# Combining InSAR and GNSS to Track Magma Transport at Basaltic Volcanoes

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## Abstract

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## 1. Introduction

## 2. The 26 May 2016 Eruption of Piton De La Fournaise Volcano

## 3. Geodetic Measurement Description

#### 3.1. InSAR Data

#### 3.2. GNSS Data

## 4. Inverse Models

#### 4.1. Inversion and Data Weighting Through the Covariance Matrix

^{th}interferogram and N is the total number of interferograms. This scaling is equivalent to applying different scaling factors of ${{\chi}^{2}}_{ref}^{i}/{\chi}_{ref}^{2}$ with $i=1,..,N$ and ${{\chi}^{2}}_{ref}^{GNSS}/{\chi}_{ref}^{2}$ to the covariance matrices of the InSAR and the GNSS data, respectively. In terms of Bayesian inference, this scaling gives the same likelihood to each of the InSAR and GNSS datasets and each of these datasets has the same contribution to the posterior probability density distribution.

#### 4.2. Static Inversion

#### 4.3. Temporal Inversion

#### 4.3.1. A Method without Any Geometric a Priori from the Static Inversion: The Ellipse Method

#### 4.3.2. A First Method with a Geometric a Priori: The Projected Disk Method

#### 4.3.3. A Second Method with a Geometric a Priori: The Subgraph Method

## 5. Results

#### 5.1. Static Inversion

#### 5.1.1. Two Model Families Which Explain the Data Equally Well

#### 5.1.2. Importance of Consistency between Time Periods Covered

#### 5.1.3. Relative Weights of Ascending Versus Descending Interferograms

#### 5.1.4. Relative Weights of InSAR Versus GNSS Data

#### 5.2. Temporal Inversion

#### 5.2.1. A Need for Geometrical a Priori to Invert for the GNSS Time Series

#### 5.2.2. Inversion of GNSS Time Series to Improve Discrimination between Families of Intrusion Geometry

## 6. Discussion

#### 6.1. Discrepancies between Independent Datasets Reveal Hidden Processes

#### 6.1.1. Discrepancies in the Covered Time Periods Reveal Pre-Eruptive Displacement

#### 6.1.2. Discrepancies in Amplitude Along the Different Los Reveal Flank Displacement

#### 6.2. Combining InSAR and GNSS for Complementary Spatial and Temporal Information

#### 6.2.1. InSAR Static Inversion Constrains the Temporal Inversion

#### 6.2.2. Advantages of the Subgraph Method for Temporal Inversions

#### 6.2.3. GNSS Temporal Inversion Solves the Conundrum of Non Unique Static Inversion

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ANR | Agence Nationale de la Recherche |

GNSS | Global Navigation Satellite System |

InSAR | Interferometric Synthetic Aperture Radar |

LOS | Line of Sight |

OVPF | Observatoire volcanologique du Piton de la Fournaise (Piton de la Fournaise Volcanological Observatory) |

Piton de la Fournaise | |

PCA | Principal Component Analysis |

SAR | Synthetic Aperture Radar |

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**Figure 1.**Piton de la Fournaise (PdF) setting. (

**A**) Location of Reunion Island; (

**B**) Shaded topography of PdF volcano and location of the GNSS monitoring network. Black and grey diamonds represent the locations of the permanent GNSS stations. The black circled diamonds refer to the 10 GNSS stations used in this study. The GITg station (black diamond to the northwest) is the reference used here for the GNSS campaign and for the epoch-by-epoch processing. The May 2016 lava flow is shown in purple. ’Dol’ marks the Dolomieu crater; (

**C**) Zoom corresponding to the area indicated by a black box in B. Black arrows represent the cumulated horizontal displacement of the 10 GNSS stations between 25 May at 17:00 and 26 May at 04:05. Standard deviations for horizontal and vertical components are 1–2 cm and 3-4 cm, respectively. Associated colored circles represent the temporal evolution of the horizontal displacement for the GNSS stations. Grey arrows represent displacements from the GNSS campaign network between 31 August 2015 and 27 May 2016, with ellipses for the 95% confidence intervals. Coordinates are in kilometers (WGS84, UTM 40S). Modified after Reference Smittarello et al. [16]

**Figure 2.**(

**Top**) Temporal evolution of baseline changes recorded between two pairs of permanent GNSS stations (SNEG and DSRG and BORG and DERG; see Figure 1 for station locations) between 29 February 2016 (date of the first SAR acquisition used in this study) and 07/06/2016 (date of the last SAR acquisition used in this study). Distance samples are daily averages. Inset corresponds to a zoom on the pre-eruptive baseline changes spanned by Sentinel–1 data. The eruption onset is shown by a red arrow; (

**Middle**) Coloured bars represent the time spanned by each of the 8 interferograms used in this study. Modified after Smittarello et al. [16]. The vertical red area represents the eruptive crisis; (

**Bottom**) Zoom. The vertical dashed line at 01:45 marks the time of the intermediate SAR acquisition by Sentinel–1. CSK = Cosmo-Skymed; S1 D1 = Sentinel–1 descending InSAR data spanning the beginining of the eruptive crisis; S1 D2 = Sentinel–1 descending InSAR data spanning the end part of the eruptive crisis; S1 A1 = Sentinel–1 ascending InSAR data spanning the pre-eruptive period; S1 A2 = Sentinel–1 ascending InSAR data spanning only the eruption.

**Figure 3.**Wrapped interferograms used in this study. (

**Left column**) Data acquired on ascending orbits; (

**Right column**) Data acquired on descending orbits; (

**First row**) Cosmo-Skymed data (X-band) spanning the whole eruptive period; (

**Second to Fourth rows**) Sentinel-1 InSAR ascending and descending data (C-band); (

**Second row**) Sentinel-1 data spanning the whole eruptive period; (

**Third row**) Sentinel-1 data spanning the first time period; (

**Fourth row**) Sentinel-1 data spanning the second time period; For Cosmo-Skymed data, fringes represent 1.5 cm LOS displacement, while for Sentinel-1 data, fringes represent 2.8 cm LOS displacement. S1 D1 = Sentinel-1 descending InSAR data spanning the beginining of the eruptive crisis; S1 D2 = Sentinel-1 descending InSAR data spanning the end part of the eruptive crisis; S1 A1 = Sentinel-1 ascending InSAR data spanning the pre-eruptive period; S1 A2 = Sentinel-1 ascending InSAR data spanning only the eruption.

**Figure 5.**Distances between connected nodes in the graph-method. Node numbers are represented in black. (

**A**) Distances from node 1 are shown in blue; (

**B**) Distance from node 9 are shown in blue and triangles with nodes at a distance of less than two edges from node 9 are colored in cyan.

**Figure 6.**Comparison between geometries of typical intrusions of family F1 (Inv04a in blue) and family F2 (Inv02a in red). The upper plots compare the geometry and fracture opening of intrusions of family F1 (

**left**) and family F2 (

**right**). Three views are shown for each subfigure, one from the east along the northern direction on the left-hand side, a map view on the right-hand side and a view from the south along the eastern direction at the bottom. The lower plot compares the eruptive fracture meshes for the two families, as seen from the east.

**Figure 7.**Methods for computing the average distance between two meshes, illustrating why ${D}_{AB}$ is slightly different from ${D}_{BA}$.

**Figure 8.**Projection of surface displacement along the LOS for satellite radar acquisition for ascending and descending orbits. The left-hand plots indicate expected displacements in the ascending and descending LOS for an inflation and an eastward flank slip. The right-hand plot shows the expected sign of the LOS projection (considered as positive when directed toward the satellite ) in the ascending and descending orbits for vectors joining a ground point to the circle.

**Figure 9.**Comparison between the intrusion determined from InSAR data covering the whole eruption (grey) and the best model intrusion geometry (blue) resulting from the temporal inversion of GNSS data at 01:45. Results for the three different methods for inverting the pressure source are shown. (

**Top**) Ellipse; (

**Middle**) Projected Disk and (

**bottom**) Subgraph methods.

**Figure 10.**Comparison of inversion results for validation test using interferogram S1 D1. Wrapped interferogram S1 D1 (

**first column**) and modeled interferograms computed with the best models determined with the Ellipse method (

**2nd column**), Projected Disk method (

**3rd column**) and Subgraph method (

**4th column**). Inversion is done with continuous GNSS data alone (

**top**) or with SAR S1 D1 data alone (

**bottom**). Black segment markss the location of the eruptive fissure. Diamonds represent the location of permanent GNSS stations.

**Figure 11.**Results of inversions of the GNSS time series for 29 time steps between 20:40 and 4:00 a.m. Green line is the Ellipse method. Blue and red dashed lines are the Projected Disk method on a priori meshes for families F1 and F2, respectively. Solid black circles are the Subgraph method for family F2. Open symbols are for inversions from the intermediate Sentinel-1 interferogram S1 D1, green star corresponds to the Ellipse method, red square to the Projected Disk on family F2 and black circle to the Subgraph method on family F2. Note that for all methods, increase in the misfit between 20:40 and 21:30 is due to the increase in the amplitude of measured displacements as shown by the sharp decrease of the relative misfit in percent Equation (1).

**Figure 12.**Location of the overpressure source determined from the temporal inversion (colors) for 14 time steps out of the 29 time steps inverted between 20:40 on 25 May and 04:00 the next day. The pressurized area is assumed to be a circular disk projected on the a priori mesh (in gray) of family F1 (

**A**) and family F2; (

**B**) or a subgraph of the a priori mesh of family F2; (

**C**). Source geometries for F1 and F2 are non unique solutions determined from the inversion of the four sets of InSAR data that cover the whole eruption.

**Table 1.**Covariance weigthing ($Cd$ is defined based on Equations (4) and (5), ’No’ means that no specific additional weighting is applied whereas ${\chi}_{ref}^{2}$ means that the misfit is scaled by the misfit of the null model), geometry of the best model (for families definition see Section 5.1.1) and % of Explained data (see Equation (9)) computed for each interferogram spanning the eruption. Numbers in bold are for data used in the inversion, values in small italics are computed a posteriori for data not used in the inversion. Inv01 was computed with InSAR data S1 A2, which does not cover the pre-eruptive period. Inv. 01 to 03 were computed without including the GNSS data. Inv02a* is the preferred model presented in Smittarello et al. [16].

Model | Covariance Weighting | Family | GNSS | InSAR | |||||
---|---|---|---|---|---|---|---|---|---|

S1D | S1 A | S1 A2 | CSKD | CSKA | Total | ||||

Inv01 | No | F1 | 78 | 97 | 81 | 75 | 97 | 83 | 93.5 |

Inv02a* | No | F2 | 73 | 95 | 82 | 72 | 95 | 82 | 92.5 |

Inv02b | No | F2 | 75 | 96 | 83 | 76 | 95 | 83 | 93.0 |

Inv03 | ${\chi}_{ref}^{2}$ | F2 | 73 | 96 | 86 | 77 | 96 | 87 | 94.1 |

Inv04a | ${\chi}_{ref}^{2}$ | F1 | 77 | 95 | 82 | 74 | 95 | 87 | 93.5 |

Inv04b | ${\chi}_{ref}^{2}$ | F2 | 71 | 96 | 83 | 72 | 96 | 85 | 93.5 |

Inv05 | No | F1 | 83 | 94 | 67 | 57 | 95 | 75 | 90.3 |

**Table 2.**Average distances (in meters) between pairs of meshes. * is the preferred model presented in Smittarello et al. [16]. Distances in bold are for models belonging to the same family.

Inversion | Inv01 | Inv02a | Inv02b | Inv03 | Inv04a | Inv04b | Inv05 | |
---|---|---|---|---|---|---|---|---|

Family | F1 | F2 | F2 | F2 | F1 | F2 | F1 | |

Inv01 | F1 | X | 129 | 124 | 109 | 90 | 139 | 132 |

Inv02a * | F2 | 121 | X | 107 | 167 | 150 | 169 | 177 |

Inv2b | F2 | 105 | 110 | X | 80 | 153 | 78 | 194 |

Inv03 | F2 | 104 | 153 | 72 | X | 157 | 51 | 202 |

Inv04a | F1 | 85 | 141 | 168 | 167 | X | 185 | 63 |

Inv04b | F2 | 116 | 144 | 61 | 46 | 164 | X | 209 |

Inv05 | F1 | 132 | 168 | 216 | 209 | 62 | 214 | X |

**Table 3.**Values of ${\chi}_{ref}^{2}$ Equation (6) and weight of each dataset in the computation of the misfit. The top three lines correspond to inversions conducted without the campaign and continous GNSS data, while the bottom three lines include the GNSS data.

GNSS | S1 D | S1 A | CSKD | CSKA | Total | |
---|---|---|---|---|---|---|

${\chi}^{2}{i}_{ref}$ | X | 1100 | 500 | 1000 | 470 | 3070 |

${\chi}^{2}{i}_{ref}/{\chi}_{ref}^{2}$ % | X | 36% | 16% | 33% | 15% | 100% |

after weighting | X | 25% | 25% | 25% | 25% | 100% |

${\chi}^{2}{i}_{ref}$ | 4800 | 1100 | 500 | 1000 | 470 | 7870 |

${\chi}^{2}{i}_{ref}/{\chi}_{ref}^{2}$ % | 61% | 14% | 6% | 13% | 6% | 100% |

after weighting | 20% | 20% | 20% | 20% | 20% | 100% |

**Table 4.**Comparison of the inverted source characteristics for different methods for tracking magma propagation and different data: continuous GNSS data from the permanent network and an intermediate Sentinel-1 interferogram S1 D1 covering the first part of the eruption (before 01:45). % explained data ($\%Ed$, see Equation (9)) computed with data used in the inversions are in bold, while those computed a posteriori using data omitted in the inversion are in small italics.

Method Data | Ellipse | Projected Disk | Subgraph | ||||
---|---|---|---|---|---|---|---|

GNSS | InSAR | GNSS | InSAR | GNSS | InSAR | ||

Overpressure (MPa) | 3.2 | 7.0 | 3.5 | 1.9 | 3.4 | 2.5 | |

Average opening (m) | 1.3 | 1.3 | 1.0 | 0.6 | 1.0 | 0.7 | |

Area (10${}^{6}$ m${}^{2}$) | 2.8 | 1.6 | 3.4 | 3.9 | 3.3 | 3.1 | |

Volume (10${}^{6}$ m${}^{3}$) | 3.6 | 2.1 | 3.5 | 2.4 | 3.3 | 2.2 | |

$\%\mathbf{Ed}$ | GNSS | 96 | 85 | 84 | 83 | 83 | 84 |

InSAR S1 D1 | 22 | 95 | 77 | 96 | 83 | 94 |

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## Share and Cite

**MDPI and ACS Style**

Smittarello, D.; Cayol, V.; Pinel, V.; Froger, J.-L.; Peltier, A.; Dumont, Q.
Combining InSAR and GNSS to Track Magma Transport at Basaltic Volcanoes. *Remote Sens.* **2019**, *11*, 2236.
https://doi.org/10.3390/rs11192236

**AMA Style**

Smittarello D, Cayol V, Pinel V, Froger J-L, Peltier A, Dumont Q.
Combining InSAR and GNSS to Track Magma Transport at Basaltic Volcanoes. *Remote Sensing*. 2019; 11(19):2236.
https://doi.org/10.3390/rs11192236

**Chicago/Turabian Style**

Smittarello, Delphine, Valérie Cayol, Virginie Pinel, Jean-Luc Froger, Aline Peltier, and Quentin Dumont.
2019. "Combining InSAR and GNSS to Track Magma Transport at Basaltic Volcanoes" *Remote Sensing* 11, no. 19: 2236.
https://doi.org/10.3390/rs11192236