1 Introduction

In geoengineering practice, appropriate aggregation of the information retrieved from spatially distributed boreholes can facilitate the subsurface analysis and mapping to support decision-making on various underground projects (e.g., [41, 52, 71] The extension of a specified geological unit (e.g., rocks, lithological strata, mineralized area) is commonly determined through borehole data shown on 2D cross sections. However, the interpretation of laterally distributed attributes from sparse vertically constrained borehole data in a geo-model, due to heterogeneity and inability to explain the spatial analysis, presents a difficult obstacle [9, 24, 43]. Consequently, the retrieved results describing the spatial distribution of an attributed parameter not only often conflict with geological knowledge [43], but also require continually updating in the later operational stage [71]. Therefore, to conceptualize the spatial patterns of subsurface characteristics, 2D geo-models need to be developed in 3D [9, 10, 29]. It then can be complemented using supplementary test data to gain more accurate inference result (e.g., [1,2,3, 58, 66]. Mathematically, a 3D geo-model expresses the digital coordinates of geological formations/objects and associated features which can be practically implemented in geoengineering site characterisations (e.g., [2, 28, 31]. This implies great advantages of 3D geo-models in analyzing the subsurface attributed information for underground utilization and urban environment. However, collecting and presenting the 3D geospatial information has always been a challenge due to a variety of data, resolutions and required details [9, 39, 58, 73].

To increase the interpretability of geospatial laterally distributed attributes, integrating different qualified borehole data with numerical algorithms [11, 42] or cognitive interpretative approaches [16, 24, 53, 53]) has been a common practice. Such methods lead to interpolate between observational points for the incorporation of expert geological knowledge. In this view, 3D interpolation of planar meshes [67], surface mesh generation [23] and triangulation algorithms [18] are considered the most appropriate alternatives for handling discrete borehole data [1]. However, applicability of these methods despite the several challenges connected to processing time and limitations in handling big data is still essential in current geoengineering modeling methods (e.g., [27, 34, 35, 49, 60].

To tackle the challenges of big data [17], the application of skilled artificial intelligence techniques (AIT) in terms of shallow/deep neural learning networks (SNLNs/DNLNs), machine learning (ML), hybrid models, and evolutionary algorithms have been emerged in a variety of geoengineering problems (e.g., [1,2,3, 32, 54, 59, 73]. The 3D computer vision of DNLNs-based models, due to characterized features [38] and thus dedicating more vigorous tool than GIS and CAD systems [21, 42, 72], have been used in geoengineering applications [1, 2, 59, 70]. However, the robustness of DNLN-based predictive models should be evaluated in terms of different accuracy performance criteria and uncertainty quantification (UQ) analysis (e.g., [5, 47, 61].

With the growth of densely-populated metropolitan areas the use of subsurface urban planning to overcome challenges associated with sustainable cities and environmental regulations in response to technology and socioeconomic demands has been highlighted [12, 46]. From a geoengineering point of view, depth to bedrock (DTB), corresponding to the thickness of the sediments above the bedrock, is a crucial factor for the proper utilization of the subsurface. In Sweden, where there is an abundance of built infrastructure (e.g., transport tunnels, roads, railways), 3D spatial analysis of the DTB is an important concern that can provide critical insights into the stability of the structures [2]. Producing sufficiently accurate 3D visualized spatial DTB models not only facilitates the interpretation of sparse geotechnical measurements, but also provides a valuable tool for identifying optimum solutions and risk assessment (e.g., [2, 3, 24,25,26, 64, 68]. However, in geoengineering projects, producing high-resolution 3D spatial DTB predictive models due to incorporated uncertainties with geotechnical-based limitations (e.g., access to an entire area, cost, distance between the soundings) not only requires different combinations of data types, but also is a critical task [1, 2, 26, 64].

The DTB can be generated using traditional geostatistical techniques. However, technical concerns such as limited computational/graphical systems adopted for big geo-data in complex problems [22, 65], non-stationarity, require for extending the random function models, difficulties in model selection that sound solutions to practical problems, parameter estimation, and propagation of uncertainty [19, 42, 44, 57] will increase the analysis time and costs for handling the big data [5]. Therefore, these techniques have often been performed in small-scale areas [69]. On the other hand, the UQ analyses are relatively under-explored topic in DNLN [5, 6, 51], where technically provide the space to look for different opportunities, new options and overlooked solutions to end results. Therefore, the UQ analyses play a pivotal role in predictive DNLN-based models because a wrong answer is not necessarily problematic, but it is an issue in the case divergence that can effectively influence on assessed accuracies during both optimization and decision-making processes. This demand with the advent of AIT technology and code developing provide powerful new applications to help meet the requirements of geo-engineers in addressing modeling obstacles.

To overcome on above illustrated challenges, developing a modern automated hybrid DNLN-UQ based scheme for producing a sufficiently accurate and high-resolution 3D visualized subsurface is highly motivated. In this paper, two distinct automated procedures were innovatively combined to follow up both model optimization and UQ analysis using randomly switched off the jointed weight database. Due to embedded inner nested loops the computational time in hyperparameters adjustment to gain the optimum model significantly can be decreased. The hybrid model as a beneficial tool for urban planning was applied on geospatial locations of 644 datasets from different soil-rock soundings and geophysical investigations in Sweden. The predictability performance then was controlled and assessed using separately fed 75,715 scanned rock surface datasets. Detailed analyses in compare with quantile regression (QR) [14] and geostatistical ordinary kriging (OK) [62] demonstrated for dedicating a high-resolution 3D spatial predictive subsurface DTB map.

2 Study area and data source

As presented in Fig. 1a, the study area is located in the south-west of Sweden, close to Gothenburg. The geotechnical data used in this study was collected along highway E20 between Bälinge and Vårgårda (Fig. 1b). The Swedish Transport Administration is planning to build a new highway close to the old E20 by considering the current problems with speeding, traffic safety and environmental factors. In the western section close to Bälinge, the proposed highway crosses rock outcrops, moraine, glacial clay, glacial deposits, and swell sediments. The bedrock along the planned road consists mainly of gneiss and it has a higher content of dark minerals such as amphibole and biotite. Pegmatite, a granitic coursed-grained rock, is found along almost the whole stretch of road route. The bedrock is general heterogeneous and complex compound. However, at the end of the route the bedrock becomes more homogenous and consists mostly of the so-called Vårgårda granite (Åsander, 2015). The fracture zones in the gneiss in this area are generally in an east–west direction. However, close to Bälinge the fracture zones are in a northwest-southeast direction, and close to Vårgårda they are more northeast-southwest. The fracture planes are generally dipping to the south, which could potentially cause stability problems if rock cuttings are performed (Åsander, 2015). Referring to Fig. 1b, a number of 644 multisource datasets from both soil/rock sounding and geophysical prospecting using the spatial geolocation coordinates were characterized and compiled. The data then was randomly split into 65%, 20% and 15% to provide train, test, and validated sets. Accordingly, before feeding to the model, the used datasets were normalized to provide same scale within [ – 1, 1] interval.

Fig. 1
figure 1

Location of project area in Sweden (a) and the superimposed of the geospatial distribution of 644 acquired test points (in red) on Google Earth image (b)

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3 Model development process

3.1 Overview of DNLN structure

The concept of AIT can be defined as a simplified imitative learnable layout of the human brain structure that aims to increase computational power through embedded connective processing elements. There are several approved advantages of AI-based systems over traditional modeling approaches, and they are increasingly being used in the field of computer vision for geoengineering applications [1, 2, 40, 50]. DNLNs are a subcategory of AIT that without being explicitly will scan data to search for combinable features for faster learning. This implies that DNLNs can perform successfully with unstructured data (different formats such as text, pictures and pdf), as well as explore new complex features that a human might miss [38]. As a result, the DNLNs provide greater capacity over ML to carry out feature engineering which leads to significantly shorten computational time. Accordingly, the analysis of UQ in DNLNs can be described in the context of input variabilities, assumptions and approximations, measurement errors, as well as sparse and imprecise data [5]. However, all sources of uncertainty using these techniques for value-based judgements may not be quantifiable [47].

As presented in Fig. 2, in a fully-connected configuration of DNLN, the output of the jth neuron in the kth hidden layer at the tth iteration, \({o}_{j}^{k}\left(t\right)\), subjected to activation function, f, is defined as:

$${o}_{j}^{k}\left(t\right)=f\left(\sum_{i=1}^{{n}_{k}-1}{w}_{ij}^{k}{x}_{i}^{k-1}\left(t\right)+{b}_{j}^{k}\right) (j\le {n}_{k})$$
(1)

where; xi is the ith signal input with a connective weight of wij to the jth neuron in the kth hidden layer. nk shows the number of neurons in the kth layer, and \({b}_{j}^{k}\) denotes the bias which shifts the summed signals received from the neuron.

Fig. 2
figure 2

Simple configuration of DNLN architecture

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The weights and biases are then updated using corresponding error of the ith neuron in the kth hidden layers,\({\rho }_{i}^{k}\left(t\right)\), to minimize the prediction error by:

$${\Delta w}_{ij}^{k}\left(t\right)={\eta }_{w}\left[\dot{f}\left({x}_{i}^{k}\left(t\right)\right)\sum_{j}{\rho }_{j}^{k+1}\left(t\right){w}_{ji}^{k+1}\left(t-1\right)\right]{{o}_{j}^{k-1}\left(t\right)+{\alpha }_{w}\Delta w}_{ij}^{k}\left(t-1\right) (k=1, 2,\dots ,m-1)$$
(2)
$${\Delta b}_{i}^{k}\left(t\right)={\eta }_{b}{[\dot{f}\left({x}_{i}^{k}\left(t\right)\right)\sum_{j}{\rho }_{j}^{k+1}\left(t\right){w}_{ji}^{k+1}\left(t-1\right)]+{\alpha }_{b}\Delta b}_{i}^{k}(t-1)$$
(3)

where; \(\dot{f}\left({x}_{i}^{k}(t)\right)\) is the first derivative of \(f\left({x}_{i}^{k}(t)\right)\) with respect to \({x}_{i}^{k}(t)\). \({\alpha }_{w}\) and \({\alpha }_{b}\) express the momentum constants that determine the influence of the past parameter changes on the current direction of movement in the parameter space, and \({\alpha }_{w}\) usually varies within [0.1, 1] intervals and is used to avoid instability in the updating procedure. \({\eta }_{w}\) and \({\eta }_{b}\) represent the learning rates.

The total error for the entire network, E(t), prediction error, \(\varepsilon \left(t\right)\), and the outcome of the lth neuron in the mth output layer, \({\widehat{y}}_{l}\), is then calculated using the updated weight by:

$$E\left(t\right)=\frac{1}{2}\sum_{j=1}^{n}{\left({d}_{j}(t)-{o}_{j}^{k}\left(t\right)\right)}^{2}$$
(4)
$$\widehat{{y}_{l}}\left(t\right)=\sum_{i=1}^{{n}_{m}-1}{w}_{ij}^{m}{x}_{i}^{m-1}\left(t\right) (l\le {n}_{\mathrm{o}})$$
(5)
$$\varepsilon \left(t\right)=y-\widehat{y}(t)$$
(6)

where; dj(t) denotes the desired output of neuron j at the tth iteration. y and no represent the actual output and the number of neurons in the output layer, respectively.

3.2 Configuring the hybrid automated model monitoring

As presented in Fig. 3, a hybrid automated approach was developed comprising two blocks and several embedded inner nested loops accompanying by switch cases for subsurface 3D modeling. The overview of given blocks (Fig. 3) independently has successfully been applied for modeling the DTB and UQ analyses of groundwater level tasks [2, 5]. However, converging two distinct automated concepts with different purposes through inner combination is a challenging task. The main technical novelty of the configured hybrid model is the inner integration of two different automated approaches using DNLN and ensemble-based structures to pursue and select the most precise topology using jointed weight database. Due to the need of DNLN for adequate computing power and data for learning [13], in Block A (Fig. 3), the procedure monitors a variety of combined internal hyperparameters as presented in Table 1. To enhance the modularity management for community support and thus improving the computational power and optimization performance, the process was coded using both Python and C +  + . Referring to Table 1, the number of hidden layers to capture the optimum DNLN as a user defined option was set for 3 in accordance with the analyzed error improvement [4, 25]. Using this approach, the monitored network performance predictability during the last and or each iteration can detect when the model is not improving, and thus further training is not called for. Referring to compiled results from different numbers of hidden layers, i.e., from 3 to 8 by increasing the number of neurons from 30 till 80, in the models with more than 4 hidden layers and 40 neurons the analyzed error did not show improvements and thereby, 3 hidden layers and 35 neurons were selected. Such integrated components then allow the system to monitor different combinations of internal hyperparameters, not only makes learning faster but also minimizes the chance of getting stuck in local minima or overfitting problems [1]. To prevent early convergence, two-step termination criteria were considered where if the root mean square error (RMSE) as the first priority did not achieve then the number of iterations (set to 1000) will replace. Accordingly, numerous captured shallows and DNLNs even with similar topology but different hyperparameters can help to improve the generalization of the selected model for new examples [2]. The UQ methods determine the calibrated confidence level of a model and thus can demonstrate biases caused by over/lack of confidence in the predictions. Therefore, evaluating the reliability and efficacy of developed DNLN-based models due to figuring inference and decision makings is an important concern before they could be applied in practice. This implies that predictability assessment of DNLN-based models for making critical decisions without considering the UQ are usually not trustworthy. Furthermore, UQ analyses are relatively under-explored topic in DNLN, especially when applied to geoengineering applications and thus automated hybridizing model to cover both optimization and uncertainty estimation is great of interest. Accordingly, to control the stability and optimality of the achieved topology, the UQ was estimated using a novel proposed ensemble-based automated randomly deactivated weight database (ARDCW) approach as presented in Block B (Fig. 3) [5]. Since the optimum topology is achieved, the weights associated with processing elements after updating and adjustments are saved as database in the form of matrix to reconcile the differences between the actual and predicted outcomes as well as direct applying on validation/unlabeled datasets for subsequent predictions. Therefore, in Block B, just the finalized weight database after automated optimization will be utilized. This implies that the deactivating approach is solely applied on the weight database of the optimum topology. Subsequently, the model is automatically retrained using switched off weight components to monitor the variation of predicted output subjected to different scenarios. Thus, the challenge of computational costs for multiple training of complex topologies through different optimizers is overcome [1].

Fig. 3
figure 3

Layout of proposed hybrid automated approach to capture the optimum topology and UQ analysis

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Table 1 Hyperparameters used to monitor the optimum model
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3.3 Results and created 3D model

Considering the combination of hyperparameters employed, the optimum model was identified among more than 2000 monitored and ranked topologies with different internal characteristics. A series of the results subjected to selected hyperparameters (Table 1) in terms of the variation of minimum RMSE for the training stage are shown in Fig. 4a. Using this process, the risk of overfitting, early convergence, and getting stuck in local minima will be minimized because even similar topologies but with different hyperparameters are examined. The overtraining problem is also detected when no improvements can be observed in the accuracy after a certain number of epochs [5]. As presented in Table 1, Figs. 3 and 4(a–d), examining different AF, changing the topologies, and residual connections were monitored to ensure preventing the vanishing gradient. Furthermore, in the automating process, the best results from three runs of each model are saved. Referring to Fig. 4a, the DNLN topology with a structure of 3–20-15–1 (input variables-neurons in hidden layer 1-neurons in hidden layer2-output) subjected to QN and Hyt can be selected as optimal. Subsequently, the predictability of the achieved topology using the randomized datasets is reflected in Fig. 4(b–d).

Fig. 4
figure 4

Identifying the optimum topology for prediction of DTB through variation of RMSE (a), and corresponding predictability using training (b), testing (c), and validation data (d) (TR training data; TE testing data; VL validation data; MAPE mean absolute percentage error)

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Creation and simulation of the 3D model using AITs is a promising tool for subsurface geoengineering purposes (e.g., [1, 3, 71, 74]. This is because the traditional method relies on the knowledge and experience of experts in the selection of assumptions, parameters, and data interpolation methods which are subjective and limited [51].

Figure 5 shows the step-by-step 3D modeling of the subsurface spatial DTB distribution for the study area compared with the real scanned data. The rock outcrops can then be identified through the overlaid generated ground surface and spatial scanned DTB, leading to a high-resolution predictive 3D subsurface model with adequate accuracy for use in geoengineering projects. Figure 5d shows the results of analyzed UQ in terms of the predicted error using the ARDCW [5] for the achieved optimum DNLN topology.

Fig. 5
figure 5

Step-by-step side view of the created 3D model for the study area comprising the a ground surface, b, c incorporating the spatial distribution of scanned and predicted DTB, and d, e estimated UQ using ARDCW using automated system

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4 Discussion and result validation

3D digital models can improve communication between engineers, constructors, and their clients, i.e., bidding, cost analysis, and listing the required resources. Since the success of 3D models is dependent to visualize resolution of subsurface complexities, in practice and before use should be carefully evaluated in terms of different accuracy performance metrics. In this study, the discussion of the generated 3D model is presented through comparison of different analytical metrics with QR and traditional geostatistical ordinary kriging (OK). However, due to the robust computational capability of AIT, the outliers are predicted more accurately than with traditional techniques and thus better performance can be expected. In addition of the randomized validation data, the performance of the presented procedure was discussed using 75,715 true bedrock surfaces obtained from laser scan with Lidar in a few areas in the northeast of the study area between Jonstorp and Hjultorp (Fig. 1). The predicted bedrock surfaces then could be compared with the scanned bedrock surfaces to evaluate the model performances.

4.1 Precision performance

Among the traditional geostatistical interpolation algorithms, the OK can serve spatial estimation when a variogram is known [62]. However, the OK due to use of the interpolation can only predict the DTB values within the range of observations, while the QR and DNLN, because of regression and considering the minimal squared error, are able to predict even outside of the tabulated domain. From a mathematical point of view, an nth order polynomial for (n + 1) DTB points can be obtained which passes exactly through all (n + 1) data. This is the main reason why in training process the OK subjected to different variogram functions showed high R2 (Table 2). Accordingly, during the training process, the regression may not be as accurate as interpolated DTB within the observed domain but may provide much better predictions in the range of below and beyond the observations. This reasonably describes why R2 of predicted data in OK varies significantly (Table 2). Moreover, since OK statistically contains several parameters that can be justified by the data, the achieved results can be attributed overfitting [20], while variability of variogram parameters resulted in different predictions based on the same dataset. This implies that the OK subjected to big data does not generalize well, and the predicted new values then provide closely response to a particular set of data.

Table 2 Summary of OK results subjected to 8 to 14 lags and different variogram functions
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Moreover, for proper evaluation, the model should never be trained with the entire dataset because this would result in too much learning and saturation [15, 56]. The results for such training would be capturing more noises in the input data that can cause wild fluctuations in the model and thus do not represent the expected trend [1]. Referring to Fig. 6a, the predicted scanned DTB using QR due to asymmetrically-weighted procedure is less affected by outliers in the data set than OK (Fig. 6b). However, estimating the parameters in QR is harder than in Gaussian or generalized regression. Figure 6c expresses that DNLN can be flexible enough to predict nonlinear patterns more accurately compared to adjusted QR and OK with total variation regularization, emphasizing practicality of the method for scanned DTB data.

Fig. 6
figure 6

Predictability of a QR, b OK, c optimum DNLN, and d calculated residuals subjected to scanned DTB

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According to the theory of estimation and testing [33], non-normal pattern especially for inappropriately-specified models or non-homogeneity of error variance can be analyzed using residual plot. Moreover, the too small values of calculated residuals provide a pattern of sufficient stability to permit valid statistical inference [33]. Figure 6d shows the random pattern of residual between scanned and predicted DTB subjected to applied models, where the results ideally should be small and unstructured in which the observed big values are due to outliers. Therefore, residuals represent the portion of the validation data not explained by the model. In this point of view, the displayed plot in Fig. 6d is a control chart to determine whether a process should be continued/adjusted to achieve the desired quality.

The predictive models are analytically dependent on the precision and accuracy of applied techniques. According to the central limit theorem, the variance of predictive models should essentially be compared for different populations to test the equal assumption. Since the scanned DTB have known standard deviation (SD), the model with closer results is more acceptable. This implies that the acceptability of the implemented method can therefore be judged on the size of the observed errors relative to a defined total allowable error (TAE) as presented in Fig. 7 [36]. Accordingly, a value of 6.811 for the TAE for 76,359 datasets corresponding to the total amount of training and scanned DTB is achieved. The TAE for the predicted DTB using the DNLN, OK and QR was then calculated as 6.49, 5.01 and 6.07, respectively. These values specify the maximum amount of error for both imprecision and bias combined that is allowed for an assay without invalidating the interpretation of a test result. Therefore, compared with the calculated residuals (Fig. 6d), the OK has exceeded from the TAE which can be interpreted as instability of this model in handling big data. The precision of TAE was then supported using SD of calculated residuals with values of 1.15, 1.68 and 0.98 for QR, OK and optimum DNLN, respectively.

Fig. 7
figure 7

Defining the TAE (SD standard deviation, SE standard error)

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According to Akaike’s information criterion (AIC) [7] that penalizes appended terms to a model, additional variables that reduce RMSE and increase R2 are not appropriate for the choice of model. The goal is to find the model that minimizes AIC and for the regression can be calculated as:

$$\mathrm{AIC}=2P+n \mathrm{log}\left(\frac{\mathrm{RSS}}{n}\right)$$
(7)

where; P is the number of variables, n denotes the number of records, and the models with k more extra variables are penalized by 2 k. For the similar training data, the AIC is an estimator of predicted error and thus can represent the quality of each model [55]. This implies that the AIC estimates the relative amount of information lost by a given model, where the less information a model loses, the higher the quality of that model. This implies that AIC deals with the trade-off between the quality of fit and the simplicity of the model, leading to both the risk of overfitting and the risk of underfitting [8]. In terms of RSS in Eq. (7), the residual sum of squares for OK, QR and optimum DNLN are 160,000, 110,899 and 81,984, respectively. In this point of view, the second algebraic term of the AIC (Eq. 7) for OK, QR and optimum DNLN models stands for 24,602, 12,549 and 2,615, respectively. Therefore, the high achieved R2 and thus low RMSE in training and validation of OK (Table 3) cannot lead to better performance than QR and optimum DNLN.

Table 3 Applied statistical error criteria to evaluate and compare the employed models
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4.2 Statistical error analysis

In geoengineering contexts, statistical analysis is an essential tool for the reliable interpretation of experimental results. Furthermore, the growing importance of decisions and opinions based on data can pivotally assess the quality of analyses and thus statistical error metrics can be used to decrease risk in decision making. Statistical analyses with a single metric, however, might be quick and simple to understand, but in complex geoengineering problems provides a very low-resolution picture of interpretation. This is because the metrics are different from loss functions. Therefore, the reasonable conclusion on statistical indices should always be drawn by analyzing multiple metrics where their results confirm each other. For example, since the coefficient of determination (R2) does not disclose correctness and information about the causation relationship in regression models, it should be supplemented with other criteria. In addition, it does not indicate the correctness of the regression model. The most common interpretation of R2 is how well the regression model fits the observed data, and thus 0.91, 0.88 and 0.82 in DNLN, QR and OK reveal that from the 75,715 data, a total of 68,901, 66,629 and 62,086 have been fitted to the regression model (Table 3). Consequently, the DNLN model provides 3.3% and 9.9% improvement as compared to QR and OK. Subsequently, RMSE gives a relatively high weight to large errors and thus is most useful when large errors are particularly undesirable. Using mean absolute percentage error (MAPE), an intuitive interpretation of the extent or importance of errors can be presented. MAPE dedicates the independence of the relative errors and thus makes it possible to compare forecast accuracy between differently-scaled data. Therefore, incorporating the RMSE and MAPE provides a quadratic loss function that can also quantify the uncertainty in a predictive model. Accordingly, using index of agreement (IA) [63], the errors are given with appropriate weighting, and are not inflated by their squared values. The variance accounted for (VAF) measures the discrepancy between a model and actual data in each running condition and is often used to verify the correctness of the output results, and thus shows how much of the variability of the data can be explained by a fitted regression model. The results of the generated 3D models in terms of MAPE, VAF, IA, R2 and RMSE, as well as general standard deviation (GSD) and difference ratio (DR), were then evaluated and are presented in Table 3. Accordingly, the model with lower MAPE, DR, GSD and RMSE and higher values of VAF, IA and R2 indicate for better performance.

4.3 Comparing 3D visualized models

3D modeling is progressively improving in its capacity to visualize geological features in detail in subsurface geoengineering applications (e.g., [9, 23, 27, 49]. Such 3D predictive geospatial models can display digitized geometric or topological objects based on the available featured factors, making it possible to capture the complexities of buildings and infrastructures in an image that closely resemble the view of the finished project. Accordingly, 3D models can assist geoengineers to select more proper structural elements and help deliver safer project design. An appropriately-tuned DNLN can easily be adapted with new data to improve the diagnostic performance and computer vision modifications. It also can be supplied with future acquired data to reflect more details of the subsurface features. With this in mind, the predictability of applied models in creating spatial 3D views supplemented by residual contours in the outlines of the area subjected to 75,715 scanned DTB are compared and presented in Fig. 8. It can be observed that the residuals between the measured/scanned and predicted bedrock surfaces are different for DNLN, QR and OK. The residuals estimated by the DNLN method have the narrowest residual interval [-3.5, 3.5 m], indicating a more accurate predicted model than the other methods.

Fig. 8
figure 8

Top view of the generated 3D model using the applied methods by incorporating the predictive models and scanned DTB data for a optimum DNLN, b QR, and c OK

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4.3.1 Estimated UQ intervals in scanned DTB

Despite the various 3D suggested advanced analysis methods in geoengineering problems (e.g., [2, 48, 72], the analyzed UQ due to used assumptions and propagated error in applied computational modeling technique can provide significant differences. Moreover, the problem associated with generalizing from a single study, difficulties in obtaining appropriate measures, and issues related to identifying the highest accuracy for model representation, should be considered [5]. In this context, prediction intervals (PI) provide an indicator for the UQ of individual estimation. The PI has advantages over point estimates because it considers the variability in the data to provide a “reasonable” range of values for an observation [45]. Therefore, PI must account for both the uncertainty in predicting the population mean, and the random variation of the individual values [37], and thus it is always wider than a confidence interval. This implies that when the statistics cannot be employed to explain the future data, the UQ for the upcoming observations can be estimated using PI based on known populations of data. One of the main advantages of using PI is that it gives a range of likely weights and thus the modeler can have a sense of how accurate the predicted weight is likely to be [30]. Figure 9 presents the estimated UQ intervals in terms of the level of 90% PI using ARDCW-DNLN, QR and OK for the scanned DTB area. The percentage of each 2 m intervals (Fig. 9) then is reflected in Table 4. According to calculated PI, the ARDCW-DNLN with estimated UQ within [0.6 m, 5.2 m] interval showed a narrower interval than OK and QR which depicts more concise results and thus higher accuracy. This implies that the bias and discrepancy of the OK, QR, and ARDCW-DNLN can be interpreted through comparison of the estimated UQ in terms of PI. The distribution of classified UQ intervals (Fig. 9) makes it possible to identify the sub-areas with higher uncertainties in which for reduction the need for more data and applicability of its corresponding in-situ test are highlighted.

Fig. 9
figure 9

Overview of the study area showing the comparative UQ analysis of the applied methods at a level of 90% PI for a OK, b and c optimum DNLN

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Table 4 Statistics of the uncertainty interval with scanned data for the compared models
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5 Concluding remarks

In the context of 3D modeling, an advanced computer vision hybrid automated DNLN-UQ framework for DTB analysis using multi source geo-data in Sweden was proposed and developed. To approve the technological advancements, the results of captured optimum DNLN in accomplishment of tasks and produced outcomes was then compared with QR and OK using a number of 75,715 scanned DTB to provide better interpretation and convenience for post-modeling analysis. According to calculated TAE, the exceeded residual of OK in predicted DTB was attributed to instability of this model in handling big data. In terms of AIC, it was demonstrated that the high achieved R2 and thus low RMSE in training and validation of OK doesn’t lead for better performance than QR and optimum DNLN. Comparison of the applied models using different statistical metrics also showed that DNLN can provide better results than QR and OK. Created 3D visualized maps subjected to 90% PI for the scanned DTB revealed a narrower UQ estimation using ARDCW-DNLN within [0.6, 5.2] and thus higher accuracy.

The presented 3D model can improve the visual analysis of more structural elements and thus better insights into the project based on the natural and man-made aspects involved, leading to a safer design process. Digital flexibility of the presented model allows to add the coordinates of the infrastructure to modify inconsistencies in the plan without requiring blueprints. This then will provide the opportunity for dedicating more applicable solutions to fix engineering difficulties right away. Using 3D models also enhances the bidding process and allows contractors to benefit from automated intelligent modeling, leading to higher quality and lower costs for construction. The adapted 3D models not only speed up the design process but also enable geoengineers and decision makers to play around with different ideas and identify potential design problems before they become actual issues. This implies that 3D modeling in construction can put all the phases of the work together and provide a realistic picture of the finished project.