We appreciate the referee's reflective and constructive feedback on our manuscript and the time and effort put into it. Below, we provide a detailed response to each comment.

Major Scientific Concerns

1. Applicability of LPD to non-fixed sampling locations and smaller-scale cases

We acknowledge the reviewer’s concern regarding the applicability of the LPD approach beyond large-scale spatial interpolation with fixed sampling locations. While the case studies in the manuscript focus on large-scale applications, the LPD method is not inherently restricted to such contexts. It is based on the local density of training points in the predictor space rather than geographic space, making it applicable across sampling strategies, including non-fixed and smaller-scale settings. Further explanation will be added to the discussion of the revised manuscript by us.

We demonstrate its applicability to local studies in the help file of the CAST package (?AOA). In this example, we use soil moisture and temperature logger data at the farm scale to show how the LPD can identify areas that are well or poorly covered by training data (in the predictor space).

2. How does RF incorporate LPD and influence tree structure?

The reviewer raises an important question about how Random Forest (RF) integrates spatial patterns and whether this necessarily leads to model improvement. While RF does not inherently encode spatial information—unless explicitly incorporated (see https://doi.org/10.5194/gmd-17-6007-2024)—it captures relationships within the predictor variables, which can include local density information.

The LPD does not influence the model itself and is not intended to be incorporated into the training process. Instead, it serves solely as a measure of training point density in the predictor data space for a new prediction location. Since it is not used during model training, it does not alter the tree structure.

Our goal is therefore not to improve model accuracy directly, but rather to enhance the reliability of predictions by identifying areas with high training coverage or by quantifying potential prediction accuracy loss related to low LPD.

We will add further clarification in the revised manuscript, emphasizing that while the LPD does not modify the tree structure (as it is not included in training) it offers a complementary measure of uncertainty.

3. Comparison with traditional spatial interpolation methods like Kriging

We appreciate the suggestion to better integrate our work within the broader context of spatial interpolation methods. However, our focus is on machine learning–based approaches, where, unlike kriging, which uses the kriging variance to quantify spatial uncertainty based on spatial autocorrelation, there is no established method for uncertainty assessment. We believe that comparing the LPD to uncertainty assessment techniques used in interpolation methods is not particularly helpful, as the models rely on different assumptions and strategies. We prefer to maintain our focus on machine learning methods in this context.

Nevertheless, we see the benefit of working out the differences between the various approaches and will elaborate on this in the discussion of the revised manuscript.

4. Identifying scenarios where LPD is most effective and areas for improvement

We fully agree that no single method is universally optimal. In our manuscript we also compare it to the dissimilarity index that was proposed earlier. To provide a clearer guideline for practitioners, we will add information to the manuscript discussing the conditions under which LPD performs best. These include:

• Cases with heterogeneous predictor distributions where training data density varies significantly.
• Applications where it is crucial to assess model applicability beyond training locations.
• Multivariate Predictor scenarios where e.g. statistical density measures like kernel density estimation fail to capture data density effectively (e.g. computational extensive for multivariate cases).

Minor Issues

1. Grammar and clarity improvements

We appreciate the reviewer’s attention to detail and will carefully review the manuscript for grammatical and clarity improvements.

2. Missing summary table for 19 bio variables

We acknowledge this oversight and will include a summary table detailing all 19 bio variables used in our study. This table will provide variable names and descriptions and will be placed in the appendix.

We are grateful for the referee’s insightful comment. We will incorporate the revisions and believe they will improve the clarity and impact of our work and strengthen the manuscript for publication.

Best regards,  

Fabian Schumacher, Christian Knoth, Marvin Ludwig, and Hanna Meyer

Dear Dr. Añel,

Thank you for your feedback and for bringing this important issue to our attention. We sincerely apologize for the mistake and any inconvenience it has caused.

We have now archived the contents of the GitHub repository referenced in our paper, which contains all scripts and data, as well as the repository containing the R package CAST on Zenodo. In the revised version of our manuscript, we will update the "Code and Data Availability" section to ensure full compliance with the journal's data policy. Specifically, we will provide links to permanent repositories containing all relevant code and data used in our study. The updated section will look like this:

The current version of the method to calculate the introduced Local Point Density (LPD) is available from the developer version of the R package CAST (https://github.com/HannaMeyer/CAST) under the GNU General Public Licence (GPL >= v2). The exact R package version of the implementation used to produce the results of this paper is CAST Version 1.0.2 which is  published on CRAN (Meyer et al., 2024b) and Zenodo (https://doi.org/10.5281/zenodo.14362793). The exact version of the simulation study and the case study including all scripts as well as the input data used to run the models and produce the results and plots described in this paper is archived on Zenodo (https://doi.org/10.5281/zenodo.14356807) under the GNU General Public Licence  (GPL v3).

In the current manuscript, CAST version 1.0.0 is cited. Since version 1.0.2, which addresses only minor issues, has now been archived, revised versions of the manuscript will reference this updated version.

We hope this update addresses the concerns raised and ensures our manuscript fully aligns with the journal's data policy. Please let us know if there is anything further we can do to resolve this matter.

Thank you for your understanding and guidance.

Best regards,

Fabian Schumacher

On behalf of all co-authors

Thank you very much for your positive and encouraging review. We appreciate your support for the publication of our work.

Thank you also for pointing out the typos. We will correct both the issues you mentioned and will make sure to carefully proofread the final version to ensure clarity and correctness.

Best regards,

Fabian Schumacher, Christian Knoth, Marvin Ludwig, and Hanna Meyer

We would like to thank the reviewer for their positive and constructive feedback. Below, we respond point-by-point to the specific comments, which helped us to further clarify and improve the manuscript.

1. "The study focuses on random forest models... wondering if LPD can perform well with other algorithms."

We agree that the behavior of different machine learning algorithms with respect to extrapolation and sensitivity to training data density can vary. To demonstrate that our LPD approach is not limited to random forest models, we have included an example using Support Vector Machines in the helpfile of the AOA function in our CAST package.  

The results show that while the overall LPD patterns remain similar across algorithms, minor differences can emerge. These are primarily due to the varying ways in which different models internally weight predictor variables. Such differences are expected, as variable importance influences both model predictions and the LPD outcome.

When we neutralize these differences - by assigning equal weights to all variables and applying the same cross-validation strategy across models - the resulting LPD patterns become identical. This is because the LPD results are driven by the location of the training data and new data in the predictor pace well as  the cross-validation design, rather than by the specific characteristics of the learning algorithm (see supplement figures).

We have added a brief discussion of this point to the revised manuscript to clarify that LPD can be used with different algorithms as it is independent of the model's internal mechanics and mainly the variable importances as well as the cross-validation folds are used for the LPD calculation - alongside the predictor values of the training data and the new data.

2. "Real-world geoscience data can show unphysical teleconnections... Will this pose a concern for the application of the LPD method?"

We use the predictor variables selected by the model - weighted by their estimated relevance - without distinguishing between causal and non-causal relationships, because the model itself does not make that distinction either. If a non-causal (i.e., potentially unphysical) variable is relevant for the model's prediction, then being close in predictor space to the training data (i.e., having a high LPD) becomes even more important - because the model is extrapolating based on a spurious relationship, which is likely to fail outside known data. Such failures are typically detectable through spatial cross-validation strategies, which we recommend using to define the similarity threshold and to assess model performance under realistic prediction scenarios.

3. "The simulation study... would be more convincing if LPD were tested on a wider range of simulated cases like non-Gaussian or multimodal responses."

Thank you for pointing this out. We agree that further testing with more complex or non-Gaussian response structures would enhance the validation. We will acknowledge in the discussion that evaluating LPD under, e.g. multimodal or skewed relationships is essential for broader generalizability. We propose this as a extension of our work and include it under “future development” in our discussion.

4. "Some technical terms (e.g., shape-constrained additive model) may be unfamiliar..."

We added brief explanations at the mention of shape-constrained additive models (Section 2.3) to ensure accessibility to a broader readership.

We once again thank the reviewer for the thoughtful feedback and are confident that the revisions have strengthened the manuscript.

Sincerely,  

Fabian Schumacher, Christian Knoth, Marvin Ludwig, and Hanna Meyer