Utilizing SAR Images to Constrain Internal Wave
ModelsChris WackermanChris Wackerman
General Dynamics [email protected]
734-480-5413
SW06 Workshop 12-14 February 2008
Overall Program Goal: Provide Characterization of Internal Waves to Constrain Forward Prediction Models
• number of waves• wave length• radii of curvature• surface current statistics• phase speed (from single image)
Spatial Map of:
Designed to measure “statistics” of IWs, not to match to individual waves
single image)
IW Future Characteristics
Modify model inputs to match
SAR characterization
now
IW Model•simple 2 layer model•full-up model
ESA 2006ENVISAT 07/23/06 C-VV
Use model inputs to predict future IW
characteristics
Hard Problem is to Characterize Surface Currents From SAR Intensity => Alternative Title: Surface Current Characterization From SAR Images via Inversion
Estimating Surface Currents and Phase Speed from SAR Imagery
• Find surface currents such that forward SAR image predictionmodel matches internal wave signature
• Use normalized observation (RCS-Mean)/Mean to avoidcalibration issues
Solve wave action balance equation
Surface current
field
Wave spectra
field
Apply 2-scale RCS model with hydro
term, breaking, PO
Forward Model
Normalized RCS Field
SAR image signature
Error
Estimate gradient of the error with respect to
surface current and phase speed
Find step in conjugate gradient direction to minimize
error
Next current estimate
Forward Model
Inverse Model
Estimating Surface Currents and Phase Speed from SAR Imagery
• Find surface currents such that forward SAR image predictionmodel matches internal wave signature
• Use normalized observation (RCS-Mean)/Mean to avoidcalibration issues
Solve wave action balance equation
Surface current gradient
field
Wave spectra
field
Apply 2-scale RCS model with hydro
term, breaking, PO
Forward Model
Normalized RCS Field
SAR image signature
Error
Estimate gradient of the error with respect to
surface current and phase speed
Find step in conjugate gradient direction to minimize
error
Next current estimate
Forward Model
Inverse Model
(((( )))) Sx
xgx Fk
A
x
uk
t
Acu
t
A====
∂∂∂∂
∂∂∂∂
∂∂∂∂
∂∂∂∂−−−−
∂∂∂∂
∂∂∂∂++++++++
∂∂∂∂
∂∂∂∂
Wave Action Balance Equation (x-variation only)
• Solved for wave scales from 1/10 to 100 meters• Uncertainties in FS terms with wind growth rate, sources/sinks of wave energy, wave equilibrium spectrum
Estimating Surface Currents and Phase Speed from SAR Imagery
• Find surface currents such that forward SAR image predictionmodel matches internal wave signature
• Use normalized observation (RCS-Mean)/Mean to avoidcalibration issues
Solve wave action balance equation
Surface current gradient
field
Wave spectra
field
Apply 2-scale RCS model with hydro
term, breaking, PO
Forward ModelRCS Model
Normalized RCS Field
SAR image signature
Error
Estimate gradient of the error with respect to
surface current and phase speed
Find step in conjugate gradient direction to minimize
error
Next current estimate
Forward Model
Inverse Model
(((( )))) (((( ))))[[[[ ]]]] (((( )))) cucucucubo dsdss,ss,sh1s,s ρρρρ++++σσσσ====σσσσ ∫∫∫∫∫∫∫∫Tilted Bragg
“Hydrodynamic” Modulations
[[[[ ]]]](((( ))))
(((( ))))sc
su
I4
22sm
2ePO s,s
cos
)0(Rk4exp ρρρρ
θθθθ
ππππσσσσ−−−−====σσσσ
(((( )))) da,0NAT
2a
BR0BR ∫∫∫∫
∞∞∞∞−−−−
σσσσσσσσ====σσσσ
Physical Optics specular reflections
Breaking waves
RCS Model
Estimating Surface Currents and Phase Speed from SAR Imagery
• Find surface currents such that forward SAR image predictionmodel matches internal wave signature
• Use normalized observation (RCS-Mean)/Mean to avoidcalibration issues
Solve wave action balance equation
Surface current gradient
field
Wave spectra
field
Apply 2-scale RCS model with hydro
term, breaking, PO
Forward Model
Error Gradient wrt. current gradient
(((( )))) (((( )))) (((( ))))
(((( ))))
sswsf
xdu
x,kfkSkS
r
rrrr
∂∂∂∂
∂∂∂∂====
(((( ))))(((( )))) (((( )))) (((( )))) (((( ))))[[[[ ]]]] (((( )))) xdsdspkSkSsG
1xE2
xdx
du
Msfsf
0o
rrrrrrr
r∫∫∫∫∫∫∫∫ −−−−++++
σσσσ====
∂∂∂∂
∂∂∂∂
Tilted Bragg Scattering
Normalized RCS Field
SAR image signature
Error
Estimate gradient of the error with respect to
surface current and phase speed
Find step in conjugate gradient direction to minimize
error
Next current estimate
Forward Model
Inverse Model
(((( ))))oxdx
du r∂∂∂∂
(((( ))))
(((( ))))(((( ))))
(((( ))))(((( ))))(((( ))))sr
sko
o
s
k
5.0kpcosxx
xdx
du
x,kf s r
r
rr
r
rr r
ββββ
++++ϕϕϕϕ−−−−δδδδ====
∂∂∂∂
∂∂∂∂“Lossy” Limit => Large
relaxation rate ββββr => no spatial “memory” in solution
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
0 50 100 150 200 250 300 350 400
Distance Across IW (m)(R
CS
-Me
an
)/M
ean
Iteration 1 Result
Iteration 2 Result
Iteration 3 Result
Iteration 4 Result
Iteration 5 Result
Observation
0.015
Previously showed that we could reconstruct input current gradient from simulated RCS values using these approximation…
-0.015
-0.010
-0.005
0.000
0.005
0.010
0.015
0 50 100 150 200 250 300 350 400
Distance Across IW (m)
Cu
rren
t G
rad
ien
t (d
ud
x)
(s**
-1)
Iteration 1 Result
Iteration 2 Result
Iteration 3 Result
Iteration 4 Result
Iteration 5 Result
Truth
Initial Guess
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
0 100 200 300 400 500 600
(RC
S-M
ean
)/M
ean
Iteration 1 Result
Iteration 2 Result
Iteration 3 Result
Observation
0.015
… although we miss the last bit of current at the end (probably
-0.015
-0.010
-0.005
0.000
0.005
0.010
0.015
0 100 200 300 400 500 600
Distance Across IW (m)
Cu
rren
t d
u/d
x (
s**
(-1))
Iteration 1 Results
Iteration 2 Result
Iteration 3 Result
Truth
Initial Guess
at the end (probably due to the approximations)
=> The current1 question is whether the forward model is generating the correct SAR image modulations for the actual surface currents and surface wave spectra
• Would like to validate the choices for the sources/sinks in the wave action balance equation from measurements of surface wave spectra over the required scales
• may be able to use buoy data for large-scale waves as
1Pardon the pun
• may be able to use buoy data for large-scale waves as the IW passes by?
• Would like to validate final model from measurements of surface currents coincident with IW SAR signature
• WamOS data may be best for this when it becomes available• OSU have graciously put out their ADCP data from the Oceanus (multiple thanks!)
Track
Oceanus ADCP Data from 08/18/06 Courtesy of OSU
Oceanus ADCP Data from 08/18/06 Courtesy of OSU
Start
Oceanus ADCP Data from 08/18/06 Courtesy of OSU
Shallowest depth is 3.5m – what is a better way to extrapolate to surface currents other than linear interpolation?
Oceanus ADCP Data from 08/18/06 Courtesy of OSU
Linear interpolation region
Oceanus ADCP Data from 08/18/06 Courtesy of OSU
Linear interpolation region
4 waves, ~ 0.5 m/s peak current, varying widths => du/dx increases left to right
Oceanus ADCP Data from 08/18/06 Courtesy of OSU
EnvisatAug 18 20062:37 GMTC-VV
N
Wind To
SAR
Look
Wind To
5 m/s
Oceanus Track (blue) and SAR Image Locations (red)
EnvisatAug 18 20062:37 GMTC-VV
Envisat0818
0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
1000 2000 3000 4000 5000
Distance (m)
Delt
a-R
CS
=> SAR Image modulations are from ±0.1 to 0.4, spacing is ~ 300 – 370 m
ERSAug 18 20093:06 GMTC-VV
ERS0818
ERSAug 18 20093:06 GMTC-VV
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
2000 3000 4000 5000 6000 7000
Distance (m)
Delt
a-R
CS
=> ½ hour later, SAR Image modulations are ~ ±0.1 to 0.3, spacing is ~ 300 – 370 m
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 1000 2000 3000 4000 5000 6000 7000 8000
Distance Along Plot (m)
(RC
S-M
ean
)/M
ean
RADARSAT 08/23 CHH
SAR Look
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0 1000 2000 3000 4000 5000 6000 7000 8000
Distance Along Plot (m)
(RC
S-M
ea
n)/
Me
an
ENVISAT 07/23 CVV
SAR Look
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0 1000 2000 3000 4000 5000 6000 7000 8000
Distance Along Plot (m)
(RC
S-M
ean
)/M
ea
n
Distance Along Plot (m) SAR Look
Wind Direction
Wind Speed 3 m/s
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0 1000 2000 3000 4000 5000 6000 7000 8000
Distance Along Plot (m)
(RC
S-M
ea
n)/
Me
an
Wind Direction
Wind Speed 5 m/s
WABE Runs
-4.0
-2.0
0.0
2.0
RC
S (
dB
)
VV Full Model
HH Full Model
VV no PO
CMOD4 RCS
HH no PO
VV With Breaking
-12.0
-10.0
-8.0
-6.0
-600 -400 -200 0 200 400 600
Distance (m)
RC
S (
dB
)
Wabe Forward Runs
1
1.5
2
2.5
3
De
lta
-RC
S
0.6 m/s peak, -0.4 m/s cp
0.6 m/s peak, 0.0 m/s cp
0.3 m/s peak, -0.4 m/s cp
0.1 m/s peak, -0.4 m/s cp
-1
-0.5
0
0.5
1
-600 -400 -200 0 200 400 600
Distance (m)
De
lta
-RC
S
Wabe Run 4 Waves
-4.0
-2.0
0.0
2.0
VV
RC
S (
dB
)
0.3 m/s peak
0.6 m/s peak
-10.0
-8.0
-6.0
-4.0
-2000 -1500 -1000 -500 0 500 1000 1500 2000
Distance (m)
VV
RC
S (
dB
)
WABE Run 4 Waves
0.5
1
1.5
2
De
lta
-RC
S V
V
0.6 m/s peak
0.3 m/s peak
-1
-0.5
0
0.5
-1000 -800 -600 -400 -200 0 200 400 600 800 1000
Distance (m)
De
lta
-RC
S V
V
Summary of WABE Run Modulations
0.5
1
1.5
2
Ma
x (
bla
ck
) o
r M
in (
red
) D
elt
a R
CS
Max Delta-RCS Modulations
Min Delta-RCS Modulations
4 wave Min Delta-RCS
4 wave Max Delta-RCS
-1
-0.5
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Peak Current (m/s)
Ma
x (
bla
ck
) o
r M
in (
red
) D
elt
a R
CS
=> With 4 wave runs, peak modulations are consistent with observations but minimum are too low
Observed Modulations
Summary• Forward model appears to be consistent with max RCS modulations
when including 4 waves, but too low for min RCS modulations– May be a noise floor problem with the SAR?
• Questions I would like to discuss during the workshop– Are the units rights for the ADCP?– What is a better way to extrapolate to a surface current?– Can I get IW modulations wavelength from the current data (i.e. do we
know the ship velocity and the IW phase speed)?– Are there other surface current observations available?– Does anyone want the SAR images and/or interpretations of them for their – Does anyone want the SAR images and/or interpretations of them for their
analysis?
• We have derived the adjoint of the WABE for RCS errors as a function of current to avoid the simplifying assumptions used in the gradient estimate
– It should converge faster and be more accurate close to the solution
• Now need to use this forward model in the new inversion and derive the current
RADARSATAug 18 200622:58 GMTC-HH
ENVISATCVV
July 23 2006
SAR Look
Line 1
Wind Direction
Wind Speed5 m/s Line 2
RADARSATCHH
August 23, ‘06
SAR Look
Wind Direction
Line 1
Wind Direction
Wind Speed3 m/s
Plot Locations
Line 2
RSI 2006
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