# Uniqueness and direct imaging method for inverse scattering by locally rough surfaces with phaseless near-field data

###### Abstract

This paper is concerned with inverse scattering of plane waves by a locally perturbed infinite plane (which is called a locally rough surface) with the modulus of the total-field data (also called the phaseless near-field data) at a fixed frequency in two dimensions. We consider the case where a Dirichlet boundary condition is imposed on the locally rough surface. This problem models inverse scattering of plane acoustic waves by a one-dimensional sound-soft, locally rough surface; it also models inverse scattering of plane electromagnetic waves by a locally perturbed, perfectly reflecting, infinite plane in the TE polarization case. We prove that the locally rough surface is uniquely determined by the phaseless near-field data generated by a countably infinite number of plane waves and measured on an open domain above the locally rough surface. Further, a direct imaging method is proposed to reconstruct the locally rough surface from the phaseless near-field data generated by plane waves and measured on the upper part of the circle with a sufficiently large radius. Theoretical analysis of the imaging algorithm is derived by making use of properties of the scattering solution and results from the theory of oscillatory integrals (especially the method of stationary phase). Moreover, as a by-product of the theoretical analysis, a similar direct imaging method with full far-field data is also proposed to reconstruct the locally rough surface. Finally, numerical experiments are carried out to demonstrate that the imaging algorithm with phaseless near-field data and full far-field data are fast, accurate and very robust with respect to noise in the data.

Key words. Inverse scattering, locally rough surface, Dirichlet boundary condition, phaseless near-field data, full far-field data.

AMS subject classifications. 35R30, 35Q60, 65R20, 65N21, 78A46

## 1 Introduction

Acoustic and electromagnetic scattering by a locally perturbed infinite plane (called a locally rough surface in this paper) occurs in many applications such as radar, remote sensing, geophysics, medical imaging and nondestructive testing (see, e.g., [3, 5, 8, 14, 11, 20]).

In this paper, we are restricted to the two-dimensional case by assuming that the local perturbation is invariant in the direction. Assume further that the incident wave is time-harmonic ( time dependence), so that the total wave field satisfies the Helmholtz equation

\hb@xt@.01(1.1) |

Here, is the wave number, and are the frequency and speed of the wave in , respectively, and represents a homogeneous medium above the locally rough surface denoted by with having a compact support in . In this paper, the incident field is assumed to be the plane wave

\hb@xt@.01(1.2) |

where is the incident direction with and is the lower part of the unit circle . This paper considers the case where a Dirichlet boundary condition is imposed on the locally rough surface. Thus, the total field vanishes on the surface :

\hb@xt@.01(1.3) |

where is the reflected wave by the infinite plane :

\hb@xt@.01(1.4) |

with and is the unknown scattered wave to be determined which is required to satisfy the Sommerfeld radiation condition

\hb@xt@.01(1.5) |

This problem models electromagnetic scattering by a locally perturbed, perfectly conducting, infinite plane in the TE polarization case; it also models acoustic scattering by a one-dimensional sound-soft, locally rough surface. See FIG. LABEL:fig6 for the geometry of the scattering problem.

The well-posedness of the scattering problem (LABEL:eq1)-(LABEL:rc) has been studied by using the variational method with a Dirichlet-to-Neumann (DtN) map in [5] or the integral equation method in [54, 58]. In particular, it was proved in [54, 58] that has the following asymptotic behavior at infinity:

\hb@xt@.01(1.6) |

uniformly for all observation directions with the upper part of the unit circle , where is called the far-field pattern of the scattered field , depending on the observation direction and the incident direction .

Many numerical algorithms have been proposed for the inverse problem of reconstructing the rough surfaces from the phased near-field or far-field data (see, e.g., [5, 8, 13, 18, 20, 21, 22, 35, 39, 40, 51, 58] and the references quoted there). For the case when the local perturbation is below the infinite plane which is called the inverse cavity problem, see [3, 38] and the reference quoted there.

In diffractive optics and radar imaging, it is much harder to obtain data with accurate phase information compared with only measuring the intensity (or the modulus) of the data [4, 6, 9, 14, 16, 25, 34, 49]. Thus it is often desirable to study inverse scattering problems with phaseless data. Inverse scattering with phaseless near-field data has been extensively studied numerically over the past decades (see, e.g., [4, 7, 9, 15, 16, 17, 25, 45, 49, 52] and the references quoted there). Recently, mathematical issues including uniqueness and stability have also been studied for inverse scattering with phaseless near-field data (see, e.g., [30, 31, 32, 33, 44, 46, 47] and the references quoted there).

In contrast to the case with phaseless near-field data, inverse scattering with phaseless far-field data is much less studied both mathematically and numerically due to the translation invariance property of the phaseless far-field data, that is, the modulus of the far-field pattern is invariant under translations of the obstacle for plane wave incidence [34, 41, 59]. The translation invariance property makes it impossible to reconstruct the location of the obstacle or the inhomogeneous medium from the phaseless far-field pattern with one plane wave as the incident field. Nevertheless, several reconstruction algorithms have been developed to reconstruct the shape of the obstacle from the phaseless far-field data with one plane wave as the incident field (see [1, 26, 27, 28, 34, 36, 37, 50]). Uniqueness has also been established in recovering the shape of the obstacle from the phaseless far-field data with one plane wave as the incident field [42, 43]. Recently, progress has been made on the mathematical and numerical study of inverse scattering with phaseless far-field data. For example, it was first proved in [59] that the translation invariance property of the phaseless far-field pattern can be broken by using superpositions of two plane waves as the incident fields for all wave numbers in a finite interval. And a recursive Newton-type iteration algorithm in frequencies was further developed in [59] to numerically reconstruct both the location and the shape of the obstacle simultaneously from multi-frequency phaseless far-field data. This method was further extended in [60] to reconstruct the locally rough surface from multi-frequency intensity-only far-field or near-field data. Furthermore, a direct imaging algorithm was recently developed in [61] to reconstruct the obstacle from the phaseless far-field data generated by infinitely many sets of superpositions of two plane waves as the incident fields at a fixed frequency. And uniqueness results have also been established rigorously in [55] for inverse obstacle and medium scattering from the phaseless far-field patterns generated by infinitely many sets of superpositions of two plane waves with different directions at a fixed frequency under certain a priori conditions on the obstacle and the inhomogeneous medium. The a priori assumption on the obstacle and the inhomogeneous medium in [55] was removed in [56] by adding a known reference ball into the scattering model. Note that the idea of adding a reference ball to the scattering system was recently used in [62] to prove uniqueness results for inverse scattering with phaseless far-field data generated by superpositions of a plane wave and a point source as the incident fields at a fixed frequency. Note further that, by adding one point scatterer into the scattering model stability estimates have been obtained in [29] for inverse obstacle and medium scattering with phaseless far-field data associated with one plane wave as the incident field under certain conditions on the obstacle and inhomogeneous medium if the point scatterer is placed far away from the scatterer. In addition, direct imaging algorithms are proposed in [29] to reconstruct the scattering obstacle from the phaseless far-field data associated with one plane wave as the incident field.

In this paper, we consider uniqueness and fast imaging algorithm for inverse scattering by locally rough surfaces from phaseless near-field data corresponding to incident plane waves at a fixed frequency. First, we prove that the locally rough surface is uniquely determined by the phaseless near-field data generated by a countably infinite number of incident plane waves and measured on an open domain above the locally rough surface, following the ideas in [58, 46]. Then we develop a direct imaging algorithm for the inverse scattering problem with phaseless near-field data generated by incident plane waves and measured on the upper part of the circle containing the local perturbation part of the infinite plane, based on the imaging function with (see the formula (LABEL:eq3) below). The theoretical analysis of the imaging function is given by making use of properties of the scattering solution and results from the theory of oscillatory integrals (especially the method of stationary phase). From the theoretical analysis result, it is expected that if the radius of the measurement circle is sufficiently large, will take a large value when is on the boundary and decay as moves away from . Based on this, a direct imaging algorithm is proposed to recover the locally rough surface from the phaseless near-field data. Further, numerical experiments are also carried out to demonstrate that our imaging algorithm provides an accurate, fast and stable reconstruction of the locally rough surface. Moreover, as a by-product of the theoretical analysis, a similar direct imaging algorithm with full far-field data is also proposed to reconstruct the locally rough surfaces with convincing numerical experiments illustrating the effectiveness of the imaging algorithm. It should be pointed out that a direct imaging method was recently proposed in [15, 16] for reconstructing extended obstacles with acoustic and electromagnetic phaseless near-field data, based on the reverse time migration technique

The remaining part of the paper is organized as follows. The uniqueness result is proved in Section LABEL:sec2 for an inverse scattering problem with phaseless near-field data. In Section LABEL:sec3, the direct imaging method with phaseless near-field data is proposed, and its theoretical analysis is given. As a by-product, the direct imaging method with full far-field data is also presented in Section LABEL:sec3. Numerical experiments are carried out in Section LABEL:sec4 to illustrate the effectiveness of the imaging method. Conclusions are given in Section LABEL:conclusion. In Appendix A, we use the method of stationary phase to prove Lemma LABEL:lem5 in Section LABEL:sec3 which plays an important role in the theoretical analysis of the direct imaging method.

We conclude this section with introducing some notations used throughout this paper. Define to be a disk centered at the origin and with radius large enough so that the local perturbation . Define , . For any and , set and let be the reflection of with respect to the -axis. Further, let , and with . Note also that if then and . Throughout this paper, the positive constants , and may be different at different places.

## 2 Uniqueness for an inverse problem

In this section, we establish a uniqueness result for an inverse scattering problem with phaseless near-field data, motivated by [46]. To this end, assume that are two locally rough surfaces, where with having a compact support in , . Further, denote by the local perturbation of and by the domain above , . For suppose that the total field is given by , where is the scattered field corresponding to the locally rough surface with its far-field pattern . Moreover, let be large enough such that the local perturbation () and let be a bounded open domain above the locally rough surfaces and . See FIG. LABEL:fig7 for the geometry of the inverse scattering problem.

We need the following result on the property of the scattered field which is also useful in the numerical algorithm in Section LABEL:sec3.

###### Lemma 2.1

Let . Then for any with large enough and the scattering solution of the scattering problem (LABEL:eq1)-(LABEL:rc) has the asymptotic behavior

\hb@xt@.01(2.1) |

with

\hb@xt@.01(2.2) | |||||

\hb@xt@.01(2.3) |

where is a constant independent of and .

Proof. The statement of this lemma follows easily from the well-posedness of the scattering problem (LABEL:eq1)-(LABEL:rc) and the asymptotic behavior (LABEL:eq96) of the scattered field (see, e.g., [58]).

We also need the following uniqueness result for the inverse scattering problem with full far-field data which is given in [58].

###### Theorem 2.2 (Theorem 4.1 in [58])

Assume that and are two locally rough surfaces and and are the far-field patterns corresponding to and , respectively. If for all and the distinct directions with and a fixed wave number , then .

We are now ready to state and prove the main theorem of this section.

###### Theorem 2.3

Assume that and are two locally rough surfaces and and are the total field corresponding to and , respectively. Let be a bounded open domain above and . If for all and the distinct directions with and a fixed wave number , then .

Proof. Fix for an arbitrary and set . Since for all , it follows from the analyticity of , , with respect to that

\hb@xt@.01(2.4) |

Noting that , , we have

\hb@xt@.01(2.5) |

Now, by Lemma LABEL:le3 we know that for ,

\hb@xt@.01(2.6) |

with

\hb@xt@.01(2.7) |

for large enough.

Write

\hb@xt@.01(2.8) |

where are real-valued functions with and . Then, by inserting (LABEL:eq28) and (LABEL:eq30) into (LABEL:eq31) we obtain that for ,

This yields

\hb@xt@.01(2.9) |

where is given by

Further, by (LABEL:eq29) we see that for ,

\hb@xt@.01(2.10) |

Substituting (LABEL:eq32) and (LABEL:eq33) into (LABEL:eq97) gives that for ,

\hb@xt@.01(2.11) |

Thus, and by (LABEL:eq34) we have that for ,

\hb@xt@.01(2.12) |

Arbitrarily fix and set and . The equation (LABEL:eq36) then becomes

\hb@xt@.01(2.13) |

Note that since and . Then we can choose such that

\hb@xt@.01(2.14) | |||

\hb@xt@.01(2.15) |

We now prove that

\hb@xt@.01(2.16) |

where we write , , for simplicity. We distinguish between the following two cases.

Case 1. is a rational number. In this case, it is easily seen that there exist with such that and . For let . Then it is easy to see that for large and . Thus, take with large in (LABEL:eq37) to obtain that

The required equality (LABEL:eq40) then follows by taking in the above equation and using (LABEL:eq38) and (LABEL:eq39).

Case 2. is an irrational number. In this case, by Kronecker’s approximation theorem (see, e.g., [2, Theorem 7.7]), we know that there exist with such that with , and . For let be defined as in Case 1. Then, similarly as in Case 1, take with large in (LABEL:eq37) to deduce that

Thus, (LABEL:eq40) also follows by letting in the above equation and using (LABEL:eq38) and (LABEL:eq39).

Finally, it follows from (LABEL:eq40) and the arbitrariness of that

for all and with . Condition (LABEL:eq41) means that the determinant of the square matrix on the left of the above matrix equation does not vanish, and so the above matrix equation only has a trivial solution, that is,

for all and with . This implies that for all and with . The required result then follows from Theorem LABEL:thm-uni-full. The proof is thus completed.

## 3 Direct imaging method for inverse problems

In this section, we consider the inverse problem: Given the incident field , to reconstruct the locally rough surface from the phaseless near-field data for all and with a fixed wave number . See FIG. LABEL:fig8 for the geometry of the inverse scattering problem. Our purpose is to develop a direct imaging method to solve this inverse problem numerically though no rigorous uniqueness result is available yet for the inverse problem.

We consider the imaging function

\hb@xt@.01(3.1) |

for . In what follows, we will study the behavior of this imaging function.

Define

\hb@xt@.01(3.2) | |||||

\hb@xt@.01(3.3) |

where

\hb@xt@.01(3.4) | |||||

\hb@xt@.01(3.5) | |||||

\hb@xt@.01(3.6) |

and

Since and , by a direct calculation (LABEL:eq3) becomes

\hb@xt@.01(3.7) |

We need the following result for oscillatory integrals proved in [15].

###### Lemma 3.1 (Lemma 3.9 in [15])

For any let be real-valued and satisfy that for all . Assume that is a division of such that is monotone in each interval , . Then for any function defined on with integrable derivative and for any ,

With the aid of Lemma LABEL:le1, we can obtain the following lemma.

###### Lemma 3.2

Let . For assume that and define

Then for all with large enough we have

\hb@xt@.01(3.8) | |||

\hb@xt@.01(3.9) |

where is a constant independent of .

Proof. We only prove (LABEL:eq80). The proof of (LABEL:eq81) is similar.

Let be small enough such that and let be large enough. Let , with , , and define for and . Then it follows that

\hb@xt@.01(3.10) |

and

\hb@xt@.01(3.11) |

We distinguish between the following two cases.

Case 1. . In this case, we rewrite (LABEL:eq8) as

\hb@xt@.01(3.12) | |||||

Set