# PUMA Applied to Time Delay Estimation for Processing GPR Data over Debonded Pavement Structures

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## Abstract

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## 1. Introduction

**Notation**: Throughout this paper, vectors and matrices are denoted by lowercase and uppercase boldface letters, respectively. ${(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}})}^{*}$, ${[\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}]}^{T}$, ${(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}})}^{H}$, ${(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}})}^{-1}$, and ${(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}})}^{\u2020}$ represent the complex conjugate, transpose, conjugate-transpose, matrix inverse, and pseudo inverse operations, respectively. $E[\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}]$ is the statistical mean. ⊗ is the Kronecker product. $vec(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}})$ is the vectorization operator. $tr(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}})$ is the trace operator. $\widehat{\mathbf{a}}$ represents the estimate of $\mathbf{a}$. $diag(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}})$ denotes a diagonal matrix. ${\mathbf{I}}_{m}$ is the $(m,m)$ identity matrix, and ${\mathbf{J}}_{m}$ is the $(m,m)$ exchange matrix with ones on its antidiagonal and zeros elsewhere. ${\left[\mathbf{a}\right]}_{m}$ refers to the ${m}^{th}$ element of any vector $\mathbf{a}$.

## 2. Data Model

- The road pavement is a stratified medium consisting of K smooth and homogeneous layers with low permittivity and negligible electrical conductivity;
- The impinging EM waves on the road pavement are assumed to be plane waves emitted from an antenna in the far field;
- Each layer contributes only one echo to the data model (single scattering assumption).

- $\mathbf{y}={[\phantom{\rule{0.166667em}{0ex}}\tilde{y}\left({f}_{1}\right),\phantom{\rule{0.166667em}{0ex}}\tilde{y}\left({f}_{2}\right),...,\tilde{y}\left({f}_{N}\right)\phantom{\rule{0.166667em}{0ex}}]}^{T}$: $(N,1)$ frequency data vector;
- $\mathbf{\Lambda}=diag(\phantom{\rule{0.166667em}{0ex}}\tilde{e}\left({f}_{1}\right),\phantom{\rule{0.166667em}{0ex}}\tilde{e}\left({f}_{2}\right),...,\tilde{e}\left({f}_{N}\right)\phantom{\rule{0.166667em}{0ex}})$: $(N,N)$ diagonal matrix whose elements are the Fourier transforms of the emitted pulse;
- $\mathbf{A}=[\phantom{\rule{0.166667em}{0ex}}\mathbf{a}\left({\tau}_{1}\right),\mathbf{a}\left({\tau}_{2}\right),...,\mathbf{a}\left({\tau}_{K}\right)\phantom{\rule{0.166667em}{0ex}}]$: $(N,K)$ mode matrix;
- $\mathbf{a}\left({\tau}_{k}\right)={[\phantom{\rule{0.166667em}{0ex}}{e}^{-j2\pi {f}_{1}{\tau}_{k}},{e}^{-j2\pi {f}_{2}{\tau}_{k}},...,{e}^{-j2\pi {f}_{N}{\tau}_{k}}\phantom{\rule{0.166667em}{0ex}}]}^{T}$: steering vector of size $(N,1)$;
- $\mathbf{s}={[\phantom{\rule{0.166667em}{0ex}}{s}_{1},{s}_{2},...,{s}_{K}\phantom{\rule{0.166667em}{0ex}}]}^{T}$: $(K,1)$ source vector containing the amplitudes of the echoes;
- $\mathbf{n}={[\phantom{\rule{0.166667em}{0ex}}\tilde{n}\left({f}_{1}\right),\tilde{n}\left({f}_{2}\right),...,\tilde{n}\left({f}_{N}\right)\phantom{\rule{0.166667em}{0ex}}]}^{T}$: $(N,1)$ noise vector with zero mean and covariance matrix ${\sigma}^{2}{\mathbf{I}}_{N}$;
- ${f}_{i}={f}_{1}+(i-1)\phantom{\rule{0.166667em}{0ex}}\Delta \phantom{\rule{0.166667em}{0ex}}f$: the frequency samples where ${f}_{1}$ is the beginning of the bandwidth, $\Delta f$ is the frequency difference between two adjacent samples, and $i=1,2,\cdots ,N$.

## 3. Time Delay Estimation Algorithms

#### 3.1. Eigendecomposition

#### 3.2. Root-MUSIC

#### 3.3. PUMA for TDE

- Initialize $\mathbf{c}$ with the least squares solution in (21): $\widehat{\mathbf{c}}={\widehat{\mathbf{c}}}_{LS}$;
- Let $\widehat{\mathbf{c}}={\widehat{\mathbf{c}}}_{WLS}$, and repeat Steps 2 and 3 until ${\parallel \widehat{\mathbf{c}}-{\widehat{\mathbf{c}}}_{WLS}\parallel}_{2}$ becomes stable.

## 4. Simulated Data

#### 4.1. Parameters for the Pavement Survey

#### 4.2. Correlation Scenarios

#### 4.2.1. Scenario 1: Uncorrelated Echoes

#### 4.2.2. Scenario 2: Full Correlation between Overlapping Echoes

#### 4.2.3. Scenario 3: Full Correlation between Echoes

#### 4.3. Computed Data Covariance Matrix

## 5. Results

#### 5.1. Evaluation Criteria

#### 5.2. Scenario 1: Uncorrelated Echoes

#### 5.3. Scenario 2: Full Correlation between the Overlapping Echoes

#### 5.4. Scenario 3: Full Correlation between Echoes

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Modified FB Covariance Matrix

## Appendix B. Modified MSSP Covariance Matrix

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**Figure 2.**(

**a**) Ricker pulse in the frequency domain with B = $3\mathrm{GHz}$; (

**b**) noiseless simulated signals over the debonded (${d}_{2}=0.5\mathrm{c}\mathrm{m}$) and the healthy regions of the pavement structure shown in Figure 1.

**Figure 3.**RMSE versus SNR, $K=3$, all echoes uncorrelated (Scenario 1), $N=150$, B = 3 $\mathrm{GHz}$.

**Figure 5.**RMSE versus SNR, $K=3$; the 2 overlapping echoes are fully correlated with each other, but uncorrelated with the surface echo (Scenario 2).

**Figure 9.**Average computational time versus the number of samples (N), fully uncorrelated signals, $K=3$, SNR = 25 dB.

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**MDPI and ACS Style**

Tchana Tankeu, B.; Baltazart, V.; Wang, Y.; Guilbert, D.
PUMA Applied to Time Delay Estimation for Processing GPR Data over Debonded Pavement Structures. *Remote Sens.* **2021**, *13*, 3456.
https://doi.org/10.3390/rs13173456

**AMA Style**

Tchana Tankeu B, Baltazart V, Wang Y, Guilbert D.
PUMA Applied to Time Delay Estimation for Processing GPR Data over Debonded Pavement Structures. *Remote Sensing*. 2021; 13(17):3456.
https://doi.org/10.3390/rs13173456

**Chicago/Turabian Style**

Tchana Tankeu, Bachir, Vincent Baltazart, Yide Wang, and David Guilbert.
2021. "PUMA Applied to Time Delay Estimation for Processing GPR Data over Debonded Pavement Structures" *Remote Sensing* 13, no. 17: 3456.
https://doi.org/10.3390/rs13173456