Earth Syst. Dynam. Discuss., https://doi.org/10.5194/esd-2018-15-AC2, 2018 © Author(s) 2018. This work is distributed under the Creative Commons Attribution 4.0 License. Interactive comment on “A mathematical approach to understanding emergent constraints” by Femke J. M. M. Nijsse and Henk A. Dijkstra Femke J. M. M. Nijsse and Henk A. Dijkstra fn235@exeter.ac.uk Received and published: 7 July 2018 We thank the referee for the careful reading and the useful comments and will adapt the manuscript accordingly. Below is a point by point reply with the referee’s comments first, followed by our reply and the changes in manuscript. 1. Comment of the referee: I see as main (yet minor) pitfalls of your paper the some- what confused presentation in Section 2 and the fact that the classification of the emer- gent constraints is also a bit unclear. Authors’ response: We will aim to clarify both issues. Changes in the text: Section 2 and 3 will be rewritten as to separate the more general derivation from the Ornstein-Uhlenbeck example. The classification of the emergent C1 constraints in section 3 will be made more clear as per the suggestions of the referee. 2. Comment of the referee: I think you should also discuss a bit more in detail the difference between considering multiple models, instead of one with parametric modu- lations. Author’s response: Suggestion followed. Changes in the text: We will address the differences between the two cases (multiple models vs parameter variation in a single model) in the revised discussion section of the paper. 3. Comment of the referee: Additionally, you might find useful a recent preprint of mine: https://arxiv.org/abs/1806.03983 where I address the problem of looking at observables as predictands and predictors. This problem is (briefly) mentioned in your paper, but maybe my preprint can be useful for discussing your results. Authors’ response: We have read the paper and the material is indeed highly interest- ing and relevant for our paper. Changes in the text: It will be used when rewriting part of section 3 on the classification of the emergent constraints. 4. Comment of the referee: Please also note the supplement to this com- ment: https://www.earth-syst-dynam-discuss.net/esd-2018-15/esd-2018-15-RC2- sup- plement.pdf Authors’ response: Many thanks for the very useful questions, remarks and sugges- tions for changes; they are discussed below. Changes in the text: All these suggestions will be taken into account in the revised paper as per the point-by-point reply below. 5. Comment of the referee: Section 2: I think you should frame response theory in general terms, and then propose this (relevant and illustrative) example. Otherwise, C2