*2.3. The Denoising Method in NSST Domain*

The NSST abandons the subsampling operation and combines the non-subsampled Laplacian pyramid transform (NSLP) with the non-subsampled directional filter. After the NSST, the size of each direction sub-band at each scale is the same as the original image, which greatly improves the image redundancy. The enhancement of the image coefficient redundancy makes the NSST have the translation invariance and avoid the spectrum aliasing and the pseudo-Gibbs phenomenon. Therefore, the NSST has a stronger data processing performance. The NSST has a more delicate representation of the scale direction and the physical significance of the transformed coefficients is clearer, which is more conducive to the processing of the coefficients. The NSST can transform the timespace domain signal *f(x)* into the NSST domain *(j*, *l*, *k)*, so the different signals can be easily separated in the NSST domain. Each NSST coefficient has its own specific frequency, direction, and position. The scale parameter *j* reflects the frequency difference; the direction parameter *l* reflects the direction difference, that is, mainly reflects the apparent velocity difference; and the position parameter *k* reflects the spatial location difference and the amplitude difference.

As shown in Figure 3, The specific steps of the coherent noise removal method of the vibrator source array seismic data in the NSST domain in this article are as follows:

**Figure 3.** The flowchart of seismic coherent noise removal method in the NSST domain.

In this article, we choose the *L*<sup>2</sup> norm to construct the threshold function. The *L*<sup>2</sup> norm of the NSST domain coefficient is:

$$
\epsilon\_{j,l} = \sqrt{\sum \mathcal{C}\_{j,l}^2} \tag{4}
$$

In Equation (4), *ej*,*<sup>l</sup>* is the *L*<sup>2</sup> norm of the NSST coefficient in the *j* scale *l* direction.

The threshold function is the adaptive threshold changing with scale and direction, which is expressed as follows:

$$T\_{j,l} = \lambda\_j \frac{\sigma \sqrt{2 \ln N} \lg(j+1)}{\lg \left( \varepsilon\_{j,l} - \min \varepsilon\_{j,l} + 10 \right)} \tag{5}$$

In Equation (5), *Tj*,*<sup>l</sup>* is the threshold function; *N* is the total number of sampling points for the seismic data; *ej*,*<sup>l</sup>* is the *L*<sup>2</sup> norm of the NSST coefficient in the *j* scale *l* direction, *j* = 1, 2, ... , *J*, *J* is the total scale of the decomposition; *σ* = median(|C|) 0.6745 , *σ* is the noise standard deviation, median( ) is a median value for all the elements in the data matrix; min( ) is minimum value function; and *λ<sup>j</sup>* is the test constant.
