A Mathematical Tutorial on SYNTHETIC APERTURE RADAR (SAR ...

A <strong>Mathematical</strong> <strong>Tutorial</strong> on
SYNTHETIC APERTURE RADAR (SAR)
Margaret Cheney
Rensselaer Polytechnic Institute and MSRI
November 2001
• developed by the engineering community
• key technology is mathematics
• unknown in mathematical community
• close connections with tomography,
integral geometry, microlocal analysis, seismic inversion
1

History
1951 Carl Wiley, Goodyear Aircraft Corp.
mid-’50s
late ’60s
first operational systems
DoD sponsorship:
U. of Illinois, U. of Michigan,
Goodyear Aircraft, General Electric,
Philco, Varian
NASA sponsorship (unclassified!)
first digital SAR processors
1978 SEASAT-A
1981 beginning of SIR (Shuttle Imaging Radar)
series
1990s
satellites sent up by many countries
SAR systems sent to Venus, Mars, Titan
2

Outline
• The wave equation
• The incident wave: beamforming
• A linearized scattering model
• The received signal
• The reconstruction method:
matched filters = backprojection
= migration
• Resolution
• The state of the art
• Current research and open problems
3

<strong>Mathematical</strong> Model
• We should use Maxwell’s equations.
• A simpler model: the scalar wave equation
(
∇ 2 − 1 )
c 2 (x) ∂2 t U(t, x) = 0
• Earth is plane x 3 = 0
• All constants = 1
4

The Incident Wave
• Field at (t, x) due to a source at the origin at time
0:
δ(t − |x|)
G 0 (t, x) =
|x|
satisfies
(
∇ 2 − ∂ 2 t
)
G 0 (t, x) = δ(t)δ(x)
• But the signal from the antenna is more complicated
.....
δ(t) → P(t) =
∫
e iωt p(ω)dω
δ(x) →
J(x)
5

Field emanating from antenna satisfies
(∇ 2 − ∂t 2 )Uin (t, x) = P(t)J(x)
Solve:
U in (t, x) = G 0 ∗ (PJ)
=
=
∫ P(t − |x − y|)
J(y)dy
|x − y|
∫ ∫ e
iω(t−|x−y|)
p(ω)J(y)dωdy.
|x − y|
In this formula:
Write point on antenna as y = γ(s) + q
For x far from the antenna, |q|

∫
Uγ(s) in (t, x) ∼
e
iω(t−|x−γ(s)|)
|x − γ(s)|) p(ω)
∫
· e iω( x−γ(s))·q ̂ J(γ(s) + q)dq dω.
Antenna beam pattern at each frequency is Fourier
transform of current density on antenna!
U in
∫ e
iω(t−|x−γ(s)|)
γ(s) (t, x) ∼ p(ω) j(ω( x ̂ − γ(s)), s) dω
|x − γ(s)|
7

Key example:
Take J = characteristic function of rectangle
j =
∫ L/2
−L/2
∫ D/2
−D/2
e iω[ ̂ x−γ(s)]·(q1 ê 1 (s)+q 2 ê 2 (s)) dq1 dq 2
∝ sinc[ω(D/2)cos θ 1 ] sinc[ω(L/2)cos θ 2 ]
where cos θ j = [ ̂ x − γ(s)] · êj and sinc x = sin x
x .
• Main lobe of beam is perpendicular to antenna
• To get angular width, set argument of sinc =
π, solve for angle.
Antenna has length L ⇒
Angular width of main lobe ≈
ωL 4π = 2λ
L
⇒ width of antenna footprint =
ωL 4π R = 2λ
L R.
• Get a more focused beam with:
– bigger antenna
– higher frequency
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Linearized scattering theory
(
∇ 2 − c −2 (x)∂ 2 t
)
U(t, x) = P(t)J(x)
(∇ 2 − ∂ 2 t )Uin (t, x) = P(t)J(x)
write U = U in + U sc , c −2 (x) = 1 + V (x), subtract:
( )
∇ 2 − ∂t
2 U sc (t, x) = V (x)∂t 2 U(t, x),
write as Lippman-Schwinger integral equation
∫
U sc (t, x) = G 0 (t − τ, x − z)V (z)∂τ 2 U(τ, z)dτdz.
9

single-scattering or Born approximation
∫ ∫
U sc (t, x) ≈ G 0 (t − τ, x − z)V (z)∂τ 2 U in (τ, z)dτdz
useful: makes inverse problem linear
not necessarily a good approximation
∫ ∫ δ(t − τ − |x − z|)
γ(s) (t, x) ≈ V (z)
|x − z|
U sc
∫ e
iω(τ−|z−γ(s)|)
·
j(ω( z ̂ − γ(s)), s)ω 2 p(ω)dωdτdz
|z − γ(s)|
≈
∫ ∫
e iω(t−(|x−z|+|z−γ(s)|))j(ω( z ̂ − γ(s)), s)˜p(ω)
dω
|z − γ(s)||x − z|
· V (z)dz
At center of antenna,
∫ ∫
Uγ(s) sc (t, γ(s)) ≈
e iω(t−2|z−γ(s)|)j(ω
(z ̂ − γ(s)), s)˜p(ω)
|z − γ(s)| 2 dω V (z)dz
10

This data is of the form
∫ ∫
D(t, s) = e iω(t−2|z−γ(s)|) a(z, s, ω)dωV (z)dz
From D, want to reconstruct V .
• D is an oscillatory integral, to which techniques
of microlocal analysis apply ( C. Nolan & M.C.)
• seismic inversion problem with constant background
but limited data
• D(t, s) depends on two variables.
Assume V (z) = Ṽ (z 1 , z 2 )δ(z 3 − h(z 1 , z 2 )).
• if a(z, y, ω) = 1, want to reconstruct V from its
integrals over spheres or circles (J. Fawcett,
L.-E. Andersson, A. Louis & E.T. Quinto, M.
Agranovsky & E.T. Quinto)
• character of problem depends on frequency content
of a
11

High frequency case
• microwave frequencies, ω ∼ 10 9 , λ ∼ cm
• can form narrow beam
• high resolution, space-borne
• doesn’t penetrate foliage
Low frequency case
• VHF: ω ∼ 10 7 , λ ∼ 10m
• cannot form beam
• penetrates foliage but resolution is low
• airborne
12

Inversion scheme: matched filter =
backprojection = migration
I(x) =
∫
P(t − 2|x − γ(s)|)D(t, s)dtds
Why is this called a matched filter?
Take V (z) = δ(z − x) in
∫ ∫
D(t, s) = e iω(t−2|z−γ(s)|) p(ω)j(ω(z ̂ − γ(s)), s)dω
·
V (z)
|z − γ(s)| 2dz
j is nearly independent of ω ⇒
D(t, s) ∝ P(t − 2|x − γ(s)|)
= data due to a “point scatterer” at position x.
In case P = δ:
The matched filter “backprojects” the data over
the sphere with radius t/2 and center γ(s).
Image at x is formed by adding data corresponding
to all spheres passing through x.
13

Why does matched filter processing give us
an image?
D(t, s) =
∫ ∫
e iω(t−2|z−γ(s)|) a(z, s, ω)dωV (z)dz = FV
Apply
I(x) =
∫ ∫
e i˜ω(t−2|x−γ(s)|) p(˜ω)d˜ωD(t, s)dtds ≈ F ∗ D
Note that phase is negative of phase of D.
Compare with Fourier inversion, Radon transform
inversion, ....
Use
∫
e it(˜ω−ω) dt = δ(˜ω − ω)
to get
I(x) =
∫
W(x, z)V (z)dz
where
W(x, z) =
∫ ∫
e 2iω(|x−γ(s)|−|z−γ(s)|) p(ω)a(z, s, ω)dωds
is the point spread function or generalized ambiguity
function of the imaging system.
14

∫
I(x) =
W(x, z)V (z)dz
W(x, z) =
∫ ∫
e 2iω(|x−γ(s)|−|z−γ(s)|) p(ω)a(z, s, ω)dωds
Main contribution comes from:
• (high frequency case) x = z only
• (low frequency case) x = z and
x = reflection of z
wavefront relation of F is
a folding canonical relation
⇒ artifacts
15

Resolution in high frequency case
How close is W to a delta function?
W(x, z) =
∫ ∫
e 2iω(|x−γ(s)|−|z−γ(s)|) p(ω)a(z, s, ω)dωds
Compare with δ(x − z) = ∫ e i(x−z)·ξ dξ.
G. Beylkin (JMP ’85) approach:
1. Do Taylor expansion of exponent:
|x − γ(s)| − |z − γ(s)| = (x − z) · Ξ(x, s, z)
near x = z, Ξ(x, s, z) ≈
̂ z − γ(s)
2. Make change of variables (ω, s) → ξ = 2ωΞ(x, s, z):
∫
W(x, z) = e i(x−z)·ξ ∂(s, ω)
p(ω(ξ))a(z, s(ξ), ω(ξ)) dξ
∂ξ
This is kernel of a pseudodifferential operator ⇒
puts singularities in the correct locations.
16

∫
W(x, z) =
e i(x−z)·ξ ∂(s, ω)
p(ω(ξ))a(z, s(ξ), ω(ξ)) dξ
∂ξ
Resolution is determined by support of p(ω(ξ)) and
a(z, s(ξ), ω(ξ)), i.e., by bandwidth and angular extent
of survey.
Resolution via Fourier transforms
∫ b
−b
e iρr dρ =
2sin br
r
corresponds to resolution 2π/b.
17

Along-track (azimuthal) resolution
W(x, z) =
∫
with ξ ≈ 2ω
e i(x−z)·ξ ∂(s, ω)
p(ω(ξ))a(z, s(ξ), ω(ξ)) dξ
∂ξ
̂ (z − γ(s)).
If flight track is γ(s) = (0, s, H) and
x − z = (0, x 2 − z 2 ,0), need only
ξ 2 ≈ 2ω z 2−γ(s) 2
R
= ω R 2(z 2 − s)
But
2max |z 2 − s| = width of antenna footprint
=
ωL 4π R = 2λ
L R
= effective length of the synthetic aperture
So max |ξ 2 | ≈ ω R 4πR
ωL = 4π L
So resolution in along-track direction is
2π
4π/L = L 2
18

Along-track resolution is L/2.
This is ...
• independent of range!
• independent of λ!
• better for small antennas!
These are all explained by noting that when a point
z stays in the beam longer, the effective aperture
for that point is larger.
In range direction, want broad frequency band ⇒
get largest coverage in ξ.
19

The state of the art (high frequency)
• Can correct for antenna motion
• interferometric SAR (two flight passes or two
antennas)
– topographic information
– changes (glacier flow, volcano movement)
• polarimetric SAR (uses full Maxwell’s equations)
• MTI = Moving Target Indicator
• SAR in wave mode: Use subaperture
information to get ocean motion
20

Current research and open problems
• Foliage-penetrating (FOPEN) SAR
– low frequencies ⇒ poor directivity ⇒
tomographic image formation from
integrals over circles or spheres
– want bare earth topography
– want to find tanks under trees
– want to estimate forest biomass,
trunk volume, tree health
• Ground-penetrating radar (GPR)
– land mines
– unexploded ordinance (UXO)
• Ultra-wideband (UWB) SAR
21

• Improved modeling
– dispersion (FOPEN)
– multiple scattering
∗ rough terrain, wave-guiding structures
∗ wave propagation in random media, clutter
• Image interpretation
– thickness of ice, species of trees,
surface roughness, . . .
– Automatic Target Recognition (ATR)
(school bus or tank?)
– sensor fusion, use of multiple frequency bands
22

• Theoretical issues
– Uniqueness theorem for backscattering
– Reconstruction for full nonlinear inverse problem
∗ limited aperture
∗ time domain (including dispersion!)
– What are theoretical limits?
23

References
• M. Cheney, “A mathematical tutorial on SAR”,
SIAM Review, SIAM Review 43 (2001) 301-
312; http://www.rpi.edu/∼cheney/sar.ps
• G.W. Stimson, Introduction to Airborne Radar,
Scitech Publishing, New Jersey, 1998.
• C. Elachi, Spaceborne Radar Remote Sensing:
Applications and Techniques, IEEE Press, New
York, 1987.
• G. Franceschetti and R. Lanari, Synthetic Aperture
Radar Processing, CRC Press, New York,
1999.
• L.J. Cutrona, “Synthetic Aperture Radar”, in
Radar Handbook, second edition, ed. M. Skolnik,
McGraw-Hill, New York, 1990.
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