Random noise attenuation using $f$-$x$ regularized nonstationary autoregression

Synthetic shot gather

para,npara,tpefpatch,fxpatch,npre,npar
Figure 4. (a) Synthetic shot gather. (b) Noisy gather. (c) Result of $f$-$x$ domain prediction (SNR=0.98). (d) Result of $t$-$x$ domain prediction (SNR=1.25). (e) Result of $f$-$x$ RNA (SNR=3.12). (f) The real part of coefficients at a given shift ${{a}_{n,i=1}}(f)$. We decimate the data in (a)-(e) for display purpose.

fxdiff,mpapatch,ndiff
Figure 5. Difference sections of $f$-$x$ domain prediction (a), $t$-$x$ domain prediction (b), and $f$-$x$ RNA (c).

Figure 4(a) shows a synthetic shot gather with four hyperbolic events, 501 traces. Some random noise is added to this gather. We do not use windows in time for this example. For $f$-$x$ RNA, the length of filter is $M=4$ and the smoothing radiuses in space and frequency axes are respectively 20 and 3, ${{r}_{x}}=20$, ${{r}_{f}}=3$. The $f$-$x$ domain prediction is implemented over a sliding window of 20 traces width with 50% overlap and the filter length is 6, $M=3$ and the $t$-$x$ domain prediction is implemented over the same sliding window and the filter length in space and time are 6 and 5 respectively. The estimated nonstationary coefficients by the proposed $f$-$x$ RNA are shown in Figure 4(f). Note that the middle coefficient is bigger than the sideward, which is because the dip of the middle is smaller than the sideward. The results of three methods are shown in Figures 4(d)- 6(d), respectively. The $f$-$x$ RNA achieves a similar result to $f$-$x$ domain and $t$-$x$ domain prediction methods. However, we use equation 11 to compute the SNRs of the results of three methods. The SNRs of three methods are 0.98 dB, 1.25 dB, 1.67 dB, respectively. The $f$-$x$ RNA can improve SNR more greatly. The $f$-$x$ RNA solves the nonstationary case by allowing the coefficients smoothly varying, while $f$-$x$ domain or $t$-$x$ domain prediction method uses windowing strategies. From the difference sections (Figure 7(a)- 5(c)), we find that $f$-$x$ domain and $t$-$x$ domain prediction methods damage more signals than $f$-$x$ RNA. If we use windows in time for this example, we can obtain better results. This example shows that $f$-$x$ RNA can be used for random noise attenuation in shot gather.

Random noise attenuation using $f$-$x$ regularized nonstationary autoregression