Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Modeling soil variability as a random field

  • 146 Accesses

  • 32 Citations


The observed variability in the spatial distribution of soil properties suggests that it is natural to describe such distribution as a random field. One of the ways to study engineering problems in such a stochastic setting is by the use of the Monte-Carlo simulation procedure. Application of this technique requires the capability to generate a large number of realizations of a given random field. A numerical procedure for the generation of such realizations in two-dimensional space is introduced as a finite difference approximation of a stochastic differential equation. The equation used was that suggested by Heine (1955). The resulting procedure is essentially similar to other autoregressive procedures used for the same purpose (Whittle, 1954; Smith and Freeze, 1979). However, contrary to these procedures, the present one is defined in terms of physically significant parameters:r 0, the autocorrelation distance;Δ, the discretization size; and the standard deviation, σ. Formulating the simulation procedure in terms of the physically significant parameters (r 0,Δ, σ) greatly simplifies the task of generating realizations that are compatable with a given soil deposit.

This is a preview of subscription content, log in to check access.


  1. Alonso, E. E. and Krizek, R. J., 1975, Stochastic formulation of soil properties: Proceedings of the 2nd International Conference on the applications of statistics and probability in soil and structural engineering: Aachen, v. 2, p. 10–32.

  2. Birnbaum, A. and Baker, R., 1977, Use of statistical methods for analysis of flexibility distribution on runways: Research report No. 77-1: Transportation Research Institute, Technion, Israel Institute of Technology, Haifa, Israel, 129 p. (in Hebrew).

  3. Gröbner, W. and Hofreiter, N., 1961, Integraltafel (Vol. 1): Springer-Verlag, Wien, 205, p.

  4. Heine, V., 1955, Models for two-dimensional stationary stochastic process: Biometrika, v. 42, p. 170–178.

  5. Journel, A. G., 1974, Geostatistics for conditional simulation of ore bodies: Econ. Geol., v. 69, p. 673–678.

  6. Lumley, J. L. and Panofsky, H. A., 1964, The structure of atmospheric turbulence: John Wiley & Sons, New York, 287 p.

  7. Mantoglou, A. and Wilson, J. L., 1981, Simulation of random fields with the turning bands method, Report no. 264, Department of Civil Engineering: Massachusetts Institute of Technology; Cambridge, Massachusetts, 199 p.

  8. Matheron, G., 1973, The intrinsic random functions and their applications: Adv. Appl. Prob., v. 5, p. 439–468.

  9. Matsuo, M., 1976, Reliability in embankment design, Report No. 550, Department of Civil Engineering: Massachusetts Institute of Technology, Cambridge, Massachusetts, 175 p.

  10. Panchev, S., 1971, Random functions and turbulence: Pergamon Press, Oxford, 444 p.

  11. Rice, S. O., 1944, Mathematical analysis of random noise: Bell Syst. Tech. Jour., v. 23, p. 282–286.

  12. Sironvalle, M. A., 1980, The random coin method: Solution of the problem of the simulation of random functions in the plane: Jour. Math. Geol., v. 12, no. 1, p. 25–32.

  13. Smith, L. and Freeze, A. R., 1979, Stochastic analysis of steady state groundwater flow in a bounded domain, 2, Two-dimensional simulations: Water Resour. Res. v. 15, no. 6, p. 1543–1559.

  14. Vanmarcke, E., 1977, Probability modelling of soil profiles: Proc. A.S.C.E. Jour. Geotech. Eng. Div., v. 103, no. GT11, p. 1227–1247.

  15. Whittle, P., 1954, On stationary processes in the plane: Biometrica, v. 41, p. 434–449.

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Baker, R. Modeling soil variability as a random field. Mathematical Geology 16, 435–448 (1984).

Download citation

Key words

  • spatial variability
  • two dimensional random field
  • Monte-Carlo simulation