signal processing for gpr
DESCRIPTION
Signal Processing for GPRTRANSCRIPT
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Master’s Thesis Defense
Model Based Signal Processing forGPR Data Inversion
Visweswaran Srinivasamurthy13th April 2005
CommitteeDr. Sivaprasad Gogineni (Chair)
Dr. Muhammad Dawood (Co-chair)Dr. Pannirselvam Kanagaratnam
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OUTLINE
IntroductionGPR ApplicationsThesis Objectives
The Inverse ProblemForward Modeling – FMCW RadarLayer Stripping ApproachThe Model Based Approach
Model Based Parameter EstimationMMSE based (Gauss-Newton)Spectral Estimation based (MUSIC)
Inversion on actual radar dataTests on Antarctic snow radar dataTests at the Sandbox labTests on Greenland Plane wave data GUI for data inversion algorithm
Conclusions & Future Work
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INTRODUCTIONGPR Applications
Ground Penetrating Radar Applications:
Ice-sheet thickness measurements, bedrock mapping (Global Warming problem)Target detection (Landmines)Non-destructive testing of engineering structuresSub-surface Characterization (Earth, Martian Surface)
Courtesy: JPL, NASA
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INTRODUCTIONConcepts
Characterization : Determining the permittivity profile of a multi-layered media
Permittivity (Dielectric Constant) : A quantity that describes the ability of a material to store electric charge.
Multi-layered structure Permittivity Profile
Radar SystemZ1 Z2 Z3 Z4
1ε
2ε3ε
4ε
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THESIS OBJECTIVES
Thesis Objectives
Develop a signal processing algorithm to
1. Enhance features of radar data (reflectivity profiles with improved resolution)
2. Estimate the permittivity profile from recorded GPR data
Electro-Magnetic (EM) Inversion
PrinciplePermittivity contrast in layered media causes reflection of incident EM Wave
ChallengesRadar return is corrupted by noise & clutterUnwanted effects due to radar system (Eg: non-linearities)Needs good understanding of EM propagation phenomenon
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OUTLINE
IntroductionGPR ApplicationsThesis Objectives
The Inverse ProblemForward Modeling – FMCW RadarLayer Stripping ApproachThe Model Based Approach
Model Based Parameter EstimationMMSE based (Gauss-Newton)Spectral Estimation based (MUSIC)
Inversion on actual radar dataTests on Antarctic snow radar dataTests at the Sandbox labTests on Greenland Plane wave data GUI for data inversion algorithm
Conclusions & Future Work
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THE GENERAL INVERSE PROBLEM
Inverse Problem: Estimation of unknown parameters given an observation
Steps for the study of an inverse problemSystem Parameterization:
Identify set of model parameters (m) which characterize the phenomenon (observation)
Observation – Radar return
Model parameters – Permittivity values
Forward Modeling:
Deduce a mathematical relationship F(m) between model parameters (m) and actual observations (Y)
Inverse Modeling:
Use forward model and observed data to infer actual values of model parameters
Y = F(m) + Noise + System effects + Clutter
Estimate m given Y
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FORWARD MODELING
Mathematical relationship between permittivities & observed radar return signalWave propagation Phenomena (1-D Plane wave approximation)
Reflection – Reflection Coefficient
( ) ( )1 1k k k k kε ε ε ε+ +Γ = − +
2
1 14 + +⎡ ⎤= +⎣ ⎦k k k k kT ε ε ε ε
2
14 R
⎛ ⎞⎜ ⎟π⎝ ⎠
Spreading
Transmission – Transmission Coefficient
Attenuation – Attenuation Coefficient
Absorption
factor
kT
kΓkB
Neglected in our analysisScattering
Conductivity, particle distribution need to be known
k
k k k jj 1
A B T=
= Γ ∏Effective amplitude of reflected signal at layer K (combined effect of , , )
1kΓ kT kB
( )1 11
2−
=
⎡ ⎤= + −⎢ ⎥
⎣ ⎦∑
L
k k k kk
z z zc
τ εC = 3x 108 m/s
Z1 – Surface height2-way time delay experienced by signal reflected from
layer K 2
Estimate using (1) and (2) recursively ε
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FORWARD MODELING Illustration - FMCW Radar
Multi-layered target
FMCW - Frequency Modulated Continuous Wave Radar
Transmits a frequency sweep – Chirp signal
Reflected signal is mixed with a copy of the transmitted signal to generate Beat Signal (IF Signal).
Beat signal is a function of time delay (beat frequency)
( ) ( )2t t 0 0V t A Cos 2 f t t= π +α +θ⎡ ⎤⎣ ⎦
b
2RBfTc
=
( ) ( ){ }k 1L 1
beat k k j 0 k k kk 0 j 1
V A T cos(2 f 2t n−−
= =
τ = Γ π τ +ατ − τ +∑ ∏
(
For multiple targets,
)beatV τ is the forward model F(m)
;bfτ∝
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FORWARD MODELING FMCW Radar
Fast Fourier Transform (FFT) of gives frequency response of the target
Plot of signal spectrum Vs distance – Range Profile( )beatV τ
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INVERSIONLAYER STRIPPING APPROACH
An elementary approach to inversionPlot signal spectrum (Range Profile) using Fast Fourier Transform (FFT)Set threshold on amplitudesLocate Amplitudes (Ak’s) and Time delays ( ) from range profilek 'sτ
rεPermittivity vector [1 3 5 2 6 ]
Depth-vector(m) Z [0.5 0.3 0.4 0.4]
Threshold
A1
A2
A3A4
1τ 2τ 3τ 4τ
Recursively use (1) and (2) from the forward model to estimate the permittivity of every layer
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LAYER STRIPPING APPROACHLimitations
Missed Peaks
False Alarms
The side-lobe masking problem- Weaker returns masked by side-lobes of stronger returns
- Windowing functions attenuate the lower frequencies that contain most of the information about the deeper structure
Layer Stripping is not very reliable to detect subtle variations in permittivity
Inappropriate thresholds
distort reconstructed profile
Missed Peak False Alarm
Distance(m)
Ref
lect
ion
Am
plitu
de
Solution:
Incorporate the underlying phenomenon into the inversion process
The Model Based Approach
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OUTLINE
IntroductionGPR ApplicationsThesis Objectives
The Inverse ProblemForward Modeling – FMCW RadarLayer Stripping ApproachThe Model Based Approach
Model Based Parameter EstimationMMSE based (Gauss-Newton)Spectral Estimation based (MUSIC)
Inversion on actual radar dataTests on Antarctic snow radar dataTests at the Sandbox labTests on Greenland Plane wave data GUI for data inversion algorithm
Conclusions & Future Work
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THE MODEL BASED ESTIMATION
Model Based Estimator An estimator which incorporates the mathematical model F(m) to estimate unknown parameters (m).
[ ] [ ] [ ]{ }1Ny....,,1y,0yY −=Given an observed data set
Regression Estimators (Data fitting or Curve fitting)
Model BasedEstimator
Fit parameters to the observation (data) - based on some criterion
, forward model F(m)
Fit m to Y
F(m) is non-linear , hence Non-linear Regression
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Least Squares Estimation
Estimate parameters based on the approach of minimizing the Mean Squared Error (MSE) between the observed data (Y) and the forward model F(m)
No assumptions are made about the data unlike other regression based estimators
For non-linear model, use Non-Linear Least Squares
THE MODEL BASED APPROACHNon-Linear Regression
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NON-LINEAR LEAST SQUARES ESTIMATION
Based on MMSE (Minimum Mean Squared Error)
( ) ( )( )∑−
=−=
1N
0n
2n,mFnYQ
Relationship between signal model F(m) and m
is non-linear
F(m) has to be linearized
How ?
The Least Squared Error Criterion is
The Gauss Newton Iterative Minimization Algorithm
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GAUSS – NEWTON METHOD
( ) ( ) ( )[ ]( )ccmc mmmFmFmF −∇+≅
2. Linearization
- matrix of partial derivatives of F(m) w.r.t m
cm
1. Initialization
0m m=
( )cm mF∇
- set of current model parameters
k 1 km m+ = + ( ) ( ) ( ) ( )1T Tk k k kH m H m H m Y F m
−−⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦
3. Updation
(Starting guess)
( )m c[H F m ]= ∇
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GAUSS – NEWTON METHODPerformance
Algorithm may yield :Global minimum convergenceLocal minimum convergenceNo convergence
No Convergence
A good starting guess yields a good estimate (A,B)
To improve convergence - Run the algorithm with multiple starting guess values
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GAUSS – NEWTON METHODPerformance - Convergence Issues
- Global minimum was reached 2/10 times
- The rest were local, non-convergence cases
- For 10 dB SNR, Global minimum was reached 1/50 times
- Convergence is dependent on SNR
- Iterative search method (Computationally inefficient)
- Convergence is not guaranteed (in spite of several starting guesses)
- Large of model parameters ( >15 ) poor convergence
Limitations
Depth(m)
Perm
ittiv
ity
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GAUSS – NEWTON METHOD
Cannot be used to invert actual radar data
Other regression based techniques are also iterative search methods and cannot guarantee global minimum convergence
Need for a more reliable estimator
Model Based Spectral Estimation Techniques
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SPECTRAL ESTIMATION BASED INVERSION
Inversion:
Estimate Frequencies Estimate Amplitudes Permittivity profile
Parametric Spectral Estimation : Using a model to estimate frequency components in a signal
Suitable for applications in which signals can be represented by complex exponential models
Radar signals consist of sinusoids embedded in noise
MUSIC Algorithm
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MUSIC
MUSIC : MUltiple SIgnal Classification
High resolution frequency estimation technique
Exploits Orthogonality of signal and Noise
Enhances valid returns and suppresses noise peaks
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MUSICFrequency Estimation
( )1
( )=
= +∑ k
Pjn
kk
x n A e w nω
Form the (M x M) autocorrelation matrix ( Rx ) of x(n)
Decompose Rx into Eigen values and Eigen vectors
Assuming x(n) consists of P complex exponentials in white noise w(n)Signal model can be written as:
i 'sλ iV 's
1 2 P P 1 M..... .....+λ ≥ λ ≥ ≥ λ ≥ λ ≥ λEigen values :
( ) ( )1
=jmusic jw
i
P eV e
ω
( )1
0( ) ; 1, 2 ,....,
Mj jk
i ik
v e v k e i p p Mω ω−
−
=
= = + +∑
1 2 P P 1 MV V ..... V V .....V+≥ ≥ ≥ ≥ ≥Eigen vectors :
‘P’ signal eigen vectors ‘M-P’ noise eigen vectors
Will yield zero at the frequencies of complex exponentials
Will yield sharp peaks at the frequencies of complex exponentialsThe frequency estimation function
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MUSICAmplitude Estimation
( )1
( )=
= +∑ k
Pjn
kk
x n A e w nω
are known from the peaks of the frequency estimation function of MUSIC
Aim is to estimate 'kA s 'k sω;
( ) ( )
1
1
1
2
1 1
1 ... 1(0) (0)(1) ... (1): : : ... : :: : : :( 1]) ( 1)...− −
⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦⎣ ⎦⎣ ⎦
k
k
jj
j N j Nk
Ax wAx e e w
x N A w Ne e
ωω
ω ω
x AS w+= .
( ) 1ˆ ˆ ˆ ˆ .−
= H HA S S S X is the Maximum Likelihood Estimator of A
( only if W is White Gaussian)
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MUSICResolution Capability
(cm)
( ) { } ( )2
0 k k k
p k2 (f 2t )
k jk 1 j 1
x n T .e w nπ τ + ατ − ατ
= == Γ +∑ ∏
Range Profiles using FFT and MUSIC
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MUSICInversion – Simulation Results
Reconstructed profile matches well with true profile
Not constrained by layer depths
Impact of SNR
Good reconstruction results up to 5 dB SNR
Does not work well below 5 dB
Actual Profile VsReconstructed Profile using MUSIC
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MUSICPerformance
Good simulation results
Can be applied on actual data (if SNR is good enough)
Computational cost (Eigen decomposition)
Good forward model is required
Gaussian Noise statistics for amplitude estimation
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OUTLINE
IntroductionGPR ApplicationsThesis Objectives
The Inverse ProblemForward Modeling – FMCW RadarLayer Stripping ApproachThe Model Based Approach
Model Based Parameter EstimationMMSE based (Gauss-Newton)Spectral Estimation based (MUSIC)
Inversion on actual radar dataTests on Antarctic snow radar dataTests at the Sandbox labTests on Greenland Plane wave data GUI for data inversion algorithm
Conclusions & Future Work
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INVERSION ON ACTUAL DATA
1. Field experiments in Antarctica using FMCW Radar
2. Sandbox tests
3. Plane Wave test in Greenland
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FMCW RADAR TEST - ANTARCTICA
Ultra Wideband FMCW Radar – Used to measure snow thickness in Antarctica
Use MUSIC to estimate the permittivity profile from measured radar data
Parameters of FMCW radar
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FMCW RADAR TEST – ANTARCTICACore Data modeling
Snow pit data(Pit 1)
Modeling the true permittivity profile
Mixture of dry snow, water & brine
Consider brine as an inclusion within a wet snow mixture
Wet snow permittivity model : Debye-like model
Brine permittivity model : Stogryn’s model
Use a mixing model for effective permittivity
Perm
ittiv
ity
Dielectric structure of the test site
Depth (meters)
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FMCW RADAR TEST – ANTARCTICAMeasured Data
FFT Range Profile(Pit 1)
• Remove antenna feed-through
• Remove system effects using calibration data
• Enhance profile using MUSIC
• Estimate unknown frequencies & amplitudes
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FMCW RADAR TEST – ANTARCTICAInversion
Range Profiles obtained using FFT and MUSICComparison of estimated beat frequencies of
core with those of FFT and MUSIC
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FMCW RADAR TEST – ANTARCTICAReconstructed Profile
Depth (meters)
Perm
ittiv
ity
Good match up until 2.15 m depth
Deviations may be due to:
(1) A discrepancy in the model representing the radar return
(2) Subtle changes in permittivity that MUSIC is not able to distinguish
(3) Error in calibration data
(4) Measurement errors
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FMCW RADAR TEST – ANTARCTICAInversion on other data sets
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SANDBOX TESTSExperiment Set-up
Wood
Styrofoam
Sand
Air
~ 30 cm
~ 3.6 cm
~ 8 cm
Dielectric stack to test inversionNetwork Analyzer ParametersRSL Sandbox facility
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SANDBOX TESTSMeasurements
Calibrate at antenna terminals
Measure S11 with Aluminum plate as target
Measure S11 with multi-layered stack arrangement
Mismatch between antenna and the cable connecting the Network Analyzer is removed by taking Sky- shot measurements
Subtract Sky shot from S11of target, plate
Use plate impulse response to remove system effect
Apply MUSIC to enhance and invert
Antenna-cable mismatch
Antenna-cable mismatch(Return from stack is buried under this signal)
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SANDBOX TESTSMeasurements
FFT Range Profile
Signal after removing sky shot, system effects
This signal can now be fed into the inversion algorithm
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SANDBOX TESTSResults
Range Profiles using MUSIC
Deviation is because of an average value of Permittivity was chosen for velocity correction- when identifying the reflecting boundaries
Perm
ittiv
ity
Distance (meters)
Reference permittivity valuesAir : 1
Wood: 2 – 6 (a value of 3 was chosen for modeling)
Styrofoam : 1.03
Sand : 2.5 – 3.5 (a value of 3 was chosen for modeling)
Problem with reconstruction of Permittivity profile
Properties of noise could not be confirmed
Layer Stripping approach was followed
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PLANE WAVE DATA INVERSIONSetup - Greenland
Surface return Internal layers
Measured data
Depth in snow (meters)
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PLANE WAVE DATA INVERSIONAnalysis
Simulated range profile of Pit using ADS
Actual radar return
Inconsistencies in measured data
Internal reflections have higher amplitudes than surface reflection
Inversion yielded very high permittivity estimates
Depth in snow (meters)
Depth in snow (meters)
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PLANE WAVE DATA INVERSIONInversion test on ADS simulated data
Range Profile using FFT on ADS data
Range Profile using MUSIC
10 dB SNR
MUSIC works well in the case of multiple reflections
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G.U.I FOR DATA INVERSION
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OUTLINE
IntroductionGPR ApplicationsThesis Objectives
The Inverse ProblemForward Modeling – FMCW RadarLayer Stripping ApproachThe Model Based Approach
Model Based Parameter EstimationMMSE based (Gauss-Newton)Spectral Estimation based (MUSIC)
Inversion on actual radar dataTests on Antarctic snow radar dataTests at the Sandbox labTests on Greenland Plane wave data GUI for data inversion algorithm
Conclusions and Future Work
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SUMMARY
Studied, simulated and analyzed inversion schemes
Layer Stripping
Gauss Newton
MUSIC yields acceptable results in simulation
Implemented the MUSIC algorithm to enhance and invert GPR data
Tested on actual radar data
Successful in Snow radar data inversion
Partly successful in Sandbox test (Enhanced Profile)
Developed a GUI for the algorithm
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FUTURE WORK
Incorporate effects of scattering due to rough surface and losses due to attenuation into the forward model
Pre-whitening filter may be used to obtain Gaussian Noise statistics(or look at techniques for amplitude estimation in colored noise)
3 - Dimensional FDTD, MOM can be used to represent forward model for better inversion results
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THANK YOU!
QUESTIONS/COMMENTS?