SlowGuess/NeurIPS_2023_OCR huggingface dataset:
The first experiment examined the impact of separate versus joint inversion of  $\mathrm{V_P}$  and  $\mathrm{V_S}$  on the accuracy of predicted Poisson ratio (Pr) maps. This process involved individually training two InversionNets on the  $\mathbb{E}^{\mathbf{FWI}}$  dataset to predict  $\mathrm{V_P}$  and  $\mathrm{V_S}$  maps, which were then used to calculate the Pr maps. The results revealed a substantial deterioration in map quality, with the independent inversion maps exhibiting significantly higher MAE and MSE, and lower SSIM values, as outlined in Table 5, compared to those reconstructed from joint inversion, shown in Table 3, especially for the complex B datasets, such as "ECFB". These findings reinforce the significance of considering the  $\mathrm{V_P}-\mathrm{V_S}$  relationship and P-S wave coupling, with the single-parameter inversion approach being deemed unviable. Detailed information on this experiment can be found in Section 6 of the supplementary materials.
Table 4: Training time in each epoch by each benchmarking method on  $\mathbb{E}^{\mathbf{FWI}}$  datasets. All the models are trained on a single GPU.
ElasticNet	ElasticGAN	ElasticTransformer
EVel Family	4m15s	2m20s	1m15s
EFault Family	8m35s	3m50s	2m30s
Table 5: Quantitative results of OPENFWI's InversionNet trained with  $\mathbb{E}^{FWI}$  data. Input z-component data and output the  $V_{P} / V_{S}$  maps independently. The performance is a benchmark of ElasticNet.
Dataset	Loss	InversionNet
VP	VS	Pr
MAE↓	RMSE↓	SSIM↑	MAE↓	RMSE↓	SSIM↑	MAE↓	RMSE↓	SSIM↑
EFVA	l1	0.0392	0.0712	0.9455	0.0239	0.0447	0.9590	0.0461	0.0885	0.8282
l2	0.0451	0.0745	0.9414	0.0251	0.0469	0.9585	0.0541	0.1039	0.8071
EFVB	l1	0.1030	0.1986	0.8260	0.0643	0.1318	0.8615	0.1290	0.2773	0.6063
l2	0.0883	0.1832	0.8453	0.0620	0.1269	0.8665	0.0905	0.1980	0.6603
ECVA	l1	0.1016	0.1699	0.7636	0.0736	0.1245	0.7837	0.1050	0.2141	0.5607
l2	0.1052	0.1730	0.7460	0.0720	0.1236	0.7798	0.1194	0.2381	0.5454
ECVB	l1	0.1854	0.3266	0.6270	0.1388	0.2405	0.6578	0.1523	0.3064	0.4695
l2	0.1820	0.3260	0.6323	0.1331	0.2344	0.6674	0.1553	0.3230	0.4648
EFFA	l1	0.0818	0.1413	0.8625	0.0584	0.1034	0.8681	0.0784	0.1585	0.6937
l2	0.0788	0.1343	0.8930	0.0946	0.1525	0.7918	0.1351	0.2485	0.6174
EFFB	l1	0.1323	0.2001	0.6790	0.0943	0.1453	0.7312	0.1180	0.2283	0.5280
l2	0.1301	0.1979	0.6808	0.0898	0.1382	0.7399	0.1124	0.2095	0.5494
ECFA	l1	0.1012	0.1638	0.8624	0.0710	0.1182	0.8485	0.0778	0.1586	0.6857
l2	0.0962	0.1663	0.8443	0.0833	0.1394	0.8106	0.1032	0.1911	0.6393
ECFB	l1	0.1702	0.2485	0.6020	0.1243	0.1807	0.6531	0.1317	0.2378	0.5091
l2	0.1745	0.2563	0.5849	0.1219	0.1775	0.6588	0.1379	0.2529	0.4841
5.2 Investigating P- and S-waves Coupling via Machine Learning
The second experiment focused on examining the interaction between P- and S-waves in the context of seismic data inversion. Two InversionNets were trained, one focusing on P-wave velocity  $(\mathrm{V_P})$  and the other on S-wave velocity  $(\mathrm{V_S})$ , while adjusting the structural characteristics of the ignored wave. This experiment, trained with the OPENFWI's InversionNet using data from  $\mathbb{E}^{\mathbf{FWI}}$ , revealed that any minor change in the disregarded wave velocity structure significantly degraded the network's performance, as evidenced in Table 6. This outcome was clearly demonstrated in the more complex datasets, such as "ECFB" test set, where changes in structure led to a substantial increase in MAE and RMSE, along with a decrease in the SSIM. For a more detailed analysis, refer to the supplementary materials.
6.1 Future Challenge
Decouple P- and S-waves: The interaction between P- and S-waves during seismic wave propagation poses a significant challenge when attempting to simultaneously determine P and S velocities. The networks described in this paper exhibit limited success in separating P- and S-waves within the seismic data. Consequently, we anticipate the development of robust methodologies that can precisely estimate both P and S velocities while effectively mitigating the interdependence between these wave components.
Generalization of data-driven methods: The elastic approximation provides a more accurate representation of field data in comparison to acoustic data. As a result, we expect the neural networks trained on the  $\mathbb{E}^{\mathrm{FWI}}$  dataset to exhibit improved resilience in handling real-world field data. However, it should be noted that there are additional physical phenomena, such as anisotropy and viscosity, which are not accounted for in the  $\mathbb{E}^{\mathrm{FWI}}$  dataset. The question of how to incorporate these phenomena into the analysis of field data remains an open and unanswered challenge.
Forward modeling: The computational expense associated with elastic forward modeling surpasses that of acoustic cases due to various factors. These include the increased memory requirements and the implementation of smaller grid sizes to counteract dispersion phenomena, among others. A detailed comparison highlighting these aspects can be found in the last section of supplementary
Table 6: Quantitative results of InversionNet trained with  $\mathbb{E}^{\mathrm{FWI}}$  data. The performance compared between testing on the same and different disregarded velocity structural datasets.
Dataset	Loss	InversionNet
VP: different VS structure	VS: different VP structure
MAE↓	RMSE↓	SSIM↑	MAE↓	RMSE↓	SSIM↑
EFVA	l1	0.1162	0.1919	0.8974	0.2040	0.2716	0.7977
l2	0.1245	0.2101	0.8849	0.2189	0.2928	0.7793
EFVB	l1	0.2098	0.3527	0.7086	0.2479	0.3285	0.7182
l2	0.1940	0.3265	0.7279	0.2540	0.3382	0.7022
ECVA	l1	0.1624	0.2590	0.7233	0.2202	0.2924	0.6794
l2	0.1678	0.2647	0.7043	0.2249	0.2987	0.6824
ECVB	l1	0.3014	0.4828	0.5206	0.3136	0.4246	0.5398
l2	0.3067	0.4900	0.5181	0.3207	0.4349	0.5341
FFA	l1	0.1741	0.2732	0.7464	0.2200	0.2865	0.7362
l2	0.1303	0.2052	0.8490	0.2228	0.2913	0.6952
FFFB	l1	0.1575	0.2291	0.6501	0.2350	0.3030	0.6480
l2	0.1627	0.2336	0.6492	0.2319	0.2982	0.6517
CFA	l1	0.1404	0.2244	0.7951	0.2327	0.3059	0.7360
l2	0.1319	0.2134	0.8159	0.2399	0.3194	0.7139
CFB	l1	0.1937	0.2798	0.5747	0.2465	0.3215	0.5785
l2	0.1928	0.2788	0.5613	0.2445	0.3183	0.5915
materials. Despite the possibility of bypassing extensive forward modeling by providing the  $\mathbb{E}^{\mathrm{FWI}}$  dataset, there remains a need to explore an efficient forward modeling algorithm to accommodate the growing volume of data in the field.
6.2 Broader Impact
Multiparameter inversion: Multiparameter inversion techniques have found wide-ranging applications across diverse scientific and engineering domains, including but not limited to geophysics, medical imaging, and material science. The introduction of  $\mathbb{E}^{\mathrm{FWI}}$  serves as a catalyst for further investigation and the pursuit of innovative methodologies in these fields. By addressing the inherent limitations and complexities associated with multiparameter inversion, this advancement encourages ongoing research and the exploration of novel solutions.
**Carbon-zero emission:** The attainment of carbon-zero emissions holds paramount significance in addressing climate change, safeguarding human well-being, and fostering sustainable development. While researchers continue to explore effective strategies towards achieving this goal, elastic FWI emerges as a promising approach that can contribute significantly. Particularly, elastic FWI plays a crucial role in assessing and developing geothermal energy resources, as well as in facilitating carbon capture and storage projects, among other applications. The introduction of  $\mathbb{E}^{\mathrm{FWI}}$  as a fundamental dataset for elastic FWI is expected to stimulate further research and innovation in this direction, thereby enhancing our understanding and capabilities in the pursuit of carbon-zero emissions.
New energy exploration: Elastic FWI can be utilized to evaluate the geological viability of potential sites for hydrogen storage, including underground formations or depleted oil and gas reservoirs. The suitability, capacity, and feasibility of a storage site heavily rely on the effectiveness of a geophysical survey and characterization approaches. With the availability of the  $\mathbb{E}^{\mathrm{FWI}}$  dataset, it would yield great potential to enhance the accuracy in characterizing the subsurface reservoir, therefore providing better identification of hydrogen storage locations.
Potential Social Impacts: Our research, while advancing inverse problems in natural science, may inadvertently support increased fossil fuel consumption if applied to optimize oil and gas drilling, raising environmental concerns. However, the same techniques can equally bolster positive initiatives, such as geothermal exploration and hydrogen storage, as highlighted in Sec 6.2. The dual potential of
our methodologies underscores the importance of their judicious application, aligning with sustainable development goals.
This paper presents  $\mathbb{E}^{\mathrm{FWI}}$ , an open-source elastic FWI dataset.  $\mathbb{E}^{\mathrm{FWI}}$  comprises eight datasets and includes benchmarks for three deep learning methods. The datasets released with  $\mathbb{E}^{\mathrm{FWI}}$ , provide diverse P-wave and S-wave velocities, specifically addressing the coupling problem encountered in multiparameter inversion. The initial benchmarks demonstrate promising results on certain datasets, while others may require further investigation. Additionally, coupling tests are conducted to provide insights into network design for multiparameter inversion problems. Furthermore, this paper discusses the future challenges that can be explored using these datasets and outlines the envisioned future advancements as  $\mathbb{E}^{\mathrm{FWI}}$  continues to evolve.
This work was funded by the Los Alamos National Laboratory (LANL) - Technology Evaluation and Demonstration (TED) program and by the U.S. Department of Energy (DOE) Office of Fossil Energy's Carbon Storage Research Program via the Science-Informed Machine Learning to Accelerate Real-Time Decision Making for Carbon Storage (SMART-CS) Initiative.
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