1. On triangular norms, metric spaces and a general formulation of the discrete inverse problem or starting to think logically about uncertainty On triangular norms, metric spaces and a general formulation of the discrete inverse problem or starting to think logically about uncertainty

2. Formulating a scientific question Such definition does not use notions such as Probability, Sampling, Filtering, Distance, McMC, Density etc.. How to formulate such quantification problem? quantifying uncertainty = quantifying lack of (human) understanding

3. An uncertainty quantification problem ? Two independent vagueness statements are given about X in terms of two pdfs What is the “combined” pdf ? what is uncertainty about X? x x pdf

4. An example photo weight scale A set of similar chairs

5. A question of conjunction: logical “and” AND10 110 000 0.60.4 0.8?? 0.2?? p1p1 p2p2 p2p2 p1p1

6. Axioms for conjunctions t-norms Which functions follow these axioms?

7. Examples of t-norms p1p1 p2p2

9. The impact of choice of conjunction p1p1 p2p2 p1p1 AND0.250.50.75 0.50.1250.250.375 AND0.250.50.75 0.50.210.370.47 AND0.250.50.75 0.50.250.5 p1p1 strong weak

10. What does the weight of a chair have to do with my reservoir? Model m: the unobservable parameters describing part of a physical system Data d: the observable parameters describing part of a physical system Reservoir: various sources of information on (d,m) Expert interpretation Physical models, rules, empirical relationships,…. The reservoir data d obs

11. Formulating the discrete inverse problem (Tarantola & Valette, 1982)

12. A third source of information: the metric density on (d,m) Cartesian variables yy xx Non-Cartesian variables rr  ss D=metric tensor

13. Formulating discrete inverse problems (Tarantola) How do we combine these three probability densities into a single probability density as a quantification of uncertainty ?

14. Application of conjunctions to densities The metric density acts as a neural element in conjunction of densities

15. Application of conjunctions to density ratios

16. Formulating the problem in metric spaces All real problems involve non-Cartesian variables with non-Euclidean distances Approximate non-Euclidean distances with Euclidean distances How? use multi-dimensional scaling

17. Recall MDS Water-cut Time, days d obs d eigen-component 1 eigen-component 2 d obs Warning common confusion MDS ≠ PCA

18. What does multi-dimensional scaling achieve?

19. Back to the problem formulation

20. Illustration of concepts Show that uncertainty quantification is dependent on the choice of t-norm Show how metric space formulation works NOT YET: show any practical methodology based on it

21. The t-norm used

22. Example by Celine Scheidt m1m1 m2m2 m 2000......

23. Metric space of model parameters Prior of model parameters

24. Metric space of data parameters Prior of data parameters

25. Modeling the “pre-theory”

26. Modeling the “theory” The “theory” is assumed exact Given a uniform distributed m what is the predicted d ?

27. Modeling the “theory”

28. Comparison of PDFs calculated from different t-norms Weaker conjunctions lead to increased uncertainty

29. Some statistics on remaining uncertainty of sum and product t-normP50 Std. dev. Product48.28.6 w = 0.7547.79.3 w = 0.547.210.5 w = 0.2547.811.9 Minimum47.713.0 t-normP50 Std. dev. Product21.72.0 w = 0.7521.82.1 w = 0.522.12.4 w = 0.2522.02.6 Minimum22.32.7 Sum Product

30. Comparison with Bayesian theory Formulations neglecting metric densities lead to paradoxes (Mosegaard, 2009) lead to inconsistencies, e.g. negative saturations (EnKf) Bayesian theory is a special case of conjunction theory Allows only limited non-linearity (Tarantola, 1982) Conditional probabilities are ill-defined (Tarantola, 1987) Often requires linear vector spaces for d and m (EnKf) Confusion with “prior”, “belief”, “updating vs revision”

31. Bayesian theory Implicit vs explicit updating models the relationship between m and some background knowledge B 0 at time t 0

32. Simple example on Bayesianism Exploration setting: Data: 2D seismic sections Do we have Deltaic (heterogeneous sand) or Aeolian (homogeneous sand) ? “A” = the reservoir is deltaic “B” = the 2D seismic data (e.g. a bright spot) Expert 1 knowledge expert provides assessment Expert 2 data expert : numerical modeling to get

33. What is the problem? Inconsistent with implicit conditioning model of Bayesianism but knowledge expert and data expert look at the same data, → their assessments are related somewhat → too small uncertainty Solution

34. Explicit conditioning

35. The  or  -model ?

36. Formulating using conjunctions Formulate the problem as a conjunction problem, not as an updating problem Updating can be part of sub-problems such as with expert 2 Conjunctions model explicitly interdependency between E 1 and E 2

37. Opinions (for debate) The most critical element of an uncertainty quantification question lies in the formulation of the problem, not necessarily its sampling The choice of conjunction = defining a way of reasoning about terms such as independence (Bayesian theory provides only one way of reasoning)

38. Final quote as far as the propositions of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. Albert Einstein as far as the propositions of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. Albert Einstein

39. Opinions (for debate) The most critical element of an uncertainty quantification question lies in the formulation of the problem, not necessarily its sampling The choice of conjunction = defining a way of reasoning about terms such as independence (Bayesian theory provides only one way of reasoning)

40. What conjunction to choose? Choosing a conjunction = choosing a way of reasoning/thinking How to deal with contradiction ? What does it mean if “something is true”, “something is false” ? For example: the minimum = weak conjunction = choosing for intuitionistic logic A negation does not mean something is false, but is refutable You can only say something is false if you have a counterexample Applies to any interpretative science (geosciences)

41. Example What is the proper Euclidean distance ?

42. Proof of axiom 4

43. The notion of conditional probability

44. All Bayesian theory is a special case of conjunction theory

45. Bayesian theory cannot deal with most non-linear problems Bayesian theory can handle this Bayesian theory cannot handle this

46. Ensemble Kalman filter

47. Additional limitation of Bayes’ rule Background knowledge can be probabilistic, but any new data cannot New concepts cannot be assimilated/integrated Anything with probability 0/1 cannot be updated