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The fuzzy partitioning Gustafson-Kessel cluster algorithm is employed for rapid and objective integration of multi-parameter Earth-science related databases. We begin by evaluating the Gustafson-Kessel algorithm using the example of a synthetic study and compare the results to those obtained from the more widely employed fuzzy c-means algorithm. Since the Gustafson-Kessel algorithm goes beyond the potential of the fuzzy c-means algorithm by adapting the shape of the clusters to be detected and enabling a manual control of the cluster volume, we believe the results obtained from Gustafson-Kessel algorithm to be superior. Accordingly, a field database comprising airborne and ground-based geophysical data sets is analysed, which has previously been classified by means of the fuzzy c-means algorithm. This database is integrated using the Gustafson-Kessel algorithm thus minimising the amount of empirical data processing required before and after fuzzy c-means clustering. The resultant zonal geophysical map is more evenly clustered matching regional geology information available from the survey area. Even additional information about linear structures, e. g. as typically caused by the presence of dolerite dykes or faults, is visible in the zonal map obtained from Gustafson-Kessel cluster analysis.

Crosshole traveltime tomography using particle swarm optimization a near-surface field example
(2012)

Particle swarm optimization (PSO) is a relatively new global optimization approach inspired by the social behavior of bird flocking and fish schooling. Although this approach has proven to provide excellent convergence rates in different optimization problems, it has seldom been applied to inverse geophysical problems. Until today, published geophysical applications mainly focus on finding an optimum solution for simple, 1D inverse problems. We have applied PSO-based optimization strategies to reconstruct 2D P-wave velocity fields from crosshole traveltime data sets. Our inversion strategy also includes generating and analyzing a representative ensemble of acceptable models, which allows us to appraise uncertainty and nonuniqueness issues. The potential of our strategy was tested on field data collected at a well-constrained test site in Horstwalde, Germany. At this field site, the shallow subsurface mainly consists of sand- and gravel-dominated glaciofluvial sediments, which, as known from several boreholes and other geophysical experiments, exhibit some well-defined layering at the scale of our crosshole seismic data. Thus, we have implemented a flexible, layer-based model parameterization, which, compared with standard cell-based parameterizations, allows for significantly reducing the number of unknown model parameters and for efficiently implementing a priori model constraints. Comparing the 2D velocity fields resulting from our PSO strategy to independent borehole and direct-push data illustrated the benefits of choosing an efficient global optimization approach. These include a straightforward and understandable appraisal of nonuniqueness issues as well as the possibility of an improved and also more objective interpretation.

Partitioning cluster algorithms have proven to be powerful tools for data-driven integration of large geoscientific databases. We used fuzzy Gustafson-Kessel cluster analysis to integrate Landsat imagery,. airborne radiometric, and regional geochemical data to aid in the interpretation of a multimethod database. The survey area extends over 3700 km(2) and is located in the Northern Cape Province, South Africa. We carefully selected five variables for cluster analysis to avoid the clustering results being dominated by spatially high-correlated data sets that were present in our database. Unlike other, more popular cluster algorithms, such as k-means or fuzzy c-means, the Gustafson-Kessel algorithm requires no preclustering data processing, such as scaling or adjustment of histographic data distributions. The outcome of cluster analysis was a classified map that delineates prominent near-to-surface structures. To add value to the classified map, we compared the detected structures to mapped geology and additional geophysical ground-truthing data. We were able to associate the structures detected by cluster analysis to geophysical and geological information thus obtaining a pseudolithology map. The latter outlined an area with increased mineral potential where manganese mineralization, i.e., psilomelane, had been located.

Inversions of an individual geophysical data set can be highly nonunique, and it is generally difficult to determine petrophysical parameters from geophysical data. We show that both issues can be addressed by adopting a statistical multiparameter approach that requires the acquisition, processing, and separate inversion of two or more types of geophysical data. To combine information contained in the physical-property models that result from inverting the individual data sets and to estimate the spatial distribution of petrophysical parameters in regions where they are known at only a few locations. we demonstrate the potential of the fuzzy c-means (FCM) clustering technique. After testing this new approach on synthetic data, we apply it to limited crosshole georadar, crosshole seismic, gamma-log, and slug-test data acquired within a shallow alluvial aquifer. The derived multiparameter model effectively outlines the major sedimentary units observed in numerous boreholes and provides plausible estimates for the spatial distributions of gamma-ray emitters and hydraulic conductivity

We integrate the information of multiple tomographic models acquired from the earth's surface by modifying a statistical approach recently developed for the integration of cross-borehole tomographic models. In doing so, we introduce spectral cluster analysis as the new core of the model integration procedure to capture the spatial heterogeneity present in all considered tomographic models and describe this heterogeneity in a fuzzy sense. Because spectral cluster algorithms analyze model structure locally, they are considered relatively robust with regard to systematically and spatially varying imaging capabilities typical for geophysical tomographic surveys conducted on the earth's surface. Using a synthetic aquifer example, a fuzzy spectral cluster algorithm can be used to integrate the information provided by 2D tomographic refraction seismic and DC resistivity surveys. The integrated information in the fuzzy membership domain is then used to derive an integrated zonal geophysical model outlining the major structural units present in both input models. We also explain how the fuzzy membership information can be used to identify optimal locations for sparse logging of additional target parameters, i.e., porosity information in our synthetic example. We demonstrate how this sparse porosity information can be extrapolated based on all tomographic input models. The resultant 2D porosity model matches the original porosity distribution reasonably well within the spatial resolution limits of the underlying tomographic models. Consecutively, we apply this approach to a field data base acquired over a former river channel. Sparse information about natural gamma radiation is available and extrapolated on the basis of the fuzzy membership information obtained by spectral cluster analysis of 2D P-wave velocity and electrical resistivity models. This field data shows that the presented parameter extrapolation procedure is robust, even if the locations of target parameter acquisition have not been optimized with regard to the fuzzy membership information.

Near-surface seismic traveltime tomography using a direct-push source and surface-planted geophones
(2009)

Information about seismic velocity distribution in heterogeneous near-surface sedimentary deposits is essential for a variety of environmental and engineering geophysical applications. We have evaluated the suitability of the minimally invasive direct-push technology for near-surface seismic traveltime tomography. Geophones placed at the surface and a seismic source installed temporarily in the subsurface by direct-push technology quickly acquire reversed multioffset vertical seismic profiles (VSPs). The first-arrival traveltimes of these data were used to reconstruct the 2D seismic velocity distribution tomographically. After testing this approach on synthetic data, we applied it to field data collected over alluvial deposits in a former river floodplain. The resulting velocity model contains information about high- and low-velocity anomalies and offers a significantly deeper penetration depth than conventional refraction tomography using surface-planted sources and receivers at the investigated site. A combination of refraction seismic and direct-push data increases resolution capabilities in the unsaturated zone and enables reliable reconstruction of velocity variations in near-surface unconsolidated sediments. The final velocity model structurally matches the results of cone-penetration tests and natural gamma-radiation data acquired along the profile. The suitability of multiple rapidly acquired reverse VSP surveys for 2D tomographic velocity imaging of near-surface unconsolidated sediments was explored.

Geophysical techniques offer the potential to tomographically image physical parameter variations in the ground in two or three dimensions. Due to the limited number and accuracy of the recorded data, geophysical model generation by inversion suffers ambiguity. Linking the model generation process of disparate data by jointly inverting two or more data sets allows for improved model reconstruction. Fully nonlinear inversion using optimization techniques searching the solution space of the inverse problem globally enables quantitative assessment of the ambiguity inherent to the model reconstruction. We used two different multiobjective particle swarm optimization approaches to jointly invert synthetic crosshole tomographic data sets comprising radar and P-wave traveltimes, respectively. Beginning with a nonlinear joint inversion founded on the principle of Pareto optimality and game theoretic concepts, we obtained a set of Pareto-optimal solutions comprising commonly structured radar and P-wave velocity models for low computational costs. However, the efficiency of the approach goes along with some risk of achieving a final model ensemble not adequately illustrating the ambiguity inherent to the model reconstruction process. Taking advantage of the results of the first approach, we inverted the database using a different nonlinear joint-inversion approach reducing the multiobjective optimization problem to a single-objective one. Computational costs were significantly higher, but the final models were obtained mutually independently allowing for objective appraisal of model parameter determination. Despite the high computational effort, the approach was found to be an efficient nonlinear joint-inversion formulation compared to what could be extracted from individual nonlinear inversions of both data sets.

Unsupervised classification techniques, such as cluster algorithms, are routinely used for structural exploration and integration of multiple frequency bands of remotely sensed spectral datasets. However, up to now, very few attempts have been made towards using unsupervised classification techniques for rapid, automated, and objective information extraction from large airborne geophysical data suites. We employ fuzzy c-means (FCM) cluster analysis for the rapid and largely automated integration of complementary geophysical datasets comprising airborne radiometric and magnetic as well as ground-based gravity data, covering a survey area of approximately 5000 km(2) located 100 km east- south-east of Johannesburg, South Africa, along the south-eastern limb of the Bushveld layered mafic intrusion complex. After preparatory data processing and normalisation, the three datasets are subjected to FCM cluster analysis, resulting in the generation of a zoned integrated geophysical map delineating distinct subsurface units based on the information the three input datasets carry. The fuzzy concept of the cluster algorithm employed also provides information about the significance of the identified zonation. According to the nature of the input datasets, the integrated zoned map carries information from near-surface depositions as well as rocks underneath the sediment cover. To establish a sound geological association of these zones we refer the zoned geophysical map to all available geological information, demonstrating that the zoned geophysical map as obtained from FCM cluster analysis outlines geological units that are related to Bushveld-type, other Proterozoic- and Karoo-aged rocks.

We strive to assess soil water content on a well-studied slow-moving hillslope in Austria. In doing so, we employ time lapse mapping of bulk electrical conductivity using a geophysical electromagnetic induction system operated at low induction numbers. This information is complemented by the acquisition of soil samples for gravimetric water content analysis during one survey campaign. Simple visual soil sample analysis reveals that the upper material in the survey area is a spatially highly variable mixture of predominately sandy, silty, clayey and organic materials. Due to this heterogeneity, classical approaches of mapping soil moisture on the basis of stationary mapping of electrical conductivity variations are not successful. Also the time-lapse approach does not allow ruling out some of the ambiguity inherent to the linkage of bulk electrical conductivity to soil water content. However, indication is found that time-lapse measurements may have supportive capabilities to identify regions of low precipitation infiltration due to high soil saturation. Furthermore, the relationship between the mean electrical conductivity averaged over a full vegetation period and an already available ecological moisture map produced by vegetation analysis is found to resemble closely the relationship observed between gravimetric soil water content and electrical conductivity during the time of sample collection except for highly organic soils. This leads us to the assumption that the relative soil moisture distribution is temporarily stable except for those areas characterized by highly organic soils.