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.

Seismic imaging is a method for generating images of the subsurface of the Earth without the need to drill down to observe it directly. Using a seismic source, high-amplitude acoustic waves are generated, which propagate through the Earth. When these waves encounter interfaces between different layers, part of their energy is reflected, traveling back upwards. At the surface, these scattered waves are recorded using an array of sensors. These recorded wavefields are then used as input for a seismic imaging algorithm, which attempts to reconstruct an accurate image of the subsurface based on these measurements.
While many different seismic imaging methods exist, this thesis focuses specifically on Full-Wavefield Migration (FWM). FWM is a least-squares migration method, based on iteratively updating the image in order to minimize the misfit between the recorded data and the forward modelled wavefield at the surface. Using this iterative approach, FWM is able to incorporate multiple scattering effects into the imaging process, increasing the accuracy of the resulting images. Also, by using explicit convolutional operators based on the one-way wave equation, the computational cost of FWMis reduced compared to alternative iterative imaging methods based on finite-difference modelling.
In this thesis, we describe a number of recent advancements to the FWM method. Our main focus is the extension of the FWM method to include converted waves. In order to take these effects into account, we need accurate reflection and transmission operators. However, the true, elastic reflection and transmission operators are notoriously non-linear, making them difficult to work with. Therefore, we introduce a novel set of approximations of these operators, which we name the extended Shuey approximations. To benchmark these approximations, we apply them in a simple, 1.5D scenario. This test shows that the extended Shuey approximations yield improved results for forward modelling and imaging, compared to the conventional Shuey approximation.
We then use these extended Shuey approximations to derive accurate reflection and transmission operators for the 2D case. Combining these operators with the existing theoretical framework of FWM we develop a robust imaging algorithm which takes converted waves into account. We then apply this algorithm, which we name elastic FWM, to two synthetic models. We show that, for these models, the elastic FWM method out-performs the conventional, acoustic FWM method. We also demonstrate that, using this method, additional information regarding the elastic medium parameters of the subsurface can be recovered.
Finally, we examine the known issue of slow convergence for the conventional, acoustic FWM algorithm. We introduce a novel preconditioner, based on approximating the pseudo-inverse using Proper Orthogonal Decomposition (POD). Using this preconditioner, we demonstrate improved convergence for the synthetic Marmousi model and a field data set from the Vøring basin in Norway.

Reflection waveform inversion (RWI) is a technique that uses pure reflection data to estimate subsurface background velocity, relying on evolving seismic images. Conventional RWI operates in a cyclic workflow, with two key components in each cycle—migration and reflection tomography. Conventional RWI may result in suboptimal background velocity estimation, partly due to limited or unresolved resolution within each component in each cycle. While gradient pre-conditioning with the reciprocal of Hessian information helps resolve this issue in both components of RWI, it becomes impractical for a large number of model parameters. One-way reflection waveform inversion (ORWI) is a reflection waveform inversion technique in which the forward modelling scheme operates in one direction (downward and then upward) via virtual parallel depth levels within the medium. Leveraging the ORWI framework, we decompose and reduce the linear Hessian operator (also known as the approximate Hessian or Gauss–Newton Hessian) into multiple smaller suboperators. In particular, the diagonal blocks of the monofrequency approximate Hessian operators, each corresponding to a single depth level within the medium, are extracted and inverted to pre-condition the corresponding monofrequency gradients in both the migration and reflection tomography components of ORWI. This depth-dependent gradient pre-conditioning transforms standard ORWI into a high-resolution, yet computationally feasible version aimed at addressing suboptimal velocity estimation, referred to as high-resolution ORWI. The effectiveness of the proposed approach is demonstrated through successful applications to synthetic data examples.

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.

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.

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.

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.

The acoustic wave equation describes wave propagation directly from basic physical laws, even in heterogeneous acoustic media. When numerically simulating waves with the wave equation, contrasts in the medium parameters automatically generate all scattering effects. For some applications - such as propagation analysis or certain wave-equation based imaging techniques - it is desirable to suppress these reflections, as we are only interested in the transmitted wave-field. To achieve this, a modification to the constitutive relations is proposed, yielding an extra term that suppresses waves with reference to a preferred direction. The scale-factor α of this extra term can either be interpreted as a penetration depth or as a typical decay time. This modified theory is implemented using a staggered-grid, time-domain finite difference scheme, where the acoustic Poynting-vector is used to estimate the local propagation direction of the wave-field. The method was successfully used to suppress reflections in media with bone tissue (medical ultrasound) and geophysical subsurface structures, while introducing only minor perturbations to the transmitted wave-field and a small increase in computation time.

Full Wavefield Migration (FWM) is a full-wavefield inversion method based on the so-called WRW model in the context of seismic imaging. This WRW model describes seismic data in terms of convolutional propagation and reflection operators in the space-frequency domain. By recursively applying these operators, multi-scattering data can be described and the FWM algorithm can find the underlying reflection properties (i.e. the ‘image’). However, the current FWM algorithm only allows for acoustic imaging. This approach disregards elastic phenomena such as wave conversions. In this report, an extension to the FWM method is given to incorporate these effects. The resulting algorithm is then applied to a simple 1.5D model, to allow for proof-of-concept numerical simulations. The results of these simulations are given and show that the proposed model can model wave conversions and other elastic effects, yielding a good proof-of-concept for further research into this extended algorithm.

Sperm that swim collectively to the fertilization site have been observed across several vertebrate and invertebrate species, with groups ranging in size from sperm pairs to massive aggregates containing hundreds of cells. Although the molecular mechanisms that regulate sperm-sperm adhesion are still unclear, aggregation can enhance sperm motility and thus offer a fertilization advantage. Here, we report a thorough computational investigation on the role of cellular geometry in the performance of sperm aggregates. The sperm head is modelled as a persistent random walker characterized by a non-trivial three-dimensional shape and equipped with an adhesive region where cell-cell binding occurs. By considering both, a simple parametric head shape and a computer reconstruction of a real head shape based on morphometric data, we demonstrate that the geometry of the head and the structure of the adhesive region crucially affects both the stability and motility of the aggregates. Our analysis further suggests that the apical hook commonly found in the sperm of muroid rodents might serve to shield portions of the adhesive region and promote efficient alignment of the velocities of the interacting cells.