Page 82 - Contributed Paper Session (CPS) - Volume 1

CPS1110 Selamawit M. et al.
2. Model Definition
The data we consider for this study is a well log from Sheringham Shoal
wind farm and we consider data only from one well. The well is discretized by
a one-dimentional grid, ∶ {1, . . . , }. The categorical facies class ∈ Ω ∶

{1, . . . , } is assigned to one of classes. The facies profile along the
discretized well is represented by = ( , . . . , )′ and the corresponding

1
data, well-log observations, is denoted by = ( , . . . , )′ . The sub-

1
surface layer prediction based on the observed well logs is solved by Bayesian
inversion and settled in a Bayesian framework

(|) = const × (|) () (1)

where, (|) is the likelihood model, () is the prior model, and ‘const’ is a
normalizing constant. The likelihood and prior models are defined and the
corresponding posterior model is developed.

Observation likelihood
We assume the likelihood model on factorial form given by

(|) = ∏ ( |κ) = ∏ ( | ) (2)

with an assumption about conditional independence and single site
dependence. Here (|) is the likelihood model and it defines the link
between the observations, and the variable of interest, .

Prior experience
The prior pdf () captures the primary knowledge we have about the
variable of interest . In our study this information can primarily be obtained
from certain geotechnical studies. Consider a first order Markov chain prior
model which allows spatial coupling in the sub-surface profile. This coupling
is represented by the conditional probability of the facies state , given all the

previous states, which only depends on the single previous state − 1, and it

is defined as

() = ( ) ∏ ( | −1 , … , ) = ( ) ∏ ( | −1 ), (3)
1
1
1

∈ −1 ∈ −1

The prior pdf () can be defined by the initial pdf = [( )] 1 ∈Ω and the
1
1
set of transition matrices −1, = [( | −1 )] −1 , ∈Ω ; ∈ . By recursion
−1

the set of marginal pdfs is defined by

′

P = [( )] ∈Ω = P −1, −1 ; ∈ . (4)

−1

71 | I S I W S C 2 0 1 9