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The present paper introduces a new approach to simulate any stationary multivariate Gaussian random field whose cross-covariances are predefined continuous and integrable functions. Such a field is given by convolution of a vector of univariate random fields and a functional matrix which is derived by Cholesky decomposition of the Fourier transform of the predefined cross-covariance matrix. In contrast to common methods, no restrictive model for the cross-covariance is needed. It is stationary and can also be reduced to the isotropic case. The computational effort is very low since fast Fourier transform can be used for simulation. As will be shown the algorithm is computationally faster than a recently published spectral turning bands model. The applicability is demonstrated using a common numerical example with varied spatial correlation structure. The model was developed to support simulation algorithms for mineral microstructures in geoscience.

The theory of uni- and multivariate random fields has been used extensively in a broad range of science and engineering disciplines such as meteorology, astrophysics and geosciences ( [

Of particular interest is the general case when the components of the multivariate vector are not independent. Numerous models of the resulting so-called cross- covariance and methods for simulation of such fields have been given in the literature (see [

A common classification is to distinguish between algorithms giving realizations which are exactly Gaussian (autoregressive, moving-average, circulant-embedding, discrete spectral simulation, see [

Another distinction is whether the approach is restricted to the simulation of the fields at regular grid locations (e.g. the ones mentioned for exactly Gaussian) or not (e.g. the ones for approx. Gaussian), the Cholesky decomposition approach ( [

Many models are also limited to certain parametric models for the cross-covariances. Such as the methods based on convolution, e.g., the so-called kernel convolution and covariance convolution approach ( [

covariance has

Recently, a very interesting ansatz without this limitation was given in [

The spectrum convolution approach proposed in this paper creates exact stationary Gaussian random fields. Such a multivariate field with values in

In a certain sense it is similar to coregionalization ( [

The outline of the present paper is as follows. Section two introduces notation and provides the theoretical frameworks and results for multivariate random fields as well as a closer look at two other approaches. This is followed by a description of the new model in section three, which contains the proof of validity. Example simulation and a comparison of the computation effort to another model is done in the second last section. The paper is topped off with a conclusion section.

This section is intended to make the reader familiar with notation, the mathematical objects used in this paper and some theoretical results.

A N-variate second-order random field

Common first and second-order characteristics are the field

which is called cross-covariance function of Z. This function satisfies positive semi- definiteness in the sense that

for all

Without loss of generality, throughout the rest of the paper, the trend m of the random fields is assumed to be constant equal to

In contrast to the univariate case, isotropy has different interpretations in the multi- variate case (see [

A multivariate Gaussian field is a random field where all finite dimensional distri- butions are normal distributions. Similar to the univariate case it is fully described by the aforementioned characteristics m and

The main theorem of this paper makes use of the famous Kolmogorov-Chentsov theorem. See [

Theorem 1 (Kolmogorov continuity) Suppose that the random field

Then there exists a version of X on the closure of T with a.s. continuous paths.

Immediately one can deduce the following result.

Corollary 1 A stationary random field

Proof. On the one hand, by continuity,

From Theorem 1 follows the statement.

The following theorem from [

Lemma 1 Suppose

exist then X is normal with mean vector b and covariance matrix C.

Also important for our main result is the next lemma.

Lemma 2 Suppose

exist then Z is Gaussian with trend m and covariance function

Proof. The finite dimensional distributions of

Also required in what follows is the following easy to prove result from linear algebra.

Lemma 3 For all integers

As introduced in the beginning there are several procedures to construct and sample random fields with predefined cross-covariance ( [

a) A common method to correlate several random fields is called coregionalization (see [

The cross-correlation of Z is now given by the positive semi-definite matrix ( [

where

b) Recently, an improved spectral turning-bands algorithm for simulating stationary multivariate Gaussian random fields was presented in [

The approach reformulated and reduced in what follows for the stationary isotropic case in

Extending the very famous result of Bochner for the multivariate case the isotropic cross-covariance at lag

where

which can be rewritten using Hankel transform (see [

where

Fix an arbitrary probability density

where

for all

As shown in [

The main part of the paper is given in what follows. The following model can be seen as a spectral variant of coregionalization (Section 2.2) but also bears analogy to the spectral turning bands method.

Let

Theorem 2 The random vector field Z on

is multivariate Gaussian with trend

For the spectrum it holds

Proof.

・ Let

Equivalently, the d-torus is obtained from the d-dimensional hypercube by gluing the opposite faces together giving a cubic domain with periodic boundaries. Because of periodicity for a continuous function

・ For

It can be written as

for

・ For the trend it follows

since

・ For the cross-covariance function of

with Lemma 3

using the transformation

which does exist since all components were assumed to be integrable. According to Lemma 2 the limit

・ Furthermore, from (3) it follows that Z is stationary and by the convolution theorem the spectrum becomes

Modeling and simulation is based on the following corollary.

Corollary 2 If a cross-covariance

using Cholesky decomposition such that Z has cross-covariance

This section is intended to describe and to demonstrate how simulation of the new model works by means of a popular example. The second part evaluates the compu- tational effort in comparison to the spectral turning bands approach.

Based on Theorem 2 and Corollary 2 one can describe a very simple simulation algorithm.

Let functions

・ Discretize the domain by a grid

・ For

・ Calculate the discrete Fourier transform

・ For each grid point

・ Apply the inverse discrete Fourier transform to

The algorithm uses the same steps as the usual approach based on Fourier transform for an arbitrary Gaussian field ( [

There are numerous models to obtain valid cross-covariance functions. The popular multivariate Matérn model ( [

Define the isotropic Matérn covariance function, that is

The spectrum is given by (see [

Here

For

is a valid function.

Let us consider the quadratic domain ^{®}.

The components of ρ are shown in

A realization of the initial univariate random fields

This short paragraph compares theoretical and practical computation times of spectral turning bands and spectrum convolution on the same machine for varying numbers of sample locations.

For the spectral turning bands method it can be deduced that the computational cost are of

summands. For a common choice of

Considering the computational costs for our spectrum convolution approach, since d and N are small numbers, matrix multiplication and Fourier transform are the main operations to be considered. Both can be performed in

and process these subsets consecutively and/or independently.

Considering a practical example performed on a single core 2 ghz laptop cpu. Recall that all code was written in Mathematica software. In

As a contribution to the topic of stochastic processes in general and to random fields in particular, a new multivariate modeling approach was presented. It allows modeling and simulation of exact stationary multivariate Gaussian random fields where also the case of isotropy is covered. It is remarkable that any such a random field can be obtained, provided the components of its cross-covariance are continuous and integrable functions. The model is easy to implement since only simulation of normal variables, matrix multiplication, Cholesky decomposition and Fourier transform are required tools. It is shown numerically that the algorithm is more than 40 times faster than spectral turning bands and it can also be modified for parallel computing but, unfortunately, it is limited to sampling in locations on a regular grid.

This work has been funded by the German Federal Ministry of Education and Research (BMBF) in research program CLIENT “International Partnerships for Sustainable Technologies and Services for Climate Protection and the Environment” project “Mineral Characterization and Sustainable Mineral Processing Strategies for the Nam Xe Rare Earth Deposits in Vietnam” (REE Nam Xe).

Teichmann, J. and van den Boogaart, K.-G. (2016) Efficient Simulation of Stationary Multivariate Gaussian Random Fields with Given Cross- Covariance. Applied Mathematics, 7, 2183- 2194. http://dx.doi.org/10.4236/am.2016.717174