Multi-parameter identification and shape reconstruction for unbounded fractal rough surfaces with tapered wave incidence

Figures & data

Figure 1. A schematic of the rough surface inverse scattering problem with the tapered wave incidence.

Figure 2. Reconstruct the surface for $\widehat{h}=0.04,\widehat{D}=1.3$ (left) and $\widehat{h}=0.1,\widehat{D}=1.3$ (right) with 1% noise.

Figure 3. Reconstruct the surface for $(\widehat{h})=0.3$, $(\widehat{D})=1.3$ (left) and $(\widehat{h})=0.4$, $(\widehat{D})=1.5$ (right) with 1% noise.

Table 1. The two-parameter $(\widehat{h},\widehat{D})$ inversion results of the fractal rough surface.

$\widehat{h}$	$\widehat{D}$	${\widehat{h}}_{num}$	${\widehat{D}}_{num}$	$N_{it}$	$Ma{x}_{er}$
0.04	1.3	0.0401	1.3059	26	$<{10}^{-3}$
0.1	1.3	0.0999	1.2981	14	$<{10}^{-3}$
0.3	1.3	0.3001	1.3003	4	$<{10}^{-3}$
0.4	1.5	0.4003	1.5001	21	$<{10}^{-3}$

Table 2. The two-parameter $(\widehat{h},\widehat{D})$ inversion results of the fractal rough surface with 1% noise level.

Noise (%)	$\widehat{h}$	$\widehat{D}$	${\widehat{h}}_{num}$	${\widehat{D}}_{num}$	$N_{it}$	$Ma{x}_{er}$
1	0.04	1.3	0.0402	1.2949	12	$<{10}^{-3}$
1	0.1	1.3	0.1000	1.2944	28	$<{10}^{-3}$
1	0.3	1.3	0.2999	1.2996	11	$<{10}^{-3}$
1	0.4	1.5	0.4005	1.5003	16	$<{10}^{-3}$

Table 3. The two-parameter $(\widehat{h},\widehat{D})$ inversion results of the fractal rough surface with 1–5% noise level.

Noise (%)	$\widehat{h}$	$\widehat{D}$	${\widehat{h}}_{num}$	${\widehat{D}}_{num}$	$N_{it}$	$Ma{x}_{er}$
1	0.3	1.3	0.2999	1.2996	11	$<{10}^{-3}$
2	0.3	1.3	0.3002	1.2986	5	$<{10}^{-3}$
3	0.3	1.3	0.3001	1.2993	9	$<{10}^{-3}$
4	0.3	1.3	0.3000	1.3012	9	$<{10}^{-3}$
5	0.3	1.3	0.3000	1.2986	8	$<{10}^{-3}$

Table 4. The two-parameter $(\widehat{h},\widehat{D})$ inversion results of the fractal rough surface with different initial values.

$({\widehat{h}}_0,{\widehat{D}}_0)$	$\widehat{h}$	$\widehat{D}$	${\widehat{h}}_{num}$	${\widehat{D}}_{num}$	$N_{it}$	$Ma{x}_{er}$
(0.2, 1.2)	0.3	1.3	0.3001	1.3003	4	$<{10}^{-3}$
(0.1, 1.2)	0.3	1.3	0.3001	1.3017	8	$<{10}^{-3}$
(0.1, 1.1)	0.3	1.3	0.3004	1.2998	7	$<{10}^{-3}$
(0.2, 1.8)	0.3	1.3	0.2998	1.3006	10	$<{10}^{-3}$
(0.4, 1.5)	0.3	1.3	0.3003	1.3009	6	$<{10}^{-3}$

Table 5. The two-parameter $(\widehat{h},\widehat{D})$ inversion results of the fractal rough surface with different survey positions.

H	$\widehat{h}$	$\widehat{D}$	${\widehat{h}}_{num}$	${\widehat{D}}_{num}$	$N_{it}$	$Ma{x}_{er}$
10	0.3	1.3	0.3001	1.3003	4	$<{10}^{-3}$
1	0.3	1.3	0.2998	1.2998	10	$<{10}^{-3}$
100	0.3	1.3	0.2996	1.3010	5	$<{10}^{-3}$

Table 6. The three-parameter $(\widehat{h},\widehat{D},K)$ inversion results of the fractal rough surface.

$\widehat{h}$	$\widehat{D}$	K	${\widehat{h}}_{num}$	${\widehat{D}}_{num}$	$K_{num}$	$N_{it}$	$Ma{x}_{er}$
0.1	1.3	0.5	0.0999	1.2904	0.5000	$30\ast 2+20$	$<{10}^{-3}$

Figure 4. Reconstruct the surface for three parameters $\widehat{h}=0.1,\widehat{D}=1.3,K=0.5.$

Figure 5. Reconstruct the surface for $\widehat{h}=0.1,\widehat{D}=1.5$ with 1% noise by single-incidence (The top), multiple-angle incidence (The middle) and multiple-frequency incidence (The bottom).

Figure 6. Reconstruct the surface for three parameters $\widehat{h}=0.1,\widehat{D}=1.3,K=0.5.$ with 1% noise.

Table 7. The two-parameter $(\widehat{h},\widehat{D})$ inversion results with multiple incidence and single incidence at a fixed frequency.

Inc	Noise (%)	$\widehat{h}$	$\widehat{D}$	${\widehat{h}}_{num}$	${\widehat{D}}_{num}$	$N_{it}$	$Ma{x}_{er}$
M-A	0	0.1	1.5	0.1000	1.5000	$5\ast 3$	$<{10}^{-6}$
S -I	0	0.1	1.5	0.1000	1.4995	35	$<{10}^{-6}$
M-A	1	0.1	1.5	0.1000	1.5020	$5\ast 3$	$<{10}^{-3}$
S -I	1	0.1	1.5	0.0987	1.4421	100	$>{10}^{-3}$

Table 8. The two-parameter $(\widehat{h},\widehat{D})$ inversion results of the fractal rough surface with multiple frequency incidence.

Inc	Noise (%)	$\widehat{h}$	$\widehat{D}$	${\widehat{h}}_{num}$	${\widehat{D}}_{num}$	$N_{it}$	$Ma{x}_{er}$
M-F	0	0.1	1.5	0.1000	1.5000	$15\ast 2$	$<{10}^{-6}$
M-F	1	0.1	1.5	0.0999	1.5021	$15\ast 3$	$<{10}^{-3}$

Table 9. The three-parameter $(\widehat{h},\widehat{D},K)$ inversion results of the fractal rough surface.

$\widehat{h}$	$\widehat{D}$	K	Noise	${\widehat{h}}_{num}$	${\widehat{D}}_{num}$	$K_{num}$	$N_{it}$	$Ma{x}_{er}$
0.1	1.3	0.5	1%	0.1002	1.2966	0.5000	$30+27$	$<{10}^{-3}$