Local averaging

The input parameters of $C$ that relate to the mean, standard deviation and spatial correlation length are assumed to be defined at the point level. Due to the finite size of each finite element, point statistical distribution must be averaged over the element. This results in reduced $\sigma_{\ln C}$ in the case of log-normal distribution and reduced $\sigma_{C}$ in the case of normal distribution. $\mu_{\ln C}$ in the first case and $\mu_{C}$ in the second case remain unaffected.

The locally-averaged standard deviations ( $\sigma_{\ln CA}$, $\sigma_{CA}$), which are used in Eqns. (26, 27), are calculated from their point values using

$$\sigma^2_{\ln CA} =\gamma \ \sigma^2_{\ln C} \sigma^2_{CA} =\gamma \ \sigma^2_{C}$$ (28)

where $\gamma$ is the variance reduction factor calculated by integration of the Markov function (22). In 1D for a finite element of side length $\alpha\theta_C$

$$\gamma=\frac{2}{\left(\alpha\theta_C\right)^2}\int_0^{\alpha\theta_C} \exp\left(-\frac{2}{\theta_C}\sqrt{x^2}\right) (\alpha\theta_C-x)dx$$ (29)

In 2D for square finite element of side length $\alpha\theta_C$

$$\gamma=\frac{4}{\left(\alpha\theta_C\right)^4}\int_0^{\alpha\thet... ...rac{2}{\theta_C}\sqrt{x^2+y^2}\right) (\alpha\theta_C-x)(\alpha\theta_C-y)dx dy$$ (30)

In 3D for hexahedral finite element of side length $\alpha\theta_C$

$$\gamma=\frac{8}{\left(\alpha\theta_C\right)^6}\int_0^{\alpha\thet... ...+y^2+z^2}\right) (\alpha\theta_C-x)(\alpha\theta_C-y)(\alpha\theta_C-z)dx dy dz$$ (31)

For the anisotropic case in 2D:

$$\gamma=\frac{4}{l^4}\int_0^{l}\int_0^{l} \exp\left[-2\sqrt{\left(... ...a_{Cx}}\right)^2 + \left(\frac{y}{\theta_{Cy}}\right)^2}\right] (l-x)(l-y)dx dy$$ (32)

and for the anisotropic case in 3D:

$$\gamma=\frac{8}{l^6}\int_0^{l}\int_0^{l}\int_0^{l} \exp\left[-2\s... ...right)^2 + \left(\frac{z}{\theta_{Cz}}\right)^2}\right] (l-x)(l-y)(l-z)dx dy dz$$ (33)

In order to calculate the variance reduction due to local averaging correctly, all elements in the mesh should be of the same size and all elements should be regular squares. If irregular elements are used, exact value of $\gamma$ is in Tochnog approximated by calculation of $\gamma$ for an equivalent square element using Eq. (31) with area equal to an average area of all elements in the mesh.

The approximate value of $\gamma$ requires that you use as much as possible elements of the same size and shape in the complete calculation domain.