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Derivation and Discussion of the SAR Migration Algorithm Within Inverse Scattering Problem: Theoretical Analysis Lianlin Li, Wenji Zhang, and Fang Li
Abstract—The analysis of synthetic aperture radar (SAR) migration developed by Gilmore et al. has been refined within the context of the inverse scattering problem, particularly the distorted-wave Born approximation (DWBA). The SAR migration algorithm can be deduced from the DWBA-based inversion formulation when the following assumptions are satisfied: 1) homogeneous and nonfrequency-dependent background medium; 2) the exploding source model; and 3) the well-resolved targets described by an orthogonal relation derived in this paper. In addition, the other contributions of this paper are as follows: 1) The removal of the ω 2 term has been clarified by the derived orthogonal relation; 2) a scale factor that balances the near–far field has been derived; and 3) a novel SAR migration algorithm for the imaging of targets embedded in a layered medium has been proposed. Index Terms—Distorted-wave Born approximation (DWBA), exploding source model, inverse scattering, SAR migration algorithm, synthetic aperture radar (SAR), well-resolved targets, ultrawideband SAR.
I. I NTRODUCTION
I
N MANY disciplines throughout science and engineering, microwave imaging has become more attractive due to its high sensitivity to the dielectric properties of the targets, which is very useful for detecting of unknown objects in a noninvasive fashion. These applications include applied geophysics, biomedical and industrial diagnostics, or subsurface sensing. In the published literatures, one may find a class of techniques, which aims at quantitatively imaging the medium parameters, and other class of techniques in which the qualitative imaging of the reflectivity is the main goal. In the following, we refer to these techniques as “inversion” and “migration,” respectively. Up to now, synthetic aperture radar (SAR) and ground penetrating radar (GPR) are two typical imaging systems for nondestructive testing. At the same time, many excellent imaging algorithms, such as the range-Doppler algorithm, frequency–
Manuscript received March 5, 2008; revised August 24, 2008, January 6, 2009, and March 29, 2009. First published September 15, 2009; current version published December 23, 2009. This work was supported by the National Natural Science Foundation of China under Grants 60701010 and 40774093. The authors are with the Institute of Electronics, Chinese Academy of Sciences, Beijing 100190, China (e-mail:
[email protected]; fli@ mail.ie.ac.cn). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TGRS.2009.2024690
wavenumber migration algorithm, matched-filter migration, chirp-scaling migration, and so on, have also been developed to identify the unknown image profile (or object function; see, e.g., [1], [2], and the references therein). Most of these algorithms identify the unknown image profile (or object function) as the inverse Fourier transform (FT) of some composite function constructed from the received signals [3]. Only for the convenience of discussion, all of these algorithms are called as the SAR algorithm as follows. The SAR algorithm usually assumes some type of physical or/and signal model and derives some imaging formulations from this. In [3], some efforts have been made to clarify these assumptions and mathematical approximations underlying these algorithms. In this paper, this analysis has been refined within the context of inverse scattering problem, particularly the distorted-wave Born approximation (DWBA). Different from the SAR migration algorithm which can only qualitatively reconstruct the location and scattering intensity of the targets, linear inversion algorithms, particularly the socalled Born-based inversion algorithm and its variants, can realize quantitatively the reconstruction of weak scatterers. Moreover, the “qualitative” geometric reconstruction (i.e., position and approximate shape) can also be achieved even when the weak scattering approximation breaks down [7], [8]. Now, a question naturally arises: What is the relationship between Born-based inversion algorithm and SAR migration algorithm? Many efforts have been made to answer this question. For example, in [8], the SAR migration algorithm has been derived from Born-based inversion algorithm when the regularization factor of Born-based inversion algorithm is specified as an infinite value. However, this operation lacks physical meaning and theoretical basis. Therefore, this problem has been reconsidered in this paper. In [3], the similarities and differences of these algorithms are also delineated, and some interesting conclusions are drawn. Several issues are worthy to be carefully studied, for example: 1) the assumption of the incident wave in the form of spherical wave without decay; 2) the neglect of the 1/R term, although the authors claim that this neglect is usually justified because the phase term in the exponential part is much more important; and 3) the removal of the ω 2 term included in [3, eq. 15]. As pointed out by the authors, there is no reasonable physical explanation for this operation. The main goal of this paper is to review and clarify monostatic SAR migration algorithm within the context of DWBA for the configuration of multimonostatic/multifrequency radar.
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domain as u(y, ω) = p˜(ω)
ω σi exp −j2 (y − yi )2 + zi2 . c i=1
N
(2)
Second, taking the FT of (2) with respect to y, one can get u(ky , ω) = p˜(ω)
N
×
σi
i=1
ω exp −j2 (y − yi )2 + zi2 + jky y dy. c (3)
Fig. 1. Physical configuration of imaging problem (from [3]).
Furthermore, some suggestions and discussions are provided. The organization of this paper is as follows. In Section II, the basic SAR imaging algorithm is outlined, and the exploding source model is discussed. In Section III, the detailed derivation of the same equation as the SAR migration algorithm from DWBA-based inversion algorithm by gradually making assumptions has been presented. In Section IV, some conclusions are briefly summarized.
II. SAR M IGRATION A LGORITHM AND THE E XPLODING S OURCE M ODEL
By employing the approximation method for solving the high oscillating integral problem, i.e., the so-called method of stationary phase (MSP), one can derive
N ω 2 u(ky , ω) = p˜(ω) σi exp −jzi 4 − ky2 − jyi ky . c i=1 (4) It is noted that, in (4), a complex-amplitude term exp(−j(π/4))/ 4k 2 − ky2 has been considered a constant and suppressed, which is only reasonable when the signal band is narrow, and both the angular field of SAR view and the look-angle of SAR are small. Finally, the SAR imaging formulation can be obtained explicitly
A. Sar Migration Algorithm In this section, the signal model and principle of SAR migration algorithm given in ([1]–[3], particularly [3]) are briefly summarized, and the symbols are borrowed from those employed in [3]. Here, we consider the problem geometry shown in Fig. 1. The radar, with an omnidirectional radiation pattern, transmits a pulse p(t) from the location (0, y, 0). The object to be imaged consists of N point targets, modeled as the so-called “exploding sources” discussed in the next section, embedded in region D, and located at position (0, yi , zi ), each with an associated “reflectivity” σi i = 1, 2, . . . , N . Then, the received signal is modeled as
u(y, t) =
N i=1
σi p t − 2
(y − yi )2 + zi2 c
∧
g(y, z) =
N
σi δ(y − yi , z − zi )
i=1
2 1 u (ky , ω(kz )) exp(−jky y−jkz z) dky dkz = 2π p˜ (ω(kz )) (5) where kz = (2(ω/c))2 − ky2 . The practical and rapid implementation of (5) can be found in lots of references, for example, [1] and [2]. B. Exploding Source Model
(1)
where c is the light speed in the free space. It should be emphasized that the spatial distribution of the wavefield excited by the source and scattered from the objects is very ambiguous, although it is claimed that the factor 1/R2 has been ignored in [1]–[3], where R = (y − yi )2 + zi2 . However, as opposed to the claim made in literatures (see, e.g., [1]–[3]), we think this factor (at least 1/R) is not ignored indeed, which will be discussed in detail in the next section. First, after taking the FT of (1) with respect to the time variable t, one can obtain the received signal in the frequency
The use of exploding source model originates from the field of seismic imaging [4] and is used extensively in GPR and SAR imaging problems later [1]–[3], [5], [6]. The typical characteristics of exploding source model are [3]–[6] as follows. 1) It is independent of incident wave, i.e., frequency, intensity, and incident direction. 2) It “explodes” and propagates toward the receiver absolutely along the propagation path of the incident wave [4]. 3) No interaction occurs between these sources, which is the same as the assumption of the Born approximation [3]. 4) The velocity of propagation in the surrounding medium is replaced by half of its true value [3]–[6].
LI et al.: DERIVATION AND DISCUSSION OF THE SAR MIGRATION ALGORITHM
In summary, according to the suggestion proposed by Leuschen and Plumb, the mathematical statement of the exploding source model can be described by the following [5], [6]:
2
{GES (r, r ; k0 )} = GFS (r, r ; 2k0 )
417
istic of targets is also nonfrequency dependent, i.e., χ(r ; k) ≡ χ(r ). Consequently, (8) can be expressed as
(6)
where k0 is the free-space wavenumber, GES is the Green’s function for the exploding source model, GFS is the free-space Green’s function, and r and r are related to the field and source points, respectively. Here, we can find that a 1/R decay term has been neglected by employing the exploding source model. III. SAR A LGORITHM W ITHIN THE C ONTEXT OF I NVERSE S CATTERING P ROBLEM The formulation of SAR migration algorithm can be derived within the context of the so-called DWBA after some assumptions (or conditions) are made (or satisfied). The DWBAbased imaging approach is based on the idea of expressing the scattering potential as a sum of a known part and an unknown part which is assumed to be small perturbation around the first dominant term. Under the Born approximation, which entails the neglect of mutual interactions, the resulting frequency-domain scattered electric field for the so-called multimonostatic/multifrequency measurement configuration is [3], [7], [8] Esca (r; ks ) = ks2 G(r, r ; ks )Eb (r ; ks , r)χ(r ; ks ) dr ,
Esca (rl ; ks ) = A(s,l) (χ(r )) where
A(s,l) (·) =
where rl is the lth sampling position of radar (l = 1, 2, . . . , L). Assume that the investigation domain D has been uniformly discretized into N subgrids with an area of (or volume for three dimension) Δ, in which the contrast χ is constant. Now, we can obtain the following linear equations: [Esca ] = [A][χ] where
⎡ ⎤ Esca,(1,1) ⎢Esca,(2,1)⎥ ⎢ ⎥ ⎢ ⎥ .. ⎢ ⎥ . ⎥ [Esca ] :=⎢ ⎢ Esca,(l,s) ⎥ ⎢ ⎥ ⎢ ⎥ .. ⎣ ⎦ .
(7)
where the footnote D is the investigation region to be imaged, ks is the sth wavenumber in free space, r = (0, y, 0) is the source/observation position, and χ(r, k) is the well-known contrast function k 2 (r) −1 χ(r, k) = 2 kb (r) where k is the wavenumber of the unknown object, kb is the wavenumber of the background medium, S is the total number of employed frequencies, and G(r, r ; ks ) is the Green’s function of the background medium. The background field Eb (r ; ks , r) generated by the radar located at r with wavenumber ks is Eb (r ; ks , r) = p˜(ks )G(r , r; ks ),
r ∈ D
where p˜(·) is the frequency-domain waveform of excited source. By the use of the reciprocity theorem, (7) becomes (8) Esca (r; ks ) = ks2 p˜(ks )G2 (r, r ; ks )χ(r ; ks ) dr . D
A. First Assumption First, let us assume that the background medium is homogeneous and nonfrequency dependent, and the electric character-
dr p˜(ks )G2 (rl , r ; ks )(·)
D
D
s = 1, 2, . . . , S
ks2
(9)
Aˆ(l,s)
(10)
Esca,(l,s) = Esca(rl ;ks)
Esca,(L,S) LS×1 ⎡ ⎤ χ1 ⎢ χ2 ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ . ⎥ [χ] :=⎢ ⎥ ⎢ χn ⎥ ⎢ . ⎥ ⎣ . ⎦ . χN N ×1 ⎡ˆ A(1,1),1 · · · Aˆ(1,1),n ⎢Aˆ(2,1),1 · · · Aˆ(2,1),n ⎢ ⎢ . .. ⎢ .. ··· . [A] =⎢ ⎢ Aˆ ˆ(l,s),n · · · A (l,s),1 ⎢ ⎢ . .. ⎣ .. ··· . Aˆ(L,S),1 · · · Aˆ(L,S),n
χn = χ(rn )
··· ··· ··· ··· ··· ···
⎤ Aˆ(1,1),N Aˆ(2,1),N ⎥ ⎥ ⎥ .. ⎥ . ⎥ ˆ A(l,s),N ⎥ ⎥ ⎥ .. ⎦ . ˆ A(L,S),N
LS×N
=Δks2 p˜(ks )G2 (rl , rn ; ks ). (s−1)L+l,n
rn ∈ D denotes the position of nth subgrid, where n = 1, 2, . . . , N , and N is the total number of discretized grids of investigation domain D. The imaging problem amounts to solving (10), which can be realized by carrying out the well-known least square technique. To this end, the norm equations of (10) should be first obtained explicitly [A]H [A][χ] = [A]H [Esca ].
(11)
Second, the solution of the obtained large-scale ill-posed problem should be solved by using some regularization technique.
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For example, through the using of the Tikhonov regularization, one has the well-known formulation of diffraction tomography for non-frequency-independent object −1 H · [A] [Esca ] [χ] = [A]H [A] + αI
(12)
where I is the unit operator, and α is a regularization factor. In [8], the Born-based imaging algorithm is derived from the Born-based inversion formulation by specifying α → ∞. Obviously, this manipulation lacks strict physical meaning and theoretical basis.
with (8), one can find that the scattered field Esca can be regarded as the expansion by these orthogonal functions. 3) The orthogonal relation is much similar to the ambiguous function (or the point spread function) of SAR system [1], [2]. Once (13) or (14) is assumed, the formulation of Born-based inversion formulation can be simplified into the Born-based migration algorithm, i.e., 1 2 2 · ks dks drp˜(ks )(G∗(rl , r ;ks )) Esca (rl ; ks ) χ(r ) = C(r ) Dobs
ks
(16) B. Second Assumption It is well known that the Born-based inversion algorithm is computationally consuming, manifested by the filling of the matrix and the inversion of the ill-posed matrix. In this section, an orthogonal relation is derived to simplify (11) and (12). In addition, by this orthogonal relation, the link between the Bornbased inversion algorithm and Born-based imaging algorithm can be reasonably constructed. From (11) and (12), it can be found that, if the orthogonal relation (or quasi-orthogonal relation) provided by (13) or (14) is justified, the matrix [Φ] := [A]H [A] is diagonal; consequently, the solution of (11) can be obtained immediately without the inverse operation of the ill-posed matrix. The required orthogonal relation is L S
ks4 |˜ p(ks )|2 |G(rl , r ; ks )G∗ (rl , r ; ks )|
C(r ) =
2
= C(r )δ(r − r ),
r , r ∈ D
⎡
⎤
(13)
or ⎢ ⎢ [Φ] = ⎢ ⎣
..
⎥ ⎥ ⎥ ⎦
. ..
.
S L
2 ks4 p˜(ks )G2 (rl , r ; ks ) .
s=1 l=1
s=1 l=1
λ1
where the summation operation has been replaced by the integral, and Dobs stands for the region where the data are collected, particularly the line z = 0 for the considered problem. As is well known, there is a so-called near–far field problem which means the intensity of the image becomes smaller when the imaging position goes further away from the radar. Consequently, as pointed by [5] and [6], a quadratic gain is needed to be applied to SAR migration algorithm. However, there lacks rigorous theoretical analysis to explain this issue. As a matter of fact, from the aforementioned discussion, it is shown that the gain needed to be added is not due to the spherical spreading pointed out in [5] and [6] but due to the norm of the orthogonal function ks2 p˜(ks )G2 (rl , r ; ks ), particularly
Therefore, factor C, called as near–far field balance factor in this paper, can play this role properly. In addition, if this factor is suppressed, (16) is absolutely identical to the formulation of the matched-filter migration algorithm [5], [6]. Remark 1: If the adjoint operator of A(s,l) is specified as
(14)
A(s,l)
H
=
L S
p˜∗ (ks ) (G∗ (rl , r ; ks ))
2
(17.1)
s=1 l=1
λN instead of
where
C(r ) =
L S
2 ks4 p˜(ks )G2 (rl , r ; ks ) ,
r ∈D
A(s,l)
H
=
S L
ks2 p˜∗ (ks ) (G∗ (rl , r ; ks ))
2
(17.2)
s=1 l=1
s=1 l=1
(15) λn =
C(rn ) . Δ
the formulations similar to (13)–(16) can be obtained explicitly L S
ks2 |˜ p(ks )|2 {G(rl , r ; ks )G∗ (rl , r ; ks )}
2
s=1 l=1
From (13) and (14), one can find three interesting conclusions. 1) If the objects to be imaged are well resolved for the considered SAR system, the orthogonal relation is (or approximately) justified [11]. Of course, no practical SAR system will satisfy (13) and (14) exactly; thus, the use of SAR imaging algorithm implies that an approximation has been made. 2) If (12) is justified, ks2 p˜(ks )G2 (rl , r ; ks ) constitutes a series of orthogonal functions; consequently, combining
= Cm (r )δ(r − r ), Cm (r ) =
S L
r , r ∈ D
2 ks2 p˜(ks )G2 (rl , r ; ks ) ,
(18) r ∈ D
s=1 l=1
χ(r ) =
1 2 · dk drl p˜∗(ks)(G∗(rl , r ;ks )) Esca(rl ; ks ), s C(r ) ks
Dobs
r ∈ D.
(19)
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exploding source model GES . As a result, (16) becomes χES(r ) =
1 2 2 ∗ · dk k ˜∗(k)[GES (rl ,r ;ks)] Esca(rl ;ks), s s dr l p C(r ) Dobs
ks
r ∈ D.
(20)
Furthermore, the use of (6) in (20) yields χES (r ) =
1 1 · C(r ) 4π
dks ks2 ks
∞ ×
dyl p˜∗ (ks )U (rl ; ks )
−∞
exp (j2ks |r − rl |) , |r − rl |
r ∈ D.
(21)
It is noted that, in [3], where the incident wavefield is assumed as a spherical wave without decay, (21) can also be derived. Obviously, this assumption is confusing and is not a good model in most cases. If the mathematical statement of exploding source model in [5] and [6] is adopted, i.e., (6), (21) can also be obtained. As a matter of fact, if the Green’s function for exploding source model GES is not introduced, (21) should be written as χ(r ) =
Fig. 2.
1 C(r )
1 4π
∞ exp (j2ks |r − rl |) 2 · dks ks dyl P ∗ (k)U (rl ; ks ) . |r − rl |2
Normalized results of (a) (13) and (b) (18).
−∞
ks
In order to justify the orthogonal relations in (13) and (18), a simple numerical simulation is provided here, shown in Fig. 2, where the x-axis stands for the position along the y-direction while the y-axis is the position along the z-direction. In this simulation, the frequency ranges from 10 to 22.4 GHz; the radar system collects the data from −0.1 to 0.1 m with a separation of 0.01 m. Fig. 2(a) shows the normalized result of (13), while Fig. 2(b) shows that of (18). From these results, the following are shown: 1) There is ambiguity near the radar and 2) the result of (18) is better than (13). Remark 2: As is mentioned earlier, the scattered field Esca can be regarded as the expansion by these orthogonal functions ks2 p˜(ks )G2 (rl , r ; ks ), which means that χ(r ) can be considered as a projection coefficient of the orthogonal component ks2 p˜(ks )G2 (rl , r ; ks ) of the scattered field Esca . Consequently, the reconstruction formulation equation (15) can be interpreted that the reconstruction of χ(r ) can be realized by judging the projection of scattered field on some component ks2 p˜(ks )G2 (rl , r ; ks ) of the scattered field.
C. Third Assumption If the assumption of exploding source model is made, the Green’s function G is replaced by the Green’s function for
2
Furthermore, if one of the terms 1/4π|r − rl | is ignored, the given equation can be reduced into χ(r ) =
1 1 C(r ) 4π ·
∞ dkks2
k
dyl P ∗ (k)U (rl ; ks )
−∞
exp (j2k|r − rl |) |r − rl |
which is identical to (21). In summary 1) If (6) is utilized, the assumption of the incident wavefield in the form of a spherical wave without decay can be avoided to obtain the formulation of SAR migration algorithm; however, it should be an open problem. 2) Comparing (21) with the back-propagation formulation used in ultrawideband SAR migration algorithm [9], one can readily find that they are identical if the term C is suppressed. By using the Weyl identical equation [12], in particular, we have (22), shown at the bottom of the next page, where kz = 4k 2 − kx2 − ky2 , and the integral is carried along the Somerfield integral path.
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Substituting (22) into (21) yields ∞ j 1 2 χES (r ) = ks dks dyl p˜∗ (ks )U (ks , rl ) C(r ) 8π 2 −∞
ks
∞ ∞ exp(−jkx x+jky (yl −y )−jkz z ) · dkx dky . kz
−∞ −∞
After some manipulations, one has ∞ j ˜ (ks , ky ) χES (r ) = dky dks ks2 · p˜∗ (ks )U C(r ) k −∞ exp −jky y −j 4ks2 −ky2 z × 4ks2 − ky2 (23) where the 2-D assumption has been made, and the definition of FT given by (24) is employed ˜ (k, xs = 0, ky , zl = 0) = 1 U 2π
∞ dyl U (k, rl ) exp(jky yl ). (24) −∞
It is noted that there is an important factor, called the near–far field balance factor in this paper ⎛ ⎞ ∞ drl 1 2⎠ 4 ⎝ C(r ) = ks |˜ p(ω)| . (25) 4π |r − rl |2 −∞
ks
If the near–far field balance factor is ignored, then (23) is identical to SAR migration algorithm. Here, as apposed to what was done in [3], the following should be pointed out: 1) For the derivation of (20), the spherical factor 1/R in (21) is not ignored and 2) the approximation MSP is not employed. Remark 3—Explanation About ks2 in the Integrand of (23): Similar to what is done in Remark 1, if the adjoint operator of A(s,l) is specified as (17), substituting (22) into (18) and (19) yields j χES (r ) = Cm (r )
∞ ks −∞
˜ (ks , ky ) dky dks p˜∗ (ks )U exp −jky y −j 4ks2 −ky2 z × 4ks2 −ky2 (26)
j exp (j2k|r − rl |) = |r − rl | 2π
=
j 2π
∞ ∞
together with
⎛ ⎞ ∞ drl 1 2 2 ⎝ ⎠ ks |˜ p(ks )| . Cm (r ) = 4π |r − rl |2 ks
Now, one can find that, if the orthogonal relation (18) and exploding source model are assumed, then (26) is identical to SAR migration algorithm without suppressing the factor ks2 . As a matter of fact, the orthogonal relation in (18) is better that in (13), which has been proven by lots of numerical simulations (see, e.g., Figs. 2 and 3). In order to justify the orthogonal relations (13) and (18) for the case of exploding source model, a simple numerical simulation is provided in Fig. 3, where the x-axis stands for the position along the y-direction while the y-axis for the
exp (jkx (xl − x ) + jky (yl − y ) + jkz |z − zl |) kz
dkx dky
exp (−jkx x − jkz z + jky (yl − y )) kz
∞ ∞
(27)
−∞
dkx dky −∞ −∞
−∞ −∞
Fig. 3. Normalized results of (a) (13) and (b) (18) for the case of exploding source model.
(22)
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position along the z-direction. All the computation conditions are identical to the example in Fig. 2. From these results, the following are shown: 1) There is no ambiguity near the radar and 2) the result of (18) is still better than (13). Remark 4—SAR Migration Algorithm for the Objects Embedded in a Layered Medium: Taking the 2-D two-layered medium, for example [10], along the same line before, we can obtain the SAR migration algorithm formulation for objects embedded in a layered medium as (28), shown at the bottom of the page, where Clayered (r ) =
ks2 |˜ p∗ (ks )|2
∞
d rl |G(rl , r ; 2ks )|
2
(29)
−∞
ks
where ka,b is the wavenumber of regions a and b, respectively; a and b stand for the upper space (free space) and lower space (lossy ground), respectively; and G(rs , r ; 2k) is the Green’s function for the layered medium. Similar to (26)–(29) can be modified as χES (r ) =
j Clayered,m (r ) ∞ ˜ (ks , ky , zl ) × dky dks p˜∗ (ks )U ks−∞
⎡ ⎤ exp −jky y +j 4ka2 −ky2 zl − 4kb2 −ky2 z ⎢ ⎥ ·⎣ ⎦ 2 2 2 2 4kb − ky + 4ka − ky (30)
Clayered,m (r ) ∞ 2 2 2 ∗ = ks |˜ p (ks )| d rl |G(rl , r ; 2ks )| .
(31)
further away from the radar. By employing the balance factor in (25), we can get a clearer image of both the near and far field targets, which are shown in Fig. 4(b).
−∞
ks
IV. C ONCLUSION
Equations (28)–(31) show that the SAR migration algorithm for objects embedded in the layered medium can be realized similar to the manner used in the SAR algorithm. Finally, an imaging result is presented to show the effectiveness of the derived near–far field balance factor. In the following simulation, the radar works in a monostatic measurement configuration and collects data from −2 to 2 m with a step of 0.1 m at a height of 0.3 m. The working frequency ranges from 0.5 to 1.5 GHz with a step of 0.1 GHz. The two identical square targets with a side of 4 cm are centered at (0, −0.1 m) and (0, −0.8 m), respectively. The unit for both the x- and y-axes in Fig. 4 is in meters. Fig. 4(a) shows the SAR migration result without the near–far field balance factor. From this figure, we can find that, even for two identical targets, the intensity of the image becomes smaller when the position of the target goes
χES (r ) =
Fig. 4. SAR migration result (a) without and (b) with the near–far field balance factor.
j Clayered (r )
∞ ks −∞
From the derivation process of SAR migration algorithm within the context of inverse scattering algorithm, we can find that the narrowband SAR migration algorithm can be deduced from DWBA-based inversion approach when the following conditions or assumptions are satisfied: 1) exploding sources model, where the Born approximation has been employed; 2) lossless and nondispersive background medium; 3) well-resolved targets; 4) neglect of the near–far field balance factor C; 2 5) neglect of the 1/ 4k − ky2 term. In the SAR migration algorithm, if the balance factor and the 1/ 4k 2 − ky2 term are not ignored, the imaging quality will be improved.
⎤ 4ka2 − ky2 zl − 4kb2 − ky2 z exp −jky y + j ⎥ ˜ (ks , ky , zl ) · ⎢ dky dks ks2 p˜∗ (ks )U ⎦ (28) ⎣ 4kb2 − ky2 + 4ka2 − ky2 ⎡
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IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 48, NO. 1, JANUARY 2010
As opposed to explanations in [3], from the aforementioned derivation of SAR migration algorithm within the context of inverse scattering problem, we can find that the factor 1/R2 (at least 1/R) is not ignored, and the approximation method for solving highly oscillating integral is also not needed. In addition, the neglect of ks2 is explained by the selection of orthogonal relation provided by (18).
[10] M. E. Peters, D. D. Blankeship, S. P. Carter, S. D. Kempf, D. A. Young, and J. W. Holt, “Along-track focusing of airborne radar sounding data from west Antarctica for improving basal reflection analysis and layer detection,” IEEE Trans. Geosci. Remote Sens., vol. 45, no. 9, pp. 2725– 2736, Sep. 2007. [11] A. J. Devaney, Super-resolution processing of multi-static data using time reversal and MUSIC, 2000. unpublished manuscript. [Online]. Available: www.ece.neu.edu/faculty/devaney [12] W. C. Chew, Waves and Fields in Inhomogeneous Media. Piscataway, NJ: IEEE Press, 1997, ch. 2.
ACKNOWLEDGMENT The authors would like to thank the reviewers for lots of constructive comments and suggestions. R EFERENCES [1] M. Soumekh, Synthetic Aperture Radar Signal Processing With MATLAB Algorithm. New York: Wiley, 1999. [2] I. G. Cumming and F. H. Wong, Digital Processing of Synthetic Aperture Radar Data. Norwood, MA: Artech House, 2005. [3] C. Gilmore, I. Jeffrey, and J. LoVetri, “Derivation and comparison of SAR and frequency–wavenumber migration within a common inverse scalar wave problem formulation,” IEEE Trans. Geosci. Remote Sens., vol. 44, no. 6, pp. 1454–1461, Jun. 2006. [4] D. Loewenthal, L. Lu, R. Roberson, and J. Sherwood, “The wave equation applied to migration,” Geophys. Prospect., vol. 24, no. 2, pp. 380–399, Jun. 1976. [5] C. J. Leuschen and R. G. Plumb, “A matched-filter-based reverse-time migration algorithm for ground-penetrating radar data,” IEEE Trans. Geosci. Remote Sens., vol. 39, no. 5, pp. 929–936, May 2001. [6] C. Leuschen and R. Plumb, “A matched-filter approach to wave migration,” J. Appl. Geophys., vol. 43, no. 2–4, pp. 271–280, Mar. 2000. [7] R. Solimene, F. Soldovieri, G. Prisco, and R. Pierri, “Three-dimensional microwave tomography by a 2-D slice-based reconstruction algorithm,” IEEE Geosci. Remote Sens. Lett., vol. 4, no. 4, pp. 556–560, Oct. 2007. [8] M. Oristagli and H. Blok, “Wavefield imaging and inversion in electromagnetic and acoustics,” Lectures from Delft University, Delft, The Netherlands, 1995. [9] R. Rau and J. H. McClellan, “Analytic models and postprocessing techniques for UWB SAR,” IEEE Trans. Aerosp. Electron. Syst., vol. 36, no. 4, pp. 1058–1074, Oct. 2000.
Lianlin Li received the B.S. degree from Xidian University, Xi’an, China, in 2001 and the Ph.D. degree from the Graduate School, Chinese Academy of Sciences, Beijing, China, in 2006. He is currently an Associate Professor with the Department of Electromagnetic Theory and Application, Institute of Electronics, Chinese Academy of Sciences. His main research interests are inverse problems, microwave imaging, compressive sensing, wave propagation, and scattering in complex media such as random media and man-made materials.
Wenji Zhang received the B.S. degree from the Huazhong University of Science and Technology, Wuhan, China, in 2004. He is currently working toward the Ph.D. degree in the Institute of Electronics, Chinese Academy of Sciences, Beijing, China. His research interests include electromagnetic inverse scattering, microwave imaging, and compressive sensing.
Fang Li received the M.S. degree from Peking University, Beijing, China, in 1981 and the Ph.D. degree from the Chinese Academy of Sciences, Beijing, in 1984. She is currently a Professor with the Department of Electromagnetic Theory and Application, Institute of Electronics, Chinese Academy of Sciences. Her research interests include electromagnetic inverse problems and wave propagation in some unconventional media, such as space plasma, metamaterials, etc.
