Abstract
This paper describes a geostatistical approach to model and visualize the space–time distribution of groundwater contaminants. It is illustrated using data from one of the world’s largest plume of trichloroethylene (TCE) contamination, extending over 23 km2, which has polluted drinking water wells in northern Michigan. A total of 613 TCE concentrations were recorded at 36 wells between May 2003 and October 2018. To account for the non-stationarity of the spatial covariance, the data were first projected in a new space using multidimensional scaling. During this spatial deformation the domain is stretched in regions of relatively lower spatial correlation (i.e., higher spatial dispersion), while being contracted in regions of higher spatial correlation. The range of temporal autocorrelation is 43 months, while the spatial range is 11 km. The sample semivariogram was fitted using three different types of non-separable space–time models, and their prediction performance was compared using cross-validation. The sum-metric and product-sum semivariogram models performed equally well, with a mean absolute error of prediction corresponding to 23% of the mean TCE concentration. The observations were then interpolated every 6 months to the nodes of a 150 m spacing grid covering the study area and results were visualized using a three-dimensional space–time cube. This display highlights how TCE concentrations increased over time in the northern part of the study area, as the plume is flowing to the so-called Chain of Lakes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aigner W, Miksch S, Mueller W, Schumann H, Tominski C (2007) Visualizing time-oriented data: a systematic view. Comput Graph 31:401–409
Andrienko N, Andrienko G, Gatalsky P (2003) Exploratory spatio-temporal visualization: an analytical review. J Vis Lang Comput 14:503–541
Archer NP, Bradford CM, Villanacci JF, Crain NE, Corsi RL, Chambers DM, Burk T, Blount BC (2015) Relationship between vapor intrusion and human exposure to trichloroethylene. J Environ Sci Health A 50(13):1360–1368
Cameron K, Hunter P (2002) Using spatial models and kriging techniques to optimize long-term groundwater monitoring networks: a case study. Environmetrics 13:629–656
Cappello C, De Iaco S, Posa D (2018) Testing the type of non-separability and some classes of space–time covariance function models. Stoch Environ Res Risk Assess 32:17–35
Cappello C, De Iaco S, Posa D (2020) Covatest: an R package for selecting a class of space–time covariance functions. J Stat Softw 94:1–42
CDC (2020) Trichloroethylene—ToxFAQs™. https://www.atsdr.cdc.gov/toxfaqs/tfacts19.pdf Accessed 6 July 2023
Cressie N, Huang HC (1999) Classes of nonseparable, spatio-temporal stationary covariance functions. J Am Stat Assoc 94:1330–1340
De Cesare L, Myers DE, Posa D (2002) FORTRAN programs for space–time modeling. Comput Geosc 28:205–212
De Iaco S (2010) Space–time correlation analysis: a comparative study. J Appl Stat 37:1027–1041
De Iaco S, Myers DE, Posa D (2001) Space–time analysis using a general product-sum model. Stat Probab Lett 52:21–28
De Iaco S, Posa D, Cappello C, Maggio S (2019) Isotropy, symmetry, separability and strict positive definiteness for covariance functions: a critical review. Spat Stat 29:89–108
Deutsch CV (1993) Kriging in a finite domain. Math Geol 25:41–52
Deutsch CV, Journel AG (1998) Geostatistical software library and user guide. Oxford University Press, New York
Dimitrakopoulos R, Luo X (1994) Spatiotemporal modeling: covariances and ordinary kriging systems. In: Dimitrakopoulos R (ed) Geostatistics for the next century. Kluwer, Dordrecht, pp 88–93
EPA (2023) TCE Consumer Fact Sheet. https://www.epa.gov/assessing-and-managing-chemicals-under-tsca/fact-sheet-trichloroethylene-tce. Accessed 6 July 2023
Fouedjio F (2017) Second-order non-stationary modeling approaches for univariate geostatistical data. Stoch Environ Res Risk Assess 31:1887–1906
Fouedjio F, Desassis N, Romary T (2015) Estimation of space deformation model for non-stationary random functions. Spat Stat 13:45–61
Gneiting T (2002) Nonseparable, stationary covariance functions for space–time data. J Am Stat Assoc 97:590–600
Gneiting T, Genton MG, Guttorp P (2007) Geostatistical space–time models, stationarity, separability and full symmetry. In: Finkenstaedt B, Isham V, Held L (eds) Statistics of spatio-temporal systems. Monographs in statistics and applied probability. Chapman & Hall/CRC Press, Boca Raton, pp 151–175
Goovaerts P (1997) Geostatistics for Natural Resources Evaluation. Oxford Univ. Press, New York
Goovaerts P (2010) Three-dimensional visualization, interactive analysis and contextual mapping of space–time cancer data. In: Painho M, Santos MY, Pundt H (eds) Proceedings AGILE 2010: the 13th AGILE international conference on geographic information science. Springer Verlag, Guimarães, Portugal
Guo L, Lei L, Zeng Z (2015) Evaluation of spatio-temporal variogram models for mapping Xco2 using satellite observations: a case study in China. IEEE J Sel Top Appl Earth Observ Remote Sens 8:376–385
Heuvelink GBM, Griffith DA (2010) Space–time geostatistics for geography: a case study of radiation monitoring across parts of Germany. Geogr Anal 42:161–179
Jacquez GM, Goovaerts P, Kaufmann A, Rommel R (2014) SpaceStat 4.0 user manual: software for the space–time analysis of dynamic complex systems, 4th ed. BioMedware, Ann Arbor. https://www.biomedware.com/files/SpaceStat_4.0_Documentation.pdf. Accessed 3 Feb 2023
Johnson PD, Goldberg SJ, Mays MZ, Dawson BV (2003) Threshold of trichloroethylene contamination in maternal drinking waters affecting fetal heart development in the rat. Environ Health Perspect 111(3):289–292
Júnez-Ferreira HE, Hernández-Hernández MA, Herrera GS, González-Trinidad J, Cappello C, Maggio S, De Iaco S (2023) Assessment of changes in regional groundwater levels through spatio-temporal kriging: application to the southern Basin of Mexico aquifer system. Hydrogeol J. https://doi.org/10.1007/s10040-023-02681-y
Kazemi H, Sarukkalige R, Shao Q (2021) Evaluation of non-uniform groundwater level data using spatiotemporal modelling. Groundw Sustain Dev 15:100659
McLean MI (2018) Spatio-temporal models for the analysis and optimisation of groundwater quality monitoring networks. PhD thesis. http://theses.gla.ac.uk/38975/
Michigan Department of Environment, Great Lakes, and Energy, Wikes Manufacturing TCE Plume (2020) https://www.michigan.gov/egle/0,9429,7-135-3311_4109_9846_30022-385691--,00.html. Accessed 3 Feb 2023
Montero JM, Fernandez-Aviles G, Mateu J (2015) Spatial and spatio-temporal geostatistical modeling and kriging. Wiley, New York
Porcu E, Furrer R, Nychka D (2021) 30 Years of space–time covariance functions. Wires Comput Stat 13:e1512. https://doi.org/10.1002/wics.1512
Rautman CA, Istok JD (1996) Probabilistic assessment of ground-water contamination: 1. Geostat Framew Ground Water 34:899–909
Remy N, Boucher A, Wu J (2008) Applied geostatistics with SGeMS: a user’s guide. Cambridge University Press, Cambridge
Rihana-Abdallah A, Pang Y (2019) Fate and transport of a TCE groundwater contamination plume. In: Proceedings of the 2019 ASEE (American Society for Engineering Education) North-Central Section conference. https://asee-ncs.org/proceedings/2019/1/71.pdf. Accessed 3 Feb 2023
Ruybal CJ, Hogue TS, McCray JE (2019) Evaluation of groundwater levels in the Arapahoe aquifer using spatiotemporal regression kriging. Water Resour Res 55:2820–2837
Sampson PD, Guttorp P (1992) Nonparametric estimation of nonstationary spatial covariance structure. J Am Stat Assoc 87:108–119
SAS Institute Inc. (2011) SAS/STAT 9.3 user’s guide. SAS Institute Inc., Cary
Tominski C, Schulze-Wollgast P, Schuman H (2005) 3D information visualization for time dependent data on maps. In: Proceedings of the ninth international conference on information visualisation. IEEE. https://doi.org/10.1109/IV.2005.3
Van Driel JN (1989) Three dimensional display of geologic data. In: Raper J (ed) Three dimensional applications in Geological Information System. Taylor & Francis, London, pp 1–9
Varouchakis EA, Hristopulos DT (2017) Comparison of spatiotemporal variogram functions based on a sparse dataset of ground-water level variations. Spat Stat 34:66
Varouchakis EA, Theodoridou PG, Karatzas GP (2019) Spatiotemporal geostatistical modeling of groundwater levels under a Bayesian framework using means of physical background. J Hydrol 575:487–498
Varouchakis EA, Guardiola-Alber C, Karatzas GP (2022) Spatiotemporal geostatistical analysis of groundwater level in aquifer aystems of complex hydrogeology. Water Resour Res 58(3):e2021WR029988
Wackernagel H (1998) Multivariate geostatistics, 2nd edn. Springer, Berlin
Acknowledgements
This research was funded by Grant 2 R44 ES031875-02 from the National Institute of Environmental Health Sciences (NIEHS). The views expressed in this publication are those of the authors and do not necessarily represent the official views of the NIEHS. A free 1-year license of the commercial software SpaceStat can be downloaded at https://biomedware.com/products/spacestat/
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Goovaerts, P., Rihana-Abdallah, A. & Pang, Y. Space–Time Distribution of Trichloroethylene Groundwater Concentrations: Geostatistical Modeling and Visualization. Math Geosci 56, 437–464 (2024). https://doi.org/10.1007/s11004-023-10107-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11004-023-10107-4