Prior distributions available in enkf
When defining a parametrization of an ECLIPSE model for use with the Ensemble Reservoir Tool, a set of prior distributions are available. These distributions can be used whenever a prior distribution is asked for in the Parametrization keywords.
Prior distributions are a central concept in Bayesian statistics.
List of priors
To set a normal (Gaussian) prior, use the keyword NORMAL. It takes two arguments, a mean value and a standard deviation. Thus, the following example will assign a normal prior with mean 0 and standard deviation 1:
NORMAL 0 1
A stochastic variable is log normally distributed if it's logarithm is normally distributed. In other words, if X is normally distributed, then exp(X) is log normally distributed. Thus, a log normal prior is suited to model positive quanties with a heavy tail (tendency to take large values). To set a log normal prior, use the keyword LOGNORMAL. It takes two arguments, the mean and standard deviation of the logarithm of the variable. Here is an example:
LOGNORMAL 0 1
This distribution is specified as
TRUNCATED_NORMAL mu std min max
This distribution is not a proper truncated normal distribution and works as follows:
- Draw random variable X ~ N(mu,std)
- Clamp X to the interval [min, max]
I.e. the truncation [min,max] should be a 'rare' event.
A stochastic variable is uniformly distributed if has a constant probability density on a closed interval. Thus, the uniform distribution is completely characterized by it's minimum and maximum value. To assign a uniform distribution to a variable, use the keyword UNIFORM, which takes a minimum and a maximum value for a the variable. Here is an example, which assigns a uniform distribution between 0 and 1 to a variable:
UNIFORM 0 1
It can be shown that among all distributions bounded below by a and above by b, the uniform distribution with parameters a and b has the maximal entropy (contains the least information). Thus, the uniform distribution should be your preferred prior distribution for robust modeling of bounded variables.
A stochastic variable is log uniformly distributed if it's logarithm is uniformly distributed on the interval [a,b]. To assign a log uniform distribution to a a variable, use the keyword LOGUNIF, which takes a minimum and a maximum value for the output variable as arguments. The example
LOGUNIF 0.00001 1
will give values in the range [0.00001,1] - with considerably more weight towards the lower limit. The log uniform distribution is useful when modeling a bounded positive variable who has most of it's probability weight towards one of the bounds.
The keyword DUNIF is used to assign a discrete uniform distribution. It takes three arguments, the number bins, a minimum and maximum value. Here is an example which creates a discrete uniform distribution on [0,1] with 25 bins:
DUNIF 25 0 1
The ERRF keyword is used to define a prior resulting from applying the error function to a normally distributed variable with mean 0 and variance 1. The keyword takes four arguments:
ERRF MIN MAX SKEWNESS WIDTH
The arguments MIN and MAX sets the minimum and maximum value of the transform. Zero SKEWNESS results in a symmetric distribution, whereas negative SKEWNESS will shift the distribution towards the left and positive SKEWNESS will shift it towards the right. Letting WIDTH be larger than one will cause the distribution to be unimodal, whereas WIDTH less than one will create a bi-modal distribution.
The keyword DERRF is similar to ERRF, but will create a discrete output. DERRF takes 5 arguments:
DERRF NBINS MIN MAX SKEWNESS WIDTH
NBINS set the number of discrete values, and the other arguments have the same effect as in ERRF.
The keyword CONST is used to assign a Dirac distribution to a variable, i.e. set it to a constant value. Here is an example of use: