Spectral simulation of vector random fields with stationary Gaussian increments in d-dimensional Euclidean spaces

Abstract

This paper addresses the problem of simulating multivariate random fields with stationary Gaussian increments in a d-dimensional Euclidean space. To this end, one considers a spectral turning-bands algorithm, in which the simulated field is a mixture of basic random fields made of weighted cosine waves associated with random frequencies and random phases. The weights depend on the spectral density of the direct and cross variogram matrices of the desired random field for the specified frequencies. The algorithm is applied to synthetic examples corresponding to different spatial correlation models. The properties of these models and of the algorithm are discussed, highlighting its computational efficiency, accuracy and versatility.

Appendices

Appendix 1: Covariance and variogram

Consider a scalar intrinsic random field $Y=\{Y(\mathbf {x}):\mathbf {x}\in \mathbb {R}^d\}$ and denote by $\gamma (\mathbf {h})$ its variogram for the separation vector $\mathbf {h}$. For given locations $\mathbf {x}$, $\mathbf {x}'$ and $\mathbf {x}_0$, let us express the covariance between the two increments $Y(\mathbf {x}) - Y(\mathbf {x}_0)$ and $Y(\mathbf {x}') - Y(\mathbf {x}_0)$ in terms of the variogram:

$$\begin{aligned} C(\mathbf {x}, \mathbf {x}', \mathbf {x}_0):=\,\,& {} \mathrm {cov}\left\{ Y(\mathbf {x})-Y(\mathbf {x}_0),\, Y(\mathbf {x}')-Y(\mathbf {x}_0)\right\} \\= \,\,& {} \mathrm {cov}\left\{ Y(\mathbf {x})-Y(\mathbf {x}'),\, Y(\mathbf {x}')-Y(\mathbf {x})\right\} + \mathrm {cov}\left\{ Y(\mathbf {x}')-Y(\mathbf {x}_0),\, Y(\mathbf {x}')-Y(\mathbf {x}_0)\right\} \\&- \mathrm {cov}\left\{ Y(\mathbf {x}')-Y(\mathbf {x}_0),\, Y(\mathbf {x})-Y(\mathbf {x}_0)\right\} + \mathrm {cov}\left\{ Y(\mathbf {x})-Y(\mathbf {x}_0),\, Y(\mathbf {x})-Y(\mathbf {x}_0)\right\} \\= & {} -2 \gamma (\mathbf {x}-\mathbf {x}') + 2\gamma (\mathbf {x}' - \mathbf {x}_0) - C(\mathbf {x}, \mathbf {x}',\mathbf {x}_0) + 2\gamma (\mathbf {x}- \mathbf {x}_0). \end{aligned}$$

Accordingly:

$$\begin{aligned} C(\mathbf {x}, \mathbf {x}^{\prime}, \mathbf {x}_0) = \gamma (\mathbf {x}- \mathbf {x}_0) + \gamma (\mathbf {x}^{\prime} - \mathbf {x}_0) - \gamma (\mathbf {x}-\mathbf {x}^{\prime}). \end{aligned}$$

Note that this covariance is unchanged by shifting the locations $\mathbf {x}$, $\mathbf {x}'$ and $\mathbf {x}_0$ by a given vector $\mathbf {h}$, i.e., $C(\mathbf {x}, \mathbf {x}^{\prime}, \mathbf {x}_0) = C(\mathbf {x}+\mathbf {h}, \mathbf {x}^{\prime}+\mathbf {h}, \mathbf {x}_0+\mathbf {h})$, showing that the intrinsic random field has second-order stationary increments. Furthermore, by taking $\mathbf {x}_0 = \mathbf {0}$ and considering that $Y(\mathbf {0}) = 0$, it ensues:

$$\begin{aligned} \mathrm {cov}\left\{ Y(\mathbf {x}),\, Y(\mathbf {x}^{\prime})\right\} = \gamma (\mathbf {x}) + \gamma (\mathbf {x}^{\prime}) - \gamma (\mathbf {x}-\mathbf {x}^{\prime}), \end{aligned}$$

which shows that the knowledge of the variogram is equivalent to that of the covariance function of the random field.

Appendix 2: Existence conditions for Example 1

The spectral density matrix $\mathbf {f}(\mathbf {u})$ defined in Eq. (10) is Hermitian because it is a real symmetric matrix. To check that it is positive semi-definite, one therefore only needs to verify that its determinant is non-negative (Horn and Johnson 1985), that is:

$$\begin{aligned} f(\mathbf {u},1,\theta _1)\cdot f(\mathbf {u},1,\theta _2) \ge \rho ^2f^2(\mathbf {u},1,\theta _{12}), \end{aligned}$$

that is:

$$\begin{aligned} \frac{\theta _1\Gamma \left( \frac{\theta _1+d}{2}\right) }{2\Gamma \left( 1-\frac{\theta _1}{2}\right) \pi ^{\theta _1}\Vert \mathbf {u}\Vert ^{\theta _1}}\cdot \frac{\theta _2\Gamma \left( \frac{\theta _2+d}{2}\right) }{2\Gamma \left( 1-\frac{\theta _2}{2}\right) \pi ^{\theta _2}\Vert \mathbf {u}\Vert ^{\theta _2}} \ge \rho ^2\frac{\theta _{12}^2\Gamma ^2\left( \frac{\theta _{12}+d}{2}\right) }{4\Gamma ^2\left( 1-\frac{\theta _{12}}{2}\right) \pi ^{2\theta _{12}}\Vert \mathbf {u}\Vert ^{2\theta _{12}}}. \end{aligned}$$

(13)

From this inequality to hold, one first needs to verify the following:

$$\begin{aligned} \frac{1}{\Vert \mathbf {u}\Vert ^{\theta _1+\theta _2}}\ge \frac{1}{\Vert \mathbf {u}\Vert ^{2\theta _{12}}}. \end{aligned}$$

(14)

If $2\theta _{12}<\theta _1+\theta _2$, inequality (14) is not satisfied for $\Vert \mathbf {u}\Vert >1$, whereas if $2\theta _{12}>\theta _1+\theta _2$, inequality (14) is not satisfied for $\Vert \mathbf {u}\Vert <1$. Therefore, one must have

$$\begin{aligned} \theta _{12}=\frac{\theta _1+\theta _2}{2}, \end{aligned}$$

in which case Eq. (14) obviously holds $\forall \mathbf {u}\in \mathbb {R}^d$. Under this condition, one obtains from Eq. (13):

$$\begin{aligned} \rho ^2\le & {} \frac{4\theta _1\theta _2}{(\theta _1+\theta _2)^2}\frac{\Gamma \left( \frac{\theta _1+d}{2}\right) \Gamma \left( \frac{\theta _2+d}{2}\right) \Gamma ^2\left( 1-\frac{\theta _1+\theta _2}{4}\right) }{\Gamma \left( 1-\frac{\theta _1}{2}\right) \Gamma \left( 1-\frac{\theta _2}{2}\right) \Gamma ^2\left( \frac{\theta _1+\theta _2}{4}+\frac{d}{2}\right) }. \end{aligned}$$

Appendix 3: Existence conditions for Example 2

The spectral density matrix $\mathbf {f}(\mathbf {u})$ defined in Eq. (12) is Hermitian positive semi-definite if its determinant is non-negative, i.e.:

$$\begin{aligned} f(\mathbf {u},b_1,\theta _1)\cdot f(\mathbf {u},b_2,\theta _2) \ge \rho ^2g^2(\mathbf {u},a_{12},\nu _{12}), \end{aligned}$$

that is:

$$\begin{aligned} \varphi (\Vert \mathbf {u}\Vert ):=\frac{(2\pi a_{12})^{2d}\Vert \mathbf {u}\Vert ^{\theta _1+\theta _2+2d}}{\left( 1+(2\pi a_{12})^2\Vert \mathbf {u}\Vert ^2\right) ^{2\nu _{12}+d}}\le \frac{\omega }{\rho ^2}, \end{aligned}$$

(15)

with

$$\begin{aligned} \omega =\frac{\theta _1\theta _2\Gamma \left( \frac{\theta _1+d}{2}\right) \Gamma \left( \frac{\theta _2+d}{2}\right) \Gamma ^2(\nu _{12})}{4\pi ^{\theta _1+\theta _2}b_1^{\theta _1}b_2^{\theta _2}\Gamma \left( 1-\frac{\theta _1}{2}\right) \Gamma \left( 1-\frac{\theta _2}{2}\right) \Gamma ^2\left( \nu _{12}+\frac{d}{2}\right) }. \end{aligned}$$

The map** $\varphi :\mathbb {R}_+\rightarrow \mathbb {R}$ is unbounded if $\theta _1+\theta _2>4\nu _{12}$, in which case inequality (15) cannot be satisfied. In the converse $(\theta _1+\theta _2\le 4\nu _{12})$, the maximum of $\varphi$ is found to be

$$\begin{aligned} \varphi _{\max }=\frac{(\theta _1+\theta _2+2d)^{(\theta _1+\theta _2)/2+d}}{(4\nu _{12}+2d)^{2\nu _{12}+d}}\times \frac{(4\nu _{12}-\theta _1-\theta _2)^{2\nu _{12}-(\theta _1+\theta _2)/2}}{(2\pi a_{12})^{\theta _1+\theta _2}}. \end{aligned}$$

 

Therefore, for the spectral density matrix $\mathbf {f}(\mathbf {u})$ to be positive semi-definite for all $\mathbf {u}\in \mathbb {R}^d$, the following necessary and sufficient conditions must be fulfilled:

$$\begin{aligned} \theta _1+\theta _2\le 4\nu _{12} \end{aligned}$$

(16)

and

$$\begin{aligned} |\rho |\le \rho _{\max } = \sqrt{\displaystyle {\frac{\omega }{\varphi _{\max }}}}. \end{aligned}$$

(17)

In the particular case when $\theta _1+\theta _2=4\nu _{12}$, then $\varphi _{\max }=(2\pi a_{12})^{-(\theta _1+\theta _2)}$, and the limit value for $|\rho|$ is given by

$$\begin{aligned} |\rho |\le \frac{\Gamma \left( \frac{\theta _1+\theta _2}{4}\right) }{2\Gamma \left( \frac{\theta _1+\theta _2}{4}+\frac{d}{2}\right) }\sqrt{\left( \frac{2a_{12}}{b_1}\right) ^{\theta _1}\left( \frac{2a_{12}}{b_2}\right) ^{\theta _2}\frac{\theta _1\theta _2\Gamma \left( \frac{\theta _1+d}{2}\right) \Gamma \left( \frac{\theta _2+d}{2}\right) }{\Gamma \left( 1-\frac{\theta _1}{2}\right) \Gamma \left( 1-\frac{\theta _2}{2}\right) }}. \end{aligned}$$