A note on analyticity properties of far field patterns

Previous Article
A geometry guided image denoising scheme
IPI Home
This Issue
Next Article
Source identification from line integral measurements and simple atmospheric models

May 2013, 7(2): 491-498. doi: 10.3934/ipi.2013.7.491

A note on analyticity properties of far field patterns

Roland Griesmaier ^1, , Nuutti Hyvönen ^2, and Otto Seiskari ^3,

1.	Mathematisches Institut, Universität Leipzig, 04009 Leipzig, Germany
2.	Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, FI-00076 Aalto
3.	Aalto University, Department of Mathematics and Systems Analysis, FI-00076 Aalto, Finland

Received October 2012 Revised December 2012 Published May 2013

Abstract
Related Papers

In scattering theory the far field pattern describes the directional dependence of a time-harmonic wave scattered by an obstacle or inhomogeneous medium, when observed sufficiently far away from these objects. Considering plane wave excitations, the far field pattern can be written as a function of two variables, namely the direction of propagation of the incident plane wave and the observation direction, and it is well-known to be separately real analytic with respect to each of them. We show that the far field pattern is in fact a jointly real analytic function of these two variables.

Keywords: partial data., Far field pattern, inverse scattering, analytic functions.

Mathematics Subject Classification: 35B30, 35R30, 35Q6.

Citation: Roland Griesmaier, Nuutti Hyvönen, Otto Seiskari. A note on analyticity properties of far field patterns. Inverse Problems and Imaging, 2013, 7 (2) : 491-498. doi: 10.3934/ipi.2013.7.491

References:

[1]	F. E. Browder, Real analytic functions on product spaces and separate analyticity, Canad. J. Math., 13 (1961), 650-656. doi: 10.4153/CJM-1961-054-1.
[2]	A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl., 16 (2008), 19-33. doi: 10.1515/jiip.2008.002.
[3]	D. L. Colton and R. Kress, "Integral Equation Methods in Scattering Theory," John Wiley & Sons, New York, 1983.
[4]	D. L. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," $2^{nd}$ edition, Springer-Verlag, Berlin, 1998.
[5]	F. Hartogs, Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veränderlichen fortschreiten, Math. Ann., 62 (1906), 1-88. doi: 10.1007/BF01448415.
[6]	L. Hörmander, "An Introduction to Complex Analysis in Several Variables," $3^{rd}$ edition, North-Holland, Amsterdam, 1990.
[7]	N. Hyvönen, P. Piiroinen and O. Seiskari, Point measurements for a Neumann-to-Dirichlet map and the Calderón problem in the plane, SIAM J. Math. Anal., 44 (2012), 3526-3536. doi: 10.1137/120872164.
[8]	A. Kirsch, The MUSIC algorithm and the factorization method in inverse scattering theory for inhomogeneous media, Inverse Problems, 18 (2002), 1025-1040. doi: 10.1088/0266-5611/18/4/306.
[9]	A. Kirsch, "An Introduction to the Mathematical Theory of Inverse Problems," $2^{nd}$ edition, Springer-Verlag, New York, 2011. doi: 10.1007/978-1-4419-8474-6.
[10]	A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems," Oxford University Press, Oxford, 2008.
[11]	S. G. Krantz, "Function Theory of Several Complex Variables," $2^{nd}$ edition, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992.
[12]	J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications," I, Springer-Verlag, New York, 1972.
[13]	A. I. Nachman, Reconstructions from boundary measurements, Ann. of Math., 128 (1988), 531-576. doi: 10.2307/1971435.
[14]	R. G. Novikov, A multidimensional inverse spectral problem for the equation $-\Delta \psi + (v(x)-Eu(x))\psi=0$, translation in Funct. Anal. Appl., 22 (1988), 263-272. doi: 10.1007/BF01077418.
[15]	A. G. Ramm, Recovery of the potential from fixed-energy scattering data, Inverse Problems, 4 (1988), 877-886. doi: 10.1088/0266-5611/4/3/020.

show all references

References:

[1]	F. E. Browder, Real analytic functions on product spaces and separate analyticity, Canad. J. Math., 13 (1961), 650-656. doi: 10.4153/CJM-1961-054-1.
[2]	A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl., 16 (2008), 19-33. doi: 10.1515/jiip.2008.002.
[3]	D. L. Colton and R. Kress, "Integral Equation Methods in Scattering Theory," John Wiley & Sons, New York, 1983.
[4]	D. L. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," $2^{nd}$ edition, Springer-Verlag, Berlin, 1998.
[5]	F. Hartogs, Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veränderlichen fortschreiten, Math. Ann., 62 (1906), 1-88. doi: 10.1007/BF01448415.
[6]	L. Hörmander, "An Introduction to Complex Analysis in Several Variables," $3^{rd}$ edition, North-Holland, Amsterdam, 1990.
[7]	N. Hyvönen, P. Piiroinen and O. Seiskari, Point measurements for a Neumann-to-Dirichlet map and the Calderón problem in the plane, SIAM J. Math. Anal., 44 (2012), 3526-3536. doi: 10.1137/120872164.
[8]	A. Kirsch, The MUSIC algorithm and the factorization method in inverse scattering theory for inhomogeneous media, Inverse Problems, 18 (2002), 1025-1040. doi: 10.1088/0266-5611/18/4/306.
[9]	A. Kirsch, "An Introduction to the Mathematical Theory of Inverse Problems," $2^{nd}$ edition, Springer-Verlag, New York, 2011. doi: 10.1007/978-1-4419-8474-6.
[10]	A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems," Oxford University Press, Oxford, 2008.
[11]	S. G. Krantz, "Function Theory of Several Complex Variables," $2^{nd}$ edition, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992.
[12]	J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications," I, Springer-Verlag, New York, 1972.
[13]	A. I. Nachman, Reconstructions from boundary measurements, Ann. of Math., 128 (1988), 531-576. doi: 10.2307/1971435.
[14]	R. G. Novikov, A multidimensional inverse spectral problem for the equation $-\Delta \psi + (v(x)-Eu(x))\psi=0$, translation in Funct. Anal. Appl., 22 (1988), 263-272. doi: 10.1007/BF01077418.
[15]	A. G. Ramm, Recovery of the potential from fixed-energy scattering data, Inverse Problems, 4 (1988), 877-886. doi: 10.1088/0266-5611/4/3/020.

[1]	Huai-An Diao, Peijun Li, Xiaokai Yuan. Inverse elastic surface scattering with far-field data. Inverse Problems and Imaging, 2019, 13 (4) : 721-744. doi: 10.3934/ipi.2019033
[2]	Jingzhi Li, Jun Zou. A direct sampling method for inverse scattering using far-field data. Inverse Problems and Imaging, 2013, 7 (3) : 757-775. doi: 10.3934/ipi.2013.7.757
[3]	Olha Ivanyshyn. Shape reconstruction of acoustic obstacles from the modulus of the far field pattern. Inverse Problems and Imaging, 2007, 1 (4) : 609-622. doi: 10.3934/ipi.2007.1.609
[4]	Francesco Demontis, Cornelis Van der Mee. Novel formulation of inverse scattering and characterization of scattering data. Conference Publications, 2011, 2011 (Special) : 343-350. doi: 10.3934/proc.2011.2011.343
[5]	Xiaoxu Xu, Bo Zhang, Haiwen Zhang. Uniqueness in inverse acoustic and electromagnetic scattering with phaseless near-field data at a fixed frequency. Inverse Problems and Imaging, 2020, 14 (3) : 489-510. doi: 10.3934/ipi.2020023
[6]	Giovanni Alessandrini, Eva Sincich, Sergio Vessella. Stable determination of surface impedance on a rough obstacle by far field data. Inverse Problems and Imaging, 2013, 7 (2) : 341-351. doi: 10.3934/ipi.2013.7.341
[7]	Qi Wang, Yanren Hou. Determining an obstacle by far-field data measured at a few spots. Inverse Problems and Imaging, 2015, 9 (2) : 591-600. doi: 10.3934/ipi.2015.9.591
[8]	Xiaosheng Li, Gunther Uhlmann. Inverse problems with partial data in a slab. Inverse Problems and Imaging, 2010, 4 (3) : 449-462. doi: 10.3934/ipi.2010.4.449
[9]	Miklós Horváth. Spectral shift functions in the fixed energy inverse scattering. Inverse Problems and Imaging, 2011, 5 (4) : 843-858. doi: 10.3934/ipi.2011.5.843
[10]	Masaru Ikehata, Esa Niemi, Samuli Siltanen. Inverse obstacle scattering with limited-aperture data. Inverse Problems and Imaging, 2012, 6 (1) : 77-94. doi: 10.3934/ipi.2012.6.77
[11]	Hanming Zhou. Lens rigidity with partial data in the presence of a magnetic field. Inverse Problems and Imaging, 2018, 12 (6) : 1365-1387. doi: 10.3934/ipi.2018057
[12]	Heping Dong, Deyue Zhang, Yukun Guo. A reference ball based iterative algorithm for imaging acoustic obstacle from phaseless far-field data. Inverse Problems and Imaging, 2019, 13 (1) : 177-195. doi: 10.3934/ipi.2019010
[13]	Deyue Zhang, Yukun Guo, Fenglin Sun, Hongyu Liu. Unique determinations in inverse scattering problems with phaseless near-field measurements. Inverse Problems and Imaging, 2020, 14 (3) : 569-582. doi: 10.3934/ipi.2020026
[14]	Suman Kumar Sahoo, Manmohan Vashisth. A partial data inverse problem for the convection-diffusion equation. Inverse Problems and Imaging, 2020, 14 (1) : 53-75. doi: 10.3934/ipi.2019063
[15]	Francis J. Chung. Partial data for the Neumann-Dirichlet magnetic Schrödinger inverse problem. Inverse Problems and Imaging, 2014, 8 (4) : 959-989. doi: 10.3934/ipi.2014.8.959
[16]	Valter Pohjola. An inverse problem for the magnetic Schrödinger operator on a half space with partial data. Inverse Problems and Imaging, 2014, 8 (4) : 1169-1189. doi: 10.3934/ipi.2014.8.1169
[17]	Li Liang. Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data. Inverse Problems and Imaging, 2015, 9 (2) : 469-478. doi: 10.3934/ipi.2015.9.469
[18]	Fioralba Cakoni, Rainer Kress. Integral equations for inverse problems in corrosion detection from partial Cauchy data. Inverse Problems and Imaging, 2007, 1 (2) : 229-245. doi: 10.3934/ipi.2007.1.229
[19]	Soumen Senapati, Manmohan Vashisth. Stability estimate for a partial data inverse problem for the convection-diffusion equation. Evolution Equations and Control Theory, 2021 doi: 10.3934/eect.2021060
[20]	Guangsheng Wei, Hong-Kun Xu. On the missing bound state data of inverse spectral-scattering problems on the half-line. Inverse Problems and Imaging, 2015, 9 (1) : 239-255. doi: 10.3934/ipi.2015.9.239

2021 Impact Factor: 1.483

Tools

Metrics

Other articles
by authors

on AIMS
Roland Griesmaier Nuutti Hyvönen Otto Seiskari
on Google Scholar
Roland Griesmaier Nuutti Hyvönen Otto Seiskari

[Back to Top]