I.S.G.E.: An Integrated Spatial Geotechnical and Geophysical Evaluation Methodology for Subsurface Investigations

Abstract

A new Integrated Spatial Geophysical and Geotechnical Evaluation (I.S.G.E) methodology has been developed to estimate the spatial distribution of geotechnical parameters using high-resolution geophysical methods. The proposed algorithm is based on fuzzy logic, and the final output is the prediction of the 2D or 3D distribution of a geotechnical parameter within a survey area. The main advantage of the developed I.S.G.E tool is that it can propagate sparse geotechnical or point information from 1D to 2D or even 3D space through a fully automatic, unbiased statistical procedure. In this study, I.S.G.E. is implemented and evaluated first using synthetic data and, afterwards, in field condition applications. The automatically derived 3D models, depicting the spatial distribution of specific geotechnical parameters, provide engineers with an additional interpretation tool for better understanding the subsurface conditions of a survey area.

1. Introduction

Standard engineering materials involve various types of metals and alloys, polymers, glass, ceramics, and different kinds of composites. Soil and rocks, whether they form part of composites or are in their natural state, are definitely among the most variable of all engineering materials [1]. As a result, geotechnical variability creates a significant degree of uncertainty, which constitutes an inherent part of any geotechnical problem. The ultimate scope of a so-called geotechnical investigation is to reduce, and ideally eliminate, this uncertainty. Briefly, the purpose of a typical geotechnical investigation can be summarized as follows: (a) determine the nature of the ground and its stratification; (b) obtain (disturbed and undisturbed) soil samples for visual identification and appropriate laboratory tests; (c) determine the depth and nature of bedrock (if encountered); (d) conduct some in situ field tests; (e) determine the ground water level and position.

A major problem faced by engineers when estimating the geological and geotechnical properties of subsurface in construction projects is the limited amount of available data and the inadequacy of the information obtained from site investigations. In an ideal world with unlimited economic resources, geotechnical engineers would be able to drill abundant boreholes and take numerous samples back to the laboratory to measure typical soil properties, such as permeability, compressibility, and shear strength. Then, based on all of this comprehensive and robust information, engineers would conduct the analysis of any geotechnical problem and be very confident in their predictions and suggested design. However, the reality is different, and geotechnical engineers must regularly deal with and rely upon very limited site investigation data.

Over the last twenty years, geophysics has achieved a noteworthy and valuable status among the available tools that enhance geotechnical analysis and design [2]. Without a doubt, geophysics can now serve as an excellent supplementary investigation tool, particularly where the use of traditional means of investigation (such as boreholes) is limited, whether due to economic or technical reasons. Various geophysical methods, such as resistivity tomography (ERT), ground-penetrating radar (GPR), seismic refraction tomography (SRT), multichannel analysis of surface waves (MASW), surface wave tomography, and magnetic and gravity methods, are widely carried out in geotechnical studies [3,4,5,6,7,8].

Although geophysical data are advantageous in imaging subsurface conditions over large areas and can reveal important information about subsurface features, the information obtained is usually qualitative from a geotechnical perspective. Many authors have tried to provide deterministic transfer functions for linking geophysical parameters to geomechanical parameters [9,10,11] or calculate empirical equations for predicting various geotechnical parameters from measured geophysical methods [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]. However, such correlation is a challenging task, since these parameters are usually strongly site-dependent and affected by many factors related, for example, to rock type, subsurface geology, water content, and permeability [28,29,30]. Additionally, the parameters measured on samples, so-called static, will differ from the dynamic ones inferred from geophysical measurements [31].

The trend in objectifying information about the condition of a structure is driving the development of tools used to appraise and integrate data from different sources that are similar and dissimilar in nature. Assessing the stability and integrity of such engineering structures often requires a multi-disciplinary approach and constructive collaboration between experts in geotechnical and environmental engineering, geophysics, hydrology, and geology. The development of new approaches regarding the integration of multiple physical property models for the identification of different geological units has been recently suggested as a new frontier in geophysical exploration [32]. Machine learning methods and data fusion techniques, applied to geophysical and geological/geotechnical data to predict soil property distribution, have attracted significant interest, with several works published in recent years [33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48].

An interesting approach to integrating geophysical methods, without the need for predefined relationships, can be achieved using statistical techniques, such as fuzzy clustering analysis [49,50,51,52,53,54,55,56]. Fuzzy C-means (FCM) is among the unsupervised classification techniques that can automatically group a set of objects into several subsets or clusters without the help of prior expert knowledge, allowing one piece of data to belong to two or more clusters [57]. Moreover, Paasche [58,59] proposes a very interesting approach for the probabilistic integration of ill-posed geophysical tomography models and logging data based on fuzzy logic. The methodology utilizes cross-borehole radar and P- and S-wave travel time tomography models, in combination with cone penetration and dielectric logging data.

In this study, a new Integrated Spatial Geophysical and Geotechnical Evaluation (I.S.G.E.) methodology is proposed to estimate the spatial distribution of geotechnical parameters using high-resolution geophysical methods. Firstly, implementation is evaluated using synthetic data before it is applied in real field conditions. The proposed algorithm is based on fuzzy logic, and the final output is the prediction of the 2D or 3D distribution of a geotechnical parameter in a survey area. The I.S.G.E. algorithm provides opportunities to fully integrate geophysical and geotechnical (G&G) data to quantitatively assess site variability and design parameters with limited geotechnical data. It is noteworthy to mention that the proposed approach does not intend to replace geotechnical investigation methodologies under any circumstances. It is complementary and intended to help maximize information deduced from limited in situ geotechnical data.

2. I.S.G.E. Methodology

The I.S.G.E. algorithm is fully automated, easy to use, and can convert interpreted geophysical models to unbiased, statistically driven geotechnical models with the incorporation of drilling information. The resulting models can be more easily understood by engineers and can give more insight into the spatial assessment of geotechnical parameters in a survey area. The main advantage is the incorporation of any available geotechnical information (borehole data and soil or rock in situ tests and laboratory measurements) in the multiparameter analysis. In this way, the extrapolation of sparse geotechnical or even point information from 1D to 2D or even 3D space can be achieved without the need for predefined analytical or empirical equations linking the measured geophysical data to the desired geotechnical parameter.

The schematic diagram of Figure 1 presents the workflow and the basic steps of the developed algorithm. The main input data for the algorithm can be (a) co-located inverted geophysical models of the survey area and (b) sparse or point geotechnical information. For a specific position (x, y, and z) of a cell in the model space, a respective parameter value for each geophysical inverted model should exist. For this reason, the same grid size and grid spacing are used for the various inverted models. This can be achieved either before inversion, using the same grid parameters for the starting model of each method (depth of model, size, and number of cells), or after inversion, using available interpolation techniques such as kriging. In the post-inversion procedure, it is recommended to use only the overlapping area of different geophysical models that exhibit high reliability (e.g., high total sensitivity, high ray density) in order to minimize uncertainty associated with interpolation. A textual (txt) file describing the x, y, and z coordinates and the values of each geophysical inverted model is required, with m rows equal to the number of model cells and (3 + v) columns, where v is the number of different available geophysical methods.

Geotechnical information derived from the site investigation program can be used as input in our algorithm. This information can be in situ geotechnical measurements in one or more boreholes, such as Standard Penetration Test (SPT) blow counts or logging measurements. Alternatively, a 1D geotechnical model at each borehole with the respective geotechnical design parameters, as suggested by the representative geotechnical experts, can be used. Two txt files are necessary in the case of incorporating borehole information. The first one contains information about the name and the coordinates of the boreholes (x, y, and elevation), and the other contains information describing the name of the boreholes and the starting and ending depth for each measured or suggested geotechnical parameter. In the ISGE algorithm, one set of borehole information (one geotechnical parameter) is used at a time for analysis.

2.1. Optimization of Input Geophysical Data

In the ISGE methodology, to reduce the uncertainty of the geophysical interpretation, a multiparameter analysis using the Fuzzy C-means (FCM) clustering technique is initially performed, following specific processing steps for the creation of a Unified Geophysical Model (U.G.M) [54,60] and the calculation of fuzzy updated geophysical models.

In general, clustering is a division of data into groups of similar objects. Each group (cluster) consists of objects that are similar to each other and dissimilar to objects of other groups. FCM is a method of clustering that allows one piece of data to belong to two or more clusters. After selecting the number of clusters, with the iterative minimization of a specific objective function, the optimum locations of the cluster centers and membership values are calculated. The membership values of a data point, which quantify its degrees of membership with respect to the various clusters, vary between zero and one; the higher the membership value, the closer the data point is to the corresponding cluster center. For any given data point, the sum of all membership values is unity. By assigning each data point to the cluster for which it has the highest membership value, the results of FCM cluster analysis can be converted to equivalent zoned models. Objective function (1) is as follows:

$$J_{FCM}=\sum _{i=1}^c\sum _{j=1}^n{\mathrm{md}}_{ij}^f {‖x_j-u_i‖}^2$$

(1)

where c is the number of clusters, n is the number of data points, and md_ij denotes the degree of membership of data point x_j to cluster i defined by its center u_i. Weighting exponent f is a fuzzification parameter, which is the degree of overlap between clusters. The minimum meaningful value of f = 1 and the solution is a hard partition. As f approaches infinity, the solution approaches its highest degree of fuzziness [57].

In particular, a multiparameter analysis using the FCM clustering technique is performed to automatically integrate information from available co-located inverted models derived from various geophysical methods in a survey area. In the U.G.M. creation methodology, the first step is the construction of a v-dimensional multiparameter space, where v is the number of the input inverted geophysical models. Each point of the multiparameter space should be characterized by the parameters of the autonomous inverted models (geophysical data input). For a specific position (x, y, and z) of a point, a respective parameter value for the inverted models should exist. After the normalization of input data to avoid scaling effects, the FCM clustering technique is performed for various numbers of clusters (c), and as a result, the optimum locations of the cluster centers and membership values are calculated.

By assigning each data point to the cluster for which it has the highest membership value, the results of the FCM cluster analysis can be converted to equivalent zoned models (hard clusters). Based on an iterative two-stage clustering procedure and the estimation of the corresponding values of certain statistical indices, such as Normalized Classification Entropy (NCE), the optimum number of clusters is selected. The derived hard cluster model and the respective maximum membership values of each cluster center are the initial output of the specific procedure.

In the next step, the updated fuzzy models can be calculated following the conventional approach proposed by Paasche [50,51] or a modified version that also takes into account the maximum and minimum values of each geophysical model to ensure the prevention of artifacts caused by extreme values in the updated geophysical models. In the alternative approach, we initially discard outlier measurements from geophysical input models using an iterative technique, and then, we define the 2^v centers (pairs of minimum and maximum values of each geophysical method) that are placed at the corner points of the multiparameter space, where v, as already mentioned, is the number of different available geophysical models. Using the total clusters (calculated from the previous procedure and the additional 2^v centers), we calculate the membership functions according to an analogue FCM cluster technique [58].

The updated fuzzy geophysical models combine the information from all geophysical methods and provide more robust results. Based on the experience of previous synthetic and real case studies [54,60,61,62], the updated fuzzy models provide a robust combination of different models’ information and are usually significantly improved in comparison with the initial inverted models regarding the delineation of subsurface conditions. For this reason, the ISGE algorithm provides the option for the updated fuzzy models to be the input geophysical models instead of the initial inverted ones.

Additionally, in the case where there is no borehole information, we can convert a selected geophysical fuzzy updated model to a desired geotechnical target using predefined equations that link geophysical to geotechnical parameters. It is obvious that the uncertainty in geophysical interpreted models and the physical or empirical relationships can introduce significant errors in the estimation of the spatial distribution of a geotechnical parameter. The predicted models should be used with caution and always as a supplement to laboratory or in situ geotechnical methodologies.

2.2. Incorporation of Geotechnical Information

For the incorporation of geotechnical information in the analysis, a data-driven approach is followed, and it is based on a modified version of the methodology proposed by Paashe [58], which does not require knowledge or assumptions about the expected relations between the tomographically imaged physical parameters and the target geotechnical parameter. We structurally integrate different geophysical models achieved by individual tomographic inversions by transforming them into fuzzy sets. In a post-inversion analysis, systems of linear equations are then set up and solved in a least-squares sense, linking the fuzzy sets and sparse information about the target parameter. The result is the prediction of the value of the desired geotechnical parameter at each center of the multiparameter space that best satisfies the equations discussed further.

We start by defining the cluster numbers (cl) of the multiparameter space based on the number of sparse values of the input geotechnical information. The user has two different options. The first uses the total number of the input geotechnical values, and the second uses the maximum number of the geophysical model’s cells that cross an available borehole. The first approach can provide more detailed models, as it uses a greater number of cluster centers, achieving, in general, a better fit in the input geotechnical data but sometimes producing noisy and unrealistic values in parts of the final model. The second one usually provides more robust results with smoother models in comparison to the first approach, sometimes lacking in resolution.

Then, we normalize the input geophysical and geotechnical parameter values between 0 and 1 in order to avoid scaling effects. Prior to normalization, the input geotechnical information is interpolated via kriging or other methods to specific grid steps with respect to the depth resolution of the input geophysical inverted models. As a next step, we calculate the membership functions according to an analogue FCM cluster technique [58] for fixed positions and the number of cluster centers (cl). The cluster centers can be either uniformly distributed in the multiparameter space or further optimized using the FCM technique. In order for all samples of the multiparameter space to be enclosed by the minimum and maximum values of the cluster centers, the 2^v centers are placed at the corner points of the multiparameter space, where v, as already mentioned, is the number of different geophysical models available.

The next step is the calibration of these fuzzy sets (the cluster centers and the respective membership values) for each cell where geotechnical data is available. The geotechnical parameters in the boreholes are used to train and calibrate our fuzzy system. This is achieved by setting up and solving an over-determined system of linear equations in the form of Equation (2):

md × T = b,

(2)

where md is the matrix of estimated membership values, b is the vector of known geotechnical values from the boreholes, and T is the vector of target geotechnical parameters to be estimated.

For instance, for a 2-dimensional multiparameter space with 10 cluster centers and 20 cells with available geotechnical information (for example, j = 201, 201, 202… 220), a system of linear equations (3) can be set up, as shown in the following set of equations.

md_(1,201) × T_(1,201) + md_(2,201) × T_(2,201) + …… + md_(10,201) × T_(10,201) = b₍₂₀₁₎
 md_(1,202) × T_(1,202) + md_(2,202) × T_(2,202) + …… + md_(10,202) × T_(10,202) = b₍₂₀₂₎
.    .    .    .    ……    .    .     = .
.    .    .    .    ……    .    .     = .
 md_(1,220) × T_(1,220) + md_(2,220) × T_(2,220) + …… + md_(10,220) × T_(10,202) = b₍₂₂₀₎

(3)

Using an iterative least-squares square method, the target parameter T that best satisfies the above equations is predicted for each center of the multiparameter space. An upper and lower limit in the least-squares technique based on the expected values of the target parameter can help provide more robust and realistic predicted geotechnical values. Finally, using the defuzzication technique proposed by Paasche [50,51], we can estimate the final predicted target parameter values.

The I.S.G.E. algorithm can also be used even if the geotechnical information is outside the survey grid. In this case, the same processing steps are followed. The only difference is that we assume that the 1D average models derived from the 3D geophysical models correspond to each of the input boreholes individually. This process is automated, as the user only needs to provide the two input files regarding the geotechnical information of boreholes outside the survey area. Of course, we suppose that there are no significant changes in the expected formation regarding their lithology and their depths. Alternatively, if sharp lateral variations exist in the area, the user could manually transfer the borehole coordinates inside the survey grid to a new position (x,y,z) that would better fit the interpreted geophysical models.

The application of the I.S.G.E. tool in synthetic and real data, which is presented in the following paragraphs, highlighted factors that have an important influence on the efficiency of the prediction models. The main factor is the quality of the input geophysical models, as well as their correlation with the target parameter. For this purpose, additional options in the newly developed algorithm are provided, such as the utilization of updated fuzzy models as input geophysical models. Moreover, the weighting of geophysical models in the initial stage of the FCM clustering technique for UGM creation is introduced, as well as in the stage of the fuzzy sets’ estimation regarding the input geophysical models and the target parameter. In addition, the I.S.G.E. algorithm provides the possibility to discard areas with lower reliability from the prediction models. In particular, each cell can be assigned to a certain zone (cluster) for which it has the highest membership value. If a cell belongs to a zone that does not match the zones defined by the cells of the available geotechnical information, the value of this cell is considered null, and it is removed from the predicted model. All of these new options improve the final result and provide users/researchers a versatile tool that can reduce the uncertainty in geotechnical investigations.

2.3. Evaluation of Output Results Using Statistical Indices

Various statistical indices are estimated for the evaluation of the output results, such as the correlation coefficient, real RMS (root mean square error), and normalized real RMS between the predicted and measured geotechnical parameters for the training data, and they can be used as quality indices for the optimum selection of the parameterization of the I.S.G.E. algorithm.

The correlation coefficient (CC) indicates the degree of the model’s correlation with reality regarding the distribution of model values. The correlation coefficient of two random variables, A and B, is a measure of their linear dependence and is calculated based on Equation (4):

$$\mathit{C}\mathit{C}={\displaystyle \frac{1}{\mathit{N}-1}}{\sum }_{\mathit{n}=1}^{\mathit{N}}({\displaystyle \frac{{\mathit{A}}_{\mathit{i}}-{\mathit{\mu }}_{\mathit{\alpha }}}{{\mathit{\sigma }}_{\mathit{\alpha }}}}\left)\right({\displaystyle \frac{{\mathit{{\rm B}}}_{\mathit{i}}-{\mathit{\mu }}_{\mathit{\beta }}}{{\mathit{\sigma }}_{\mathit{\beta }}}}).$$

(4)

A denotes the values of the predicted model, B denotes the real given values at the borehole, μ_α and σ_α are the mean and standard deviation of A, respectively, and μ_β and σ_β are the mean and standard deviation of B.

The real RMS (RMSr) is an index showing how the amplitudes of the predicted model values converge towards the amplitudes of the actual measured values at each borehole. The real RMS is calculated based on Equation (5):

$$\mathit{R}\mathit{M}\mathit{S}\mathit{r}=\sqrt{{\displaystyle \frac{1}{\mathit{N}}}{\sum }_{\mathit{n}=1}^{\mathit{N}}{\left|{\mathit{A}}_{\mathit{n}}-{\mathit{B}}_{\mathit{n}}\right|}^2},$$

(5)

where A denotes the values of the predicted model, and B denotes the real given values at each borehole.

Normalizing RMSr facilitates the comparison between datasets or models with different scales. The normalized Rms, NRMSr, is calculated based on Equation (6):

$$\mathit{N}\mathit{R}\mathit{M}\mathit{S}\mathit{r}=\mathit{R}\mathit{M}\mathit{S}\mathit{r}/(\mathit{max}\left(\mathit{A}\right)-\mathit{min}\left(\mathit{A}\right)).$$

(6)

The equation results in a value between 0 and 1 (0–100%), where values closer to 0 represent better-fitting models.

3. Application of I.S.G.E. to a Synthetic Dataset

A feasibility study is the most efficient way to assess the capability of our algorithm to resolve a possible geological/geotechnical scenario. In this study, the geotechnical scenario is inspired by a real case study concerning the subsurface lithological formation that was faced during the design and construction phase of an underground hazardous waste repository at the Lavrion Technological and Cultural Park (LTCP) in Greece (Figure 2). The underground hazardous waste repository of LTCP is the first of its kind in Greece and among the few underground repositories worldwide [63].

The construction of realistic synthetic datasets simulating the subsurface conditions in the Lavrion area gives us the possibility to evaluate the effectiveness of the ISGE algorithm concerning this type of structure. The main lithological formations that were drilled by the investigation boreholes are (a) clay, (b) greenschist–phyllite, (c) graphic schist, (d) alterations of schist–limestone, (e) limestone of tectonic cover, and (f) limestone marble. Based on the available data, we created a simplified ground truth lithological/geotechnical model, with a significant graphic schist intrusion in the limestone marble (a problem that was encountered during the construction phase) approximately at the middle of the 2D section.

In order for our synthetic dataset to be as realistic as possible, the 2D variance for each lithological unit was calculated based on the statistical indices (mean, maximum, minimum, and standard deviation) of three different geotechnical parameters (unconfined compressive strength—UCS; point load test strength—Is; elastic modulus—E ) derived from the analysis of the actual measurements in eight boreholes drilled during the design phase of the underground hazardous waste repository. Due to the absence of measured parameters for the clay formation, we manually assigned the lowest value corresponding to all geotechnical parameters. As an example, we present the 2D ground truth model regarding the spatial distribution of the unconfined compressive strength (UCS) in Figure 3.

Using an empirical relationship that correlates the unconfined compressive strength with resistivity [64], we first obtain the equivalent resistivity model and simulate the application of the 2D resistivity tomography method. Using a 2D resistivity forward code [65], we calculated the synthetic apparent resistivity for a mixed dipole–dipole and Wenner–Schlumberger array for a 72-electrode configuration with 5 m interstation distance. Figure 3 depicts the apparent resistivity map after adding 5% Gaussian noise. These measurements were used as input in the Resd2dinv v3.54 [66] resistivity inversion software.

As input data in the I.S.G.E. algorithm, we used the resistivity inverted model and two synthetic boreholes approximately at the edges of the model, as shown in Figure 4. For the synthetic boreholes, we follow two different approaches. The first one uses the values of the ground truth model with a 1 m step regarding the depth resolution. The second approach, which gave more robust results and is presented in Figure 4, uses the target geotechnical parameter’s mean values (

Table 1. Statistical parameters of the lithological units used as modeling input, with UCS being the unconfined compressive strength.

Lithology	UCS (Mpa)
	Min	Max	Mean	Median	STD
clay	-	-	-	-	-
Greenschist–Phyllite	8	80	43	48	28
Graphic Schist	22	70	43	38	19
Alteration of Schist–Limestone	17	68	39	35	26
Limestone of tectonic cover	38	83	62	61	12
Limestone Marble	39	117	71	75	128

) for each lithological unit as input data. The second approach can approximately correspond to a possible 1D representative geotechnical model in each borehole that could probably be suggested by geotechnical experts based on available information from the site investigation program. As we can observe in the comparison of the I.S.G.E.-predicted section of the UCS distribution (Figure 4) with the ground truth model (Figure 3), the effectiveness of the algorithm is significantly dependent on the resolution of the applied geophysical method. In any case, it is very difficult to obtain the actual variance in each geotechnical unit of the ground truth model.

However, when we compare it with the equivalent ground truth geotechnical model that uses the mean value of the target parameter for each geotechnical unit, the images are in very good agreement. Similar results were also obtained for the other geotechnical model parameters (Is and E) that were tested. It is noteworthy that the proposed methodology takes advantage of the capability of geophysical methods to efficiently retrieve the relative spatial distribution of a measured geophysical parameter rather than its absolute value.

4. Application of I.S.G.E. Tool at a Controlled Test Site

The evaluation of the proposed methodology in real conditions was carried out at a test site at the Campus of the National Technical University of Athens (Figure 5). In the adjacent area of the test site, a geotechnical survey was implemented in 2009, and it is related to the construction of new university facilities. Based on the geotechnical data, three main lithological formations are expected in the broader area of the test site: (a) a gravelly sand layer of 4.3 m thickness, (b) a clayey sand layer of 3.9 m thickness, and (c) a conglomerate (rock formation) up to 12 m depth (end of the boreholes). In addition, a “cut-and-cover” tunnel, which serves university facilities, crosses the test site. Based on the construction plans provided by the University’s Technical Department, the reinforced concrete shell is 2 m high and 2 m wide, and its bottom is buried at an approximate depth of 4 m. Regarding borehole information, the 1D geotechnical model for the new University Building area is available, incorporating the values put forward by the geotechnical experts for cohesion (c’), the friction angle (φ₀), the modulus of elasticity of soils (Es), the Poisson ratio (v), the dry unit weight γd, the total unit weight (γt), and Standard Penetration Test (SPT) blow counts (N values).

Moreover, an integrated geophysical investigation was carried out in the same area, providing more insight into the subsurface conditions. In this context, a 3D resistivity tomography with various electrode arrays was conducted, as well as 3D seismic acquisition for the exploitation of first arrivals and surface waves. Both the resistivity and seismic datasets were acquired using a fully 3D approach exploiting geophysical equipment with a single 3D acquisition scheme. Resistivity measurements were acquired by a single setup of 72 electrodes in a pure 3D layout (3 m inter-electrode distance), using the Syscal Pro system of Iris Instruments with dipole–dipole, Wenner–Schlumberger, and pole–pole arrays. For seismic acquisition, an almost rectangular dense grid of 60 vertical 4.5 Hz geophones was deployed, with an interstation distance of about 3 m. A 60-channel seismograph (StrataView Geometrics) was used for acquiring data generated by 68 seismic shots, performed with a weight drop (GISCO ESS-MINI-BASIC Electronic Seismic Source). A sampling rate of 2000 sps with a record length of 1024 ms was applied for all shot gathers. A total of 34 different shot positions around and inside the grid ensured an azimuthally acceptable coverage of the sources in relation to the various geophone pairs. More details about the acquisition of geophysical data can be found in Leontarakis et al. [67]. The final 3D resistivity model was calculated based on the MOST technique [68] by inverting and processing data from different resistivity measurement arrays (two-, three-, and four-electrode arrays). Res3dinv v2.14 software was used for the inversion of resistivity data. The analysis of the seismic dataset initially resulted in a 3D P-wave velocity (Vp) model of the area using the P-wave first arrival travel time tomography method via the psTomo 3D raytracing tomography code [69] and cell sizes of 1 m in the x, y, and z directions. In addition, a 3D shear-wave velocity (Vs) model was estimated using a surface wave tomography approach proposed by Orfanos et al. [70] via the analysis of the same seismic dataset. As a result, the input data for the I.S.G.E algorithm consist of (a) the 3D geophysical tomographic models (P-wave velocity, S-wave velocity, and resistivity models) and (b) the 1D geotechnical model assessed by borehole data, assuming that they are representative of the test site area.

4.1. Multiparameter Analysis for the Creation of a Unified Geophysical Model

In the processing workflow of the I.S.G.E. methodology (Figure 1), an option for optimizing the geophysical input data using the UGM approach exists. This approach leads to the estimation of updated fuzzy models for each geophysical method that can be used as input geophysical models. In this study, we further evaluate the effectiveness of the UGM approach by estimating the correlation coefficient between the average 1D model of each geophysical method and the distribution of the 1D geotechnical model for each target parameter. We conclude that the updated fuzzy geophysical models (Figure 6) generally have a higher correlation with the geotechnical parameters compared to the initial inverted geophysical models. This is clearly shown in

Table 2. The value of the correlation coefficient (the absolute value) between the initial and fuzzy updated geophysical models of Vp, Vs, and resistivity ρ with available geotechnical data.

Correlation Coefficient Values (%)
	Initial Inverted				Updated Fuzzy
	Vp	Vs	ρ	Average	Vp	Vs	ρ	Average
Cohesion	85	86	44	72	85	84	80	83
Friction angle	30	44	7	27	37	39	27	34
Elastic modulus	63	72	28	54	67	67	58	64
Poisson ratio	63	72	27	54	67	67	58	64
SPT	87	76	53	72	81	79	83	81

, where the correlation between geophysical models (initial and updated) and the available geotechnical data is presented for various geotechnical parameters. We observe an improvement in both the individual geophysical models and in the average correlation coefficient across all models.

The main advantage of this approach is that we can obtain an unbiased integrated model and the respective fuzzy updated models of each geophysical model through a fully automated statistical procedure. The application of more than one geophysical method in the same survey area is very beneficial as it reduces the uncertainty of the interpretation. It is very important to have an integrated model delineating the main features of the subsurface. For this reason, in the case where no borehole information exists, in the I.S.G.E. algorithm, the option of UGM model creation is used by default.

4.2. I.S.G.E. Application with Incorporation of Geotechnical Information

When borehole data exist in a survey, the I.S.G.E. algorithm, with the incorporation of available geotechnical information into the multiparameter analysis process, aims to estimate the spatial distribution of geotechnical parameters. This is achieved by utilizing both geotechnical and geophysical methods, without the need for predefined equations linking the measured geophysical parameters to the desired geotechnical parameters. As input geophysical models in the I.S.G.E. algorithm, the updated fuzzy ones were used in this case study, and the output is the spatial distribution of a target geotechnical parameter. The output for two different geotechnical parameters (SPT and Cohesion) is presented as an example in Figure 6. As can be observed in both cases, it is possible to assess their 3D distribution by exploiting the point information (1D) of the representative geotechnical model. The predicted values are in very good agreement with the given values in the borehole (training data), with correlation coefficients for the SPT and Cohesion values of 95% and 88%, respectively.

In this case study, although the boreholes are outside the survey layout of geophysical measurements, with the I.S.G.E. algorithm, it was still possible for the useful geotechnical information to be incorporated in the analysis. Moreover, the utilization of the fuzzy updated models as input data has significantly improved the final result. They resulted in predicted output models with stronger correlations for the training data and lower values for the RMS and NRMS statistical indices between the predicted values and the real values in the boreholes (

Table 3. Comparison of the values of three statistical indices (correlation coefficient, RMSr, and NRMSr) between the predicted values and the real values in the boreholes using the initial inverted geophysical models or the updated fuzzy ones as input data.

Statistical Indices Values
	Initial Inverted			Updated Fuzzy
	CC (%)	RMSr	NRMSr (%)	CC (%)	RMSr	NRMSr(%)
Cohesion	87	5.35	22	88	4.97	19
Friction angle	73	1.53	41	75	1.45	29
Elastic modulus	81	7.3	34	81	7	27
Poisson ratio	81	0.012	34	81	0.011	28
SPT	86	7	26	95	4.6	17

). This also suggested the need to add to our algorithm the option of weighting the input models based on their effectiveness in the initial stage or even in the stage of creating fuzzy sets. This can be achieved by estimating the correlation coefficient between the values of the geophysical models and the target parameter in the respective boreholes and their incorporation in the analysis as a weight factor.

5. Implementation of I.S.G.E. in a Challenging Urban Environment at the Center of Athens

Considering the importance of subsurface soil characterization in urban areas in geotechnical and geological engineering projects, our main aim was to apply and evaluate the proposed methodology to a convenient real geotechnical application in the city of Athens that could meet specific requirements. These requirements concern (a) the availability of at least two boreholes inside the survey grid of the geophysical measurements; (b) reliable laboratory measurements and in situ geotechnical tests; (c) a challenging environment simulating the difficulties that can be faced by geophysical methods when applied in urban areas; and (d) an adequate divergence in stratigraphy and lithology conditions between the two or more available boreholes in the possible survey area. Having in mind the above requirements, a suitable location was selected. The main target was to give an answer to a specific engineering problem related to the rehabilitation and construction of new facilities in an area located at the center of Athens, where an existing building is lying. The lack of free space, the presence of an existing building, the cover of the surrounding space with concrete, tiles, or even asphalt in some areas, as well as the numerous possible sources of noise (cables, traffic, etc.), make the application of any geophysical method very challenging in such an environment (Figure 7). A great challenge for us was not only the very difficult conditions for acquiring the necessary geophysical data with respect to the signal-to-noise ratio (S/N) but also revealing the subsurface formation and its geotechnical properties under the existing building.

5.1. Description of Survey Area

In the context of the rehabilitation works for the existing building, a geotechnical investigation was carried out for the assessment of subsurface conditions. Based on the lithological formations revealed by two boreholes of 15 m depth, as well as the geotechnical in situ and laboratory tests, a representative 2D geotechnical cross-section (Figure 7) was constructed by the geotechnical experts, along with the proposed geotechnical design parameters (

Table 4. Geotechnical design parameters of the survey area.

Geotechnical Design Parameters	UnitI	Unit IIa	UnitIIb	Unit IIc
Unit Weight γ (KN/m3)	20	23	24	21
Friction Angle φ’ (ο)	28	31	36	33
Cohesion c’ (KN/m2)	17	20	400	10
Elastic Modulus E (Mpa)	8.3	20	450	50

). According to these results, a superficial sandy clay dipping to the north overlays a gravelly sandy clay layer containing lenses of conglomerate. Below these, at a depth of about 7.5 m, a layer of conglomerate with increasing thickness to the south overlays a layer of well-graded clayey gravels. Finally, a possible conglomerate bedrock could be located at a depth of 14–15 m, but it is not fully supported by the boreholes’ sampling. However, the design of this cross-section is highly based on the subjective judgment and the experience of the geotechnical engineer, with a higher degree of uncertainty as the distance from the boreholes increases.

5.2. Quasi-3D Vs Velocity Estimation Using Common-Mid-Point Cross-Correlation Stacking Surface Wave Tomography

Surface wave tomography can be an attractive alternative as a non-destructive investigation tool in geotechnical applications for calculating both in-depth and lateral variations of shear-wave velocity in a survey area [70]. Shear-wave velocity can be directly calculated by surface wave analysis and constitutes an important characterization parameter in geotechnical standards that is related to the elastic shear modulus. For this purpose, an almost arbitrary but dense layout of 60 vertical 4.5 Hz geophones was deployed, with a mean inter-station distance of about 5–6 m, covering the available free space as much as possible, as shown in Figure 8. A 60-channel seismograph, StrataView by Geometrics, was used for acquiring data generated by a weight drop (GISCO ESS-MINI-BASIC) seismic source. A sampling rate of 0.5 ms with a record length of 1024 ms was used for all shot gathers. A total of 37 different shot positions around and inside the grid ensured an azimuthally acceptable coverage of the sources with respect to the various geophone pairs. The exact coordinates of sources and receivers positions were measured using an RTK GPS Leica system.

After building the final frequency-dependent group and phase velocity maps according to Leontarakis et al. [67], a joint group and phase 1D inversion was applied to every pair of estimated local dispersion curves. Figure 9 depicts the derived quasi-3D Vs model of the survey area (mean RMS error less than 5%), along with the depth section crossing the two available boreholes G1 and G2. As can be observed, the main lithological formations are well delineated, providing the geotechnical engineers with an auxiliary interpretation tool for the assessment of subsurface conditions using a 3D approach.

5.3. Application of I.S.G.E. Algorithm

The fully automated I.S.G.E algorithm is implemented in this case study to predict the spatial distribution of the survey area’s geotechnical design parameters. The prediction models are calculated for each geotechnical parameter separately, incorporating the available geotechnical information in the analysis. The 3D Vs model was exclusively used as the geophysical input data in the algorithm, as the very difficult acquisition conditions in the survey area only allowed this tomography method to be applied. As borehole input data, the values of the design parameters for each geotechnical zone drilled by the G1 and G2 boreholes were used. Additionally, the in situ measured SPT N values at each geotechnical zone were also used. In Figure 10, the predicted 3D models for two different geotechnical parameters and the respective sections crossing G1 and G2 are presented as an example. The predicted models of the target parameters in both cases propagate the point information (1D) of the borehole’s geotechnical model to 3D space and provide reasonable information in comparison with the expected subsurface conditions. The values of the predicted model have a strong correlation with the given values in the borehole, with mean correlation coefficients for the friction angle and SPT values of 79% and 97%, respectively. RMSr and NRMSr are 1.6 and 22% for the friction angle and 8 and 10% for the SPT values, respectively.

The final 3D model can be provided as a simple ASCII file for further analysis with geotechnical software, or a specific subset of the main 3D cube can be selected and analyzed. In Figure 11, the cross-sections that represent the line between the two boreholes are presented, providing the engineer with a fully automated unbiased statistical prediction model for the friction angle and SPT. The interpreter is able to get an idea about the lateral variation of a desired geotechnical parameter that he or she has defined in a borehole. Specifically, in this study, because of the application of 3D Common-Mid-Point Cross-Correlation Stacking Tomography, it was possible to obtain a reasonable prediction model even under the existing building, despite the lack of free space to place geophones.

6. Validation and Effectiveness Assessment of I.S.G.E. Algorithm

A new Integrated Spatial Geophysical and Geotechnical evaluation tool (I.S.G.E.) is proposed in this study, and it was initially evaluated via a synthetic dataset and further tested at a test site under real field conditions using a 3D integrated geophysical investigation.

The synthetic data example shows that, using the specific methodology, it was possible to reconstruct (with only two boreholes) a prediction model in very good agreement with the mean geotechnical ground truth model (Figure 4). We also observed that the effectiveness of the algorithm is significantly dependent on the resolution of the applied geophysical method. In addition, the implementation of the I.S.G.E algorithm in a challenging urban area before the rehabilitation of an existing building in the city of Athens, Greece, showed very encouraging results, providing 3D geotechnical models with good agreement with the available boreholes (Figure 5 and Figure 6).

The derived 3D models extrapolate the point geotechnical information from 1D to 3D, without requiring predefined equations linking the measured geophysical parameters to the desired geotechnical parameters. The predicted models depict the spatial distribution of the target geotechnical parameters, even under the existing building (Figure 10, Figure 11 and Figure 12). Nevertheless, the aforementioned diagram deals with the prediction’s efficiency corresponding to the “training data” of our automatic algorithm. It is very important for the I.S.G.E. algorithm, which is based on the accurate geophysical model’s spatial distribution, to be able to predict models that are well correlated with the available geotechnical data that are used in the procedure. However, some key questions remain about the performance of the algorithm in new, unexplored positions regarding drilling. The questions raise issues, such as the following: “What could be the degree of correlation between I.S.G.E. results and a new borehole?” or “What is the level of uncertainty in our approach?”

In order to further quantify the effectiveness of our technique, we consider the G1 borehole as “known” and evaluate the effectiveness of the I.S.G.E. algorithm in predicting the results of the G2 borehole and vice versa. In Figure 13, we present the results for two different geotechnical parameters, the friction angle and the SPT blow counts, after only using the information from the G1 borehole as “training data”. The G2 borehole data serves as the ground truth model for comparison with the output of the I.S.G.E. algorithm. As we can observe, the results are very encouraging since the algorithm was able to predict the target parameters in the G2 borehole with adequate correlation in both cases.

After estimating (a) the correlation between each geotechnical parameter and each geophysical model for each borehole, (b) the correlation of the I.S.G.E. output with the respective geotechnical input data for the “training data”, and (c) the correlation of the I.S.G.E.’s predicted output with the “ground truth” borehole for each geotechnical parameter, we constructed two different plots (Figure 13: upper and lower panel) that focus on the influence of the correlation between training data and input geotechnical models, as well as the correlation between input geophysical models and input geotechnical models, on the accuracy and reliability of the I.S.G.E. algorithm’s output.

7. Discussion and Conclusions

A new Integrated Spatial Geophysical and Geotechnical evaluation (I.S.G.E) method was developed to estimate the spatial distribution of geotechnical parameters using high-resolution geophysical methods. The proposed algorithm is based on fuzzy logic, and the final output is the prediction of the two- or three-dimensional distribution of a geotechnical parameter in a survey area. The ISGE methodology translates geophysical models into engineering information, enabling engineers to examine how subsurface conditions continue between boreholes.

Despite its effectiveness, the I.S.G.E. methodology also exhibits certain constraints. The quality of the input data (geophysical models and geotechnical measurements), as well as the degree of correlation between the measured geophysical and the target geotechnical parameter, is critical to the method’s effectiveness. As expected, the stronger the correlation between the geophysical measurements and the target parameter, the more accurate the predictions. Errors or inconsistencies in input data can significantly impact the predicted model. Furthermore, the resolution of the input geophysical inverted models directly constrains the achievable resolution of the predicted geotechnical parameters. The method also assumes relatively uniform subsurface conditions that are adequately represented by the available boreholes, which may not be valid in areas with complex or rapidly changing geology.

It should be emphasized that this methodology is not intended to replace geotechnical investigations under any circumstances. Instead, it serves as an additional interpretation tool that can be used in several ways. Specific subsets or cross-sections can be extracted from the 3D output and directly compared with cross-sections derived from geotechnical investigations. In this way, engineers can confirm, refine, or expand their interpretations. The I.S.G.E. output can also be exported for numerical modelling in standard engineering software. Because the 3D spatial variation of geotechnical properties strongly influences the interaction between an engineering structure and the ground, a clearer understanding of this variation can reduce both uncertainty and cost. Conversely, a poorly scoped geotechnical investigation and inadequate reporting can greatly increase project costs.

Applying I.S.G.E. to additional case studies will broaden our knowledge and database regarding the method’s uncertainty. This will allow us to produce more confident trends in diagrams such as Figure 13, providing us with a better overview of the associated uncertainty. The correlation coefficient index already offers insight into the similarity between observed and predicted data distributions. Moreover, incorporating statistical indices such as the root mean square (RMS) error and the normalized RMS error in I.S.G.E. output plots for the training data can further quantify the uncertainty associated with the predicted parameter values. Future research will focus on addressing several issues, including the differences in resolution and scale between geophysical and geotechnical data, as well as the presence of minor artifacts that may arise from overfitting during the calibration process. Additionally, the incorporation of uncertainty from both geophysical and geotechnical methods into the analysis will be examined. Finally, it would be of great interest to compare the predicted models derived from the I.S.G.E. methodology with those generated by other artificial intelligence and data fusion approaches. Such comparisons, conducted on the same datasets and supported by drilling verification, would offer valuable insights into the relative performance and reliability of each method.

The fully automated output of the I.S.G.E algorithm, which uses an unbiased statistical process based on fuzzy logic, provides engineers with an additional interpretation tool for understanding subsurface conditions and reducing spatial uncertainty within the area under investigation. By translating geophysical data into geotechnical parameters that engineers can readily interpret, the method promotes the integration of geophysical surveys into standard geotechnical investigation programs.