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information increases, converging on the underlying distribution of variability. Figure 4.3 shows
this evolution in a general, one-dimensional case.
Figure 4.3. Distribution of physical variability assuming perfect information, or no uncertainty,
and predictive Bayesian distributions based on limited data, and limited data with professional
judgment, showing the decrease in information entropy with increased information availability.
The general predictive Bayesian analytical model that represents the unconditional probability of
a particle of oil at X x, y is the following expression, in which the integration is over an
uncertain parameter space:
J
f( x, y, t )
j
f j ( x, y , t |
x, j
,
y, j
,
x, j
,
y, j
,
j
) L Ci | μ, σ, ρ, γ
μ σ ρ γ
(4.1)
j 1
in which f j (X, t | μ j , σ j ,
parameters mean μ j
sunken oil fraction
j
) is the j-th Gaussian patch given knowledge of the range of
μ x μ y j , standard deviation σ j
j
, with
j
σ x σ y j , correlation coefficient
1 to maintain conservation of mass; μ j
j
, and
v j t ; v j is the
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