Abstract
A computer program called jk3d for 3-D ordinary kriging interpolation of scattered data has been developed. A specific feature of the code is that differences of the quality of the measured data are taken into account by weighting them according to the assumed error of the data. The code is demonstrated on subsurface temperature measurements with different measurement errors. One of the main problems when ordinary kriging is used for 3-D subsurface data is a lack of sufficient data for computing the necessary variograms. A feasible procedure to obtain such a variogram is discussed. The final result of the 3-D interpolated temperature data shows, that the quality of the interpolation increases substantially if the measurement error is taken into account. The code uses a modified GSLIB like input. Although not all functions of the GSLIB kriging library are supported, all substantial functions are available. jk3d is LGPL licensed, open source. The current version is available at http://sourceforge.net/projects/jk3d.html.
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Agemar T, Schellschmidt R, Schulz R (2012) Subsurface temperature distribution in germany. Geothermics 44(0):65–77. doi:10.1016/j.geothermics.2012.07.002
Bächler D, Kohl T, Rybach L (2003) Impact of graben-parallel faults on hydrothermal convection—Rhine Graben case study. Phys Chem Earth 28(9–11):431–441
Bonte D, Guillou-Frottier L, Garibaldi C, Lopez S, Bourgine B, Lucazeau F, Bouchot V (2010) Subsurface temperature maps in French sedimentary basins: new data compilation and new interpolation. Bull Soc Gol Fr 181(4):377–390
Bredehoeft J, Papadopulos I (1965) Rates of vertical groundwater movement estimated from the earth’s thermal profile. Water Res Res 1:325–328
Budrikaité L, Dučinskas K (2005) Modelling of geometric anisotropic spatial variation. In: Proceedings of the 10th international conference of mathematical modelling and analysis (MMA’05)&CMAM2. Technika, Trakai, pp 361–366
Chilès J, Delfiner P (1999) Geostatistics: modeling spatial uncertainty. Wiley, New York
Chilès JP, Gable R (1984) Three dimensional modelling of a geothermal field. Geostatistics for natural resources characterization, Part 2. D. Reidel Publish. Comp, Hingbam, pp 587–598
Cressie N (1988) Spatial prediction and ordinary kriging. Math Geol 20(4):405–421
Davis JC (2002) Statistics and data analysis in geology, 3rd edn. Wiley, New York
Delhomme J (1978) Kriging in the hydrosciences. Adv Water Resour 1(5):251–266. doi:10.1016/0309-1708(78)90039-8, http://www.sciencedirect.com/science/article/pii/0309170878900398
Deming D (1989) Application of bottom-hole temperature corrections in geothermal studies. Geothermics 18:775–786
Deutsch CV, Journel AG (1998) GSLIB: geostatistical software library and users’s guide, 2nd edn. Oxford University Press, Oxford
Garibaldi C, Guillou-Frottier L, Lardeaux JM, Bonte D, Lopez S, Bouchot VPL (2010) Relationship between thermal anomalies, geological structures and fluid flow: new evidence in application to the Provence basin (southeast France). Bull Soc Gol Fr 181(4):363–376
Goutorbe B, Lucazeau E, Bonneville A (2007) Comparison of several BHT correction methods: a case study on an Australian data set. Geophys J Int 170:913–922
Haenel R, Zoth G (1982) Temperature measurements and determination of heat flow density. The Urach geothermal project. Schweizerbart’sche Verlagsbuchhandlung, Stuttgart, pp 81–88
Huenges E, Kohl T, Kolditz O, Bremer J, Scheck-Wenderoth M, Vienken T (2013) Geothermal energy systems: research perspective for domestic energy provision. Environ Earth Sci 70(8):3927–3933. doi:10.1007/s12665-013-2881-2
Isaaks EH, Srivastava RM (1989) An introduction to applied geostatistics. Oxford University Press, Oxford
Journel A, Huijbregts C (1978) Mining geostatistics. Academic Press, London
Kitanidis P (1997) Introduction to geostatistics, applications in hydrogeology. Cambridge University Press, Cambridge
Luijendijk E, ter Voorde M, van Balen R, Verweij H, Simmelink E (2011) Thermal state of the Roer Valley Graben, part of the European Cenozoic Rift System. Basin Res 23(1):65–82. doi:10.1111/j.1365-2117.2010.00466.x
Mansure AJ, Reiter M (1979) A vertical groundwater movement heat flow correction. J Geophys Res 84(B7):3490–3496
Matheron G (1965) Les variables régionalisées et leur estimation: une application de la théorie des fonctions aléatoires aux sciences de la nature. Masson, Paris
Pollack HN, Hurter SJ, Johnson JR (1993) Heat flow from the earth’s interior: analysis of the global data set. Rev Geophys 31(3):267–280. doi:10.1029/93RG01249
Pribnow D, Schellschmidt R (2000) Thermal tracking of upper crustal fluid flow in the Rhine Graben. Geophys Res Lett 27(13):1957–1960
Rühaak W (2006) A Java application for quality weighted 3-D interpolation. Comput Geosci 32(1):43–51. doi:10.1016/j.cageo.2005.04.005
Rühaak W, Rath V, Clauser C (2010) Detecting thermal anomalies within the Molasse Basin, southern Germany. Hydrogeol J 18:1897–1915. doi:10.1007/s10040-010-0676-z
Wackernagel H (2003) Multivariate geostatistics: an introduction with applications. Springer, Berlin
Wessel P, Smith WHF (1998) New, improved version of generic mapping tools released. EOS Trans Am Geophys Union 79(47):579–579. doi:10.1029/98EO00426
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For plotting the temperature maps the free software GMT (Wessel and Smith 1988) was used.
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Rühaak, W. 3-D interpolation of subsurface temperature data with measurement error using kriging. Environ Earth Sci 73, 1893–1900 (2015). https://doi.org/10.1007/s12665-014-3554-5
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DOI: https://doi.org/10.1007/s12665-014-3554-5