Limiting stochastic operations for stationary spatial processes
A natural extrapolation of stochastic operations (continuity and differentiation) already described in time domain (one-dimensional case) is established for spatial processes (two- or three-dimensional case). If stationarity decision is assumed, the continuity and differentiability (in the mean square sense) of a spatial process depends on the continuity and differentiability of the correlation function at the origin. Spatial processes described by stationary random functions are not continuous (in the mean square sense) when the covariance function presents a nugget effect, and they are not differentiable when the same covariance function is described by a spherical or an exponential covariance (models which are often used in geostatistics).
Key wordsmean square convergence nugget effect stationary random functions stochastic convergence
Unable to display preview. Download preview PDF.
- Adler, R. J., 1981, The Geometry of Random Fields: John Wiley & Sons, New York, 280 p.Google Scholar
- Cramer, H., and Leadbetter, M. R., 1967, Stationary and Related Stochastic Processes: John Wiley & Sons, New York, 348 p.Google Scholar
- Journel, A. G., and Huijbregts, C. J., 1981, Mining Geostatistics: Academic Press, London, 600 p.Google Scholar
- Mandelbrot, B. B., 1975, Stochastic Models for the Earth's Relief, the Shape and the Fractal Dimension of the Coastlines, and the Number-Area Rule for Islands: Proc. Nat. Acad. Sci. USA, v. 72, p. 3825–3828.Google Scholar
- Matern, B., 1960, Spatial Variation; Comm. Swed. Forestry Res. Inst., v. 49, p. 1–144.Google Scholar
- Panda, D. P., 1977, Statistical Properties of Thresholded Images: T. R. 559, Computer Science Centre, University of Maryland, College Park.Google Scholar
- Papoulis, A., 1965, Probability, Random Variables, and Stochastic Processes: McGraw-Hill, New York, 583 p.Google Scholar
- Priestley, M. B., 1981, Spectral Analysis and Time Series: Academic Press, New York, 653 p.Google Scholar
- Robinson, E. A., 1967, Statistical Communication and Detection: Griffin, London.Google Scholar
- Yaglom, A. M., 1962, An Introduction to the Theory of Stationary Random Functions: Prentice Hall, Englewood Cliffs, New Jersey.Google Scholar