Approaches Towards a Probabilistic Assessment of
Geotechnical Parameter Distributions Relying on
Geophysical Imaging
Hendrik PAASCHEUFZ – Helmholtz Centre for Environmental Research, Permoserstr. 15, 04318 Leipzig, Germany, hendrik.paasche@ufz.de
Abstract. Geophysical imaging techniques, such as seismic crosshole tomographic methods, provide spatially continuous
information about physical parameter variations in the ground. This physical parameter information, e.g., distribution of seismic velocities, is frequently used to estimate spatial distributions of geotechnical target parameters that could only be calibrated by sparse measurements in boreholes. The employed geophysical model generation procedure is usually deterministic, which does not allow for transducing the ambiguity inherent to the geophysical imaging procedure into the prediction of spatial geotechnical target parameter distributions. Here, we review self-organizing geophysical inversion of tomographic data allowing for data-driven ambiguity assessment of geophysical models. Opposite to deterministic geophysical imaging resulting in a single geophysical model this approach delivers ensembles of different but equally plausible geophysical models. We use this probabilistic information about geophysical model uncertainty and employ fuzzy cluster analysis for propagating the geophysical model generation uncertainty into probabilistic inference of spatial geotechnical parameter distributions exemplary illustrated by generating 2D scenarios of sleeve friction distribution.
Keywords. Self-organizing inversion, ambiguity, cone penetration test, crosshole tomography, cluster analysis, prediction,
probabilistic, sleeve friction
1. Introduction
Geotechnical engineers employ highly specialized techniques to gather information about subsurface properties relevant for the utilization of the ground for construction work. Usually, relevant information underlying educated decisions of the geotechnical engineer is collected by probing the ground using invasive techniques, such as drilling boreholes or direct push probing. In doing so, high-resolution information about various mechanical, chemical or hydrological properties of the probed ground are collected providing detailed information about vertical property variability at distinct locations. A full 2D or 3D imaging of relevant geotechnical property variations is usually not possible.
Geophysical tomography offers the unique capability to image the subsurface spatially continuously. Physical parameter variations, such as propagation velocities of mechanical waves, are imaged and provide information about the
spatial heterogeneity of the ground described by physical parameters. Unfortunately, the limited number and accuracy of geophysical data usually result in non-uniqueness of the geophysical model generation procedure. In most cases, the inversion of geophysical data is regularized using mathematical constrains, such as damping or smoothness, and carried out deterministically. This means that one geophysical model that compromises the geophysical observations and constrains is achieved at the end of the model generation procedure. Numeric assessment of the ambiguity inherent to geophysical tomographic imaging is usually not possible.
Theoretical and empirical studies have shown that physical parameters, which can be imaged by geophysical tomography are often linked to geotechnical property variations of relevance for the geotechnical engineer. However, parameter relations are usually site specific, often non-linear or even non-unique. Nevertheless, many deterministic transfer functions have been developed to convert
© 2015 The authors and IOS Press. This article is published online with Open Access by IOS Press and distributed under the terms of the Creative Commons Attribution Non-Commercial License. doi:10.3233/978-1-61499-580-7-898
physical parameters into geotechnical properties and vice versa. Selection of a suitable transfer function is usually difficult and requires on-site calibration. If deterministic geophysical model generation is combined with deterministic transfer functions to convert the imaged physical parameter variations into geotechnical property information, this will be deterministic, too. Here, we illustrate a different approach building on fully non-linear inversion of geophysical tomographic data resulting in ensembles of plausible geophysical models. We describe the spatial heterogeneity of all models in a fuzzy sense and calibrate the fuzzy pattern using sparse sleeve friction information achieved from a direct push cone penetration test (CPT). Finally, we generate thousands of different scenarios for the spatial distribution of sleeve friction all honoring the underlying information provided by the ensembles of geophysical tomographic models.
2. Fully non-Linear Geophysical Tomography A number of techniques have been developed to invert geophysical tomographic data sets non-deterministically. The computational effort is usually significantly higher compared to deterministic approaches. In most cases, geophysicists strive to reduce the computational effort by constraining the maximal complexity of the physical parameter variability in the ground by describing the subsurface either by a few polygons or a limited number of layers (e.g., Velis, 2001; Roy et al. 2005). Such subsurface parameterization requires a priori knowledge about the spatial heterogeneity of the subsurface, which must be guessed in most cases. Nevertheless, these approaches generally allow for assessing the tomographic ambiguity as function of limited model parameterization (e.g., Tronicke et al. 2012). The transdimensional tomography of Bodin et al. (2012) is exceptional, since it employs a self-adapting triangular mesh allowing for data-driven highly flexible subsurface parameterization. Deterministic transfer functions can be set up and used to convert the resultant ensembles of geophysical models into ensembles of geotechnical target parameters (e.g., Rumpf and Tronicke, 2014).
2.1. Self-Organizing Inversion
In this study, we employ a fully non-linear inversion procedure using particle swarm optimization (PSO; Kennedy and Eberhart, 1995) when searching the geophysical model solution space. We formulate the inverse problem in a fuzzy domain initially setting expected upper and lower boundaries for physical parameter values in the tomogram. During the inversion procedure, these settings can be refined or moved based on learning of the inversion. The model area is parameterized using a finite difference mesh consisting of rectangular grid cells with lateral dimensions not exceeding the physical spatial resolution limit of the tomographic data set. When searching for plausible geophysical models explaining the observations we strive to minimize a data objective function, e.g., in the form of an rms-error measuring the distance between data vector and model response as well as a structural constrain applied in the fuzzy domain avoiding the occurrence of implausible solutions, e.g., a checkerboard-like variability of physical property values. Both objectives are balanced using a game theoretic decision strategy (e.g., Paasche and Tronicke, 2014). When selecting the final models, the data objective is the only decision criterion, which allows creating an ensemble of final models with equal rms-error but variable model parameter amplitudes and spatial heterogeneity.
3. The Field Site and Database 3.1. Field Site
The field site is located 30 km south of Berlin. The German Federal Institute for Materials and Testing maintains the area. In 2004, a small test site for non-destructive testing of foundations was established comprising several concrete pile foundations and a number of PVC-cased boreholes reaching down to approximately 17 m (Figure 1). The local geology is characterized by glacial and fluvioglacial sediments of prominent sand and gravel composition with interbedded thin layers of medium gravel and organic deposits below 8 m depth (Linder et al., 2010).
Figure 1. Location of boreholes, concrete piles and cone
penetration tests. CPT location at x=2.75 m corresponds to DP1. CPT location at x=8 m corresponds to DP2 (Linder et al., 2010).
3.2. Seismic Crosshole Tomography
P- and S-wave crosshole tomographic data were recorded between boreholes BH2, BH3, and BH5 (Figure 1). P-wave data were acquired using a sparker source and 24 hydrophones placed below the ground water level with vertical source and receiver spacing of 0.25 m. S-wave data were recorded using a borehole impactor source and two 5-component borehole geophones pneumatically coupled to the borehole casing. Vertical source and receiver spacing was 0.5 m. Dominant frequencies in the P- and S-wave experiments were 750 Hz and 200 Hz, respectively. First cycle onsets have been picked
Figure 2. P-wave velocity distributions of 30 equivalent 2D models achieved by fully non-linear self-organizing inversion. Each
2D model consists of rectangular velocity cells with 0.5 m vertical side length and 1 m lateral side length. The 30 black lines in the left panel show the P-wave velocity distribution at the left model edge between x=0 m and x=1 m. The left model edge with x=0 m corresponds to BH2 (see Figure 1).
Figure 3. S-wave velocity distributions of 30 equivalent 2D models achieved by fully non-linear self-organizing inversion. Each
2D model consists of rectangular velocity cells with 0.5 m vertical side length and 1 m lateral side length. The 30 black lines in the left panel show the S-wave velocity distribution at the left model edge between x=0 m and x=1 m. The left model edge with x=0 m corresponds to BH2 (see Figure 1).
resulting in a P-wave and S-wave crosshole tomographic data set (Linder et al., 2010). We employed self-organizing inversion and parameterized the ground into rectangular grid cells with horizontal and vertical side lengths of 1 m and 0.5 m, respectively. Thus, a 2D geophysical velocity model comprises 23 grid cell rows in vertical direction and 11 cell columns in horizontal directions. BH2 corresponds to x=0 m, and BH5 corresponds to x=10.9 m in the tomographic model. We calculated 30 P-wave and 30 S-wave velocity models. All P-wave and S-wave models are illustrated in Figures 2 and 3, respectively. 3.3. Cone Penetration Tests
Cone penetration tests have been carried out at two locations between BH2 and BH3 at x=2.75 m, as well as between BH3 and BH5 at x=8.0 m (Linder et al., 2010). Direct push sleeve friction data are shown in Figure 4 with a vertical sample distance of 1 cm.
4. Inference of Spatial Distribution Scenarios for Sleeve Friction
4.1. Describing the Tomographic Heterogeneity Fuzzy partitioning cluster analysis is an iterative technique that enables the integrated description of multiple geophysical models by a fuzzy membership matrix and a corresponding cluster center matrix (e.g., Paasche et al. 2006). The elements of the membership matrix are dimensionless. The number of rows equals the number of clusters and the number of columns equals the number of samples present in an n-dimensional parameter space with orthogonal axes. Here, we span a two-dimensional parameter space with P-wave and S-wave velocities normalized to the interval [0 1] scaled along the two orthogonal axes (Figure 5). We populate the parameter space considering every possible pair of P- and S-wave models resulting in 900 different P- and S-wave velocity model combinations. Each of these model pairs comprises 23 x 11 grid cells with P- and S-wave information. The parameter space illustrated in Figure 5 shows the distribution of all 900 x 253
P- and S-wave velocity samples. For describing the information in this parameter space in a fuzzy sense we define 23 cluster centers. Four of them are located at the corners of the two-dimensional parameter space according to minimal and
Figure 4. Sleeve friction data.
maximal velocities. The other 19 cluster centers are randomly positioned. The 23 normalized P- and S-wave velocities defining the position of the cluster centers are the row vectors of the cluster center matrix, which is of dimension 23 x 2. When knowing the cluster center locations in the parameter space we can compute the corresponding membership matrix assuming spherical clusters as used in a fuzzy c-means cluster algorithm (Höppner et al. 1999). The membership matrix comprises 23 rows and 900 x 253 columns. Every sample present in the parameter space is described by a 23 element column vector in the fuzzy domain whose elements sum to unity. Element (1,1) of the membership matrix describes the degree of membership of sample one (top left corner in the velocity grid in the model domain) defined by P- and S-wave models one and one, respectively, to cluster one with its mean values defined by the first row in the center matrix.
Figure 5. Parameter space spanned by all 30 x 30 P- and
S-wave velocity models. Black crosses indicate a sample position and gray circles illustrate the positions of the cluster centers.
4.2. Calibrating the Attribute Matrix
The cluster center matrix describes the physical attributes (P- and S-wave velocity) in relation to the dimensionless fuzzy membership information stored in the membership matrix. We assume that either one or both geophysical data sets capture information that is somehow related to the geotechnical target parameter, which is sleeve friction in our example. We calibrate the fuzzy information in the membership matrix by adding a column to the center matrix assigning mean values of sleeve friction to each cluster center. In doing so, we set up and solve a linear system of equations of the type Ax=b. The column vector x comprises 23 unknown elements of sleeve friction values typical for each cluster center. The first 23 elements of column vector b comprise 23 sleeve friction values taken from sleeve friction logging data shown in Figure 4. For example, we use 23 different samples of the log at x=2.75 m which fall in the 23 different grid cells of the geophysical tomogram at x=2.75 m. This set of 23 sleeve friction calibration data is repeatedly added to b resulting in a column vector with 900 x 23 elements. The coefficient matrix A is a 20700 x 23 matrix containing in the first 23 rows the membership information of the
23 grid cells at x=2.75 m for the first P- and S-wave velocity model pair. When solving A-1b=x we can add x as an additional column to the center matrix assigning a typical sleeve friction value to the 23 cluster centers. Here, we achieved this typical sleeve friction values by calibrating dimensionless pattern information expressed in the form of fuzzy cluster memberships with sparse sleeve friction calibration data. Note that an averaging over all geophysical models is performed when solving the over-determined system of equations.
4.3. Inference of 2D Sleeve Friction
By using the fuzzy membership information as linear weighting coefficients for the 23 sleeve friction values added to the attribute matrix we can infer 900 different scenarios for 2D distributions of sleeve friction on the scale of the tomographic models. Differences between these 900 2D sleeve friction models are related to the ambiguity of the geophysical model generation procedure. Furthermore, we face a scaling problem when calibrating the membership information with sleeve friction information, since multiple (noise-affected) sleeve friction readings in the logged data fall within each grid cell. Here, we recursively run the calibration procedure described in section 4.2 selecting different samples for calibration of the membership information of a distinct model grid cell. Thus, we can propagate high-frequency information from the logs (including the data noise) into the inference of 2D sleeve friction scenarios. Additionally, we do this for the position at x = 8 m employing the second sleeve friction log and the membership information at this location. Figure 6 illustrates the results of 90000 calculations of 2D sleeve friction models. Such data-driven probabilistic inference of spatial distributions of sparsely measured parameters may improve the decision making process of engineers heavily relying on subsurface imaging techniques and their inherent limitations not accessible when employing deterministic strategies.
Figure 6. Illustration of the probability density function of 90000 inferred scenarios of 2D sleeve friction distribution.
5. Conclusions
We used fully non-linear self-organizing inversion to achieve two ensembles of 30 equivalent geophysical models describing the ambiguity of crosshole P-wave and S-wave velocity distributions according to the performed crosshole tomographic experiments. We transformed the information of the geophysical models into a fuzzy domain expressing the model heterogeneity in a fuzzy sense. We calibrated the fuzzy matrix describing geophysical heterogeneity in the form of cluster memberships by using sparse information about sleeve friction achieved from cone penetration tests. Repeated calibration and utilization of all P- and S-wave models resulted in the inference of a large number of plausible 2D sleeve friction models. This approach does not require knowledge about or definition of the distinct relationships between the geotechnical target parameter and the underlying geophysical tomograms.
6. Acknowledgements
We are grateful to the German Federal Institute for Materials and Testing for access to the field site. We thank Marko Dubnitzki, Steffen Linder (both University of Potsdam), and Thomas Vienken (UFZ Leipzig) for support during the field work.
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