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A Gamma-function Model for
D-variate Spectra and Cross-spectra
for Large Scale Frequency Domain
Simulation of Stationary Random
Functions in Rn
Leon E. Borgmanl and John W. Kern2
Abstract.--Spatially correlated random fields are used to model phenomena
ranging in scale from description of the distribution of fibers in a sheet of
paper to simulation of the forces of ocean waves along a beach. To make use
of h s framework with computationally efficient frequency domain methods,
characterization of auto- and cross-spectra are required. Univariate and crossspectral models based on the gamma probability density function are proposed
for d-variate random vectors in n-dimensional space with geometric
anisotropy. The corresponding class of spatial covariance functions is also
derived. Modeling the second order structure between pairs of random
variables is shown to simplify the usually tedious lag domain modeling of
cross-covariance functions.
INTRODUCTION
Spatially correlated random fields have been used to model physical
phenomena ranging in scale from description of the distribution of fibers in a sheet of
paper (Ahuja and Schachter, 1983), to simulation of the forces of ocean waves along
a beach (Borgman et al. 1994). Prediction techniques usually known as kriging,
(Krige 195l), Matheron (1971) and Journel and Huigbregts (1 978) are used to
produce best linear unbiased predictions at unsampled locations conditioned on
known data . In contrast, conditional simulation methods (Journel and Isaaks 1984),
(Borgman et a1.1984, 1993), are used to produce an ensemble of realizations of the
random process where the correlation structure and natural variability of the original
'process are preserved and known data are interpolated.
Conditional simulations can be produced through space domain or frequency
domain methods. However, for large scale regional simulations (100 by 100 grids or
more), the space domain methods are slow and require large amounts of computer
memory to execute. To overcome the limitations of space domain simulation
' Statistician, Department of Statistics and Geology/Geophysics, University of Wyoming, Laramie WY.
Statistician, Western Ecosystems Technology Inc. Cheyenne WY.
techruques, the faster less memory-intensive frequency domain methods were
developed, allowing simulation of larger spatially correlated random fields at much
greater speeds.
To obtain the spectral density models required for frequency domain techniques,
covariance functions developed for use in space domain applications such as the
spherical model, hole effect model or members of the Matern class (Matern 1960),
have typically been Fourier transformed to frequency domain. In general, these
spectral density models have little intuitive connection with space domain processes
and in some cases demonstrate pathologies in practice.
With the exception of Goulard and Voltz (1992), there has been little development
of cross covariance/spectral density models for simulation of multivariate correlated
random fields. Since few environmental or ecological problems are univariate in
nature, this lack of available multivariate models has limited application of conditional
simulation techniques to these areas.
We propose a class of univariate and multivariate spectral/covariance models
suitable for simulation of regional scale processes. The class of models is flexible and
easily interpreted using properties of the individual univariate processes..
UNIVARIATE SPECTRAL MODEL WITH INTUITIWC PARAMETERS
The prcposed models will be developed for second order stationary random functions
(Cressie 1991, page 53). Given any d dimensional covariance hnction C(h)
corresponding to a second order stationary random function such
that [I C(h) 1 dh < m , there exists a d-dimensional spectral density function given
by the Fourier transform (Bochner 1955)
A spectral density k t i o n S(f) is isotropic if S(f)=G(I f 1) for some scalar hnction G,
and radial if there exists some d dimensional linear transformation A, such that
S(f)=G(I MI). In other words a space with radial spectral density fimction can be
deformed through linear transformation to an isotropic space.
We propose an anisotropic radial spectral model with an ellipsoidal base, and radial
fhction based on the gamma probability density
where p represents fi-equency scaled in ellipsoidal coordinates. A subset of the family
of gamma radial hnctions is given in figure 1.
Intuitive Motivation of Parameters
At large scales, many physical processes exhibit nearly-periodic behavior (i.e. the
human eye observes repetition although the pattern also has a random component).
Examples include topography of mountainous terrain, vertical height of ocean waves,
recurrence of vegetative patches and
porosity of fractured media. Simulations
of these processes at large scales should
include these "pseudo-periodicities".
The mode frequency of the gamma
spectral density hnction controls the
periodicity of a simulation. If the
spectrum is very peaked, (ar>>l), most
of the variation will be concentrated in
oscillations at a single frequency. The
Frequency f
simulation will be nearly perfectly
periodic like an egg carton. In contrast, Figure 1. Family of radial gamma spectral
if the spectrum is very flat ( a = I), the models with mode frequency 0.2 and a
corresDondinp: simulated data will ranging from 1.5 to 5.5 and p = l/(a-I). These
I
u
--
models all have dominant wavelength 5 units
contain
at
with varying amounts of high frequency noise.
frequencies (white noise). Anisotropies
controlled by the ellipsoidal base of a
radial spectrum as in figure (2) cause "ridges" in simulated data. Few physical
processes are isotropic (one rarely sees round mountains) and hence this feature is
essential to realistic simulations. The axes of the ellipsoidal base are easily chosen or
estimated from data. Finally the scaling constant Y, may be chosen to give simulated
data the proper variance. Using the gamma spectral density model in frequency
domain, these properties are easiiy controlled to produce conditional or unconditional
simulations. The same properties are implicit in space domain methods although not
dirrectly controlled by the practitioner.
COVARIANCE FUNCTIONS CORRESPONDING TO RADIAL SPECTRA
Let S(f) be a radial spectral density hnction with radial fimction G, corresponding
to a second order stationary random fbnction with f~ Rn (n dimensional Euclidean
space), and ellipsoidal base given by f TBf = 1 , whereB = VL V T ,is the
eigenvalueieigenvector decomposition of B. Making the substitutions
+ = L " ~ v ~ ~I J, J = I ~ / " ~ a n d V~"~r,inequation(l)gives
h=
The n-dimensional integral in equation (3), can be reduced to a one dimensional
Hankel transform through introduction of the hyper-spherical coordinate
transformation (Miller 1964 ). Computing n- 1 integrals gives
d m ,
is a function of
where R =
h, and J, is the Bessel function of order
v. This is the Hankel transform of order
(n-2)12 which is tabled for many
common hnctions by Ditkin and
Prudnikov (1965 p. 432). Note that C
is a fbnction of I L -lI2v
T hI and hence
must be radial with an ellipsoidal base.
Further, the covariance hnction is given
a
of
by the Hankel transform of the radial
fbnction . These results for second Figure 2. Two dimensional radial gamma
order stationary random functions can spectral density function with 3 to 1
be combined with the converse from anisotropy.
Taheri (1980) and Hagen (1982).
In summary, we have that for S(f) and C(h), the spectral densitylcovariance pair with
radial functions G and g respectively; S(f) is radial with ellipsoidal base f T ~=f 1 if
and only if C(h) is radial with ellipsoidal base h T ~ - ' =
h 1, and
= 4.0
if and only if
m
Rdio
Anisdmpy: 3:1
The Gamma Spectral Model for n Dimensional Space
These results can be used to derive the gamma and other spectral density
covariance pairs in n dimensions. Using equation (5) and the Hankel transform of the
gamma radial fbnction (Ditkin and Prudnikov 1965, p.439, eq. 11.38), the n
dimensional covariance hnction corresponding to the gamma spectral model is given
by
where pP(z), is the Legendre Spherical function (Stegun p. 332, eq. 8.1.2) and the
v
scaling constant Kn =
02*(
2pa+"-' F(a +n- 1) r(n/2) nn" )-', is obtained by
forcing the integral of the spectrum over
the full frequency space to be 02. A
subset of the class of radial functions is
plotted in figure (3). It should also be
noted that the class of covariance
functions is quite flexible containing
members with shapes ranging from that
of the Gaussian to the hole effect model.
Widely varying model types can be
~.........,.........,.........,....,,,,,,,.,......,
chosen through parameter estimation
Scaled Distance h h ,
alone as opposed to fitting several
models from separate classes followed by
Figure 3. Family of covariance functions
ad hoc procedures for choosing the best corresponding to the gamma spectral model
class of models.
with a ranging from 0.01 to 0.61 in increments
dm
-0.50
0.00
0.20
0.40
0.60
0.80
1.00
of 0.2 and from 1 to 46 in increments of 5.
MULTI-VARIATE CROSSSPECTRA IN IRn
To simulate multivariate (vector) random processes with frequency domain
methods, models of cross spectra are required. In general these models are obtained
through modeling the cross covariance in lag domain. This is sometimes
unsatisfactory in that the cross covariance is typically not symmetric about zero and
does not have any a-priori expected form. However, with some simplifying
assumptions, a reasonable cross spectral model can be obtained which is intuitive in
nature. In particular, it is assumed that the coherence and phase are approximately
constant in the range of frequencies with significant power in the univariate spectra.
Let x be an n dimensional vector in Rn,and V,(x), k=l,2,3.. .d be a set of d
intercorrelated second order stationary random functions. The cross covariance
function between the ijthpair of random functions is
a rJh )
=
E{[v,(~)- P ~ ] [ v ,+h)
( ~ - I$])
(8)
For i = j, this is the auto covariance function. The cross spectrum for any pair is
S@
=
J ~ , ( h ) e -i2nf Th dh
= c$f)
-iq,(f)
n
(9)
where ci and q,are the co- and quad-spectra. In general, C&h) may not be symmetric
about the origin, and hence S&f) will be complex valued. However, since C,(h) is real
valued, the co-spectrum is even and the quad-spectrum is odd (Borgman, 1993). The
coherence and phase are defined in terms of the co- and quad-spectra as
These relations can be looked at as a polar transformation with angle $e and radius
r =
. Solving for c, and q, gives
.-/,
where the argument f has been suppressed (although assumed to be present) for
notational convenience. Note that if S, is zero, then the co and quad- spectra are also
zero. Therefore values for y, and +e are required only for those frequencies where
the individual univariate spectra are nonzero. The proposed model is
( y , , / m e i m q , for
F~ < f T ~ f ( ~flu2 o
where the phase and coherence are assumed constant in the range of frequencies
where the univariate spectra are non-zero, and where FL and Fu are the lower and
upper bounds on that region The model is defined piecewise to allow the quadspectrum to be odd. The sets on the right side of equation (12) are denoted Q' and
a-respectively. These sets could be generalized to any half space.
To apply the model, estimates of S,, S,, y,, and 4 are required. The marginal
spectra may be estimated and modeled independently using appropriate spectral
models. The gamma is proposed here due to its flexibility in shape and ease of
application, although other spectral models may be used.
Given that each of the d marginal spectra have been estimated or reasonably chosen,
the coherence and phase remain to be estimated. By expressing the cross-covariance
hnctions in terms of the inverse Fourier transform of the cross-spectra, the phase and
coherence parameters (assumed constant) may be estimated in lag domain. The
Fourier transform of the cross spectra may be expressed as a sum of sine and cosine
integrals C, ( h) = y ,cos(@,) Ic(h ) + y ,sin($,) Is( h ),where I, and 4 are given by
Ic J J ~ c o s ( 2 s r f Th) df;
n
I$ = / J G s i n ( 2 n f
and
Th)df - / , / G s i n ( 2 n f
(13)
Th)df:
If Cij(h) has been estimated possibly with some form of bin averaging or
nonparametric smoothing at a set of N lags (h;, k=l,2,3,.. .,N), estimates of the
phase and coherence may be obtained from the N linear equations in ~,cos$,, and
y 'J. . sin+q,
These equations can be solved by least squares to obtain estimates of gi, and gi,. The
estimated phase and coherence are given by
An example of a cross correlation
function corresponding to the proposed
gamma model is plotted in figure (4).
Each of the marginal processes have
univariate gamma spectra with ellipsoidal
bases. The maximum correlation is
located 270" counter clockwise from the
first lag domain
axis h,, which
corresponds to a constant phase shift of
270 . The minimum correlation is
negative and located
from the
ma~klum~0ITelation.This type of Cross
0
Figure 4. Cross covariance function for 2
correlated anisotropic random functions with
radial gamma spectral models, coherence 1.0
and phase
2700.
correlation function could be used to simulate the relation between rainfall and
surface runoff where a directional lag would be anticipated in the spatial correlations.
DISCUSSION
Frequency domain conditional simulation techniques allow the user to directly
control pseudo-periodicities anisotropy and high frequency noise in simulated data
sets. The univariate gamma spectral density model allows the user to easily control
these characteristics by directly identifying parameters with physical properties. The
gamma model has been applied by and Peacock and Kern (1995) to model hydraulic
conductivity in the Powder river basin in Wyoming at regional scales.
The proposed multivariate models provide a Framework for estimating cross spectra
or equivalently cross-correlation functions in which complicated lag domain functions
are simplified to a pair of band limited functions of frequency which are typically zero
over most of the frequency domain. Further research into this area should include
investigation of non-constant functional forms for the phase and coherence.
Frequency domain methods should be of particular interest to geographers and
environmental scientists handling large data sets in GIs systems. Large scale
simulations at computational speeds obtained with frequency domain methods are not
feasible with space domain procedures.
ACKNOWLEDGMENTS
We are grateful for hnding of this work which came from a cooperative agreement
between the state of Wyoming Department of Environmental Quality, Wyoming State
Engineers Office, the United States Office of Surface Mines, the United States Bureau
of Land Management and the University of Wyoming.
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BIOGRAPHICAL SKETCH
Leon E. Borgman is a professor of statistics and geology and geophysics at the
University of Wyoming, Laramie, Wyomilig. He has specialized in application of
stationary random function theory to a wide range of environmental science,
hydrgeologic and oceanographic/civil engineering problems.
John W. Kern is a consulting statistician with Western Ecosystems Technology Inc.,
Cheyenne, Wyoming. John has primarily specialized in application of stationary
random function theory to environmental, hydrogeological and ecological problems.