We thank the commenter for his positive overall assessment.

He makes many well-founded specific comments for improving the textual and graphic clarity, as well as the referencing. We will take them to heart.

We thank the reviewer for their helpful comments.

We agree with point 1: the paper would be well served by moving much of the adjoint-state derivation into an appendix.

Regarding the comments on figures in point 2: we agree that it would be more useful for the regularized and unregularized examples to share the same ground truth; it is no problem to do this. We also agree about adding more linkage between these diagrams and the rest of the text: both by qualitative discussion and by reporting the various error metrics that correspond the fields shown, both before and after calibration. The figure should also be moved closer to its point of first reference.

Regarding point 3: in our view one of the most interesting conclusions of our work is that measurements must be at least 4x as dense as the scale of contiguous features one wishes to delineate, with denser measurements providing increasingly better performance. And although some combinations of measurements performed somewhat better than others, this basic pattern was shared by all the different combinations of measurement types we considered, as shown in Fig. 4. Though this information was contained in the original draft, we think it could benefit from being communicated more clearly, as it is so important.

We appreciate the reviewer's remarks that the paper is well written and take comfort in the fact that they did not identify any methodological errors. That said, we take exception to the overall gist of their review and believe that there may have been confusion about the nature of our paper’s contribution.

We view our work as basic science concerning the information content of point hydraulic measurements, but we think the reviewer may have understood it as a benchmarking study aimed at evaluating calibration method performance.  This interpretation would explain some remarks that we otherwise find difficult to account for. The Zimmerman paper they mention (DOI: 10.1029/98WR00003) is a benchmarking study comparing the performance of geostatistical inversion schemes but does not otherwise address our key themes.

Discussing two reviewer remarks in particular:

A) The reviewer indicates that they struggle to see what is novel in our study. Here are two major original features:

1. The principal novelty of our study, as we stress in our introductory material, is the investigation of the connection between spatial scale of features in the hydraulic conductivity field and our ability to characterize them from a given density of point measurements. As far as we know, this has never been systematically investigated and no one has previously quantified the relationship between amount learned and relative measurement density, as we have. Our key result---that feature identification becomes possible above 4x measurement scale, with quality increasing in power law fashion with increasing feature scale above this threshold---has no real precedents.
2. The Karhunen-Loeve basis function framework we develop for assessing calibration accuracy without requiring the imposition of experimenter-specified a priori spatial zonation or subjective weights on the objective function is itself novel and can be applied (with suitable modifications) to analysis of a variety of different field recovery problems.

B) The reviewer remarks that the flow regime we employ in our analysis---steady flow in a mildly heterogeneous multi-Gaussian conductivity field---is too straightforward for their liking, and they write that the manuscript would only be worthwhile to publish if it considered a "tougher problem" with, e.g., an esoteric non-Gaussian statistical correlation structure, or high variance, or transient flow. This seems strange and perhaps represents a misunderstanding of what we are studying.

If our work were an algorithm benchmarking study where we were trying to select the most robust approach from amongst candidates, we could see the merit in choosing a "tough" benchmark. But our work is not a benchmarking study; rather, it concerns the information content of hydraulic measurements. This basic science question is best answered using a canonical flow model that exhibits the physics we wish to study, employing realistic parameters and standard assumptions but otherwise shedding needless complexity. Therefore, the more niche modeling scenarios suggested by the reviewer are counterproductive for our purposes.

We stress: our study is not "about" any specific flow configuration; rather we are studying how reliably groundwater flow transports information about conductivity features to remote measurement locations. Given that the feature scale / identifiability relationship has never been quantified in any scenarios at all, we believe that choosing a common flow scenario as a basis for our analyses is a good choice.

We were surprised to see the flow scenario chosen described as a “little interesting case”, given that the assumptions made regarding the flow model are all common ones in the literature used in, conservatively, hundreds of papers. Concerning two mentioned decisions:

1. Multivariate Gaussian random fields are overwhelmingly used in hydrologic Monte Carlo studies, and the log-variance we employed is similarly well within the normal range of values considered. Such models are of course idealizations; the working hypothesis in any Monte Carlo analysis of this sort is that the precise details of the conductivity correlation structure are of at most secondary significance compared to the meta-parameters of interest (in our case correlation length). To the extent this is true, the specific choice is of no impact. To the extent structural details are impactful, why would selection of a less common structure be superior and render the work publishable? We cannot see the rationale behind this.
2. Also consider the suggestion that we employ a transient model. In nature, at field scale, it is common to find flow that admits a quasi-steady model, and flow that requires a transient model. These lead to different calibration problems, and both are relevant; it is not clear why the reviewer considers the transient option superior. In any event it would not be feasible for us to study the latter systematically. Performing the high-quality systematic study we already present under steady conditions consumed 1000s of compute hours. Moving to a transient problem would require an explosion of computational resources, as we would now be inverting in the <conductivity> x <storativity> x <flow history> space, and with a much slower-to-run model. And we would still have to make arbitrary decisions constraining the flow history.

Ultimately, we do not consider the negative comments well founded.