Random variables and random functions

Let us start with an example related to groundwater hydrology.

• We wish to study the steady state response of a confined heterogeneous aquifer, under conditions of water withdrawal. We measure the hydraulic conductivity at various points in the aquifer for confidence in the results. Hence it is convenient to consider modeling the aquifer as a medium with random characteristics.

To this aim, we perform a pumping test: withdraw water from a well at specified rates and observe water drawdown at different wells. By means of pumping test theory, calculate transmissivities at the different well locations (points in space). (Note that the number of observation wells is generally small due to the high drilling costs).

The possible outcomes in our experiment (i.e. the corresponding profiles of the transmissivities) are functions which assume a value at each point in the space which we are working in. Let $S \subset$   $^3$ be the mathematical representation of this space (the domain of the aquifer). Then the elements of $\Omega$ are the functions $\omega : S \rightarrow$    where

$\omega(x)$ = transmissivity at point $x$ ( $\omega \in \Omega$).

and we assume that these functions are continuous ( $\Omega = C^1(S)$). A more classical notation for the random transmissivity at point $x$ when the profile is $\omega$ would be:

$$T(x\omega) \equiv \omega(x)$$

$\Omega$ is made up of continuous functions. Let $h:S\rightarrow$ be the piezometric head ( $h\in C^2(S)$). The aquifer is confined and the steady state mass balance equation together with Neumann (no flow) boundary conditions is:

$$\frac{\partial}{\partial x_i} \left( T_{ij}(\vec{x}, \omega) \frac{\partial h}{\partial x_j} \right) = q(x) S$$ (1.2)

$${\partial u \over \partial n} = 0 \partial S$$ (1.3)

where $q(x)$ represents the rate of water withdrawal or injection per unit time and unit volume.

The solution to this boundary value problem is a function

$$h : S \times \Omega \rightarrow$$   

For each point $x \in S$ we determine a function $\omega \mapsto h(x, \omega)$ which represents the piezometric level at the point $x$ if the transmissivity rofile in the medium is $\omega \in \Omega$.

The piezometric head is a random function of the random variable $\omega$. Note that a random variable (RV) is also a function from the sample set to the real axis ( $\omega : S \rightarrow$   ), while the random function $h(x,\omega)$ is a function $S \times \Omega h \rightarrow$   . In other words, $h(x_o,\omega)$ and $h(x_1,\omega)$ are two random variables if $x_o\neq x_1$; $h(x,\omega_o)$ is a realization of $h(x,\omega)$, i.e. the profile (function of $x$) of $T$ when (among all possible observations) only $\omega_o(x)$ is assumed.

In general we are interested in real functions $X$ defined in the space of outcomes, and in associating a probability with all the events of the type

• The value of $X$ is larger than $b \in$   .
• The value of $X$ is not larger than $a \in$   .
• The value of $X$ falls in the interval $[a, b]$, etc.

These events will be denoted by means of the symbols $(X > b)$, $(X \le a)$, $(a \le X \le b)$, etc.