A backscattering model based on corrector theory of homogenization for the random Helmholtz equation

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October 2019, 24(10): 5377-5407. doi: 10.3934/dcdsb.2019063

A backscattering model based on corrector theory of homogenization for the random Helmholtz equation

Wenjia Jing ^1, and Olivier Pinaud ^2,

1.	Yau Mathematical Sciences Center, Tsinghua University, No.1 Tsinghua Yuan, Beijing 100084, China
2.	Department of Mathematics, Colorado State University, Fort Collins, CO 80525, USA

Received May 2018 Revised November 2018 Published October 2019 Early access April 2019

This work concerns the analysis of wave propagation in random media. Our medium of interest is sea ice, which is a composite of a pure ice background and randomly located inclusions of brine and air. From a pulse emitted by a source above the sea ice layer, the main objective of this work is to derive a model for the backscattered signal measured at the source/detector location. The problem is difficult in that, in the practical configuration we consider, the wave impinges on the layer with a non-normal incidence. Since the sea ice is seen by the pulse as an effective (homogenized) medium, the energy is specularly reflected and the backscattered signal vanishes in a first order approximation. What is measured at the detector consists therefore of corrections to leading order terms, and we focus in this work on the homogenization corrector. We describe the propagation by a random Helmholtz equation, and derive an expression of the corrector in this layered framework. We moreover obtain a transport model for quadratic quantities in the random wavefield in a high frequency limit.

Keywords: Sea ice, random Helmholtz equation, homogenization, correctors.

Mathematics Subject Classification: 35R30, 60H15, 35Q60.

Citation: Wenjia Jing, Olivier Pinaud. A backscattering model based on corrector theory of homogenization for the random Helmholtz equation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5377-5407. doi: 10.3934/dcdsb.2019063

References:

[1]	S. Armstrong, T. Kuusi and J.-C. Mourrat, The additive structure of elliptic homogenization, Invent. Math., 208 (2017), 999-1154. doi: 10.1007/s00222-016-0702-4.
[2]	G. Bal, J. B. Keller, G. Papanicolaou and L. Ryzhik, Transport theory for waves with reflection and transmission at interfaces, Wave Motion, 30 (1999), 303-327. doi: 10.1016/S0165-2125(99)00018-9.
[3]	G. Bal and K. Ren, Transport-based imaging in random media, SIAM Applied Math., 68 (2008), 1738-1762. doi: 10.1137/070690122.
[4]	G. Bal, Central limits and homogenization in random media, Multiscale Model. Simul., 7 (2008), 677-702. doi: 10.1137/070709311.
[5]	G. Bal, J. Garnier, S. Motsch and V. Perrier, Random integrals and correctors in homogenization, Asymptot. Anal., 59 (2008), 1-26.
[6]	G. Bal and W. Jing, Fluctuations in the homogenization of semilinear equations with random potentials, Comm. Partial Differential Equations, 41 (2016), 1839-1859. doi: 10.1080/03605302.2016.1238482.
[7]	A. Bensoussan, J.-L. Lions and G. C. Papanicolaou, Asymptotic analysis for periodic structures, In Studies in Mathematics and its Applications, 5, North-Holland Publishing Co., Amsterdam-New York, 1978.
[8]	E. Bolthausen, On the central limit theorem for stationary mixing random fields, Ann. Probab., 10 (1982), 1047-1050. doi: 10.1214/aop/1176993726.
[9]	F. Castella, The radiation condition at infinity for the high-frequency Helmholtz equation with source term: A wave-packet approach, J. Funct. Anal., 223 (2005), 204-257. doi: 10.1016/j.jfa.2004.08.008.
[10]	P. C. Y. Chang, J. G. Walker and K. I. Hopcraft, Ray tracing in absorbing media, Journal of Quantitative Spectroscopy & Radiative Transfer, 96 (2005), 327-341. doi: 10.1016/j.jqsrt.2005.01.001.
[11]	G. Ciraolo and R. Magnanini, A radiation condition for uniqueness in a wave propagation problem for 2-D open waveguides, Mathematical Methods in the Applied Sciences, 32 (2009), 1183-1206. doi: 10.1002/mma.1084.
[12]	W. Dierking, Sea ice monitoring by synthetic aperture radar, Oceanography, 26 (2013), 100-111. doi: 10.5670/oceanog.2013.33.
[13]	R. Figari, E. Orlandi and G. Papanicolaou, Mean field and Gaussian approximation for partial differential equations with random coefficients, SIAM J. Appl. Math., 42 (1982), 1069-1077. doi: 10.1137/0142074.
[14]	E. Fouassier, High frequency limit of Helmholtz equations: Refraction by sharp interfaces, J. Math. Pures Appl. (9), 87 (2007), 144-192. doi: 10.1016/j.matpur.2006.11.002.
[15]	P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), 323-379. doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO; 2-C.
[16]	A. Gloria, S. Neukamm and F. Otto, Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics, Invent. Math., 199 (2015), 455-515. doi: 10.1007/s00222-014-0518-z.
[17]	K. M. Golden, M. Cheney, K. H. Ding, A. K. Fung, T. C. Grenfell, D. Isaacson, J. A. Kong, S. V. Nghiem, J. Sylvester and D. P. Winebrenner, Forward electromagnetic scattering models for sea ice, IEEE Transactions on Geoscience and Remote Sensing, 36 (1998), 1655-1674. doi: 10.1109/36.718637.
[18]	Y. Gu and G. Bal, Random homogenization and convergence to integrals with respect to the Rosenblatt process, J. Differential Equations, 253 (2012), 1069-1087. doi: 10.1016/j.jde.2012.05.007.
[19]	M. Hairer, E. Pardoux and A. Piatnitski, Random homogenisation of a highly oscillatory singular potential, Stoch. Partial Differ. Equ. Anal. Comput., 1 (2013), 571-605. doi: 10.1007/s40072-013-0018-y.
[20]	B. Holt, P. Kanagaratnam, S. P. Gogineni, V. C. Ramasami, A. Mahoney and V. Lytle, Sea ice thickness measurements by ultrawideband penetrating radar: First results, Cold Regions Science and Technology, 55 (2009), 33-46. doi: 10.1016/j.coldregions.2008.04.007.
[21]	W. Jing, Limiting distribution of elliptic homogenization error with periodic diffusion and random potential, Anal. PDE, 9 (2016), 193-228. doi: 10.2140/apde.2016.9.193.
[22]	D. Khoshnevisan, Multiparameter Processes, Springer Monographs in Mathematics. Springer-Verlag, New York, 2002. An introduction to random fields. doi: 10.1007/b97363.
[23]	S. M. Kozlov, Averaging of differential operators with almost periodic rapidly oscillating coefficients, Mat. Sb. (N.S.), 107 (1978), 199-217,317.
[24]	P.-L. Lions and T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana, 9 (1993), 553-618. doi: 10.4171/RMI/143.
[25]	L. Miller, Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary, J. Math. Pures Appl. (9), 79 (2000), 227-269. doi: 10.1016/S0021-7824(00)00158-6.
[26]	J. Nevard and J. B. Keller, Homogenization of rough boundaries and interfaces, SIAM J. Appl. Math., 57 (1997), 1660-1686. doi: 10.1137/S0036139995291088.
[27]	G. C. Papanicolaou and S. R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, In Random Fields, Vol. I, II (Esztergom, 1979), volume 27 of Colloq. Math. Soc. János Bolyai, pages 835-873. North-Holland, Amsterdam, 1981.
[28]	K. R. Parthasarathy, Probability Measures on Metric Spaces, Probability and Mathematical Statistics, No. 3. Academic Press, Inc., New York-London, 1967.
[29]	L. Ryzhik, G. Papanicolaou and J. B. Keller, Transport equations for waves in a half space, Comm. PDE's, 22 (1997), 1869-1910. doi: 10.1080/03605309708821324.
[30]	M. Shokr and N. Sinha, Sea Ice, Physics and Remote Sensing, Wiley, 2015.
[31]	M. R. Vant, R. O. Ramseier and V. Makios, The complex dielectric constant of sea ice at frequencies in the range 0.1-40 GHz, Journal of Applied Physics, 49 (1978), 1264-1280. doi: 10.1063/1.325018.

show all references

References:

[1]	S. Armstrong, T. Kuusi and J.-C. Mourrat, The additive structure of elliptic homogenization, Invent. Math., 208 (2017), 999-1154. doi: 10.1007/s00222-016-0702-4.
[2]	G. Bal, J. B. Keller, G. Papanicolaou and L. Ryzhik, Transport theory for waves with reflection and transmission at interfaces, Wave Motion, 30 (1999), 303-327. doi: 10.1016/S0165-2125(99)00018-9.
[3]	G. Bal and K. Ren, Transport-based imaging in random media, SIAM Applied Math., 68 (2008), 1738-1762. doi: 10.1137/070690122.
[4]	G. Bal, Central limits and homogenization in random media, Multiscale Model. Simul., 7 (2008), 677-702. doi: 10.1137/070709311.
[5]	G. Bal, J. Garnier, S. Motsch and V. Perrier, Random integrals and correctors in homogenization, Asymptot. Anal., 59 (2008), 1-26.
[6]	G. Bal and W. Jing, Fluctuations in the homogenization of semilinear equations with random potentials, Comm. Partial Differential Equations, 41 (2016), 1839-1859. doi: 10.1080/03605302.2016.1238482.
[7]	A. Bensoussan, J.-L. Lions and G. C. Papanicolaou, Asymptotic analysis for periodic structures, In Studies in Mathematics and its Applications, 5, North-Holland Publishing Co., Amsterdam-New York, 1978.
[8]	E. Bolthausen, On the central limit theorem for stationary mixing random fields, Ann. Probab., 10 (1982), 1047-1050. doi: 10.1214/aop/1176993726.
[9]	F. Castella, The radiation condition at infinity for the high-frequency Helmholtz equation with source term: A wave-packet approach, J. Funct. Anal., 223 (2005), 204-257. doi: 10.1016/j.jfa.2004.08.008.
[10]	P. C. Y. Chang, J. G. Walker and K. I. Hopcraft, Ray tracing in absorbing media, Journal of Quantitative Spectroscopy & Radiative Transfer, 96 (2005), 327-341. doi: 10.1016/j.jqsrt.2005.01.001.
[11]	G. Ciraolo and R. Magnanini, A radiation condition for uniqueness in a wave propagation problem for 2-D open waveguides, Mathematical Methods in the Applied Sciences, 32 (2009), 1183-1206. doi: 10.1002/mma.1084.
[12]	W. Dierking, Sea ice monitoring by synthetic aperture radar, Oceanography, 26 (2013), 100-111. doi: 10.5670/oceanog.2013.33.
[13]	R. Figari, E. Orlandi and G. Papanicolaou, Mean field and Gaussian approximation for partial differential equations with random coefficients, SIAM J. Appl. Math., 42 (1982), 1069-1077. doi: 10.1137/0142074.
[14]	E. Fouassier, High frequency limit of Helmholtz equations: Refraction by sharp interfaces, J. Math. Pures Appl. (9), 87 (2007), 144-192. doi: 10.1016/j.matpur.2006.11.002.
[15]	P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), 323-379. doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO; 2-C.
[16]	A. Gloria, S. Neukamm and F. Otto, Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics, Invent. Math., 199 (2015), 455-515. doi: 10.1007/s00222-014-0518-z.
[17]	K. M. Golden, M. Cheney, K. H. Ding, A. K. Fung, T. C. Grenfell, D. Isaacson, J. A. Kong, S. V. Nghiem, J. Sylvester and D. P. Winebrenner, Forward electromagnetic scattering models for sea ice, IEEE Transactions on Geoscience and Remote Sensing, 36 (1998), 1655-1674. doi: 10.1109/36.718637.
[18]	Y. Gu and G. Bal, Random homogenization and convergence to integrals with respect to the Rosenblatt process, J. Differential Equations, 253 (2012), 1069-1087. doi: 10.1016/j.jde.2012.05.007.
[19]	M. Hairer, E. Pardoux and A. Piatnitski, Random homogenisation of a highly oscillatory singular potential, Stoch. Partial Differ. Equ. Anal. Comput., 1 (2013), 571-605. doi: 10.1007/s40072-013-0018-y.
[20]	B. Holt, P. Kanagaratnam, S. P. Gogineni, V. C. Ramasami, A. Mahoney and V. Lytle, Sea ice thickness measurements by ultrawideband penetrating radar: First results, Cold Regions Science and Technology, 55 (2009), 33-46. doi: 10.1016/j.coldregions.2008.04.007.
[21]	W. Jing, Limiting distribution of elliptic homogenization error with periodic diffusion and random potential, Anal. PDE, 9 (2016), 193-228. doi: 10.2140/apde.2016.9.193.
[22]	D. Khoshnevisan, Multiparameter Processes, Springer Monographs in Mathematics. Springer-Verlag, New York, 2002. An introduction to random fields. doi: 10.1007/b97363.
[23]	S. M. Kozlov, Averaging of differential operators with almost periodic rapidly oscillating coefficients, Mat. Sb. (N.S.), 107 (1978), 199-217,317.
[24]	P.-L. Lions and T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana, 9 (1993), 553-618. doi: 10.4171/RMI/143.
[25]	L. Miller, Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary, J. Math. Pures Appl. (9), 79 (2000), 227-269. doi: 10.1016/S0021-7824(00)00158-6.
[26]	J. Nevard and J. B. Keller, Homogenization of rough boundaries and interfaces, SIAM J. Appl. Math., 57 (1997), 1660-1686. doi: 10.1137/S0036139995291088.
[27]	G. C. Papanicolaou and S. R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, In Random Fields, Vol. I, II (Esztergom, 1979), volume 27 of Colloq. Math. Soc. János Bolyai, pages 835-873. North-Holland, Amsterdam, 1981.
[28]	K. R. Parthasarathy, Probability Measures on Metric Spaces, Probability and Mathematical Statistics, No. 3. Academic Press, Inc., New York-London, 1967.
[29]	L. Ryzhik, G. Papanicolaou and J. B. Keller, Transport equations for waves in a half space, Comm. PDE's, 22 (1997), 1869-1910. doi: 10.1080/03605309708821324.
[30]	M. Shokr and N. Sinha, Sea Ice, Physics and Remote Sensing, Wiley, 2015.
[31]	M. R. Vant, R. O. Ramseier and V. Makios, The complex dielectric constant of sea ice at frequencies in the range 0.1-40 GHz, Journal of Applied Physics, 49 (1978), 1264-1280. doi: 10.1063/1.325018.

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