A 1-D time-varying median filter for seismic random, spike-like noise elimination

Signal-to-noise ratio (SNR) estimation using the stack method

A simple definition of the SNR was introduced by Liu and Li (1997). Window $D$, a part of the seismic record, can be chosen for SNR analysis:

$$D = [x_{i,j}]_{M \times N} \qquad (0< M \le N_x, 0< N \le N_t)\;.$$ (2)

Further assumptions are: waveform, amplitude, and phase of seismic wavelet in window $D$ keep stable in respect to distance ``$i$''; noise is ``zero mean'' randomly distributed, along with survey-line direction being independent (decorrelated) of the signal, so that

$$x_{i,j} = s_j + n_{i,j}\;$$ (3)

$$\sum_{i=1}^{M} n_{i,j} = 0\;,$$ (4)

where $s_j$ is amplitude of signal, and $n_{i,j}$ is amplitude of noise. These assumptions generally imply a limitation to this method, but they can be satisfied if the local window is chosen in the stable signal region of the seismic section. So if the signal energy in the window is

$$E_S = M\sum_{j=1}^{N} s_{j}^2 = \frac{1}{M} \sum_{j=1}^{N}(\sum_{i=1}^{M}x_{i,j})^2\;,$$ (5)

the noise energy can be calculated by

$$E_N = \sum_{j=1}^{N}\! \sum_{i=1}^{M}x_{i,j}^2 - E_S\;.$$ (6)

Finally, a decibel expression of the SNR is estimated as

$$SNR = \frac{E_S}{E_N} = 10\,\log_{10}{\left(\frac{\displa... ...\sum_{j=1}^{N}\big(\sum_{i=1}^{M}x_{i,j}\big)^2}\right)}\;.$$ (7)

A 1-D time-varying median filter for seismic random, spike-like noise elimination