Physics-guided full waveform inversion using Encoder-Solver convolutional neural networks

Seismic wave forward modelling is a crucial method for studying the propagation characteristics of seismic waves in subsurface media and is a key component of full waveform inversion. Compared to time‐domain forward modelling, frequency‐domain forward modelling offers advantages such as not being constrained by stability limits and reducing the dimension of the solution space. However, forward algorithms based on the rotation coordinate system in the frequency domain cannot adapt to situations with unequal spatial sampling intervals. To enhance the adaptability of the forward modelling algorithm in the frequency domain, we derived a 21‐point finite‐difference scheme based on the average derivative method and calculated the difference coefficients and dispersion conditions. Additionally, to address the significant computational cost in frequency domain forward modelling, we developed an improved 25‐point finite‐difference scheme. The improved 25‐point format is more accurate than the conventional 25‐point format. Building on this foundation, we applied the two derived differential schemes to full waveform inversion to synthesize the shot records of the inversion data. Additionally, we introduced a frequency compensation factor into the gradient processing, which effectively compensates for the deep layer while suppressing noise in the shallow gradient field. Finally, we demonstrated the effectiveness of our approach through a full waveform inversion application on the Marmousi model showcasing its capability in invertig fine subsurface structures.

In most full-waveform inversion (FWI) problems, sufficient prior information is available to constrain the velocity of certain parts of the model, e.g., the water column or, in some cases, near-surface velocities. We take advantage of this situation and develop a fast Schur-complement-based forward modeling and inversion approach by partitioning the velocity model into two parts. The first part consists of the constrained zone that does not change during the inversion, whereas the second part is the anomalous zone to be updated during the inversion. For this decomposition, we partially factorize the governing system of linear equations by computing a Schur complement for the anomalous zone. The Schur complement system is then solved to compute the fields in the anomalous zone, which are then back substituted to compute the fields in the constrained region. For each successive modeling steps with new anomalous zone velocities, the corresponding Schur complement is easily computed using simple algebra. Because the anomalous part of the model is comparatively smaller than the whole model, considerable computational savings can be achieved using our Schur approach. Additionally, we showed that the Schur complement method maintains the accuracy of standard frequency-domain finite difference formulations, but this comes at a slightly higher peak memory requirement. Our FWI workflow shows reduced runtime by 15%–57% depending upon the depth of the water column without losing any accuracy compared to the standard method.

Ultrasound computed tomography (USCT) is an emerging imaging modality that holds great promise for breast imaging. Full-waveform inversion (FWI)-based image reconstruction methods incorporate accurate wave physics to produce high spatial resolution quantitative images of speed of sound or other acoustic properties of the breast tissues. However, FWI reconstruction is computationally expensive, which limits its application in a clinical setting. This contribution investigates using the use of a convolutional neural network (CNN) to learn a mapping from USCT data to speed of sound estimates. The CNN was trained using a supervised approach that employed a large set of anatomically and physiologically realistic numerical breast phantoms (NBPs) and simulated USCT measurements. Once trained, this CNN can then be evaluated for real-time FWI image reconstruction from USCT data. The performance of the proposed method was assessed and compared against FWI using a hold-out sample of 41 NBPs and corresponding USCT images. Accuracy was measured using relative mean square error (RMSE) and structural self-similarity index measure (SSIM). This numerical experiment demonstrates that a supervised learning model can achieve accuracy comparable to FWI, while significantly reducing computational time and memory requirements.

When reliable a priori information is not available, it is difficult to correctly predict near-surface S-wave velocity models from Rayleigh waves through existing techniques, especially in the case of complex geology. To tackle this issue, we have developed a new method: two-grid genetic-algorithm Rayleigh-wave full-waveform inversion (FWI). Adopting a two-grid parameterization of the model, the genetic algorithm inverts for unknown velocities and densities at the nodes of a coarse grid, whereas the forward modeling is performed on a fine grid to avoid numerical dispersion. A bilinear interpolation brings the coarse-grid results into the fine-grid models. The coarse inversion grid allows for a significant reduction in the computing time required by the genetic algorithm to converge. With a coarser grid, there are fewer unknowns and less required computing time, at the expense of the model resolution. To further increase efficiency, our inversion code can perform the optimization using an offset-marching strategy and/or a frequency-marching strategy that can make use of different kinds of objective functions and allows for parallel computing. We illustrate the effect of our inversion method using three synthetic examples with rather complex near-surface models. Although no a priori information was used in all three tests, the long-wavelength structures of the reference models were fairly predicted, and satisfactory matches between “observed” and predicted data were achieved. The fair predictions of the reference models suggest that the final models estimated by our genetic-algorithm FWI, which we call macromodels, would be suitable inputs to gradient-based Rayleigh-wave FWI for further refinement. We also explored other issues related to the practical use of the method in different work and explored applications of the method to field data.

Geophysical imaging utilising the Ground Penetrating Radar (GPR) is a staple for observing the subsurface in high resolution. It produces images of the distribution of dielectric permittivity and electrical conductivity of the surveyed subsurface region. To produce these images, the data is supplied to an inversion algorithm. This algorithm iteratively improves the estimates of what the subsurface looks like. Each iteration is based on how close the response produced by the estimate is to the data.To evaluate the response from estimate models, we require an efficient forward solver of the Maxwell's equations. It can be performed in time-domain, as very capably done via gprMax, or in frequency-domain. Working in frequency-domain allows for spectral analysis of subsurface response, choosing particular frequencies of interest, and unique parallelisation and inversion strategies. For wide-band antennas used in a GPR survey, we are developing an efficient frequency domain forward modelling code in 3D. Simulating in 3D allows us to account for the antenna's complete radiation pattern. Moreover, to account for the wide band data, the forward solver will be equipped with a custom meshing algorithm that creates separate meshes for different frequency ranges of the data spectrum. Care needs to be taken that adequate boundary conditions are implemented in the forward solver for accurate domain truncation.We will present the capabilities and features of the forward solver with synthetic test cases. Supplying our forward engine to a suitable Full Waveform Inversion (FWI) strategy that maximises the adaptable meshing and frequency domain evaluations could result in high performance. The exact strategy for such an FWI will be defined and tested such that it leads to accurate and efficient reconstruction of the electrical subsurface properties.

A developed nearly analytic discrete method for forward modeling in the frequency domain

<p>The existing conventional inversion algorithm does not provide satisfactory results due to the complexity of propagated wavefield though the gas cloud. Acoustic full waveform inversion has been developed and applied to a realistic synthetic offshore shallow gas cloud feature with Student-t approach, with and without simultaneous sources encoding. As a modeling operator, we implemented the grid based finite-difference method in frequency domain using second order elastic wave equation. Jacobin operator and its adjoint provide a necessary platform for solving full waveform inversion problem in a reduced Hessian matrix. We invert gas cloud model in 5 frequency band selected from 1 to 12 Hz, each band contains 3 frequencies. The inversion results are highly sensitive to the misfit. The model allows better convergence and recovery of amplitude losses. This approach gives better resolution then the existing least-squares approach. In this paper, we implement the full waveform inversion for low frequency model with minimum number of iteration providing a better resolution of inversion results.</p>

Full Waveform Inversion (FWI) is a seismic imaging technique that utilizes the entire content of seismic trace. This method aims to minimizes differences between real and synthetic seismograms. FWI consists of three main steps: forward modelling to simulate synthetic seismograms, adjoint simulation to calculate the gradient misfit, and optimization to obtain an appropriate model perturbation. FWI is an extremely non-linear inversion technique and computationally intensive since it involves many model parameters. In the context of geophysics, finding parameter models that describe the structure of the earth’s subsurface is the main goal. Geophysical measurements in the field only obtain data that is influenced by the parameter model, in this case seismic recording data, therefore mathematical techniques are studied and applied to determine the correct parameter model from the data obtained, also known as the inversion process. In this study, main research area is on optimization step which include mathematical approach to find perturbation model effectively. The Newton method can be used in the optimization step of FWI. However, its implementation is not promising due to high computational cost in calculating Hessian matrix. To reduce computation cost, an identity matrix is chosen to replace the Hessian matrix, as it usually done in the steepest descent method. Since FWI has a high level of non-linearity, steepest descent method requires many iterations to minimize the objective function. Therefore, this study uses a conjugate gradient method to make the data misfit minimized more quickly. Apart from that, the Line search technique is used to find an appropriate step-length at each iteration on both methods. The FWI workflow is applied to the synthetic acoustic velocity model. We conduct FWI using steepest descent and conjugate gradient methods. The results show that the conjugate gradient method requires fewer iterations than the steepest descent method to minimize the objective function. FWI model result using conjugate gradient method is closer with the true model compared to steepest descent method, showed on misfit result and model image. By comparing these methods we can see that conjugate gradient method can be chosen for application on FWI optimization step for better result on seismic imaging. On the industry scale, better results provide less uncertainty on the model parameters that useful for energy exploration or engineering.

Full waveform inversion (FWI) iteratively improves the accuracy of the model by minimizing the discrepancies between the predicted and the observed data. However, FWI commonly suffers from cycle skipping when the initial model is poor, leading to an erroneous result. To mitigate this problem, we propose deep-learning-backed adaptive waveform inversion (DL-AWI), which introduces a deep twin neural network to precondition the waveforms and compare the ratio of two signals with a zero-lag spike, thereby enhancing the stability of the inversion process. DL-AWI can project the synthetic and observed signals into an extended latent space via several convolutional neural networks (CNNs) with shared weights, which can accelerate the data matching. Compared with classic FWI methods, the proposed DL-AWI provides a wider space for model updates, significantly decreasing the risk of being trapped in local minima. We use synthetic and field examples to validate its efficiency in subsurface model inversion, and the results show that DL-AWI is robust even when a poor initial model is provided.

In this paper we proposed an accurate and efficient numerical algorithm for solving the acoustic wave equation in three-dimensional heterogeneous media. Numerical solution of the wave equation has been used in various science and engineering applications, such as the seismic full waveform inversion (FWI) problem. FWI is a computationally intensive procedure, in which the acoustic wave equation is numerically solved (forward modelling) repeatedly during the iterative process. Therefore, efficiency and accuracy of the numerical method for solving the acoustic wave equation is critical in the success of seismic full waveform inversion. The new method is obtained by combining the Pade approximation and a novel algebraic manipulation with the Alternative Directional Implicit (ADI) method. Numerical experiments have shown that the new method is accurate, efficient and stable.

To exploit Hessian information in full-waveform inversion (FWI), the matrix-free truncated Newton method can be used. In such a method, Hessian-vector product computation is one of the major concerns due to the huge memory requirements and demanding computational cost. Using the adjoint-state method, the Hessian-vector product can be estimated by zero-lag crosscorrelation of the first-/second-order incident wavefields and the second-/first-order adjoint wavefields. Different from the implementation in frequency-domain FWI, Hessian-vector product construction in the time domain becomes much more challenging because it is not affordable to store all of the time-dependent wavefields. The widely used wavefield recomputation strategy leads to computationally intensive tasks. We have developed an efficient alternative approach to computing the Hessian-vector product for time-domain FWI. In our method, discrete Fourier transform is applied to extract frequency-domain components of involved wavefields, which are used to compute wavefield crosscorrelation in the frequency domain. This makes it possible to avoid reconstructing the first- and second-order incident wavefields. In addition, a full-scattered-field approximation is proposed to efficiently simplify the second-order incident and adjoint wavefield computation, which enables us to refrain from repeatedly solving the first-order incident and adjoint equations for the second-order incident and adjoint wavefields (re)computation. With our method, the computational time can be reduced by 70% and 80% in viscous media for Gauss-Newton and full-Newton Hessian-vector product construction, respectively. The effectiveness of our method is also verified in the frame of a 2D multiparameter inversion, in which our method almost reaches the same iterative convergence of the conventional time-domain implementation.

An accurate earth model is key to any successful depth imaging effort. Full waveform inversion is an advanced velocity model building process which uses the full two way wave equation. Existing methods use a ray tracing approach to distribute velocity errors within the model. In this presentation, we will show examples of seismic data processed with the latest technology, including earth model building with full waveform inversion. Introduction The industry has moved to using two-way wave equation migrations(commonly known as reverse time migration), especially in areas of complex geology such as the salt bodies in the Gulf of Mexico and offshore Brazil. The velocity model, including velocity anisotropy, is key to any depth migration effort. The natural next step is to use the two-way wave equation for velocity model building. One of the most advanced tools for velocity model building using the two-way wave equation is Full Waveform Inversion(FWI). Full-waveform inversion uses computer intensive forward modeling of the seismic measurement combined with residual wavefield back propagation to iterate to a final velocity model, which can provide greater detail than tomographic ray tracing approaches. Velocity inversion using full waveform inversion has been used on a number of projects around the world including the GOM, Brazil and the North Sea.

Full waveform inversion (FWI) is a technique used to estimate subsurface model parameters by minimizing the difference between observed and calculated seismic data. Sparsity-promoting regularization are useful tools for traditional FWI methods to tackle complex subsurface structures. Since the traditional regularization techniques can only impose some fixed priors, it is necessary to develop a regularization strategy to obtain more flexible priors. In this way, we develop a structural sparse representation method that exploits the non-local self-similarity prior of the model, which is achieved by grouping similar patches using graph matching operators and a dynamic group selection strategy. A group-based dictionary is trained with the aim of providing the best sparse representation of complex features and variations in the entire model perturbation. The dynamic selection strategy of the training method can balance computational efficiency and inversion accuracy by constantly updating and retaining groups during the processing. In addition, two loop algorithm framework is utilized to enhance the robustness and the efficiency of the proposed method. Numerical experiments are presented to demonstrate that the proposed method outperforms the total variation regularization method and the adaptive dictionary learning with non-local self-similarity in terms of robustness and resolution. This structural sparsity-promoting regularization is incorporated into the FWI problem through a two-loop algorithm framework, enhancing the robustness and efficiency of FWI results.

plex shallow subsurface. This workflow incorporates dispersion curve inversion, traveltime tomography and full waveform inversion into a single framework to optimize the advantages of each. An accurate near-surface velocity is derived using the dispersion curve inversion technique that overcomes limitations of traditional traveltime tomography. This velocity is then incorporated into layer-stripping full waveform or RMO inversion schemes by “freezing” the ultra-shallow part of the model in order to retrieve the deeper velocity structure. We illustrate the workflow on a test data set from Inner Mongolia, China, whose long-offsets and low-frequency content provide the necessary input for both the dispersion curve and full waveform inversion techniques. Using this data set, we demonstrate the efficacy of the integrated methodology in determining both the macro-velocity model and near-surface velocities in structurally complex geological setting.

Summary Full Waveform Inversion (FWI) is now a well-established technology widely used to derive migration velocities in complex settings where conventional travel time tomography is inadequate. Most seismic contractors have developed their own flavours of FWI workflows which differ in several ways; they utilize different codes with their own strengths and weaknesses, utilize different parts of the recorded wavefield and follow different update-strategies. When running a production FWI project several complex choices must be made and their combined impact is difficult to predict. This paper investigates some of the impacts of different FWI approaches by comparing the results from a benchmark study where 3 seismic processing contractors used the same 2D data set to run FWI free of tomography. It was found that each of them have a distinctive piece of technology that helps to overcome specific challenges in a FWI production flow. Here we show and discuss their results, as well as their route map to get there.