SYNTHETIC-APERTURE RADAR IMAGING AND WAVEFORM DESIGN FOR DISPERSIVE MEDIA By Jos´ e H´ ector Morales B´ arcenas A Thesis Submitted to the Graduate Faculty of Rensselaer Polytechnic Institute in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Major Subject: MATHEMATICS Approved by the Examining Committee: Professor Margaret Cheney, Thesis Adviser Professor Joyce McLaughlin, Member Professor Birsen Yazici, Member Professor Isom Herron, Member Professor David Isaacson, Member Dr. Trond Varsolt, Member Rensselaer Polytechnic Institute Troy, New York August 2008 (For Graduation December 2008)
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SYNTHETIC-APERTURE RADAR IMAGING ANDWAVEFORM DESIGN FOR DISPERSIVE MEDIA
3.1 Comparisons of the image reconstruction variances by the different in-put signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
v
LIST OF FIGURES
1.1 Background permittivity medium εr with a perturbation εT . . . . . . . . 5
2.1 Radar scene and the transmitted waveform. . . . . . . . . . . . . . . . 20
2.2 In Figs. 2.2(a) and 2.2(c) we show the noiseless data projections gener-ated by one single scatterer. The dotted lines over these plots indicateslices on the projections shown in Figs. 2.2(b) and 2.2(d), respectively.In the dispersive case 2.2(c) the projections have been smoothed outby attenuation and Fig. 2.2(d) shows clearly the presence of the edgetransients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 3D surface plots of the image reconstructions from noiseless data for thesame two scatterers separated by ∆x = 20 cm. Note that the scale onthe vertical axis is different in the two plots. We have normalized theamplitudes with respect to the non-dispersive case image. In 2.3(d), wenote the smoothness and the attenuation of the image due to dispersion. 25
2.4 Cross-sectional comparisons of the reconstructed positions of two differ-ent scatterer’s in different media. In the dispersive medium, 2.4(d) and2.4(f), the two scatterers positions have merged into a single one. In2.4(e) we see that the minimum resolvable distance between the scatter-ers is /1 cm. In the last two plots, 2.4(e) and 2.4(f), we have increasedthe number of sampled points. . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 Profile of the real and imaginary parts of the complex-valued refractiveindex with a relaxation time parameter τ = 10.1 ps. . . . . . . . . . . . 27
3.1 Profile of the function a0 (Eqn. (3.26)) when y0 = 0, r = 100 m, h = 10m, and nI is the imaginary part of the square-root of the Fung-Ulabypermittivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 Profile of Wλ at different values of λ : λ1 < λ2 < λ3. Other parametersare fixed as follows: σTn = 1, y0 = 0, r = 100 m, h = 10 m, and nI isthe imaginary part of the square-root of the Fung-Ulaby permittivity. . 39
3.3 Function Wλ vs. λ. Observe how the amplitude of Wλ increases consid-erably and how its peak moves to higher frequencies when we decreasethe value of λ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4 Function Wλ vs. the target-to-noise ratio σTn = ST/Sn for a fixed valueof λ. The amplitude of Wλ is reduced slightly and its peak moves tolower frequencies when we decrease the value of σTn. We have normal-ized the Wλ amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.7 Input signals used in the numerical simulations. . . . . . . . . . . . . . 44
3.8 Profile of the real and imaginary parts of the complex-valued refractiveindex with a relaxation time parameter τ = 8 ns. . . . . . . . . . . . . . 45
3.9 Ten reconstructions from the Brilluoin precursor waveform input signalfor different realizations of noise when the target-to-noise ratio = 1. . . 46
3.10 Ten reconstructions from the windowed sinusoidal waveform input sig-nal for different realizations of noise when the target-to-noise ratio =1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.11 Ten reconstructions from the optimal waveform input signal for differentrealizations of noise when the target-to-noise ratio = 1. . . . . . . . . . 48
3.12 Average of the ten reconstructions from the three input signals for dif-ferent realizations of noise. . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.13 Surface of the average of the reconstruction from the precursor inputsignal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.14 Surface of the average of the reconstruction from the windowed sinu-soidal input signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.15 Surface of the average of the reconstruction from the optimal input signal. 52
3.16 Slice on the average of the reconstructions at position y = 0. Thescatterers are located at x = −1.5 and x = 1.5. . . . . . . . . . . . . . . 53
vii
ACKNOWLEDGMENT
I am very grateful to my academic advisor, professor Margaret Cheney, for her
guidance, support and patience throughout these years. I wish to thank Trond
Varslot who offered helpful advice and comments in the preparation of this work.
Much of his co-operation is reflected on these pages.
This work was supported by the Air Force Office of Scientific Research 1 under
agreement number FA9550-06-1-0017.
The beginning of my graduate studies would not have been possible without
financial support from the National Council for Sciences and Technology (CONA-
CyT) from Mexico.
I would like to acknowledge my friends Luigi Vanfretti and Jesus Adrian
Espınola in the enterprise of completing this dissertation.
Finally and most of all I would like to thank my wife, Patricia Hernandez, for
her love, constant support and daily encouragement through all these years in the
United States. In the preparation of this work I received practical and analytical
help thanks to her excellent mathematical skills.
1Consequently the U.S. Government is authorized to reproduce and distribute reprints for Gov-ernmental purposes notwithstanding any copyright notation thereon. The views and conclusionscontained herein are those of the author and should not be interpreted as necessarily represent-ing the official policies or endorsements, either expressed or implied, of the Air Force ResearchLaboratory or the U.S. Government.
viii
ABSTRACT
In this dissertation we develop a method for synthetic-aperture radar (SAR) imag-
ing through a dispersive medium and we provide a method to obtain the optimal
waveform design for imaging.
We consider the case when the sensor and scatterers are embedded in a ho-
mogeneous dispersive material, and the scene to be imaged lies on a known surface.
We use a linearized (Born) scalar scattering model, and allow the flight path of the
radar antenna to be an arbitrary smooth curve.
We formulate our filtered back-projection imaging algorithm in a statistical
framework where the measurements are polluted with thermal noise. We assume
that we have prior knowledge about the power-spectral densities of the scene and
the noise.
We test our algorithms when the scene consists of point-like scatterers located
on the ground. The position of the targets is well resolved when the target-to-noise
ration is relatively small. For relatively large noise levels, the position of the targets
are still well resolved employing the optimal waveform as an input signal in the
reconstruction algorithm.
We show the results of simulations in which the dispersive material is modeled
with the Fung-Ulaby equations for leafy vegetation. However, the method is also
applicable to other dielectric materials where the dispersion is considered relevant
in a frequency range of the transmitted signals.
ix
CHAPTER 1
WAVE PROPAGATION THROUGH A DISPERSIVE
MEDIUM
If the speed of wave propagation in a medium depends on frequency, the medium
is said to be dispersive. Most materials are dispersive to some extent [1]; however,
most current work on radar imaging neglects the effect of dispersion. For most radar
systems, this is a reasonable assumption, because a) the dispersion is very weak in
dry air, and b) the frequency bands of most radar systems are not wide enough for
dispersive effects to be important. However, for imaging radar systems, the range
resolution is proportional to the bandwidth. As a consequence, high-resolution
systems require a broadband pulses, and for such pulses, the issue of dispersion may
become important. Therefore, it is of interest to develop synthetic-aperture radar
(SAR) imaging algorithms that account for dispersive wave propagation [2].
SAR image formation is closely related to the solution of an inverse scattering
problem. The time-domain direct and inverse scattering problems for dispersive
media have been extensively studied by many authors [3, 4, 5, 6, 7, 9, 10]. However,
most of this work is for the one-dimensional case. An optimization-based recon-
struction of a parametrized multidimensional dispersive medium was demonstrated
in [11]. A SAR image formation method for propagation through the ionosphere
was recently developed in [8]. A SAR imaging method for imaging through a weak
dispersive layer was developed in [2], where the wave propagation was modeled in
terms of a Fourier integral operator (FIO).
This dissertation is based on SAR imaging methods developed in [29] and
[23] for non-dispersive media. On the other hand, this work also complements the
work developed in [22] and [39] in a statistical setting where the measurements are
corrupted with noise and clutter.
In this dissertation we will develop an algorithm for forming SAR images of
objects that are embedded in a known homogeneous dispersive background. Al-
though we follow the general strategy of [2], we find that our linearized forward
1
2
model is not a FIO, and thus our imaging algorithm does not follow from the FIO
theory.
The dissertation is divided into three chapters. In Chapter 1 we provide the
mathematical model for scattering in a dispersive medium, and define the direct and
inverse problem. In Chapter 2, we show how the inverse problem is solved approx-
imately by means of a filtered back-projection reconstruction method. Numerical
examples in Sec. 2.1 illustrate the implementation of the algorithm. Finally chapter
3 is dedicated to answering the question: Given a particular dispersive material
model, what is the optimal waveform for imaging?
1.1 Electromagnetic Scattering Model
In this section we discuss the wave equations that govern the evolution of
electromagnetic fields in the presence of scatterers in a dispersive medium.
1.1.1 Maxwell’s Equations and Constitutive Relations
Our starting point is the Maxwell equations, which are given in differential
form by [12]
∇ ·D(x, t) = ρ(x, t) (1.1a)
∇ ·B(x, t) = 0 (1.1b)
∇×E(x, t) = −∂tB(x, t) (1.1c)
∇×H(x, t) = ∂tD(x, t) + J(x, t) (1.1d)
Here, D is the electric displacement, B is the magnetic induction, E is the electric
field, H is the magnetic field, ρ is the charge density, J is the current density,
x = (x1, x2, x3) ∈ R3 represents Cartesian coordinates, t ∈ R is time, and ∇· and
∇× denote divergence and curl operators, respectively.
In order to complete this set of equations, we need to add some constitutive
3
relations. We will use the following relations [3, 13]:
In the non-dispersive case, we replace nR(ωn) in (2.48) by the constant nR(ω0)
(see step (f) in 2.1.1).
(b) We discretize the filter (2.39), Qoptmn, as follows
Qoptmn =
anm(zις)Pn|anm(zις)|2|Pn|2Jnm(zις) + Sn/ST
. (2.49)
where anm(zις) = a(ωn, sm, zι, zς), Pn = P (ωn) and Jnm(zις) = J(ωn, sm, zι, zς).
The Jacobian J is shown in Appendix C for flat topography and a circular
flight path.
(c) We use normally-distributed white noise with constant spectrum Sn, and a
constant reflectivity spectrum ST for the scatterers.
(d) Finally, we form the image of the scatterer positions on a square region in
z-space where (zι, zς) = z.
Figure 2.3 displays surface plots of the image reconstructions of two scatterers
embedded in a homogeneous non-dispersive and a dispersive medium, respectively.
The scatterers are separated by a distance of 40 cm and their locations are well
resolved by the algorithm. The wiggles on the base of the peaks in Fig. 2.3(d) are
typical “star pattern” artifacts associated with back-projection reconstructions [30].
In Fig. 2.4, we show the effect of decreasing the distance between scatterers.
We display only transversal cuts of the (not shown) full reconstructed images. In
Fig. 2.4(a), we show a slice of Fig. 2.3(c); in Fig. 2.4(b) we display a slice of figure
2.3(d). In Figs. 2.4(c) and 2.4(d) the distance between the scatterers is 14 cm.
24
Fast−Time (s)
Slow−T
ime
(rad)
Non−Dispersive Material
0 1 2 3 4 5 6 7 8x 10−8
0
1
2
3
4
5
6
(a) Noiseless data projections.
2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5x 10−8
−4
−3
−2
−1
0
1
2
3
4x 10−7
Fast−Time (s)
V/m
Back−Scattered Wavefield
(b) Projections slice.
Fast−Time (s)
Slow−T
ime
(rad)
Dispersive Material
0 1 2 3 4 5 6 7 8x 10−8
0
1
2
3
4
5
6
(c) Noiseless data projections.
2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5x 10−8
−3
−2
−1
0
1
2
3
x 10−9
Fast−Time (s)
V/m
Brillouin Precursors
(d) The leading and trailing edge transientsare called Brillouin precursors.
Figure 2.2: In Figs. 2.2(a) and 2.2(c) we show the noiseless data pro-jections generated by one single scatterer. The dotted linesover these plots indicate slices on the projections shown inFigs. 2.2(b) and 2.2(d), respectively. In the dispersive case2.2(c) the projections have been smoothed out by attenua-tion and Fig. 2.2(d) shows clearly the presence of the edgetransients.
25
However, we can still distinguish two peaks in both cases. We see that the resolution
in the dispersive case is approximately 14 cm. From Fig. 2.4(e) we see that the
maximum resolution for the non-dispersive case is reached around 1 cm. In the
dispersive case, Fig. 2.4(f), we see only one point. Notice that in both cases noise is
not present in the data, but the dispersion smoothes the images. Observe the relative
decrement in amplitude for the dispersive case relative to the non-dispersive one.
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(a) Non-dispersive material
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(b) Dispersive material
(c) Non-dispersive material (d) Dispersive material
Figure 2.3: 3D surface plots of the image reconstructions from noiselessdata for the same two scatterers separated by ∆x = 20 cm.Note that the scale on the vertical axis is different in the twoplots. We have normalized the amplitudes with respect to thenon-dispersive case image. In 2.3(d), we note the smoothnessand the attenuation of the image due to dispersion.
Figure 2.4: Cross-sectional comparisons of the reconstructed positions oftwo different scatterer’s in different media. In the dispersivemedium, 2.4(d) and 2.4(f), the two scatterers positions havemerged into a single one. In 2.4(e) we see that the mini-mum resolvable distance between the scatterers is /1 cm. Inthe last two plots, 2.4(e) and 2.4(f), we have increased thenumber of sampled points.
27
108 109 1010 1011 1012 1013 1014 10150
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
frequency (Hz)
Fung−Ulaby Model Dielectric
nRnI
Figure 2.5: Profile of the real and imaginary parts of the complex-valuedrefractive index with a relaxation time parameter τ = 10.1 ps.
2.2 Concluding Remarks
In this chapter, we have developed a formula for forming SAR images through
a temporally dispersive medium. For a given waveform, we derived a filtered-back-
projection reconstruction method, the filter of which was optimally designed to
minimize the reconstruction error in the mean-square sense.
We showed reconstructions from this method for the case of a transmitted
waveform consisting of a square-wave-modulated sine wave.
In the next chapter, we address the issue of optimal waveform design for imag-
ing in dispersive media.
CHAPTER 3
WAVEFORM DESIGN FOR DISPERSIVE MEDIA
A problem of particular interest is the determination of the structural form of the
input pulse that is the best for imaging in a dispersive medium [34]. In this chap-
ter, in a statistical framework, we provide an optimal waveform design theory for
synthetic-aperture imaging through dispersive media. From previous work [22] and
[39], we follow a variational approach to obtain an optimal waveform in terms of
minimising the mean square error in the resulting image.
The strategy is as follows. After the optimal filter Qopt is substituted into the
functional (2.36), this functional depends on the signal power spectrum amplitude
only. From the extrema of the functional we get the optimal power spectrum. We
impose the constraint that the total power of the input signal is constant along the
entire antenna flight path trajectory.
The final objective in this chapter is to find an optimal waveform and not only
its power spectrum. To this end, we perform a spectral decomposition getting the
minimum-phase and causal signal with the initial optimal power spectrum. This
spectral decomposition method is know as the Kolmogorov spectral decomposition.
In the previous chapter, we suppressed the possible dependence of the wave-
form on the slow-time parameter s. Here, however, we explicitly include the possi-
bility that different waveforms could be transmitted from different positions on the
flight path, and consequently the power spectrum amplitude depends on both ω and
s.
28
29
3.1 Determination of the Power Spectrum Amplitude of the
Waveform
The departure point is the functional given by Eqn. (2.36). If we substitute
the optimal filter Qopt into (2.36), we get a new functional
∆(P ) =
∫Ωy
dωds
∫Γ
dyST (ω, s)/J(ω, s,y)(
ST (ω,s)Sn(ω,s)
J(ω, s,y)|a(ω, s,y)|2|P (ω, s)|2 + 1) , (3.1)
where |P |2 is the power spectrum that we want to determine, J is the Jacobian that
comes from the Stolt change of variables (2.13), a is the geometrical spreading factor
(2.8), which is a real-valued and non-negative function (we will drop the absolute
value sign), ST and Sn are the target and noise power spectrum respectively (see
Sec. 2.0.2), the domain Ωy is defined in (2.16) and Γ ⊂ R2 is the surface where we
form images.
In terms of the spectral density functions, we define the (frequency-dependent)
target-to-noise ratio as follows:
σTndef=STSn
(target-to-noise ratio). (3.2)
The noise-to-target ratio is defined by σnT = Sn/ST
We perform a variational derivative of Eqn. (3.1) with respect to |P |2, subject
to the constraint that the total transmitted energy along the flight path is a constant
MP ; i.e., ∫Ωy
dωds|P (ω, s)|2 = MP . (3.3)
Thus, the cost functional E(λ, P ) is defined as follows:
E(λ, P )def= ∆(P ) + λ
(∫Ωy
dωds|P (ω, s)|2 −MP
). (3.4)
The parameter λ is the Lagrange multiplier and is constant with respect to the
variables (ω, s). Note that E depends explicitly on |P |2. Then, we simplify the cal-
culations writing W = |P |2, where W = W (ω, s). Therefore, we minimize E(λ,W )
30
with respect to W :
0 =d
dε
∣∣∣∣ε=0
E(λ,W + εWε), (3.5)
where ε is a parameter and Wε(ω, s) is an admissible function. From Eqn. (3.5) we
obtain the necessary condition for the existence of the optimal W :
0 = −∫
Ωy
dωds
(WεSTσTn
∫Γ
dya2
(σTnJa2W + 1)2
)+ λ
∫Ωy
dωdsWε. (3.6)
Eqn. (3.6) must be valid for any arbitrary continuous function Wε(ω, s). Note that
the two terms on the right-hand side of Eqn. (3.6) are constant. We rewrite this
expression as follows:
0 =
∫Ωy
dωds Wε Λ, (3.7)
where
Λ(ω, s) = λ− STσTn∫
Γ
dya2
(σTnJa2W + 1)2. (3.8)
From (3.7) we conclude that, if Λ is a continuous function of (ω, s) ∈ Ωy, it must
be identically zero 6. Therefore, from Eqn. (3.8) we have that
λ = STσTn
∫Γ
dya2
(σTnJa2W + 1)2. (3.9)
Note that λ must be positive.
Next we develop an approach to obtain a formula for W in the case when
the surface Γ, on which we form images, is considered small in the sense that the
integrand does not vary too much on Γ. We observe that the integrand of the
right-hand side of (3.9) is a function of (ω, s) and y; i.e.,
Because L(ω, s,y) is everywhere non-negative and continuous in the region Γ, we
6See [37] Ch. IV, §3.4.
31
can approximate the integral in Eqn. (3.9) by the Mean Value Theorem 7. Thus,∫Γ
dyL(ω, s,y) = L0(ω, s,y0) U, (3.11)
where U =∫
Γdy is the area of the region Γ ⊂ R2, L0 = L(ω, s,y)|y=y0
and y0 ∈ Γ.
Then Eqn. (3.9) becomes
λ =STσTna
20U
(σTnJ0a20W + 1)2
, (3.12)
where a0 = a(ω, s,y0) and J0 = J(ω, s,y0).
From (3.12), we solve for W in terms of λ:
Wλ =1
σTnJ0a0
(σTn
√SnU
λ− 1
a0
). (3.13)
The function Wλ depends on the functions a0 and J0 (see (2.8) and (C.2)), which
both depend on the complex-valued index of refraction n(ω). Explicitly,
a0 =ω2e−ϑI(ω)|rs,y0 |
(16π3|rs,y0|)2, and
J0 =
∣∣∣∣vp(ω)vg(ω)
ω
∣∣∣∣ |rs,y0 |4|ψ(y0) · γ(s)− r2|
,
(3.14)
where ψ is the ground surface defined in Subsection (1.1.2), γ(s) is the antenna
flight trajectory introduced in Sec. 1.1.3, |rs,y0 | is the distance between the ground
point y0 and the antenna position, r is the radius of the antenna circular flight path
trajectory, vg and vp are the group and the phase velocity respectively [See (C.3)],
ϑI(ω) = 2ωnI(ω)/c0 and nI(ω) is the imaginary part of the complex-valued index
of refraction (see Fig. 3.2).
Observe that a0 is a well-behaved function of s if the distance |rs,y0 | is a
smooth and bounded function of s. On the other hand, when the frequency ω → 0
and ω → ∞, the term −1/a0 in (3.13) diverges to −∞ (see below Fig. (3.1)).
However, Wλ must be a positive quantity to be considered a meaningful power
7See [38] Ch. IV, §5.
32
spectrum amplitude |P |2. Thus, from (3.13) we have that
σTn
√SnU
λ>
1
a0
, (3.15)
which implies that the Lagrange multiplier λ must satisfy the inequality
0 < λ < Uσ2Tn(ω, s)Sn(ω, s)a2
0(ω, s,y0). (3.16)
This expression (3.16) is equivalent to the expression (68) in [39]. The positiveness
condition of Wλ restricts the values of (ω, s) to some finite sub-region of 8
Ω0 = (ω, s) : ω ∈ (ωmin, ωmax) ⊂ R+ and s ∈ (0, 2π] ⊂ R, (3.17)
where we use the same notation as in (2.16). In practice, outside of the interval
(ωmin, ωmax), Wλ can be approximated by zero (see the numerical simulation section
below).
3.1.1 Determination of the Lagrange multiplier with the help of the
total energy
Now, we determine the Lagrange multiplier λ from the expression (3.13) and
with the help of the constraint (3.3). We integrate both sides of (3.13) with respect
to ω and s to get that
MPdef=
∫Ω0
dωdsWλ =
√U
λ
∫Ω0
dωds
√Sn
J0a0
−∫
Ω0
dωds1
σTnJ0a20
. (3.18)
We solve (3.18) for√U/λ to obtain
√U
λ=MP +
∫Ω0
dωds 1σTnJ0a2
0∫Ω0
dωds√Sn
J0a0
,
λ = U
( ∫Ω0
dωds√Sn
J0a0
MP +∫
Ω0dωds 1
σTnJ0a20
)2
.
(3.19)
8Notice that we change Ωy by Ω0 in (3.3), because the expressions are evaluated at a particularpoint y = 0.
33
For fixed values of Sn and ST , we observe that the relation between λ and the total
energy MP is MP ∝ 1/√λ. Thus, if we decrease λ we increase the total energy and
vice versa. From (3.16) we see that λ is bounded above, which implies that the total
energy MP is bounded below:1√λmax
. MP , (3.20)
where λmax is provided in (3.16). This means that there exists a minimum total
energy MP that guaranties that the function Wλ is positive in the region Ω0. With
the help of the Lagrange multiplier, we can adjust the power constraint (3.18) to a
desired value. Actually, we use this approach in the numerical simulations, where
we look for a λ that allows to Wλ satisfies the power constraint.
Consider the case in which Wλ is positive. We rewrite (3.13) as follows
Wλ
Sn=
1
J0a0
(√U
Snλ− 1
STa0
). (3.21)
When λ→ 0, the value of the ratio Wλ/Sn increases without bound. This solution
for Wλ means that we could transmit high power to compensate for the noise level.
When λ→ λmax, the ratio Wλ/Sn decreases its value monotonically to zero.
3.2 Getting the Full Waveform via Spectral Factorization
We have obtained a formula for the optimal power spectrum W (ω); now we
need to find a causal waveform p(t) whose Fourier transform P (ω) satisfies
W (ω) = |P (ω)|2. (3.22)
The problem of factoring the power spectrum W (ω) in this manner is the same as
the problem of finding the transfer function of a desired minimum-delay filter, i.e.,
the transfer function and its inverse are causal and stable [40].
Spectral decomposition is used to recover the minimum-phase complex-valued
(causal) signal given its power spectrum amplitude only. Thus, the transfer function
34
P (ω) of a desired filter may be expressed as
P (ω) =√W (ω)eiΘ(ω), (3.23)
where Θ(ω) is the desired phase spectrum. The method is due to Kolmogorov (see
[41]).
We use of the z-transform where we define z = eiω. Under this transform P (ω)
becomes P (z). Since P (z) has no singularities or zeros within the unit circle, then
ln P (z) is analytic within this region 9. On the unit circle, the spectrum is specified
as follows
P (z)P (1/z) = W (z) = eK(z) = eC(1/z)+C(z) = eC(1/z)eC(z). (3.24)
Given the spectrum W (z) for each value on the unit circle, we could deduce the log
spectrum K(z) = ln W (z) at each point on the unit circle:
K(z) = ln[S(z)] = C(1/z) + C(z). (3.25)
This is the answer we have been looking for. Given K(z) for all real values of ω,
we could inverse transform to the time domain, obtaining the two-sided function
u(t) = c(−t) + c(t). Setting to zero the coefficients at negative times eliminates
c(−t), leaving just c(t); whose Fourier transform is C(z). And we know that the
exponential of C(z) gives P (z) with a causal p(t). This method is implemented
below in the next section.
3.3 Numerical Simulations
We calculate numerically the optimal waveform and its power spectrum fol-
lowing the procedure above. Then, we use that waveform for imaging. At the end
of this section we compare the variance, Eqn. (2.28), of the image reconstructions
from the optimal, the Brillouin precursor and the square-wave-modulated sinusoidal
9Since this spectrum can be zero at some frequencies, and since the logarithm of zero is infinite,there is a pitfall. When the logarithm of zero arises during the computation, it is replaced by thelog of a small number (see [41]).
35
waveforms.
In our simulations, we consider a simplified scenario in which the amplitude
of the spectral density functions of noise Sn and the target ST are constants. The
spectral density function of the target is proportional to the target reflectivity T
and we assume that the noise spectral density is proportional to ST , i.e., Sn = βST ,
where β is a constant (see Sec. 2.1).
We choose the ground point of reference that appears in the Eqn. (3.11) to
be y0 = (0, 0). With this value of y0, the formulas (3.14) no longer depend on the
slow-time parameter s because the distances from the antenna flight trajectory to
the point y0 are the same:
a0 =ω2e−ϑI(ω)
√r2+h2
(16π3)2(r2 + h2), and
J0 =
∣∣∣∣vp(ω)vg(ω)
ω
∣∣∣∣ √r2 + h2
4r2.
(3.26)
The point y0 is located exactly beneath the center of the circular flight trajectory
of radius r = 100 m and an altitude of h = 10 m. The area of reconstruction is
U = 100 m2. The two point-like scatterers are located on the ground at (−1.5, 0)
and (1.5, 0), respectively. In Fig. 3.1 we show the profile of the function a0(ω,y0).
When we uniformly discretize the slow-time parameter s, the total energy MP
is then equally distributed in Ns firings (Ns is the number of points on the antenna
flight path trajectory). Consequently, on the plots, for a fixed λ, the (positive) area
under the graph of |P (ω, s)|2 for every s is proportional to a fraction (MP/Ns) of
the total energy.
The complex-valued refractive index n(ω) used in the simulations is provided
by the Fung-Ulaby model dielectric (see Appendix A.3.2). We show the profile of
this refractive index in Fig. 3.8.
In Fig. 3.2 we show the behavior ofWλ when we change the Lagrange multiplier
value λ. There are values of λ that give us Wλ < 0; this is the case for λ3. On the
other hand, there are λ for which Wλ has both positive and negative values; this is
the case for λ1. The intermediate case is λ2, when we get that Wλ ≤ 0. The relation
between the λ values is the following: λ1 < λ2 < λ3.
36
In Fig. 3.3 we show how the function Wλ changes as λ → 0. Note that the
peak of Wλ moves to higher frequencies when λ decreases.
In Fig. 3.4 we show how the function Wλ changes when we change the target-
to-noise ratio σTn. We keep ST constant and change the noise amplitude Sn only.
When we decrease σTn, i.e., when we increase the noise amplitude, it is interesting
to note that the normalized amplitude of Wλ decreases slightly but its peak moves
notably toward lower frequencies.
In Fig. 3.5 we show the procedure to obtain a causal waveform from the am-
plitude of its power spectrum. Firstly, given a value for the Lagrange multiplier
λ, we obtain the function Wλ(ω), shown in Fig. 3.5(a). We replace the negative
values by zeros (zero-padding) to get the power spectrum prototype |P (ω)|2, shown
in Fig. 3.5(b). We take the logarithm ln(|P |2) and, by means of the inverse Fourier
transform, we obtain a time domain representation of this quantity. We set to zero
the non-causal part in the time domain. Then, we apply the Fourier transform to
get an expression in the frequency domain and exponentiate to invert the logarithm.
Finally, we apply the inverse Fourier transform to return to the time-domain, ob-
taining a causal waveform, shown in Fig. 3.5(c). In Fig. 3.5(d) is shown the power
spectrum from Fig. 3.5(c), which is indeed an approximation of the power spectrum
shown in Fig. 3.5(b).
In Fig. 3.6 we show a detail of the optimal waveform 3.5(c). Note that the
wavelength of this wave-field is shortened and its amplitude increases at the begin-
ning of the plot.
In Fig. 3.7 we show the three different input signals that we employ for imaging.
In Fig. 3.8 we show the profile of the real and imaginary parts of the complex-valued
index of refraction when the relaxation time parameter τ = 8 ns. The noticeable
change of the index of refraction takes place in the vicinity of 0.1 GHz, around this
value are located the carrier frequencies of the input signals.
In Figs. 3.9, 3.10 and 3.11 we show ten reconstructions of the three different
input signals at different realization of noise with the same target-to-noise ratio
σTn = 1.
In Fig. 3.12 we show the image reconstruction averages that come from each
37
input signal. When we employ the square-wave-modulated sinusoidal and the op-
timal waveforms, we can distinguish the two point-like scatterers located on the
ground. In Figs. 3.13, 3.14 and 3.15 we show surfaces of this reconstructions. In
Fig. 3.16 we show a slice of each average image at y = 0 position, where we observe
the difference between reconstructions when we employ different input signals.
In Chapter 2 we determined the optimal filter Q, and in this chapter we
determine the optimal waveform so that the MSE ∆(Q,P ) is minimized. From
Eqn. (2.23) and (2.21) we can see that
V(Q,P ) =
∫dz 〈|E(z)− 〈E(z)〉|2〉
=
∫dz 〈|I(z)− 〈I(z)〉|2〉.
(3.27)
We show the normalized variance V in Table 3.3 for the three different input signals.
The variance calculated with the optimal waveform input signal is the smallest one.
Table 3.1: Comparisons of the image reconstruction variances by the dif-ferent input signals.
3.4 Concluding Remarks
For a large amount of power (small λ), the optimal waveform spectrum Wλ
shifts to higher frequencies. If the target-to-noise ratio is larger in a particular
frequency band, then this is also where the waveform power spectrum is large.
For a given noise level on the forward data and when we employ the windowed
sinusoidal and the optimal waveforms, we can resolve the position and reflectivity
amplitude of the two point-like scatterers on the ground (see Fig. 3.16). However,
from the variances reported in Table 3.3, we observe that imaging with the optimal
38
0 0.5 1 1.5 2 2.5 3x 109
0
0.5
1
1.5
2
2.5
3
3.5x 1026
frequency (Hz)
a 0(ω)
Figure 3.1: Profile of the function a0 (Eqn. (3.26)) when y0 = 0, r = 100m, h = 10 m, and nI is the imaginary part of the square-rootof the Fung-Ulaby permittivity.
input signal we obtain reconstructions that spread out less from its mean value than
if we employ the other two input signals. In order to affirm that this is because of
the optimal waveform “robustness”, we need to explore more on the effect of other
levels of noise.
We define the total energy constraint MP in (3.3) on a region given in (3.17).
We found that the Lagrange multiplier λ is related to this total energy in (3.19).
This expression (3.19) comes from the fact that the function Wλ is non-negative only
on the region Ω0 (see Fig. 2.1). Outside of Ω0, Wλ is negative. This means that if
Wλ is an approximation of a power spectrum (PS) |P (ω, s)|2, then the corresponding
real-valued and causal signal p(t) must be determined via the spectral factorization
using the frequencies in Ω0. From this point of view, the zero-padding of Wλ should
39
0 2 4 6 8 10 12x 108
−1.5
−1
−0.5
0
0.5
1
1.5x 10−20
frequency (Hz)
Wλ
λ3λ2λ1
Figure 3.2: Profile of Wλ at different values of λ : λ1 < λ2 < λ3. Otherparameters are fixed as follows: σTn = 1, y0 = 0, r = 100 m,h = 10 m, and nI is the imaginary part of the square-root ofthe Fung-Ulaby permittivity.
be considered another approximation to |P (ω, s)|2 (see Fig. 3.5).
On the other hand, we do not restrict the frequency domain of the precursor
and the windowed sinusoid waveforms in the same way as we do with the optimal
waveform. For the comparisons in this work to hold, we design the other waveforms
in a way that their carrier frequencies remain in the same region Ω0. Thus, the
comparison between different signals, with the same MP , is valid.
40
0 5 10 15
x 108
0
0.5
1
1.5
x 10−19 λ = 0.01
f (Hz)
Wλ
0 0.5 1 1.5 2
x 109
0
0.5
1
1.5
2
x 10−16 λ = 1e−05
f (Hz)
Wλ
0 1 2 3
x 109
0
0.5
1
1.5
2
2.5
3
3.5
x 10−6 λ = 1e−15
f (Hz)
Wλ
0 1 2 3 4
x 109
0
1
2
3
4
5x 10
9 λ = 1e−30
f (Hz)
Wλ
0 2 4 6
x 109
0
1
2
3
4
5
6
7x 10
39 λ = 1e−60
f (Hz)
Wλ
0 2 4 6
x 109
0
2
4
6
8x 10
59 λ = 1e−80
f (Hz)
Wλ
Figure 3.3: Function Wλ vs. λ. Observe how the amplitude of Wλ increasesconsiderably and how its peak moves to higher frequencieswhen we decrease the value of λ.
41
5.2 5.4 5.6 5.8 6
x 109
0
0.2
0.4
0.6
0.8
1
σTn
= 1e+06
f (Hz)
Wλ
5.2 5.4 5.6 5.8 6
x 109
0
0.2
0.4
0.6
0.8
1
σTn
= 1e+04
f (Hz)
Wλ
5.2 5.4 5.6 5.8 6
x 109
0
0.2
0.4
0.6
0.8
1
σTn
= 1
f (Hz)
Wλ
5.2 5.4 5.6 5.8 6
x 109
0
0.2
0.4
0.6
0.8
1
σTn
= 1e−04
f (Hz)
Wλ
5.2 5.4 5.6 5.8 6
x 109
0
0.2
0.4
0.6
0.8
1
σTn
= 1e−06
f (Hz)
Wλ
5.2 5.4 5.6 5.8 6
x 109
0
0.2
0.4
0.6
0.8
1
σTn
= 1e−09
f (Hz)
Wλ
Figure 3.4: Function Wλ vs. the target-to-noise ratio σTn = ST/Sn for afixed value of λ. The amplitude of Wλ is reduced slightly andits peak moves to lower frequencies when we decrease thevalue of σTn. We have normalized the Wλ amplitude.
42
0 2 4 6 8 10 12 14 16 18 20x 108
−0.5
0
0.5
1
frequency (Hz)
Wλ
negative values
(a)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 109
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
frequency (Hz)
|P(ω
)|2
"zero"−padding
(b)
0 5 10 15x 10−8
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
time (s)
p(t)
(c)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 109
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
frequency (Hz)
|P(ω
)|2
(d)
Figure 3.5: Minimum-phase optimal waveform via spectral factorizationfor Fung-Ulaby model dielectric.
Figure 3.7: Input signals used in the numerical simulations.
45
106 107 108 1090
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
frequency (Hz)
Fung−Ulaby Model Dielectric
nRnI
Figure 3.8: Profile of the real and imaginary parts of the complex-valuedrefractive index with a relaxation time parameter τ = 8 ns.
46
!5 0 5
!5
0
5!5 0 5
!5
0
5
!5 0 5
!5
0
5!5 0 5
!5
0
5
!5 0 5
!5
0
5!5 0 5
!5
0
5
!5 0 5
!5
0
5!5 0 5
!5
0
5
!5 0 5
!5
0
5!5 0 5
!5
0
5
Figure 3.9: Ten reconstructions from the Brilluoin precursor waveforminput signal for different realizations of noise when the target-to-noise ratio = 1.
47
−5 0 5
−5
0
5−5 0 5
−5
0
5
−5 0 5
−5
0
5−5 0 5
−5
0
5
−5 0 5
−5
0
5−5 0 5
−5
0
5
−5 0 5
−5
0
5−5 0 5
−5
0
5
−5 0 5
−5
0
5−5 0 5
−5
0
5
Figure 3.10: Ten reconstructions from the windowed sinusoidal wave-form input signal for different realizations of noise whenthe target-to-noise ratio = 1.
48
−5 0 5
−5
0
5−5 0 5
−5
0
5
−5 0 5
−5
0
5−5 0 5
−5
0
5
−5 0 5
−5
0
5−5 0 5
−5
0
5
−5 0 5
−5
0
5−5 0 5
−5
0
5
−5 0 5
−5
0
5−5 0 5
−5
0
5
Figure 3.11: Ten reconstructions from the optimal waveform input signalfor different realizations of noise when the target-to-noiseratio = 1.
49
Precursor WF
x
y
−5 0 5
−5
0
5
Sinusoidal WF
xy
−5 0 5
−5
0
5
Optimal WF
x
y
−5 0 5
−5
0
5
0.511.522.533.54
x 10−14
2
4
6
8
10
x 10−14
246810121416
x 10−14
Figure 3.12: Average of the ten reconstructions from the three input sig-nals for different realizations of noise.
50
Figure 3.13: Surface of the average of the reconstruction from the pre-cursor input signal.
51
Figure 3.14: Surface of the average of the reconstruction from the win-dowed sinusoidal input signal.
52
Figure 3.15: Surface of the average of the reconstruction from the opti-mal input signal.
53
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8x 10−13
x−position (m)
Precursor WFSinusoidal WFOptimal WF
Figure 3.16: Slice on the average of the reconstructions at position y = 0.The scatterers are located at x = −1.5 and x = 1.5.
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[6] Fuks P, Karlsson A and Larson G 1994 Direct and inverse scattering fromdispersive media Inverse Problems 10 55571
[7] Oughstun K E and Sherman G C 1994 Electromagnetic Pulse Propagation inCausal Dielectrics Springer-Verlag Berlin
[8] Tsynkov S V 2007 On SAR Imaging through the Earth Ionosphere, to bepublished
[9] Bui D D 1995 On the well-posedness of the inverse electromagnetic scatteringproblem for a dispersive medium Inverse Problems 11 835-863
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[11] Gustafsson M 2000 Wave Splitting in Direct and Inverse Scattering ProblemsPhD Thesis Department of Applied Electronics Electromagnetic Theory,Lund University, Sweden
[12] Jackson J D 1999 Classical Electrodynamics 3rd Edition John Wiley & SonsUSA
[13] Sihvola A 1999 Electromagnetic Mixing Formulas and Applications TheInstitution of Electrical Engineers, London UK
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[14] Lide D R 2004 CRC Handbook of Chemistry and Physics 85th Ed CRC Press,Boca Raton 8-141
[15] Skolnik M I 1962 Introduction to Radar Systems McGraw-Hill, NY USA
[16] Tsang L et al 2000 Scattering of Electromagnetic Waves–Theories andApplications, John Wiley & Sons, Inc. NY USA
[17] Fung A K and Ulaby F T 1978 A scatter model for leafy vegetation IEEETrans. Geosci. Electron. 16 281-6
[18] El-Rayes M A and Ulaby F T 1987 Microwave Dielectric Spectrum ofVegetation-Part I: Experimental Observations IEEE Transactions onGeoscience and Remote Sensing Vol GE-25, No 5 September
[19] Ulaby F T and El-Rayes M A 1987 Microwave Dielectric Spectrum ofVegetation Part II: Dual-Dispersion Model IEEE Transactions on Geoscienceand Remote Sensing Vol GE-25 No 5 September
[20] Milonni, P. W. Fast Light, Slow Light and Left-Handed Light. Series in Opticsand Optoelectronics. Institute of Physics, 2005. Bristol, UK.
[21] Cheney M 2001 A Mathematical Tutorial on Synthetic Aperture Radar SIAMReview Vol 43 No 2 301-312
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[35] Cartwright N A and Oughstun K E 2007 Uniform Asymptotics Applied toUltrawideband Pulse Propagation SIAM Review Vol 49, No 4, 628-648
[36] Epstein C L 2003 Introduction to the Mathematics of Medical imagingPearson, NJ USA
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[39] Varslot, T., Yarman, C. E., Cheney, M. and Yazici, B. 2007 A variationalApproach to Waveform Design for Synthetic-Aperture Imaging. InverseProblems and Imaging. Vol. 1, No. 3, 577-592.
[40] Robinson, E. A. and Treitel, S. 1980. Geophysical Signal Analysis. 1st. Ed.Prentice-Hall, New Jersey, USA.
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APPENDIX A
SOME DISPERSIVE MATERIALS
A.1 Debye model.
Water is an example of a ubiquitous dispersive medium [13]. Between 0 and
20 C, water’s relative permittivity ranges from 88 to 80.1 in contrast to the value
of 1 for vacuum and approximately 1.00054 for air [14]. Water molecules absorb
microwaves and other radio wave frequencies and attenuate radar signals. Different
frequencies attenuate at different rates, such that some components of air are opaque
to some frequencies and transparent to others. Therefore, in its various forms,
depending on whether it is vapor or liquid, water affects strongly the permittivity
of the medium [15]. The Debye model is generally good for polar molecules such as
water in microwave regime [13, 16]. The Debye model is
εr(ω) = ε∞ +εs − ε∞1− iωτ
, (A.1)
where for water, typical values are as follows. The zero-frequency relative permit-
tivity is εs = 80.35, the infinite-frequency relative permittivity is ε∞ = 1.00 and the
relaxation time is τ = 8.13 ps.
We assume that (A.1) satisfies
∂βωf(ω) ≤ Cβ(1 + ω2)−(1+|β|)/2, (A.2)
where f(ω) = (εs − ε∞)/(1 − iωτ) for every non-negative integer β. The large-ω
decay of f(ω) implies that f(ω) appears only in the remainder terms of large-ω
asymptotic calculations [13].
57
58
A.2 Lorentz model.
The Lorentz model, which is a harmonic-oscillator model, is generally good
for solid materials. The permittivity for a Lorentz medium is given by
εr(ω) = ε∞ −ω2p
ω2 − ω20 + 2iων
, (A.3)
where again ε∞ is the high-frequency permittivity of the material. The other pa-
rameters are the plasma frequency ωp, the resonance frequency ω0, and the damping
amplitude ν [13].
A.3 Models for vegetation
Vegetation is a dispersive medium, because water is dispersive and is a major
constituent of leaves and living wood [13][16].
A.3.1 Brown-Curry-Ding model (100 MHz - 10 GHz).
This is a model for a sparse random medium such as the branches and tree
trunks of a forest. A modification due to Ding of the Brown-Curry model is
εr(ω) =
[ε∞ +
b
1 + (ω/ωc)2
]+ i
[(ω/ωc)b
1 + (ω/ωc)2+
a
(2πω)0.96
]. (A.4)
Here ε∞ ≈ 4.5, b is a constant that depends on the density and fractional volume
of wood, a = 1.5 × 109 (respectively 4 × 109) when the electric field is parallel
(perpendicular) to the wood grain and 2πωc is the temperature-dependent frequency
that is roughly 20 GHz at 25 C [2].
For time-domain continuity, the permittivity ε(ω) must decay at high fre-
quencies faster than ω−1 [12]. The Brown-Curry-Ding model does not obey this
condition.
A.3.2 Fung-Ulaby model (8 - 17 GHz).
Roughly speaking, vegetation could be considered a simple mixture of dry
matter, air and water. In our numerical examples, we considered the model for
complex-valued permittivity developed by Fung and Ulaby [17]. In this model, the
59
permittivity of the vegetation (taken to be a combination of water and some solid
material) was estimated by a mixing formula for two-phase mixtures [18] [19]. The
effective relative Debye-type permittivity is given by
εr,eff(ω) = vlεl + (1− vl),
εl = εR(ω) + i εI(ω),
εR(ω) = 5.5 +em − 5.5
1 + τ 2ω2,
εI(ω) =(em − 5.5)τω
1 + τ 2ω2,
em = 5 + 51.56vm.
(A.5)
where vl fractional volume occupied by leaves, vw water volume fraction within a
“typical leaf”, and τ empirical relaxation time of water molecules.
The relaxation time τ is governed by the interaction of the water molecules
with its environment and by the temperature T . For pure water at 20 C, τ ≈10.1× 10−12 s. This is the value that we employ in Sec. 2.1. However, when water
is mixed with other materials, that is the case of bulk vegetation and water, its
response to the electric field changes. If the water molecules are under the influence
of non-electrical forces, such as mechanical or chemical forces, its response to an
applied wave-field is impeded by these forces, which has the equivalent effect of
increasing the relaxation time τ (see for details [18], [19] and [13]).
APPENDIX B
THE METHOD OF STATIONARY PHASE
In this appendix we show how to use the method of stationary phase to approximate
the multiple integral (2.31). This integral is of the type:
∆T (P,Q) =
∫eiφ(x)u(x)dx. (B.1)
The stationary phase method deals with integrals of the form (B.1) in which the
integration is over a compact setK. We consider only the case in which the phase has
only non-degenerate critical points in K. We say that xl ∈ Rn is a non-degenerate
critical point of φ if φ′(xl) = 0 and det(φ′′(xl)) 6= 0, where φ′′(x) =(∂2φ(x)∂xj∂xk
)1≤j,k≤n
is the Hessian of φ at x. The stationary phase theorem states that if u ∈ C∞0 ⊂ Rn,
and φ has only non-degenerate critical points, then as λ → ∞ the leading-order
approximation to (B.1) is obtained from the estimate (See Proposition 2.3 in [24]):∣∣∣∣∣∫eiλφ(x)u(x)dx−
∑l
Rlu(xl)λ−n/2eiλφ(xl)
∣∣∣∣∣ ≤ Cλ−n/2∑|κ|≤n+3
(supK|∂κu(x)|
),
(B.2)
where λ ≥ 1, κ is a multi-index, where
Rl =(2π)n/2eiπsgn(φ′′(xl))/4
|detφ′′(xl)|1/2, (B.3)
and where xl denote the critical points in K, and where sgn(·) denotes the signature
of the Hessian, i.e., the number of positive eigenvalues minus the number of negative
ones.
To put (2.31) into the form (B.1), we first make the change of variables ξ =
λξ, ξ′ = λξ′, where λ = |ξ|. This change of variables converts the phase to φ =
60
61
λ[(z − y) · ξ − (z − y′) · ξ′]. The integrand u of (B.2) (with x↔ (z, ξ′)) is then
u(λξ, λξ′,y,y′, z
)=
1
λ4
[Q(λξ, z
)A(λξ,y
)J(λξ, z,y
)− 1]
×[Q(λξ′, z
)A(λξ′,y′
)J(λξ′, z,y′
)− 1
](B.4)
We apply the stationary phase analysis to the z and ξ′ integrals, so that n in (B.2)
is 4; the leading-order term is (2.34).
The right side of (B.2) decays more rapidly than λ−2 provided u satisfies a
symbol estimate as in [24]. This is the case because a) the flight path is smooth;
b) for the operating frequencies of the radar, the permittivity εr is smooth; and c)
P is assumed to satisfy symbol estimates. Although derivatives with respect to ξ′
give rise to factors of λ, because u involves P , whose derivatives decay increasingly
rapidly in ω (ω being proportional to |ξ| = λ), these extra factors of λ do not lead
to an overall worsening of the decay rate in λ. For the calculation of (2.34), we
assume that Q is smooth; then smoothness of (2.40) can be checked, so that the
assumption is consistent with the result.
The more rapid large-λ decay of the error term corresponds to its being
smoother than the leading-order term.
APPENDIX C
BEYLKIN DETERMINANT
The Stolt change of variables (2.13) introduce the Jacobian
J(ω, s,y) =
∣∣∣∣∂(s, ω)
∂ξ
∣∣∣∣ . (C.1)
This Jacobian is calculated when the topography is flat and the antenna flight
path trajectory is circular of radius r and height h (the center of this trajectory is
given by (0, 0, h)). We obtain the following expression
J(ω, s,y) =
∣∣∣∣vp(ω)vg(ω)
ω
∣∣∣∣ |ψ(y)− γ(s)|4|ψ(y) · γ(s)− r2|
(C.2)
where
vg(ω) =c0
nR(ω) + ω dnR(ω)dω
,
vp(ω) =c0
nR(ω)
nR(ω) = Re√εr(ω),
ψ(y) = (y1, y2, 0),
γ(s) = (r cos(s), r sin(s), h).
(C.3)
If we do not take into account a dispersive medium, i.e., nR =√εr is constant,
then the Jacobian becomes
J(ω, s,y) =
∣∣∣∣ c20
ωεr
∣∣∣∣ |ψ(y)− γ(s)|4|ψ(y) · γ(s)− r2|
. (C.4)
In the numerical simulations we take into account the dispersive medium and
that the ground position of reference is the center of the scene y0 = (0, 0), the