American Institute of Mathematical Sciences

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August  2022, 16(4): 997-1017. doi: 10.3934/ipi.2022009

A non-iterative sampling method for inverse elastic wave scattering by rough surfaces

Tielei Zhu and  Jiaqing Yang ,

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi, 710049, China

*Corresponding author: Jiaqing Yang

Received  June 2021 Revised  January 2022 Published  August 2022 Early access  March 2022

Fund Project: The research of JY is supported by NNSF of China Grants No. 12122114 and 11771349

Consider the two-dimensional inverse elastic wave scattering by an infinite rough surface with a Dirichlet boundary condition. A non-iterative sampling method is proposed for detecting the rough surface by taking elastic field measurements on a bounded line segment above the surface, based on reconstructing a modified near-field equation associated with a special surface, which generalized our previous work for the Helmholtz equation (SIAM J. Imag. Sci. 10(3) (2017), 1579-1602) to the Navier equation. Several numerical examples are carried out to illustrate the effectiveness of the inversion algorithm.

Keywords: Inverse scattering, elastic wave, Navier equation, rough surface, sampling method.
Mathematics Subject Classification: Primary: 35P25, 35Q74; Secondary: 74J20, 74J25.
Citation: Tielei Zhu, Jiaqing Yang. A non-iterative sampling method for inverse elastic wave scattering by rough surfaces. Inverse Problems and Imaging, 2022, 16 (4) : 997-1017. doi: 10.3934/ipi.2022009
References:
[1]

T. Arens, Uniqueness for elastic wave scattering by rough surfaces, SIAM J. Math. Anal., 33 (2001), 461-476.  doi: 10.1137/S0036141099359470.

[2]

T. Arens, Existence of solution in elastic wave scattering by unbounded rough surfaces, Math. Methods Appl. Sci., 25 (2002), 507-528.  doi: 10.1002/mma.304.

[3]

G. Bao, J. Gao and P. Li, Analysis of direct and inverse cavity scattering problems, Numer. Math. Theor. Meth., 4 (2011), 335-358.  doi: 10.4208/nmtma.2011.m1021.

[4]

G. Bao and P. Li, Near-field imaging of infinite rough surfaces, SIAM J. Appl. Math., 73 (2013), 2162-2187.  doi: 10.1137/130916266.

[5]

G. Bao and P. Li, Near-field imaging of infinite rough surfaces in dielectric media, SIAM J. Imag. Sci., 7 (2014), 867-899.  doi: 10.1137/130944485.

[6]

G. Bao and J. Lin, Imaging of local surface displacement on an infinite ground plane: The multiple frequency case, SIAM J. Appl. Math., 71 (2011), 1733-1752.  doi: 10.1137/110824644.

[7]

G. Bao and J. Lin, Near-field imaging of the surface displacement on an infinite ground plane, Inverse Probl. Imag., 7 (2013), 377-396.  doi: 10.3934/ipi.2013.7.377.

[8]

G. Bao and L. Zhang, Shape reconstruction of the multi-scale rough surface from multi-frequency phaseless data, Inverse Problems, 32 (2016), 085002, 16 pp. doi: 10.1088/0266-5611/32/8/085002.

[9]

C. Burkard and R. Potthast, A time-domain probe method for three-dimensional rough surface reconstructions, Inverse Probl. Imag., 3 (2009), 259-274.  doi: 10.3934/ipi.2009.3.259.

[10]

F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory: An Introduction, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-31230-7.

[11]

F. Cakoni, D. Colton and P. Monk, The Linear Sampling Method in Inverse Electromagnetic Scattering, SIAM, 2011. doi: 10.1137/1.9780898719406.

[12]

S. N. Chandler-Wilde and P. Monk, Existence, uniqueness, and variational methods for scattering by unbounded rough surfaces, SIAM J. Math. Anal., 37 (2005), 598-618.  doi: 10.1137/040615523.

[13]

M. Ding, J. Li, K. Liu and J. Yang, Imaging of local rough surfaces by the linear sampling method with near-field data, SIAM J. Imag. Sci., 10 (2017), 1579-1602.  doi: 10.1137/16M1097997.

[14]

J. Elschner and G. Hu, Elastic scattering by unbounded rough surfaces, SIAM J. Math. Anal., 44 (2012), 4101-4127.  doi: 10.1137/12086203X.

[15]

J. Elschner and G. Hu, Elastic scattering by unbounded rough surfaces: Solvability in weighted Sobolev spaces, Appl. Anal., 94 (2015), 252-279.  doi: 10.1080/00036811.2014.887695.

[16]

G. Hu, X. Liu, B. Zhang and H. Zhang, A non-iterative approach to inverse elastic scattering by unbounded rigid rough surfaces, Inverse Problems, 35 (2019), 025007, 20 pp. doi: 10.1088/1361-6420/aaf3d6.

[17]

V. D. Kupradze, Potential Methods in the Theory of Elasticity, Israeli Program for Scientific Translations, Jerusalem, 1965.

[18]

A. Lechleiter, Factorization Methods for Photonics and Rough Surfaces, Ph.D thesis, Universitätsverlag Karlsruhe, 2008.

[19]

J. Li, X. Liu, B. Zhang, and H. Zhang, The Nyström method for elastic wave scattering by unbounded rough surfaces, preprint, arXiv: 2108.02600.

[20]

J. Li and G. Sun, A nonlinear integral equation method for the inverse scattering problem by sound-soft rough surfaces, Inverse Probl. Sci. Eng., 23 (2015), 557-577.  doi: 10.1080/17415977.2014.922077.

[21]

J. Li, G. Sun and B. Zhang, The Kirsch-Kress method for inverse scattering by infinite locally, Appl. Anal., 96 (2017), 85-107.  doi: 10.1080/00036811.2016.1192141.

[22]

C. D. Lines and S. N. Chandler-Wilde, A time domain point source method for inverse scattering by rough surfaces, Computing, 75 (2005), 157-180.  doi: 10.1007/s00607-004-0109-8.

[23]

X. Liu, B. Zhang and H. Zhang, A direct imaging method for inverse scattering by unbounded rough surfaces, SIAM J. Imag. Sci., 11 (2018), 1629-1650.  doi: 10.1137/18M1166031.

[24]

X. Liu, B. Zhang and H. Zhang, Near-field imaging of an unbounded elastic rough surface with a direct imaging method, SIAM J. Appl. Math., 79 (2019), 153-176.  doi: 10.1137/18M1181407.

[25]

P. A. Martin, On the scattering of elastic waves by an elastic inclusion in two dimensions, Q. J. Mech. Appl. Math., 43 (1990), 275-291.  doi: 10.1093/qjmam/43.3.275.

[26]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000.

[27]

A. Meier, T. Arens, S. N. Chandler-Wilde and A. Kirsch, A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces, J. Integral Equ. Appl., 12 (2000), 281-321.  doi: 10.1216/jiea/1020282209.

[28]

H. Zhang and B. Zhang, A novel integral equation for scattering by locally rough surfaces and application to the inverse problem, SIAM J. Appl. Math., 73 (2013), 1811-1829.  doi: 10.1137/130908324.

show all references

References:
[1]

T. Arens, Uniqueness for elastic wave scattering by rough surfaces, SIAM J. Math. Anal., 33 (2001), 461-476.  doi: 10.1137/S0036141099359470.

[2]

T. Arens, Existence of solution in elastic wave scattering by unbounded rough surfaces, Math. Methods Appl. Sci., 25 (2002), 507-528.  doi: 10.1002/mma.304.

[3]

G. Bao, J. Gao and P. Li, Analysis of direct and inverse cavity scattering problems, Numer. Math. Theor. Meth., 4 (2011), 335-358.  doi: 10.4208/nmtma.2011.m1021.

[4]

G. Bao and P. Li, Near-field imaging of infinite rough surfaces, SIAM J. Appl. Math., 73 (2013), 2162-2187.  doi: 10.1137/130916266.

[5]

G. Bao and P. Li, Near-field imaging of infinite rough surfaces in dielectric media, SIAM J. Imag. Sci., 7 (2014), 867-899.  doi: 10.1137/130944485.

[6]

G. Bao and J. Lin, Imaging of local surface displacement on an infinite ground plane: The multiple frequency case, SIAM J. Appl. Math., 71 (2011), 1733-1752.  doi: 10.1137/110824644.

[7]

G. Bao and J. Lin, Near-field imaging of the surface displacement on an infinite ground plane, Inverse Probl. Imag., 7 (2013), 377-396.  doi: 10.3934/ipi.2013.7.377.

[8]

G. Bao and L. Zhang, Shape reconstruction of the multi-scale rough surface from multi-frequency phaseless data, Inverse Problems, 32 (2016), 085002, 16 pp. doi: 10.1088/0266-5611/32/8/085002.

[9]

C. Burkard and R. Potthast, A time-domain probe method for three-dimensional rough surface reconstructions, Inverse Probl. Imag., 3 (2009), 259-274.  doi: 10.3934/ipi.2009.3.259.

[10]

F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory: An Introduction, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-31230-7.

[11]

F. Cakoni, D. Colton and P. Monk, The Linear Sampling Method in Inverse Electromagnetic Scattering, SIAM, 2011. doi: 10.1137/1.9780898719406.

[12]

S. N. Chandler-Wilde and P. Monk, Existence, uniqueness, and variational methods for scattering by unbounded rough surfaces, SIAM J. Math. Anal., 37 (2005), 598-618.  doi: 10.1137/040615523.

[13]

M. Ding, J. Li, K. Liu and J. Yang, Imaging of local rough surfaces by the linear sampling method with near-field data, SIAM J. Imag. Sci., 10 (2017), 1579-1602.  doi: 10.1137/16M1097997.

[14]

J. Elschner and G. Hu, Elastic scattering by unbounded rough surfaces, SIAM J. Math. Anal., 44 (2012), 4101-4127.  doi: 10.1137/12086203X.

[15]

J. Elschner and G. Hu, Elastic scattering by unbounded rough surfaces: Solvability in weighted Sobolev spaces, Appl. Anal., 94 (2015), 252-279.  doi: 10.1080/00036811.2014.887695.

[16]

G. Hu, X. Liu, B. Zhang and H. Zhang, A non-iterative approach to inverse elastic scattering by unbounded rigid rough surfaces, Inverse Problems, 35 (2019), 025007, 20 pp. doi: 10.1088/1361-6420/aaf3d6.

[17]

V. D. Kupradze, Potential Methods in the Theory of Elasticity, Israeli Program for Scientific Translations, Jerusalem, 1965.

[18]

A. Lechleiter, Factorization Methods for Photonics and Rough Surfaces, Ph.D thesis, Universitätsverlag Karlsruhe, 2008.

[19]

J. Li, X. Liu, B. Zhang, and H. Zhang, The Nyström method for elastic wave scattering by unbounded rough surfaces, preprint, arXiv: 2108.02600.

[20]

J. Li and G. Sun, A nonlinear integral equation method for the inverse scattering problem by sound-soft rough surfaces, Inverse Probl. Sci. Eng., 23 (2015), 557-577.  doi: 10.1080/17415977.2014.922077.

[21]

J. Li, G. Sun and B. Zhang, The Kirsch-Kress method for inverse scattering by infinite locally, Appl. Anal., 96 (2017), 85-107.  doi: 10.1080/00036811.2016.1192141.

[22]

C. D. Lines and S. N. Chandler-Wilde, A time domain point source method for inverse scattering by rough surfaces, Computing, 75 (2005), 157-180.  doi: 10.1007/s00607-004-0109-8.

[23]

X. Liu, B. Zhang and H. Zhang, A direct imaging method for inverse scattering by unbounded rough surfaces, SIAM J. Imag. Sci., 11 (2018), 1629-1650.  doi: 10.1137/18M1166031.

[24]

X. Liu, B. Zhang and H. Zhang, Near-field imaging of an unbounded elastic rough surface with a direct imaging method, SIAM J. Appl. Math., 79 (2019), 153-176.  doi: 10.1137/18M1181407.

[25]

P. A. Martin, On the scattering of elastic waves by an elastic inclusion in two dimensions, Q. J. Mech. Appl. Math., 43 (1990), 275-291.  doi: 10.1093/qjmam/43.3.275.

[26]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000.

[27]

A. Meier, T. Arens, S. N. Chandler-Wilde and A. Kirsch, A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces, J. Integral Equ. Appl., 12 (2000), 281-321.  doi: 10.1216/jiea/1020282209.

[28]

H. Zhang and B. Zhang, A novel integral equation for scattering by locally rough surfaces and application to the inverse problem, SIAM J. Appl. Math., 73 (2013), 1811-1829.  doi: 10.1137/130908324.

Figure 1.  Elastic wave scattering by unbounded rough surfaces
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Figure 2.  A chosen Lipschitz domain $ D_{R, h} $
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Figure 3.  A chosen Lipschitz domain $ \Omega_{a, b} $
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Figure 4.  Local perturbation of the plane $ x_2 = 0.5 $
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Figure 5.  Reconstruction of the locally rough surface $ \Gamma $ given in Example 1 from data with no noise (a), 2% noise (b) and 5% noise (c) by the indicator function (54)
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Figure 6.  Reconstruction of the locally rough surface $ \Gamma $ given in Example 2 from data with no noise (a), 2% noise (b) and 5% noise (c) by the indicator function (54)
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Figure 7.  Reconstruction of the locally rough surface $ \Gamma $ given in Example 3 from data with no noise (a), 2% noise (b) and 5% noise (c) by the indicator function (54)
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Figure 8.  Reconstruction of the locally rough surface $ \Gamma $ given in Example 4 from data with $ a = 5 $ (a), $ a = 7.5 $ (b) and $ a = 10 $ (c) by the indicator function (54)
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Figure 9.  Reconstruction of the locally rough surface $ \Gamma $ given in Example 5 from data with no noise (a), 2% noise (b) and 5% noise (c) by the indicator function (54)
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Figure 10.  Reconstruction of the locally rough surface $ \Gamma $ given in Example 6 from data with $ R = 20 $ (a), $ R = 40 $ (b) and $ R = 60 $ (c) by the indicator function (54)
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Table 1.  Numerical solutions of $G^{ sc}(x, z, p;R)$ with different $R$
$R$ $G^{ sc}(x^{(1)}, z, p;R)$ $G^{sc}(x^{(2)}, z, p;R)$ $G^{sc}(x^{(3)}, z, p;R)$
$10^2$ $1.0{\text{e}}-2\cdot(1.3750+0.7988{\rm i}) \\ 1.0{\text{e}}-3\cdot(-9.2804+4.6422{\rm i})$ $1.0{\text{e}}-2\cdot(1.5892+0.9119{\rm i}) \\ 1.0{\text{e}}-3\cdot(-5.6177-3.3897{\rm i})$ $1.0{\text{e}}-2\cdot(1.5368+0.8773{\rm i}) \\ 1.0{\text{e}}-2\cdot(-0.1188-1.2583{\rm i})$
$10^3$ $1.0{\text{e}}-5\cdot(-4.6964+3.1245{\rm i})\\ 1.0{\text{e}}-4\cdot(-5.8867-7.7026{\rm i})$ $1.0{\text{e}}-5\cdot(-3.6553+2.5430{\rm i})\\ 1.0{\text{e}}-4\cdot(-5.8972-7.7034{\rm i})$ $1.0{\text{e}}-5\cdot(-2.6095+1.9516{\rm i})\\ 1.0{\text{e}}-4\cdot(-5.9051-7.7043{\rm i})$
$10^4$ $1.0{\text{e}}-7\cdot(0.0569+1.3741{\rm i})\\ 1.0{\text{e}}-5\cdot(-4.2793+9.0659{\rm i})$ $1.0{\text{e}}-7\cdot(0.0848+1.1093{\rm i})\\ 1.0{\text{e}}-5\cdot(-4.2786+9.0665{\rm i})$ $1.0{\text{e}}-8\cdot(1.1240+8.4444{\rm i})\\ 1.0{\text{e}}-5\cdot(-4.2780+9.0669{\rm i})$
$10^5$ $1.0{\text{e}}-9\cdot(-0.4012-1.0360{\rm i})\\ 1.0{\text{e}}-5\cdot(1.1725+1.0492{\rm i})$ $1.0{\text{e}}-10\cdot(-1.5029-7.2349{\rm i})\\ 1.0{\text{e}}-5\cdot(1.1725+1.0492{\rm i})$ $1.0{\text{e}}-10\cdot(1.0061-4.1102{\rm i})\\ 1.0{\text{e}}-5\cdot(1.1726+1.0492{\rm i})$
$10^6$ $1.0{\text{e}}-11\cdot(0.0181+1.2834{\rm i})\\ 1.0{\text{e}}-6\cdot(7.9495-0.4940{\rm i})$ $1.0{\text{e}}-12\cdot(0.0330-7.2675{\rm i})\\ 1.0{\text{e}}-6\cdot(7.9495 - 0.4940{\rm i})$ $1.0{\text{e}}-12\cdot(-0.1152+1.7009{\rm i})\\ 1.0{\text{e}}-6\cdot(7.9495-0.4940{\rm i})$
$R$ $G^{ sc}(x^{(4)}, z, p;R)$ $G^{sc}(x^{(5)}, z, p;R) $ $G^{sc}(x^{(6)}, z, p;R) $
$10^2$ $1.0{\text{e}}-2\cdot(1.2156-0.6900{\rm i})\\ 1.0{\text{e}}-2\cdot(0.3021-2.1052{\rm i})$ $1.0{\text{e}}-3\cdot(6.7691+3.7704{\rm i})\\ 1.0{\text{e}}-2\cdot(0.6058-2.7001{\rm i})$ $1.0{\text{e}}-4\cdot(1.6784-0.7840{\rm i})\\ 1.0{\text{e}}-2\cdot(0.7233-2.9150{\rm i})$
$10^3$ $1.0{\text{e}}-5\cdot(-1.5604+1.3531{\rm i})\\ 1.0{\text{e}}-4\cdot(-5.9101-7.7055{\rm i})$ $1.0{\text{e}}-6\cdot(-5.0901+7.5026{\rm i})\\ 1.0{\text{e}}-4\cdot(-5.9124-7.7068{\rm i})$ $1.0{\text{e}}-6\cdot(5.4321+1.4604{\rm i})\\ 1.0{\text{e}}-4\cdot(-5.9120-7.7083{\rm i})$
$10^4$ $1.0{\text{e}}-8\cdot(1.3992+5.7956{\rm i})\\ 1.0{\text{e}}-5\cdot(-4.2777+9.0671{\rm i})$ $1.0{\text{e}}-8\cdot(1.6737+3.1466{\rm i})\\ 1.0{\text{e}}-5\cdot(-4.2776+9.0672{\rm i})$ $1.0{\text{e}}-8\cdot(1.9479+0.4977{\rm i})\\ 1.0{\text{e}}-5\cdot(-4.2777+9.0671{\rm i})$
$10^5$ $1.0{\text{e}}-10\cdot(3.5152-0.9855{\rm i})\\ 1.0{\text{e}}-5\cdot(1.1726+1.0492{\rm i})$ $1.0{\text{e}}-10\cdot(6.0243+2.1393{\rm i})\\ 1.0{\text{e}}-5\cdot( 1.1726+1.0492{\rm i})$ $1.0{\text{e}}-10\cdot(8.5334+5.2640{\rm i})\\ 1.0{\text{e}}-5\cdot(1.1726+1.0492{\rm i})$
$10^6$ $1.0{\text{e}}-12\cdot(-0.2635-3.8658{\rm i})\\ 1.0{\text{e}}-6\cdot(7.9495-4.9404{\rm i})$ $1.0{\text{e}}-12\cdot(-0.4117-9.4324{\rm i})\\ 1.0{\text{e}}-6\cdot(7.9495-0.4940{\rm i})$ $1.0{\text{e}}-11\cdot(-0.0560-1.4999{\rm i})\\ 1.0{\text{e}}-6\cdot(7.9495-0.4941{\rm i})$
$R$ $G^{ sc}(x^{(7)}, z, p;R)$ $G^{sc}(x^{(8)}, z, p;R)$ $G^{sc}(x^{(9)}, z, p;R)$
$10^2$ $1.0{\text{e}}-3\cdot(-6.4395-3.9529{\rm i}) \\ 1.0{\text{e}}-2\cdot(0.6266-2.7036 {\rm i})$ $1.0{\text{e}}-2\cdot(-1.1845-0.7152{\rm i}) \\ 1.0{\text{e}}-2\cdot(0.3364-2.1128{\rm i})$ $1.0{\text{e}}-2\cdot(-1.5090-0.9115{\rm i}) \\ 1.0{\text{e}}-2\cdot(-0.0834-1.27712{\rm i})$
$10^3$ $1.0{\text{e}}-5\cdot(1.5950-0.4567{\rm i}) \\ 1.0{\text{e}}-4\cdot(-5.9087-7.7099{\rm i})$ $1.0{\text{e}}-5\cdot(2.6452-1.0550{\rm i}) \\ 1.0{\text{e}}-4\cdot(-5.9028-7.7118{\rm i})$ $1.0{\text{e}}-5\cdot(3.6925-1.6461{\rm i}) \\ 1.0{\text{e}}-4\cdot(-5.8941-7.7138{\rm i})$
$10^4$ $1.0{\text{e}}-8\cdot(2.2226-2.1510{\rm i}) \\ 1.0{\text{e}}-5\cdot(-4.2781+9.0669{\rm i})$ $1.0{\text{e}}-8\cdot(2.4981-4.7993{\rm i}) \\ 1.0{\text{e}}-5\cdot(-4.2786+9.0666{\rm i})$ $1.0{\text{e}}-8\cdot(2.7749-7.4469{\rm i}) \\ 1.0{\text{e}}-5\cdot(-4.2794+9.0660{\rm i})$
$10^5$ $1.0{\text{e}}-9\cdot(1.1042+0.8839{\rm i}) \\ 1.0{\text{e}}-5\cdot(1.1726+1.0492{\rm i})$ $1.0{\text{e}}-9\cdot(1.3551+1.1514{\rm i}) \\ 1.0{\text{e}}-5\cdot(1.1726+1.0492{\rm i})$ $1.0{\text{e}}-9\cdot(1.6060+1.4639{\rm i}) \\ 1.0{\text{e}}-5\cdot(1.1726+1.0492{\rm i})$
$10^6$ $1.0{\text{e}}-11\cdot(-0.0708-2.0566{\rm i}) \\ 1.0{\text{e}}-6\cdot(7.9495-0.4940{\rm i})$ $1.0{\text{e}}-11\cdot(-0.0856-2.6132{\rm i}) \\ 1.0{\text{e}}-6\cdot(7.9495-0.4940{\rm i})$ $1.0{\text{e}}-11\cdot(-0.1004-3.1699{\rm i}) \\ 1.0{\text{e}}-6\cdot(7.9495-0.4940{\rm i})$
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$R$ $G^{ sc}(x^{(1)}, z, p;R)$ $G^{sc}(x^{(2)}, z, p;R)$ $G^{sc}(x^{(3)}, z, p;R)$
$10^2$ $1.0{\text{e}}-2\cdot(1.3750+0.7988{\rm i}) \\ 1.0{\text{e}}-3\cdot(-9.2804+4.6422{\rm i})$ $1.0{\text{e}}-2\cdot(1.5892+0.9119{\rm i}) \\ 1.0{\text{e}}-3\cdot(-5.6177-3.3897{\rm i})$ $1.0{\text{e}}-2\cdot(1.5368+0.8773{\rm i}) \\ 1.0{\text{e}}-2\cdot(-0.1188-1.2583{\rm i})$
$10^3$ $1.0{\text{e}}-5\cdot(-4.6964+3.1245{\rm i})\\ 1.0{\text{e}}-4\cdot(-5.8867-7.7026{\rm i})$ $1.0{\text{e}}-5\cdot(-3.6553+2.5430{\rm i})\\ 1.0{\text{e}}-4\cdot(-5.8972-7.7034{\rm i})$ $1.0{\text{e}}-5\cdot(-2.6095+1.9516{\rm i})\\ 1.0{\text{e}}-4\cdot(-5.9051-7.7043{\rm i})$
$10^4$ $1.0{\text{e}}-7\cdot(0.0569+1.3741{\rm i})\\ 1.0{\text{e}}-5\cdot(-4.2793+9.0659{\rm i})$ $1.0{\text{e}}-7\cdot(0.0848+1.1093{\rm i})\\ 1.0{\text{e}}-5\cdot(-4.2786+9.0665{\rm i})$ $1.0{\text{e}}-8\cdot(1.1240+8.4444{\rm i})\\ 1.0{\text{e}}-5\cdot(-4.2780+9.0669{\rm i})$
$10^5$ $1.0{\text{e}}-9\cdot(-0.4012-1.0360{\rm i})\\ 1.0{\text{e}}-5\cdot(1.1725+1.0492{\rm i})$ $1.0{\text{e}}-10\cdot(-1.5029-7.2349{\rm i})\\ 1.0{\text{e}}-5\cdot(1.1725+1.0492{\rm i})$ $1.0{\text{e}}-10\cdot(1.0061-4.1102{\rm i})\\ 1.0{\text{e}}-5\cdot(1.1726+1.0492{\rm i})$
$10^6$ $1.0{\text{e}}-11\cdot(0.0181+1.2834{\rm i})\\ 1.0{\text{e}}-6\cdot(7.9495-0.4940{\rm i})$ $1.0{\text{e}}-12\cdot(0.0330-7.2675{\rm i})\\ 1.0{\text{e}}-6\cdot(7.9495 - 0.4940{\rm i})$ $1.0{\text{e}}-12\cdot(-0.1152+1.7009{\rm i})\\ 1.0{\text{e}}-6\cdot(7.9495-0.4940{\rm i})$
$R$ $G^{ sc}(x^{(4)}, z, p;R)$ $G^{sc}(x^{(5)}, z, p;R) $ $G^{sc}(x^{(6)}, z, p;R) $
$10^2$ $1.0{\text{e}}-2\cdot(1.2156-0.6900{\rm i})\\ 1.0{\text{e}}-2\cdot(0.3021-2.1052{\rm i})$ $1.0{\text{e}}-3\cdot(6.7691+3.7704{\rm i})\\ 1.0{\text{e}}-2\cdot(0.6058-2.7001{\rm i})$ $1.0{\text{e}}-4\cdot(1.6784-0.7840{\rm i})\\ 1.0{\text{e}}-2\cdot(0.7233-2.9150{\rm i})$
$10^3$ $1.0{\text{e}}-5\cdot(-1.5604+1.3531{\rm i})\\ 1.0{\text{e}}-4\cdot(-5.9101-7.7055{\rm i})$ $1.0{\text{e}}-6\cdot(-5.0901+7.5026{\rm i})\\ 1.0{\text{e}}-4\cdot(-5.9124-7.7068{\rm i})$ $1.0{\text{e}}-6\cdot(5.4321+1.4604{\rm i})\\ 1.0{\text{e}}-4\cdot(-5.9120-7.7083{\rm i})$
$10^4$ $1.0{\text{e}}-8\cdot(1.3992+5.7956{\rm i})\\ 1.0{\text{e}}-5\cdot(-4.2777+9.0671{\rm i})$ $1.0{\text{e}}-8\cdot(1.6737+3.1466{\rm i})\\ 1.0{\text{e}}-5\cdot(-4.2776+9.0672{\rm i})$ $1.0{\text{e}}-8\cdot(1.9479+0.4977{\rm i})\\ 1.0{\text{e}}-5\cdot(-4.2777+9.0671{\rm i})$
$10^5$ $1.0{\text{e}}-10\cdot(3.5152-0.9855{\rm i})\\ 1.0{\text{e}}-5\cdot(1.1726+1.0492{\rm i})$ $1.0{\text{e}}-10\cdot(6.0243+2.1393{\rm i})\\ 1.0{\text{e}}-5\cdot( 1.1726+1.0492{\rm i})$ $1.0{\text{e}}-10\cdot(8.5334+5.2640{\rm i})\\ 1.0{\text{e}}-5\cdot(1.1726+1.0492{\rm i})$
$10^6$ $1.0{\text{e}}-12\cdot(-0.2635-3.8658{\rm i})\\ 1.0{\text{e}}-6\cdot(7.9495-4.9404{\rm i})$ $1.0{\text{e}}-12\cdot(-0.4117-9.4324{\rm i})\\ 1.0{\text{e}}-6\cdot(7.9495-0.4940{\rm i})$ $1.0{\text{e}}-11\cdot(-0.0560-1.4999{\rm i})\\ 1.0{\text{e}}-6\cdot(7.9495-0.4941{\rm i})$
$R$ $G^{ sc}(x^{(7)}, z, p;R)$ $G^{sc}(x^{(8)}, z, p;R)$ $G^{sc}(x^{(9)}, z, p;R)$
$10^2$ $1.0{\text{e}}-3\cdot(-6.4395-3.9529{\rm i}) \\ 1.0{\text{e}}-2\cdot(0.6266-2.7036 {\rm i})$ $1.0{\text{e}}-2\cdot(-1.1845-0.7152{\rm i}) \\ 1.0{\text{e}}-2\cdot(0.3364-2.1128{\rm i})$ $1.0{\text{e}}-2\cdot(-1.5090-0.9115{\rm i}) \\ 1.0{\text{e}}-2\cdot(-0.0834-1.27712{\rm i})$
$10^3$ $1.0{\text{e}}-5\cdot(1.5950-0.4567{\rm i}) \\ 1.0{\text{e}}-4\cdot(-5.9087-7.7099{\rm i})$ $1.0{\text{e}}-5\cdot(2.6452-1.0550{\rm i}) \\ 1.0{\text{e}}-4\cdot(-5.9028-7.7118{\rm i})$ $1.0{\text{e}}-5\cdot(3.6925-1.6461{\rm i}) \\ 1.0{\text{e}}-4\cdot(-5.8941-7.7138{\rm i})$
$10^4$ $1.0{\text{e}}-8\cdot(2.2226-2.1510{\rm i}) \\ 1.0{\text{e}}-5\cdot(-4.2781+9.0669{\rm i})$ $1.0{\text{e}}-8\cdot(2.4981-4.7993{\rm i}) \\ 1.0{\text{e}}-5\cdot(-4.2786+9.0666{\rm i})$ $1.0{\text{e}}-8\cdot(2.7749-7.4469{\rm i}) \\ 1.0{\text{e}}-5\cdot(-4.2794+9.0660{\rm i})$
$10^5$ $1.0{\text{e}}-9\cdot(1.1042+0.8839{\rm i}) \\ 1.0{\text{e}}-5\cdot(1.1726+1.0492{\rm i})$ $1.0{\text{e}}-9\cdot(1.3551+1.1514{\rm i}) \\ 1.0{\text{e}}-5\cdot(1.1726+1.0492{\rm i})$ $1.0{\text{e}}-9\cdot(1.6060+1.4639{\rm i}) \\ 1.0{\text{e}}-5\cdot(1.1726+1.0492{\rm i})$
$10^6$ $1.0{\text{e}}-11\cdot(-0.0708-2.0566{\rm i}) \\ 1.0{\text{e}}-6\cdot(7.9495-0.4940{\rm i})$ $1.0{\text{e}}-11\cdot(-0.0856-2.6132{\rm i}) \\ 1.0{\text{e}}-6\cdot(7.9495-0.4940{\rm i})$ $1.0{\text{e}}-11\cdot(-0.1004-3.1699{\rm i}) \\ 1.0{\text{e}}-6\cdot(7.9495-0.4940{\rm i})$
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