Abstract
Geochemical surfaces are reconstructed by interpolating geochemical measurements obtained from stream-water and stream-sediment samples. The geographical region that influences (and therefore is represented by) the value of a geochemial sample is its topographic catchment area. However, standard convention is to treat and to record the stream sample in the database as a point location, and to reconstruct geochemical surfaces utilizing conventional point interpolation procedures. These interpolation procedures assume, generally, that a data point exerts geographical influence away from itself in all directions, and that influence declines with distance away from that data point. Conventional interpolation procedures are poorly suited for reconstructing geochemical surfaces from stream samples; they do not take into account the true geographic area that geochemical sample points represent (topographic catchments). In this paper we propose a method of interpolation which assumes that data points are representative of their topographic catchment areas. Experimental data indicates that a surface reconstruction procedure which preserves the areal character of geochemical samples provides results more meaningful than surfaces reconstructed using more conventional interpolation techniques.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akima, H., 1978, A method of bivariate interpolation and smooth surface fitting for irregularly distributed data points: ACM Trans. Mathematical Software, v. 4, no. 2, p. 148–159.
Bartels, R. H., Beatty, J. C., and Barsky, B. A., 1987, An introduction to splines for use in computer graphics & geometric modeling: Morgan Kaufmann Pub., Inc., Los Altos, California, 476 p.
Boneva, L., Kendall, D., and Stefanov, J., 1971, Spline transformations: Jour. Roy. Stat. Soc., Ser. B., v. 33, p. 1–70.
Bonham-Carter, G. F., and Agterberg, F. P., and Wright, D. F., 1988, Integration of gold datasets for gold exploration in Nova Scotia: Photogrammetric Engineering and Remote Sensing, v. 54, no. 11, p. 1582–1592.
Briggs, I. C., 1974, Machine contouring using minimum curvature: Geophysics, v. 39, no. 1, p. 39–48.
Burrough, P. A., 1987, Principles of geographical information systems for land resources assessment (2nd ed.): Clarendon Press, Oxford, 193 p.
Crain, I. K., 1970, Computer interpolation and contouring of two-dimensional data: a review: Geoexploration, v. 8, no. 8. p. 71–86.
David, M., 1977, Geostatistical ore reserve estimation: Developments in Geomathematics 2: Elsevier Sci. Publ. Co., Amsterdam, 364 p.
DuChateau, P., and Zachmann, D. 1989, Applied partial differential equations: Harper and Row, New York, 620 p.
Dyn, N., and Wahba, G., 1982, On the estimation of functions of several variables from aggregate data: SIAM Jour. Math. Anal., v. 13, no. 1, p. 134–152.
Dyn, N., Wahba, G., and Wong, W. H., 1979, Comments to “Smooth pycnophylactic interpolation for geographical regions,” by W. Tobler, Jour. Am. Stat. Assoc., v. 74, no. 367, p. 530–535.
Fulton, S. R., Ciesielski, P. E., and Schubert, W. H., 1986, Multigrid methods for elliptic problems: a review: Monthly Weather Review, v. 114, p. 943–959.
Hodgson, M. E., 1989, Searching methods for rapid grid interpolation: Prof. Geographer, v. 41, no. 1, p. 51–61.
Hutchinson, M. F., 1989, A new procedure for gridding elevation and stream line data with automatic removal of spurious pits: Jour. Hydrology, v. 106, no. 3–4, p. 211–232.
Journel, A. G., 1985, Geostatistics: models and tools for the earth sciences: Math. Geology, v. 18, no. 1, p. 119–140.
Keller, C. P., Shasko, M., and Gartrell, M., 1989, Genesis: random landscape generation software: Spatial Sci. Lab., Univ. Victoria, Victoria, BC, Canada, unpubl. computer program.
Krige, D. J., 1976, A review of the development of geostatistics in South Africa,in Guarascio, M., and others, eds., Advanced geostatistics in the mining industry: D. Reidel Publ. Co., Dordrecht, Holland, p. 279–293.
Lam, N. S., 1981, The reliability problem of spatial interpolation models: Modelling and Simulation, v. 12, p. 869–876.
Matheron, G., 1986, Philipian/Watsonian high (flying) philosophy, “Letter to the Editor”: Math. Geology, v. 18, no. 5, p. 503–504.
MacEachran, A. M., and Davidson, J. V., 1987, Samping and isometric mapping of continuous geographic surfaces: Am. Cartographer, v. 14, no. 4, p. 299–330.
Peuker, T. K., Fowler, R. J., Little, J. J., and Mark, D. M. 1978, The triangulated irregular network: Proc. Am. Soc. Photogrammetry: Digital Terrain Models (DTM) Symposium (St. Louis, Missouri, p. 516–540.
Philip, G. M., and Watson, D. F., 1985, Matheronian geostatistics—Quo Vadis?: Math. Geology, v. 18, no. 1, p. 93–117.
Philip, G. M., and Watson, D. F., 1987, Probabilism in geological data analysis: Geol. Mag., v. 124, no. 6, p. 577–583.
Press, W. H., Flannery, B. P., Teukolsky, S. A., and Veterling, W. T., 1988, Numerical recipes in C, the art of scientific programming: Cambridge Univ. Press, Cambridge, 735 p.
Rogers, P. J., Bonham-Carter, G. F., and Ellwood, D. J., 1988, Regional stream sediment geochemistry and exploration of a mature area: target selection by catchment basin analysis, Cobequid Highlands, Nova Scotia (abst): Integrating Technology and Geoscience Applications, National Academy of Sciences, p. 24–25.
Schoenburg, I., 1973, Cardinal spline interpolation (Philadelphia): SIAM, Mon. No. 12, p. 115–119.
Sibson, R., 1981, A brief description of natural neighbour interpolation,in Barnett, V., ed., Interpreting Multivariate Data: John Wiley & Sons, Chichester, p. 21–36.
Smith, G. D., 1978, Numerical solution of partial differential equations (2nd ed.): Oxford Univ. Press, Oxford, 304 p.
Smith, W. H. F., and Wessel, P., 1990, Gridding with continuous curvature splines in tension: Geophysics, v. 55, no. 3, p. 293–505.
Swain, C. J., 1976, A FORTRAN IV program for interpolating irregularly spaced data using the difference equations for minimum curvature: Computers & Geosciences, v. 1, no. 4, p. 231–240.
Tobler, W. F., 1979, Smooth pycnophylactic interpolation for geographical regions: Jour. Am. Stat. Assoc, v. 74, no. 367, p. 519–530.
Tobler, W. R., 1992, Unpublished documentation for pycnophylactic interpolation: Univ. California, Santa Barbara, unpubl. computer program.
Watson, D. F., 1992, Contouring, a guide to the analysis and display of spatial data: Pergamon Press, Oxford, 321 p.
Watson, D. F., and Philip, G. M., 1984, Triangle based interpolation: Math. Geology, v. 16, no. 8, p. 779–795.
Watson, D. F., and Philip, G. M., 1987, Neighbourhood-based interpolation: GEOBYTE, v. 2, no. 2, p. 12–16.
Zoraster, S., 1987, Honouring discontinuities and other surface features during grid processing on vector computers: Cartographica, v. 24, no. 4, p. 37–48.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bartier, P.M., Keller, C.P. Interpolation for geochemical surface reconstruction incorporating topographic catchment definitions. Math Geol 28, 253–273 (1996). https://doi.org/10.1007/BF02083200
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02083200