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.