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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 & 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.  Google Scholar

[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.  Google Scholar

[3]

G. Bal and K. Ren, Transport-based imaging in random media, SIAM Applied Math., 68 (2008), 1738-1762.  doi: 10.1137/070690122.  Google Scholar

[4]

G. Bal, Central limits and homogenization in random media, Multiscale Model. Simul., 7 (2008), 677-702.  doi: 10.1137/070709311.  Google Scholar

[5]

G. Bal, J. Garnier, S. Motsch and V. Perrier, Random integrals and correctors in homogenization, Asymptot. Anal., 59 (2008), 1-26.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

E. Bolthausen, On the central limit theorem for stationary mixing random fields, Ann. Probab., 10 (1982), 1047-1050.  doi: 10.1214/aop/1176993726.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[12]

W. Dierking, Sea ice monitoring by synthetic aperture radar, Oceanography, 26 (2013), 100-111.  doi: 10.5670/oceanog.2013.33.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

D. Khoshnevisan, Multiparameter Processes, Springer Monographs in Mathematics. Springer-Verlag, New York, 2002. An introduction to random fields. doi: 10.1007/b97363.  Google Scholar

[23]

S. M. Kozlov, Averaging of differential operators with almost periodic rapidly oscillating coefficients, Mat. Sb. (N.S.), 107 (1978), 199-217,317.  Google Scholar

[24]

P.-L. Lions and T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana, 9 (1993), 553-618.  doi: 10.4171/RMI/143.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28] K. R. Parthasarathy, Probability Measures on Metric Spaces, Probability and Mathematical Statistics, No. 3. Academic Press, Inc., New York-London, 1967.   Google Scholar
[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.  Google Scholar

[30]

M. Shokr and N. Sinha, Sea Ice, Physics and Remote Sensing, Wiley, 2015. Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

G. Bal and K. Ren, Transport-based imaging in random media, SIAM Applied Math., 68 (2008), 1738-1762.  doi: 10.1137/070690122.  Google Scholar

[4]

G. Bal, Central limits and homogenization in random media, Multiscale Model. Simul., 7 (2008), 677-702.  doi: 10.1137/070709311.  Google Scholar

[5]

G. Bal, J. Garnier, S. Motsch and V. Perrier, Random integrals and correctors in homogenization, Asymptot. Anal., 59 (2008), 1-26.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

E. Bolthausen, On the central limit theorem for stationary mixing random fields, Ann. Probab., 10 (1982), 1047-1050.  doi: 10.1214/aop/1176993726.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

W. Dierking, Sea ice monitoring by synthetic aperture radar, Oceanography, 26 (2013), 100-111.  doi: 10.5670/oceanog.2013.33.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

D. Khoshnevisan, Multiparameter Processes, Springer Monographs in Mathematics. Springer-Verlag, New York, 2002. An introduction to random fields. doi: 10.1007/b97363.  Google Scholar

[23]

S. M. Kozlov, Averaging of differential operators with almost periodic rapidly oscillating coefficients, Mat. Sb. (N.S.), 107 (1978), 199-217,317.  Google Scholar

[24]

P.-L. Lions and T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana, 9 (1993), 553-618.  doi: 10.4171/RMI/143.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28] K. R. Parthasarathy, Probability Measures on Metric Spaces, Probability and Mathematical Statistics, No. 3. Academic Press, Inc., New York-London, 1967.   Google Scholar
[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.  Google Scholar

[30]

M. Shokr and N. Sinha, Sea Ice, Physics and Remote Sensing, Wiley, 2015. Google Scholar

[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.  Google Scholar

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