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Page last updated: 30/04/2004

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Alberto Bellin

Hydrogen: a random field generator


Unix Click on the following link to download the package:
Installation (instructions)
DOS, WINDOWS Click on the following link to download the package:
Installation (instructions)
HYDROGEN PostScript User's Manual - manual.ps, adding.ps

The program
HYDRO_GEN is a computer code for generating a distributed attribute z(x), where x denotes the spatial coordinate, which is modelled as a Random Space Function (RSF) Z(x) with assigned covariance function. In the actual implementation the code generates two-dimensional RSFs with six types of spatial correlation: 1) discrete covariance function, read from a file; 2) exponential covariance function; 3) Gaussian covariance function; 4) Whittle isotropic covariance function; 5) Mizell isotropic covariance function (type B) [Mizell et al., 1982]; 6) self-similar or self-affine (fractal) random fields. Downloading instructions are shown at the end of this page.

The methodology
The mathematical foundations of the method are described in the paper by Bellin and Rubin [1996] which discusses also accuracy and efficiency of unconditional generation of RSFs. The generation of unconditional and conditional self-similar random fields is discussed in the paper by Rubin and Bellin [1997]. The reader is referred to the above two papers for a detailed discussion of the methodology and the applications The proposed algorithm consists in generating the Z field discretely over a pre-determined arbitrary grid. The generation technique resembles the Sequential Indicator Simulation (SIS) method, proposed by [Gomez-Hernandez and Srivastava, 1990] . Hydro_gen and SIS have in common some general concepts but HYDRO_GEN introduces significant modifications that make it faster and more accurate. Our starting point is the generation of a realization z of the random field at the node x0 where local data are not available. This is accomplished using a standard random generator with the unconditional mean <Z> and the unconditional variance var[Z] used as target statistics. For fractal fields the variance is estimated using the value of the semivariogram at the lag r equal to the field dimension. Once z(x0) is generated, it is considered as a datum and it will be used to condition the Z values which will be generated subsequently at neighboring nodes. At the next step, generation of a realization at a nearby point x1 is considered. This time Z(x1) is conditioned on the previously generated x(x0 ), using the Gaussian conditioning procedure. The procedure continues with the generation of a realization at the third and the subsequent grid nodes while conditioning at each step on a selected number of previously generated z values. For single-scale RSFs only values of z inside a suitable search neighborhood are used for conditioning. Due to the fast decreasing of correlation with the distance the dimensions of the search neighborhood are limited to few integral scales. In fact the influence on the value assumed by Z at the generation node of the previously generated values vanishes as the distance from the generation node increases.
The density of the grid bears significantly on the computational burden but on the other hand accurate simulations often require very fine discretizations at least in some portions of the domain. To introduce high resolution, Hydro_gen allows for multi-stage grid refinement. The grid can be refined to any a-priori determined level of discretization. Grid refinement is performed as follows. First, realizations of Z at the nodes of a coarse initial grid are determined following the methodology outlined above. Then, the grid is refined by increasing the density of nodes, and new Z values are generated at the additional nodes. Since large-scale spatial correlation was already taken care of at the first step, the additional Z values are computed conditioning only on the four nearest nodes, thus utilizing the screening effect often employed in geostatistical applications.

  • References
    Bellin A., Y. Rubin, Hydro_gen: A Spatially Distributed Random Field Generator for Correlated Properties, Stochastic Hydrology and Hydraulics, 10(4), 253-278, 1996.
  • Gomez Hernandez, J. J., and R. M. Srivastava, Isim3d: An ansi-c three-dimensional multiple indicator conditional simulator program, Computers & Geosciences, 16(4), 395-440, 1990.
  • Rubin Y., and A. Bellin, Conditional Simulation of Geologic media with Evolving Scales of Heterogeneity, In: Scale Dependence and Scale Invariance in Hydrology, ed. G. Sposito, Cambridge University Press, 1997 (in press).

Examples of application

Covariance function
Color Table
Exponential Isotropic
Exponential Anisotropic (anisotropy ratio e=10)
Gaussian Isotropic
Gaussian Anisotropic (anisotropy ratio e= 10)
Self Similar
Self Affine