# Quantitative Evaluation for the Internal Defects of Tree Trunks Based on the Wavefield Reconstruction Inversion Using Ground Penetrating Radar Data

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## Abstract

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## 1. Introduction

- We applied WRI to quantitatively evaluate the health condition of tree trunks. In the proposed algorithm, the grouped multi-frequency strategy is used to avoid cycle skipping, and a variation projection is applied to make WRI computationally tractable.
- Comparing the performance of traditional FWI and WRI under different conditions in detail. The results indicate that in contrast with traditional FWI, WRI can reduce nonlinearity and dependence on the initial model. In addition, appropriate frequency strategy and mesh generation method are important for reasonable WRI results.
- To address the deficiency that most methods can only locate defects, the proposed WRI can accurately depict the shape, location, and properties of internal defects and structures in tree trunks. The potential of WRI in tree detection was explored through numerical and field cases. The successful results indicate that WRI has prospects in tree trunk detection.

## 2. Theoretical Background

#### 2.1. GPR Finite-Element Frequency Domain Simulation

_{z}(A/m

^{2}) are the electric field density and current density in the frequency domain, respectively. α

_{x}, α

_{y}, β are anisotropic medium parameters, k = sqrt(ꞷ

^{2}εμ-iꞷσμ) is wavenumber in the frequency domain, ꞷ and I = sqrt(−1) represents the angular frequency and the imaginary unit, respectively. The model parameters ε (F/m), σ (S/m), and μ (H/m) are the dielectric constant, conductivity, and permeability of the tree trunk, respectively. In this paper, we assume that the permeability is constant.

**A**(ꞷ,ε,σ) is the Helmholtz operator which represents a series of partial differential equations (PDEs), and

**u**(ꞷ) and

**q**(ꞷ) represent the wavefield vector and source vector, respectively. For the sake of simplicity, we will abbreviate

**A**(ꞷ,ε,σ),

**u**(ꞷ), and

**q**(ꞷ) as

**A**(ε,σ)

**, u**and

**q**. Considering the irregular bark, the FEFD based on perfectly matching layers (PML) is adopted to discrete the model space and conduct the forward simulation [39].

#### 2.2. GPR-WRI in the Frequency Domain

**m**= [m

_{1},m

_{2},…,m

_{M}]

^{T}represents the model parameters (containing the dielectric constant ε and conductivity σ) at each element of the finite-element mesh. M is two times the number of elements. Here, the bound constraints are considered in the inversion. The dielectric constant models are bounded by the lower bound 1 and the upper bound 81, and the conductivity models are bounded by the lower bound 0.

**P**is an operator that extracts the modeled wavefield at the receiver positions, and

**d**is the observed data at the same receiver. The objective function above is an optimization problem constrained by the wave equation (Equation (2)). It indicates

**A**(

**m**)

^{−1}

**q**is subject to Equation (2) and represents the modeled wavefield

**u**in Equation (3).

_{p}is the positive penalty parameter. The first term is the data misfit, and the second is the wave equation term. Because WRI with a small penalty parameter will become the optimized problem in weakly constrained, it has the ability to reduce the nonlinearity and dependence on the initial model [31,40].

_{p}increases, the wave equation term is more tightly constrained, and WRI is close to the conventional FWI. If λ

_{p}decreases, the inversion will expand the search space in loose constraints by the wave equation [41].

_{ω}and N

_{s}represent the number of frequency and source, respectively.

**m**:

**G**= ∂

**A**(

**m**)/∂

**m**is the diffraction matrix. In this paper, we update the optimization problem via the limited-memory Broyden–Fletcher–Goldfarb–Shanno algorithm (L-BFGS) algorithm [42].

#### 2.3. The Grouped Multi-Frequency Strategy

_{1}, ꞷ

_{2}, …, ꞷ

_{i}, …, ꞷ

_{Nꞷ}). Then, sorting the data from low-frequency to high-frequency, and dividing the sorted data into Q subsequences. Each subsequent C

_{i}(i = 1, 2, …, Q) is a subset of the total inversion frequency sequence. Next, the first subsequence on the initial model until the iteration termination condition is inverted, and the inversion resulting model at the previous subsequence is used as the new initial model of the next subsequence. In detail, the frequency component of each subsequence can be described as follows:

## 3. Numerical Examples

_{1}(550, 575, 600, 625, 650, 675 MHz) and C

_{2}(700, 750, 800, 850, 900, 950 MHz). We set a misfit of Φ

^{(0)}/Φ

^{(k)}≤ 1 × 10

^{−5}or the maximum iterations in each frequency subsequence as the iteration termination condition. Specifically, each frequency subsequence maximum iteration is set as 100.

**m**

_{true}and

**m**

_{inv}denote the true model and the inverted model.

#### 3.1. Effect of the Penalty Parameter

_{p}= 1, λ

_{p}= 100, and λ

_{p}= 10,000. The total inversion frequency sequence is subsampled to two subsequences (Q = 2) to conduct inversion. Figure 4 shows the inversion results and detailed profile of FWI and WRI with three different penalty parameters. Below each figure is the detailed profile for the while dotted curve in Figure 3a,b, where black, blue, and red curves represent the parameter value from the true, initial, and reconstructed models, respectively. Figure 4a–d are reconstructed models of the dielectric constant, which are in accordance with the true model shown in Figure 3a,b. From the results, we can obtain the location, size, and shape of internal defects. Some details such as the crack depth and the cambium width are depicted well. Additionally, the specific value fits well with the true value in the profile. It indicates that FWI and WRI both have the potential to describe the dielectric constant of tree trunks. Figure 4e–h are the conductivity results and have a lower resolution than the dielectric constant results. Additionally, the comparisons of FWI and WRI are visible. For the FWI result, the shape of the hollow occurs distortion and the disturbances hinder the identification of defects and structure. WRI with larger λ

_{p}mitigates the disturbances and artifacts, but the conductivity value of decay is a little different from the true model. From the results shown in Figure 4, as the penalty parameter λ

_{p}decreases, the reconstructed model is closer to the true model and the disturbance mitigates obviously.

_{p}. Moreover, as λ

_{p}decreases, equation solutions and time reduce continuously.

_{p}= 1) has less misfit and terminates iteration fastest. While in Figure 6b,c, they show the comparison of model reconstruction error convergence curves under different inversion methods. Regardless of dielectric constant and conductivity, WRI with λ

_{p}= 1 or λ

_{p}= 100 appears with smaller final errors and numbers of iterations. In Figure 6c, what is interesting is that the blue and green curves (represent FWI and WRI with λ

_{p}= 10,000) first rise and then fall. The possible reason is that they have stronger nonlinearity and dependence on the initial model. The continuous decline of the red curve proves that WRI with a small penalty parameter can reduce the nonlinearity and dependence on the initial model.

_{p}increases, which is consistent with the principle. As λ

_{p}decreases, the resolution and efficiency of WRI increase. Specifically, when λ

_{p}= 1 in this model, the cost time used WRI is reduced to one-half of the cost time used FWI. Furthermore, the PDE solutions used WRI are reduced to one-third of the PDE solutions used FWI, which means WRI greatly improves computational efficiency. The improvement indicates that WRI has promising potential to detect trunk defects efficiently and delicately.

#### 3.2. Effect of the Initial Model

_{p}= 1.

_{p}= 1 appears with the smallest final error and number of iterations in Figure 9a–c.

#### 3.3. Effect of the Frequency Strategy

_{p}= 1.

#### 3.4. Effect of the Grid Generation Method

## 4. Field Example

_{p}= 1.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Diagram of (

**a**) tree trunk detection using GPR, (

**b**) GPR signal propagation inside tree trunk with hollow.

**Figure 3.**True tree trunk model of (

**a**) dielectric constant and (

**d**) conductivity; the first initial model (IM1) of (

**b**) dielectric constant and (

**e**) conductivity; the second initial model (IM2) of (

**c**) dielectric constant and (

**f**) conductivity. The red symbols “○” indicate the sources, and the red symbols “×” indicate receivers.

**Figure 4.**Reconstructed models of different inversion methods on IM1: (

**a**,

**e**) FWI, (

**b**,

**f**) WRI with λ

_{p}= 10,000, (

**c**,

**g**) λ

_{p}= 100, and (

**d**,

**h**) λ

_{p}= 1. Profiles at a radius of 0.28 m along the true models (black), initial models (blue), and reconstructed models (red) are shown below the reconstructed models. The upper layer is the dielectric constant result, and the lower layer is the conductivity result.

**Figure 5.**Reconstruction Errors, PDE solves, and cost time of different inversion methods on IM1. From left to right are the results of FWI, WRI (λ

_{p}= 10,000), WRI (λ

_{p}= 100), and WRI (λ

_{p}= 1).

**Figure 6.**The curves of different inversion methods on IM1: (

**a**) misfit curve, (

**b**) dielectric constant error convergence curve, (

**c**) conductivity error convergence curve. The blue, green, yellow, and red curves denote FWI, WRI (λ

_{p}= 10,000), WRI (λ

_{p}= 100), and WRI (λ

_{p}= 1), respectively.

**Figure 7.**Reconstructed models of different inversion methods on IM2: (

**a**,

**b**) FWI and (

**c**,

**d**) WRI. The upper layer is the dielectric constant result, and the lower layer is the conductivity result.

**Figure 8.**Profiles at a radius of 0.28 m along the true models (black), initial models (blue), reconstructed models of FWI (green), reconstructed models of WRI (red), (

**a**) dielectric constant, and (

**b**) conductivity.

**Figure 9.**The curves of different inversion methods on IM2: (

**a**) misfit curve, (

**b**) dielectric constant error convergence curve, (

**c**) conductivity error convergence curve. The blue and red curves denote FWI, and WRI (λ

_{p}= 1), respectively.

**Figure 10.**Reconstructed models of WRI with different frequency strategies on IM2: (

**a**,

**e**) S, (

**b**,

**f**) B1, (

**c**,

**g**) B2, and (

**d**,

**h**) C. Profiles at a radius of 0.28 m along the true models (black), initial models (blue), and reconstructed models (red) are shown below the reconstructed models. The upper layer is the dielectric constant result, and the lower layer is the conductivity result.

**Figure 11.**Reconstruction errors, PDE solves, and cost time of WRI with different frequency strategies on IM2. The results of WRI(S), WRI(B1), WRI(B2), and WRI(C) are from left to right.

**Figure 12.**The curves of WRI with different frequency strategies on IM2: (

**a**) misfit curve, (

**b**) dielectric constant error convergence curve, (

**c**) conductivity error convergence curve. The blue, green, yellow, and red curves denote WRI(S), WRI(B1), WRI(B2), and WRI(C), respectively.

**Figure 13.**Tree trunk model with different grid generation methods: (

**a**,

**d**) true model discretized by irregular triangular simulation grid, (

**b**,

**e**) IM1 discretized by irregular triangular reconstructed grid, (

**c**,

**f**), IM1 discretized by regular quadrilateral reconstructed grid. The upper layer is the dielectric constant result, and the lower layer is the conductivity result.

**Figure 14.**Reconstructed models of (

**a**,

**c**) irregular triangular reconstructed grid and (

**b**,

**d**) regular quadrilateral reconstructed grid. The upper layer is the dielectric constant result, and the lower layer is the conductivity result.

**Figure 17.**The comparison between the observed field data and the calculation data of the final inversion results at three frequencies (f1 = 99.2 MHz, f2 = 218.3 MHz, and f3 = 416.7 MHz): (

**a**) amplitude; (

**b**) phase.

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## Share and Cite

**MDPI and ACS Style**

Feng, D.; Liu, Y.; Wang, X.; Ding, S.; Xu, D.; Yang, J.
Quantitative Evaluation for the Internal Defects of Tree Trunks Based on the Wavefield Reconstruction Inversion Using Ground Penetrating Radar Data. *Forests* **2023**, *14*, 912.
https://doi.org/10.3390/f14050912

**AMA Style**

Feng D, Liu Y, Wang X, Ding S, Xu D, Yang J.
Quantitative Evaluation for the Internal Defects of Tree Trunks Based on the Wavefield Reconstruction Inversion Using Ground Penetrating Radar Data. *Forests*. 2023; 14(5):912.
https://doi.org/10.3390/f14050912

**Chicago/Turabian Style**

Feng, Deshan, Yuxin Liu, Xun Wang, Siyuan Ding, Deru Xu, and Jun Yang.
2023. "Quantitative Evaluation for the Internal Defects of Tree Trunks Based on the Wavefield Reconstruction Inversion Using Ground Penetrating Radar Data" *Forests* 14, no. 5: 912.
https://doi.org/10.3390/f14050912