The phenomenon of elastic wave conversions, where acoustic, pressure (P-) waves are converted to elastic, shear (S-) waves and vice-versa, is commonly disregarded in seismic imaging. This can lead to lower quality images in regions with strong contrasts in elastic parameters. While a number of methods exist that do take wave conversions into account, they either deal with P and S waves separately, or are prohibitively computationally expensive, as is the case for elastic full-waveform inversion. In this paper an alternative approach to taking converted waves into account is presented by extending full wavefield migration (FWM) to account for wave conversions. FWM is a full-wavefield inversion method based on explicit, convolutional, one-way propagation and reflection operators in the space–frequency domain. By applying these operators recursively, multiscattering data can be modelled. Using these operators, the FWM algorithm aims to reconstruct the reflection properties of the subsurface (i.e. the ‘image’). In this paper, the FWM method is extended by accounting for wave conversions due to angle-dependent reflections and transmissions using an extended version of Shuey’s approximation. The resulting algorithm is tested on two synthetic models to give a proof of concept. The results of these tests show that the proposed extension can model wave conversions accurately and yields better inversion results than applying conventional, acoustic FWM.

Structural information about the subsurface near the borehole can be obtained from reconstructed conductivity distributions. These distributions may be reconstructed via the inversion of deep-sensing electromagnetic induction log data. Unfortunately, these complex media often display anisotropy and structural variations in both horizontal and vertical directions, making the three-dimensional inversion computationally demanding and ill-posed. To address these challenges, we introduce a sequential inversion strategy of deep-sensing electromagnetic induction logging data that is measured while drilling. For the inversion at each logging position, we employ a matrix-free implementation of the adjoint integral equation method and a quasi-Newton algorithm. To tackle the ill-posed nature of the problem, we regularize the inverse problem by employing a multi-dimensional inversion parameter technique that shifts from zero- to three-dimensional parameterization. The model derived from the inversion of the data at multiple positions is incrementally integrated by utilizing the sensitivity data at each logging position. To validate our approach, we tested our method on simulated data using an anisotropic model. These experiments show that this approach produces a good reconstruction of the true conductivity for the whole track while only doing the inversion at a single position at a time.

As seismic migration is increasingly applied to more and more complex media, more sophisticated imaging techniques are required to generate accurate images of the subsurface. Currently, the best results for imaging are achieved by least-squares migration methods, such as least-squares reverse time migration and full-wavefield migration (FWM). These methods iteratively update the image to minimize the misfit between the forward modelled wavefield and the recorded data at the surface. However, a key challenge for these techniques is the speed of convergence. To accelerate the speed of convergence, pre-conditioning is commonly applied. The most common pre-conditioner is the reciprocal of the Hessian operator. However, this operator is computationally expensive to calculate, making it difficult to apply directly. In this paper, we present a novel, alternative, pre-conditioner for FWM. This pre-conditioner is based on applying Galerkin projections to a linear system, which projects the system onto a set of known basis vectors. To find an appropriate set of basis vectors for this approach we apply proper orthogonal decomposition (POD) to a set of partial solutions of the linear system. The resulting method gives an approximation to the pseudo-inverse based on these basis vectors. To test this technique, which we name model-order reduced FWM (MOR-FWM), we apply it to the synthetic Marmousi model as well as to field data from the Vøring basin in Norway. For these examples, we show that MOR-FWM yields an improved data-misfit compared to the standard FWM approach. In addition, we show that the result for the field data case can be improved by normalizing the partial solutions before applying POD.

As seismic imaging moves towards the imaging of more complex media, properly modelling elastic effects in the subsurface is becoming of increasing interest. In this context, elastic wave conversion, where acoustic, pressure (P-) waves are converted into elastic, shear (S-) waves, is of great importance. Accounting for these wave conversions, in the framework of forward and inverse modelling of elastic waves, is crucial to creating accurate images of the subsurface in complex media. The underlying mechanism of wave conversion is well understood and described by the Zoeppritz equations. However, as these equations are highly nonlinear, approximations are commonly used. The most well-known of these approximations is Shuey’s approximation. However, this approximation only holds for small angles and small contrasts, making it insufficient for realistic forward and inverse modelling scenarios, where angles and contrasts may be large. In this paper we present a novel set of approximations, based on Taylor expansions of the Zoeppritz equations, which we name the extended Shuey approximations. We examine the quality of these approximations to the Zoeppritz equations and compare them to existing approximations described in literature. We then apply these extended Shuey approximations to the elastic full-wavefield modelling algorithm for a simple, synthetic, 1.5-D example, where we show that we can accurately model the P- and S-wavefields in a forward modelling case. Finally, we apply our approximations to the elastic full-wavefield migration algorithm for a simple, synthetic, 1.5-D example, where we show that we can recover an accurate image in an inverse modelling case.

Inversion of electromagnetic induction log while drilling provides structural information around borehole for geosteering decision process. Imaging a complex three-dimensional (3D) anisotropic formation is very challenging due to the significant cost of the forward solver. To overcome this challenge, we have developed an inversion algorithm using an adjoint Integral Equation Method for logging while drilling scenarios in anisotropic media with several inexpensive approaches in this study. We use a quasi-Newton method with L-BFGS algorithm and implement a matrix-free adjoint Integral Equation method for efficient update of inverted model parameters. In addition, we propose the use of limited number of iterations for solving the adjoint equation as an approximation to the gradient for a faster computation time. To regularize the inverted models, we adopt the use of multiplicative regularization and weighted L2 total variation. Additionally, we impose priori information in our formulations, such as 2D parameterization and the range of conductivity values. We show a numerical example of logging while drilling inversion on a 2.5D faulted anisotropic formation. Numerical experiments show that our inversion workflow shows a good structural agreement of the inverted model within the sensitive range of the tool.

The deployment of electromagnetic (EM) induction tools while drilling is one of the standard routines for assisting the geosteering decision-making process. The conductivity distribution obtained through the inversion of the EM induction log can provide important information about the geological structure around the borehole. To image the 3-D geological structure in the subsurface, 3-D inversion of the EM induction log is required. Because the inversion process is mainly dependent on forward modelling, the use of a fast and accurate forward modelling tool is essential. In this paper, we present an improved version of the integral equation (IE) based modelling technique for general anisotropic media with domain decomposition pre- conditioning. The discretized IE after domain decomposition equals a fixed-point equation that is solv ed iterativ ely with either the block Gauss-Seidel or Jacobi pre-conditioning. Within each iteration, the inverse of the block matrix is computed using a Krylov subspace method instead of a direct solver. An additional reduction in computational time is obtained by using an adaptive relative residual stopping criterion in the iterativ e solv er. Using this domain decomposition scheme, numerical experiments show computation time reductions by factors of 1.97-2.84 compared to solving the full-domain IE with a GMRES solver and a contraction IE pre- conditioner. Additionally, the reduction of memory requirement for covering a large area of the induction tool sensitivity enables acceleration with limited GPU memory. Hence, we conclude that the domain decomposition method is improving the efficiency of the IE method by reducing the computation time and memory requirement.

The phenomenon of wave conversions, where acoustic, pressure (P) waves are converted to elastic, shear (S) waves is commonly disregarded in seismic imaging, which can lead to lower-quality images in regions with strong reflectors. While a number of methods exist which do take wave conversions into account, most deal with P- and S-waves separately, rather than in using a single, unified, framework. Elastic Full-Waveform Inversion (FWI), on the other hand, which does offer a unified framework for all elastic effects, is prohibitively computationally expensive in many cases.

We present an alternative approach by extending Full Wavefield Migration (FWM) to account for wave conversions. Full Wavefield Migration describes seismic data in terms of convolutional propagation and reflection operators in the space-frequency domain. By applying these operators recursively, multi-scattering data can be modelled and inverted. Using Shuey’s approximation to constrain the number of parameters necessary to describe the full, elastic, reflection and transmission operators, we present an elastic FWM algorithm which accounts for wave conversions.

The resulting algorithm is tested on a synthetic model to give a proof of concept. The results show that the proposed extension can model wave conversions accurately and yields better inversion results than applying conventional, acoustic FWM.

Multi-modality or multi-physics imaging is gaining interest beacuase it overcomes the limitations of a single imaging modality, as each modality typically suffers from its own application specific limitations. Different imaging techniques are developed to combine the outcome of both modalities; varying from image fusion up to the usage of prior information obtained from one imaging modality and used as input for the other one. In this work an alternative approach is presented. The method employs a multi-physics Born inversion algorithm where structural similarity is used as regularization parameter to align the acoustic and electromagnetic contrast interfaces with each other. To align the interfaces the gradients of the acoustic and electromagnetic contrast functions are used. Two approaches are tested successfully on a synthetic profile; one where the cross-product of the two gradients and one where the gradient differences are considered. Both approaches work but the gradient-difference approach outperforms the cross-gradient one. Overall, it is shown that multi-physics Born inversion both approaches reveals details in the electromagnetic contrast function that would have been missed by only employing electromagnetic inversion. This improvement is obtained at the cost of an increase in computational complexity compared to single modality inversion.

We present an iterative nonlinear inverse scattering algorithm for high-resolution acoustic imaging of density and velocity variations. To solve the multi-parameter nonlinear direct scattering problem, the acoustic wave equation for inhomogeneous media in the frequency domain is transformed into a vectorial integral equation of the Lippmann-Schwinger type for the combined pressure and pressure-gradient field. To solve the multi-parameter nonlinear inverse scattering problem, we use the Newton-Kantorovich method in conjunction with matrix-free representations of the Fréchet derivative operators and their adjoints. The approximate Hessian information that is accounted for in our iterative solution of the (nonlinear) multi-parameter inverse scattering problem is essential for the mitigation of multi-parameter cross talk effects. Numerical examples related to seismic and medical ultrasound breast imaging illustrate the performance of the new algorithm for multi-parameter acoustic imaging.

Full Wavefield Migration (FWMig) is an inversion-based seismic imaging modality that incorporates multiple reflections via one-way wave propagation. The flexible Full Wavefield Modelling (FWMod) engine that undergirds FWMig can be extended to address both compressional and converted waves. To take care of the angle-dependent nature of reflection and transmission coefficients, a vast number of unknown subsurface parameters has to be estimated in the FWMig process, especially when elastodynamic wave propagation is considered. This can easily result in a significant null space, potentially hampering the underlying inversion procedure. To restrain the number of unknown parameters, we propose an efficient new parameterization for FWMod by expanding reflection and transmission coefficients in Legendre polynomials, providing us with an orthonormal basis that is expected to benefit FWMig. With the aid of a numerical experiment in a two-dimensional layered elastic medium, we show that a relatively small number of only three or four Legendre polynomials per coefficient per gridpoint is sufficient to model pre-critical seismic data. We prospect that our methodology can be extended to include (spatially-varying) reflector dips, so that it might eventually be used for FWMig in laterally-varying two- and three-dimensional elastic media.

Real-time 3D imaging of the induction log is essential for improving the decision-making process in geosteering. To fulfill this need, we investigated various strategies for reducing the computational cost for 3D modelling of induction logs using the integral equation (IE) method, including the use of iterative Krylov solver, convolution with FFT algorithm, contraction IE formulation, computation acceleration with GPUs, and domain decomposition. We present two cases example to demonstrate the implementation of IE with these strategies. In the first case, we show that the application of domain decomposition allows one to only discretize the inhomogeneous domain and save the computation cost in the case of isolated domains. We present a logging while drilling scenario on a complex model for the second case. Our implementation of the efficient IE on GPUs enables significant acceleration and allows the computation of 3D forward modelling within less than two minutes for each local 3D simulation domain with approximately two million grid cells on a laptop. The implementation of domain decomposition formulation shows a different arrangement of solving IE by decomposing the domain.

Objective. The image reconstruction of ultrasound computed tomography is computationally expensive with conventional iterative methods. The fully learned direct deep learning reconstruction is promising to speed up image reconstruction significantly. However, for direct reconstruction from measurement data, due to the lack of real labeled data, the neural network is usually trained on a simulation dataset and shows poor performance on real data because of the simulation-to-real gap.Approach. To improve the simulation-to-real generalization of neural networks, a series of strategies are developed including a Fourier-transform-integrated neural network, measurement-domain data augmentation methods, and a self-supervised-learning-based patch-wise preprocessing neural network. Our strategies are evaluated on both the simulation dataset and real measurement datasets from two different prototype machines.Main results. The experimental results show that our deep learning methods help to improve the neural networks' robustness against noise and the generalizability to real measurement data.Significance. Our methods prove that it is possible for neural networks to achieve superior performance to traditional iterative reconstruction algorithms in imaging quality and allow for real-time 2D-image reconstruction. This study helps pave the path for the application of deep learning methods to practical ultrasound tomography image reconstruction based on simulation datasets.

Quantitative images showing the speed of sound proffle of the breast may be obtained by employing full-waveform inversion (FWI) methods on the measured data. These reconstruction methods work well for both dense and normal breasts. Contrast source inversion (CSI) is a frequency domain FWI method. In literature, many examples of successful application of CSI for breast imaging can be found. However, all these works are based on simulated data. In this work, we will present our first results obtained with employing CSI on experimental data. CSI was developed by Delft University of Technology and the experimental data was provided by FUJIFILM Healthcare Corporation. The experimental data is obtained using a ring-shaped transducer which scans a breast-mimicking gelatine phantom. Our initial results obtained with CSI look promising; all inclusions within the phantom are accurately reconstructed.

Conventional Full Wavefield Migration (FWM) is a full-wavefield inversion method based on recursively applying one-way convolutional propagation and reflection operators in the space-frequency domain at every depth level. Therefore, it struggles to model diving waves and image steep reflectors accurately. In this paper, the Interface Contrast imaging technique, an imaging technique based on the scattering integral developed in the context of medical ultrasound, is presented and used to provide a natural omni-directional extension to the conventional FWM method. The resulting algorithm is applied to a synthetic 2D model featuring a steep reflector. The results of these simulations are given and show that the technique can successfully image steep reflectors. This result yields a proof-of-concept for further research into this algorithm, where including internal scattering is a top priority.

Imaging by inversion of acoustic or electromagnetic wave fields have applications in a wide variety of areas, such as non-destructive testing, biomedical applications, and geophysical exploration. Unfortunately, each modality suffers from its own application-specific limitations, typically being difficulties in distinguishing different materials/tissues from each other in the case of acoustic wave fields and a low spatial resolution in the case of electromagnetic wave fields. To exploit the advantages of both imaging modalities, we present a Born inversion method where we use an additive regularization term based on structural similarity between the acoustic and electromagnetic contrast. To validate our approach, we compare separate with joint inversion results for one particular example. The results for this example clearly show that separate inversion succeeds in reconstructing the acoustic contrast, but fails to properly reconstruct the electromagnetic contrast. Fortunately, with the joint inversion method, both the acoustic and electromagnetic contrast functions are reconstructed successfully.

With full waveform inversion (FWI) all available information enclosed in the recorded wavefield - including multiple scattering, dispersion, and diffraction - is used to obtain accurate images showing quantitative information of the tissue parameters. These non-linear inversion methods are implemented either in the time or in the frequency domain. Unfortunately, selecting which implementation should be used for a specific problem is not trivial. To ease the selection process, we compare the performance of one time-domain inversion (TDI) and one frequency-domain inversion (FDI) - also known as Contrast Source Inversion - to provide insights into the strengths and weaknesses of each FWI method. In this contribution, we investigate the effect of the (i) bandwidth, (ii) problem complexity, (iii) number of sources and receivers, and (iv) initial speed-of-sound model on the performance of each FWI method by comparing the resulting reconstructions. Both methods are tested for the same configuration: a 2-D tomographic scan of a cancerous breast model. To avoid an inverse crime, TDI is tested on synthetic data obtained using a frequency-domain forward solver and CSI on data from a time-domain forward solver.

Reverse time migration (RTM) is a method developed for performing non-invasive physical imaging. It aims to generate diagnostic images based on forward and backward modeling of waves, which can be solved with different type of techniques. A widely used one is Finite Differences (FD) due to its low computational cost and its simplicity, as compared to other methods, to implement. In this work we show how to implement RTM with curved coordinates using FD. In particular we focus on the application of curved coordinates to improve the quality of the resulting images and reduce the computational cost of RTM. Here, we apply a strategy to curve coordinates during the wave propagation modeling stage for a modified version of the Marmousi model. The results show that it is completely feasible the redistribution of grid nodes to locally improve quality of the migrated models and reduce the number of grid nodes at the same time. We also found that typical issues associated to the imaging condition and FD scheme remain.

Deep neural networks have proven to excel classical medical image reconstruction techniques. Some networks are based on fully connected (FC) layers to achieve domain transformation such as from the data acquisition domain to the image domain. However, FC layers result in huge numbers of parameters which take a lot of GPU memory. Hence, they do not scale well, and the overall performance is limited. For ultrasound computed tomography (USCT) application, we propose a memory-efficient convolutional network that reconstructs images from the frequency domain to image domain with much less parameters compared with multilayer perceptron, by using data-driven learning. Extensive experiments demonstrate that our method achieves high reconstruction quality. It improves the structural similarity measure (SSIM) from 0.73 to 0.99 when compared with state-of-the-art reconstruction methods in this field while reduces 2/3 parameters when compared with deep neural network with FC layers to reconstruct images from frequency domain to image domain.

Full-waveform inversion methods are widely used for breast imaging to provide quantitative information about the tissues such as the speed of sound. Full-waveform inversion faces a cycle-skipping problem when applied to large contrasts or large spatial domains. This problem leads to erroneous image reconstructions. To overcome this problem, low frequency measurements or good starting models are needed. In this work, we propose and test a frequency-hopping technique for our full-waveform inversion method (Contrast Source Inversion) to avoid the cycle-skipping problem. For a synthetic example, we first show the effect of cycle skipping in the reconstructed image. Next, we show for the same example that the implementation of the proposed frequency-hopping method solves this problem.

Finite difference is a well-suited technique for modeling acoustic wave propagation in heterogeneous media as well as for imaging and inversion. Typically, the method aims at solving a set of partial differential equations for the unknown pressure field by using a regularly spaced grid. Although finite differences can be fast and cheap to implement, the accuracy of the solution is always restricted by the computational resources. This is a fundamental key point to treat when dealing with large-scale problems. In this work, we present and test a method that uses a non-uniform distribution of grid points to improve on accuracy or to reduce the required computational resources. The applied grid is generated through a coordinate transformation. Differential geometry and generalized coordinates are used to handle and analyze the effect of using a non-uniform grid. Results obtained with the presented method show that the applied transformation as well as the number of points-per-wavelength influences the stability and dispersion in the solution. We exploit this observation to locally improve the accuracy of our simulations. The work presented in this paper allows us to conclude that differential geometry for finite differences can be used to reduce dispersion and hence improve the accuracy when modeling acoustic wave propagation in heterogeneous media. In addition, it can be used to avoid oversampling through the optimization of the number of grid nodes required to have an accurate solution or just honor to the boundaries.