 + γx[k]2,

so we can rewrite equation 8 as

t[j, k]− t[j, k] =x[k]2

2t[j, k](γ − γ) . (9)

Thus, the change in NMO time that results from a smallchange in velocity is proportional to offset squared andinversely proportional to time.

To reflect this proportionality, we choose a weight-ing function w[j, k] that has a similar dependency onoffset and time:

w[j, k] = a + bc[j]x[k]2

t[j, k], (10)

where a and b are parameters to be determined, and c[j]is calculated as the ratio of the zero-offset time to theaverage offset squared:

c[j] =τ [j]NXk

x[k]2. (11)

Multiplying by c[j] ensures that b is unitless.

The relative values of the parameters a and b inequation 10 effectively determine how the far offsets areweighted. In cases where we expect large weights forthe farthest offsets, the ratio of b to a must approachinfinity. To satisfy this condition more easily, we choose

a = 1− b, (12)

so that

w[j, k] = 1− b + bc[j]x[k]2

t[j, k]. (13)

In addition, we allow b values only between zero andone. Bounding b ensures that the weighting function isnon-negative, which is sufficient for weighted semblanceto remain normalized between zero and one.

After substituting equation 13 for w[j, k] in equa-tions 6, we have for weighted semblance

sw[i] =Wrq[i]

2

Wrr[i]Wqq[i], (14)

where

Wrq[i] = (1− b)Crq[i] + bBrq[i],

Wrr[i] = (1− b)Crr[i] + bBrr[i],

Wqq[i] = (1− b)Cqq[i] + bBqq[i], (15)

where Crq[i], Crr[i], and Cqq[i] are defined in equations4, and Brq[i], Brr[i], and Bqq[i] are defined as

Brq[i] ≡X

j

h[i− j]X

k

c[j]x[k]2

t[j, k]r[j]q[j, k],

Brr[i] ≡X

j

h[i− j]X

k

c[j]x[k]2

t[j, k]r[j]2,

Bqq[i] ≡X

j

h[i− j]X

k

c[j]x[k]2

t[j, k]q[j, k]2. (16)

Weighted semblance is now a function of the parameterb.

Note that although the weighting function is de-rived from the hyperbolic moveout equation, we do notmake any assumptions about how the seismic data areNMO-corrected. Because semblance is calculated afterNMO correction, we are free to use any moveout equa-

158 S.Luo & D.Hale

tion, hyperbolic or non-hyperbolic, to correct the data.Our method for increasing resolution works in eithercase.

2.5 Increasing resolution

To increase the resolution of semblance spectra, we min-imize semblance with respect to b. Recall that in thecase where the trial velocity equals the correct velocity,semblance is calculated along what are assumed to beconstant trace amplitudes, i.e., amplitude is indepen-dent of trace number.

If amplitude q[j, k] is independent of trace index k,then q[j, k] = r[j]/N can be pulled out of the summationover k in equations 4 and equations 16. Then, semblanceis unity, regardless of the weighting function. Becausesemblance peaks where sc[i] = 1 are not influenced bythe weighting function, we can increase the resolution ofsemblance spectra by minimizing semblance away fromthe peaks.

To minimize semblance sw[i] for any time index i,we set the first derivative with respect to b equal to zero:

dsw(b)

db= 0. (17)

Solving this equation, we find that semblance as a func-tion of b has two stationary points:

b1 =Crq[i]

Crq[i]−Brq[i], (18)

b2 =

„1 +

2Crq[i]Brr[i]Bqq[i]−Brq[i]A[i]

2Brq[i]Crr[i]Cqq[i]− Crq[i]A[i]

«−1

, (19)

where

A[i] = Crr[i]Bqq[i] + Cqq[i]Brr[i]. (20)

A typical plot of sw(b) is shown in Figure 2. Note thatone stationary point is a local minimum while the otheris a local maximum. Also, note that stationary pointb1 always gives a semblance of zero. Although Figure 2shows b1 as a local minimum and b2 as a local maximum,this is not always the case. Depending on the values ofequations 4 and equations 16, in some cases b1 may bea local maximum and b2 a local minimum.

When calculating weighted semblance, we choosethe stationary point that corresponds to the local min-imum. Let us define

Rrq[i] ≡Crq[i]

Crq[i]−Brq[i],

Rrr[i] ≡Crr[i]

Crr[i]−Brr[i],

Rqq[i] ≡Cqq[i]

Cqq[i]−Bqq[i]. (21)

These ratios give the b values of the zero and the twodiscontinuities in the plot of semblance as a functionof b. Moreover, their relative values determine which ofthe two stationary points is a local minimum. It can be

Figure 2. Plot of semblance as a function of b.

shown that b2 corresponds to a local minimum if either

Rrr[i] < Rrq[i] < Rqq[i], (22)

or

Rqq[i] < Rrq[i] < Rrr[i]. (23)

Thus, if b is between zero and one, we minimize sem-blance by choosing stationary point b2 in cases whereeither inequality 22 or inequality 23 holds, and by choos-ing stationary point b1 in all other cases.

If b is not between zero and one, we simply choosethe minimum value of sw(0) and sw(1). We choose theminimum because we are increasing resolution by min-imizing semblance.

3 RESULTS

To illustrate the action of the weighting function w[j, k]on the resolution of semblance spectra, we compareweighted semblance to conventional semblance for syn-thetic CMP gathers and for a field CMP gather fromthe North Viking Graben.

3.1 Synthetic gather

For all synthetic data examples, the CMP gathers havecable length 3 km, receiver group interval 50 m, and aRicker wavelet peak frequency of 25 Hz.

The first CMP gather consists of a series of syn-thetic primary reflections with linearly increasing NMOvelocities. The velocity increases from 2 km/s at zero-offset time τ = 0 s to 3 km/s at τ = 4 s. Figure 3adepicts the CMP gather, and Figure 3b depicts the bvalues used in the weighting function w[j, k]. In the con-ventional and weighted semblance spectrum shown inFigures 3c and 3d, respectively, the contour lines marks = 0.1 and s = 0.4. Note the spread in spectral am-plitude across a range of velocities in the conventionalsemblance spectra. In comparison, in the weighted sem-blance spectrum, both the spread in amplitude and thearea enclosed by the contour lines have decreased.


(a) (b)

(c) (d)

Figure 3. Synthetic CMP gather (a), plot of b values (b),conventional (c) and weighted (d) semblance spectrum.

We can directly compare semblance peaks by plot-ting semblance as a function of trial velocity for a cho-sen zero-offset time. Figure 4 depicts this plot for thefirst synthetic CMP gather at zero-offset time τ = 3.2s. In the figure, we see that minimizing semblance hasreduced the semblance values at velocities away fromthe peak. As a result, the weighted semblance peak issharper than the conventional semblance peak.

3.2 Synthetic gather with multiples

We add a second set of reflections to the synthetic CMPgather shown in Figure 3a to simulate interfering multi-ples. The second set of reflections have NMO velocitiesthat increase linearly from 1.98 km/s at zero-offset timeτ = 0 s to 2.70 km/s at τ = 4 s.

Figure 5a depicts the CMP gather, and Figure 5bdepicts a plot of the b values used in the weighting func-

Figure 4. Plot of semblance as a function of trial velocityat τ = 3.2 s.

tion w[j, k]. Figures 5c and 5d depict the conventionaland weighted semblance spectrum, respectively. As con-firmed by the semblance curve in Figure 6, the weightedsemblance spectrum affords higher resolution as it bet-ter distinguishes the two sets of semblance peaks. Again,minimizing semblance has reduced the semblance valuesat velocities away from the peaks.

Note that the weighted semblance peaks havesmaller amplitude compared to the conventional sem-blance peaks. This is a result of minimizing semblance.A necessary assumption for this minimization was thatthe NMO-corrected trace amplitudes are constant forthe correct trial velocity. For our synthetic data, andfor field data especially, this assumption is incorrect.Thus, in minimizing semblance, we actually expect thepeak amplitudes to decrease in most cases.

3.3 Synthetic gather with multiples and noise

Next we consider a synthetic gather contaminated byadditive noise. For this example, we added bandlimitedrandom noise to the CMP gather shown in Figure 5awith a signal-to-noise ratio of 1. Here, the signal-to-noiseratio is computed as the ratio of the root-mean-square(rms) amplitude of the signal to the rms amplitude ofthe noise.

Figure 7a depicts the noise-contaminated syntheticCMP gather, and Figure 7b plots the b values used in theweighting function. Figure 7c depicts the conventionalsemblance spectrum, and Figure 7d depicts the weightedsemblance spectrum.

Again, we see an increase in resolution and a de-crease in overall amplitude going from weighted to con-

160 S.Luo & D.Hale

(a) (b)

(c) (d)


ventional semblance. However, because the conventionalsemblance peaks have relatively low amplitudes to be-gin with, the reduction in amplitude of the weightedsemblance peaks has almost completely eliminated thes = 0.4 contour line in Figure 7d.

3.4 Viking Graben example

Our final example compares conventional and weightedsemblance for a CMP gather taken from a 2D seismicdataset from the North Viking Graben. The cable lengthis 3 km, and the offset sampling interval is 50 m. Themultiples in the data have been suppressed in order tomake the semblance peaks easier to identify.

Figure 8a depicts the CMP gather, while Figure8b depicts a plot of the b values used in the semblanceweighting function. Figure 8c shows the conventional

Figure 6. Plot of semblance as a function of trial velocityat τ = 3.2 s.

semblance spectrum, and Figure 8d shows the weightedsemblance spectrum.

In the weighted semblance spectrum, the spread insemblance associated with the near offsets has been re-duced, and the decrease in the area enclosed by the con-tours indicates that the semblance peaks are sharper aswell.

4 CONCLUSION

Weighting terms in the semblance calculation that aresensitive to changes in velocity increases the resolutionof semblance spectra. Our implementation of weightedsemblance increases resolution by using a weightingfunction to minimize semblance while maintaining anormalized semblance value bounded between zero andone.

Implementing the weighted semblance calculationrequires a small change to the conventional semblanceimplementation. This change increases the cost of calcu-lating semblance. However, the cost is still comparableto that of conventional semblance because the compu-tational complexity of calculating weighted semblanceremains on the order of Nx × Nt × Nv, where Nx, Nt,and Nv are the number of offset, time, and velocity sam-ples, respectively.

Weighted semblance increases the resolution ofsemblance spectra for synthetic data consisting of iso-lated and interfering events and for field seismic dataas well. Using weighted semblance to obtain a higherresolution semblance spectra can improve the accuracyof NMO velocity estimates and velocity models, which


(a) (b)

(c) (d)


in turn can improve the quality of seismic images of thesubsurface.

ACKNOWLEDGEMENTS

Thanks to Mobil for providing the Viking Graben seis-mic data. Thanks also to our sponsors and colleagues atthe Center for Wave Phenomena at the Colorado Schoolof Mines. Thanks especially to Ken Larner, Jeff Godwin,and Diane Witters for helping review and revise this pa-per.

REFERENCES

Celis, V., and Larner, K., Selective-correlation velocityanalysis:, M.Sc. thesis, CWP-434, Center for WavePhenomena, Colorado School of Mines, 2002.

(a) (b)

(c) (d)

Figure 8. Viking Graben CMP gather (a), plot of b values(b), conventional (c) and weighted (d) semblance spectrum.

Hale, D., Structure-oriented smoothing and sem-blance:, Technical Report CWP-635, Center for WavePhenomena, Colorado School of Mines, 2009.

Neidell, N. S., and Taner, M. T., 1971, Semblance andother coherency measures for multichannel data: Geo-physics, 36, 482–297.

Taner, M. T., and Koehler, F., 1969, Velocity spec-tra – digital computer derivation and applications ofvelocity functions: Geophysics, 34, 859–881.

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