P1: FLF

Mathematical Geology [mg] PL091-875 October 26, 1999 3:26 Style file version June 30, 1999

Mathematical Geology, Vol. 32, No. 1, 2000

Letter to the Editor

Note and Corrections to “Automatic Modeling of (Cross)Covariance Tables Using Fast Fourier Transform”

by T. Yao and A. G. Journel

Yao and Journel (1998) presented a method for joint modeling of multivariate(cross)covariances based on Bochner’s theorem (Bochner, 1949) and discrete FastFourier Transform (FFT) (Bracewell, 1986). The basic paradigm consists of avoid-ing any analytical models, replacing them by a set of discrete (cross)covariancetables/volumes that have been checked to be jointly positive definite. This check isbest done in the spectral domain on FFT’s of the original sample covariance tablescompleted for missing values.

The authors acknowledge that the qualifier “automatic” in the original title ismisleading. The algorithm still requires the user to define a certain number of op-erating parameters, e.g., various window widths, similar to the tolerance/averagingparameters required to calculate a sample variogram. These parameters would beset from the user’s prior experience depending on the specific layout of the sampledata available.

The proposed method frees modeling from the constraints of the linear modelof (co)regionalization (LMC). In particular, there is no need to specify the number,type, and anisotropy of the component covariances shared by all attribute (cross)covariances.

The impact of the input parameters on the final (cross)covariance tables can beevaluated by plotting some line sections of these tables, and (1) checking that theymake physical/geological sense and (2) comparing them to the original samplecovariances, similar to what is done to evaluate a traditional variogram modelby comparing visually how it fits the sample variogram. Numerical measures ofgoodness of fit can also be defined.

Sensitivity analysis of the input parameters can also be performed, e.g., byrunning a limited test kriging/simulation using alternative covariance tables result-ing from alternative sets of input parameters. Again, this is similar to what onewould do to check a traditional variogram model.

147

0882-8121/00/0100-0147$18.00/1C© 2000 International Association for Mathematical Geology

P1: FLF

Mathematical Geology [mg] PL091-875 October 26, 1999 3:26 Style file version June 30, 1999

148 Letter to the Editor

We would like to take advantage of this short note to make two correctionson the paper:

1. The labels NS and EW have been inadvertently exchanged in all cross-section plots.

2. The cross-sections displayed in Figure 12 were obtained with a smoothingwindow size 1× 1, not 7× 7.

Thanks to Ricardo Olea and Vera Pawlowsky for their dedication in reviewingthe paper and trying to reproduce all the results. The computer codes we passedto them were subsequently considerably improved and are available upon demandfrom tingting@pangea.stanford.edu.

REFERENCES

Yao, T., and Journel, A. G., 1998, Automatic modeling of (cross)covariance tables using Fast FourierTransform: Math. Geology, v. 30, no. 6, p. 589–615.

Bochner, S., 1949, Fourier transform: Princeton University Press, London, 219 p.Bracewell, R., 1986, The Fourier transform and its application: Mc-Graw Hill, Singapore, 474 p.

Tingting YaoMobil Technology Company13777 Midway RoadDallas, Texas 75244