Packages
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 twodimensional
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) selfsimilar or selfaffine (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 selfsimilar 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 predetermined
arbitrary grid. The generation technique resembles the Sequential
Indicator Simulation (SIS) method, proposed by [GomezHernandez
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 x_{0} 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(x_{0}) 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(x_{0} ), 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 singlescale 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 multistage
grid refinement. The grid can be refined to any apriori 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 largescale
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), 253278, 1996.
 Gomez Hernandez, J. J., and R. M. Srivastava, Isim3d: An
ansic threedimensional multiple indicator conditional simulator
program, Computers & Geosciences, 16(4), 395440, 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 

