Synthetic aperture radar tomography sampling criteria and ...callen58/826/Tan2009... · of-sight of radar and azimuth plane[6]. When the center of the elevation synthetic is equivalent

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Synthetic aperture radar tomography sampling

criteria and three-dimensional range migration

algorithm with elevation digital spotlighting

TAN WeiXian1,2,3†, HONG Wen1,2, WANG YanPing1,2, LIN Yun1,2,3 & WU YiRong1,2

1 State Key Laboratory of Microwave Imaging Technology, Beijing 100190, China;2 Institute of Electronics, Chinese Academy of Sciences, Beijing 100190, China;3 Graduate School, Chinese Academy of Sciences, Beijing 100190, China

Based on the general geometric model of multi-baseline Synthetic Aperture Radar Tomography (To-moSAR), the three-dimensional (3-D) sampling criteria, the analytic expression of the 3-D Point SpreadFunction (PSF) and the 3-D resolution are derived in the 3-D wavenumber domain in this paper. Con-sidering the relationship between the observation geometry and the size of illuminated scenario, a 3-DRange Migration Algorithm with Elevation Digital Spotlighting (RMA-EDS) is proposed. With this algo-rithm 3-D images of the area of interest can be directly and accurately reconstructed in the 3-D spaceavoiding the complex operations of 3-D geometric correction. Finally, theoretical analyses and simu-lation results are presented to demonstrate the shift-varying property of the 3-D PSF and the spatial-varying property of the 3-D resolution and to demonstrate the validity of the 3-D RMA-EDS.

synthetic aperture radar tomography (TomoSAR), three-dimensional (3-D) SAR imaging, 3-D sampling criteria, 3-D resolution,

3-D range migration algorithm with elevation digital spotlighting (RMA-EDS)

1 Introduction

The conventional two-dimensional (2-D) synthetic

aperture radar (SAR) only provides 2-D images

along the slant range and the azimuth direc-

tion, which represents the projection of the three-

dimensionally (3-D) distributed objects over the 2-

D plane. Therefore, it has the disadvantages of

shading, foreshortening and layover due to the in-

fluence of relative geometry between radar and the

investigated scene and other factors. With Inter-

ferometric SAR[1] (InSAR) technique, it is feasible

to generate the digital elevation model (DEM) of

the illuminated area. However, the elevation sam-

pling number and spacing are both limited, severe

elevation ambiguity occurs, and only the surface

height information is obtainable. That is to say,

the distribution of the objects in 3-D space remains

unknown.

Received January 25, 2008; accepted June 20, 2008

doi: 10.1007/s11432-009-0003-2†Corresponding author (email: [email protected])

Supported by the National Science Fund for Distinguished Young Scholars (Grant No. 60725103), the National Natural Science Foundation of

China (Grant No. 60602015), the National Key Laboratory Foundation (Grant No. 9140C1903030603) and the Knowledge Innovation Program

of Chinese Academy of Sciences (Grant No. 07QNCX-1154)

Sci China Ser F-Inf Sci | Jan. 2009 | vol. 52 | no. 1 | 100-114

Tomographic imaging technique was originally

developed in the field of medical imaging. Since the

middle of the 1980s, Farhat[2,3], Jakowatz[4], Knaell

et al.[5] have analyzed the relationship between to-

mographic imaging and SAR imaging. Since the

late 1990s, Reigber et al.[6] extended tomographic

imaging concept to 3-D SAR imaging, proposed the

basic concept multi-baseline SAR tomography (To-

moSAR) imaging and reconstructed the first air-

borne TomoSAR 3-D imagery. However, the pro-

posed 3-D range migration algorithm (RMA) was

deduced without considering the influence of the

3-D sampling spacing. In addition, the second syn-

thetic aperture was considered to be formed in the

normal direction perpendicular to the line-of-sight

and the azimuth direction, the center of the eleva-

tion synthetic aperture was symmetrical about the

scene in the elevation direction in the TomoSAR

geometry.

Contemporaneously, She et al.[7] reconstructed

the 3-D satellite SAR image with ERS-1 (European

Remote Sensing satellite) data using beamforming

method in elevation focusing. Later on, Fornaro et

al.[8,9] developed the spectral estimation techniques

and inversion methods with long-term spaceborne

ERS-1/2 data. Pi et al.[10] also developed a spec-

tral estimation technique for tomographic imaging.

Recently, Fortuny et al.[11] have developed the 3-

D SAR near field imaging algorithms and provided

a lot of significant results for SAR 3-D imaging

technique. A version of 3-D RMA was proposed to

verify the planar scanning geometry. But the algo-

rithm implies that the middle of the elevation syn-

thetic aperture is symmetrical about the scene in

the elevation direction. The relationship between

the size of the ground-range and elevation direction

is not considered. Since 2005, Frey et al.[12] have

been engaged in the time-domain back-projection

algorithm for TomoSAR experimental data focus-

ing.

In this paper, we introduce a more general geo-

metric model for TomoSAR imaging, propose the

3-D range migration algorithm with elevation digi-

tal spotlighting (RMA-EDS) for the general geome-

try and extend the existing 3-D RMA developed by

Reigber[6], Fortuny[11] and Soumekh[13]. The paper

is organized as follows. In section 2, we introduce

the general imaging geometry. In section 3, we de-

rive the 3-D sampling criteria, and present the 3-D

Point Spread Function (PSF) and the 3-D resolu-

tion of TomoSAR system. In section 4, we make

a review of the work in refs. [6, 11, 13], and pro-

pose a 3-D RMA-EDS algorithm which is suitable

for processing the chirped and stepped frequency

TomoSAR data with less elevation sampling num-

ber. In section 5, we conduct the simulation exper-

iment to validate the 3-D RMA-EDS and illustrate

the capabilities of removing the layover and fore-

shortening in the traditional 2-D imaging geome-

try. Then we test the quality parameters of the

3-D TomoSAR image and reveal the shift-varying

property of the PSF and the spatial-varying prop-

erty of the 3-D resolution of the TomoSAR system.

Finally, in section 6, we draw some conclusions and

outline the future work.

2 TomoSAR imaging geometry

We consider the imaging scenario in a 3-D spatial

domain OXY Z, where X-axis, Y -axis and Z-axis

refer to the ground-range, azimuth (flight track)

and elevation direction, respectively. The illumi-

nated area is located at one side of the flight direc-

tion, as shown in Figure 1. A synthetic aperture,

which relates to the variation of the instantaneous

aspect angle of the radar antenna azimuth beam,

is formed with the movements of the radar in the

azimuth direction. An additional synthetic aper-

ture, which relates to the variation of the off-nadir

angle, is formed in the elevation direction (Z) with

the multi-pass flight, multi-baseline or antenna ar-

ray. Note that the elevation synthetic aperture is

perpendicular to ground-range and azimuth OXY

plane in this geometric model, but not to the line-

of-sight of radar and azimuth plane[6]. When the

center of the elevation synthetic is equivalent to

zero, the geometry is simplified into the model

mentioned in refs. [6, 11, 13], and the model can

be considered as Down-Looking SAR[14] when the

X-axis and Z-axis are used to denote elevation and

ground-range direction, respectively.

For convenience, we define the coordinates sys-

TAN WeiXian et al. Sci China Ser F-Inf Sci | Jan. 2009 | vol. 52 | no. 1 | 100-114 101

tem PUVW of target Pn in Figure 1, which re-

flects the relationship between target Pn and the

observation angle. The origin of PUVW is set at

the position of Pn, and PUVW varies with the

target Pn, U -axis refers to the average line-of-sight

direction pointing to Pn, V -axis is parallel to Y -

axis, and W -axis is perpendicular to PUV plane.

That the center of elevation synthetic aperture is

equivalent to any values is the first point in this pa-

per which is distinguished from that of the existing

geometries[6,11,13].

Figure 1 SAR tomography imaging geometry.

3 TomoSAR signal modeling and sam-pling criteria

3.1 TomoSAR signal model

Provided that the targets are aspect-angle inde-

pendent in the 3-D spatial domain with the size

of X0 × Y0 × Z0 (ground-range swath × azimuth

length × elevation height). The point target Pn is

located at (xn, yn, zn) with the reflectivity function

δn(xn, yn, zn), which is assumed to be frequency in-

variant. The SAR sensor is positioned at (x, y, z)

and is flying on the x = 0 plane with a constant

effective velocity vA. The center of the target zone

is (Xc, 0, 0). Therefore, xn ∈ [(Xc −X0/2), (Xc +

X0/2)], yn ∈ [−Y0/2, Y0/2], zn ∈ [−Z0/2, Z0/2].

The center points of the elevation synthetic aper-

ture LH are located at the line x=0, z=Zc. If the

linear frequency-modulated (FM) chirped pulses

p(t) with bandwidth B are transmitted, the ideal

echoed signal from Pn is

s(t, x = 0, y, z) = δn(xn, yn, zn)p(t− 2Rn/C), (1)

where

Rn =√

(Xc + x′n)2 + (y − yn)2 + (Zc + z′ − zn)2

is the range between the sensor and the target,

y is the sampling in the azimuth direction and

y ∈ [(yn − LA/2), (yn + LA/2)], C is the speed

of light, LA is the azimuth synthetic aperture size

determined by the −3 dB azimuth beam width

ΨA. z′ is the sampling along the elevation direc-

tion and z′ = z − Zc, z′ ∈ [−LH/2, LH/2]. x

′n =

xn − Xc, x′n ∈ [−X0/2,X0/2]. After the carrier

frequency has been removed, the slant range fre-

quency matched filtering is performed via

H1(ft) =

∫

[p(t− 2Rc/C) exp(−j2πfct)]

× exp(−j2πftt)dt

∗

, (2)

where Rc =√

X2c + Z2

c is the reference range, ft

is the slant range frequency corresponding to the

slang range sampling time t. p(t) = exp(j2πfct +

jπKt2

r )rect(t/T ), if |t| 6 T/2, rect[t/T ] = 1, else

rect[t/T ] = 0. Then the filtered function presented

in the slant range wavenumber domain is

SRC(Kω, y, z′) = δn exp−j2[Rn −Rc]Kω, (3)

whereKω = 2π(fc+ft)/C is the wavenumber of the

transmit signal andKω ∈ [Kω min,Kω max],Kω min =

2π(fc − B/2)/C,Kω max = 2π(fc + B/2)/C,Kc =

2πfc/C.

The echoed signal in slant range wavenumber do-

main from the whole target area can be denoted by

the linear sum of the backscattering information of

different targets in slant range wavenumber domain

according to the Born approximation,

SSRC(Kω, y, z′) =

∑

n

SRC(Kω, y, z′). (4)

3.2 TomoSAR sampling criteria

Both data acquisition and signal processing require

that the data should be discretized for SAR system.

To discretize the radar signal without aliasing, the

Nyquist sampling criteria should be satisfied. In

this paper, the Nyquist sampling criteria for To-

moSAR is analyzed.

3.2.1 Slant range sampling. The wavenumber

support of the echoed signal SSRC(Kω, y, z′) in the

102 TAN WeiXian et al. Sci China Ser F-Inf Sci | Jan. 2009 | vol. 52 | no. 1 | 100-114

range direction is Ωs = [2Kω min, 2Kω max]. In or-

der to perform Fourier analysis without aliasing,

the phase difference between the two points should

satisfy 2(Kω max−Kω min)∆R 6 2π, where ∆R is

the distance between the two targets in the slant

range direction. We have

fs =1

∆t=

C

2∆R>

[

C(Kω max −Kω min)

2π

]

= B,

(5)

where fs and ∆t are the slant range sampling fre-

quency and sampling time interval. In practice,

slant range oversampling ratio is usually set be-

tween 1.1 and 1.2[15] to provide a guard band. As

for the stepped frequency system[11], the frequency

sampling spacing ∆f is governed by

∆f 6C

2

√

X20 + Z2

0

cos[ψA/2]

−1

. (6)

3.2.2 Azimuth sampling. In TomoSAR, the in-

stantaneous wavenumber observed along the Y -

axis is

Ky =∂

∂y[arg(SRC(Kω, y, z

′))]

= − 2Kω sin θn(z′) sinφn(y), (7)

where θn = arcsin√

(Xc + x′n)2 + (y − yn)2/Rn

is the instantaneous incidence angle at differ-

ent heights z′ and φn = arcsin[y − yn]/√

(Xc + x′n)2 + (y − yn)2 is the azimuth instanta-

neous aspect angle.

Therefore, the azimuth wavenumber support

band of (4) is Ωy = [−2Kω sin θn(z′) sin φn(ΨA/2),

2Kω sin θn(z′) sin φn(ΨA/2)], where φn(ΨA/2) is

the largest azimuth aspect angle of target Pn with

respect to the sensor. The whole support band

BY in the azimuth direction is approximately equal

to 4Kω sin(ΨA/2); therefore, the azimuth Nyquist

sampling spacing is

∆y 6π

2Kω max sin(ΨA/2). (8)

Hence, the pulse repetition frequency (PRF) is

PRF =βvA

π2Kω max sin(ΨA/2), (9)

where β ∈ (1.1, 1.4)[15] is the azimuth oversampling

ratio. The influences of the range and azimuth am-

biguity ratio[13,15] are not considered here.

3.2.3 Elevation sampling. In TomoSAR, the

imaging scenario is illuminated by the sensor with

different incidence angles, which is similar to the

Spotlight SAR[4] mode along the elevation direc-

tion. The instantaneous elevation wavenumber is

Kz′(Kω,Ky; z′)

=∂

∂zargFTy[SSRC(Kω, y, z

′)]

= −√

4K2ω −K2

y cos θn(z′), (10)

where FTy indicates the Fourier transform (FT)

of the signal SSRC(Kω, y, z′). In accordance with

(10), the support band of FTy[SSRC(Kω, y, z′)] in

the elevation wavenumber domain is

ΩZ =⋃

Ky

⋃

x′n,z′

[−2Kω cos θn(z′)

2Kω cos θn(z′)]. (11)

In (10) and (11), the average elevation Doppler

centroid of Pn changes with its spatial position so

that the whole elevation Doppler support is spread

by the variant elevation Doppler centroid. The

spread support band observed in the elevation di-

rection becomes

BZ = (2Kω max cos θn min−2Kω min cos θn max), (12)

where θn max and θn min are the maximum and mini-

mum instantaneous incidence angle from the sensor

to the target area. Therefore, the elevation Nyquist

sampling spacing is determined via

∆z0 6πRmin

[Kω max(LH + Z0)], (13)

where Rmin ≈ (Xc −X0/2) if |Zc| 6 (Z0 + LH)/2,

else

Rmin ≈√

(Xc −X0/2)2 + (|Zc| − (Z0 + LH)/2)2.

However, the echoed signal sampled with the

above sampling spacing has already included re-

dundant information. With a reference signal

H3(Kω,Ky, z′),

H3(Kω,Ky, z′)

= exp

j√

Xc + [Zc + z′ − zn]2

√

4K2ω −K2

y

, (14)

the elevation sampling spacing could be re-

duced without damaging any spectrum informa-


tion of the targets. Then multiply the signal

SSRC(Kω,Ky, z′) (azimuth FT of SSRC(Kω, y, z

′))

with H3(Kω,Ky, z′) and take the FT in the eleva-

tion direction for the every given Ky considering

z′ ≪√

X2c + Z2

c and zn ≪√

X2c + Z2

c , we can get

SSAFRC(Kω,Ky,Kz′)

∼∑

n

exp(−jRre

√

4K2ω −K2

y)

· sin c

LH

π

[

Kz′ − ε√

4K2ω −K2

y

]

, (15)

where Rre = (√

[Xc + x′n]2 + [Zc − zn]2 − Rc) and

ε = zn/√

[Xc + x′n]2 + [Zc − zn]2 − Zc/Rc.

Thus the whole elevation bandwidth BZC is re-

duced by multiplying with H3(Kω,Ky, z′), that is

BZC

≈ 2Z0Kω max/√

[Xc + x′n]2 + Z2

c + 2π/LH

≈ 2Z0Kω max/√

[Xc + x′n]2 + Z2

c , (16)

which is much smaller than the support BZ of

SSAFRC(Kω,Ky, z′). So the elevation sampling

spacing can be reduced to ∆z.

∆z 62π

BZC

=π√

(Xc −X0/2)2 + Z2c

[Kω maxZ0]. (17)

The elevation oversampling ratio is set between

1.1 and 1.2 to provide a guard band. Sampling with

(17), the elevation sampling number is reduced, but

some additional steps are needed to recovery the

spectrum of the targets accurately. The problem

is to be discussed in section 4. If the elevation

sampling distance is greater than 2π/BZC , it will

cause severe elevation ambiguity. It is possible to

gain the information of the strong backscattering

targets with the inversion methods, spectral esti-

mation technique and so on[8−10].

3.3 TomoSAR point spread function

The TomoSAR point spread function (PSF) and

the 3-D resolution are analyzed in this subsection.

Assume that the elevation sampling distance sat-

isfies the restriction in (13). Take elevation and

azimuth FT, in accordance with the method of sta-

tionary phase (MSP)[16], the 3-D wavenumber sig-

nal is

S(Kω,Ky,Kz′)

= δn(xn, yn, zn)A2 expj2KωRc

× exp−j[Kz′(−Zc + zn) +Kyyn]

× exp−jxn

√

4K2ω −K2

y −K2z′, (18)

where A2 ≈ −j4πKωxn(4K2ω − K2

y − K2z′)−1 is a

slowly fluctuating amplitude function which does

not play any important role in the following anal-

ysis. Hence, the influence of this term is not con-

sidered.

To analyze the point spread function (PSF) of

target Pn, a Cartesian coordinate system PUVW

of the target Pn has been defined in Figure 1.

The elevation Doppler centroid of target Pn is ap-

proximately −2Kω cos(ξnc), and it changes with

the position of the targets. Assuming that the

azimuth Doppler centroid is equivalent to zero,

the translation relation in wavenumber domain be-

tween OXY Z coordinates and PUVW coordinates

is denoted by

KX

KY

KZ

=

sin ξnc 0 cos ξnc

0 1 0

− cos ξnc 0 sin ξnc

KU

KV

KW

, (19)

where ξnc = arctanxn/(Zc − zn) if (Zc − zn) 6= 0,

else ξnc = π/2.

Eqs. (18) and (19) imply that the 3-D spatial

wavenumber support of target Pn is a part of a

“hollow sphere” with radii 2Kω min and 2Kω max,

and the received signal of target Pn is the 3-D

spherical band-pass signal, as shown in Figure 2(a).

The size of spherical surface is proportional to the

size of azimuth and elevation synthetic aperture.

When the “hollow sphere” is projected onto the

KU -axis and KV -axis plane, the support band BUn

in the KU -axis direction is an annulus spanning a

wavenumber of BUn = 2[Kω max −Kω min].

When the 3-D “hollow sphere” is projected onto

the KU -axis and KW -axis plane in Figure 2(b), the

support band BV n in the KV -axis direction is an-

other annulus with BV n = BY ≈ 4Kω sin(ΨA/2).

When the 3-D “hollow sphere” is projected

onto the V -axis and W -axis plane, it is an-

other annulus which spans a spatial wavenum-

ber of BWn in the W -axis direction with BWn ≈

2KωncLH sin(ξnc)r−1hn , where rhn = [(Xc + x′

n)2 +

(Zc − zn)2]0.5,Kωnc is the wavenumber center with

respect to every azimuth wavenumber Ky, given by

Kωnc =1

2

√

4K2ω −K2

y |max +√

4K2ω −K2

y |min

.

(20)


Figure 2 SAR tomography imaging geometry. (a) 3-D wavenumber support; (b) support projected onto OXZ plane.

As for the narrow band signal, the spherical re-

gion of the support is not very apparent. The sup-

port of the target in the 3-D wavenumber domain

could be viewed as an approximately regular cube.

Combining (18) with (19), the 3-D PSF can be ex-

pressed as

PSF (un, vn, wn;xn, yn, zn)

∼ sinc

BUnun

2π

sinc

BV nvn

2π

sinc

BWnwn

2π

, (21)

where the sinc function is defined as sinc(x) =

sin(πx)/(πx). The translation relation between

OXY Z and PUVW is stated as

un

vn

wn

=

sin ξnc 0 − cos ξnc

0 1 0

cos ξnc 0 sin ξnc

x− xn

y − yn

z − zn

.

(22)

As shown in Figure 3(a), the PSF of target Pn is

a 3-D sinc-like function. Figures 3(b)–3(d) are the

slices of the PSF on the different slices in 3-D space

with dynamic range 30 dB. The impulse response

width (IRW) is defined as −3 dB width (resolu-

tion) of the main lobe of the impulse response, and

the 3-D resolution is approximated by

ρUn ≈ [0.886C]/[2B],

ρV n ≈ [0.886π]/[2Kω min sin(ΨA/2)],

ρWn ≈ [0.886πrhn]/[Kω minLH sin(ξnc)],

(23)

where ρUn, ρV n and ρWn represent the resolution of

target Pn along the U -axis, V -axis and W -axis in

the 3-D spatial domain, respectively.

The transform (22) shows that the orientation

of the 3-D PSF relates to the target coordinates

and the PSF at elevation and azimuth plane is

generated by rotating the 3-D sinc-like function

by ξnc. Therefore, the PSF of target Pn in To-

moSAR system is shift-varying. From (23), we

can see that the resolutions are dependent on the

position of the target and affected by the imag-

ing geometry. Therefore, the resolutions of target

Pn in TomoSAR system are spatial-varying. The

detailed description of shift-varying and spatial-

varying property will be demonstrated with the

simulation in section 5. The shift-varying and

spatial-varying property of the 3-D PSF of To-

moSAR is the second point in this paper which is

distinguished from the work of Reigber et al.[6,11,13].

4 3-D range migration algorithm with ele-vation digital spotlighting

In this section, 3-D range migration algorithm with

elevation digital spotlighting (RMA-EDS) which is

distinct from refs. [6,11,13] is presented, as shown

in Figure 4. The 3-D RMA-EDS can operate for

any case of Zc and the elevation sampling with

(17). In addition, the algorithm is suitable for

processing the chirped and stepped-frequency SAR

data.


Figure 3 Pn’s PSF processed by the method[11] under the condition (13). (a) PSF in the 3-D space; (b) PSF at OXY slice; (c) PSF

at OY Z slice; (d) PSF at OXZ slice.

Figure 4 Block diagram representation of RMA-EDS for the

general TomoSAR geometry with elevation sampling restriction

(17).

As for image formation, the imaging geometry is

extended to a more general case in which the cen-

ter of elevation aperture Zc is arbitrary, and the 3-

D image reconstructed accurately with RMA-EDS

needs no geometric correction. As for SAR sys-

tem, the raw data volume becomes smaller with

the sampling spacing (17) and the data acquisi-

tion time is shorter, the elements of antenna ar-

ray becomes much less and the configuration of the

system gets more compact. For accurately recon-

structing the 3-D image without geometric correc-

tion, the following problems should be considered:

• Usually, the center of the elevation synthetic

aperture is not symmetrical about the illuminated

scenario in the elevation direction.

• The elevation Doppler centroid updates with

the position of the target and spreads the elevation

spectrum in the elevation wavenumber domain.

• The elevation sampling with (17) will cause

the spectral aliasing if the elevation FT is used di-

rectly.

• The ground-range swath of the illuminated

area is usually much wider than the height of the

illuminated area.

In the following subsections, the main operations

will be formulated or described according to the

signal flow in the processing chain in Figure 4. The

RMA-EDS is the third point in this paper which is

distinct from that of the existing algorithms[6,11,13].

4.1 Slant range and azimuth processing

In this step, slant range matched filtering and az-

imuth FT is performed.

As for the linear FM pulse SAR system, the

matched filter H1(ft) in the slant range frequency

domain is performed first. As for the stepped fre-

quency SAR system, only a phase compensation

function H2(Kω) multiplication with the reference

range Rc is carried out,


H2(Kω) = exp[j2KωRc], (24)

where Kω ∈ [Kω min,Kω max]. Then the processed

signal in the slant range wavenumber domain could

be written as

S1(Kω, y, z′) = δn exp−j2[Rn −Rc]Kω, (25)

instead of considering the form of the transmitted

signal.

The azimuth and elevation 2-D FT[6,11,13] could

not be performed directly because the elevation

support is much larger than the elevation sampling

rate. Therefore, only the azimuth FT could be

performed before the elevation spectral alias-free

recovery; that is,

S2(Kω,Ky, z′) = δnA3 expj2RcKω

× exp−jKyyn

− jRs

√

4K2ω −K2

y, (26)

where Rs =√

[Xc + x′n]2 + [Zc + z′ − zn]2 and

A3 = exp(−jπ/4)(4K2ω −K2

y)−1.

4.2 Elevation digital spotlighting

processing

This step is to perform the elevation wavenumber

alias-free recovery under the elevation sampling re-

stricted condition (17). As shown in Figure 5, the

ground-range swath is much larger than the height

of the target, and the equivalent height RH is usu-

ally higher than the real height Z0. The eleva-

tion Doppler centroid −2Kω cos(ξnc) of target Pn

spreads the whole elevation Doppler support.

Step 1. The elevation spectrum extension is

conducted by zero-interleaving with the factor γ.

γ =

⌈

RH

Z0

⌉

=

⌈

(cos θn min − cos θn max)

Z0/√

(Xc − 0.5X0)2 + Z2c

⌉

, (27)

where ⌈∗⌉ represents the smallest integer that is

larger than or equal to ∗.

Note that zero-interleaving is different from that

used in ref. [18], where it is utilized to remove the

whole azimuth aliasing. Here it is used to extend

the spectrum due to the elevation Doppler centroid

variation but not the sampling spacing with (17).

Step 2. Zero-padding is performed consid-

ering the relation between the length of elevation

synthetic aperture and the height of the area of in-

terest to avoid the circular convolution aliasing[13].

Step 3. After the multiplying the elevation

signal with H3(Kω,Ky, z′), the elevation spectrum

is reduced to ∆z−1, where z′ is the elevation sam-

pling after the above operations.

Figure 5 The ground-range and elevation plane in TomoSAR

geometry.

Step 4. The elevation and slant range signal

are transformed to the polar domain through the

slant range inverse FT and the elevation FT. The

spotlight filter which is similar to that of ref. [13]

is introduced.

H4(Kω,Ky,Kz′)

=

1 if |Rn sin(θc + θn) −Xc| 6 (X0/2),

&|Rn cos(θc + θn) − Zc| 6 (Z0/2),

0 else,

(28)

where θn represents the instantaneous incidence,

and Rn is the slant range, θc = arccos(Zc/Rc).

With this filter, the area of interest can be recon-

structed without any ambiguities through limiting

t and θn.

Step 5. Perform the slant range FT and

zero-padding the elevation. Then the alias-free

S3(Kω,Ky, z′′) is recovered via the multiplication

of the elevation signal with H5(Kω,Ky, z′′) and el-

evation inverse FT,

H5(Kw,Ky, z′′) = exp(j2RcKω)

× exp(−j√

4K2ω −K2

y

√

X2c + (Zc + z′′)2), (29)

where z′′n = (n − NH/2 − 1)∆z0, n = 1, 2, · · · , NH

and NH = 2⌈max(γZ0, LH)/∆z0⌉.


Step 6. Finally, a phase function should be

multiplied to transform the elevation band-pass

signal to the low-pass signal S4(Kω,Ky, z′′) before

the elevation Fourier transforms, and the filter is

H6(Kw,Ky, ξnc) = exp[jKωncz′′(Zc/Rc)]. (30)

Step 7. After performing the elevation

Fourier transform of S4(Kω,Ky, z′′), we have

S5 (Kω,Ky,Kz′′) ∼ exp[−j(Kz′′(zn − Zc))]

× exp[−j(Xc + x′n)

√

4K2ω −K2

y −K2z′′ ]

⊗ σ(KZ) × exp[−jKyyn + j2KωRc], (31)

where KZ = Kz′′ + KωncZc/Rc,Kz′′ = 2π(n −

NH/2 − 1)/(∆z0NH), n = 1, 2, · · · , NH . ⊗ and σ

indicate the convolution operation and the delta

function, respectively.

4.3 3-D wavenumber filtering

Combining the above operations, the echoed sig-

nal is transformed to the 3-D wavenumber do-

main. Then the 3-D matched filtering in the 3-D

wavenumber domain is performed, that is,

H7(Kω,Ky,Kz′′)

= expjXc

√

4K2ω −K2

y −K2Z

× exp−j2RcKω−jZc(Kz′′ +KωncZc/Rc).(32)

Subsequently, with (32) the filtered signal of tar-

get Pn in the 3-D wavenumber is obtained.

S6 (Kω,Ky,Kz′′)

= S5(Kω,Ky,Kz′′)H7(Kω,Ky,Kz′′)

∼ δn(xn, yn, zn) exp−j[KZzn +Kyyn]

× exp−jx′n

√

4K2ω −K2

y −K2Z. (33)

4.4 3-D STOLT mapping

3-D STOLT mapping is performed in 3-D

wavenumber domain to transform the signal in the

sphere coordinate into the uniformly sampling sig-

nal in the 3-D Cartesian coordinate (KX ,KY ,KZ)

for inverse FT. Eq. (33) is redefined as

S6(Kω, Ky,Kz′′) ∼ δn(xn, yn, zn)

× exp−jznKZ(Kω,Ky,Kz′′)

× exp−jynKY (Kω,Ky,Kz′′)

× exp−jx′nKX(Kω,Ky,Kz′′), (34)

where

KX =√

4K2ω −K2

y − [Kz′′ +KnωcZc/Rc]2,

KY =Ky,

KZ =[Kz′′ +KnωcZc/Rc].

(35)

Eq.(35) indicates the 3-D STOLT mapping[6],

which is the extension of 2-D STOLT

mapping[17,19]. With the 3-D STOLT mapping

in the 3-D wavenumber domain, the nonorthog-

onality of the received signal axes is rectified by

mapping the signal in (Kω,Ky,K′′z ) domain into

the signal S7(KX ,KY ,KZ) in (KX ,KY ,KZ) do-

main; in other words, the 3-D STOLT mapping

essentially replaces wavenumber in the direction of

wave propagation by wavenumber observed in the

ground-range (X), azimuth (Y ) and elevation (Z)

direction.

S7(KX ,KY ,KZ)

≈∑

|KX−l∆KX |6NIX∆KX

|KY −m∆KY |6NIY ∆KY

|KZ−n∆KZ |6NIZ∆KZ

S′6(Kω,Ky,Kz′′), (36)

where NIX , NIY and NIZ denote the half-length of

the sinc interpolation kernel used for STOLT map-

ping. ∆Kω,∆Ky and ∆Kz′′ are the wavenumber

sampling spacings of S6(KX ,KY ,KZ).

S′6(Kω, Ky,Kz′′)

= S6(Kωl,Kym,Kz′′n)

× sinc3KX ,KY ,KZ, (37)

where Kωl,Kym and Kz′′n are the discretized

wavenumber of S6(Kωl,Kym,Kz′′n)

sinc3 KX ,KY ,KZ

= sinc[KX − l∆KX ]/∆KX

× sinc[KY −m∆KY ]/∆KY

× sinc[KZ − n∆KZ ]/∆KZ. (38)

During the course of STOLT interpolation, the

number of the samples in 3-D wavenumber domain

KXKYKZ grid is

NX = 2⌈(KX max −KX min)/(2∆KX)⌉,

NY = 2⌈(KY max −KY min)/(2∆KY )⌉,

NZ = 2⌈(KZ max −KZ min)/(2∆KZ)⌉,

(39)

where the spatial wavenumber spacing ∆KX 6

2π/X0,∆KY 6 2π/Y0,∆KZ 6 2π/(γZ0),KX min,


KX max,KY min,KX max,KZ min and KZ max are the

minimum and maximum value of KX ,KY and KZ ,

respectively.

As in (38), the interpolation kernel consist of

three sinc functions. For 3-D mapping, one method

is to utilize two times 1-D sinc interpolation

(The first interpolation is performed in azimuth

and slant range wavenumber domain. Another

one is conducted in ground-range and elevation

wavenumber domain). The 3-D mapping in (36)

is based the evenly spaced data, the interpolation

along KY and KZ direction could be skipped when

∆KY = 2π/(NY ∆y) and ∆KZ = 2π/(NZγ∆z).

Therefore, the most direct 3-D STOLT mapping

could be obtained by 1-D sinc interpolation[15,17,19]

along KX direction through considering the az-

imuth wavenumber Ky and elevation wavenumber

Kz′′ at the same time.

4.5 3-D inverse FT and image space

determination

After the 3-D STOLT interpolation, the 3-D SAR

image can be obtained with 3-D inverse Fourier

transform. The reconstructed target area is

f(xn, yn, zn)

∼∑

n

δnPSF (un, vn, wn;xn, yn, zn)

∼∑

n

δn(xn, yn, zn)sinc

(

BUnun

2π

)

× sinc

(

BV nvn

2π

)

sinc

(

BWnwn

2π

)

. (40)

The ground-range, azimuth and elevation sam-

ples of the reconstructed 3-D image correspond to

the above uniformly spaced samples in the 3-D spa-

tial domain OXY Z

xn = Xc + (l −NX/2 − 1)∆X ,

yn = (m−NY /2 − 1)∆Y ,

zn = −Z0/2 + (n−NZ − 1)∆Z ,

(41)

where l = 1, 2, . . . , NX ,m = 1, 2, . . . , NY and

n = 1, 2, . . . , NZ .∆X = 2π/(NX∆KX), ∆Y =

2π/(NY ∆KY ), and ∆Z = 2π/(NZ∆KZ) are the

evenly spaced samples.

As the step of zero-interleaving has been per-

formed with the factor γ to recover the elevation

spectrum, the actual samples of the reconstructed

image in the 3-D spatial domain should be reduced

to NZ × NY × [NZγ−1], where [NZγ

−1] represents

the integer part of NZγ−1. The focused 3-D image

is generated in ground-range (X-axis), azimuth (Y -

axis) and elevation (Z-axis) direction, so the geo-

metric correction is not needed in the reconstruc-

tion.

4.6 Discussion of the approximation

The main operations used in the 3-D RMA-EDS al-

gorithm are the Fourier transform, elevation spec-

trum alias-free recovery and STOLT mapping. As

for the Fourier transform, the only approxima-

tions in the algorithm lie in the slant range Fourier

transform, azimuth Fourier transform and eleva-

tion Fourier transform. The transforms are based

on the assumption of a high time-bandwidth prod-

uct in slant range direction, azimuth and elevation

direction. Usually in the TomoSAR system, these

approximations could be ignored. As for eleva-

tion spectrum alias-free recovery, the data colleted

with sampling spacing (17) has included redun-

dant information. With the aforementioned steps,

the elevation spectrum could be recovered without

any approximation. As for the STOLT mapping,

2-D or 1-D wavenumber interpolation is used in

wavenumber domain, and the signal in the Sphere

coordinates wavenumber domain could be mapped

to Cartesian coordinates accurately.

5 TomoSAR simulation experiments

In the above sections, the sampling criteria, PSF

of TomoSAR and the 3-D RMA-EDS are analyzed.

In this section several simulation experiments are

conducted to demonstrate the 3-D resolution ca-

pabilities, the shift-varying properties of the point

spread function of TomoSAR, the spatial-varying

property of the 3-D resolution and so on. For sim-

plicity, the echoed raw data is generated by (4).

In order to test the validity of RMA-EDS, the sys-

tem parameters are chosen as listed in Table 1. In

addition, the TomoSAR experiment has been per-

formed in microwave anechoic chamber and pro-

cessed with this algorithm[20].

For a 200 m×100 m×40 m (ground-range, az-

imuth, elevation) target zone, the elevation sam-


pling spacing is about 0.62 m according to (17) if

the 3-D RMA is used directly. The antenna aper-

ture angle is 0.575, so the azimuth antenna size is

about 2.99 m. With the 3-D RMA-EDS, the eleva-

tion sampling spacing can be set at 2.03 m. Hence,

the sampling number in the elevation direction is

51 when the volumetric height is 40 m.

Table 1 Parameters used in the simulation experiment

Parameter name Value

Center frequency (GHz) 10.00

Signal bandwidth (MHz) 100.00

Elevation aperture length (m) 101.55

Center of elevation aperture (m) 5000.41

Azimuth resolution (m) 1.32

⊥Line-of-sightresolution (m) 1.30

Line-of-sight resolution (m) 1.33

Sampling number of azimuth 226

Sampling number of elevation 51

Azimuth sampling spacing (m) 1.2030

Elevation sampling spacing (m) 2.03

Range step frequency size (MHz) 0.40

Center of elevation target zone (m) 0

Azimuth length of the targets area (m) [−100 100]

Elevation height of the targets area (m) [−20 20]

Center of ground-range target zone (m) 5000.00

Ground-range width of the targets area (m) [−100 100]+ 5000

As shown in Figure 6, in order to exhibit 3-D

resolution capability and overcome the foreshort-

ening and layover effects with TomoSAR imaging

technique, the relative positions of 168 point tar-

gets are given. 3 point targets are evenly located

in the line x = 4910, z = −18 with azimuth inter-

val 24.00 m in ‘A’ area in Figure 6(c) and (d). 18

point targets are evenly located at the plane z − 6

(x − 4913) = 0 with azimuth interval 6.00 m and

ground-range interval 3 m in ‘B’ area in Figure 6(c)

and (d). 8 point targets are symmetrically located

in the circle z = −18, (x − 4952)2 + y2 = 242 with

angle interval 45, 7 point targets are evenly lo-

cated in the line x = 4952, y = 0 with elevation

interval 6 m in ‘C’ area in Figure 6(c) and (d). 28

point targets are evenly distributed at the plane

z − (x − 4982) = 0 with azimuth interval 6.00 m

and ground-range interval 3 m in ‘D’ area in Figure

6(c) and (d). 28 point targets are distributed at the

plane z− 0.25(x− 5024) = 0 with azimuth interval

6 m and ground-range interval 4.00 m, correspond-

ing to the elevation interval 1.00 m in ‘E’ area in

Figure 6(c) and (d). 76 point targets are evenly lo-

cated in the edges of the rectangular parallelepiped

with azimuth length 48.00 m, ground-range width

36.00 m and height 36.00 m shown in ‘F’ area in

Figure 6(d).

The conventional SAR data collecting geometry

has a property of cylindrical symmetry. Therefore,

the targets distributed in the elevation direction

could not be resolved in the conventional 2-D imag-

ing geometry with the method[13] of 2-D SAR fo-

cusing. Meanwhile, the serious foreshortening and

layover effects occur in the conventional 2-D imag-

ing geometry, as shown in Figure 7.

(1) Layover. As shown in Figure 7, the posi-

tion of target ‘B’ area on the 2-D SAR image is

reversed with that of target ‘A’ area for the in-

fluence of layover. In addition, the targets are lo-

cated at the center line of ‘C’ area perpendicular to

the horizontal plane, and the phenomenon of lay-

over becomes the most serious. The target plane

z − (x − 4982) = 0 is approximately perpendicu-

lar to the line-of-sight of the SAR sensor, and the

distances between the targets and the sensor are

approximately equal, so 28 point targets are super-

posed into 9 points. The amplitude of the super-

posed target becomes stronger with the increasing

number of the targets, e.g. ‘G’ and ‘H’ are stronger

than that of other 7 targets, as shown in Figure 7.

(2) Foreshortening. One example is the target

‘C’ area as shown in Figure 6(a), two targets are

located in the circle in the ground-range direction

with distance 34.94m in between, while in the cor-

responding 2-D SAR image as shown in Figure

7(a), the distance is only about 23.81 m. An-

other example is the target ‘E’ area, whose slope

is about 14 and is much less than the incidence

angle. Therefore, the foreshortening occurs, and

the large area consisting of large number of tar-

gets is reduced to a small area, and the imaging

area is brighter than the area nearby, as shown in

Figure 7. Also the size of the area 12.70 m (slant

range)×48.00 m (azimuth) in 2-D SAR image is

much smaller than the real area 24.74 m (slant

range)×48.00 m (azimuth). Note that both lay-

over and foreshortening occur in target ‘F’. They

are more difficult to be resolved. Therefore, the


Figure 6 The targets and the projections onto OXY , OY Z and OXZ. (a) The targets in 3-D space; (b) projections onto OY Z plan;

(c) projections onto OXY plane; (d) projections onto OXZ plane.

Figure 7 The 2-D SAR images obtained at different height. (a) 2-D SAR image at z=4948.82 m; (b) 2-D SAR image at z=1969.13

m.

conventional SAR system could not avoid the

above problems due to the observation geometry,

and cannot resolve the targets in the 3-D spatial

domain.

With the 3-D RMA-EDS, the 3-D structure of

the targets can be reconstructed and resolved. As

shown in Figure 8(a), the 3-D image is displayed

at 3.01dB contour. As expected, the whole space

structure is very consistent with the real situa-

tion in Figure 6(a). The image is reconstructed

in OXY Z space, and thereby, it needs no geomet-

ric correction after focusing. In Figure 6(b), the

projected targets in the solid black boxes are very

clear, while in Figure 8(b) the projected targets in

the solid black boxes are blurred since the line-of-

sight resolution and the ⊥Line−of−sight resolution are

about 1.30 m and 1.48 m, respectively. Because the

ground-range position of these targets is different,

they can still be resolved in the 3-D SAR space, as

shown in Figure 8(a), (c) and (d).

The expanded 3-D SAR image of target ‘C’ area

is shown in Figure 9(a), and Figure 9(b) and (c)

displays the details of target ‘C’ area at different

layers with a dynamic range of 20 dB.

As shown in Figure 3, the PSF of the tar-

get is a 3-D sinc-like function, and the recon-

struction of the PSF is performed without any

weighted window. In this paper, the IRW, the

peak sidelobe ratio (PSLR) and the integrated side-

lobe ratio (ISLR) are measured to verify the per-

formance and focusing accuracy of the 3-D RMA-

EDS. Three point targets are set in another simu-

lation to demonstrate the property of the 3-D PSF,

e.g. P1(3080.00 m, 24.00 m, 12.50 m), P2(5000.00

m, 0 m, 0 m) and P3(7920.00 m, −24.00 m, −12.50

m). The theoretical 3-D IRW is calculated with

(24) and shown in Table 2.

On the one hand, the IRWs of the target varies

with the position of target and the measured re-

sults shown in Table 2 are consistent with the the

oretical value listed in Table 3, that is to say, the

resolution of the TomoSAR is spatial-varying. On


Figure 8 The 3-D SAR image and the projections on the different planes. (a) The 3-D TomoSAR image; (b) projection onto the

OY Z plane; (c) projection onto the OXY plane; (d) Projection onto the OXZ plane.

Figure 9 The 3-D SAR image of ‘C’ and the layers at different planes. (a) Target ‘C’ area; (b) layer at x =4951.8 m; (c) layer at

z = −17.56 m.

Table 2 Theoretical IRWs of different targets in the 3-D spatial domain

Target position (xn, yn, zn)Ideal IRW (m)

U V W

P1(3080.00 m, 24.00 m, 12.50 m) 1.33 1.33 1.45

P2(5000.0 m, 0 m, 0 m) 1.33 1.33 1.30

P3(7920.0 m, −24.0 m, −12.5 m) 1.33 1.33 1.45

Table 3 Image quality parameters measured in PUV W coordinates

Target position (xn, yn, zn)IRW (m) PSLR (dB) ISLR (dB)

U V W U V W U V W

P1(3080.00 m, 24.00 m, 12.50 m) 1.35 1.34 1.42 −13.13 −13.54 −13.21 −10.16 −10.62 −10.16

P2(5000.00 m, 0 m, 0 m) 1.37 1.33 1.30 −13.22 −13.42 −14.23 −10.10 −10.51 −10.18

P3(7920.00 m, −24.00 m, −12.50 m) 1.33 1.34 1.43 −13.81 −13.27 −15.32 −10.55 −10.35 −10.61


Figure 10 The SPFs (top) of target ‘P1’, ‘P2’ and ‘P3’, the slices (bottom) of the corresponding PSFs of the targets at OXZ plane.

(a) Target ‘C’ area; (b) layer at x =4951.85 m; (c) layer at z = −17.56 m.

the other hand, the orientation of the 3-D SPF of

different targets relates to nc, as shown in Figure

10. The rotation angles of SPF of targets ‘P1’, ‘P2’

and ‘P3’ at ground-range and elevation plane are

31.63, 45.00 and 57.74, respectively. Thus the

PSF of the target is shift-varying.

6 Conclusion and future work

In this paper, we have provided a treatment of sam-

pling criteria, point spread function (PSF) and 3-

D reconstruction algorithm issues about the To-

moSAR imaging technique theoretically.

(1) The 3-D sampling criteria of TomoSAR is

derived in the 3-D wavenumber domain and then

verified both numerically and experimentally.

(2) The shift-varying property of the PSF of To-

moSAR and the spatial-varying property of the 3-

D resolution for the case of narrow band system

are demonstrated with numerous simulation exper-

iments and analyses in 3-D wavenumber domain.

(3) The 3-D RMA-EDS, which is suitable for

any case of Zc, the elevation sampling spacing with

(17), the case in which ground-range swath of the

illuminated area is wider than the height of the

area, the chirped and stepped-frequency SAR sys-

tem, has been proposed and verified with the sim-

ulation and the measured image quality parame-

ters. In addition, with 3-D RMA-EDS, the 3-D

target area can be reconstructed without any am-

biguities. The focused 3-D SAR image needs no

geometric correction and can be displayed directly

in the 3-D spatial domain.

(4) Compared to the conventional SAR tech-

nique, TomoSAR imaging has the capabilities of

providing the 3-D image of the area of interest.

The simulation experiment of multi-baseline To-

moSAR is performed, and the results processed by

RMA-EDS are presented to demonstrate that the

foreshortening and layover that seriously affect the

interpretation of the 2-D SAR image could be re-

moved very well.

The main limitation of the proposed algorithm

is the need for assumption of the azimuth constant

effective velocity and elevation uniformly sampling.

Therefore, the problems of sampling nonunifor-

mity, irregularity, sparsity and motion compensa-

tion technique in TomoSAR imaging that occur

frequently in practice are not addressed in this pa-

per. In addition, as the imaging scenario is illu-

minated by multi-baseline or array antenna, the

coherence of the backscattering information of the

targets at different heights or different incidences

is worth studying.

The authors would like to thank the anonymous reviewers and

the readers for their valuable comments and suggestions.


1 Goldstein R M, Zebker H A. Interferometric radar measure-

ments of ocean surface currents. Nature, 1987, 328(20): 707–

709

2 Chan C K, Farhat N H. Frequency swept tomographic imag-

ing of three-dimensional perfectly conducting objects. IEEE

Trans Antennas Propag, 1981, AP-29(2): 312–319

3 Farhat N H, Werner C L, Chu T H. Prospects for three-

dimensional projective and tomographic imaging radar net-

works. Radio Sci, 1984, 19(5): 1347–1355

4 Jakowatz C V Jr, Wahl D E, Eichel P H, et al. Spotlight-

mode synthetic aperture radar: a signal processing approach.

Boston: Kluwer Academic Publishers, 1996

5 Knaell K K, Cardillo G P. Radar tomography for the gen-

eration of three-dimensional images. IEE Proc-Radar Sonar

Navig, 1995, 142(2): 54–60

6 Reigber A, Moreira A. First demonstration of airborne SAR

tomography using multibaseline L-band data. IEEE Trans

Geosci Remote Sens, 2000, 38(5): 2142–2152

7 She Z S, Gray D A, Bogner R E, et al. Three-dimensional

space-borne synthetic aperture radar (SAR) imaging with mul-

tiple pass processing. Int J Remote Sens, 2002, 23(20): 4357–

4382

8 Fornaro G, Lombardini F. Three-dimensional multipass SAR

focusing: experiments with long-term spaceborne data. IEEE

Trans Geosci Remote Sens, 2005, 43(4): 702–714

9 Fornaro G, Serafino F. Imaging of single and double scatter-

ers in urban areas via SAR tomography. IEEE Trans Geosci

Remote Sens, 2006, 44(12): 3497–3505

10 Li J, Min Rui, Pi Y M. SAR stereo imaging algorithm based

on amplitude and phase estimation. In: 1st Asian and Pacific

Conference on Synthetic Aperture Radar Proceedings. Huang-

shan: IEEE Press, 2007. 385–388

11 Lopez-Sanchez J M, Fortuny-Guasch J. 3-D Radar imaging us-

ing range migration techniques. IEEE Trans Antennas Propag,

2000, 48(5): 728–737

12 Frey O, Morsdorf F, Meier E. Tomographic processing of multi-

baseline P-band sar data for imaging of a forested area. In:

Proceedings of IGARSS. Barcelona: CIMNE, 2007. 156–159

13 Soumekh M. Synthetic aperture radar signal processing with

matlab algorithms. New York: John Wiley & Sons, 1999

14 Weiß M, Ender J, Peters O, et al. An airborne radar for

three dimensional imaging and observation-technical realisa-

tion and status of ARTINO. In: 6th European Conference on

Synthetic Aperture Radar. Dresden: Deutschen Nationalbib-

liografie, 2006

15 Cumming I G, Wong F H. Digital processing of synthetic aper-

ture radar data: algorithms and implementation. Boston:

Artech House, 2005

16 Stamnes J J. Waves, rays, and the method of stationary phase.

Optics Express, 2002, 10(16): 740–751

17 Cafforio C, Prati C, Rocca F. SAR data focussing using seis-

mic migration techniques. IEEE Trans Aerosp Electron Syst,

1991, 27(2): 195–207

18 Prati C, Rocca F. Focusing SAR data with time-varying

doppler centroid. IEEE Trans Geosci Remote Sens, 1992,

30(3): 550–559

19 Stolt R H. Migration by Fourier transform techniques. Geo-

phys, 1978, 43(1): 23–48

20 Tan W X, Wang Y P, Hong W, et al. SAR Three-dimensional

imaging experiments with microwave anechoic chamber SAR

data. In: Proceedings of Asia Pacific Microwave Conference.

Bangkok: Bangkok University Press, 2007. 1423–1426


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