# 3D Probabilistic Modelling and Uncertainty Analysis of Glacial and Post-Glacial Deposits of the City of Saguenay, Canada

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{s}value of the top 30 m of soil (V

_{s,}

_{30}), the average V

_{s}of all of the soil deposits (V

_{s,avg}) and the fundamental site period (T

_{0}) or frequency (f

_{0}) [13,14]. In Eastern Canada, Rosset et al. [15] developed three different models for the Montreal region using predictive equations for V

_{s}as a function of depth: a single-layer model based on the total thickness of soft soils, a four-layer model based on geological and geotechnical information from borehole data and a composite model that included the characteristics of the two former models. Nastev et al. [16] in the Ottawa and St. Lawrence Valleys and Foulon et al. [17] in the Saguenay City region assigned a typical Vs depth function for post-glacial sediments and a single V

_{s}value to glacial sediments and bedrock units. These studies used a deterministic 3D geological model for mapping the spatial distribution of V

_{s,avg}and ${T}_{0}$. They analysed the uncertainty propagated to site parameters using the first-order, second-moment approach, and they considered only the statistical uncertainty related to the V

_{s}of soil deposits. This approach ignores the spatial uncertainties related to the 3D geological model. Considering the uncertainties related to the type and thickness of the soil layers certainly helps the development of reliable seismic hazard maps.

## 2. Methodology

## 3. Applied Geostatistical Methods

#### 3.1. Spatial Interpolation

#### 3.2. Spatial Variation

_{α}and the number of data pairs within distance h in the respective direction. In practice, the tolerance for distance h and its direction is specified. The direction of the separation vectors becomes irrelevant when the directional tolerance increases sufficiently. An omnidirectional variogram is a useful starting tool for structural analysis and provides the prerequisite information for calculating the directional variograms, whilst a directional variogram reveals the anisotropy pattern and the direction of the maximum and minimum spatial continuities [20]. Equation (1) is applied for continuous variables, whilst an indicator variogram is calculated for categorical variables by substituting indicator data $i\left({u}_{\alpha};k\right)$ for K indicators as follows:

#### 3.3. Uncertainty of Spatial Interpolation

#### 3.4. Stochastic Simulation

- (i)
- Transformation of soil types to K indicator variables$$i\left({u}_{\alpha};k\right)=\left(\right)open="\{">\begin{array}{c}1ifcategorykprevailsatlocationu,k=1,\dots ,K.\\ 0otherwise\end{array}$$
- (ii)
- Determination of indicator variograms to model the spatial continuity of the indicator soil types;
- (iii)
- Simulation of the soil types honouring field observation at sampled locations (conditional simulation) in a sequential and reproducible manner.

## 4. Saguenay City DatPreparation and Analysis

#### 4.1. Geologic Framework of the Study Area

^{2}, with a population of 147,100. It has a hilly topography and lies in the southern portion of the E–W-trending Saguenay graben. The regional seismic activity of this region was reassessed following the 1988 M6.0 Saguenay earthquake. The epicentre of this intraplate earthquake with a midcrustal depth of 29 km was 35 km south of the downtown area [29]. The earthquake’s secondary effects included liquefaction, rock falls and landslides observed within a distance of 200 km from the epicentre [30].

- Till: This glacial sediment is located at the base of the stratigraphic soil column; it is compact and semiconsolidated. Till is the most widespread soil unit in the study area and ranges in thickness from a few meters to >10 m at certain locations. In the highlands, the till veneer is frequently discontinuous and results in areas of rock outcrops. Most of the till outcrops are assumed to be less than 1 m thick on the geological map [33]. With the exception of rock outcrops, till continuously covers the bedrock elsewhere, representing an important assumption in the 3D modelling approach.
- Gravel: This coarse sediment is mainly of glaciofluvial and alluvial origin; it consists of gravel, sand and sometimes till. This unit is occasional in the region, often in contact with till or sand units.
- Clays: These fine post-glacial sediments are the most present soil type by volume in the study area. They are composed mainly of silt, silty clays and clay. They have a thickness of up to 10 m and may attain a maximum thickness of >100 m in the lowlands.
- Sand: This group consists mainly of coarse glaciomarine deltaic and prodeltaic sediments and alluvial sands composed of sand and gravely sands.
- Other sediments: This extremely heterogeneous category comprises all the remaining sediments; it mainly includes loose post-glacial sediments consisting of alluvium, floodplain sediments, organic sediments and occasional landslide colluvium that can be classified into sand, clay and gravel on the basis of grain size distribution.

#### 4.2. Input Data and Analysis

- Borehole logs: The database contains 3524 borehole logs distributed over the study area [34]. A total of 2402 boreholes are sufficiently deep to reach the bedrock. The remaining 1122 boreholes that do not reach the bedrock indicate that the bedrock is deeper than the borehole depth, and a groundwater-bearing layer is possibly encountered in the coarse soil deposits.
- Virtual logs: A total of 26 geological cross-sections distributed over the region were developed on the basis of expert opinion in previous geological studies [34]. These cross-sections include 973 virtual logs distributed in a regular spatial pattern at a distance of ~500 m to improve the data coverage mainly in the lowland areas (Figure 6).
- Rock outcrops: During the geographic information system processing of the surficial geology map, additional 1033 data points were introduced to indicate rock outcrops. Located within the bedrock polygons, they improve the realistic spatial variability of the sediment thickness.
- Till veneer: Till sediments cover most of the study area. Till outcropping areas, with a thickness equal to or less than 1.0 m, are located in the highlands and are referred to as a till veneer. In these areas, the till thickness is fixed to 1 m, and the till outcrop polygons are modelled with a mesh of 75 m, generating an additional 42,649 points with a known thickness.

**Figure 6.**Complete set of the available observation points, including borehole logs, rock outcrops and shallow till data.

#### 4.3. Modelling Spatial Variation: Variogram Analysis

## 5. Results

#### 5.1. Construction of the Total Soil Thickness Map (Depth to Bedrock)

#### 5.1.1. Spatial Interpolation

#### 5.1.2. Validation

#### 5.2. Determination of the Till Thickness Map

#### 5.3. 3D Modelling of Discontinuous Soil Layers

#### 5.4. Thickness Maps of Discontinuous Soil Layers

_{0}), the geometry (soil thickness) and shear wave velocity (${V}_{{s}_{i}}$) of each soil layer are important variables. Therefore, the 3D model must be transformed into a set of 2D thickness maps to obtain the thickness of the individual discontinuous soil units. Thus, the thickness mean and variance of each block are computed on the basis of the discrete probability distribution of the random categorical variable (${X}_{i}$) with an event probability ${p}_{i}$ as follows:

## 6. Conclusions

_{0}), which are important factors in seismic hazard assessment.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${\tilde{C}}_{00}$ | Variance of point values |

${\tilde{C}}_{ij}$ | Covariance between measured samples |

${\tilde{C}}_{i0}$ | Covariance between measured and unknown values |

EBK | Empirical Bayesian Kriging |

f_{o} | Fundamental site frequency of vibration |

$i\left({u}_{\alpha};k\right)$ | Binary indicator value at location ${u}_{\alpha}$and for category k |

$\mathrm{ME}$ | Mean error |

$\mathrm{MSE}$ | Mean standardised error |

$\mathrm{MSSE}$ | Mean square standardised error |

$\mathrm{RMSE}$ | Root mean square error |

SIS | Sequential indicator simulation |

TIN | Triangulated irregular network |

T_{o} | Fundamental site period of vibration |

u | Coordinates vector |

${V}_{s}$ | Shear wave velocity |

${V}_{s,30}$ | Average shear wave velocity of the top 30 m |

${V}_{S,avg}$ | Average shear wave velocity of the entire soil deposit |

${w}_{i}{w}_{j}$ | Kriging weights |

Z(${u}_{\alpha}$) | Random variable at location ${u}_{\alpha}$ |

$\widehat{\gamma}\left(h\right)$ | Experimental variogram |

${\widehat{\gamma}}_{I}\left(h;k\right)$ | Indicator variogram for category k |

${\tilde{\sigma}}_{k}^{2}$ | Error variance of kriging |

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**Figure 3.**Phase II: workflow of the spatial interpolation of total soil thickness and thickness of the continuous basal layer.

**Figure 4.**Phase III: methodology for determining soil thickness map(s) and associated uncertainties of discontinuous soil layers using geostatistical simulation.

**Figure 7.**Thickness distributions of soil deposits as observed in borehole logs: (

**a**) total soil thickness, depth to rock; (

**b**) total soil thickness, including rock outcrops; (

**c**) till sediments; (

**d**) till sediments following replacement of outliers. The black line represents the normal distribution curve.

**Figure 8.**Example of variogram modelling: a nugget and two spherical nested structures are fitted on an experimental sample variogram. R

_{1}and R

_{2}refer to the ranges of the two nested models.

**Figure 9.**Total soil thickness maps obtained with: (

**a**) EBK, (

**b**) TIN; and (

**c**) the map of the kriging standard deviation (${\tilde{\sigma}}_{k}^{}$) EBK. The areas with till or rock outcrops are excluded and indicated with white background.

**Figure 10.**Thickness error distributions for the test set of 1122 boreholes not reaching the bedrock estimated by (

**a**) EBK and (

**b**) TIN.

**Figure 12.**(

**a**) Plan and (

**b**) cross-section of one SIS realisation of sand, clay and gravel. The thickness of the till unit shown in the cross-section is determined in Section 5.2.

**Figure 13.**Stratigraphic cross-sections: (

**a**) deterministic based on expert opinion (modified from CERM-PACES [34]); (

**b**) soil units with the highest probability of occurrence based on conditional SIS; individual probabilities of occurrence for (

**c**) clay, (

**d**) sand and (

**e**) gravel obtained from a set of 100 conditional SIS; and (

**f**) total standard deviation (σ

_{h}) of the thickness computed by using the probability of categorical distribution.

**Figure 14.**Spatial distribution of the weighted thickness and associated spatial standard deviation (σ

_{h}) for (

**a**,

**b**) clay, (

**c**,

**d**) sand, (

**e**,

**f**) gravel and (

**g**,

**h**) total post-glacial deposits.

Geological Unit | Real Borehole Data (%) | Virtual Logs (%) |
---|---|---|

Clay | 53.60% | 58.54% |

Gravel | 6.80% | 2.06% |

Sand | 35.66% | 18.37% |

Till | 3.94% | 21.03% |

Variables | Number of Structures | Model Properties Structure 1 | Model Properties Structure 2 | ||||
---|---|---|---|---|---|---|---|

Model Type | Anisotropy Axis (a _{max}, a_{med}, a_{min}) | Model Parameters | Model Type | Anisotropy Axis (a _{max}, a_{med}, a_{min}) | Model Parameters | ||

Clay | 2 | Sp. | (135°,45°,90°) | Nugget: 0.01 R _{1}: (375,212.5,75)Sill _{1} *: 0.18 | Ex. | (135°,45°,90°) | R_{2}: (12825,4275,75)Sill _{2} *: 0.05 |

Sand | 2 | Sp. | (135°,45°,90°) | Nugget: 0.02 R _{1}: (412.5187.5,62.5)Sill _{1} *: 0.17 | Sp. | (0°,0°,90°) | R_{2}: (12375,12375,62.5)Sill _{2} *: 0.03 |

Gravel | 2 | Sp. | - | Nugget: 0.01 R _{1}: (150,150,150)Sill _{1} *: 0.026 | Ga. | (0°,0°,90°) | R_{2}: (4600,4600,150)Sill _{2} *: 0.015 |

_{max}, a

_{med}and a

_{min}refer to the azimuths of the three principal axes of the anisotropy.

ME (m) | RMSE (m) | MSE | MSSE |
---|---|---|---|

0.05 | 8.94 | 0.01 | 0.94 |

Thickness Error | TIN | EBK |
---|---|---|

Mean (m) | 12.2 | 11.8 |

Sum (m) | 3889.8 | 3682.6 |

Error count (boreholes) | 318 | 313 |

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**MDPI and ACS Style**

Salsabili, M.; Saeidi, A.; Rouleau, A.; Nastev, M.
3D Probabilistic Modelling and Uncertainty Analysis of Glacial and Post-Glacial Deposits of the City of Saguenay, Canada. *Geosciences* **2021**, *11*, 204.
https://doi.org/10.3390/geosciences11050204

**AMA Style**

Salsabili M, Saeidi A, Rouleau A, Nastev M.
3D Probabilistic Modelling and Uncertainty Analysis of Glacial and Post-Glacial Deposits of the City of Saguenay, Canada. *Geosciences*. 2021; 11(5):204.
https://doi.org/10.3390/geosciences11050204

**Chicago/Turabian Style**

Salsabili, Mohammad, Ali Saeidi, Alain Rouleau, and Miroslav Nastev.
2021. "3D Probabilistic Modelling and Uncertainty Analysis of Glacial and Post-Glacial Deposits of the City of Saguenay, Canada" *Geosciences* 11, no. 5: 204.
https://doi.org/10.3390/geosciences11050204