1 Master’s Thesis Defense Model Based Signal Processing for GPR Data Inversion Visweswaran Srinivasamurthy 13 th April 2005 Committee Dr. Sivaprasad Gogineni (Chair) Dr. Muhammad Dawood (Co-chair) Dr. Pannirselvam Kanagaratnam 2 OUTLINE  Introduction  GPR Applications  Thesis Objectives  The Inverse Problem  Forward Modeling – FMCW Radar  Layer Stripping Approach  The Model Based Approach  Model Based Parameter Estimation  MMSE based (Gauss-Newton)  Spectral Estimation based (MUSIC)  Inversion on actual radar data  Tests on Antarctic snow radar data  Tests at the Sandbox lab  Tests on Greenland Plane wave data  GUI for data inversion algorithm  Conclusions & Future Work 3 INTRODUCTION GPR Applications Ground Penetrating Radar Applications: ¾ Ice-sheet thickness measurements, bedrock mapping (Global Warming problem) ¾ Target detection (Landmines) ¾ Non-destructive testing of engineering structures ¾ Sub-surface Characterization (Earth, Martian Surface) Courtesy: JPL, NASA 4 INTRODUCTION Concepts 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 System Z 1 Z 2 Z 3 Z 4 1 ε 2 ε 3 ε 4 ε 5 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 Æ Electro-Magnetic (EM) Inversion Principle Permittivity contrast in layered media causes reflection of incident EM Wave Challenges  Radar return is corrupted by noise & clutter  Unwanted effects due to radar system (Eg: non-linearities)  Needs good understanding of EM propagation phenomenon 6 OUTLINE  Introduction  GPR Applications  Thesis Objectives  The Inverse Problem  Forward Modeling – FMCW Radar  Layer Stripping Approach  The Model Based Approach  Model Based Parameter Estimation  MMSE based (Gauss-Newton)  Spectral Estimation based (MUSIC)  Inversion on actual radar data  Tests on Antarctic snow radar data  Tests at the Sandbox lab  Tests on Greenland Plane wave data  GUI for data inversion algorithm  Conclusions & Future Work 7 THE GENERAL INVERSE PROBLEM Inverse Problem: Estimation of unknown parameters given an observation Steps for the study of an inverse problem  System 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 8 FORWARD MODELING Æ Mathematical relationship between permittivities & observed radar return signal Wave propagation Phenomena (1-D Plane wave approximation)  Reflection – Reflection Coefficient ( ) ( ) 11kk k k k ε εεε ++ Γ= − + 2 11 4 ++ ⎡⎤ =+ ⎣⎦ kkkkk T εε ε ε 2 1 4R ⎛⎞ ⎜⎟ π ⎝⎠ ¾ Spreading  Transmission – Transmission Coefficient  Attenuation – Attenuation Coefficient ¾ Absorption factor k T k Γ k B Neglected in our analysis ¾ Scattering Conductivity, particle distribution need to be known k kkk j j1 AB T = =Γ ∏ Effective amplitude of reflected signal at layer K (combined effect of , , ) 1 k Γ k T k B () 11 1 2 − = ⎡ ⎤ =+ − ⎢ ⎥ ⎣ ⎦ ∑ L kkkk k zzz c τ ε C = 3x 10 8 m/s Z 1 – Surface height 2-way time delay experienced by signal reflected from layer K 2 Estimate using (1) and (2) recursively ε 9 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) ( ) ( ) 2 tt 0 0 Vt ACos2 ft t= π +α +θ ⎡⎤ ⎣⎦ b 2RB f Tc = () ( ) {} k1 L1 beat k k j 0 k k k k0 j1 VATcos(2f2tn − − = = τ= Γ π τ+ατ −τ + ∑ ∏ ( For multiple targets, ) beat V τ is the forward model F(m) ; b f τ ∝ 10 FORWARD MODELING FMCW Radar  Fast Fourier Transform (FFT) of gives frequency response of the target  Plot of signal spectrum Vs distance – Range Profile ( ) beat V τ [...]... convergence Need for a more reliable estimator Model Based Spectral Estimation Techniques 20 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... Profile Vs Reconstructed Profile using MUSIC 26 MUSIC Performance 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 27 OUTLINE Introduction GPR Applications Thesis Objectives The Inverse Problem Forward Modeling – FMCW Radar Layer Stripping Approach The... Algorithm 21 MUSIC MUSIC : MUltiple SIgnal Classification High resolution frequency estimation technique Exploits Orthogonality of signal and Noise Enhances valid returns and suppresses noise peaks 22 MUSIC Frequency Estimation P Signal model can be written as: x ( n ) = ∑ Ak e jnω k Assuming x(n) consists of P complex exponentials in white noise w(n) + w (n) k =1 Form the (M x M) autocorrelation matrix... METHOD Performance Algorithm may yield : Global minimum convergence Local minimum convergence No convergence No Convergence A good starting guess yields a good estimate (A,B) To improve convergence - Run the algorithm with multiple starting guess values 18 GAUSS – NEWTON METHOD Performance - Convergence Issues - Global minimum was reached 2/10 times - The rest were local, non-convergence cases - For 10... STRIPPING APPROACH An elementary approach to inversion Plot signal spectrum (Range Profile) using Fast Fourier Transform (FFT) Set threshold on amplitudes Locate Amplitudes (Ak’s) and Time delays ( τk 's) from range profile Permittivity vector Depth-vector(m) Z εr [1 3 5 2 6] [0.5 0.3 0.4 0.4] A1 A3 A4 Threshold Recursively use (1) and (2) from the forward model to estimate the permittivity of every layer... contain most of the information about the deeper structure Reflection Amplitude Missed Peaks Missed Peak False Alarm Distance(m) Layer Stripping is not very reliable to detect subtle variations in permittivity Solution: Incorporate the underlying phenomenon into the inversion process The Model Based Approach 12 OUTLINE Introduction GPR Applications Thesis Objectives The Inverse Problem Forward Modeling... 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 15 NON-LINEAR LEAST SQUARES ESTIMATION Based on MMSE (Minimum Mean Squared Error) The Least Squared Error Criterion is N −1 Q = ∑ (Y (n ) − F (m , n )) 2 n=0 Relationship between signal model F(m) and m is non-linear... data GUI for data inversion algorithm Conclusions & Future Work 13 THE MODEL BASED ESTIMATION Model Based Estimator An estimator which incorporates the mathematical model F(m) to estimate unknown parameters (m) Regression Estimators (Data fitting or Curve fitting) Fit parameters to the observation (data) - based on some criterion Given an observed data set Y = { y[0 ], y[1], , y[N − 1] } , forward model... M) autocorrelation matrix ( Rx ) of x(n) Decompose Rx into Eigen values λ i 's and Eigen vectors Vi 's Eigen values : λ1 ≥ λ 2 ≥ ≥ λ P ≥ λ P +1 ≥ .λ M Eigen vectors : V1 ≥ V2 ≥ ≥ VP ≥ VP +1 ≥ .VM ‘P’ signal eigen vectors M −1 ( ) = ∑ v (k ) e vi e jω k =0 i − jkω ‘M-P’ noise eigen vectors ; i = p + 1, p + 2 , , M Will yield zero at the frequencies of complex exponentials ( ) Pmusic e jω The frequency... Estimation MMSE based (Gauss-Newton) Spectral Estimation based (MUSIC) Inversion on actual radar data Tests on Antarctic snow radar data Tests at the Sandbox lab Tests on Greenland Plane wave data GUI for data inversion algorithm Conclusions & Future Work 28 INVERSION ON ACTUAL DATA 1 Field experiments in Antarctica using FMCW Radar 2 Sandbox tests 3 Plane Wave test in Greenland 29 FMCW RADAR TEST - . The Model Based Approach  Model Based Parameter Estimation  MMSE based (Gauss-Newton)  Spectral Estimation based (MUSIC)  Inversion on actual radar data  Tests on Antarctic snow radar data . Plane wave data  GUI for data inversion algorithm  Conclusions & Future Work 14 THE MODEL BASED ESTIMATION Model Based Estimator An estimator which incorporates the mathematical model F(m). Spectral Estimation based (MUSIC)  Inversion on actual radar data  Tests on Antarctic snow radar data  Tests at the Sandbox lab  Tests on Greenland Plane wave data  GUI for data inversion algorithm 