# Subsoil Reconstruction in Geostatistics beyond Kriging: A Case Study in Veneto (NE Italy)

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

^{−1}–10

^{−2}m/s).

^{3}/s). This regional context affects our area of study, contributing to its heterogeneity.

#### Markovian Categorical Prediction (MCP)

**h**is a multidimensional lag from any observed location s. The transition probability ${t}_{ij}\left(h\right)$ is the element in the i-th row and in the j-th column of the matrix T(h) such that:

## 3. Data Analysis

## 4. Transiogram Analysis

## 5. Simulation Results

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Pyrcz, M.J.; Deutsch, C.V. Geostatistical Reservoir Modeling, 2nd ed.; Oxford University Press: Oxford, UK, 2014; p. 448. ISBN 978-0199731442. [Google Scholar]
- Koltermann, C.E.; Gorelick, S.M. Heterogeneity in sedimentary deposits: A review of structure-imitating, process-imitating, and descriptive approaches. Water Resour. Res.
**1996**, 32, 2617–2658. [Google Scholar] [CrossRef] - De Marsily, G.; Delay, F.; Teles, V.; Schafmeister, M.T. Some current methods to represent the heterogeneity of natural media in hydrogeology. Hydrogeol. J.
**1998**, 6, 115–130. [Google Scholar] [CrossRef] - De Marsily, G.; Delay, F.; Gonçalvès, J.; Renard, P.; Teles, V.; Violette, S. Dealing with spatial heterogeneity. Hydrogeol. J.
**2005**, 13, 161–183. [Google Scholar] [CrossRef] - Al-Khalifa, M.A.; Payenberg, T.H.D.; Lang, S. Overcoming The Challenges of Building 3D Stochastic Reservoir Models Using Conceptual Geological Models: A Case Study. In Proceedings of the PE Middle East Oil and Gas Show and Conference, Manama, Bahrain, 11–14 March 2007; pp. 1–12. [Google Scholar]
- Comunian, A.; Renard, P.; Straubhaar, J.; Bayer, P. Three-dimensional high resolution fluvio-glacial aquifer analog—Part 2: Geostatistical modeling. J. Hydrol.
**2011**, 405, 10–23. [Google Scholar] [CrossRef] [Green Version] - Dell’Arciprete, D.; Bersezio, R.; Felletti, F.; Giudici, M.; Comunian, A.; Renard, P. Comparison of three geostatistical methods for hydrofacies simulation: A test on alluvial sediments. Hydrogeol. J.
**2012**, 20, 299–311. [Google Scholar] [CrossRef] - Marini, M.; Felletti, F.; Beretta, G.P.; Terrenghi, J. Three Geostatistical Methods for Hydrofacies Simulation Ranked Using a Large Borehole Lithology Dataset from the Venice Hinterland (NE Italy). Water
**2018**, 10, 844. [Google Scholar] [CrossRef] [Green Version] - Haldorsen, H.H.; Chang, D.M. Notes on stochastic shales; from outcrop to simulation model. In Reservoir Characterization; Elsevier: Amsterdam, The Netherlands, 1986; pp. 445–485. [Google Scholar]
- Viseur, S. Stochastic Boolean Simulation of Fluvial Deposits: A New Approach Combining Accuracy with Efficiency. In Proceedings of the SPE Annual Technical Conference and Exhibition, Houston, TX, USA, 3–6 October 1999. [Google Scholar]
- Vargas-Guzmán, J.A.; Al-Qassab, H. Spatial conditional simulation of facies objects for modeling complex clastic reservoirs. J. Pet. Sci. Eng.
**2006**, 54, 1–9. [Google Scholar] [CrossRef] - Matheron, G.; Beucher, H.; De Fouquet, C.; Galli, A.; Guerillot, D.; Ravenne, C. Conditional Simulation of the Geometry of Fluvio-Deltaic Reservoirs. In Proceedings of the SPE Annual Technical Conference and Exhibition, Dallas, TX, USA, 27–30 September 1987. [Google Scholar]
- Armstrong, M. Plurigaussian Simulations in Geosciences; Springer: Berlin/Heidelberg, Germany, 2011; ISBN 3642196071. [Google Scholar]
- Journel, A.G. Nonparametric estimation of spatial distributions. J. Int. Assoc. Math. Geol.
**1983**, 15, 445–468. [Google Scholar] [CrossRef] - Trevisani, S.; Fabbri, P. Geostatistical modeling of a heterogeneous site bordering the Venice lagoon, Italy. Ground. Water
**2010**, 48, 614–623. [Google Scholar] [CrossRef] - Dalla Libera, N.; Fabbri, P.; Mason, L.; Piccinini, L.; Pola, M. A local natural background level concept to improve the natural background level: A case study on the drainage basin of the Venetian Lagoon in Northeastern Italy. Environ. Earth Sci.
**2018**, 77, 487. [Google Scholar] [CrossRef] - Fabbri, P. Probabilistic Assessment of Temperature in the Euganean Geothermal Area (Veneto Region, NE Italy). Math. Geol.
**2001**, 33, 745–760. [Google Scholar] [CrossRef] - Schwarzacher, W. The use of Markov chains in the study of sedimentary cycles. J. Int. Assoc. Math. Geol.
**1969**, 1, 17–39. [Google Scholar] [CrossRef] - Luo, J. Transition Probability Approach to Statistical Analysis of Spatial Qualitative Variables in Geology; Springer: Boston, MA, USA, 1996; pp. 281–299. [Google Scholar]
- Carle, S.F.; Fogg, G.E. Transition probability-based indicator geostatistics. Math. Geol.
**1996**, 28, 453–476. [Google Scholar] [CrossRef] - Carle, S.F.; Fogg, G.E. Modeling Spatial Variability with One and Multidimensional Continuous-Lag Markov Chains. Math. Geol.
**1997**, 29, 891–918. [Google Scholar] [CrossRef] - Lee, S.Y.; Carle, S.F.; Fogg, G.E. Geologic heterogeneity and a comparison of two geostatistical models: Sequential Gaussian and transition probability-based geostatistical simulation. Adv. Water Resour.
**2007**, 30, 1914–1932. [Google Scholar] [CrossRef] - Weissmann, G.S.; Carle, S.F.; Fogg, G.E. Three-dimensional hydrofacies modeling based on soil surveys and transition probability geostatistics. Water Resour. Res.
**1999**, 35, 1761–1770. [Google Scholar] [CrossRef] [Green Version] - Weissmann, G.S.; Fogg, G.E. Multi-scale alluvial fan heterogeneity modeled with transition probability geostatistics in a sequence stratigraphic framework. J. Hydrol.
**1999**, 226, 48–65. [Google Scholar] [CrossRef] - Miall, A.D. Markov chain analysis applied to an ancient alluvial plain succession. Sedimentology
**1973**, 20, 347–364. [Google Scholar] [CrossRef] - Hattori, I. Entropy in Markov chains and discrimination of cyclic patterns in lithologic successions. J. Int. Assoc. Math. Geol.
**1976**, 8, 477–497. [Google Scholar] [CrossRef] - Jef Caers, T.Z. Multiple-point Geostatistics: A Quantitative Vehicle for Integrating Geologic Analogs into Multiple Reservoir Models. AAPG Memoir
**2005**, 383–394, ISSN: 02718529. [Google Scholar] - Chugunova, T.L.; Hu, L.Y. Multiple-Point simulations constrained by continuous auxiliary data. Math. Geosci.
**2008**, 40, 133–146. [Google Scholar] [CrossRef] - Comunian, A.; Renard, P.; Straubhaar, J. 3D multiple-point statistics simulation using 2D training images. Comput. Geosci.
**2012**, 40, 49–65. [Google Scholar] [CrossRef] [Green Version] - Mariethoz, G.; Renard, P. Reconstruction of Incomplete Data Sets or Images Using Direct Sampling. Math. Geosci.
**2010**, 42, 245–268. [Google Scholar] [CrossRef] [Green Version] - Strebelle, S. Conditional simulation of complex geological structures using multiple-point statistics. Math. Geol.
**2002**, 34, 1–21. [Google Scholar] [CrossRef] - Emery, X.; Lantuéjoul, C. Can a Training Image Be a Substitute for a Random Field Model? Math. Geosci.
**2014**, 46, 133–147. [Google Scholar] [CrossRef] - Breslow, N.E.; Clayton, D.G. Approximate Inference in Generalized Linear Mixed Models. J. Am. Stat. Assoc.
**1993**, 88, 9–25. [Google Scholar] - Cao, G.; Kyriakidis, P.C.; Goodchild, M.F. A multinomial logistic mixed model for the prediction of categorical spatial data. Int. J. Geogr. Inf. Sci.
**2011**, 25, 2071–2086. [Google Scholar] [CrossRef] - Csiszar, I. Why Least Squares and Maximum Entropy? An Axiomatic Approach to Inference for Linear Inverse Problems. Ann. Stat.
**1991**, 19, 2032–2066. [Google Scholar] [CrossRef] - Christakos, G. A Bayesian/maximum-entropy view to the spatial estimation problem. Math. Geol.
**1990**, 22, 763–777. [Google Scholar] [CrossRef] - Bogaert, P. Spatial prediction of categorical variables: The Bayesian maximum entropy approach. Stoch. Environ. Res. Risk Assess.
**2002**, 16, 425–448. [Google Scholar] [CrossRef] - Bogaert, P.; D’Or, D. Estimating Soil Properties from Thematic Soil Maps. Soil Sci. Soc. Am. J.
**2002**, 66, 1492–1500. [Google Scholar] [CrossRef] - D’Or, D.; Bogaert, P.; Christakos, G. Application of the BME approach to soil texture mapping. Stoch. Environ. Res. Risk Assess.
**2001**, 15, 87–100. [Google Scholar] [CrossRef] - D’Or, D.; Bogaert, P. Spatial prediction of categorical variables with the Bayesian Maximum Entropy approach: The Ooypolder case study. Eur. J. Soil Sci.
**2004**, 55, 763–775. [Google Scholar] [CrossRef] - Allard, D.; D’Or, D.; Froidevaux, R. An efficient maximum entropy approach for categorical variable prediction. Eur. J. Soil Sci.
**2011**, 62, 381–393. [Google Scholar] [CrossRef] - Huang, X.; Wang, Z.; Guo, J. Prediction of categorical spatial data via Bayesian updating. Int. J. Geogr. Inf. Sci.
**2016**, 30, 1426–1449. [Google Scholar] [CrossRef] - Allard, D.; Comunian, A.; Renard, P. Probability Aggregation Methods in Geoscience. Math. Geosci.
**2012**, 44, 545–581. [Google Scholar] [CrossRef] - Sartore, L.; Fabbri, P.; Gaetan, C. spMC: An R-package for 3D lithological reconstructions based on spatial Markov chains. Comput. Geosci.
**2016**, 94, 40–47. [Google Scholar] [CrossRef] [Green Version] - R Development Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2019; Volume 2. [Google Scholar]
- De Supinski, B.; Klemm, M. OpenMP Technical Report 7: Version 5.0 Public Comment Draft EDITORS; Austin, TX 78746, USA. 2018. Available online: https://www.openmp.org/wp-content/uploads/openmp-TR7.pdf (accessed on 5 March 2020).
- Fontana, A.; Mozzi, P.; Bondesan, A. Alluvial megafans in the Venetian-Friulian Plain (north-eastern Italy): Evidence of sedimentary and erosive phases during Late Pleistocene and Holocene. Quat. Int.
**2008**, 189, 71–90. [Google Scholar] [CrossRef] - Carraro, A.; Fabbri, P.; Giaretta, A.; Peruzzoa, L.; Tateo, F.; Tellini, F. Effects of redox conditions on the control of arsenic mobility in shallow alluvial aquifers on the Venetian Plain (italy). Sci. Total Environ.
**2015**, 532, 581–594. [Google Scholar] [CrossRef] - Carraro, A.; Fabbri, P.; Giaretta, A.; Peruzzo, L.; Tateo, F.; Tellini, F. Arsenic anomalies in shallow Venetian Plain (Northeast Italy) groundwater. Environ. Earth Sci.
**2013**, 70, 3067–3084. [Google Scholar] [CrossRef] - Fabbri, P.; Gaetan, C.; Zangheri, P. Transfer function-noise modelling of an aquifer system in NE Italy. Hydrol. Process.
**2011**, 25, 194–206. [Google Scholar] [CrossRef] - Fabbri, P.; Piccinini, L. Assessing transmissivity from specific capacity in an alluvial aquifer in the middle Venetian plain (NE Italy). Water Sci. Technol.
**2013**, 67, 2000–2008. [Google Scholar] [CrossRef] [PubMed] - Vorlicek, P.A.; Antonelli, R.; Fabbri, P.; Rausch, R. Quantitative hydrogeological studies of the Treviso alluvial plain, NE Italy. Q. J. Eng. Geol. Hydrogeol.
**2004**, 37, 23–29. [Google Scholar] [CrossRef] - Fabbri, P.; Piccinini, L.; Marcolongo, E.; Pola, M.; Conchetto, E.; Zangheri, P. Does a change of irrigation technique impact on groundwater resources? A case study in Northeastern Italy. Environ. Sci. Policy
**2016**, 63, 63–75. [Google Scholar] [CrossRef] - Fabbri, P.; Ortombina, M.; Piccinini, L. Estimation of Hydraulic Conductivity Using the Slug Test Method in a Shallow Aquifer in the Venetian Plain (NE, Italy). AQUA Mundi
**2012**, 3, 125–133. [Google Scholar] - Journel, A.G.; Deutsch, C.V. Entropy and spatial disorder. Math. Geol.
**1993**, 25, 329–355. [Google Scholar] [CrossRef] - Ahrens, J.; Geveci, B.; Law, C. ParaView: An End-User Tool for Large Data Visualization. In The Visualization Handbook; Academic Press: Cambridge, MA, USA, 2005; pp. 717–731. [Google Scholar] [CrossRef]

**Figure 1.**Geographical location of studied area and cross section of Venetian Plain (Fabbri et al., 2011, modified). Map coordinates are shown in metric units, Monte Mario/Italy Zone 1.

**Figure 2.**(

**a**) Location of 13 piezometers (red dots); simulated area (red rectangle); traces of cross sections (yellow lines) (image from Google Earth); (

**b**) stratigraphy of Ps3 with example of lithologic reduction to 5 materials; (

**c**) core box of Ps3 piezometer.

**Figure 3.**Normalized frequency distribution of lithologies (

**a**); boxplot thickness for every lithology along direction Z (depth) (

**b**); extension of the lithologies along the directions X (

**c**) and Y (

**d**).

**Figure 4.**Experimental transition probabilities with respect to the distance (h) (black circles) and theoretical transition probabilities modeled by Markov chains (red lines) along the directions Z, X and Y).

**Figure 6.**Markovian-type Categorical Prediction (MCP) 3D simulation on 91 layers every 5 cm in depth from 53 to 48.5 m a.s.l.; cross sections A, B, C of MCP 3D simulation and simplified stratigraphies of the 13 boreholes.

**Table 1.**Mean of thickness along Z, and its extension along the directions X and Y for the considered lithologies.

Lithology | Thickness Z (m) | Extension X (m) | Extension Y (m) |
---|---|---|---|

Clay (A) | 0.27 | 21.15 | 20.67 |

Gravel (G) | 0.70 | 32.99 | 37.74 |

Silt (L) | 0.35 | 26.55 | 33.13 |

Silty Sand (LS) | 0.14 | 23.73 | 31.83 |

Sand (S) | 0.38 | 31.31 | 37.33 |

**Table 2.**Embedded Transition Probabilities (ETP) along the directions Z (depth), X (longitude) and Y (latitude).

Z | A | G | L | LS | S | X | A | G | L | LS | S | Y | A | G | L | LS | S |

A | - | 0.29 | 0.12 | 0.18 | 0.41 | A | - | 0.64 | 0.14 | 0.05 | 0.18 | A | - | 0.66 | 0.11 | 0.00 | 0.23 |

G | 0.14 | - | 0.36 | 0.11 | 0.39 | G | 0.24 | - | 0.31 | 0.10 | 0.36 | G | 0.08 | - | 0.69 | 0.08 | 0.15 |

L | 0.14 | 0.37 | - | 0.20 | 0.26 | L | 0.20 | 0.40 | - | 0.10 | 0.30 | L | 0.08 | 0.70 | - | 0.00 | 0.23 |

LS | 0.07 | 0.14 | 0.17 | - | 0.62 | LS | 0.11 | 0.47 | 0.09 | - | 0.34 | LS | 0.06 | 0.47 | 0.26 | - | 0.21 |

S | 0.20 | 0.18 | 0.23 | 0.36 | - | S | 0.04 | 0.45 | 0.41 | 0.10 | - | S | 0.15 | 0.42 | 0.40 | 0.03 | - |

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**MDPI and ACS Style**

Fabbri, P.; Gaetan, C.; Sartore, L.; Dalla Libera, N.
Subsoil Reconstruction in Geostatistics beyond Kriging: A Case Study in Veneto (NE Italy). *Hydrology* **2020**, *7*, 15.
https://doi.org/10.3390/hydrology7010015

**AMA Style**

Fabbri P, Gaetan C, Sartore L, Dalla Libera N.
Subsoil Reconstruction in Geostatistics beyond Kriging: A Case Study in Veneto (NE Italy). *Hydrology*. 2020; 7(1):15.
https://doi.org/10.3390/hydrology7010015

**Chicago/Turabian Style**

Fabbri, Paolo, Carlo Gaetan, Luca Sartore, and Nico Dalla Libera.
2020. "Subsoil Reconstruction in Geostatistics beyond Kriging: A Case Study in Veneto (NE Italy)" *Hydrology* 7, no. 1: 15.
https://doi.org/10.3390/hydrology7010015