the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 25 Jun 2025
TensorWeave 1.0: Interpolating geophysical tensor fields with spatial neural networks
Abstract. Tensor fields, as spatial derivatives of scalar or vector potentials, offer powerful insight into subsurface structures in geophysics. However, accurately interpolating these measurements – such as those from full-tensor potential field gradiometry – remains difficult, especially when data are sparse or irregularly sampled. We present a physics-informed spatial neural network that treats tensors according to their nature as derivatives of an underlying scalar field, enabling consistent, high-fidelity interpolation across the entire domain. By leveraging the differentiable nature of neural networks, our method not only honours the physical constraints inherent to potential fields but also reconstructs the scalar and vector fields that generate the observed tensors. We demonstrate the approach on synthetic gravity gradiometry data and real full-tensor magnetic data from Geyer, Germany. Results show significant improvements in interpolation accuracy, structural continuity, and uncertainty quantification compared to conventional methods.
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Notice on discussion status
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Preprint
(25959 KB)
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
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- BibTeX
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- Final revised paper
Journal article(s) based on this preprint
Interactive discussion
Status: closed
-
CEC1: 'Comment on egusphere-2025-2345 - No compliance with the policy of the journal', Juan Antonio Añel, 24 Jul 2025
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AC1: 'Reply on CEC1', Akshay Kamath, 14 Aug 2025
Dear Editor,
In line with the journal’s data-sharing policy, we have made the dataset and code publicly available. The Geyer dataset used in the manuscript is provided as a Joblib pickle in ./Datasets/ of the repository (https://github.com/k4m4th/tensorweave). A reproducible Jupyter notebook to replicate the interpolation results is available at ./Notebooks/geyer_single_model.ipynb. The repository is actively maintained and additional notebooks are being added. We will include a “Data availability” section in the revised manuscript that points to this repository (and the exact release/commit used).
Kindest regards,
Akshay Kamath (on behalf of the authors)Citation: https://doi.org/10.5194/egusphere-2025-2345-AC1
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AC1: 'Reply on CEC1', Akshay Kamath, 14 Aug 2025
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RC1: 'Comment on egusphere-2025-2345', Italo Goncalves, 24 Jul 2025
The manuscript presents a novel methodology for modelling tensor gradiometry data based on neural networks, achieving better results by incorporating physical constraints.
The text is very well-written, with clear explanations, thorough validation experiments and nice figures. I would ask only for a few clarifications and additions to the discussion:
- It would be interesting to point to the reference that coined the term "neural field".
- What was the activation function used in the network? How its choice impacts the results?
- The constraints-as-data approach is effective in practice, but perhaps it would be more desirable to encode the Laplace constraint within the model itself. Any comments on how this could be accomplished? Perhaps with a physically derived activation function and/or constraints on the network's weights. The references below contain examples in the context of Gaussian processes.
- Regarding uncertainty estimation, perhaps it would be simpler to implement a Bayesian neural network, which would incorporate uncertainty by resampling the RFF weights at each iteration of training. The MLP weights could remain deterministic if desired.
- In principle the Laplace constraint could be imposed to RBF as well, as the usual radial basis are differentiable. This would allow a fairer comparision of the models. Many works model conservative fields with RBF and Gaussian processes, but to my knowledge they only have gradient constraints.
Minor details:
- line 153 - missing parenthesis
- line 303 - revise citation command
- Figure 2 - figure shows (sin, cos) features instead of (sin + phase) as described in the text.
Rerefences:
JIDLING, C. et al. Linearly constrained Gaussian processes. [s.l: s.n.]. Disponível em: <http://arxiv.org/abs/1703.00787>.
LANGE-HEGERMANN, M. Linearly Constrained Gaussian Processes with Boundary Conditions. v. 130, 2020.
Citation: https://doi.org/10.5194/egusphere-2025-2345-RC1 -
AC2: 'Reply on RC1', Akshay Kamath, 25 Aug 2025
Dear Ítalo Gonçalves,
We thank you for your time and effort reviewing the submitted manuscript, and are pleased that you appreciated our results. We have incorporated your suggestions into the revised manuscript, as detailed in the following pages. Please note that to facilitate the evaluation of our revision, the line numbers of the reviewers’ comments refer to the originally submitted manuscript while line numbers of our responses refer to our revised manuscript.
Kindest regards,
Akshay Kamath (on behalf of the authors)
-
RC2: 'Comment on egusphere-2025-2345', Anonymous Referee #2, 26 Jul 2025
This paper develops a neural-fields approach for interpolating geophysical tensor fields (e.g. for gravity gradiometry and tensor magnetics). A scalar potential is parameterized by the weights of a neural network, and the loss function that is used for training is comprised of two terms, a first term which penalizes misfit from the measured data and a second term that penalizes the trace of the Hessian (which should be zero for harmonic fields). The authors present both synthetic and field examples. The premise of the paper is interesting, and it is good to see the development of interpolation approaches that better respect the physics. The major critiques I have of this paper are: the math is a bit loose and should be made more rigorous to help with clarity, the discussion of the training process is quite sparse and would benefit from expansion, and I have a few big picture questions:
- First: the comparison is done with fairly simple methods (e.g. RBFs), but it is common to use equivalent sources for processing (e.g. this would be standard with a Falcon AGG survey). At a minimum, equivalent sources should be discussed as an approach, but ideally, including a comparison with equivalent-source-based interpolation would be valuable
- In practice, the full tensor may not be measured in a survey. For example, it is common for gravity gradiometry to only measure two components, e.g. the Falcon system only measures Gne and Guv, and the rest of the tensor is computed from these values. How would you handle this in your method?
Some more specific questions
- As a comment throughout, it would be helpful to have equation numbers to be able to reference
- The equations in sections 3.1 & 3.2 are confusing. The notation is mixed: I understand lowercase bold as vectors, should v_i not be a vector? or is it meant to be an entry of v (in which case the first term W*r is still a vector, so that would need to be indexed? What is the size of r? is it 3N X 1 or N X 3 ?
- Also, it would be helpful to clarify that the number of Fourier features (M) is the same as the number of frequencies
- These questions follow from the point about equations in sections 3.1 and 3.2. I don’t understand how the sizes in the equation at line 127 work. Ws is size M x N, but now r is in R2, does r only have 2 entries, or is it N x 2 ? I suspect you are treating it as N x 2. Do you then add these together or take a dot product to collapse it to a vector? is N the total number of points? or is N just 3 because it is a 3D vector. Clarifying this would help with the rest of the math in this section
- Section 3.4 - Network architecture: are you using activation functions between the MLP layers? if so, what are they?
- It would be helpful to state the full training problem in section 3.4 – e.g. what are you minimizing over, presumably the weights in W, and you are summing this over all available data points
- In section 4, it would be useful to show the loss curves and provide some discussion of the training process – e.g. how many iterations? What do you use for an optimization algorithm? What was the stopping criterion used? Was it the number of iterations or a threshold value for the loss? What are the final values for each component of the loss (e.g. the data fit vs. Laplacian loss – it would also be nice to see how these evolve as a function of iteration)
Other details
- line 25: define Random Fourier Features before the acronym (RFF)
- the definition of equation 110 is a bit of abuse of notation in defining Ws; it would be cleaner to state Ws = 1/l_s W
- equation at line 123, if you want to stick with vectors as boldface, I suggest bolding k-hat. This equation is a bit odd, because rxy is in R2, and it implies that rz = [0, 0, rz] when you add them together, so the sizes don’t match. I get what you mean, but you might be better off stating r = [rxy, rz] or similar
Citation: https://doi.org/10.5194/egusphere-2025-2345-RC2 -
AC3: 'Reply on RC2', Akshay Kamath, 25 Aug 2025
Dear Reviewer,
We thank you for your time and effort reviewing the submitted manuscript, and are pleased that you appreciated our results. We have incorporated your suggestions into the revised manuscript, as detailed in the following pages. Please note that to facilitate the evaluation of our revision, the line numbers of the reviewers’ comments refer to the originally submitted manuscript while line numbers of our responses refer to our revised manuscript.
Kindest regards,
Akshay Kamath (on behalf of the authors)
-
AC3: 'Reply on RC2', Akshay Kamath, 25 Aug 2025
-
RC3: 'Comment on egusphere-2025-2345', David Nathan, 04 Aug 2025
The paper presents a novel approach to interpolating full tensor gradiometric data, offering a clear theoretical foundation and demonstrating compelling improvements over truncated radial basis function (RBF) interpolation. The manuscript is well written, with strong motivation and a logical flow from theory to results. The examples provided effectively highlight the advantages of the proposed method in preserving spatial continuity and enhancing interpolation accuracy.
Given that interpolated potential field data often serve as input for geophysical inversion, further discussion of the ENF approach’s implications in this context would strengthen the manuscript. Specifically, the observation noted in line 293 suggests potential limitations when applying the method in ensemble-based inversion frameworks, such as the ensemble Kalman inversion. These methods rely on statistical assumptions and error covariance structures that could be influenced by interpolation artifacts or over-smoothing. A brief exploration of how ENF interpolation might influence inversion performance and uncertainty propagation would add valuable context for practitioners.
Minor corrections:
Line 123: \hat{k} needs to be defined
Line 388-389: Please update the reference Laloy et al., 2013. I was unable to find any source for it
Citation: https://doi.org/10.5194/egusphere-2025-2345-RC3 -
AC4: 'Reply on RC3', Akshay Kamath, 25 Aug 2025
Dear David Nathan,
We thank you for your time and effort reviewing the submitted manuscript, and are pleased that you appreciated our results. We have incorporated your suggestions into the revised manuscript, as detailed in the following pages. Please note that to facilitate the evaluation of our revision, the line numbers of the reviewers’ comments refer to the originally submitted manuscript while line numbers of our responses refer to our revised manuscript.
Kindest regards,
Akshay Kamath (on behalf of the authors)
-
AC4: 'Reply on RC3', Akshay Kamath, 25 Aug 2025
Interactive discussion
Status: closed
-
CEC1: 'Comment on egusphere-2025-2345 - No compliance with the policy of the journal', Juan Antonio Añel, 24 Jul 2025
Dear authors,
Unfortunately, after checking your manuscript, it has come to our attention that it does not comply with our "Code and Data Policy".
https://www.geoscientific-model-development.net/policies/code_and_data_policy.htmlYour manuscript does not contain a Data Availability section, and it must include it. In such section, you must include the links and DOIs for the repositories containing all the data used to perform your work, both input and output data.
Moreover, I would like to ask you to clarify in your manuscript the version numbers of the software and libraries that you have used for your work.
Please, reply to this comment with the requested information as soon as possible, and include it in any future version of your manuscript. Also, note that if you do not fix these issues, we will not be able of publishing your manuscript.
Juan A. Añel
Geosci. Model Dev. Executive Editor
Citation: https://doi.org/10.5194/egusphere-2025-2345-CEC1 -
AC1: 'Reply on CEC1', Akshay Kamath, 14 Aug 2025
Dear Editor,
In line with the journal’s data-sharing policy, we have made the dataset and code publicly available. The Geyer dataset used in the manuscript is provided as a Joblib pickle in ./Datasets/ of the repository (https://github.com/k4m4th/tensorweave). A reproducible Jupyter notebook to replicate the interpolation results is available at ./Notebooks/geyer_single_model.ipynb. The repository is actively maintained and additional notebooks are being added. We will include a “Data availability” section in the revised manuscript that points to this repository (and the exact release/commit used).
Kindest regards,
Akshay Kamath (on behalf of the authors)Citation: https://doi.org/10.5194/egusphere-2025-2345-AC1
-
AC1: 'Reply on CEC1', Akshay Kamath, 14 Aug 2025
-
RC1: 'Comment on egusphere-2025-2345', Italo Goncalves, 24 Jul 2025
The manuscript presents a novel methodology for modelling tensor gradiometry data based on neural networks, achieving better results by incorporating physical constraints.
The text is very well-written, with clear explanations, thorough validation experiments and nice figures. I would ask only for a few clarifications and additions to the discussion:
- It would be interesting to point to the reference that coined the term "neural field".
- What was the activation function used in the network? How its choice impacts the results?
- The constraints-as-data approach is effective in practice, but perhaps it would be more desirable to encode the Laplace constraint within the model itself. Any comments on how this could be accomplished? Perhaps with a physically derived activation function and/or constraints on the network's weights. The references below contain examples in the context of Gaussian processes.
- Regarding uncertainty estimation, perhaps it would be simpler to implement a Bayesian neural network, which would incorporate uncertainty by resampling the RFF weights at each iteration of training. The MLP weights could remain deterministic if desired.
- In principle the Laplace constraint could be imposed to RBF as well, as the usual radial basis are differentiable. This would allow a fairer comparision of the models. Many works model conservative fields with RBF and Gaussian processes, but to my knowledge they only have gradient constraints.
Minor details:
- line 153 - missing parenthesis
- line 303 - revise citation command
- Figure 2 - figure shows (sin, cos) features instead of (sin + phase) as described in the text.
Rerefences:
JIDLING, C. et al. Linearly constrained Gaussian processes. [s.l: s.n.]. Disponível em: <http://arxiv.org/abs/1703.00787>.
LANGE-HEGERMANN, M. Linearly Constrained Gaussian Processes with Boundary Conditions. v. 130, 2020.
Citation: https://doi.org/10.5194/egusphere-2025-2345-RC1 -
AC2: 'Reply on RC1', Akshay Kamath, 25 Aug 2025
Dear Ítalo Gonçalves,
We thank you for your time and effort reviewing the submitted manuscript, and are pleased that you appreciated our results. We have incorporated your suggestions into the revised manuscript, as detailed in the following pages. Please note that to facilitate the evaluation of our revision, the line numbers of the reviewers’ comments refer to the originally submitted manuscript while line numbers of our responses refer to our revised manuscript.
Kindest regards,
Akshay Kamath (on behalf of the authors)
-
RC2: 'Comment on egusphere-2025-2345', Anonymous Referee #2, 26 Jul 2025
This paper develops a neural-fields approach for interpolating geophysical tensor fields (e.g. for gravity gradiometry and tensor magnetics). A scalar potential is parameterized by the weights of a neural network, and the loss function that is used for training is comprised of two terms, a first term which penalizes misfit from the measured data and a second term that penalizes the trace of the Hessian (which should be zero for harmonic fields). The authors present both synthetic and field examples. The premise of the paper is interesting, and it is good to see the development of interpolation approaches that better respect the physics. The major critiques I have of this paper are: the math is a bit loose and should be made more rigorous to help with clarity, the discussion of the training process is quite sparse and would benefit from expansion, and I have a few big picture questions:
- First: the comparison is done with fairly simple methods (e.g. RBFs), but it is common to use equivalent sources for processing (e.g. this would be standard with a Falcon AGG survey). At a minimum, equivalent sources should be discussed as an approach, but ideally, including a comparison with equivalent-source-based interpolation would be valuable
- In practice, the full tensor may not be measured in a survey. For example, it is common for gravity gradiometry to only measure two components, e.g. the Falcon system only measures Gne and Guv, and the rest of the tensor is computed from these values. How would you handle this in your method?
Some more specific questions
- As a comment throughout, it would be helpful to have equation numbers to be able to reference
- The equations in sections 3.1 & 3.2 are confusing. The notation is mixed: I understand lowercase bold as vectors, should v_i not be a vector? or is it meant to be an entry of v (in which case the first term W*r is still a vector, so that would need to be indexed? What is the size of r? is it 3N X 1 or N X 3 ?
- Also, it would be helpful to clarify that the number of Fourier features (M) is the same as the number of frequencies
- These questions follow from the point about equations in sections 3.1 and 3.2. I don’t understand how the sizes in the equation at line 127 work. Ws is size M x N, but now r is in R2, does r only have 2 entries, or is it N x 2 ? I suspect you are treating it as N x 2. Do you then add these together or take a dot product to collapse it to a vector? is N the total number of points? or is N just 3 because it is a 3D vector. Clarifying this would help with the rest of the math in this section
- Section 3.4 - Network architecture: are you using activation functions between the MLP layers? if so, what are they?
- It would be helpful to state the full training problem in section 3.4 – e.g. what are you minimizing over, presumably the weights in W, and you are summing this over all available data points
- In section 4, it would be useful to show the loss curves and provide some discussion of the training process – e.g. how many iterations? What do you use for an optimization algorithm? What was the stopping criterion used? Was it the number of iterations or a threshold value for the loss? What are the final values for each component of the loss (e.g. the data fit vs. Laplacian loss – it would also be nice to see how these evolve as a function of iteration)
Other details
- line 25: define Random Fourier Features before the acronym (RFF)
- the definition of equation 110 is a bit of abuse of notation in defining Ws; it would be cleaner to state Ws = 1/l_s W
- equation at line 123, if you want to stick with vectors as boldface, I suggest bolding k-hat. This equation is a bit odd, because rxy is in R2, and it implies that rz = [0, 0, rz] when you add them together, so the sizes don’t match. I get what you mean, but you might be better off stating r = [rxy, rz] or similar
Citation: https://doi.org/10.5194/egusphere-2025-2345-RC2 -
AC3: 'Reply on RC2', Akshay Kamath, 25 Aug 2025
Dear Reviewer,
We thank you for your time and effort reviewing the submitted manuscript, and are pleased that you appreciated our results. We have incorporated your suggestions into the revised manuscript, as detailed in the following pages. Please note that to facilitate the evaluation of our revision, the line numbers of the reviewers’ comments refer to the originally submitted manuscript while line numbers of our responses refer to our revised manuscript.
Kindest regards,
Akshay Kamath (on behalf of the authors)
-
AC3: 'Reply on RC2', Akshay Kamath, 25 Aug 2025
-
RC3: 'Comment on egusphere-2025-2345', David Nathan, 04 Aug 2025
The paper presents a novel approach to interpolating full tensor gradiometric data, offering a clear theoretical foundation and demonstrating compelling improvements over truncated radial basis function (RBF) interpolation. The manuscript is well written, with strong motivation and a logical flow from theory to results. The examples provided effectively highlight the advantages of the proposed method in preserving spatial continuity and enhancing interpolation accuracy.
Given that interpolated potential field data often serve as input for geophysical inversion, further discussion of the ENF approach’s implications in this context would strengthen the manuscript. Specifically, the observation noted in line 293 suggests potential limitations when applying the method in ensemble-based inversion frameworks, such as the ensemble Kalman inversion. These methods rely on statistical assumptions and error covariance structures that could be influenced by interpolation artifacts or over-smoothing. A brief exploration of how ENF interpolation might influence inversion performance and uncertainty propagation would add valuable context for practitioners.
Minor corrections:
Line 123: \hat{k} needs to be defined
Line 388-389: Please update the reference Laloy et al., 2013. I was unable to find any source for it
Citation: https://doi.org/10.5194/egusphere-2025-2345-RC3 -
AC4: 'Reply on RC3', Akshay Kamath, 25 Aug 2025
Dear David Nathan,
We thank you for your time and effort reviewing the submitted manuscript, and are pleased that you appreciated our results. We have incorporated your suggestions into the revised manuscript, as detailed in the following pages. Please note that to facilitate the evaluation of our revision, the line numbers of the reviewers’ comments refer to the originally submitted manuscript while line numbers of our responses refer to our revised manuscript.
Kindest regards,
Akshay Kamath (on behalf of the authors)
-
AC4: 'Reply on RC3', Akshay Kamath, 25 Aug 2025
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Samuel T. Thiele
Hernan Ugalde
Bill Morris
Raimon Tolosana-Delgado
Moritz Kirsch
Richard Gloaguen
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
(25959 KB) - Metadata XML
Dear authors,
Unfortunately, after checking your manuscript, it has come to our attention that it does not comply with our "Code and Data Policy".
https://www.geoscientific-model-development.net/policies/code_and_data_policy.html
Your manuscript does not contain a Data Availability section, and it must include it. In such section, you must include the links and DOIs for the repositories containing all the data used to perform your work, both input and output data.
Moreover, I would like to ask you to clarify in your manuscript the version numbers of the software and libraries that you have used for your work.
Please, reply to this comment with the requested information as soon as possible, and include it in any future version of your manuscript. Also, note that if you do not fix these issues, we will not be able of publishing your manuscript.
Juan A. Añel
Geosci. Model Dev. Executive Editor