Non-conditional Simulation in one Dimension

In some special cases (as with the spherical or the exponential model) it is possible to express the one-dimensional covariances $C^{(1)}(s)$ in the form of a convolution product of a function $f(u)$ and its transpose $f(-u)$:

Therefore a random function in one dimension can sometimes be simulated by using the method of moving average. This will be shortly illustrated with the spherical model.

The covariance function is defined by

$C^{(1)}(s)$ is computed via the derivative

which means,

This function may be written as convolution product of the function

with its transpose. We have, which is easily to be verified,

In practice we would proceed as follows: Generate at equidistant points $x_{i-k}, /ldots,$ $x_{i},/ldots,x_{i+k}$ with distance $b$ uniformly distributed random numbers $t_{i-k},/ldots,t_{i+k}$ in the interval $[-a/2, a/2]$. The distance $kb$ should be at least $a/2$. Then a realization of $Y$ (the one-dimensional random function) is produced through linear combination (moving average)

where of course, $f(kb) = 0$ for $/vert kb /vert > a/2$. Because of the discretization ($b$ should not be too large) the distribution properties of $y_{i}$ should be carefully checked.