# Full-Waveform Inversion of Time-Lapse Crosshole GPR Data Using Markov Chain Monte Carlo Method

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Probabilistically Formulated Inversion

**d|m**) represents the modeling error due to the defective forward method or an defective parameterization. ${\mu}_{D}\left(\mathit{d}\right)$ describes the homogeneous state of information. It ensures that when the coordinate system changes, parameterization is invariant. In most cases, ${\mu}_{D}\left(\mathit{d}\right)$ can be assumed to a constant.

#### 2.2. Forward Model Based on Waveform

#### 2.3. Local Sampling Based on Extended Metropolis Algorithm

- (1)
- Giving a current model ${\mathit{m}}_{cur}$, it conforms the a priori probability density. Generating a perturbation in the current model ${\mathit{m}}_{cur}$, we get a candidate model, ${\mathit{m}}_{pro}$.
- (2)
- Decide whether to accept the proposed model ${\mathit{m}}_{pro}$, the probability of acceptance is measured by the ratio of likelihood function:

- Starting in the current model, ${\mathit{m}}_{cur}$, which is the result of the first inversion, the range size of the target area is $\mathbf{\nabla}{\mathit{m}}_{loc}$. A new model candidate, ${\mathit{m}}_{pro}$, which samples in the $\mathbf{\nabla}{\mathit{m}}_{loc}$ only using the sequential Gibbs sampler.
- The proposed model is accepted with probability ${P}_{acc}=min(1,\frac{L\left({\mathit{m}}_{pro}\right)}{L\left({\mathit{m}}_{cur}\right)})$.
- If ${\mathit{m}}_{pro}$ is accepted, we use ${\mathit{m}}_{pro}$ instead of ${\mathit{m}}_{cur}$. Consequently, the proposed model takes the place of the current model, ${\mathit{m}}_{cur}$ = ${\mathit{m}}_{pro}$. Otherwise, the random walker stays at a location in ${\mathit{m}}_{cur}$, and ${\mathit{m}}_{cur}$ is counted again.

#### 2.4. Time-Lapse Inversion-Double Difference Strategy

## 3. Synthetic Examples

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The double difference strategy. ${\mathit{d}}_{obs1}$,${\mathit{d}}_{obs2}$ are the corresponding observational data at different times ${T}_{1}$,${T}_{2}$. The models ${\mathit{m}}_{T1}$ and ${\mathit{m}}_{T2}$ are the inversion result at time ${T}_{1}$,${T}_{2}$. The result of time-lapse inversion can be obtained by subtracting ${\mathit{m}}_{T1}$ from ${\mathit{m}}_{T2}$.

**Figure 2.**Recording geometry; sources (red crosses) and receivers (black dots) are represented by a connecting black line.

**Figure 3.**Noise-free waveforms and noisy waveforms: the signal-to-noise ratios from the top are 8, 15, 24, 32 and 40, respectively.

**Figure 4.**Initial model ${\mathit{m}}_{T1}$, monitor model ${\mathit{m}}_{T2}$, and the perturbation.

**Figure 7.**Five waveform traces. Blue dotted curves: simulated waveforms calculated on a posteriori model. Red curves: the observed waveform data.

**Figure 8.**(

**a**) Full sampling inversion model of ${T}_{2}$, and (

**b**) the perturbation of the full sampling inversion. (

**c**) Local sampling inversion model of ${T}_{2}$, and (

**d**) the perturbation of the local sampling inversion.

**Figure 10.**(

**a**) The design perturbation model histogram, (

**b**) perturbation of all sampling histograms, and (

**c**) perturbation of local sampling histogram.

Method | Mean | Variance |
---|---|---|

Design perturbation | 1.25 | 0.43 |

All sampling | 0.75 | 0.72 |

Local sampling | 1.21 | 0.47 |

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**MDPI and ACS Style**

Wang, S.; Han, L.; Gong, X.; Zhang, S.; Huang, X.; Zhang, P.
Full-Waveform Inversion of Time-Lapse Crosshole GPR Data Using Markov Chain Monte Carlo Method. *Remote Sens.* **2021**, *13*, 4530.
https://doi.org/10.3390/rs13224530

**AMA Style**

Wang S, Han L, Gong X, Zhang S, Huang X, Zhang P.
Full-Waveform Inversion of Time-Lapse Crosshole GPR Data Using Markov Chain Monte Carlo Method. *Remote Sensing*. 2021; 13(22):4530.
https://doi.org/10.3390/rs13224530

**Chicago/Turabian Style**

Wang, Shengchao, Liguo Han, Xiangbo Gong, Shaoyue Zhang, Xingguo Huang, and Pan Zhang.
2021. "Full-Waveform Inversion of Time-Lapse Crosshole GPR Data Using Markov Chain Monte Carlo Method" *Remote Sensing* 13, no. 22: 4530.
https://doi.org/10.3390/rs13224530