Convolutional Sparse Coding for Noise Attenuation of Seismic Data

Zhaolun Liu
KAUST, zhaolun.liu@kaust.edu.sa

Figure 1: Training phase	Figure 2: Denoising phase

1 Objective

To learn how to use convolutional sparse coding to attenuate the noise in seismic data.

2 Reading Materials

Please read the SEG Beijing abstract and PPT for theory of seismic denoising.

3 Prerequisites

MATLAB

4 Theory

The convolutional sparse coding (CSC) problem can be defined as finding the optimal ${ \bf {d}}$ and ${ \bf {z}}$ that minimize the following objective function:

$$\argmin_{{ \bf {d}},{ \bf {z}}} \frac{1}{2}\lVert { \bf {x}}-\sum... ...f {d}}_k*{ \bf {z}}_k\rVert_2^2+ \beta\sum_{k=1}^K\lVert{ \bf {z}}_k\rVert_1,$$ (1)

where ${ \bf {x}}$ is an $m\times n$ image in vector form, ${ \bf {d}}_k$ refers to the $k$-th $d\times d$ filter in vector form, ${ \bf {z}}_k$ is vector of sparse coefficients with size $(m+d-1)\times(n+d-1)$, $\beta$ controls the $l_1$ penalty, and $*$ denotes the 2D convolution operator.

The noise attenuation method by CSC can be divided into the training phase and the denoising phase. The seismic data with a relatively high signal-to-noise ratio are chosen for training to get the learned basis functions. Then we use all (or a subset) of the basis functions to attenuate the random or coherent noise in the seismic data.

5 Procedure

• Download the codes and unzip it.
• Change your Matlab working directory.
• Type "training_phase" to do the training phase. The learned filters will be saved as "filters.mat". The result is shown in Figure 1.
• Type "denoise" to do the denoise phase. It will load "filters.mat" and do denoising. The result is shown in Figure 2.

6 Exercise

• Play with the parameters: a) the size and the number of the filter in the training phase, b) the coefficient of l1 norm $\beta$ in the denoising phase to see how the denoising results are going to change.
• Compare the filters of CSC to those of the first layer of CNN. What's the similarity and difference between them?