Width estimation of a subsurface water-filled crack using internal multiple reflections from electromagnetic scattering model of a cylinder

Figures & data

Figure 1. (a) A dielectric cylinder impinged by a plane wave. (b) Illustration of ray path for a dielectric cylinder impinged by a plane wave. (c) Propagation processes of internal multiple reflections.

Figure 2. (a) Ratio of path length to the diameter (L/d) with respect to incident angles. (b) (L/d) with respect to relative permittivities. (c) (t/τ) with respect to incident angles. (d) (t/τ) with respect to relative permittivities.

Figure 3. Backscattering field of a cylinder with and .

Table 1. Theoretic and simulated values of normalized time delay.

m	2	3	4	5	6
Theoretic	1.12	2.11	3.11	4.11	5.11
Simulated	1.00	2.15	3.15	4.27	5.29

Figure 4. (a) The SPWV distribution of the electric field from cylinder with d = 4 cm and . (b) From the cylinder with d = 4 cm, . (c) From the cylinder with d = 4 cm, . (d) From the cylinder with d = 4 cm, .

Figure 5. (a) The SPWV distribution of the electric field from cylinder with d = 8 cm and . (b) From the cylinder with d = 8 cm, . (c) From the cylinder with d = 8 cm, . (d) From the cylinder with d = 8 cm, .

Table 2. Width estimation of the cylinder with diameter 4 cm.

εr	m	1	2	3	4	5	6
10	△tm (ns)	–	1.00	–	–	–	–
	de (cm)	–	9.49	–	–	–	–
33	△tm (ns)	–	–	0.78	1.85	1.65	1.90
	de (cm)	–	–	4.07	9.66	8.62	9.03
55	△tm (ns)	1.36	1.21	1.07	1.21	0.61	1.14
	de (cm)	5.50	4.89	4.33	4.89	2.48	4.60
81	△tm (ns)	1.36	1.15	1.21	1.43	1.08	1.68
	de (cm)	4.56	3.80	4.06	4.76	3.60	5.60

Table 3. Width estimation of the cylinder with diameter 8 cm.

εr	m	1	2	3	4	5
10	△tm (ns)	–	0.94	0.76	–	–
	de (cm)	–	8.92	7.21	–	–
33	△tm (ns)	1.60	1.59	1.44	1.53	1.44
	de (cm)	8.36	8.30	7.52	7.99	7.52
55	△tm (ns)	1.90	2.05	1.98	2.05	1.93
	de (cm)	7.69	8.31	7.99	8.29	7.81
81	△tm (ns)	2.35	2.44	2.51	2.50	2.44
	de (cm)	7.83	8.13	8.37	8.33	8.13

Figure 6. (a) Experiment scenario for water-filled crack in free space. (b) The SPWV distribution of the impulse responses from water-filled plastic bottle. (c) From water-filled PVC pipe.

Table 4. Estimated width for cracks in free space.

d (cm)	m	1	2	3
3.0	△tm (ns)	0.78	1.17	0.82
	de (cm)	2.59	3.91	2.73
3.5	△tm (ns)	1.03	0.97	1.07
	de (cm)	3.42	3.23	3.57

Figure 7. (a) Experiment scenario for subsurface water-filled crack. (b) The SPWV distribution of the impulse responses from water-filled PVC pipe. (c) From water-filled plastic bottle with d = 5.7 cm.

Table 5. Estimated width for subsurface cracks.

d (cm)	m	1	2	3
3.5	△tm (ns)	0.93	1.21	0.92
	de (cm)	3.10	4.03	3.07
5.7	△tm (ns)	1.84	1.66	1.75
	de (cm)	6.13	5.53	5.83