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1
Synthetic Aperture Radar Processing
SAR and IFSAR
Giorgio FranceschettiUniversita’ Federico II, Napoli, Italy
2
REFERENCE TEXT
Giorgio Franceschetti, Riccardo Lanari, SYNTHETIC APERTURE RADAR
PROCESSING TECHNIQUES, CRC Press, BOCA RATON, Fla, 1999
3
4
Table of content
• Basic concepts• Processing• Interferometric SAR
5
Basic Concepts
• Geometry• Response in Time Domain• Response in Frequency Domain
6
I. Geometry
1. Cylindrical reference coordinate system
7
2. Boresight strip mode illumination geometry
8
3. Boresight acquisition geometry in the (x,r) plane
9
II. SAR response in time-domain
1. Transmitted waveform
for the chirp case
( ) ( ) ( )
τ−
−ω= nn
ttttPtjtf rectexp1
( ) ( )
−
α=− 2
2exp nn ttjttP
10
2. Received waveform from a unitary scattered at after heterodine
with
( ) [ ]rxxwcRtt
cRttP
cRjrttxxf n
n
nnn ,
2
rect22exp,, 2 −
τ
−−
−−
ω−=−−
( )22 xxrR n −+=
),,( ϑrx
11
3. Change of Coordinates
( ) rxxrrRR
ttt
tcr
n
n
−−+=−=∆
−=′
′=′
22
2
4. Normalize
Rx ∆, LrX 0λ=
r 2τc
to
to
12
5. Received waveform
with
( ) ( ) ( )rrrxxgrxdxdrrxh ,,,, −′−′γ=′′ ∫∫
( ) ( )
λτ
π−γ→γ rcjrxrx 2exp,,
( ) [ ]rxxwRLjaRcXrrR
cXrrPrrrxxg ,2exp2rect2,, 2 −′
∆
λ−
∆
τ−−′
∆
τ−−′=−′−′
2LXa π=
13
III. SAR response in frequency-domain1. SAR signal spectrum
where
is the SAR transfer function( ) ( ) ( )[ ] ( )[ ]rrjxxjrrrxxgrdxdrG −′η−−′ξ−−′−′′′=ηξ ∫∫ expexp,,,,
( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )∫∫
∫∫∫∫∫∫
ηξη−ξ−γ=
′η−′ξ−−′−′′′γ=
′η−′ξ−′′′′=ηξ
rGrjxjrxdxdr
rjxjrrrxxgrdxdrxdxdr
rjxjrxhrdxdH
,,expexp,
expexp,,,
expexp,,
2. Simplest case
( ) ( ) ( ) ( ) ( ) ( ) ( )ηξηξΓ=ηξη−ξ−γ=ηξ ∫∫ ,,,expexp,, GGrjxjrxdxdrH
14
Processing• Asymptotic evaluation of integrals• Transfer Function expansion• Narrow Focus SAR• Wide Focus SAR
15
I. Asymptotic evaluation of integrals
( ) ( ) ( )[ ]∫ Ω=Ω tqjtdtfI exp
1. Type of integral
with 1>>Ω
2. Asymptotic evaluation
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )[ ] ( ) ( ) ( )[ ]sss
sss
sss
s
tqjtfqjttqjtqjtdtfI
tqtttqtqtq
Ω′′Ω
−′′ΩΩΩ
=′−′′
+≈
∫ exp2~2
expexp~
0,!2
2
2
π
16
3. Examples
( ) [ ] [ ]
−
−= ∫− b
jb
jbqqjdqF2
exp2
rect~expexp2
22/1
2/1
ηηηη
17
II. Transfer function evaluation
( )
fff
rXcpr
XcR
fRLapR
cXRLapp
RcXrrq
xxp
∆=
ω∆π
=ε
τ−+
τ
=∆
τ∆π
εη+∆
λ+ξ=∆
τη+∆
λ+ξ=Ψ
∆τ
−−′=
−′=
222
21222
2
22
1
1. Convenient evaluation formLet
we have
( ) ( )[ ] [ ] [ ] ( ) [ ]∫∫ η−Ψ−=ηξ qqPqjdqrpwpjdprG rectexp,exp,, 21
18
For a chirped signal
( ) ( )[ ] [ ] ( )[ ] [ ]∫ ∫ Ψ−Ψ−=ηξ qqjdqrpwpjdprG rectexp,exp,, 22
1
with
( )
( )
τπ
ηελ
ξ
η
fbb
RLapp
bqqq
∆=
+∆+=Ψ
−=Ψ
2121
22
19
2. Asymptotic evaluation
( )( )
( )[ ]
( )[ ]rja
wbab
rja
wbbrra
rG
,,exp22
rect
,,exp22
rect~,,
2
2
0
ηξξηπ
ηξξηπηξ
Ψ−
−
≈
Ψ−
−
with
( )
ξ≈
ξ−
ξ−
η
ε+
λλ+
η
ε+
λ
−η
=ηξΨ
aaw
bLa
rrL
rr
bLa
br
2rect
2
212
212
4,,
2
222
00
22
20
3. Simplest SAR transfer function expression
Expansion around leads to0=ξ
η
ε+
λξ−
η
ε+λ
≈ξ−
η
ε+
λb
a
Lb
Lab
La
214
/2
122
122
222
and
with
Accordingly
( )
η
ε+′
ξ−
η=
η
ε+
ξ−
η≈ηξΨ
bab
brrab
r
2144
2144
,,22
0
22
rraa 0=′
( )
′
ξ
ξ
η−
η≈ηξ
aj
abj
brG
4exp
2rect
4exp
2rect,,
22
21
4. Resolution
fffbr
LXL
ax
rrbxxarxrdxd
GHrjxjdd
baGH
GH
/2/1
22/
)]'(sinc[)]'(sinc[),(
),(*),(]'exp[]'exp[21
2rect
2rect),(),(*),(
),(),(),(
∆→
∆==∆
→==∆
−−=
=
Γ=
Γ=
∫∫∫∫
λτ
π
π
γ
ηξηξηξηξπ
ηξηξηξηξ
ηξηξηξ
22
III. Transfer function expansion
1. SAR transfer function recalled
( )( )
( )[ ]
( )[ ]
( ) 222
00
22
2
2
0
212
212
4,,
,,exp22
rect
,,exp22
rect~,,
ξ−
η
ε+
λλ+
η
ε+
λ
−η
=ηξΨ
ηξΨ−
ξ−
ηπ
≈
ηξΨ−
ξ−
ηπ
ηξ
bLa
rrL
rr
bLa
br
rja
wbab
rja
wbbrra
rG
23
2. Transfer function phase expansion around 0=η
Let( ) ( ) ),()(,,, 00 ηξ−+ηξΨ=ηξΨ Krrr
with
( ) ( )
( ) 2222
0
22222
00
212
212,
212
212
4,,,
0
ξ−
η
ε+
λλ+
η
ε+
λ
−=ηξ
ξ−
η
ε+
λλ+
η
ε+
λ
−η
=ηξΨ=ηξΨ
bLaL
bL
raK
bLaL
bLa
br
r
24
( ) ( ) ( ) ( ) 2, ηξζ+ηξ+ξµ=ηξ vK
we have
with
( )
( )
( )2
0
2
0
20
2
24
24
4
εξ
−≈ξζ
εξ≈ξ
ξ−≈ξµ
bar
barv
ar
25
III. Narrow focus SAR
1. Simplest approximation
Assume 0=ζ==µ v
( ) ( )ηξ→ηξ ,,, 0GrG
2. SAR processingFrom
( ) ( ) ( )ηξΓηξ=ηξ ,,, 0GH
we get
( ) ( ) ( )ηξηξ=ηξΓ ,,, *0GH
26
3. Examples Reference Image
27
Uncompensated term( )rξµ
28
Uncompensated term( ) rv ηξ
29
IV. Wide focus SAR
1. Starting point
( ) ( ) ( ) ( ) ( )( )[ ]( ) ( )[ ]ηξ+ηξΓηξ=
ηξ+η−ξ−γηξ=ηξ ∫∫,,,
,expexp,,,
0
0
KG
rKjxjrxdxdrGH
with( ) ( ) ( )ξη+ξµ≈ηξ vK ,
Accordingly( ) ( ) ( )( ) ( )[ ] ( ) ( ) ( )[ ]( ) ( )ξ+=ξΩ
ξµ+ξΩηξΓηξ=ξµ+ξ+ηξΓηξ≈ηξv
GvGH1
,,1,,, 00
30
2. Range processing
( ) ( )[ ] [ ] ( ) ( )[ ]
( ) ( ) ( )
ξΩ′
ξγ
ξΩ′
ξµ−≈
ξµ+ηξΩξΓ′ηηπ
=ηξηξ ∫−
rrj
rjdGHFT
,ˆexp
,exp21,, *
01
31
3. Simplest case
0=v so that 1=Ω
( ) ( ) ( )[ ]ξµ+ηξΓηξ=ηξ ,,, 0GH
32
4. Chirp scaling
( )( )DBBA
DBDCDB
+=+=
−=Ω
11
33
Range processing
34
Full processing
35
Interferometric SAR
• Basic principles• Decorrelation effects• Phase Unwrapping techniques• Examples
36
I. Basic principles1. Geometry
Single imaging sensor Dual imaging sensor
37
Relations
lead to
( ) ( )ϑ′′−=
β−ϑ′′−+′=′δ+′
cossin2222
rHzrllrrr
( ) ( )β−ϑ′ϑ′′
−≈
β−ϑ′′
′δ+′−ϑ′′=
′δ∂ϑ′∂
ϑ′∂∂
=′δ∂
∂cos
sincos
sinlr
rlrrr
rz
rz
2. Interferometric phase
( ) ( ) ( )[ ] rrrxrxrx ′δλπ
=′δ+′′γ′′γ=′′ϕ4,ˆ,ˆPh, *
21
so that
( )
ϕπ
ϑ′′λ−=
ϕ∆π
ϑ′′λ−=
πϕλ
∆β−ϑ′
ϑ′′−=∆
⊥
⊥
lrz
lr
lrz
4sin
4sin
4cossin
38
In order to compute the interferometric phase, the followingsteps must be implemented(a) A couple of SLC images must be generated
39
(b) The two images must be registered
for infinite bandwidth SAR
( ) ( ) ( )[ ] ( )[ ]
( ) ( ) ( ) ( )[ ] ( )[ ]rrrbxxarrjrxdxdrrx
rrbxxarjrxdxdrrx
δ−−′⋅−′
δ+
λπ
−γ=′′γ
−′−′
λπ
−γ=′′γ
∫∫
∫∫
sincsinc4exp,,ˆ
sincsinc4exp,,ˆ
2
1
( ) ( )
( ) ( )
′
λπ
−′δ−′′γ=′′γ
′
λπ
−′′γ=′′γ
rjrrxrx
rjrxrx
4exp,,ˆ
4exp,,ˆ
2
1
and registration is required
( ) ( ) ( ) ( )
′δ+′
λπ
−′′γ=′δ+′′γ→′′γ rrjrxrrxrx 4exp,,ˆ,ˆ 22
40
Interferometric fringes
41
II Decorrelation effects
1. Misregistration
accuracy of 1/20 of pixel is required
2. Spatial decorrelation
The same cell is imaged from two different looking directions
ϑ∆λ
′′
=⊥ cotan2r
rl c c⊥⊥ ≤ ll,
42
3. Doppler centroid decorrelationThe same cell is imaged from two different squint angles
43
4. Temporal decorrelation
Present only in dual pass systems
5. Overall decorrelation estimate
( )021
22
21
*21 exp
]|~[|]|~[|]~~[ ϕγγ
γγχ jkqqp
EEE
===
Cross-correlation factor
coherency map
44
III. Topographic mapping
1. Ideal Phase Unwrapping (PhU)procedure
π
−πx
ϕm
2π 2π−π
x
ϕm
π
π
−πx
s
π
−πx
ϕ
45
2. Discontinuities presence
(a) Non-ambiguous phase jump (b) Ambiguous phase jump
46
IV. Phase unwrapping techniques
1. Local procedure
2. Global procedures: Least mean square method
[ ] ( ) ( )[ ] min→−∇⋅−∇= ∫∫S
dSL ss ϕϕϕ
47
3. Global procedure: Green’s identity method
[ ]
( ) ( )
( ) ( ) ( ) ( ) ( )
SrnrrgrdcrrrgdSr
rrrrg
ngdcggdS
S c
cS
∈′
∂−′∂
+∇⋅−′∇−=′
−′=−′∇
∂∂
=∇⋅∇+∇
∫∫ ∫
∫∫∫
ϕϕϕ
δ
ϕϕϕ
2
2
48
4. Local and global PhU techniques
( ) ( ) ( )
( ) ( ) ( )
⋅+=
⋅⋅
+⋅−=
∫∫
∫∫∫
rs
nrnr
sr
ˆ,,210
2ˆˆ
,ˆˆ
,2
ˆ0
02
0
2
00
2
0
M
M
rM
MM
Mr
rdrrd
rrrdr
rrdrd
θθϕθπ
ϕ
πθϕθθ
πθϕ
π
ππ
49
V. Examples: Sardinia, Italy
Interferometric SAR DataDEM
⇒
Amplitude Phase Coherence
Raw data ESA copyright
50
This short course is over:I do hope all of you enjoyed it.