The ``Turning Bands'' Method

Consider the line $D$, in the three-dimensional space $R^3$, as shown on Figure 7.2. Consider the one-dimensional random function $Y(/boma{x}_{D}$) defined on line $D$; this random function is second-order stationary, with a zero expectation, $E(Y(/boma{x}_{D})) = 0$ and a one-dimensional covariance $C^{(1)}(/boma{h}_{D}).$

Let $/boma{x}_{D}$, be the projection of any point $/boma{x}$ onto the line $D$, and consider the three-dimensional random function defined by $Z_{1}(/boma{x}) = Y(/boma{x}_{D}), / /forall {/boma x} /in R^3$. This random function $Z_{1}(/boma{x})$ is second-order stationary, with a zero expectation and a three-dimensional covariance equal to

where $/boma{h}_{D}$ is the projection of the vector $/boma{h}$ onto the line $D$.

In practice, to produce a realization of $Z_{1}(/boma{x})$, the value $z_{1}(/boma{x}_{D})$, simulated at the point $/boma{x}_{D}$, on the line $D$, is given to all points inside the slice (or the band) centered on the plane $/{{/boma x}_D = constant/}$ perpendicular at ${/boma x}_D$ to $D$, as illustrated in Figure 7.2. The thickness of this slice is the equidistance between the simulated values on the line $D$.

Figure 7.2: Geometry of the ``Turning band''-methode.

We then consider $N$ lines $D_{1},/ldots,D_{N}$ corresponding to the directions of the unit vectors $k_1,/ldots, k_N$ uniformly distributed over the unit sphere. On each line $D_i$ a realization $y({/boma x}_{D_i})$ of a random function $Y({/boma x}_{D_i})$ is generated isomorphic to $Y({/boma x}_{D})$, the $N$ random functions $/{Y({/boma x}_{D_i}), i=1,/ldots,N/}$ being independent. Using the method outlined above, a three-dimensional realization $z_i({/boma x})= y({/boma x}_{D_i}), / /forall {/boma x} /in R^3$, is made to correspond to each one-dimensional realization $y({/boma x}_{D_i})$.

A final value is then assigned to each point $x$ by taking the sum of the $N$ contributions from the $N$ lines:

(As the $N$ lines can be deduced from each other by rotation, so can the slices (or bands) which they define in the three-dimensional space, whence the name ``turning bands'' method.)

The resulting realization $z_s({/boma x})$ is a realization of a three-dimensional random function $Z_s({/boma x}) = Z_s(u,v,w)$ which is of course, of second-order stationary and has a zero expectation. The covariances are treated in practice in the way that the three-dimensional covariance $C(r)$ is fixed and the one-dimensional covariance $C^{(1)}(s)$ to be simulated on each of the $N$ lines is thus given by the derivative

where $C(s)$ denotes the parametric representation of the covariance function of $Z$ in direction $D$.

It is thus always possible to determine the one-dimensional covariance $C^{(1)}(s)$ to be imposed on the random function $Y({/boma u}_i)$ simulated on the lines $D_i$. By turning these lines and their corresponding bands in $R^3$, the required three-dimensional simulation $z_s(u, v, w)$, with the specified isotropic covariance $C(r)$ is obtained.

We remark that the number of lines $N$ chosen in practice is often 15.