# Unilateral ARMA processes on a square net by the herringbone method

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## Abstract

Because multidimensional ARMA processes have great potential for the simulation of geological parameters such as aquifer permeability, it was important to resolve which of two proposed alternative methods should be used for determining the two-dimensional weighting parameter, φ″, for a unilateral ARMA (1, 0) process on a square net. Practical simulations demonstrates that the correct formulation is: φ″=ρ_{10}/(1+*ρ* _{10} ^{2} where ρ_{r,s} is the correlation between lattice points at lags*r* and s. When the simulations are performed with correlations of 0.8 or more a residual bias was detected which was found to be caused by a difference in the variance between the one- and two-dimensional models. This can be rectified by modifying the two- dimensional model as follows: z_{ij}=φ″(z_{i−1, j} + z_{i, j−1}) + λa_{ij} where*λ*^{2}=1/(1 +*ρ* _{10} ^{2} ).

## Key words

autoregressive processes multidimensional time series herringbone method spatial models## Preview

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## References

- Aroian, L. A., and Sharp, W. E., 1988, Isotropic Statistical Space Series on a Square Net: Belgian J. Oper. Res. Stat. Comput. Sci. [JORBEL], v. 28, n. 3, p. 3–13.Google Scholar
- Box, G. E. P., and Jenkins, G. M., 1970,
*Time Series Analysis, Forecasting, and Control*: Holden-Day, San Francisco, 553 p.Google Scholar - Ernst, W. G., 1969,
*Earth Materials*: Prentice-Hall, Englewood Cliffs, New Jersey, 149 p.Google Scholar - IMSL, 1989, IMSL STAT/Library, Version 1.1: International Mathematical and Statistical Libraries, Houston, Texas.Google Scholar
- Martin, R. J., 1990a, “The Generation of Multidimensional Autoregressive Series by the Herringbone Method” by W. E. Sharp and Leo A. Aroian, and the “Letters to the Editor” concerning this paper by Zekâi Şen, and by W. E. Sharp and Leo A. Aroian: Math. Geol., v. 22, n. 1, p. 147–148.Google Scholar
- Martin, R. J., 1990b, Unilateral ARMA Lattice Processes: Belgian J. Oper. Res., Stat. Comp. Sci. [JORBEL], v. 30, n. 2, p. 49–61.Google Scholar
- Martin, R. J., 1991, Comment on “Spatial Simulation of Geological Variables” by Zekâi Şen: Math. Geol., v. 23, n. 5, p. 789–791.Google Scholar
- Martin, R. J., 1992, Comment on the Letter to the Editor “Spatial Simulation of Geological Variables” by Zekâl Şen: Math. Geol., v. 24, n. 3, p. 345–346.Google Scholar
- Şen, Z., 1989, “The Generation of Multidimensional Autoregressive Series by the Herringbone Method” by W. E. Sharp and Leo A. Aroian: Math. Geol., v. 21, n. 2, p. 267–268.Google Scholar
- Şen, Z., 1990, Spatial Simulation of Geological Variables: Math. Geol., v. 22, n. 2, p. 175–188.Google Scholar
- Şen, Z., 1991, Spatial Simulation of Geological Variables: Math. Geol., v. 23, n. 6, p. 887–890.Google Scholar
- Sharp, W. E., 1982, Estimation of Semivariograms by the Maximum Entropy Method: Math. Geol., v. 14, p. 457–474.Google Scholar
- Sharp, W. E., and Aroian, L. A., 1985, The Generation of Multidimensional Autoregressive Series by the Herringbone Method: Math. Geol., v. 17, n. 1, p. 67–79.Google Scholar
- Sharp, W. E., and Aroian, L. A., 1989, Reply to Comment by Zekâi Şen: Math. Geol., v. 21, n. 2, p. 269.Google Scholar