Spectral Corrected Semivariogram Models
- PDF / 443,109 Bytes
- 9 Pages / 432 x 648 pts Page_size
- 45 Downloads / 159 Views
Spectral Corrected Semivariogram Models1 Michael J. Pyrcz2 and Clayton V. Deutsch3 Fitting semivariograms with analytical models can be tedious and restrictive. There are many smooth functions that could be used for the semivariogram; however, arbitrary interpolation of the semivariogram will almost certainly create an invalid function. A spectral correction, that is, taking the Fourier transform of the corresponding covariance values, resetting all negative terms to zero, standardizing the spectrum to sum to the sill, and inverse transforming is a valuable method for constructing valid discrete semivariogram models. This paper addresses some important implementation details and provides a methodology to working with spectrally corrected semivariograms. KEY WORDS: nested structures, kriging, stochastic simulation, geostatistics, Fourier transform.
INTRODUCTION The random function paradigm of semivariogram based geostatistics depends heavily on the calculation and fitting of a reasonable semivariogram model. The inference step is largely automatic once a decision of stationarity is taken and a semivariogram model is chosen. This paper is aimed at the determination of a valid semivariogram model. A valid semivariogram model is one that is conditionally non-negative definite and that does not lead to numerical artifacts due to instability. The conventional method of modeling semivariograms by nested structures is well established (Journel and Huijbregts, 1978). While this provides a workable mechanism for modeling most semivariograms, there are some cases that are not well fit with this framework. Figure 1 shows an example structure commonly observed in experimental semivariograms that is not easy to fit with the conventional structures. The largely unexplored suite of valid models, known as 1Received
22 March 2005; accepted 8 March 2006; Published online: 31 January 2007. Stratigraphy, Earth Science R & D, Chevron Energy Technology Company, Houston, TX 77002 USA; e-mail: [email protected] 3Department of Civil and Environmental Engineering, University of Alberta, Edmonton, AB Canada, T6G 2W2; e-mail: [email protected] 2Quantitative
891 C 2006 International Association for Mathematical Geology 0882-8121/06/1100-0891/1
892
Pyrcz and Deutsch
Figure 1. An example semivariogram that is not well fit by nested sets of traditional semivariogram models.
geometric semivariograms, is explored in a companion paper (Pyrcz and Deutsch, in press). The covariance is related to the semivariogram under second order stationarity, C(h) = σ 2 − γ (h), and covariances will be referred to when it is standard practice to call on the covariance and not the semivariogram.
SPECTRAL CORRECTED SEMIVARIOGRAM MODELS Fitting an arbitrary function to experimental semivariogram points, γ (h), does not guarantee a valid model for subsequent estimation and simulation. Spectral correction offers an efficient means to correct arbitrary fitted semivariogram to be conditional negative definite. Bochner’s theorem defines the general form of a co
Data Loading...
