Probability fields revisited in the context of ensemble Kalman filtering

• Unconditional probability field (UPF) vs. conditional probability field (CPF).
• CPF outperforms UPF.
• In UPF the correlation between probabilities and state variable is weak.
• In CPF the correlation between probabilities and state variable is strong.
• Application of the EnKF in which the parameter being updated are probabilities.

SummaryHu et al. (2013) proposed an approach to update complex geological facies models generated by multiple-point geostatistical simulation while keeping geological and statistical consistency. Their approach is based on mapping the facies realization onto the spatially uncorrelated uniform random numbers used by the sequential multiple-point simulation to generate the facies realization itself. The ensemble Kalman filter was then used to update the uniform random number realizations, which were then used to generate a new facies realization by multiple-point simulation. This approach has not a good performance that we attribute to the fact that, being the probabilities random and spatially uncorrelated, their correlation with the state variable (piezometric heads) is very weak, and the Kalman gain is always small. The approach is reminiscent of the probability field simulation, which also maps the conductivity realizations onto a field of uniform random numbers; although the mapping now is done using the local conditional distribution functions built based on a prior statistical model and the conditioning data. Contrary to Hu et al. (2013) approach, this field of uniform random numbers, termed a probability field, displays spatial patterns related to the conductivity spatial patterns, and, therefore, the correlation between probabilities and state variable is as strong as the correlation between conductivities and state variable could be. Similarly to Hu et al. (2013), we propose to use the ensemble Kalman filter to update the probability fields, and show that the existence of this correlation between probability values and state variables provides better results.