# Seismic Constrained Gravity Inversion: A Reliable Tool to Improve Geophysical Models Away from Seismic Information

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- a local (smaller than $1\xb7{10}^{5}$ m × $1\xb7{10}^{5}$ m) area to be investigated;
- a good geological knowledge of the area available from previous studies;
- several seismic profiles or even 3D seismic available, properly constraining the shallowest geological structures and units;
- well logs to constrain density distribution in the shallowest layers;
- Moho and basement depths known with poor spatial resolution and accuracy.

- a large (greater than $1\xb7{10}^{5}$ m × $1\xb7{10}^{5}$ m) area to be investigated;
- a general geological knowledge of the area available from literature;
- few seismic profiles available, giving sparse information on the shallowest geological structures and units;
- few, or even no, constraints for the density distribution;
- Moho and basement depths known only from global models (such as the CRUST1.0 model [5]) but with poor spatial resolution and accuracy.

## 2. Methods

## 3. The Synthetic Case Study

- 1.
- we compute a thickness of each geological unit;
- 2.
- we estimate an empirical covariance of the thickness of each layer and interpolate it by means of Gaussian theoretical covariance function;
- 3.
- for each layer, we compute a new random thickness with the same stochastic characteristics of the original dataset by classical triangular decomposition of the covariance matrix [13];
- 4.
- we apply the same procedure to the first layer, namely the bathymetry;
- 5.
- finally, starting from the new bathymetry and thicknesses, we build a set of simulated seismic profiles with the same stochastic characteristics as the initial one.

## 4. Results

#### 4.1. Test 1: The Two-Layer Problem (Moho Estimate)

#### 4.2. Scenario 1: Advanced Modelling of Deepest Layers (Basement and Moho Estimate)

#### 4.3. Scenario 2: Exploration in Frontier Areas (Shallowest Layers Fixed in Few Points)

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Simulation of Non-Stationary Random Fields

- 1
- we compute the empirical variogram, using the available interpreted seismic profiles data as observations;
- 2
- we fit the empirical variogram by means of a proper theoretic variogram function;
- 3
- knowing the theoretical variogram, we can estimate a map of the depth of the considered geological unit and its predicted accuracy by means of a kriging solution;
- 4
- we randomly select on the obtained map a set of points;
- 5
- we extract a random value on each of the points obtained at step 4 from a Gaussian probability function with a STD equal to the one predicted by the kriging map on that specific location;
- 6
- we spatially correlate the extracted sample by applying a kriging procedure on the random points with the same variogram function estimated at step 2.

**Figure A1.**Available seismic profiles observing the bathymetry over the considered area (

**a**). Empirical and theoretical variograms (

**b**). Estimated bathymetry from the seismic profiles (

**c**) and predicted accuracy in terms of STD (

**d**). Random realisation of possible deformation (

**e**). Possible realisation of bathymetry coherent with seismic data (

**f**).

## References

- Reguzzoni, M.; Sampietro, D. GEMMA: An Earth crustal model based on GOCE satellite data. Int. J. Appl. Earth Obs. Geoinf.
**2015**, 35, 31–43. [Google Scholar] [CrossRef] - Sampietro, D.; Capponi, M. Gravity for lIthosphere Architecture Determination and Analysis: The Central Eastern Mediterranean Case Study. Geophys. Prospect.
**2021**. [Google Scholar] [CrossRef] - Ebbing, J.; Braitenberg, C.; Götze, H.J. Forward and inverse modelling of gravity revealing insight into crustal structures of the Eastern Alps. Tectonophysics
**2001**, 337, 191–208. [Google Scholar] [CrossRef] - Zingerle, P.; Pail, R.; Gruber, T.; Oikonomidou, X. The combined global gravity field model XGM2019e. J. Geod.
**2020**, 94, 66. [Google Scholar] [CrossRef] - Laske, G.; Masters, G.; Ma, Z.; Pasyanos, M. Update on CRUST1. 0—A 1-degree global model of Earth’s crust. Geophys. Res. Abstr.
**2013**, 15, 20132658abstrEGU. [Google Scholar] - Sampietro, D.; Sanso, F. Uniqueness theorems for inverse gravimetric problems. In VII Hotine-Marussi Symposium on Mathematical Geodesy; Springer: Berlin/Heidelberg, Germany, 2012; pp. 111–115. [Google Scholar]
- Marchetti, P.; Sampietro, D.; Capponi, M.; Rossi, L.; Reguzzoni, M.; Porzio, F.; Sansò, F. Lithological constrained gravity inversion. A Bayesian approach. In Proceedings of the 81st EAGE Conference and Exhibition 2019, London, UK, 3–6 June 2019; pp. 1–5. [Google Scholar]
- Sansó, F.; Sampietro, D. Analysis of the Gravity Field: Direct and Inverse Problems; Birkhäuser: Basel, Switzerland, 2021. [Google Scholar]
- Robert, C.; Casella, G. Monte Carlo Statistical Methods; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Geman, S.; Geman, D. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell.
**1984**, PAMI-6, 721–741. [Google Scholar] [CrossRef] [PubMed] - Wackernagel, H. Multivariate Geostatistics: An Introduction with Applications; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2003. [Google Scholar]
- Sampietro, D.; Capponi, M.; Mansi, A.; Gatti, A.; Marchetti, P.; Sansò, F. Space-Wise approach for airborne gravity data modelling. J. Geod.
**2017**, 91, 535–545. [Google Scholar] [CrossRef][Green Version] - Davis, M.W. Production of conditional simulations via the LU triangular decomposition of the covariance matrix. Math. Geol.
**1987**, 19, 91–98. [Google Scholar] - Nagy, D. The gravitational attraction of a right rectangular prism. Geophysics
**1966**, 31, 362–371. [Google Scholar] [CrossRef] - Sampietro, D.; Capponi, M. Practical tips for 3D regional gravity inversion. Geosciences
**2019**, 9, 351. [Google Scholar] [CrossRef][Green Version] - Oldenburg, D.W. The inversion and interpretation of gravity anomalies. Geophysics
**1974**, 39, 526–536. [Google Scholar] [CrossRef] - Ebadi, S.; Barzaghi, R.; Safari, A.; Bahroudi, A. Evaluation of different gravimetric methods to Moho recovery in Iran. Ann. Geophys.
**2019**, 62, GD561. [Google Scholar] [CrossRef] - Reguzzoni, M.; Sampietro, D.; Rossi, L. The gravimetric contribution to the Moho estimation in the presence of vertical density variations. Rend. Lincei Sci. Fis. E Nat.
**2020**, 31, 69–81. [Google Scholar] [CrossRef]

**Figure 1.**Depth to basement and Moho depth of the synthetic model (upper images). Example of two profiles (AA’ and BB’) on the synthetic 3D model. Solid black lines represent discontinuity between different geological units, colours represent density in kg/m${}^{3}$. W stands for oceanic water, P Plio-Quaternary sediments, M Messinian salt, T Tortonian sediments, O Oligocene sediments, C Cretaceous sediments, J Jurassic sediemnts, PJ Pre-Jurassic sediments.

**Figure 2.**Gravitational effect of the synthetic model (

**left**), and simulated observation error (

**right**).

**Figure 3.**Example of two profiles (AA’ and BB’ in Figure 1) for the two-layer problem inversion. Solid black lines represent discontinuity between different geological units, dashed lines represent uncertainty, colours represent density in kg/m${}^{3}$.

**Figure 4.**Two-layer problem inversion: Residuals between the gravitational effect of the true and a priori model (

**left**) and of the true and the a posteriori model (

**right**).

**Figure 5.**Two-layer problem inversion: Moho estimated by the Bayesian inversion (

**left**) and from classical Parker–Oldenburg inversion (

**right**).

**Figure 6.**Example of two profiles (AA’ and BB’ in Figure 1) for Scenario 1 inversion:. Solid black lines represent discontinuity between different geological units, dashed lines represent uncertainty, colours represent density in kg/m${}^{3}$.

**Figure 7.**Scenario 1 inversion: Difference between the gravitational effect of the true and a priori model (

**left**) and of the true and the a posteriori model (

**right**).

**Figure 9.**Differences between true and a priori model of Scenario 2 inversion for the base of the Messinian layer (

**left**) and for the Moho (

**right**). Black lines represent the simulated seismic profiles.

**Figure 10.**Example of two profiles (AA’ and BB’ in Figure 1) for Scenario 2 inversion. Solid black lines represent discontinuity between different geological units, dashed lines represent uncertainty, colours represent density in kg/m${}^{3}$.

**Figure 11.**Scenario 2 inversion: Difference between the gravitational effect of the true and a priori model (

**left**) and of the true and the a posteriori model (

**right**).

**Figure 12.**Example of two profiles (AA’ and BB’ in Figure 1) of the estimated model for Scenario 2 inversion. Cyan lines represent the geological horizons of the true model, black lines represent the horizons estimated by the inversion, dashed black lines represent the horizons of the a priori model and colours represent density in kg/m${}^{3}$.

**Figure 14.**Density distribution at a depth of 5000 m from the true (

**a**), a priori (

**b**) and a posteriori models (

**c**).

Layer | Average Depth [m] | Depth STD [m] | Correlation Length [m] |
---|---|---|---|

Bathymetry | 2010 | 730 | 24,000 |

Base Plio-Quaternary | 2220 | 710 | 23,000 |

Base Mess. Salt | 3550 | 970 | 26,000 |

Base Tortonian | 4210 | 1460 | 29,000 |

Base Oligocene | 4990 | 1550 | 26,000 |

Base Creataceous | 5800 | 1480 | 28,000 |

Base Jurassic | 6920 | 1680 | 26,000 |

Basement | 11,610 | 2960 | 46,000 |

Moho | 28,360 | 1730 | 31,000 |

Layer | Average Density [kg/m${}^{3}$] | Density Gradient [kg/m${}^{4}$] |
---|---|---|

Water | 1030 | 0 |

Plio-Quaternary | 2220 | 0 |

Mess. Salt | 2160 | 0 |

Tortonian | 2260 | 0 |

Oligocene | 2400 | 0 |

Creataceous | 2480 | 0 |

Jurassic | 2550 | 0 |

Pre-Jurassic | 2620 | 0.005 |

Continental Crust | 2670 | 0.012 |

Mantle | 3300 | 0 |

**Table 3.**Density values and accuracy of the a priori models for Scenario 1 and 2 inversions (Section 4.2 and Section 4.3).

Layer | Average Density [kg/m${}^{3}$] | Density STD [kg/m${}^{3}$] |
---|---|---|

Water | 1030 | 0 |

Plio-Quaternary | 2206 | 10 |

Mess. Salt | 2172 | 10 |

Tortonian | 2251 | 10 |

Oligocene | 2411 | 10 |

Creataceous | 2473 | 10 |

Jurassic | 2549 | 10 |

Pre-Jurassic | 2614 | 22 |

Continental Crust | 2680 | 31 |

Mantle | 3321 | 31 |

**Table 4.**Scenario 2 inversion: Statistics of the differences between true, a priori and a posteriori models.

Layer | A-PRIORI vs. True Model | A-POSTERIORI vs. True Model | ||
---|---|---|---|---|

Mean [m] | STD [m] | Mean [m] | STD [m] | |

Water | 0 | 0 | 0 | 0 |

Base Plio-Quaternary | 0 | 100 | 0 | 85 |

Base Mess. Salt | 150 | 659 | 47 | 378 |

Base Tortonian | 323 | 742 | 94 | 393 |

Base Oligocene | 316 | 883 | 168 | 409 |

Base Cretaceous | 370 | 861 | 309 | 418 |

Base Jurassic | 250 | 930 | 707 | 517 |

Basement | 80 | 1718 | 267 | 818 |

Moho | 361 | 1736 | 837 | 825 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sampietro, D.; Capponi, M.
Seismic Constrained Gravity Inversion: A Reliable Tool to Improve Geophysical Models Away from Seismic Information. *Geosciences* **2021**, *11*, 467.
https://doi.org/10.3390/geosciences11110467

**AMA Style**

Sampietro D, Capponi M.
Seismic Constrained Gravity Inversion: A Reliable Tool to Improve Geophysical Models Away from Seismic Information. *Geosciences*. 2021; 11(11):467.
https://doi.org/10.3390/geosciences11110467

**Chicago/Turabian Style**

Sampietro, Daniele, and Martina Capponi.
2021. "Seismic Constrained Gravity Inversion: A Reliable Tool to Improve Geophysical Models Away from Seismic Information" *Geosciences* 11, no. 11: 467.
https://doi.org/10.3390/geosciences11110467