can see in Figure 4(c), after applying the filter, noise is removed and the layers are more visible, however, the many of the boundaries have become murky. Structures with low amplitude are most affected by denoising, as opposed to the high-amplitude structures which are more visible, and therefore, reconstruction of the layers has not been done efficiently.
6.3. Applying the AR Filter in the Wavelet Domain
Here we have used the undecimated discrete wavelet transform in applying the autoregressive filter to the noise section in the wavelet domain. We note that here, the applied filter is linear which is a great advantage in noise reduction due to the linearity of the wavelet space.
The denoised section by using this method is shown in Figure 4(d) which looks promising, since the layer boundaries are distinguishable and there is logical smooth trend throughout the section. Both the low and high amplitude structures are more well-defined compared to the autoregressive filter in the f-x domain. We also note that the layer trends are efficiently reconstructed.
6.4. Applying the Method on Real Data
Here, we apply the linear regressive model to real data first in the f-x and then in the wavelet domain in order to remove random noise and eventually compare the results. We show traces #450 onward for better comparison in Figure 6.
We first, suppress the noise in the f-x domain using the autoregressive filter on the noisy real data shown in Figure 5(a) where the boundaries and traces (especially after trace #450) are difficult to distinguish and the bulges are faded in the noise. As shown in Figure 6(c), the autoregressive filter in the f-x domain has properly reduced the noise and the layer trends are visible.
As mentioned before, the noise is random and does not have a specific source. As we can see in Figure 5(a), the right (and specially lower right) portion of the section is very noisy as noise has covered all the trends and layers. Figure 5(c) shows the denoised section after applying the autoregressed filter in the wavelet domain. As we can see in Figure 5(c), the trends are well-defined, especially in the lower right corner of the section and the previously vague parts can now be distinguished to a much greater extent. Therefore the method has successfully reduced the noise.
6.5. Comparing the Application of AR in the Wavelet and f-x Domains
Considering the previous sections on wavelet and f-x domains, as well as the better performance of the filter in the wavelet domain, here we compare denoising on synthetic and real data in both domains. By comparing Figure 5(b) and Figure 5(c), we notice the better performance of the filter in denoising the synthetic data in the wavelet domain. Especially, in the later arrival times which correspond to greater depths, noise reduction is more evident. We note that the boundaries are better represented and are more distinguishable. Also, by comparing
Figure 5. The GPR data section; (a) raw data; (b) after applying autoregression in the f-x domain; (c) after applying autoregression in the wavelet domain.
Figure 6. The GPR data section from trace 450 to the end; (a) raw data; (b) after applying autoregression in the f-x domain; (c) after applying autoregression in the wavelet domain.
Figure 6(b) and Figure 6(c), we can see that again the wavelet domain has had a much better performance in reducing the noise and retrieving the real signal (the layer trends are more visible). Overall, the autoregressive filter in the wavelet domain had done a better job in retrieving the signal due to the linearity of the wavelet domain.
As discussed above, various factors such as phone networks, power posts, utility poles, etc. cause contamination in the GPR data. Since the goal of this study was to increase the signal to noise ratio, a method was chosen to damage the signal as least as possible while reducing the noise. In this study, the noise suppression procedure was applied to both synthetic and real GPR data in f-x and wavelet domains through using autoregressive filter. As we see, noise reduction improves interpretation of data and Autoregressive filter bears good results in both f-x and wavelet domains. Which means that Linear regression in the wavelet domain leads to better results, compared to those of the f-x domain, due to the local nature of the wavelet transform and the imposed linearity on the events on different scales.
We should also note that in contrast with the f-x domain, the autoregressive filter is linear just as the wavelet domain which is why the filter does not work as well in the f-x domain.
Conflicts of Interest
The authors declare no conflicts of interest.
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