3.1. Layer Boundaries
Sedimentary sequences are formed by deposition and erosion. If the depositional conditions change, a new layer is formed. We first consider the case of a continuous layer surface in the model area. The layer boundary, described in 2D by a line, separates two layers. The elevation of this line is determined at a few points by drilling or sounding and predicted in between. In the following, the borehole data in
Figure 4 are interpolated, and the uncertainties of the predicted layer boundary are exemplarily quantified using sequential Gaussian simulation (SGSIM).Then, 2D pixels or 3D volume pixels (voxels) are used to analyze and present the uncertainties.
Figure 4 shows the results of deterministic interpolation using the RBF technique with a linear and a cubic function.
Figure 5 compares the predicted layer boundary resulting from kriging and SGSIM. In
Figure 5a, besides the results of kriging interpolation, the boundaries obtained in the individual, along with the mean elevation, are shown. In
Figure 5b, the kriging variance is compared with the variance calculated from 500 simulated layer boundaries. As shown by [
16], the global entropy becomes quasi-stationary after a certain number of simulations. Therefore, the minimum number of simulations must be equal to or larger than the number required to achieve the stationary entropy value. Additionally, as indicated in [
18], a substantially larger number of simulations is required for precise quantile values at a high confidence level. In the geological context, these statistical uncertainties are less significant than the uncertainties resulting from the variography, the adopted simulation grid, and the presence of unidentified subsoil structures.
Figure 5 shows that the kriging variance consistently remains lower than the variance observed in the geostatistical simulations. Kriging, which operates as a linear regression of covariances, characterizes the kriging variance as a representation of unaccounted variance or as an indicator of the smoothing effect (for a comprehensive mathematical explanation, refer to [
4,
14]). The kriging variance is proportional to the distance from the sampled points. However, for a comprehensive assessment of spatial uncertainty, methods such as SGSIM are mandatory, to mitigate the smoothing effect inherent in kriging and reproduce the semivariogram and histogram in each simulation run. Subsequently, these simulations form the basis for quantifying the uncertainty associated with the prediction.
By categorizing pixels based on their relative position with respect to the layer boundary, it becomes possible to compute a corresponding probability.
Figure 5c shows the variability for every pixel. At the mean elevation of the simulations, the variability reaches its highest values. There, the prediction of the occurring layer is very uncertain, while in the vertical direction above or below the mean, the variability decreases with the distance. The variability below and above the mean of the simulation is low, meaning that no erosion gully or ridge is likely to be encountered.
3.2. Lenticular Bedding
Lenticular bedding develops through different sedimentary processes (e.g., spatial and temporal changes in the depositional environment and water flow patterns) that result in lens-like layers within a larger body of soil and rock. The manual modeling of lens-like layers is highly laborious, and the uncertainty of the predictions is relatively large in comparison to the case of continuous layer boundaries. Even more than for the location of a continuous layer surface, the lens’s model depends on the expert’s interpretation. Alternatively, deterministic interpolation methods and geostatistical simulations can also be used to implicitly model lenses and discontinuous soil layers.
RBF requires fewer computational steps than kriging and can be used for large models with large amounts of data. It has been implemented in Leapfrog
TM by ARANTZ
TM and is numerically efficient [
1]. It does not require geostatistical analysis or specification of the range of correlation but is thus less flexible and less adaptable to specific geologic conditions. Kriging estimates closely resemble those of RBF interpolations. However, as shown in
Figure 6, the kriging variance in a point of the model only depends on the distance of this point to the nearest borehole data; the shape of the lenses does not influence the kriging variance (
Figure 6c).
Figure 7 shows the results of a geostatistical simulation using the SISIM method.
Figure 7a depicts the results of one simulation,
Figure 7b the mean value of 500 simulations (expected value), and
Figure 7c the variability. The layer boundaries fit the data perfectly along the borehole profiles. At the same time, a random component can be seen in areas where no information is available based on the small-scale heterogeneity. As can be seen, the relative probability of occurrence or variability of the boundaries can be more realistically assessed using the geostatistical simulation than using the kriging variance; along the expected layer boundaries in
Figure 7b, the uncertainty is the highest (high variability).
By examining
Figure 6a,b, one can realize that interpolation methods generate smoother lens boundaries, while rough boundaries are predicted by the geostatistical simulation (
Figure 7b). Though smoothness in the geostatistical simulation can be somewhat improved by increasing the number of simulations, in our opinion, the ultimate aim of the simulation is not to achieve a flawlessly smooth model but rather to quantify uncertainties effectively. Nevertheless, mathematical algorithms can be developed to smooth the boundaries of the predicted lenses.
3.3. Case Study—Combination of Simulation Methods
Figure 8 shows the workflow for a probabilistic model located in the Munich region. In the Neogene, sands and clays were deposited (lenticular bedding). During the glaciation of the Alps in the Quaternary, these deposits were eroded, and the so-called gravel plane was deposited [
19]. The coexisting sedimentation of sands and clays generated discontinuous layer boundaries. For the expected formation geometry, the indicator-based interpolation and SISIM are better suited [
15]. The surface of the Neogene formation was modeled with the SGSIM, as SISIM is not able to simulate interfaces such as faults or erosion surfaces reliably [
12].
In the presented approach, the depth-dependent relative fraction of Clay (Cl) and Sand (Sa) facies was used as prior information to deal with the anisotropy. For oil reservoir modeling, soft data from seismic investigations often proves valuable for parameter inference and could complement simulation efforts.
Figure 8 shows the workflow for combining realizations from 3D SISIM and 2D SGSIM. First, the 2D SGSIMs need to be mapped to 3D by classifying the voxels above and below the layer boundary. Then, the models for the Quaternary and Neogene can be combined.
Figure 9 shows 3D visualizations of a subsoil model for a construction site in Munich, including the boreholes and the enclosing walls of a construction pit.
Figure 9a shows the most probable soil type and
Figure 9b the entropy of the prediction as an indicator of the uncertainty. The vanishing entropy near the boreholes indicates the low uncertainty in these regions. On the contrary, the entropy increases with increasing the distance to the boreholes. The accuracy of the input data is an essential prerequisite for the quality of the prediction. An inaccurate or biased soil description in the borehole inevitably compromises the meaningfulness of the model.
The model can be applied to the optimization of the horizontal excavation support, the detection of potential openings in the clay layer affecting the vertical water tightness of the construction pit, the estimation of water discharge, and the evaluation of the risk of a hydraulic failure water ingress into the excavation. It also helps in planning additional subsoil investigations and estimating the volume of excavated materials for different scenarios, among other applications [
21]. Though this is beyond the scope of this contribution, cross-validation and plausibility checks of the subsoil model by experts would be mandatory before it is used. Validation should include both the subsoil prediction and the uncertainty (e.g., [
21] for an illustrative example).
The present study elucidates the potential advantages of integrating uncertainty into subsoil digitalization. Visualization of the uncertainties related to the subsoil characterization is essential for assessing geotechnical risks, helping decision-makers and experts develop and implement efficient mitigation measures, e.g., optimizing the number of additional subsoil investigations and ultimately yielding more sustainable project outcomes.
3.4. Incorporation in BIM
In the construction industry context, geotechnical engineering is responsible for developing a BIM discipline model containing specific subsoil information [
22]. A discipline model does not contain all the available information, and it is a model specifically prepared for integration into the overall BIM (coordination) model [
23]. The primary role of the overall model lies in facilitating interdisciplinary integration and verifying and defining dependencies among the trades involved in the construction project, such as collision checks. However, the widespread adoption of consolidating all subdomain models into a coordination model within the DACH region (Germany, Austria, Switzerland) is still pending [
24].
The discipline model for the subsoil comprises a digital terrain model (DTM) and incorporates data from site investigations (e.g., boreholes, surveys, predicted geological models, etc.). Furthermore, it should encompass layer information and soil properties of the subsoil based on geotechnical reports [
25]. The subsoil discipline model includes geometric information (layers, lenses) and semantic data of geotechnical properties. The discipline model for the subsoil complements and builds upon the “working model” or “domain model”, containing all “working data” used for the geotechnical report [
23]. The discipline models validity depends on the latter’s inclusion [
22].
In Germany, according to [
26], the domain model for the subsoil serves as a communication interface between the geotechnical expert and the project stakeholders. This standard governs the domain model independently of the BIM methodology. While certain parts of the geotechnical report are transferred to the discipline BIM model, other information, such as geotechnical calculations and other related measures not performed by the geotechnical expert, may be excluded. The information transferred from the domain model to the BIM discipline model is project-specific. It is based on the requirements of all stakeholders and the long-term application of the BIM model [
23]. The complexity of the attributes and property transfer presents challenges and may lead to discrepancies with the original geotechnical report. As a result, the geotechnical report will remain indispensable for the foreseeable future, as certain information may not be effectively captured with equivalent quality within the discipline model [
27]. It is outside the scope of the present study to summarize and discuss the present developments of, e.g., the Eurocode [
28]. The following implementation fulfills the requirements of current design codes and presents a practical solution for the incorporation of prediction uncertainties in BIM.
Open BIM represents a collaborative approach to the design, construction, and operation of buildings and infrastructure. It emphasizes the interoperability of digital data between various software applications and stakeholders in the AEC industry. Open BIM’s core is the industry foundation classes (IFC) data model, which serves as a standardized format for exchanging information about building elements and their properties. IFC facilitates the seamless sharing of 3D models, drawings, and other project-related data, ensuring that all project participants can work cohesively, irrespective of the software they utilize. This interoperability enhances efficiency, reduces errors, and fosters a more transparent and collaborative environment in construction. While IFC versions 4.3 and 4.4 provide entities tailored to geologic and geotechnical features, they are not yet established as an ISO standard. This emphasizes the importance of relying on earlier IFC versions for practical applications. The established IFC schema versions aligning with [
29] and [
30] encompass IFC versions 2.3 and 4.0.
Currently, practical scenarios, including recent bid calls in pilot projects, underscore the significance of adhering to schema version 2.3. This particular emphasis on generating geotechnical models within schema version 2.3 stems from the realization that, while version 4.4 is set to introduce enhancements and specific geoscience entities, the practical integration of these versions into commercial software and industry workflows will still take time. This delay can be attributed to the intricacies of transitioning to new schema versions and ensuring compatibility with existing tools. From this standpoint, utilizing schema versions 2.3 and 4.0 emerges as a pragmatic solution, to promptly convey crucial information to planners and stakeholders. This approach recognizes the practical timeline for the industry to embrace the improvements offered by schema versions 4.3 and 4.4, while ensuring alignment with current practices and available tools. In versions 2.3 and 4.0 of IFC, no designated entities exist for boreholes, soil, prediction uncertainty, or groundwater.
Consequently, a need arises to align the components of the geotechnical subsoil model with the existing entities and framework. This approach has been previously employed, for instance, in the interface of Leapfrog
TM or in [
31]. However, it is worth mentioning that a reference implementation of this mapping currently needs to be included. The development of such an implementation falls outside the scope of the present study, but it is part of our future plans.
A Python interface based on the implementation of [
31] was used to convert boreholes and the layer boundaries to the IFC schema 2.3. The method presented in
Section 3.4 relies on a discretization approach utilizing rectangular voxels. Currently, voxels are not implemented in IFC versions 2.3 and 4.0, but they are expected to be implemented in IFC 4.4 [
32], which is currently in the software deployment phase. In general, voxels pose a significant challenge to widely used planning software such as Revit
TM. Each individual voxel block necessitates a separate geometry construction, consuming substantial storage space, often in the order of hundreds of megabytes to gigabytes for medium-sized models (250,000 voxels). This poses a formidable challenge for BIM viewers, in efficiently rendering these complex 3D representations.
Furthermore, the analytical tools within BIM viewers need to be better-suited for handling such models. Unlike more advanced tools such as ParaView, which enables the display of diverse information for categories and prediction variability using the same voxels, this capability is lacking in conventional BIM software(version. 9.2.0.128), e.g., Revit
TM. Consequently, voxels of the same category (e.g., soil type) obtained from the simulations are consolidated into 3D structures representing layers (see
Figure 10) or classes of prediction variability (see
Figure 11).
Figure 10 shows the model from
Section 3.3 displayed in the BIM viewer BIMVision
® and the corresponding IFC structure. Each soil type can be selected separately, and information from the geotechnical report can be referenced. In this way, the subsoil model complements the geotechnical report.
The introduction of a novel classification, as outlined in
Table 1, acknowledges the intrinsic variability in prediction results by assigning discrete grades to different ranges of prediction variability. These discrete categories enable the effective communication of the level of confidence one can place in the 3D model’s predictions at a particular point in the model.
In this classification scheme, the boundaries of each range are strategically defined to demarcate transitions in prediction variability. A prediction variability lower than 15% is categorized as good. The fair classification corresponds to prediction variabilities within the 15–30% range, indicating an acceptable level of precision. As the variability drops into the 30–45% range, the classification shifts to “poor”, signifying a reduced precision. The very poor classification, encompasses a prediction variability above 45%.
Figure 11 shows an example of an IFC-BIM subsoil model, showing the variability classes of the prediction.
Where information is available near the boreholes, the green colors indicate a low variability and prediction uncertainty. Red colors occur where the depth of the interface between two adjacent layers is very uncertain. The model can be sliced in any direction, providing the planner or geotechnical expert valuable with insights concerning the uncertainty of the prediction.