Application of Machine Learning Methods for Gravity Anomaly Prediction

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper is timely and relevant, particularly in the context of advancing geophysical modeling through machine learning techniques. The authors demonstrate sound methodology and provide a detailed performance assessment, employing both statistical and spatial analyses.

However, I recommend that the authors include references to recent and ongoing work from the GGOS Focus Area on Artificial Intelligence for Geodesy (AI4G), particularly the AI for Gravity Field and Mass Change initiative. This international effort explores the use of machine learning and AI algorithms—including generative models and gap-filling techniques—for gravity field analysis. Citing relevant findings or methodologies from this initiative would strengthen the scientific context of the study and situate it more clearly within the current global research landscape. Relevant information is available at: https://ggos.org/about/org/fa/ai-for-geodesy/

Author Response

Comments 1: [The paper is timely and relevant, particularly in the context of advancing geophysical modeling through machine learning techniques. The authors demonstrate sound methodology and provide a detailed performance assessment, employing both statistical and spatial analyses. However, I recommend that the authors include references to recent and ongoing work from the GGOS Focus Area on Artificial Intelligence for Geodesy (AI4G), particularly the AI for Gravity Field and Mass Change initiative. This international effort explores the use of machine learning and AI algorithms—including generative models and gap-filling techniques—for gravity field analysis. Citing relevant findings or methodologies from this initiative would strengthen the scientific context of the study and situate it more clearly within the current global research landscape. Relevant information is available at: https://ggos.org/about/org/fa/ai-for-geodesy/]

Response 1: [Dear Reviewer, we would like to express our sincere gratitude for your positive evaluation of our manuscript “Application of Machine Learning Methods for Gravity Anomaly Prediction” submitted to Geosciences (MDPI). We highly appreciate your support and encouraging comments regarding the quality of our work. Regarding your suggestion to include references to the work of the GGOS Focus Area on Artificial Intelligence for Geodesy (AI4G), specifically the AI for Gravity Field and Mass Change initiative, we acknowledge its relevance. However, due to time constraints and the limited access to detailed methodological information of the referenced project, we were not able to sufficiently study or incorporate it meaningfully into our revised manuscript. Thank you for your valuable recommendation. We are confident that these studies will further enhance the dynamics of our future scientific research focused on the application of artificial intelligence technologies to the study of the Earth's gravitational field. We hope for your understanding on this matter.

Once again, thank you very much for your review and your valuable feedback.

Sincerely, Dr. Yerzhankyzy

Corresponding Author]

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

Dear Authors,

I have just completed the review of your article. The study presents a clear and precise method of interpolation as an alternative to the classical kriging techniques. The proposed approach, based on machine learning, is applied to a gravity dataset. The article focuses on improving the interpolation process in order to minimize errors in geoid estimation, particularly in areas lacking real gravity measurements. It is well written and supported by a solid methodological framework.

Minor comments have been provided throughout the manuscript.

My major remarks are related to the discussion and conclusions:

In your concluding remarks, you state that the machine learning (ML) approach does not properly resolve interpolation in mountainous regions. More specifically, the misfit between the kriging and ML methods appears larger in these areas compared to valleys or low-altitude terrains.

-First, you mention this limitation but do not discuss it in detail. While you suggest further investigation, I believe it would be helpful to offer an initial hypothesis or suggestion at this stage.

-Second, I have concerns regarding the computation of the Bouguer anomaly. In your map, the distribution of the Bouguer anomaly appears inconsistent, which seems to be due to a strong anti-correlation with topography. Ideally, the Bouguer anomaly should show minimal (or no) correlation with terrain morphology, as it should reflect gravity variations caused by subsurface mass rather than elevation changes. As discussed in previous studies (e.g., Caratori Tontini et al., 2008, and references therein), the choice of reduction density used in the Bouguer correction is critical in managing this correlation.

This suggests that your ML approach could potentially perform well (perhaps even better than kriging) also in mountainous regions if the Bouguer anomaly were correctly computed. In your case, the mountain regions show sharp horizontal gravity gradients, likely due to uncompensated topographic effects that are not well handled by the ML method. I believe this may be the root cause of the issue.

Therefore, I recommend including a discussion of this aspect in the manuscript and considering possible approaches to address this issue.

Best Regards

Comments for author File: Comments.pdf

Author Response

Comments 1: [In your concluding remarks, you state that the machine learning (ML) approach does not properly resolve interpolation in mountainous regions. More specifically, the misfit between the kriging and ML methods appears larger in these areas compared to valleys or low-altitude terrains.

-First, you mention this limitation but do not discuss it in detail. While you suggest further investigation, I believe it would be helpful to offer an initial hypothesis or suggestion at this stage.]

Response 1: [

We sincerely thank you for your valuable comments on our manuscript entitled “Application of Machine Learning Methods for Gravity Anomaly Prediction” submitted to Geosciences (MDPI), and we address each of your points in detail below.

1. As you correctly noted in your comment, we indicated that the machine learning (ML) approach does not properly resolve interpolation in mountainous regions. We hypothesize that several limitations affected the performance of the machine learning models in mountainous regions:

First, we employed simple Bouguer anomalies, which do not include terrain corrections, due to the incomplete availability of terrain correction data for a significant portion of the study area. An additional analysis of the relationship between prediction errors and terrain corrections, presented in Figure 6e, shows that errors systematically increase with higher terrain correction values. In regions where terrain corrections were absent (assigned a value of zero), prediction errors remained relatively low and stable. However, in areas where terrain corrections were substantial—typically corresponding to mountainous regions—larger deviations and a higher concentration of outliers were observed, particularly for the Exponential GPR model.

Second, the same grid spacing was applied across the entire study area, regardless of terrain complexity. Consequently, the data density in mountainous areas was relatively low compared to flat regions, leading to reduced model training efficiency and increased interpolation errors in areas with rapid changes in elevation.

Third, algorithmic sensitivity: although all tested machine learning models demonstrated the ability to predict gravity anomalies to varying degrees of accuracy, notable differences in performance were observed. The Fine Gaussian SVR and Bagged Trees models showed comparatively lower predictive capabilities across both lowland and mountainous regions. This behavior can be attributed to inherent algorithmic characteristics: SVR models are known to be highly sensitive to data noise and require careful tuning of kernel parameters, which makes them less robust in complex and heterogeneous datasets. Bagged Trees, while effective in reducing variance through ensemble learning, are primarily suited for discrete classification tasks and may struggle to accurately model continuous and subtle variations typical of geophysical fields such as gravity anomalies.

In contrast, Exponential GPR demonstrated significantly better predictive performance. This can be explained by the algorithm’s intrinsic strengths: GPR is a non-parametric, probabilistic model capable of capturing complex nonlinear relationships, while simultaneously providing uncertainty estimates for each prediction. Its ability to flexibly model spatial variability made it particularly effective in smooth, continuous regions where gravity changes gradually. However, GPR’s performance showed sensitivity to data sparsity and abrupt topographic variations, as indicated by the presence of a few extreme outliers in mountainous areas. These outliers are consistent with the Bayesian nature of GPR, which, under conditions of limited and irregularly distributed training data, can lead to amplified prediction uncertainties. Therefore, while GPR generally outperformed other methods in terms of predictive accuracy, its flexibility also made it more vulnerable to producing rare but higher-magnitude deviations in complex terrains.]

Comments 2: [-Second, I have concerns regarding the computation of the Bouguer anomaly. In your map, the distribution of the Bouguer anomaly appears inconsistent, which seems to be due to a strong anti-correlation with topography. Ideally, the Bouguer anomaly should show minimal (or no) correlation with terrain morphology, as it should reflect gravity variations caused by subsurface mass rather than elevation changes. As discussed in previous studies (e.g., Caratori Tontini et al., 2008, and references therein), the choice of reduction density used in the Bouguer correction is critical in managing this correlation.]

Response 2: [

2. In this study, simple Bouguer anomaly values were sourced directly from the original ground-based observation points that underlie historical gravity maps at a scale of 1:200,000. These values were adopted in their original form, without any modifications or additional reductions. Most historical gravity maps in Kazakhstan were compiled using Simple Bouguer anomalies (σ = 2.67 g/cm³), which do not incorporate terrain corrections. As a result, terrain correction data are unavailable for a significant portion of the study area and, more broadly, for much of Kazakhstan.

To ensure consistency with the original dataset, we therefore used Simple Bouguer anomalies (σ = 2.67 g/cm³) throughout this study. As a result, the observed anti-correlation between Bouguer anomalies and topography can be attributed to the use of simple Bouguer anomalies without terrain correction.

The availability of terrain-corrected datasets is often limited—particularly in studies based on historical data. Consequently, evaluating machine learning performance using Simple Bouguer anomalies remains both relevant and necessary, as it realistically reflects the data limitations encountered in many regional-scale gravity studies.

The Discussion section has been revised to reflect the raised concerns, and all comments have been thoroughly considered in the updated version.]

Comments 3: [ According to peer-review-46024243.v2.pdf  notices]

Response 3: [We sincerely thank you for your thorough reading and for providing additional comments directly within the manuscript. These in-text annotations were highly valuable for improving the clarity, consistency, and overall structure of the paper. All of your comments and suggestions have been carefully addressed and corrected in the revised version of the manuscript.]

Thank you very much for your review and your valuable feedback.

Author Response File: Author Response.docx

Reviewer 3 Report

Comments and Suggestions for Authors

This study presents a comprehensive comparison of machine learning (ML) methods (SVR, GPR, Ensemble of Trees) and traditional Kriging interpolation for predicting Bouguer gravity anomalies in southeastern Kazakhstan. The work addresses a relevant geophysical challenge and provides valuable insights into the applicability of ML in regions with complex terrain. The methodological framework is robust, and the results are supported by rigorous statistical and spatial error analyses. However, several areas require clarification or improvement to strengthen the scientific contribution and practical implications of the research.

1. It is recommended that the potential bias or error associated with relying on historical maps (1:200,000 scale) be discussed. For example, how does the spatial resolution of these maps affect model training and generalization?
2. Show more details on hyperparameter optimization and data preprocessing. Was automated optimization (e.g., grid search, Bayesian methods) employed?
3. Why was kriging chosen as the only traditional method?
4. Explain why Exponential GPR produces larger outliers than Kriging. Is this due to overfitting in smooth regions or sensitivity to sparse data in mountains?
5. Revise Figure 4 legends to ensure color scales and labels are fully visible. The names and values of the axes in Figure 5 should be clear.
6. Correct inconsistent subscript notation (e.g., "G₀" vs. "G0" in Table 1).
7. Try to explain or discuss whether the ML models’ poor performance in mountainous areas stems from insufficient training data density, feature selection, or inherent algorithm limitations.

Author Response

Dear Reviewer,

 

We sincerely thank you for your valuable comments on our manuscript entitled “Application of Machine Learning Methods for Gravity Anomaly Prediction” submitted to Geosciences (MDPI), and we address each of your points in detail below.

Comments 1: [It is recommended that the potential bias or error associated with relying on historical maps (1:200,000 scale) be discussed. For example, how does the spatial resolution of these maps affect model training and generalization?]

Response 1: [In this study, gravity anomaly data were obtained directly from the original ground-based observation points that were initially used in the construction of historical gravity maps at a scale of 1:200,000. This approach was intentionally chosen to avoid potential inaccuracies associated with digitizing printed maps, where extracted values might correspond to interpolated rather than actual measured data. By relying on the original observation points, the dataset used for training the machine learning model preserves the intrinsic accuracy of gravity surveys conducted at various scales, which served as the basis for the compilation of the 1:200 000-scale gravity map.]

Comments 2: [Show more details on hyperparameter optimization and data preprocessing. Was automated optimization (e.g., grid search, Bayesian methods) employed?]

Response 2: [Data preprocessing included feature normalization to ensure consistent contribution from input features (latitude, longitude, elevation, normal gravity). The regression modeling was performed using the MATLAB^® environment (version R2024a). Hyperparameter optimization within Regression Learner was conducted automatically using built-in Bayesian optimization methods. This automated procedure systematically explored optimal hyperparameter combinations to minimize model error (RMSE, MAE), thus enhancing predictive accuracy and model generalization. Additionally, 10-fold cross-validation was employed to robustly evaluate model performance and mitigate potential overfitting. We have included this part in section 2 «Materials and Methods»]

Comments 3: [Why was kriging chosen as the only traditional method?]

Response 3: [The primary objective of our study was to evaluate the effectiveness of machine learning (ML) methods for predicting gravity anomalies.  Conducting a detailed comparative analysis of traditional interpolation methods was not within the scope of this work. Based on a number of studies, we decided to choose Kriging method as a reliable and theoretically well-founded method and is widely regarded as the benchmark among traditional interpolation techniques in gravity research.]

Comments 4: [Explain why Exponential GPR produces larger outliers than Kriging. Is this due to overfitting in smooth regions or sensitivity to sparse data in mountains?]

Response 4: [Preliminary analysis suggests that the larger outliers observed in Exponential Gaussian Process Regression (GPR) may be attributed to a combination of factors, including sensitivity to sparse training data and complex topography. While our findings point to this explanation, we emphasize that this remains a working hypothesis, and further experiments are needed to fully validate the underlying causes.

GPR is a non-parametric, probabilistic model capable of capturing complex nonlinear relationships while simultaneously providing uncertainty estimates for each prediction. Its ability to flexibly model spatial variability made it particularly effective in smooth, continuous regions where gravity changes gradually. However, the model’s flexibility also made it sensitive to data sparsity and abrupt topographic variations, as indicated by the presence of a few extreme outliers in mountainous areas. These deviations are consistent with the Bayesian nature of GPR, which—under conditions of limited and irregularly distributed training data—can lead to amplified prediction uncertainties. In contrast, Kriging constrains predictions based on spatial autocorrelation, producing smoother, more conservative interpolation results even under challenging terrain conditions. Therefore, while GPR generally outperformed other methods in terms of predictive accuracy, it also exhibited a higher susceptibility to rare but significant errors in complex regions.

The Discussion section has been revised to reflect the raised concerns, and all comments have been thoroughly considered in the updated version.]

Comments 5: [Revise Figure 4 legends to ensure color scales and labels are fully visible. The names and values of the axes in Figure 5 should be clear.]

Response 5: [The color scales and labels in Figure 4 have been adjusted for full visibility, and the axis names and values in Figure 5 have been clarified as requested. The revised figures have been updated accordingly in the manuscript.]

Comments 6: [Correct inconsistent subscript notation (e.g., "G₀" vs. "G0" in Table 1).]

Response 6: [We have carefully reviewed and corrected the subscript notation throughout the manuscript, including Table 1, to ensure consistency. The corrected version has been incorporated into the revised manuscript.]

Comments 7: [Try to explain or discuss whether the ML models’ poor performance in mountainous areas stems from insufficient training data density, feature selection, or inherent algorithm limitations.]

Response 7: [Several limitations affected the performance of the machine learning models, particularly in mountainous regions:

First, we employed simple Bouguer anomalies without applying full terrain corrections, primarily due to the incomplete availability of terrain correction data for a significant portion of the study area. The availability of such corrections is often limited—particularly in studies relying on historical datasets.

An additional analysis of the relationship between prediction errors and terrain corrections, presented in Figure 6e, shows that errors systematically increase with higher terrain correction values. In regions where terrain corrections were absent (assigned a value of zero), prediction errors remained relatively low and stable. However, in areas where terrain corrections were substantial—typically corresponding to mountainous regions—larger deviations and a higher concentration of outliers were observed, particularly for the Exponential GPR model.

Second, the same grid spacing was applied across the entire study area, regardless of terrain complexity. Consequently, the data density in mountainous areas was relatively low compared to flat regions, leading to reduced model training efficiency and increased interpolation errors in areas with rapid changes in elevation.

Third, algorithmic sensitivity: although all tested machine learning models demonstrated the ability to predict gravity anomalies to varying degrees of accuracy, notable differences in performance were observed. The Fine Gaussian SVR and Bagged Trees models showed comparatively lower predictive capabilities across both lowland and mountainous regions. This behavior can be attributed to inherent algorithmic characteristics: SVR models are known to be highly sensitive to data noise and require careful tuning of kernel parameters, which makes them less robust in complex and heterogeneous datasets. Bagged Trees, while effective in reducing variance through ensemble learning, are primarily suited for discrete classification tasks and may struggle to accurately model continuous and subtle variations typical of geophysical fields such as gravity anomalies.

In contrast, Exponential GPR demonstrated significantly better predictive performance. This can be explained by the algorithm’s intrinsic strengths: GPR is a non-parametric, probabilistic model capable of capturing complex nonlinear relationships, while simultaneously providing uncertainty estimates for each prediction. Its ability to flexibly model spatial variability made it particularly effective in smooth, continuous regions where gravity changes gradually. However, GPR’s performance showed sensitivity to data sparsity and abrupt topographic variations, as indicated by the presence of a few extreme outliers in mountainous areas. These outliers are consistent with the Bayesian nature of GPR, which, under conditions of limited and irregularly distributed training data, can lead to amplified prediction uncertainties. In contrast, Kriging, by design, constrains predictions based on spatial autocorrelation, leading to smoother, more conservative interpolation results even under challenging conditions. Therefore, while GPR generally outperformed other methods in terms of predictive accuracy, its flexibility also made it more vulnerable to producing rare but higher-magnitude deviations in complex terrains.

The Discussion section has been revised to reflect the raised concerns, and all comments have been thoroughly considered in the updated version.]

Thank you very much for your review and your valuable feedback.

Sincerely,

Dr. Yerzhankyzy

Corresponding Author

 

Author Response File: Author Response.docx