Unveiling the Mechanism of Continuous Representation Full-Waveform Inversion: A Wave Based Neural Tangent Kernel Framework

Ruihua Chen; Yisi Luo; Bangyu Wu; Deyu Meng

cs.LGMar 23, 2026

Unveiling the Mechanism of Continuous Representation Full-Waveform Inversion: A Wave Based Neural Tangent Kernel Framework

Authors: Ruihua Chen, Yisi Luo, Bangyu Wu, Deyu Meng

Organizations: School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi, China · Macao Institute of Systems Engineering, Macau University of Science and Technology, Taipa, Macao

Abstract

Full-waveform inversion (FWI) estimates physical parameters in the wave equation from limited measurements and has been widely applied in geophysical exploration, medical imaging, and non-destructive testing. Conventional FWI methods are limited by their notorious sensitivity to the accuracy of the initial models. Recent progress in continuous representation FWI (CR-FWI) demonstrates that representing parameter models with a coordinate-based neural network, such as implicit neural representation (INR), can mitigate the dependence on initial models. However, its underlying mechanism remains unclear, and INR-based FWI shows slower high-frequency convergence. In this work, we investigate the general CR-FWI framework and develop a unified theoretical understanding by extending the neural tangent kernel (NTK) for FWI to establish a wave-based NTK framework. Unlike standard NTK, our analysis reveals that wave-based NTK is not constant, both at initialization and during training, due to the inherent nonlinearity of FWI. We further show that the eigenvalue decay behavior of the wave-based NTK can explain why CR-FWI alleviates the dependency on initial models and shows slower high-frequency convergence. Building on these insights, we propose several CR-FWI methods with tailored eigenvalue decay properties for FWI, including a novel hybrid representation combining INR and multi-resolution grid (termed IG-FWI) that achieves a more balanced trade-off between robustness and high-frequency convergence rate. Applications in geophysical exploration on Marmousi, 2D SEG/EAGE Salt and Overthrust, 2004 BP model, and the more realistic 2014 Chevron models show the superior performance of our proposed methods compared to conventional FWI and existing INR-based FWI methods.

Figures & tables

Figure 1: (a) The acquisition process of observed seismic data obtained from the receivers placed on the surface; (b) FWI is a nonlinear inverse problem that inverts the subsurface models from observed seismic data. The CR-FWI framework synthesizes seismic data by solving the wave equation 1 , where the model $m({{\bf x}})$ is represented by a coordinate-based neural network $m_{\theta}({{\bf x}})$ , e.g., INR.

Figure 2: (a) Inversion results using conventional FWI (i.e., Multiscale ADFWI ( Liu et al., 2025a ) ), existing INR-based FWI (i.e., IFWI ( Sun et al., 2023a ) ), and our proposed IG-FWI methods with smooth (after 500 epochs) and constant initial models (after 1500 epochs) on the 2004 BP model; (b) Eigenvalue spectrum comparison of wave-based NTK using different FWI methods.

Figure 3: Pipeline of CR-FWI. CR-FWI employs (a) implicit neural representation, (b) low rank tensor function, or (c) multi-grid parametric encoding to represent the velocity parameter model and integrate the wave equation in a loop.

Figure 4: Pipeline of the proposed INR-Grid hybrid representation for FWI. This method combines encoding features from INR and MPE, which are then fused using an MLP network.

Figure 5: (a) Mean and standard deviation of the Frobenius norm of the wave-based NTK at initialization. (b) Relative Frobenius norm during training. (c) Eigenvalues decay using different methods.

Method	Marmousi Model					Overthrust Model		Salt Model		2004 BP Model		Computing Resource
Method	Smooth	Constant	G-Noise	F-Cutoff	S-Shots	Smooth	Constant	Smooth	Constant	Smooth	Constant	Time	Params	Memory
ADFWI	0.2132	1.1522	0.5422	0.4975	0.4730	0.0592	1.3364	0.6462	0.6337	0.1003	0.4281	1.89	94.00	17.09
ADFWI-TV	0.2449	0.9386	0.3875	0.3764	0.3547	0.1154	1.3142	0.5041	0.5318	0.0995	0.4259	1.92	94.00	17.09
WECI-FWI	0.2977	1.2537	0.6366	0.3220	0.3337	0.1182	1.5526	0.3032	1.0293	0.0951	0.3847	3.12	94.00	17.56
MS-FWI	0.1683	1.1313	0.3547	0.4198	0.3325	0.0612	1.0802	0.5358	0.9234	0.0672	0.4667	2.25	94.00	17.09
IFWI	0.1907	0.9474	0.3374	0.3358	0.3483	0.0683	0.6432	0.1201	0.7412	0.0719	0.1412	1.88	50.05	17.48

Table 1: Performance comparison (MSE) of conventional FWI (ADFWI, ADFWI-TV, WECI-FWI, MS-FWI) and CR-FWI (IFWI, WinFWI, LR-FWI, MPE-FWI, IG-FWI) methods. Best results are highlighted in bold , and second-best are underlined . Complete results see Appendix B.6.1 .

Figure 6: Conventional FWI and CR-FWI performance comparison using different initial velocity models on different velocity model datasets. More results are in the Appendices from B.6.2 to B.6.6 .

Figure 7: Convergence curves of velocity model MSE using different FWI methods and different datasets with constant initial models. (a) Marmousi. (b) Overthrust. (c) Salt body. (d) 2004 BP

Figure 8: Generated velocity visualizations corresponding to different iterations using IFWI, MPE-FWI, and IG-FWI methods on 2004 BP model. More detailed results are in Appendix B.6.7 .

Figure 9: Ablation study for frequency, grid resolution, and weighting factor.

Methods	InvNet	ADFWI	IFWI	LR-FWI	IG-FWI
FaltVel	36.78	21.30	11.72	6.65	6.94
CurveVel	94.16	57.07	16.87	9.89	7.93
FaltFault	113.18	48.23	46.59	19.59	28.84
CurveFault	100.09	99.14	65.89	60.23	24.61
Style	53.76	27.66	48.25	14.56	17.08

Table 2: MSE of data-driven FWI and CRFWI. Best results are highlighted in bold , and second-best are underlined .

Appendix figures & tables31 assets

Supplementary material from the paper’s appendix.

Appendix

Category	Abbreviation	Method Name
Conventional FWI Methods	ADFWI ( Liu et al., 2025a )	Automatic differentiation-based FWI
	ADFWI-TV ( Esser et al., 2018 )	ADFWI with total variant regularization
	WECI-FWI ( Song et al., 2023 )	Weighted envelope correlation-based FWI
	MS-FWI ( Fichtner et al., 2013 )	Frequency-based multiscale inversion FWI
Existing CR-FWI Methods	IFWI ( Sun et al., 2023a )	INR-based FWI with SINRE’s activation
Existing CR-FWI Methods	WinFWI ( Yang & Ma, 2025 )	INR-based FWI with Gabor wavelet activation

Appendix

Table 3: Conventional FWI, existing CR-FWI, and proposed CR-FWI methods.

Figure 10: Datasets and Initial velocity model, including smooth, linear, and constant initial models.

Figure 11: Observed seismic data using different velocity models, including (a) Marmousi model, (b) SEG/EAGE Overthrust model, (c) SEG/EAGE Salt model, and (d) 2004 BP model.

Figure 12: (a) 1D velocity model. (b) observed seismic data. (c) Ricker source function.

Metric	Method	Initial velocity model			Gaussian noise		Frequency cutoff		Shots number
Metric	Method	Smooth	Linear	Constant	$4\sigma_{0}$	$8\sigma_{0}$	4 Hz	6 Hz	9 shots	5 shots
SSIM $\uparrow$	ADFWI	0.6260	0.2689	0.0549	0.2167	0.1401	0.2594	0.2287	0.2499	0.1969
	ADFWI-TV	0.5466	0.2735	0.1503	0.2628	0.2456	0.2607	0.2462	0.2718	0.2289
	WECI-FWI	0.4239	0.3638	0.0220	0.1193	0.0948	0.3581	0.3475	0.3240	0.2926
	MS-FWI	0.7093	0.6559	0.0952	0.5274	0.4218	0.5298	0.4193	0.5362	0.4798
	IFWI	0.6510	0.6173	0.5252	0.5410	0.4607	0.5587	0.5124	0.5802	0.4234

Appendix

Table 4: Comparisons between different conventional FWI (i.e., ADFWI, ADFWI-TV, WECI-FWI, and MS-FWI) and CR-FWI (i.e., IFWI, WinFWI, LR-FWI, MPE-FWI, and IG-FWI) methods across different evaluation metrics for the 2D Marmousi model. Here, bold represents the best evaluation result, while underline represents the second-best evaluation

Metric	Method	2004 BP model			Salt model			Overthrust model
Metric	Method	Smooth	Linear	Constant	Smooth	Linear	Constant	Smooth	Linear	Constant
SSIM $\downarrow$	ADFWI	0.5104	0.2391	0.0920	0.1717	0.3770	0.2334	0.7334	0.4699	0.0324
	ADFWI-TV	0.5042	0.2573	0.0963	0.2778	0.5272	0.3315	0.4497	0.3098	0.1035
	WECI-FWI	0.4392	0.2111	0.0973	0.3290	0.2525	0.0877	0.4958	0.1413	0.0931
	MS-FWI	0.6872	0.4157	0.0493	0.2590	0.4515	0.3020	0.7033	0.6802	0.0456
	IFWI	0.6168	0.5807	0.5533	0.5882	0.5177	0.4189	0.7124	0.6849	0.6398

Appendix

Table 5: Comparisons between conventional FWI (i.e., ADFWI, ADFWI-TV, WECI-FWI, and MS-FWI) and CR-FWI (i.e., IFWI, WinFWI, LR-FWI, MPE-FWI, and IG-FWI) methods for the other challenging models. Here, bold represents the best evaluation result, while underline represents the second-best evaluation.

Figure 13: Conventional FWI, INR-based FWI, and our proposed FWI methods ( bold ) performance comparison using different initial velocity models (i.e., smooth, linear, and constant) on the 2D Marmousi model.

Figure 14: Conventional FWI, INR-based FWI, and our proposed FWI methods ( bold ) performance comparison using a linear initial model with different seismic data quality on the 2D Marmousi model.

Figure 15: Conventional FWI, INR-based FWI, and our proposed FWI methods ( bold ) performance comparison using different initial velocity models (i.e., smooth, linear, and constant) on the 2D SEG/EAGE Overthrust model.

Figure 16: Conventional FWI, INR-based FWI, and our proposed FWI methods ( bold ) performance comparison using different initial velocity models (i.e., smooth, linear, and constant) on the 2D SEG/EAGE Salt model.

Figure 17: Conventional FWI, INR-based FWI, and our proposed FWI methods ( bold ) performance comparison using different initial velocity models (i.e., smooth, linear, and constant) on the 2004 BP model.

Figure 18: Generated velocity model corresponding to different iterations with conventional FWI, IFWI, MPE-FWI, IG-FWI, and LR-FWI using Marmousi model

Figure 19: Generated velocity model corresponding to different iterations with conventional FWI, IFWI, MPE-FWI, IG-FWI, and LR-FWI using Overthrust model

Figure 20: Generated velocity model corresponding to different iterations with conventional FWI, IFWI, MPE-FWI, IG-FWI, and LR-FWI using Salt model

Figure 21: Generated velocity model corresponding to different iterations with conventional FWI, IFWI, MPE-FWI, IG-FWI, and LR-FWI using 2004 BP model

Figure 22: (a) Initial source wavelet and (b) its corresponding normalized spectrum, as provided by Chevron Oil Company for FWI.

Figure 23: Inversion Results of 2014 Chevron blind test model. Left: (a) 2014 Chevron initial velocity model. (b) Inversion results using multiscale ADFWI ( Fichtner et al., 2013 ; Liu et al., 2025a ) . (c) Inversion results using INR-based FWI ( Sun et al., 2023b ) . (d) Inversion results using our proposed IG-FWI; Right: (a) Velocity profiles of the well log, initial model, and inverted models using different FWI methods. (b) Comparison of a common-source shot gather between the true and modeled records. Here $I$ denotes simulated data and $O$ denotes the observed data.

Figure 24: (a) 3D acquisition systems. (b) Representative seismic data.

Method	Normalized MSE $\downarrow$	SSIM $\uparrow$	Memory (GB)	Time (s/it)
ADFWI	0.1469	0.5576	15.89	13.82
IFWI	0.1302	0.4634	30.61	15.08
IG-FWI	0.1151	0.5729	22.84	14.61
LR-FWI	0.0968	0.5914	16.01	14.09

Appendix

Table 6: Performance comparison of different FWI methods on 3D Overthrust model. Here, bold represents the best evaluation result, while underline represents the second-best evaluation.

Figure 25: 3D SEG/EAGE Overthrust model test using a smooth initial model. (a) Ground truth. (b) Smooth initial model. (c) ADFWI. (d) INR-based FWI. (e) IG-FWI. (f) LR-FWI.

Figure 26: OpenFWI datasets and noisy seismic data input.

Figure 27: Inversion results on FlatVel dataset using different FWI methods.

Figure 28: Inversion results on FlatFault dataset using different FWI methods.

Figure 29: Inversion results on CurveVel dataset using different FWI methods.

Figure 30: Inversion results on CurveFault dataset using different FWI methods.

Figure 31: Inversion results on Style dataset using different FWI methods.

Figure 32: Inversion results on the Canadian Foothills model with irregular topography

Figure 33: Inversion results on the Overthrust model with a near-surface low-speed layer

Figure 34: Inversion results using LRG-FWI method.

Method	MPE-FWI	LR-FWI	LRG-FWI	IG-FWI
Marmousi (MSE $\downarrow$ )	2.2266	0.2893	0.1995	0.2961
Overthrust (MSE $\downarrow$ )	1.2887	0.1592	0.1354	0.5724
Salt body (MSE $\downarrow$ )	1.1008	0.2368	0.2179	0.1883

Appendix

Table 7: Comparisons with LRG-FWI and other CR-FWI methods.

Figure 35: Framework of our theoretical proof.

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Full-waveform inversion (FWI) estimates unknown parameters in the wave equation from limited boundary measurements. Recent advances in neural reparameterized FWI (NeurFWI) demonstrate that representing the parameters using a neural network can reduce the reliance on the high-quality initial model and wavefield data, at the cost of slow high-resolution convergence. However, its underlying theoretical mechanism remains unclear. In this study, we establish the neural sensitivity kernel (NSK) and the wave tangent kernel (WTK) to analyze their convergence behavior from both model and data domains. These theoretical frameworks show that the neural tangent kernel (NTK) induced by neural representation adaptively modulates the original sensitivity and wave tangent kernels. This modulation leads to several key outcomes, i.e., the spectral filtering effect, the gradient wavenumber modulation, and the wave frequency bias, connecting the convergence behavior of NeurFWI with the eigen-structures of NSK and WTK. Building on these insights, we propose several enhanced NeurFWI methods with tailored eigen-structures in NSK and WTK to improve inversion performances and efficiency. We numerically validate these theoretical claims and the proposed methods in seismic exploration, and firstly extend their application to medical imaging.

Abstract

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