Wavenumber-aware diffusion sampling to regularize multiparameter elastic full waveform inversion

SUMMARY

Multiparameter elastic full waveform inversion (EFWI) provides a more realistic depiction of the subsurface models than the standard acoustic approximation. In practice, however, the significant additional cost and interdependence between the unknown parameters (cross-talks) hinder the application of such algorithms. Diffusion model-based regularization can be used to improve the inversion results while simultaneously injecting prior information into the solution. The main challenge here is how to inject such priors into the EFWI iterations that can better complement the solution’s evolution. To address this challenge, we incorporate a model wavenumber continuation process into a diffusion model-based regularization contribution to multiparameter EFWI. To do so, we promote a sampling strategy such that at the early iteration, the proposed regularization updates account for the low wavenumber component more and increase progressively with the iteration. We first train the diffusion model on elastic moduli images in an unsupervised manner and incorporate the trained model during the EFWI inversion. We deliberately use single-component measurements, which is the most common acquisition scenario, during the inversion to demonstrate the effectiveness of our regularization. At the inference stage, the proposed framework provides more accurate solutions with negligible additional computational cost compared to several conventional regularization algorithms.

1 INTRODUCTION

Multiparameter elastic full waveform inversion (EFWI) attempts to directly invert the elastic properties of the Earth from seismic recordings. For seismic imaging, having access to such a process might provide high-resolution velocity models that enhance the seismic image quality in areas with high-contrast elastic bodies. For energy exploration, this might alleviate the need to perform sophisticated processing steps for reservoir characterization to perform the more common post-migration and angle gathers inversions. For imaging the deep structures of the Earth, multiparameter FWI provides more information as it incorporates more accurate assumptions about the medium.

Recently, successful applications of multiparameter FWI have been reported ranging from exploration (Vigh et al. 2014; Wang et al. 2021; Routh et al. 2023) to global scales (Bozdağ et al. 2016; Lei et al. 2020; Cui et al. 2024). While these applications possess their own unique challenges amidst the differences in scales, they share the same theoretical foundation dictated by the FWI process. Fundamentally, FWI involves a data-driven iterative process in which the solution is sought by minimizing the observed and simulated seismic data (Lailly & Bednar 1983; Tarantola 1984). Consequently, obtaining good quality seismic recording becomes a prerequisite that often dictates the inverted results. Unfortunately, the limited seismic recording coverage (i.e. illumination), the unknown observation noise and the band-limited nature of our recorded data result in data with limited or no sensitivity to certain scales of the velocity model. These scales include the infamous middle model wavenumber components between the low wavenumbers controlling wave propagation and high wavenumbers causing scattering.

Such phenomenon, otherwise known as scale separation (Claerbout 1985; Wu & Toksöz 1987; Jannane et al. 1989; Mora 1989), has also been at the heart of many seismic processing and model building workflows. To mitigate the missing middle wavenumber components, typical seismic processing workflows include a two-step process in which a specific algorithm is used alternately to process the low and high wavenumber components of the model. Typically, the low wavenumber is kept fixed, and migration-based algorithms (McMechan 1982; Baysal et al. 1983; Loewenthal & Mufti 1983) are employed to update the high wavenumber components of the model. Then, the process is followed by, in turn, keeping the high wavenumber fixed while the low wavenumber is updated by traveltime- (Bishop et al. 1985; Farra & Madariaga 1988; Stork 1992; Billette & Lambaré 1998; Taillandier et al. 2009; Waheed et al. 2015; Waheed & Alkhalifah 2017) or velocity analysis-based algorithms (Gardner et al. 1974; Al-Yahya 1989; MacKay & Abma 1992; Zhu et al. 1998).

Instead of treating the low and high wavenumber components separately, FWI attempts to recover the complete wavenumber components of the medium. Theoretically, in a perfect setting where the sources and receivers completely enclose the medium, we can reliably obtain the middle model wavenumber information of the subsurface (Virieux & Operto 2009). In practice, however, the lack of low-frequency components in the data and the missing ultra-long offset acquisition setup make the FWI updates fail to accurately progress from the (low wavenumber) initial to the (high wavenumber) desired model.

To mitigate this, based on the migration-based traveltime inversion concept (Chavent et al. 1994), FWI has also been extended to solely update the low wavenumber components by assuming a known high wavenumber model/image (Xu et al. 2012; Ma & Hale 2013; Wu & Alkhalifah 2015). By explicitly treating the scale separation issue, Zhou et al. (2015) combined the concept of reflection FWI and diving waves FWI, termed joint FWI (JFWI), to update the high and low wavenumber components alternately. However, in the context of multiparameter EFWI, the workflows mentioned above significantly increase the already computationally expensive EFWI process.

Alternatively, different sources of middle wavenumber components can be injected during FWI without explicit scale separation. One potential solution to partially fill the missing middle wavenumber is in the form of wavenumber continuation, which can be viewed from either data (Bunks et al. 1995; Pratt et al. 1996) and model standpoint (Almomin & Biondi 2013; Tang et al. 2013; Alkhalifah 2014). Typically, we perform several cycles of FWI in which we hope the highest inverted wavenumber components for a given cycle form sufficiently low wavenumber components for the next.

Another potentially more efficient source of filling the missing middle wavenumber components is regularization (Alkhalifah 2016). Theoretically, regularization is introduced to inject desirable properties (known prior) about the inverted subsurface while simultaneously reducing the solution space during the inversion. In practice, such information is usually introduced by enforcing (by way of additional constraints) specific desirable properties: smoothness (Tikhonov 1963), blockiness (Strong & Chan 2003), etc. In the context of multiparameter EFWI, besides the cross-talk issue, the need to account for different magnitudes of the elastic parameters poses a significant challenge when performing such regularization techniques. The emerging deep learning-based solutions have also demonstrated regularization capabilities for EFWI. Zhang & Alkhalifah (2020, 2022) utilize a supervised deep neural network to inject borehole-derived facies information into EFWI. Similarly, Li et al. (2021) promotes using borehole information to improve the resolution of EFWI by utilizing a supervised neural network in a probabilistic manner. While shown to possess expressive learning capabilities (e.g. include various types of prior information), deep learning models trained in this fashion require adequate training data pairs.

To address the need for suitable individual weights, Taufik et al. (2024) promoted the use of the denoising diffusion model (Ho et al. 2020) to replace the proximal gradient solvers when performing regularization in EFWI. Unlike the previous supervised learning framework, diffusion models alleviate the need to generate training data pairs and thus can be straightforwardly deployed for field data applications. By doing so, they showed that they could partially mitigate the cross-talk issue, resulting in a better model reconstruction and better data fitting. One potential problem that might arise is that the denoising progress inherently treats all wavenumber components simultaneously. With wavenumber continuation in mind, we ask ourselves how can we progressively inject the appropriate prior information during the diffusion-regularized EFWI efficiently?.

To address this issue, we resort to the iterative latent variable refinement (ILVR) sampling technique (Choi et al. 2021) to control the wavenumber updates, such that we progressively move from low to high wavenumber updates. Instead of only performing conditional sampling about the FWI updates during the inversion, we introduce an additional degree of freedom to ensure certain smoothness levels are met during the reverse diffusion sampling. Practically, we can ensure that in the early EFWI iteration, the diffusion samples match the smooth background of the EFWI updates and progressively increase the wavenumber content along the EFWI iteration.

We organize this paper in the following manner. First, we briefly present the fundamental principles of training diffusion models before explaining how such models can be utilized as regularization for multiparameter EFWI. We continue the theoretical development by introducing a novel diffusion sampling considering the parameter updates’ wavenumber component. To demonstrate the applicability of our framework, we show the different geological situations to which our framework can adapt. Then, we analyse the performance of our framework in light of challenging ocean-bottom cable data from the Volve field. Finally, we conclude by benchmarking with different regularization techniques, highlighting the significant accuracy improvement and the negligible additional cost of the proposed framework.

2 THEORY

In this section, we will first describe diffusion models and then share their role as a regularizer for EFWI. Afterwards, we introduce our wavenumber-aware diffusion regularization for EFWI.

2.1 Generative diffusion models

There are two important components during the training of a diffusion model, namely the forward and reverse processes (Fig. 1). The forward process tries to accumulate (using a Markov process) a known level of noise such that samples at the last time step |${\bf x}_N$| (⁠|$t=\lbrace 0,\dots , i,\dots , N\rbrace$|⁠), a transformation of the original image |${\bf x}_0$| becomes samples belonging to a Gaussian distribution. This can be formally written as a Gaussian translation

with |$0\lt \beta _1,\dots ,\beta _N\lt 1$| is a sequence of positive noise scales. This results in forward updates given by

Conversely, during the reverse process, the goal is to estimate the denoised sample distribution q that includes a neural network, |$\mathbf {s}_{\theta }$| (defined as the score function), such that |$q\approx p_{\theta }$|⁠. Thus, the diffusion training aims to find the optimal weight |$\theta$| to perform this reverse operation. Specifically, a variational discrete Markov chain is used such that the Gaussian translation becomes

and resulting in the reverse updates

Thus, the diffusion training aims to find the optimal weight |$\theta$| for a neural network to perform this approximation. Using a re-weighted variant of the estimated lower bound (ELBO), we seek

where a simple change of variable (⁠|$\alpha :=\Pi _{j=1}^i(1-\beta _j)$|⁠) is performed to the forward distribution p such that |$p_{\alpha _i}\left(\mathbf {x}_i \mid \mathbf {x}_{i-1}\right)=\mathcal {N}\left(\mathbf {x}_i ; \sqrt{\alpha _i} \mathbf {x}_{i-1}, (1-\alpha _i) \mathbf {I}\right)$|⁠.

2.2 Diffusion-regularized EFWI

For the forward modelling and adjoint calculations for EFWI, we resort to the velocity-stress 2-D isotropic elastic wave equation in the form of

where |$\vec{\mathbf {v}}=(v_x,v_z)$| are the horizontal and vertical particle velocity fields, |$\boldsymbol{\sigma }=(\boldsymbol{\sigma }_{xx},\boldsymbol{\sigma }_{zz},\boldsymbol{\sigma }_{xz})$| are the stress fields, |$f=(f_{\boldsymbol{\sigma }_{xx}},f_{\boldsymbol{\sigma }_{zz}})$| are the source terms, |$\rho$| is density, |$\lambda$| and |$\mu$| are the Lame parameters. Matrices |${\bf C}$| and |${\bf D}$| represent the isotropic elastic tensor in Voigt notation and is a collection of spatial differential operators in 2-D, defined as

Thus, the inverse problem aims to solve the following problem (6)

in which the relationship between the measured data |${\bf y}=\lbrace \vec{\mathbf {v}},\boldsymbol{\sigma }\rbrace$| and the unknown elastic parameters |${\bf x}=\lbrace \lambda ,\mu ,\rho \rbrace$| is facilitated by the first-order system of elastic wave equations, summarized in the modelling operator |$\mathcal {A}$| with an additional unknown measured noise |$\epsilon$|⁠. The solution to equation (8) can be obtained by solving an optimization problem in the form of

where a data-fidelity term |$\mathcal {D}$| ensures the estimated data obey the measured ones, a regularization function |$\mathcal {R}$| is introduced to inject prior information into the estimates. A popular choice in the literature for |$\mathcal {D}$| is a simple least-square (⁠|$L_2$|⁠) norm. For regularization, a smoothness-preserving or roughness-preserving function is often used (Tikhonov 1963).

Taufik et al. (2024) demonstrated that for a general form of |$\mathcal {R}$| that includes non-smooth functions, we could replace the proximal solvers (e.g. the fast iterative shrinkage thresholding algorithm) with the reverse diffusion process. In essence, instead of relying on non-distributional denoising algorithms (Ho et al. 2020) to replace the proximal solver, Venkatakrishnan et al. (2013) utilized a diffusion model that inherently learns the distribution (and the relationship) of the individual elastic parameters. In summary, we can essentially extend the standard FWI process, granted that we have sufficiently trained the diffusion model (Table 1).

2.3 Wavenumber-aware sampling for diffusion-regularized EFWI

Though has been demonstrated to handle the cross-talk issue partially (Taufik et al. 2024), on its inception, the regularization is performed by utilizing the denoising property (due to the denoising diffusion probabilistic models sampler) of such models. Thus, the updated model is calculated by sampling in the diffusion model conditioned by the adjoint-state updates. Although this process has significantly improved compared to conventional regularization (e.g. total-variation regularization), the denoising process does not fully resemble the progressive (wavenumber) updates associated with FWI. This is because, during the inversion, we usually start with smooth models to get the kinematic part of the wave propagation correct before updating the scattering components. Thus, when denoising is performed during the inversion, such a process affects both the models’ high and low wavenumber components. In the context of FWI, progressively updating the model based on the wavenumber components might significantly improve the inversion results (Bunks et al. 1995; Alkhalifah 2016).

To modify the diffusion updates to account for progressive wavenumber updates, we utilize a simple extension to the reverse conditional sampling using the iterative latent variable refinement (ILVR) technique (Chung et al. 2023). The proposed algorithm is summarized in Table 2. As we can see, modification to the standard diffusion-regularized EFWI is straightforward and represented by the blue-coloured lines. Neglecting these lines will result in the original approach of incorporating diffusion models to regularize FWI (Wang et al. 2023; Taufik et al. 2024). The bottommost blue-coloured line denotes the most critical component of the ILVR sampling. We first need to initialize a filter |$\Phi _N(.)$| with a downsampling factor N to perform this sampling. Small N forces the diffusion samples (⁠|${\bf x}$|⁠) to match the high wavenumber component of the reference samples (⁠|${\bf y}$|⁠). This results in relatively minor variations between the samples and the reference. In EFWI, we use the adjoint-state updates as the reference. A small N intuitively indicates that the regularized updates inherent more features from the adjoint-state updates than the diffusion updates. Consequently, we can progressively modify the downsampling factor as we proceed with the EFWI iterations. For example, we can start with large N to make the updates match the low wavenumber background and decrease N at the later stage to increase the high wavenumber information from the diffusion regularization.

The subsampling filter |$\Phi _N(.)$| is essential in controlling the wavenumber component updates. Fig. 2 shows that the ILVR sampling essentially performs a background matching procedure between the FWI updates and diffusion model samples. The subsampling factor N controls the smoothness of such a procedure. Smaller N corresponds to a high wavenumber background information that, in turn, produces diffusion samples closely resembling the (reference) FWI updates.

3 NUMERICAL EXPERIMENTS

The performance of our new diffusion sampling to regularize EFWI is attested on three examples. The first two involve synthetic data simulation in which we analyse two different geological settings: salt and thrust tectonics. Then, in the last example, we demonstrate the applicability of our framework to work with ocean-bottom cable data from the Volve field. In the first two examples, only the vertical particle velocity is used during the inversion, while the OBC measures the pressure field.

Table 3 summarizes the details for each EFWI parameter for the following cases. In all of the EFWI experiments, the PyTorch wave propagation package, Deepwave (Richardson 2023), is used to perform the multiparameter EFWI. A single NVIDIA A100 graphical processing unit is utilized in the following experiments. To allow the readers to evaluate the novel aspects of our framework, we have made the code available at https://github.com/DeepWave-KAUST/ilvrefwi.

3.1 Diffusion model training

Throughout the following three EFWI examples, we perform two diffusion model training that differ only in the velocity model distribution. We combine various open-source velocity models hosted by the Society of Exploration Geophysicists (SEG) wiki for the synthetic Overthrust and Volve field experiments. This resulted in 27 727 |$V_p,V_s$|⁠, and |$\rho$| 2-D images. Then, we slightly modify the density values to account for salt body characteristics in the synthetic BP 2004 model to train during the second diffusion training. We utilize the transfer learning approach to accelerate the training of the second diffusion model dedicated to the salt example.

The diffusion model architecture is kept the same for these two training. Specifically, the denoising diffusion probabilistic model sampling (Ho et al. 2020) is used during the reverse and forward processes. The diffusion model takes three-channel images of size 256×256 and utilizes sinusoidal positional embedding to project the discretized (of size 1000) time vector to higher dimensional space. The model utilizes convolutional blocks with a channel size 128, attention blocks with attention heads of size 4, channel multipliers of size [1, 2, 4, 8, 16] and a single residual block. To control the noise level |$\beta$|⁠, a linear scheduler is used with a value between 1e-4 and 2e-2. An Adam optimizer with a learning rate of 1e-5 is used to optimize the training for 45 epochs and a batch size of 10. The training of the diffusion models takes around 134 min for each epoch on a single GPU. However, once trained, it can be used, as we will see, in various EFWI applications at negligible additional cost. This includes the application of field data.

3.2 Modified BP 2004 synthetic data

To understand the performance of our framework in improving high-contrast elastic bodies (e.g. salt), we first utilized the modified BP 2004 model. We perform the EFWI process utilizing the smoothed version of the true model (rightmost column of Fig. 4) as the initial model (rightmost column of Fig. 3). Vertical component noise-free synthetic data are used as the observed data for EFWI. As shown in Fig. 3, the proposed framework (leftmost column) significantly improves the inversion when compared to non-regularized EFWI (second column from the right). Also, the proposed ILVR sampling produces better model reconstruction than the original diffusion-regularized EFWI (second column from the left). The proposed framework better delineates the salt boundary and obtains a better volume.

Compared to other regularization schemes, shown in Fig. 4, our framework produces the best model reconstruction, mostly around the top of the salt where the seismic rays are mainly dominated. More importantly, as depicted in Fig. 7(a), our framework stabilizes the original diffusion-regularized EFWI while obtaining the best convergence rate compared to conventional regularization schemes. Finally, we summarize the overall performance of our framework against several inversion strategies in Table 4. Our framework produces the best model reconstruction for three quantitative metrics while maintaining a computational cost similar to that of conventional algorithms.

3.3 Modified overthrust synthetic data

To analyse the robustness against noise and complex geological structures, we add relatively strong Gaussian noise to the synthetic data generated from the modified Overthrust model. The smooth initial model (rightmost column in Fig. 5) is extracted from the true model (rightmost column in Fig. 6) via a simple Gaussian smoothing. As depicted in Fig. 5, conventional EFWI (without regularization) fails to converge. This is mainly because the added Gaussian noise significantly deteriorates the single-component data used during the inversion. Introducing diffusion regularization in this case can substantially improve the inverted elastic moduli. More importantly, the diffusion (prior) samples have more influence towards convergence than the FWI updates. This results in relatively similar results between the proposed framework and the original diffusion-regularized EFWI (two leftmost columns in Fig. 5). Compared to other regularization schemes, diffusion model-based regularization still provides the best model reconstruction (Fig. 6) while delivering better data fitting (Fig. 7b).

3.4 Volve field data

For the field example, we utilize the ocean-bottom cable (OBC) data from the Volve field. The field commenced in 2008 before finally ceasing operation in 2016. The 3-D data were acquired by Statoil in 2010, and we consider a 2-D line extracted from the survey. Several processing steps are performed before using it for EFWI to enhance the data quality (Ravasi et al. 2015, 2016; Alfarhan et al. 2024). After removing the multiples using a multidimensional deconvolution algorithm, we perform denoising, vector fidelity corrections, anti-aliasing filtering and scaling by |$\sqrt{t}$| to account for geometrical spreading when performing 3-D to 2-D conversion. After applying these processing steps, we perform low-pass filtering with a maximum frequency of 7 Hz. The main objective of the acquisition was to map the reservoir zones around 3 km depth. The challenge in applying EFWI to this data is twofold. First, the maximum offset to the depth of interest ratio is relatively small. This makes the deeper areas of interest relatively poorly illuminated and, consequently, results in small EFWI updates. Secondly, the observed data, even after such thorough pre-processing steps, are still contaminated with spurious events, which can be observed as diagonal events dipping in conflicting directions with the first-arrivals (Figs 10 and 11).

On top of the already challenging EFWI, these additional challenges present a more substantial reason for performing regularization. As shown in Fig. 8, the inverted models without regularization (second column from the right) show poorly resolved models that do not resemble the expected sediment layers—employing diffusion model-based regularization results in better model reconstruction. Compared to the original diffusion-regularized EFWI, the proposed framework clearly shows significant well-illuminated areas (i.e. the shallow part of the model). As can be seen from the |$V_p/V_s$| ratio (Fig. 8b), the proposed framework produces higher |$V_p/V_s$| values around the shallow part of the model. Compared to other regularization schemes, the diffusion model-based regularization clearly provides a better model reconstruction. To highlight the importance of EFWI here, we also show results from the acoustic FWI without regularization (the rightmost column in Fig. 9) in which only |$V_p$| is inverted. At the same time, the other elastic parameters are computed using empirical formulations. The acoustic FWI includes a lot of high wavenumber artefacts, possibly caused by the inability of the acoustic assumption to match all the wave modes in elastic data.

To validate our findings from the reconstructed models further, we analyse the data fitting quality from different points of view. First, we compare the generated shot gathers and analyse the relevant reflections. As shown in Figs 10(d) and 11(d), the diffusion model-based regularization provides the best data fitting quality. Comparing the proposed framework and the original diffusion-regularized EFWI reveals that the former offers better consistency around the first arrivals. The arrows highlight the improvements. To better appreciate the improvements, we show the individual traces corresponding from two different source locations in Figs 12 and 13.

To conclude our data quality analysis, we perform imaging using the inverted velocity models and share the angle-domain common image gathers (ADCIGs) (Biondi & Symes 2004), shown in Figs 14 and 15, to further justify our findings. The same conclusion can be drawn that the diffusion model-based regularization provides the best ADCIGs amongst several inversion scenarios. Comparing the proposed framework and the original diffusion-regularized EFWI reveals that the latter possesses spurious ‘frowny’ concaved downward events near 1 km depth at 7 km offset (Fig. 14b). This hints at a relatively poor kinematic reconstruction, specifically higher inverted velocities than needed to flatten the angle gathers. This finding further confirms that the difference up shallow in the proposed framework (Fig. 8b) resulted in a better kinematic representation of the model. The conventional regularization schemes, on the other hand, are clearly contaminated by these frowny events even in the deeper part of the model (Fig. 14d).

4 DISCUSSIONS

Wavenumber continuation strategies can potentially improve the performance of FWI in scenarios where severe scale separation occurs. This work facilitates a novel approach to conduct wavenumber continuation from a model standpoint as part of injecting prior information into the inversion. The basis for our development comes from the suboptimal denoising procedure inherent in the diffusion-regularized EFWI, which treats the wavenumber components uniformly. The proposed ILVR-regularized EFWI provides a mechanism to control the contribution between the FWI and diffusion model updates. We have demonstrated that by focusing only on the low wavenumber components during the early EFWI iterations and progressively increasing it at the later iterations, we can inherently perform the wavenumber continuation without running several EFWI cycles.

The proposed framework demonstrates a better convergence than the original diffusion-regularized EFWI (Taufik et al. 2024) in scenarios where the FWI updates dominate the inversion. However, in scenarios where significant prior information is needed to guarantee convergence, the differences between the two frameworks are almost negligible. Intuitively, this is because the proposed framework relies more on the background (reference) model coming from the EFWI updates. Therefore, the two frameworks become nearly similar in scenarios where the prior information is the main source of information for the model. This trade-off is subject to the acquisition setup of the problem and, more importantly, the subsampling factor (N) evolution throughout the EFWI iterations. Compared to other regularization techniques, the proposed framework requires an overhead cost to train the diffusion model while adding negligible additional cost during the (inference) EFWI process. It is worth noting, as we saw that the necessary diffusion training can be performed only once granted that the subsequent EFWI process does not involve severe differences in its geological setting (i.e. moving from salt to non-salt application). For example, we only perform a single diffusion model training for the synthetic Overthrust and the Volve field examples. Without anomalous geological bodies, like salts, we can reliably utilize the same training data set as the prior information for EFWI.

The proposed ILVR-based framework enables us to control the wavenumber components of the diffusion samples. The strength of the wavenumber component is primarily controlled by the chosen subsampling factor N. To analyse its influence in practice, we show the wavenumber spectrum images coming from two different N values, |$N=4$| (Fig. 17) and |$N=8$| (Fig. 18). By analysing the spectrum of each reverse diffusion sample, we can see significant differences around the regions that are bounded by the dashed boxes (Figs 17b and 18b), which are not present in the spectrum produced without ILVR sampling (Fig. 16). This is mainly because we cannot control specific wavenumber components of the diffusion samples. In contrast, using reference in the proposed ILVR-based EFWI indicates that we can inject certain wavenumber components during EFWI. By comparing Figs 17(b) and 18(b), we can see that the value of N dictates the differences; they are more focused towards the centre (shrinking) with larger N. Intuitively, a large N number corresponds to the lower wavenumber components and, thus, areas centred around the middle of the wavenumber spectrum images.

As discussed, there might be cases where the proposed solution does not provide meaningful improvements over the original diffusion-regularized EFWI. While we argue that this can happen mainly for specific use cases, other potential solutions as an extension of our work might produce significant improvements in the scenarios above. One intuitive extension is to use a completely different mechanism than denoising when training the diffusion model. Bansal et al. (2022) demonstrated that denoising is just one mechanism that can be used during diffusion training. They showed several other means that might as well produce similar samples to the one generated from the original denoising diffusion model. In our work context, deblending might yield better regularization updates that naturally obey the wavenumber continuation.

Moreover, in this work, we consider informing only the elastic properties distributions as priors for EFWI. Thus, after training the diffusion model, we perform conditional sampling about the previous EFWI iterates. When other means of information are available, that is, well-log data, facies distribution, porosity distribution, etc., we can include that information during the diffusion training. By doing so, we essentially utilize the conditional capability of the diffusion model during the inference (Wang et al. 2024). Intuitively, having access to such information will yield better EFWI convergence and might even alleviate the cross-talk issues. Additionally, while we focus our demonstrations on exploration scale applications, extending our framework to deal with broad-band seismic recordings is relatively straightforward. The main difference will be the choice of the subsampling factor N. In such applications, we might want to focus on choosing N so that it incorporates more of the low-frequency components of the observed data. Compared to exploration scale applications, broad-band seismic recordings provide more low-frequency components of the velocity model that, in turn, makes the inversion strategy more confident in utilizing the low-frequency component of the data in the early iterations.

Finally, here we consider performing wavenumber continuation from a model standpoint. It is worth noting that our framework is orthogonal to other wavenumber continuation algorithms. For example, we can combine the proposed framework with the frequency continuation (multiscale FWI) approach (Bunks et al. 1995) for better model reconstruction. We deliberately neglect such extension to focus more on what type of improvements our framework produces compared to standard EFWI. At the same time, it significantly reduces the computational cost required by the multiscale FWI.

5 CONCLUSIONS

We proposed a novel diffusion sampling to make the regularization function progressively update the elastic moduli based on their wavenumber components. Thus, at the early stage of the multiparameter EFWI iteration, the diffusion updates account more for the smooth kinematic part of the model and allow for higher model wavenumber components to contribute at the later stage. By doing so, we essentially perform wavenumber continuation in the model domain without having to perform the often expensive scale separation between low and high wavenumber inversion, which requires multiple EFWI cycles. To do this, we resort to a simple modification facilitated by the iterative latent variable refinement sampling. The beauty of this modification is that it can be instantly deployed to any trained diffusion model without retraining. The proposed sampling provides a more flexible diffusion-regularized EFWI by introducing an extra degree of freedom, which more naturally resembles the FWI mode evolution in practice. Compared to the previous diffusion model regularization and other regularization techniques, the proposed framework offers better elastic moduli that fit the observed data better. The proposed method is generic and can be applied to other diffusion samplers without additional computational cost. Although we only implemented it in this work into a denoising-based diffusion EFWI, other diffusion samplers can also be extended using the proposed algorithm.

ACKNOWLEDGMENTS

We thank King Abdullah University of Science and Technology (KAUST) and the DeepWave Consortium sponsors for their support. We are grateful to Mustafa Alfarhan for his help on the acoustic FWI experiment of the Volve field data. We also thank the Seismic Wave Analysis group for the supportive environment and Equinor for releasing the field data set. This work utilized the resources of the Supercomputing Laboratory at KAUST in Thuwal, Saudi Arabia.

DATA AVAILABILITY

Codes and data needed to reproduce the results presented here will be made available at https://github.com/DeepWave-KAUST/ilvrefwi. The data are available from the corresponding author upon request.

REFERENCES