Direct surface waves

Direct surface waves are always present and have a high amplitude, and so are tackled separately and first. Direct surface waves can especially cause problems by obscuring echo data that arrives very soon after emission [10]. Since these waves are at a single angle in the B-scan (90°) coincident with the x-axis (the scan position), the median of each horizontal line is subtracted from the line. Then, on an individual pixel basis along each line, any magnitudes greater than the original pixel magnitude are forced to the original pixel magnitude, but the sign remains as that of the subtracted value.

Consider a series of horizontal lines of increasing amplitude contaminated by AWGN, as de- picted by figure 6.9a. The increasing amplitude of the horizontal lines manifests as a darkening gradient down the image as each horizontal line gets a progressively higher amplitude. The ideal way to remove the lines in this case would be to subtract the mean or median along each horizontal line, leaving only the noise. These horizontal lines represent surface waves, and where direct surface waves are strong, this subtraction removes them very effectively. This is because most interfering signals will be much weaker and will not exist along the entire line, and hence are averaged out.

Now consider the case of a weak horizontal line with a strong interfering angled line crossing it. Figure 6.10a represents this by treating a series of horizontal lines of the same amplitude as a uniform background; each horizontal line of the background is the same. The vertical line in figure 6.10a represents an angled line in a B-scan, which is crossing each of the horizontal lines that makes up the background of the scan. The amplitude of the vertical line varies, such that each horizontal line can be considered to be crossed by an angled line of a different amplitude. Despite being narrow, at the bottom of the figure the relatively high amplitude of the vertical line results in it having a much higher mean along the horizontal direction than the horizontal line, and subtracting the mean of each horizontal line rather than the median results in a DC offset. For example, consider if the background had an amplitude of 1 at each pixel, and each horizontal line had 10 pixels. If the vertical line was a single pixel wide, and had an amplitude of 2, then the median along a horizontal segment of the image would be 1 and the mean would be 1.1 (calculated from(1×2 + 9×1)/10), an increase of only 10%, with the median being the correct

value to subtract in order to remove the background without a DC offset. If the vertical line had an amplitude of 10, the median would remain at 1, whereas the mean would be 1.9 (calculated from(1×10 + 9×1)/10). Now the increase is 90%, and again of course the median is the correct

value to subtract in order to remove the background without a DC offset.

The magnitude (magnitude in this case is the amplitude without the sign) of any point along a horizontal line, due to the subtraction of the median, is not allowed to be larger than the original value, minimising the impact of strongly interfering signals increasing the median or signal gaps from being made into strong signals due to the subtraction. Such a restriction on the subtraction of the median minimises banding whilst still allowing other signals superimposed on the direct surface wave to appear after removal. Figure 6.10b again depicts horizontal lines (as in figure 6.10a), but now with a gap or a signal destructively superimposed, which the vertical line represents and hence has a reduced amplitude relative to the background. Removal of the background is still desirable, but not at the expense of creating a larger problem; subtracting the median without limiting the magnitude range of the result will lead to a large amplitude at a gap. With such a limitation in place, after subtraction the values along the vertical line would be the negative of the line until the line value reached below zero. After this point, the value at the line would simply equal the current line value. This causes a relatively small error if the line is a destructively interfering signal, but prevents a much larger error if the line is an attenuation of the horizontal line at that point. For example, consider if the background had an amplitude of 10 at each pixel, and each horizontal line had 10 pixels. If the vertical line was a single pixel wide, and had an amplitude of 1 (due to attenuation of the horizontal line at that point), then the median along a horizontal segment of the image would be 10. If this were subtracted, the vertical line pixel would now have an amplitude of -9, effectively creating a signal where none should exist. With the restriction in place, the pixel is limited to an amplitude in the range 1 to -1, and takes the value -1 in this case, a minor error from the correct result of 0. If the result should actually be -9 due to the vertical line representing a destructively interfering signal instead of an attenuation of the background, the error is clearly larger, and the vertical line pixel will effectively have been attenuated by the processing. There is no error if a vertical line is constructively interfering as in the previous case (figure 6.10a). It is considered better to lose signals in the small number of cases where a destructively interfering signal (vertical line in this case) has a consistently smaller amplitude than a background signal (horizontal line in this case) over the length of the vertical line, than to create signals where the background is attenuated. Without performing the processing described, it would not be possible

(a) X (pixel) Y (pixel) −100 −50 0 50 100 −50 0 50 100 Amplitude (arbitrary) 0 0.5 1 1.5 2 (b) X (pixel) Y (pixel) −100 −50 0 50 100 −50 0 50 100 Amplitude (arbitrary) 0 0.5 1 1.5 2

Figure 6.9: Image (a) has a vertical intensity gradient,A, linearly increasing fromA= 1toA= 2.

AWGN has been added withx= 0andσ= 0.2. Image (b) has the mean subtracted for lines above

the threshold (γ= 0.14). Thresholding based on the median would result in a similar problem in

this case. The mean and the median are identical except due to the influence of noise.

(a) X (pixel) Y (pixel) −100 −50 0 50 100 −100 −50 0 50 100 Amplitude (arbitrary) 0 0.2 0.4 0.6 0.8 1 (b) X (pixel) Y (pixel) −100 −50 0 50 100 −100 −50 0 50 100 Amplitude (arbitrary) −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4

Figure 6.10: Image (a) has a background amplitudeA= 0.01and a line seven pixels wide of initial

amplitude0 linearly increasing to 1 along the length. This represents (but does not resemble) a

segment of a parabola in a B-scan interfering with a surface wave. Where the line is strong, the largest contribution to the mean is from the line, whereas the median is unaffected by such a thin line. Image (b) has a background amplitudeA= 0.4and a line seven pixels wide of initial amplitude 0 linearly decreasing to−1 along the length. This represents either an attenuation, which is only

relevant up to where the combined amplitudeA= 0, or a signal destructively interfering.

to process that scan region for other purposes, and therefore the destructively interfering signal would be lost under those conditions anyway; better the signal be attenuated than lost completely. Usually, in practical cases, at some point, the interfering signal will reach an edge of the horizontal line, and after that point the interfering signal will have a greater amplitude, reducing the severity of the error. Consider if the horizontal line had an amplitude of 1 at each pixel, and the vertical line pixel an amplitude of -8; after subtraction of the median (equal to 1), the vertical line pixel will still have an amplitude of -8 due to the restriction, rather than the ideal case -9. However, the error is only -1 (approximately 10%).

Thresholding introduces artefacts (figure 6.9b), and hence is not used for subtracting horizontal lines.