Summary

The quality of seismic data plays a crucial role in geological interpretation. Actual seismic data usually carry a lot of noise, which makes the formation blurred and the fault structure unclear. Non-local mean filter method (NLM) can effectively suppress random noise, but its computational efficiency is low, so it has limitations in the application of large-scale seismic data processing. This paper presents a fast adaptive NLM algorithm, which uses the central symmetric data integration algorithm to improve the computational efficiency of the NLM method, and uses the similarity standard deviation to estimate the uniformity to adaptively adjust the filter parameters to further improve the denoising effect. Therefore, the improved non-local mean filtering method can effectively improve the computational efficiency and enhance the effect of noise suppression. Finally, the feasibility and effectiveness of the method are verified by model data and actual data.

**Abstract**

The quality of seismic data plays a critical role in geological interpretation.However,the real seismic data usually contain a lot of noise,leading to fuzzy strata and unclear fault structures.The non-local means (NLM) filtering algorithm can effectively suppress random noise,but its computational efficiency is low.Therefore,it has limitations when being applied to large-scale seismic data processing.This study proposed a fast adaptive NLM algorithm,for which the computational efficiency was improved using the centrosymmetric data integration algorithm and the filtering parameters were adaptively adjusted using the standard deviation of similarity to estimate the homogeneity,thus further improving the noise attenuation effect.Therefore,the modified NLM filtering algorithm can effectively improve computational efficiency and enhance the noise attenuation effect.Furthermore,the feasibility and effectiveness of the algorithm were verified using model data and actual data.

**Preface**

Noise suppression plays a vital role in seismic data processing, and the effect of denoising will directly affect seismic data interpretation. In order to effectively suppress the random noise of seismic data and improve the signal-to-noise ratio of seismic data, many scholars at home and abroad have proposed different methods, which are roughly divided into five categories in this paper. The first is the method based on mathematical transformation, such as Fourier transform [1], Radon transform[2-3 ] , Curvelet transform [ 4⇓ - 6 ] , Dreamlet transform [ 7 ] and Wavelet transform [ 8-9 ] ; the second is Methods based on predictive filtering, such as FX predictive filtering [ 10-11 ] and adaptive predictive methods [ 12 ] ; the third is based on non-local mean (NLM) filtering methods, such as traditional NLM methods [ 13-14 ] and self- adaptive Adapted to NLM filtering [15 - 16 ] ; the fourth is based on rank reduction method, such as matrix rank reduction method [ 17 ⇓ - 19 ] and tensor rank reduction method [ 20 - 21 ] ; the fifth is based on artificial intelligence method, such as data Powering compact frameworks [ 22 ] and machine learning [ 23 ⇓ ⇓ ⇓ - 27 ] .

Different from the predictive filtering method and the reduced-rank filtering method, the NLM method is not based on the linear assumption, so when dealing with curved events, this method can effectively protect the effective signal and suppress random noise. The traditional NLM method originated from image random noise suppression processing [ 28 ] , and then Bonar successfully introduced this method to seismic data noise suppression processing [ 13 ] . However, traditional NLM methods also have certain limitations in application. Compared with methods such as matrix rank reduction or predictive filtering, this method takes a long time to calculate and is inefficient when dealing with large seismic data. In order to solve this problem, the predecessors proposed block NLM method [ 28 ] , parallel block NLM method [ 29 ] , NLM method based on random projection algorithm [ 30 ] , down-sampling fast NLM method [ 31 ] , variable window The fast NLM method [ 32 ] et al. However, in order to improve the calculation efficiency, the above method does not completely traverse each data point in the calculation process, so the calculation accuracy may be sacrificed. And the fast NLM method based on data integration algorithm [ 33 ], which is equivalent to traversing all data points in the calculation process in principle, thus avoiding the possibility of sacrificing accuracy while increasing the calculation speed. When the traditional NLM method deals with actual seismic data, the selected filter parameter value is usually a constant. In order to further improve the denoising effect, predecessors used the structure tensor algorithm[34 - 35 ] , the matrix eigenproperty algorithm [ 36 ] , Gray correlation analysis algorithm [ 37 ] and other methods can choose different filter parameters for different regions to improve the denoising effect, but it will significantly increase the amount of calculation. The denoising effect of NLM method is greatly affected by the selection of filter parameters. If the parameters are too large, the details will be lost and the image will be blurred. If the parameters are too small, the noise cannot be completely suppressed. Therefore, some scholars adaptively calculate the filtering parameters through random noise estimation of local data, such as high-pass filtering method [ 38 ] and minimum variance estimation method [ 15-16 ] . The above methods can reduce the influence of filtering parameters on the calculation speed while improving the denoising effect of the traditional NLM method, but there is still room for improvement in the calculation efficiency of this method.

This paper proposes a seismic random noise suppression method based on fast adaptive non-local mean filter, which can quickly and effectively suppress seismic random noise. First, in order to improve the computational efficiency of the NLM method, a central symmetric data integration algorithm is proposed in this paper, which effectively reduces the computational cost. Secondly, by calculating the similarity of two neighborhood windows, this paper presents a method to adaptively adjust the distribution of filtering parameters by using the uniformity. Finally, the validity and practicability of the method are verified by model data and actual data.

**1.1 Non-local mean filtering method**

The noise data D noise ( t , x ) can be expressed by the following formula:

Dnoise(t,x)=Dtrue(t,x)+n,

(1)

In the formula: D true ( t , x ) means noise-free data; n means random noise. Suppose ( D s , D s ) represents the search window radius, ( d s , d s ) represents the radius of the neighborhood window, and D denoise ( t , x ) represents the denoised data, by doing D true ( t , x ) The weighted average calculation can get the D denoise ( t , x ) data at each point [ 13 , 28 ] . Therefore, D denoise ( t ,x ) can be expressed as:

Ddenoise(t,x)= 1Z(t,x)

· ∑i,j∈Ω[exp(−∥V(t,x)−V(i,j)∥22h2)Dnoise(i,j)]

,

(2)

In the formula, the filter parameter h is a constant, which is the main parameter to control the denoising level.∥V(t,x)−V(i,j)∥22

Used to measure the similarity between two neighborhood windows, namely

∥V(t,x)−V(i,j)∥22

= 1(2ds+1)2

·∑z1,z2∈Z

‖Dnoise(t+z1,x+z2)- Dnoise(i+z1,j+z2)∥∥22

,

(3)

In the formula, ℤ={z1,z2∈Z:|z1|≤ds,|z2|≤ds}

. In formula (2),1Z(t,x)

is a normalization factor, used to ensure that the weighting coefficient exp(-∥V(t,x)−V(i,j)∥22h2

) is equal to 1. Therefore, Z ( t , x ) can be expressed as:

Z(t,x)= ∑i,j∈Ω

exp(- ∥V(t,x)−V(i,j)∥22h2

),Ω={i,j∈Ω:|i−t|≤Ds,|j−x|≤Ds}

。

(4)

The NLM method can effectively deal with curved events, but the computational cost is high. Assuming that the total number of points in D noise ( t , x ) is N = N t N x , the amount of similarity calculation between two square neighborhood windows is d 2 =(2ds+1)2

. For each data point in D noise ( t , x ), it is necessary to calculate D 2 =(2Ds+1)2

The computational complexity of NLM filtering is O ( ND 2 d 2 ), and the amount of calculation is huge. Therefore, the traditional NLM method will be constrained by computational efficiency obviously when dealing with large real seismic data.

1.2 Central symmetric data integration algorithm

In order to improve the computational efficiency of the NLM method, the predecessors proposed a data integration algorithm to speed up the NLM method [ 33 , 39 ] . The conventional algorithm is calculating ‖ V ( t , x )- V ( i , j )∥22

When , it is necessary to traverse two adjacent windows first, calculate the square of the difference between each pair of points, and finally add the differences of all points in the neighborhood. The data integration algorithm needs to calculate the square of the difference of all data points first, and then calculate the integral of the difference. Therefore, the algorithm needs to construct an integral matrix S t ( t 1 , x 1 ), which can be expressed by integrating the difference matrix s ( t , x ):

s(t,x)= ∥∥Dnoise(t,x)−Dnoise(t+r1,x+r2)∥∥22

, ℝ= {r1,r2∈R;|r1|≤Ds,|r2|≤Ds}

,S t (t 1 ,x 1 )=∑t,x∈N

s(t,x),

(5)

ℕ= {t,x∈N;1≤t≤t1,1≤x≤x1}

,

(6)

∥V(t,x)−V(i,j)∥22

= 1(2ds+1)2

· [St(t+ds,x+ds)+St(t-ds-1,x-ds-1)-St(t+ds,x-ds-1)-St(t-ds-1,x+ds)],

(7)

From formula (5) ~ formula (7),∥V(t,x)−V(i,j)∥22

The amount of calculation is greatly reduced, and the computational complexity of the NLM method based on the data integration algorithm is reduced to O ( ND 2 ).

By observing formula (5), it is found that the established difference matrix s( t , x ) is centrally symmetric, so this paper presents a centrally symmetric data integration algorithm, which uses the mathematical symmetry of data integration to improve the computational efficiency of the NLM method . suppose

⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪s(t,x)[+r]=∥∥Dnoise(t,x)−Dnoise(t+r1,x+r2)∥∥22,s(t,x)[−r]=∥∥Dnoise(t,x)−Dnoise(t−r1,x−r2)∥∥22,R={r1,r2∈R:|r1|≤Ds,|r2|≤Ds}。

(8)

Further, the above formula can be deduced as:

s (t−r1,x−r2)[+r]

= ∥∥Dnoise(t−r1,x−r2)−Dnoise(t,x)∥∥22

=s(t,x)[-r],

(9)

According to this, the value of s ( t , x ) [- r ] can be expressed as the central symmetric value of s ( t , x ) [+ r ] , and in the calculation process, it is possible to avoid constructing s ( t , x ) [- r ] matrix, thus reducing the amount of calculation by half, and substituting equation (9) into equation (6),

St(t1−r1,x1−r2)[+r]

=St(t1,x1)[−r]

,

(10)

Substituting formula (10) into formula (7), we get:

∥V(t,x)−V(i,j)∥22[−r]

= 1(2ds+1)2[St(t+ds−r1,x+ds−r2)[+r]+St(t−ds−r1−1,x−ds−r2−1)[+r]

- St(t+ds−r1,x−ds−r2−1)[+r]−St(t−ds−r1−1,x+ds−r2)[+r]]

。 (11)

(11)

By using formula (8) ~ formula (11), it is possible to avoid establishing difference matrix s ( t , x ) [- r ] and integral matrix S t(t1,x1)[−r]

. Therefore the distance term∥V(t,x)−V(i,j)∥22

The amount of calculation can be further reduced, and the computational complexity of NLM is reduced to O ( ND 2 / 2).

**1.3 Homogeneity Estimation**

The selection of the filter parameter h in formula (2) is crucial to the denoising effect of the NLM method. A large number of seismic data show that the effective signal energy of different formations is quite different, and even the distribution of random noise is not completely random, so the value of the filter parameter h is usually difficult to determine. Compared with using the same filter parameter h for seismic data in the entire region, using a parameter matrix h to control the denoising level of data in different regions can improve the noise suppression effect. The adaptive algorithm based on the minimum variance estimation can adaptively select the filter parameter h [15-16 ] . This method considers that the estimation of the parameter h is the standard deviation of the noise σ , which satisfies the following formula

E ∥V(t,x)−V(i,j)∥22

= ∥V0(t,x)−V0(i,j)∥22

+ 2s 2 ,

(12)

Among them, V0 represents noise-free data . The estimation of the adaptive filter parameter matrix h 2 can be expressed as

h2(t,x)≈σ2=min(E ∥V(t,x)−V(i,j)∥22

)/2 。

(13)

By using the method of minimum variance estimation, the filter parameter values within the entire data region can be adjusted. Figure 1 shows the distribution of a filter parameter h 2 estimated from noisy data, as shown in Figure 1c, the filter parameter h 2 obtained by using the minimum variance estimation method is relatively large at the edge of the effective signal structure, while the background is uniform The filter parameter values of the region are relatively small.

In order to further improve the suppressing effect of random noise, the filter parameter h should be relatively large in the uniform area , and should be relatively small in the structure edge area [ 35 ] . Therefore, this paper presents a method to adaptively estimate the uniformity and adjust the distribution of filtering parameters by using the similarity standard deviation algorithm. Tuning filter parameters by utilizing the natural exponential functionh2adp

distribution of , and adaptively estimateh2adp

the value of

h2adp

(t,x)= h2

exp (1−2STDsmax(STDs))

,

(14)

In the formula, ST Ds ( t , x ) represents the mean square error of the similarity,

STDs(t,x)= stdi,j∈Ω(∥V(t,x)−V(i,j)∥22)

, Ω={i,j∈Ω;|i−t|≤Ds,|j−x|≤Ds}

。

(15)

When the similarity standard deviation value ST Ds ( t , x ) is large, it means that the data in the search area changes greatly, that is, it is located at the edge of the data structure; when the similarity standard deviation value is small, it means that the data in the search area does not change significantly. That is, the data is located in a uniform region. In addition, the calculation of the standard deviation of similarity in this paper is by using the calculated distance between adjacent windows in the NLM method∥V(t,x)−V(i,j)∥22

to fulfill. Therefore, compared with other structure-oriented algorithms, the calculation cost of adaptive filtering parameters is almost negligible.

Based on the fast adaptive non-local mean filtering method proposed in this paper, the calculation efficiency of the traditional NLM method is further improved by introducing the central symmetric data integration algorithm. At the same time, the similarity standard deviation is used to estimate the uniformity, and the adaptive filtering parameter adjustment is realized. Effectively improve the denoising effect and calculation efficiency.

Model data test

In seismic data noise suppression processing, the signal-to-noise ratio ( SNR ) [ 40 ] , peak signal-to-noise ratio ( PSNR ) and mean square error ( MSE ) [ 41 ] are usually used to quantitatively analyze the seismic data denoising effect of the method, which is defined as as follows:

SNR=20lg(∥Dtrue∥2∥∥Ddenoise−Dtrue∥∥2)

,

(16)

PSNR=20lg (255MSE√)

,

(17)

MSE= ∥∥Ddenoise−Dtrue∥∥2NxNt

,

(18)

In the formula: D true and D denoise represent the original noise-free data and noise-suppressed data respectively; N = N t N x represents the total number of points of the noise data D noise ( t , x ). The traditional NLM method [ 13 ] and the NLM method based on minimum variance estimation [ 15 ] are compared with the fast adaptive NLM method in this paper, and the effectiveness of the method in this paper is verified from two aspects of computational efficiency and accuracy by using two model seismic data.

Model data experiment 1 is a simple synthetic seismic section, including two linear events and two curved events. Figure 2 a is the 60-channel noise-free model data, Figure 2 b is the noise-added data, and the signal-to-noise ratio is -4.55 dB. Noise results and corresponding residual profiles. Since the FX predictive filtering method [ 10 ] and rank reduction method [ 18 ] are based on linear assumptions, there are errors in the processing of curved events, while the calculation accuracy of the NLM method for curved events is significantly higher than that of the above two methods ( Fig. 2c ~h). Since it is difficult to see the difference between the three NLM methods from the figure, the SNR , PSNR and MSE of the denoising results of the above methods are calculated respectively , as shown in Table 1 . It can be seen that the SNR and PSNR obtained by the method in this paper are higher than other methods, and the MSE is significantly lower than other methods. The suppression effect of this method on random noise is better than other methods mentioned above.

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