# On the Potential of 3D Transdimensional Surface Wave Tomography for Geothermal Prospecting of the Reykjanes Peninsula

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## Abstract

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## 1. Introduction

## 2. Transdimensional Surface Wave Tomography

#### 2.1. Model Parameterization

#### 2.2. The Likelihood

#### 2.3. Forward Modeling

#### 2.4. The Prior

#### 2.5. Reversible Jump McMC

## 3. Experiment Setup and Computational Performance Tests

#### 3.1. Ray Path Update and Computational Cost

#### 3.2. Block Models

#### 3.3. Sensitivity Kernels

#### 3.4. Additive Noise and Modeling Errors

#### 3.5. Modeling and Inversion Parameters

## 4. Results and Discussion

#### 4.1. Coarse Block Model

#### 4.2. Intermediate Block Model

#### 4.3. Fine Block Model

#### 4.4. Chain Statistics and Convergence

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Geographical locations of 83 seismic stations of the RARR. The left and bottom axes indicate spherical coordinates, whereas the right and the top axes display UTM coordinates. Stations locations are depicted as colored triangles. Colors indicate the elevation of the corresponding station. Elevation information was lacking for five stations that are depicted with white triangles.

**Figure 2.**Two random Voronoi parameterizations. The first uses 33 Voronoi cells (

**a**), and the second uses 704 Voronoi cells (

**b**). Both cell locations and the assigned velocities are generated randomly.

**Figure 3.**Depth to frequency conversion of surface wave velocities using the modal approximation method. (

**a**) Surface wave velocity model generated randomly using Voronoi cells. (

**b**) Computed phase velocities corresponding to each depth profile using the modal approximation method.

**Figure 4.**Flowchart of the transdimensional Markov chain Monte Carlo algorithm used in this work. Here, a total of M samples is drawn from the posterior.

**Figure 5.**Decrease in computation time as a function of the frequency at which ray paths are updated. The speed up is depicted for three different FMM resolutions.

**Figure 6.**Three different synthetic block models to test the 3D transdimensional algorithm. (

**a**) Coarse block model, (

**b**) Intermediate block model, (

**c**) Fine block model. Inverted yellow triangles indicate the locations of the stations of the RARR. (See also Figure 1).

**Figure 7.**Surface wave sensitivity analysis of one of the depth profiles of each block model. The left column displays shear wave velocity as a function of depth for (

**a**) the coarse block model, (

**d**) the intermediate block model, and (

**g**) the fine block model. The middle column (

**b**,

**e**,

**h**) displays the corresponding sensitivity of the Rayleigh waves for different periods (1/f) as a function of depth. On the right (

**c**,

**f**,

**i**), the corresponding phase velocity dispersion is shown as a function of frequency.

**Figure 8.**Two tests to analyze modeling errors. (

**a**) Relative modeling errors while computing travel times through the fine block model of Figure 6c using three different grid resolutions. (

**b**) Modeling errors for computing travel times through the randomly generated model of Figure 3a using three different grid resolutions. Relative error due to the additive random Gaussian noise based on Equation (4) with ${a}_{j}=0.04$ and ${b}_{j}=0.1$ is also included in green line on both (

**a**,

**b**).

**Figure 9.**Transdimensional tomographic models estimated from noise-free synthetic travel times through the coarse block model in Figure 6a. The true block model: (

**a**) horizontal slice at the surface of the model including station locations, (

**b**) vertical cross-section at easting of 410 km, and (

**c**) vertical cross-section at northing of 7090 km. The vertical cross-sections are indicated with black dashed lines in (

**a**). (

**d**–

**f**) Pointwise averaged velocities calculated from post-burn-in retained samples. (

**g**–

**i**) Standard deviation (model uncertainty) calculated from post-burn-in retained samples.

**Figure 10.**Transdimensional tomographic models estimated from synthetic travel times containing Gaussian random noise for the coarse block model in Figure 6a. The true block model: (

**a**) horizontal slice at the surface of the model including station locations, (

**b**) vertical cross-section at easting of 410 km, and (

**c**) vertical cross-section at northing of 7090 km. The vertical cross-sections are indicated with black dashed lines in (

**a**). (

**d**–

**f**) Pointwise averaged velocities calculated from post-burn-in retained samples. (

**g**–

**i**) Standard deviation (model uncertainty) calculated from post-burn-in retained samples.

**Figure 11.**Transdimensional tomographic models for the noise-free synthetic travel times through the intermediate block model in Figure 6b. The true block model: (

**a**) horizontal slice at the surface of the model including station locations, (

**b**) vertical cross-section at easting of 405 km, and (

**c**) vertical cross-section at northing of 7095 km. The vertical cross-sections are indicated with black dashed lines in (

**a**). (

**d**–

**f**) Pointwise averaged velocities calculated from post-burn-in retained samples. (

**g**–

**i**) Standard deviation (model uncertainty) calculated from post-burn-in retained samples.

**Figure 12.**Transdimensional tomographic models for the synthetic travel times containing Gaussian random noise for the intermediate block model in Figure 6b. The true block model: (

**a**) horizontal slice at the surface of the model including station locations, (

**b**) vertical cross-section at easting of 405 km, and (

**c**) vertical cross-section at northing of 7095 km. The vertical cross-sections are indicated with black dashed lines in the horizontal slice. (

**d**–

**f**) Pointwise averaged velocities calculated from post-burn-in retained samples. (

**g**–

**i**) Standard deviation (model uncertainty) calculated from post-burn-in retained samples.

**Figure 13.**Transdimensional tomography results of the noise-free synthetic travel times for the fine block model in Figure 6c. The true block model: (

**a**) horizontal slice at the surface of the model including station locations, (

**b**) vertical cross-section at easting of 402 km, and (

**c**) vertical cross-section at northing of 7097 km. The vertical cross-sections are indicated with black dashed lines in the horizontal slice of (

**a**). (

**d**–

**f**) Pointwise averaged velocities calculated from post-burn-in retained samples. (

**g**–

**i**) Standard deviation (model uncertainty) calculated from post-burn-in retained samples.

**Figure 14.**Transdimensional tomography results of the synthetic travel times containing Gaussian random noise for the fine block model in Figure 6c. The true block model: (

**a**) horizontal slice at the surface of the model including station locations, (

**b**) vertical cross-section at easting of 402 km, and (

**c**) vertical cross-section at northing of 7097 km. The vertical cross-sections are indicated with black dashed lines in the horizontal slice of (

**a**). (

**d**–

**f**) Pointwise averaged velocities calculated from post-burn-in retained samples. (

**g**–

**i**) Standard deviation (model uncertainty) calculated from post-burn-in retained samples.

**Figure 15.**Variation of the misfit, the number of cells (dimensionality of model space), and the noise hyper-parameters during ten chains of McMC sampling for the intermediate block model in the noise-free experiment (

**a**–

**d**) and the experiment with additive Gaussian noise (

**e**–

**h**). Each color represents a sampling chain. Noise hyper-parameters are plotted for a single frequency. The black lines in (

**g**–

**h**) represent the actual values. Histograms of the posterior distribution for the retained models for each parameter are shown as insets of each panel.

**Table 1.**Modeling and inversion parameters. Parameters are the same for the noise-free experiment and the experiment with synthetic additive noise.

Modeling and Inversion Parameters | Coarse Block Model | Intermediate Block Model | Fine Block Model |
---|---|---|---|

Model dimension (km) | 120 × 70 × 20 | 120 × 70 × 10 | 120 × 70 × 10 |

Number of grid points in generating synthetic data | 241 × 141 × 81 | 241 × 141 × 81 | 241 × 141 × 81 |

Number of grid points in McMC sampling | 121 × 71 × 41 | 121 × 71 × 41 | 121 × 71 × 41 |

Number of grid points for calculating post-burn-in pointwise average | 241 × 141 × 41 | 241 × 141 × 41 | 241 × 141 × 41 |

Valid range of shear wave velocity (km/s) | 1.5–6 | 1.5–6 | 1.5–6 |

Valid range of noise hyper parameter a | ${10}^{-5}$–1 | ${10}^{-5}$–1 | ${10}^{-5}$–1 |

Valid range of noise hyper parameter b | 0–2 | 0–2 | 0–2 |

Proposal width for a move step. $Md$ is the model dimension. | $0.07\ast Md$ | $0.06\ast Md$ | $0.05\ast Md$ |

Velocity proposal width (km/s) | 0.4 | 0.4 | 0.3 |

Proposal width for a | ${10}^{-3}$ | ${10}^{-3}$ | ${10}^{-3}$ |

Proposal width for b | ${10}^{-2}$ | ${10}^{-2}$ | ${10}^{-2}$ |

Thinning level | 200 | 200 | 200 |

Ray path update step | 200 | 200 | 200 |

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**MDPI and ACS Style**

Rahimi Dalkhani, A.; Zhang, X.; Weemstra, C. On the Potential of 3D Transdimensional Surface Wave Tomography for Geothermal Prospecting of the Reykjanes Peninsula. *Remote Sens.* **2021**, *13*, 4929.
https://doi.org/10.3390/rs13234929

**AMA Style**

Rahimi Dalkhani A, Zhang X, Weemstra C. On the Potential of 3D Transdimensional Surface Wave Tomography for Geothermal Prospecting of the Reykjanes Peninsula. *Remote Sensing*. 2021; 13(23):4929.
https://doi.org/10.3390/rs13234929

**Chicago/Turabian Style**

Rahimi Dalkhani, Amin, Xin Zhang, and Cornelis Weemstra. 2021. "On the Potential of 3D Transdimensional Surface Wave Tomography for Geothermal Prospecting of the Reykjanes Peninsula" *Remote Sensing* 13, no. 23: 4929.
https://doi.org/10.3390/rs13234929