GlobalTomo: A global dataset for physics-ML seismic wavefield modeling and FWI

Diversity of the earth models

Our earth model design follows established principles in geophysical studies. The random 3D earth model is constructed using Latin Hypercube Sampling with a spherical harmonics parameterization, which serves as a complete basis function. This approach provides sufficient representation of complex and diverse 3D structures, as described in Section 2.2. We set the maximum perturbation range between -10% and 10%, ensuring that the synthetic models remain within a realistic range relative to observed earth structures. This range is ample, as most prior 3D global tomography studies report perturbations of only a few percent. Testing extreme values would lack geophysical interpretability and fail to offer meaningful insights [1-3]. We will clarify this point further in the revised manuscript.

How to avoid bias in earth models

We agree with the reviewer’s point. This is a challenging scientific question and is precisely why we design synthetic models that try to closely mimic real Earth scenarios. This approach is aligned with prior work in large-scale earthquake simulation on supercomputers, like those recognized with the Gordon Bell Prize [1]. By testing the inversion performance on these synthetic models, we assess whether the inverse results are within an acceptable range. If these results align well, it provides confidence that similar accuracy can be expected with real data. that real data will yield comparable outcomes. Given the inherently ill-posed nature of this problem, our objective is not an absolute solution but rather to achieve increasingly refined inverse results that can support meaningful geophysical interpretation.

[1] Fu H, He C, Chen B, et al. 18.9-Pflops nonlinear earthquake simulation on Sunway TaihuLight: enabling depiction of 18-Hz and 8-meter scenarios. Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis.

Consider dispersion like in numerical solvers

Thank you for your constructive feedback. We acknowledge the importance of dispersion, especially in the context of inverse problems. Our work represents an initial step in addressing global tomography through established machine learning tools (MLP, CNN, Transformers, and Physics-informed Neural Operators) and evaluation methods (Relative L2 Loss, Pearson Correlation, etc.). Our current approach prioritizes achieving a rapid approximation via machine learning operators (6000 times faster than the numerical solvers as shown in our experiments), even at the expense of precise PDE solution fidelity. Thus, we can focus on an efficient and broad exploration of the large parameter space in inverse problems to obtain better tomography (Section 3.2.2). We appreciate your suggestion and look forward to incorporating more comparative metrics from numerical PDE literature to assess ML-based forward modeling in future work.

Out-of-distribution problems

Thank you for this valuable suggestion regarding out-of-distribution (OOD) generalizability. We acknowledge the importance of exploring OOD cases in ML studies; however, given the uniqueness of Earth as our subject, creating OOD scenarios falls outside our primary objective of achieving accurate Earth tomography. Our design relies on well-established geological studies [1-3] that provide a credible distribution of Earth's internal properties, ensuring our model is grounded in recognized physical and structural characteristics. Expanding to OOD cases would involve hypothetical or artificially generated distributions, potentially misaligned with realistic geological relevance.

[1] Dziewonski A M, Woodhouse J H. Global images of the Earth's interior. Science, 1987.

[2] French S W, Romanowicz B A. Whole-mantle radially anisotropic shear velocity structure from spectral-element waveform tomography. Geophysical Journal International, 2014.

[3] Tromp J. Seismic wavefield imaging of Earth’s interior across scales. Nature Reviews Earth & Environment, 2020.

Data fit in inversion

In our study, we explore two types of inversion methods.

The first method involves training a forward model, which we then use as a proxy for gradient-based optimization. This approach resembles traditional Gauss-Newton or other inversion methods that fit the observed data with the forward model, but with a key difference: rather than using a numerical solver, which can be computationally slow, our forward model is learned directly from data. This data-driven approach provides a more efficient alternative.

The second method we employ is a direct inversion network, which directly predicts the model structure from observed data in a single step. Unlike traditional methods, this approach does not fit the observed data per se; instead, it learns the underlying distribution of structure-data pairs using simulated examples.

We would like to further make sure that our response addresses your concerns regarding Earth model diversity, potential biases, forward modeling dispersion, and inversion techniques. Our research makes significant contributions to solving the long-standing challenges in global seismic tomography through advanced ML approaches. We believe our work will facilitate both geophysical tomography capabilities and machine learning methodologies for forward and inverse problems. We welcome any additional questions you may have about our work. Based on these clarifications, we respectfully request a raising of the score.

Thank you for your recognition of our dataset's structure and our efforts in establishing benchmarks using multiple ML methods for both forward and inverse problems. For your questions:

(a) For generating different examples, do we always perturb the same 1D background model or different? How is the data generation algorithm create different varying geological settings in the dataset? How do we measure the variability in different velocity structures? (b) In the model configuration section, it is mentioned that the model 3D structure is generated by perturbing a 1D background model from -10% to 10%. Given a 1D background model, how are number of layers decided per model example and how the perturbation and parameterization (using spherical harmonics) work to create a synthetic example?

The current geophysical community already reach a consensus on the 1D profile of real Earth structure. The 1D earth profile is well-constrained and based on the huge amount of historical traveltime and ray data. Here we choose the widely accepted 1D earth PREM model [1] and fix it throughout. Our focus here is to find the fine-grained 3D structure of Earth. This approach aligns with standard practices in the field, where current 3D Earth models are typically built upon an initial 1D model, followed by 3D perturbations. Importantly, variations in the choice of the initial 1D model have minimal impact on the resulting 3D tomography [2]. Beyond this, the 3D model structure is constructed by spherical harmonics at each predefined layer in the PREM model (Section 2.2 Parameterization, Line 200-205). Each layer has a clear geophysical meaning representing a fundamental interface such as the 410km and 660km discontinuity, and the core-mantle boundary (Appendix C.2, Line 951-957). The random 3D model is granted with the best variability through a Latin Hypercube Sampling of the coefficients (Line 213). The perturbation range is set from -10% to 10% to ensure that the distribution of the synthetic models do not lie far away from the real earth scenario. This allowed range is large enough as most previous 3D global tomography results show perturbation of only a few percent only. Testing beyond this range would be geophysically unrealistic and lack meaningful interpretation [3-5].

[1] Dziewonski, A. M., & Anderson, D. L. Preliminary reference Earth model. Physics of the earth and planetary interiors, 1981.

[2] Tromp J. Seismic wavefield imaging of Earth’s interior across scales. Nature Reviews Earth & Environment, 2020.

[3] French S W, Romanowicz B A. Whole-mantle radially anisotropic shear velocity structure from spectral-element waveform tomography. Geophysical Journal International, 2014.

[4] Thrastarson, S., van Herwaarden, D. P., Noe, S., Josef Schiller, C., & Fichtner, A. REVEAL: A global full‐waveform inversion model. Bulletin of the Seismological Society of America, 2024.

[5] Lei, W., Ruan, Y., Bozdağ, E., Peter, D., Lefebvre, M., Komatitsch, D., ... & Pugmire, D. Global adjoint tomography—model GLAD-M25. Geophysical Journal International, 2020.

The results section is not very clear to understand especially due to lack of enough evidence for certain claims.

Thank you for your reminder. Due to the limited pages, we have put our training details and hyperparameters in Appendix E and more visualization of forward and inversion results in Appendix H. Regarding how to incorporate physical constraints in the training, we have referred the readers to the paper [1] in Section 3.1 Baselines (Line 303). The numbers of examples in three ties are listed in table 1, indicated by sample number.

[1] Wang S, Wang H, Perdikaris P. Learning the solution operator of parametric partial differential equations with physics-informed DeepONets[J]. Science advances, 2021.

Would it be possible to have some visualizations of some datasets from all tiers along with experimental results for forward and inverse modeling for tier-2 and tier-3 dataset?

Applying the baseline to larger dataset tiers is challenging considering the scale of the problem and the computational cost. In this work, we focus on establishing the workflow for the general tomography task. We have released the dataset and baselines and hope to attract more ML researchers. We are keen to apply the Real Earth dataset in the next step.

We appreciate your recognition of the dataset's potential benefit to the community, as well as your positive assesment on our presentation.

I think the machine learning models are kind of simple. Based on my experience, MLPs seem not to be the best choices for 3D problems compared with other methods you chose in the paper (CNN or transformer). But the results show that MLP / InversionMLP performs best in most cases. I am concerned whether all methods are well-trained and under the best hyperparameters.

We have try our best to optimize our models using hyperparamter search techniques. While MLP is a fundamental machine learning model, its strong performance in our study aligns with its ability to generalize across broader scientific domains. A similar observation was made in the Climsim dataset for climate prediction [1], the best paper in NeurIPS Dataset and Benchmark Track 2024, where MLP also outperformed more complex models like CNN, RPN, and cVAE. As we noted in Section 3.2.1, MLP's advantage lies in predicting the output in a vector space rather than a point space, as used in methods like DeepONet and GNOT. Thus, the model parameters of MLPs will be 20 times more than those in the point-based models. These point-based models, despite their sophistication, may struggle with convergence on large-scale problems, even with 10 times the training steps required by MLP (see Appendix E.3 for training details). Additionally, given the sparse spatial-temporal structure of wavefield and seismogram data, localized feature extraction—which CNNs excel at—is less critical in this case. MLP, with its fully connected layers, effectively captures global patterns in our seismic data even when the signals are sparse.

[1] Yu S, Hannah W, Peng L, et al. ClimSim: A large multi-scale dataset for hybrid physics-ML climate emulation. Advances in Neural Information Processing Systems, 2024.

Could you compare the performance of these methods in scenarios of out-of-distribution?

Thank you for this valuable suggestion regarding out-of-distribution (OOD) generalizability. We acknowledge the importance of exploring OOD cases in ML studies; however, given the uniqueness of Earth as our subject, creating OOD scenarios falls outside our primary objective of achieving accurate Earth tomography. Our design relies on well-established geological studies [1-3] that provide a credible distribution of Earth's internal properties, ensuring our model is grounded in recognized physical and structural characteristics. Expanding to OOD cases would involve hypothetical or artificially generated distributions, potentially misaligned with realistic geological relevance.

[1] Dziewonski A M, Woodhouse J H. Global images of the Earth's interior. Science, 1987.

[2] French S W, Romanowicz B A. Whole-mantle radially anisotropic shear velocity structure from spectral-element waveform tomography. Geophysical Journal International, 2014.

[3] Tromp J. Seismic wavefield imaging of Earth’s interior across scales. Nature Reviews Earth & Environment, 2020.

In line 396, the paper says “To evaluate the flexibility of FWI”. Why is this FWI? I think this section is talking about forward modeling. Could you clarify whether this section is indeed about forward modeling rather than FWI?

Thank you for your reminder. This section is focused on forward modeling at higher resolution. The reference to FWI is intended to highlight the significance of conducting higher-resolution forward modeling, which is to achieve better inversion results in FWI across different resolution scales. We will change the sentence to "To evaluate seismic modeling across different resolution scales".

We appreciate your acknowledgment of our contributions in addressing key gaps in seismic data modeling and inversion, as well as your constructive feedback on areas for further improvement.

Wavefield Estimation. Question 1.

We appreciate the reviewer’s question regarding the inclusion of the full wavefield in the dataset. The primary motivation for incorporating the full wavefield is twofold:

1. Physics-informed training: Including the full wavefield allows the training process to incorporate rich, multi-scale physical constraints (obey the wave equation) that could regularize the inversion outcomes. By leveraging the complete spatiotemporal data, the model can learn physics-informed patterns in wave propagation throughout the subsurface layers. In contrast, purely fitting the subsurface data may fail to ensure physical consistency.
2. Interpretability and scientific insight: Access to the complete wavefield enables us to interpret model outputs in relation to the underlying geophysical processes, enhancing the scientific interpretability of the learned representations, and avoidng potential black-box mapping and data overfitting. Training with the full wavefield allows for step-by-step insights into how subsurface structures influence wave signals, which ultimately manifest in surface seismograms. This interpretability is crucial for geological researchers to assess the reliability of inversion outputs, particularly when only surface data is available during actual deployment.

It remains unclear whether the dataset, or its three tiers, adequately represents different distributions or if they cover a range of scenarios that would facilitate generalization. Question 2.

The dataset’s design is informed by geological studies [1-3] that accurately reflect Earth's internal structural parameters, specifically through the use of spherical harmonics. These harmonics, sampled using Latin Hypercube Sampling from -10% to 10% (Line 191), ensure comprehensive coverage across the entire design space of interest, thereby grounding the model in well-recognized physical and structural distributions . As our goal is to inverse our unique Earth's structure, we didn't consider out-of-distribution (OOD) generalization since OOD cases would involve hypothetical or artificially generated distributions, potentially misaligned with realistic geological relevance. Regarding the diversity of the three tiers, they incorporate a range of physical processes and equations that capture the dynamic difference of seismic scenarios, thereby enhancing our dataset’s applicability in various seismology applications. We will provide improved illustrations in the main text and table 1 to better showcase the dataset’s diversity.

[1] Dziewonski A M, Woodhouse J H. Global images of the Earth's interior. Science, 1987.

[2] French S W, Romanowicz B A. Whole-mantle radially anisotropic shear velocity structure from spectral-element waveform tomography. Geophysical Journal International, 2014.

[3] Tromp J. Seismic wavefield imaging of Earth’s interior across scales. Nature Reviews Earth & Environment, 2020.

Additionally, I did not find any generalization tests conducted with direct inversion methods, as outlined in Section 3.2.2. Question 3.

There seems to be some misunderstanding. The results for direct inversion methods in Section 3.2.2 are indeed evaluated on an unseen test set, using the same 90% training and 10% test split as our other experiments to assess generalization. We will make this clearer in the revised manuscript.

In geophysics, global geophysics and exploration or controlled geophysics are two different branches. These authors published a global seismic data which aims at large scale or global scales. Open FWI are aimed at exploration especially in oil and gas. I think this global dataset is very important for earthquake prediction, natural hazards.