Detection of anomalies inside hollow metal cylindrical structures

Figures & data

Figure 1. Metal cylindrical hollow structure.

Figure 2. Section of a metal cylindrical hollow structure.

Figure 3. System geometry and cylindrical reference system (z, θ, x).

Figure 4. Circular anomaly as a constant radial reduction.

Figure 5. Lookup-table with different curves β_i(f) for different pairs (l_a, r_ap)_i. Note: l_a varies from 0 to 12 m with steps of 1 m; r_ap varies from 0% to 10% with steps of 1%. The curves are not equally spaced and the mapping is strongly non-linear.

Figure 6. β_i(f) curves as a function of normalized frequency, for some (l_a, r_a)_i pairs. Section length l_s = 12 m.

Figure 7. β_i(f) curves as a function of normalized frequency, for some (l_a, r_a)_i pairs. Section length l_s = 24 m.

Figure 8. Absolute value of the transmitted waveform (after demodulation).

Figure 9. Spectrum amplitude of the transmitted signal shown in Figure 8.

Figure 10. Spectrum phase of the transmitted signal shown in Figure 8.

Figure 11. Absolute value of the echoes from the joints at coordinates x_i and x_j: l_a = 0 m, r_ap = 0%.

Figure 12. Absolute value of the echoes from the joints at coordinates x_i and x_j: l_a = 2 m, r_ap = 3%.

Figure 13. Spectra phase of the signals in Figure 11.

Figure 14. Spectra phase of the signals in Figure 12.

Figure 15. Phase of the spectra ratio of the joints at coordinates x_i and x_j: l_a = 0 m, r_ap = 0%.

Figure 16. Phase of the spectra ratio of the joints at coordinates x_i and x_j: l_a = 2 m, r_ap = 3%.

Figure 17. Mean value of the inverted anomaly length versus SNR. Exact solution: l_a = 2 m and r_ap = 3%.

Figure 18. SD of the inverted anomaly length versus SNR. Exact solution: l_a = 2 m and r_ap = 3%.

Figure 19. Mean value of the inverted percentage radius reduction versus SNR. Exact solution: l_a = 2 m and r_ap = 3%.

Figure 20. SD of the inverted percentage radius reduction versus SNR. Exact solution: l_a = 2 m and r_ap = 3%.

Table 1. Inverted parameters–synthetic data: regular (a) and deformed (b) elements.

Example	la(m)	rap (%)
(a), no noise	0	0	0	0
(b), no noise	2	3	2	3
(b), SNR = 10 dB	2	3	2.3 ± 3.8	2.9 ± 2.5
(b), SNR = 20 dB	2	3	2.0 ± 0.5	3.0 ± 0.7

Figure 21. Experimental setup for the detection of the circular anomaly.

Figure 22. Real data for a regular section. The arrows indicate the starting time of the echoes from the joints.

Figure 23. Real data for a deformed section. The arrows indicate the starting time of the echoes from the joints.

Figure 24. Absolute value of the echoes from the joints at coordinates x_i = 84 m and x_j = 108 m: real data with regular section.

Figure 25. Absolute value of the echoes from the joints at coordinates x_i = 84 m and x_j = 108 m: real data with deformed section.

Figure 26. Spectra phase of the signals in Figure 24.

Figure 27. Spectra phase of the signals in Figure 25.

Figure 28. Phase of the spectra ratio of the joints echoes at x_i = 84 m and x_j = 108 m: real data with regular section.

Figure 29. Phase of the spectra ratio of the joints echoes at x_i = 84 m and x_j = 108 m: real data with deformed section.

Table 2. Inverted parameters–real data: regular (a) and deformed (b) elements.

Example	la(m)	rap (%)
(a)	0	≅0	0.35	0.35
(b)	≅3	≅5	3.85	4.90