Introduction

Estimating many parameters using few samples (realizations) leads to sampling errors and spurious correlations.
In practice, this leads to parameters being updated more than they should have been.
Localization is a way to try and fix this and is in widespread use in industry today.
The basic idea is really Tobler's first law of geography
which states that "everything is related to everything else, but near things are more related than distant things."
We can exploit this by only using near-by observations when updating subsets of parameters.

The algorithm is something like this:
- Pick a subset of adjacent parameters
- Pick observations that are "in the vicinity" of this subset
"in the vicinity" can mean many different things!
- Do the update using only these observations

Here's a schematic representation courtesy of asch2016:

This is called domain localization, local analysis or distance based localization.
It is possible to do distance based localization in ERT but the configuration is prohibitively complicated.
The method relies on multiple parameters that need to be input by the user, and good values for these parameters are difficult to estimate.
As an alternative, Evensen proposes something called adaptive localization which is what we are currently working on implementing and testing.

The adaptive localization algorithms is something along these lines:
- Pick a subset of adjacent parameters
- Pick responses that are highly correlated to the parameters
- What "highly correlated" means is not obvious. More on this later.
- An assumption made was that if a single parameter is highly correlated to observations,
then all parameters in subset are also highly correlated.
We were not sure whether this assumption was a good idea, and came up with a slighly different proposal.
More on this later.
- Do the update using only observations that correspond to these responses

As mentioned, an assumption was made that adjacent parameters are correlated to the same responses,
which may be try on a layer-by-layer basis but not so across layers.
Also, it requires information of where parameters are located in relation to each other.
This information is not currently available in ERT (we might want to do something about this though).

We therefore suggest performing the update on each and every parameter.
Yes, we suggest looping through each and every parameter, calculate its correlation with responses,
pick the ones that are highly correlated and do the update.
That sounds computationally expensive, does it not?
It is expensive, but also highly parallelizable and is the solution that makes the fewest assumptions.
How much optimization will be necessary to be able to do this on large fields with many parameters is currently not known.

Status

• Evensen has tested this solution on REEK and is happy with the results.
• We have proposed a simple method to automatically calculate the correlation threshold based on ensemble size.
  Evensen is looking into other, more complex methods of calculating said threshold.
  If succesful, this would mean that adaptive localization needs no user-input - which would be awesome!
• Figure out a way to "prove" that the method is at least as good or better than distance based localization,
  and of course, that it's better than no localization at all.
• Test effect of tapering
  Details omitted for brevity