AVO-Friendly Velocity Analysis Based on the High-Resolution ...

Citation: Zhang, C.; Fan, L.; Chen, G.;

Li, J. AVO-Friendly Velocity Analysis

Based on the High-Resolution

PCA-Weighted Semblance. Appl. Sci.

2022, 12, 6098. https://doi.org/

10.3390/app12126098

Academic Editors: Paolo Mauriello

and Domenico Patella

Received: 7 May 2022

Accepted: 13 June 2022

Published: 15 June 2022

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applied sciences

Article

AVO-Friendly Velocity Analysis Based on the High-ResolutionPCA-Weighted SemblanceChunlin Zhang 1,*, Liyong Fan 2, Guiting Chen 3,* and Jijun Li 4

1 PetroChina Research Institute of Petroleum Exploration and Development, Beijing 100083, China2 PetroChina Changqing Oilfield Company, Xi’an 710017, China; [email protected] School of Information and Communication Engineering, University of Electronic Science and Technology

of China, Chengdu 610054, China4 School of Geosciences, China University of Petroleum (East China), Qingdao 266580, China; [email protected]* Correspondence: [email protected] (C.Z.); [email protected] (G.C.)

Abstract: Velocity analysis using the semblance spectrum can provide an effective velocity modelfor advanced seismic imaging technology, in which the picking accuracy of velocity analysis issignificantly affected by the resolution of the semblance spectrum. However, the peak broadeningof the conventional semblance spectrum leads to picking uncertainty, and it cannot deal with theamplitude-variation-with-offset (AVO) phenomenon. The well-known AB semblance can process theAVO anomalies, but it has a lower resolution compared with conventional semblance. To improvethe resolution of the AB semblance spectrum, we propose a new weighted AB semblance based onprincipal component analysis (PCA). The principal components or eigenvalues of seismic events arehighly sensitive to the components with spatial coherence. Thus, we utilized the principal componentsof the normal moveout (NMO)-corrected seismic events with different scanning velocities to constructa weighting function. The new function not only has a high resolution for velocity scanning, but it isalso a friendly method for the AVO phenomenon. Numerical experiments with the synthetic andfield seismic data sets proved that the new method significantly improves resolution and can providemore accurate picked velocities compared with conventional methods.

Keywords: seismic velocity; AB semblance spectrum; AVO phenomenon; principal componentanalysis; spatial coherence

1. Introduction

Velocity analysis is an essential step in seismic exploration, which can estimate anappropriate velocity model for subsequent seismic signal processing and imaging. Thenormal-moveout (NMO)-based velocity analysis with the semblance spectrum [1,2] iswidely applied in industry due to its straightforward implementation and relatively minorcomputational cost. The semblance spectrum uses a series of scanning velocities to applyNMO corrections with hyperbolic trajectories [3,4] and then measures the spatial coherenceof the corrected seismic events to obtain the corresponding semblances. When an accuratevelocity is used, the corrected seismic events have a strong spatial coherence [5], and thepeak value will appear in the semblance spectrum. The velocity values corresponding to thepeaks in the semblance spectrum are then picked for velocity modeling [6,7]. The resolutionof the semblance spectrum determines its ability to distinguish and pick individual peaks,and the high-resolution semblance can significantly improve the accuracy of the velocitymodeling [8–10].

Advanced velocity modelling methods [11–14], such as full waveform inversion [15],usually require the NMO-based semblance spectrum to provide an effective initial model.However, due to insufficient resolution, the peak broadening of the conventional semblancespectrum leads to picking uncertainty, which makes it difficult to provide an effective initialmodel. To improve the resolution of the semblance spectrum, many scholars have improved

Appl. Sci. 2022, 12, 6098. https://doi.org/10.3390/app12126098 https://www.mdpi.com/journal/applsci

Appl. Sci. 2022, 12, 6098 2 of 13

the coherence measurement in the semblance spectrum [16] and proposed a new semblancebased on the coherence measurement of the covariance matrix. The covariance matrix of theNMO-corrected seismic events is sensitive to far-offset traces, which improves the ability todistinguish individual peaks [17–19] and to develop a new coherence measurement basedon the bootstrapped differential semblance. The bootstrapping approach estimates statisti-cal properties of the NMO-corrected seismic events with a coherency estimator, which cansignificantly improve the resolution of the semblance spectrum. In addition, many scholarshave developed the semblance spectra through the singular value decomposition (SVD) ofthe data matrix [20–23]. The SVD-based semblance is sensitive to the spatial coherency ofthe NMO-corrected seismic events, thus improving the ability to distinguish individualpeaks. On the other hand, many scholars have constructed the weighting functions toimprove the resolution. The weighting functions are usually sensitive to the coherency ofthe NMO-corrected seismic events. [23] constructed a weighting function by measuringthe local similarity between the reference data and each trace. This similarity-weightedfunction can significantly improve the resolution of semblance. Then, [24] applied thesimilarity-weighted function to the simultaneous-source data, and achieved better resultsthan the traditional methods.The similarity-weighted function requires an appropriatereference trace as a judging criterion. However, how to select a reliable reference traceneeds further study [25].

Coherence measurement in the semblance spectrum assumes that there is no vari-ation of amplitudes or phases along with seismic events. However, the amplitudes ofseismic events that exhibit the amplitude-variation-with-offset phenomenon are usuallyvariable [26]. The conventional semblance spectrum based on this assumption cannotdeal with the amplitude variation, and the energy clusters in the conventional semblancespectrum will be lost, making it impossible to pick up the effective velocity value. To over-come the problem that the conventional spectrum cannot deal with AVO anomalies, [27]introduced a trend-based semblance operator to account for the effect of AVO anomalieson the semblance spectrum. This semblance operator, referred to as the AB semblance.Ref. [28] provided an explicit form of AB semblance through the minimization of thetrend-fitting parameters, and demonstrated a significant improvement in dealing withthe AVO phenomenon. However, the resolution of standard AB semblance is lower thanclassical methods, and its large peak-broadening leads to inaccurate picking. Then, thebootstrapping-weighted operator and AB semblance were combined to construct an AVO-friendly bootstrapped differential semblance [29]. This method has a high resolution andcan handle the AVO regions. Besides, the AVO-friendly weighting term can also improvethe resolution of AB semblance [9]. This method measures local similarity between thereference trace and other traces in a time window. The local-similarity method is sensitiveto the seismic curvature of the far-offset data, thus significantly improving the resolution ofAB semblance.

In this paper, we propose a new weighting operator based on the principal componentsanalysis to improve the resolution of the AB semblance. Due to the NMO-corrected seismicgather having strong spatial coherence or low-rank attributes, the first principal componentis much larger than other principal components, and it has an overwhelming advantagein the composition of the NMO-corrected seismic gather. The first principal componentrepresents the components with strong coherence in the data set, and these componentscome from the NMO-corrected seismic events with an accurate velocity. In comparison,the other principal components represent the non-coherence data sets and they come fromthe seismic events that have not been completely corrected. Thus, we used the distributioncharacteristics of the principal components to construct the new weighting function. Thenew function is sensitive to the curvature of seismic events at far-offset traces. Whenthe seismic events are not fully corrected, the weighting function will not enhance theintensity of the semblance spectrum. By combining the new weighting function related tothe principal components and trend-based AB coefficients, the new method has a higherresolution than that of the conventional methods, and it can handle the AVO phenomenon.

Appl. Sci. 2022, 12, 6098 3 of 13

Numerical experiments with the synthetic and field data examples proved that the newmethod has a better performance in velocity analysis than conventional semblances.

2. Method2.1. Review of Conventional AB Semblance

The NMO-corrected seismic event can be denoted as a vector a ∈ RN . Assumingthat there is no variation of amplitudes along with seismic events, the reference can beregarded as a constant vector c ∈ RN whose elements are all equal to 1. Then the coherencecoefficient of vectors a and c can be defined as

γ2(a, c) =a·c|a||c| =

(N∑

i=1ai)

2

NN∑

i=1a2

i

(1)

where a is the amplitude of the NMO-corrected seismic event. Considering that a se-ries of scanning velocities is used for NMO correction, the semblance spectrum can beexpressed as:

S(it, υk) =

it+w/2∑

i=it−w/2(

N∑

j=1ak(i, j))

2

Nit+w/2

∑i=it−w/2

(N∑

j=1a2

k(i, j))(2)

where ak(i, j) is the amplitude of NMO-corrected CMP gather at the time index i and tracenumbers j and N are the number of traces. S(it, υk) represents the coherence coefficient atthe time index it with a scanning velocity υk and w is the length of the window along withthe time axis.

The reference vector c is not a constant sequence when the AVO anomalies exist. It wassupposed that the reference vector has a trend b(i, j) = A(i) + B(i)θ(j) [27], where A(i) andB(i) represent the AVO intercept and gradient, respectively. θ(j) is a known function can bereplaced by the offset x(j) and b(i, j) is the seismic amplitude of the Shuey approximationat offset x(j). The coefficients A(i) and B(i) can be estimated from the least-squares fitting:

b(i, j) = arg minA(i),B(i)

(N

∑j=1

a(i, j)− A(i)− B(i)x(j)). (3)

Here Appendix A gives the exact expressions for solving A(i) and B(i). Then, sub-stituting Equation (3) into Equation (2), we obtain the well-known AB semblance with ananalytic form.

SAB(it, υk) =

it+w/2∑

i=it−w/2(

N∑

j=1ak(i, j)bk(i, j))

2

Nit+w/2

∑i=it−w/2

N∑

j=1a2

k(i, j)N∑

j=1b2

k(i, j)(4)

The AB semblance uses a trend-based reference b(i, j) to replace the constant referencec. This trend function represents the fully corrected seismic events with the classic AVOanomalies. Compared with the conventional semblance, the AB-semblance can recover theenergy clusters in the AVO region. However, the trend function significantly reduces theresolution of the AB semblance spectrum, and even the resolution of the AB semblancespectrum is lower than that of the conventional spectrum.

2.2. High-Resolution AB Semblance with a PCA-Based Weighting Function

Principal component analysis is a common multivariate analysis method, and it hasbeen widely used in data mining, pattern recognition, machine learning, and image com-

Appl. Sci. 2022, 12, 6098 4 of 13

pression [29–31]. Principal component analysis regroups the original relevant indicatorsinto a new set of unrelated comprehensive indicators. For the NMO-corrected seismicevent, if the true scanning velocity is adopted, the NMO-corrected seismic event will becompletely flattened, and the data matrix will have a strong spatial coherence. Thus, thefirst principal component has the absolute advantage compared with other components. Ifthe wrong scanning velocity is adopted, the NMO-corrected seismic event is not flattened,then the first principal component of the data matrix is not overwhelming.

We denote the amplitude of NMO-corrected gather as a matrix A ∈ RN×M, then thecovariance matrix of A is

CA=(A−∼A)(A−

∼A)T = ∑

iλiui, (5)

where∼A is the mean value of matrix A and λi represents the eigenvalue of the (i)-th principal

component and λ1 ≥ λ2 ≥ · · · ≥ λk. ui represents the corresponding eigenvectors of thecovariance matrix. If the matrix CA is a low-rank matrix or its column vectors have a stronglinear coherence with each other, then the first principal component λ1 is much larger thanthe others, and the remains quickly decrease to zero. For example, Figure 1a shows a seismicevent at time 0.24 s with the velocity υ = 1950 m/s. We used a series of scanning velocitiesto perform NMO corrections, then analyzed the corresponding eigenvalues of the principalcomponents. As shown in Figure 1b, the eigenvalue of the first principal component hadthe absolute advantage when the true velocity υ = 1950 m/s was used, and the remainingeigenvalues decreased rapidly. If we used the velocity υ = 1940 m/s with a slight deviationto perform NMO correction, the corresponding first principal component no longer has theabsolute advantage compared with others, indicating that the first principal component ishighly sensitive to the variation of scanning velocity. Thus, we define two weighting factors

β1 =λ1

∑kj=1 λj

, (6)

andβ2 =

λ1

λ2, (7)

to improve the resolution of the semblance spectrum. We demonstrated two weightingfactors β1 and β2 with different scanning velocities in the above case (Figure 1b). As shownin Figure 2, the weighting factors β1 and β2 both had a narrow pulse at true velocityυ = 1950 m/s. As long as the scanning velocity deviated slightly from the true value, thetwo weighting factors β1 and β2 decreased rapidly, indicating that the weighting factorswere very sensitive to changes in velocity.

Then, we combined factors β1 and β2 to construct a new weighting function w(it, υi).The new weighting function is defined as

w(it, υi) =λ2

1

λ2k∑

j=2λj + ε

, (8)

where ε is a stability factor for avoiding division by zero. Since w(it, υi) is the product ofthe two weighting factors β1 and β2, its sensitivity to velocity is double compared to thesingle weighting factor. Then, for different scanning velocities υi, the weighting functionsw(it, υi) become a vector.

w(it) = (w(it, υ1), w(it, υ2), ..., w(it, υN)), (9)

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where the vector w(it) represents the weights at time it with a series of scanning velocities.We denote the maximum weight as wmax(it) = max(w(it)). Then, the new semblance canbe defined as

SPCA(it, υk) =w(it, υk)SAB(it, υk)

wmax(it)=

w(it, υk)it+w/2

∑i=it−w/2

(N∑

j=1ak(i, j)bk(i, j))

2

wmax(it)Nit+w/2

∑i=it−w/2

N∑

j=1a2

k(i, j)N∑

j=1b2

k(i, j)(10)

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value, the two weighting factors 1

and

2 decreased rapidly, indicating that the

weighting factors were very sensitive to changes in velocity.

(a) (b)

Figure 1. A seismic event at time 0.24 s with a velocity 1950 m/s = . (a) CMP gather and (b) The

corresponding principal components (eigenvalues) of the NMO-corrected CMP gathers with differ-

ent scanning velocities.

(a) (b)

Figure 2. The weighting factors 1 and 2 varying with the scanning velocity. (a) Represents

the weighting factor 1 varying with the scanning velocity and (b) represents the weighting factor

2 .

Then, we combined factors 1

and 2

to construct a new weighting function

( , )iw it . The new weighting function is defined as

Figure 1. A seismic event at time 0.24 s with a velocity υ = 1950 m/s. (a) CMP gather and (b) Thecorresponding principal components (eigenvalues) of the NMO-corrected CMP gathers with differentscanning velocities.

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value, the two weighting factors 1

and

2 decreased rapidly, indicating that the

weighting factors were very sensitive to changes in velocity.

(a) (b)

Figure 1. A seismic event at time 0.24 s with a velocity 1950 m/s = . (a) CMP gather and (b) The

corresponding principal components (eigenvalues) of the NMO-corrected CMP gathers with differ-

ent scanning velocities.

(a) (b)

Figure 2. The weighting factors 1 and 2 varying with the scanning velocity. (a) Represents

the weighting factor 1 varying with the scanning velocity and (b) represents the weighting factor

2 .

Then, we combined factors 1

and 2

to construct a new weighting function

( , )iw it . The new weighting function is defined as

Figure 2. The weighting factors β1 and β2 varying with the scanning velocity. (a) Represents theweighting factor β1 varying with the scanning velocity and (b) represents the weighting factor β2.

Appl. Sci. 2022, 12, 6098 6 of 13

Here, we divide weights w(it, υk) by their maximum value wmax(it) to normalizethe new semblance from 0 to 1, which is convenient for the automatic velocity pickingalgorithm. The new semblance is a combination of the standard AB semblance SAB(it, υk)and the weighting function. The AB semblance SAB(it, υk) can handle with AVO anomalies,and the weighting function improves the resolution of AB semblance spectrum significantly.For example, Figure 3 shows the semblances of different methods at time 0.24 s in the abovecase. It can be seen that the proposed PCA-weighted semblance had a narrow pulse atscanning velocity υ = 1950 m/s, indicating that it had the highest resolution comparedwith other methods.

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2

1

2

2

( , )=i k

j

j

w it

=

+,

(8)

where is a stability factor for avoiding division by zero. Since ( , )iw it is the product

of the two weighting factors 1

and 2 , its sensitivity to velocity is double compared

to the single weighting factor. Then, for different scanning velocities i , the weighting

functions ( , )iw it

become a vector.

1 2( ) ( ( , ), ( , ),..., ( , ))Nit w it w it w it =w, (9)

where the vector ( )itw represents the weights at time it with a series of scanning ve-

locities. We denote the maximum weight as max ( ) max( ( ))w it it= w . Then, the new sem-

blance can be defined as

/22

k

/2 1

/22 2max

max

/2 1 1

( , ) ( ( , ) ( , ))( , ) ( , )

( , )( )

( ) ( , ) ( , )

it w N

k k

i it w jk AB kPCA k it w N N

k k

i it w j j

w it a i j b i jw it S it

S itw it

w it N a i j b i j

+

= − =

+

= − = =

= =

(10)

Here, we divide weights ( , )kw it by their maximum value max ( )w it to normal-

ize the new semblance from 0 to 1, which is convenient for the automatic velocity picking

algorithm. The new semblance is a combination of the standard AB semblance

( , )AB kS it and the weighting function. The AB semblance ( , )AB kS it

can handle with

AVO anomalies, and the weighting function improves the resolution of AB semblance

spectrum significantly. For example, Figure 3 shows the semblances of different methods

at time 0.24 s in the above case. It can be seen that the proposed PCA-weighted semblance

had a narrow pulse at scanning velocity 1950 m/s = , indicating that it had the highest

resolution compared with other methods.

Figure 3. Comparison of the semblances of different methods at time 0.24 s. Where the blue curve

represents the AB-semblance, the red curve represents the conventional semblance, and the yellow

curve represents the proposed PCA-weighted semblance.

Figure 3. Comparison of the semblances of different methods at time 0.24 s. Where the blue curverepresents the AB-semblance, the red curve represents the conventional semblance, and the yellowcurve represents the proposed PCA-weighted semblance.

3. Experiments3.1. Synthetic Data

To analyze the performance of the new method in velocity analysis, we comparedand analyzed the resolution of proposed semblance, conventional semblance, and ABsemblance with a synthetic CMP gather. As shown in Figure 4a, we generated a CMPgather with the class-II AVO anomalies [28] by the inverse normal moveout, where the AVOanomalies have the polarity reversal seismic amplitudes. We performed NMO correctionswith a series of scanning velocities, and calculated the different semblance spectra on theNMO-corrected data. Then, we picked the velocity values corresponding to the peaks insemblance spectra. Figure 4b shows the conventional semblance of the synthetic CMPgather, in which the solid line represents the picked velocities. It can be seen that theenergy clusters of conventional semblance were lost in the AVO regions (time 1.5 s to 3.0 s),resulting in inaccurate velocity picking. Figure 4c shows that the energy clusters of theAB semblance were restored in the AVO regions. However, the size of the energy clusterbecame larger, indicating a drop in resolution. Figure 4d shows that the resolution of ournew method is significantly improved, which is of great benefit to avoiding the multiplicityof the auto-picking algorithm, and the energy clusters of new method are also strong in theAVO region.

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3. Experiments

3.1. Synthetic Data

To analyze the performance of the new method in velocity analysis, we compared

and analyzed the resolution of proposed semblance, conventional semblance, and AB

semblance with a synthetic CMP gather. As shown in Figure 4a, we generated a CMP

gather with the class-II AVO anomalies [28] by the inverse normal moveout, where the

AVO anomalies have the polarity reversal seismic amplitudes. We performed NMO cor-

rections with a series of scanning velocities, and calculated the different semblance spectra

on the NMO-corrected data. Then, we picked the velocity values corresponding to the

peaks in semblance spectra. Figure 4b shows the conventional semblance of the synthetic

CMP gather, in which the solid line represents the picked velocities. It can be seen that the

energy clusters of conventional semblance were lost in the AVO regions (time 1.5 s to 3.0

s), resulting in inaccurate velocity picking. Figure 4c shows that the energy clusters of the

AB semblance were restored in the AVO regions. However, the size of the energy cluster

became larger, indicating a drop in resolution. Figure 4d shows that the resolution of our

new method is significantly improved, which is of great benefit to avoiding the multiplic-

ity of the auto-picking algorithm, and the energy clusters of new method are also strong

in the AVO region.

(a) (b) (c) (d)

Figure 4. Semblance spectra of the synthetic CMP gather with the class-II AVO anomalies. From left

to right: (a) uncorrected CMP gather on which all methods of semblances were applied, (b) conven-

tional semblance, (c) AB semblance, and (d) PCA-weighted semblance.

We applied each set of picked velocities to correct the movement of the synthetic

CMP gather as a form of quality control. Figure 5 shows the velocity values picked up

from different semblances, where the black curve represents the true values of velocities,

the green curve represents the picked velocities from conventional semblance, the cyan

curve represents the AB semblance, and the red curve represents our new method. It is

clear that the picked velocities of conventional semblance had obvious errors in the AVO

region. Conventional semblance is an incompetent method for the AVO phenomenon.

The results of AB semblance show a significant improvement in dealing with the AVO

phenomenon. However, the accuracy of AB semblance is inferior to the conventional

Figure 4. Semblance spectra of the synthetic CMP gather with the class-II AVO anomalies. From left toright: (a) uncorrected CMP gather on which all methods of semblances were applied, (b) conventionalsemblance, (c) AB semblance, and (d) PCA-weighted semblance.

We applied each set of picked velocities to correct the movement of the synthetic CMPgather as a form of quality control. Figure 5 shows the velocity values picked up fromdifferent semblances, where the black curve represents the true values of velocities, thegreen curve represents the picked velocities from conventional semblance, the cyan curverepresents the AB semblance, and the red curve represents our new method. It is clearthat the picked velocities of conventional semblance had obvious errors in the AVO region.Conventional semblance is an incompetent method for the AVO phenomenon. The resultsof AB semblance show a significant improvement in dealing with the AVO phenomenon.However, the accuracy of AB semblance is inferior to the conventional method because thetrend function of AB semblance is insensitive to the velocity. The red curve reveals that theproposed method has the highest accuracy compared with other methods, and its accuracyin the AVO region is still good. Figure 6 shows the NMO-corrected gathers by the pickedvelocities from different semblances. It can be seen that the NMO-corrected gather of theconventional method (Figure 6b) suffered from both over-correction and under-correctionproblems in the far-offset traces, especially for the AVO area. Clearly, the seismic eventswith the AVO anomalies were better flattened in the AB semblance (Figure 6c) comparedwith the conventional semblance. In the shallow region of NMO-corrected gather, Figure 6dshows that the seismic events were flattened completely, indicating that the new methodcan provide more accurate picked velocities to perform NMO correction.

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method because the trend function of AB semblance is insensitive to the velocity. The red

curve reveals that the proposed method has the highest accuracy compared with other

methods, and its accuracy in the AVO region is still good. Figure 6 shows the NMO-cor-

rected gathers by the picked velocities from different semblances. It can be seen that the

NMO-corrected gather of the conventional method (Figure 6b) suffered from both over-

correction and under-correction problems in the far-offset traces, especially for the AVO

area. Clearly, the seismic events with the AVO anomalies were better flattened in the AB

semblance (Figure 6c) compared with the conventional semblance. In the shallow region

of NMO-corrected gather, Figure 6d shows that the seismic events were flattened com-

pletely, indicating that the new method can provide more accurate picked velocities to

perform NMO correction.

Figure 5. Picked velocity values from different semblances, where the black curve represents the

true values of velocities, the green curve represents the picked velocities from the conventional sem-

blance, the cyan curve represents the AB semblance, and the red curve represents our new method.

(a) (b) (c) (d)

Figure 6. NMO-corrected gathers using different picked velocities from the above semblances. From

left to right: (a) uncorrected CMP gather, (b) NMO-corrected CMP using the picked velocities from

the conventional semblance spectrum, (c) NMO-corrected CMP gather from the AB semblance spec-

trum, and (d) NMO-corrected CMP gather from the PCA-weighted semblance spectrum.

Figure 5. Picked velocity values from different semblances, where the black curve represents the truevalues of velocities, the green curve represents the picked velocities from the conventional semblance,the cyan curve represents the AB semblance, and the red curve represents our new method.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 8 of 13

method because the trend function of AB semblance is insensitive to the velocity. The red

curve reveals that the proposed method has the highest accuracy compared with other

methods, and its accuracy in the AVO region is still good. Figure 6 shows the NMO-cor-

rected gathers by the picked velocities from different semblances. It can be seen that the

NMO-corrected gather of the conventional method (Figure 6b) suffered from both over-

correction and under-correction problems in the far-offset traces, especially for the AVO

area. Clearly, the seismic events with the AVO anomalies were better flattened in the AB

semblance (Figure 6c) compared with the conventional semblance. In the shallow region

of NMO-corrected gather, Figure 6d shows that the seismic events were flattened com-

pletely, indicating that the new method can provide more accurate picked velocities to

perform NMO correction.

Figure 5. Picked velocity values from different semblances, where the black curve represents the

true values of velocities, the green curve represents the picked velocities from the conventional sem-

blance, the cyan curve represents the AB semblance, and the red curve represents our new method.

(a) (b) (c) (d)

Figure 6. NMO-corrected gathers using different picked velocities from the above semblances. From

left to right: (a) uncorrected CMP gather, (b) NMO-corrected CMP using the picked velocities from

the conventional semblance spectrum, (c) NMO-corrected CMP gather from the AB semblance spec-

trum, and (d) NMO-corrected CMP gather from the PCA-weighted semblance spectrum.

Figure 6. NMO-corrected gathers using different picked velocities from the above semblances. Fromleft to right: (a) uncorrected CMP gather, (b) NMO-corrected CMP using the picked velocities from theconventional semblance spectrum, (c) NMO-corrected CMP gather from the AB semblance spectrum,and (d) NMO-corrected CMP gather from the PCA-weighted semblance spectrum.

3.2. Field Data

We used a field seismic data set to verify the effectiveness of the proposed method.Firstly, we used a 2-D real-world seismic gather (Figure 7a) to apply the velocity analysis,and the corresponding semblance spectra are shown in Figure 7b–d. It can be seen that theproposed method has a high sensitivity in velocity scanning (Figure 7d) and the energyclusters in our new semblance were easy to be picked out, thus reducing the uncertainty ofvelocity analysis. The experiment with the 2-D real-world seismic gather proved that theproposed method is effective in the field data processing.

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3.2. Field Data

We used a field seismic data set to verify the effectiveness of the proposed method.

Firstly, we used a 2-D real-world seismic gather (Figure 7a) to apply the velocity analysis,

and the corresponding semblance spectra are shown in Figure 7b–d. It can be seen that

the proposed method has a high sensitivity in velocity scanning (Figure 7d) and the en-

ergy clusters in our new semblance were easy to be picked out, thus reducing the uncer-

tainty of velocity analysis. The experiment with the 2-D real-world seismic gather proved

that the proposed method is effective in the field data processing.

(a) (b) (c) (d)

Figure 7. 2-D field data and the corresponding semblance spectra, where the solid curves in the

semblance represent the picked velocities. From left to right: (a) uncorrected CMP gather on which

all methods of semblance were applied, (b) conventional semblance, (c) AB semblance, and (d) PCA-

weighted semblance.

To further verify the performance of the new method, we use a historic 2D line from

the Gulf of Mexico [9] to analyze the resolution of different semblances. As shown in Fig-

ure 8, the real-world data had 250 CMP points. We obtained different semblances under

the same scanning parameters. Figure 9. shows the obtained semblances displayed with a

3-D form, it can be seen that the conventional semblance lost some energies in the shallow

layer (Figure 9a), but the corresponding energies were recovered in the AB semblance

(Figure 9b). Unfortunately, the peak broadening of the AB semblance spectrum led to

picking uncertainty. Figure 9c shows the result of our new method, it can be seen that the

resolution of the new method was significantly improved, and the intensity of energy

cluster was stronger than that of the conventional method.

Figure 7. 2-D field data and the corresponding semblance spectra, where the solid curves in thesemblance represent the picked velocities. From left to right: (a) uncorrected CMP gather on whichall methods of semblance were applied, (b) conventional semblance, (c) AB semblance, and (d) PCA-weighted semblance.

To further verify the performance of the new method, we use a historic 2D line fromthe Gulf of Mexico [9] to analyze the resolution of different semblances. As shown inFigure 8, the real-world data had 250 CMP points. We obtained different semblances underthe same scanning parameters. Figure 9. shows the obtained semblances displayed with a3-D form, it can be seen that the conventional semblance lost some energies in the shallowlayer (Figure 9a), but the corresponding energies were recovered in the AB semblance(Figure 9b). Unfortunately, the peak broadening of the AB semblance spectrum led topicking uncertainty. Figure 9c shows the result of our new method, it can be seen thatthe resolution of the new method was significantly improved, and the intensity of energycluster was stronger than that of the conventional method.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 10 of 13

Figure 8. A field data with 250 CMP gathers from the Gulf of Mexico.

(a)

(b)

Figure 8. A field data with 250 CMP gathers from the Gulf of Mexico.

Appl. Sci. 2022, 12, 6098 10 of 13

Appl. Sci. 2022, 12, x FOR PEER REVIEW 10 of 13

Figure 8. A field data with 250 CMP gathers from the Gulf of Mexico.

(a)

(b)

Appl. Sci. 2022, 12, x FOR PEER REVIEW 11 of 13

(c)

Figure 9. Semblances of the field date from the Gulf of Mexico: (a) conventional semblance, (b) AB

semblance, and (c) PCA-weighted semblance.

4. Conclusions

In this study, we present a new weighted semblance spectrum to improve the reso-

lution of the NMO-based velocity analysis. The proposed method utilizes principal com-

ponent analysis to measure the spatial coherence or low-rank property of the NMO-cor-

rected data, and construct a new weighting function through the ratio between principal

components (eigenvalues). The new weighting function was highly sensitive to slight

changes of seismic events when we used different scanning velocities to apply NMO cor-

rection, so our method has higher resolution than the conventional method. In addition,

the proposed method still performed well for the AVO anomalies, and the energy clusters

of new semblance spectrum were also strong in the AVO region, which is of great benefit

in avoiding the multiplicity of the auto-picking algorithm. Numerical experiments on the

synthetic and field data sets also proved that the new method has a better performance in

velocity analysis and provides more accurate picked velocities than those of conventional

methods.

Author Contributions: Conceptualization, C.Z. and G.C.; methodology, C.Z. and G.C.; software,

G.C.; validation, L.F.; investigation, L.F. and J.L.; writing—original draft preparation, C.Z. and G.C.;

writing—review and editing, J.L.; project administration, C.Z.; funding acquisition, C.Z. All authors

have read and agreed to the published version of the manuscript.

Funding: This research was funded by the Prospective and Basic Research Project of CNPC

(2021DJ0503), the Strategic Priority Research Program of the Chinese Academy of Sciences

(XDA14010403), the National Natural Science Foundation of China (No. 42172145), and the China

National Science and Technology Major Project (2016ZX05007-002).

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.

Data Availability Statement: The data presented in this study are available on request from the

corresponding author.

Conflicts of Interest: The authors declare no conflict of interest.

Appendix A

The well-known Shuey approximation linearly approximates the offset dependence

of seismic reflectivity. The Shuey approximation is expressed as

( , ) ( ) ( ) ( )b i j A i B i x j= + , (A1)

Figure 9. Semblances of the field date from the Gulf of Mexico: (a) conventional semblance, (b) ABsemblance, and (c) PCA-weighted semblance.

Appl. Sci. 2022, 12, 6098 11 of 13

4. Conclusions

In this study, we present a new weighted semblance spectrum to improve the res-olution of the NMO-based velocity analysis. The proposed method utilizes principalcomponent analysis to measure the spatial coherence or low-rank property of the NMO-corrected data, and construct a new weighting function through the ratio between principalcomponents (eigenvalues). The new weighting function was highly sensitive to slightchanges of seismic events when we used different scanning velocities to apply NMO cor-rection, so our method has higher resolution than the conventional method. In addition,the proposed method still performed well for the AVO anomalies, and the energy clustersof new semblance spectrum were also strong in the AVO region, which is of great benefitin avoiding the multiplicity of the auto-picking algorithm. Numerical experiments onthe synthetic and field data sets also proved that the new method has a better perfor-mance in velocity analysis and provides more accurate picked velocities than those ofconventional methods.

Author Contributions: Conceptualization, C.Z. and G.C.; methodology, C.Z. and G.C.; software,G.C.; validation, L.F.; investigation, L.F. and J.L.; writing—original draft preparation, C.Z. and G.C.;writing—review and editing, J.L.; project administration, C.Z.; funding acquisition, C.Z. All authorshave read and agreed to the published version of the manuscript.

Funding: This research was funded by the Prospective and Basic Research Project of CNPC (2021DJ0503),the Strategic Priority Research Program of the Chinese Academy of Sciences (XDA14010403), theNational Natural Science Foundation of China (No. 42172145), and the China National Science andTechnology Major Project (2016ZX05007-002).

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.

Data Availability Statement: The data presented in this study are available on request from thecorresponding author.

Conflicts of Interest: The authors declare no conflict of interest.

Appendix A

The well-known Shuey approximation linearly approximates the offset dependence ofseismic reflectivity. The Shuey approximation is expressed as

b(i, j) = A(i) + B(i)x(j), (A1)

The residuals squared between corrected amplitudes a(i, j) and Shuey approximationmodel b(i, j) is

F = ∑Nj=1 (a(i, j)− b(i, j))2 = ∑N

j=1 (a(i, j)− A(i)− B(i)x(j))2, (A2)

To find optimal approximation A(i) and B(i), the derivatives of F with respect to A(i)and B(i) are equal to zero:

∂F∂A(i)

= −2∑Nj=1 (a(i, j)− A(i)− B(i)x(j)) = 0, (A3)

∂F∂B(i)

= −2∑Nj=1 (a(i, j)− A(i)− B(i)x(j))x(j) = 0, (A4)

Expanding and rearranging Equations (A3) and (A4), we get a system of two equations

NA(i) + B(i)∑Nj=1 x(j) = ∑N

j=1 a(i, j)A(i)∑N

j=1 x(j) + B(i)∑Nj=1 x2(j) = ∑N

j=1 a(i, j)x(j), (A5)

Appl. Sci. 2022, 12, 6098 12 of 13

If we express Equation (A5) in matrix form and rearrange it, we obtain

(A(i)B(i)

)=

(N ∑N

j=1 x(j)∑N

j=1 x(j) ∑Nj=1 x2(j)

)−1(∑N

j=1 a(i, j)∑N

j=1 a(i, j)x(j)

), (A6)

Solving for A(i) and B(i), we get

A(i) =∑N

j=1 x(j)∑Nj=1 a(i, j)x(j)−∑N

j=1 a(i, j)∑Nj=1 x2(j)(

∑Nj=1 x(j)

)2− N∑N

j=1 x2(j), (A7)

and

B(i) =∑N

j=1 x(j)∑Nj=1 a(i, j)− N∑N

j=1 a(i, j)x(j)(∑N

j=1 x(j))2− N∑N

j=1 x2(j). (A8)

Substituting Equations (A7) and (A8) into Equation (3) provides an explicit measureof the Shuey approximation. Then the AB semblance can be expressed as:

SAB(it, υk) =∑it+w/2

i=it−w/2

(∑N

j=1 ak(i, j)bk(i, j))2

∑it+w/2i=it−w/2 ∑N

j=1 a2k(i, j)∑N

j=1 b2k(i, j)

. (A9)

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