Probabilistic distributions

The section summarises mathematical formulation of the so-called random finite element method, as described, e.g. in [6].

Distribution of a random variable (e.g., $C$) is controlled by these basic parameters: parameters of the statistical distribution (typically mean value $\mu_C$ and standard deviation $\sigma_C$) and so-called correlation length $\theta_C$ that controls spatial variability of variable $C$.

Two probabilistic distributions are available in Tochnog: normal distribution and log-normal distribution. Probability function $P(C)$ of normal distribution is defined as:

$$P(C)=\frac{1}{\sigma_C\sqrt{2\pi}}\exp\left[-\frac{\left(C-\mu_C\right)^2}{2\sigma_C^2}\right]$$ (18)

where $\mu_C$ is a mean value and $\sigma_C$ is standard deviation. Probability function $P(C)$ of log-normal distribution is defined as:

$$P(C)=\frac{1}{C\sigma_{\ln C}\sqrt{2\pi}}\exp\left[-\frac{\left(\ln C-\mu_{\ln C}\right)^2}{2\sigma_{\ln C}^2}\right]$$ (19)

Quantities $\mu_{\ln C}$ and $\sigma_{\ln C}$ may be calculated from $\mu_{C}$ and $\sigma_{C}$ using

$$\sigma_{\ln C} =\sqrt{\ln\left[1+\left(\frac{\sigma_C}{\mu_C}\right)^2\right]} \mu_{\ln C} =\ln \mu_C-\frac{1}{2}\sigma^2_{\ln C}$$ (20)