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The airborne transient electromagnetics method has been widely used in geophysical exploration in recent years, but it still faces challenges in balancing the accuracy and efficiency of electromagnetic data interpretation. As far as forward modelling is concerned, the higher the dimension of the geophysical model, the higher the accuracy of data interpretation, but, correspondingly, the more computing resources need to be consumed, which will greatly reduce investigation efficiency and practicability. In this paper, the spectral element method is first introduced for solving the 2.5‐dimensional forward modelling of the airborne transient electromagnetic system, which has a smaller computing scale than three‐dimensional modelling and is closer to the actual geological structure than either one‐ or two‐dimensional modelling. In the forward algorithm, a non‐uniform quadrilateral structured mesh is adopted to simplify the computing scale, and the Talbot algorithm rather than Gaver–Stefest algorithm is applied to the inverse Laplace transform to improve the numerical precision of this conversion. Moreover, we use parallel computing technology to improve the algorithm efficiency while keeping satisfactory accuracy. The study shows that, whether a low‐resistivity or high‐resistivity layered geophysical model, the numerical solutions of the proposed spectral element method forward algorithm agree well with the analytical solutions of the corresponding models; furthermore, the key factors affecting the accuracy of the numerical solution are analysed by experiments. Finally, we successfully applied it to the 2.5‐dimensional geoelectric model simulation. [ABSTRACT FROM AUTHOR]

The critical angle plays a crucial role in data processing for refraction seismology. In the context of three‐dimensional data, the critical angle exhibits azimuthal dependence, particularly in the presence of an anisotropic model. In this paper, we propose a method to determine the critical angle (phase angle) and analyse the sensitivity of the critical angle to the model parameters and the available azimuthal range for both transversely isotropic medium with a vertical symmetry axis and orthorhombic models. For more complex orthorhombic models, the critical angle can be computed while considering changes in azimuth. In a numerical example, we apply sensitivity analysis to examine the existence of the critical angle and determine its corresponding values to variations in model parameters and the azimuthal range. Additionally, we conduct computations of the critical angle for the simplified acoustic and elliptical orthorhombic models. This analysis can be extended to encompass all model parameters for different wave modes (pure and converted waves), and they provide generalized predictions for the available range of data in seismic data processing applications. [ABSTRACT FROM AUTHOR]

In this paper, we present a frequency‐domain volume integral method to model the microseismic wavefield in heterogeneous anisotropic‐elastic media. The elastic wave equation is written as an integral equation of the Lippmann–Schwinger type, and the seismic source is represented as a general moment tensor. The actual medium is split into a background medium and a scattered medium. The background part of the displacement field is computed analytically, but the scattered part requires a numerical solution. The existing matrix‐based implementation of the integral equation is computationally inefficient to model the wavefield in three‐dimensional earth. An integral equation for the particle displacement is, hence, formulated in a matrix‐free manner through the application of the Fourier transform. The biconjugate gradient stabilized method is used to iteratively obtain the solution of this equation. The integral equation method is naturally target oriented, and it is not necessary to fully discretize the model. This is very helpful in the microseismic wavefield computation at receivers in the borehole in many cases; say, for example, we want to focus only on the fluid injection zone in the reservoir–overburden system and not on the whole subsurface region. Additionally, the integral equation system matrix has a low condition number. This provides us flexibility in the selection of the grid size, especially at low frequencies for given wave velocities. Considering all these factors, we apply the numerical scheme to three different models in order of increasing geological complexity. We obtain the elastic displacement fields corresponding to the different types of moment tensor sources, which prove the utility of this method in microseismic. The generated synthetic data are intended to be used in inversion for the microseismic source and model parameters. [ABSTRACT FROM AUTHOR]

Considerably different from sandstones, sands have point‐like contact of grains. In contrast, sandstones have cementation between grains, which has diminished the very narrow gap at contact of grains. In this paper, ultrasonic S‐wave in brine‐saturated glass beads and Ottawa sands is used to detect the very narrow space at contact of grains. The data of the dry beads/sands and brine are inputted into Biot theory to yield phase velocity of the saturated beads/sands. By fitting the theoretical velocity with the ultrasonic measurement, phase velocity, the quality factor and S‐wave permeability are determined as functions of frequency. The predicted ultrasonic quality factors appear to be very close to that of water‐saturated Berea sandstone. Our previous study showed that for Berea sandstone, the low‐frequency S‐wave permeability is approximately half of Darcy permeability. However, for glass beads and Ottawa sands, the S‐wave permeabilities at low frequencies are one‐order magnitude lower than Darcy permeabilities. This well shows that S‐wave permeability of the beads/sands is associated with the very narrow gap at contact of grains which is successfully detected by ultrasonic S‐wave in the indirect way. [ABSTRACT FROM AUTHOR]

Constant Q, also called frequency‐independent Q, is a phenomenological assumption often used in practice for dissipative wave propagation in both forward and inverse seismic modelling. A constant Q dissipative model generally results in a temporal convolution‐type wave equation, which is computationally costly to solve by a time‐domain numerical modelling method. In this paper, we use the recently proposed weighting function method to derive the nearly constant Q wave equations in differential form for viscoelastic transversely isotropic media. These wave equations explicitly involve specified Q parameters, which is convenient for time‐domain waveform modelling and inversion. As numerical examples, we analyse the nearly constant Q behaviour of the proposed model , nearly constant Q wave reflection and wave propagation in a heterogeneous case. [ABSTRACT FROM AUTHOR]

The accuracy of an explicit traveltime‐offset approximation affects the results of the velocity analysis that plays a crucial role in the seismic data processing. Seismic anisotropy is important to account for large offset and azimuth since it can provide detailed information comparing with the isotropic assumption. For the perturbation‐based method, different traveltime approximation forms result in different accuracy. It is necessary to find the optimal traveltime approximation for specific signs and magnitudes of the anellipticity and different offset ranges. In this paper, a series of perturbation‐based traveltime approximations from different acoustic assumptions and parameterizations are specified. The perturbation coefficients are derived from the corresponding acoustic eikonal equations. We test the accuracy of the defined perturbation approximations in four homogeneous transversely isotropic medium with vertical symmetry axis (VTI) medium with a different set of anisotropy parameters and one multilayered transversely isotropic medium with vertical symmetry axis medium. The sensitivity in anellipticity for different approximations is also analysed. We find that different approximation achieves different accuracy under the circumstance of specific offset range and the value of anellipticity. Therefore, the optimal approximation form with higher accuracy can be selected based on different offset ranges and the magnitudes of anellipticity. [ABSTRACT FROM AUTHOR]

ABSTRACT: This paper presents, in a tutorial form, some analytical inversion techniques for the Goupillaud‐layered earth model. Finding the reflection coefficients from the reflection seismogram is the inverse problem for the model. For this reason we present a thorough description of the inverse problem for the Goupillaud model, two solutions to solve the inverse scattering problem using linear discrete equations and the solution obtained using the classic dynamic deconvolution method. The inversion is achieved using Robinson's polynomials Pk(z), Qk(z), Ak(z) and their reverse polynomials, as well as some properties of the model (the Lorentz transformation and the Einstein subtraction formula). The method of dynamic deconvolution, which makes the inversion of the model very simple computationally, is based on the physical structure of the reflection seismogram. We present the classic dynamic deconvolution algorithm for the non‐free‐surface Goupillaud model to show that the dynamic deconvolution method can provide efficient discrete procedures for the inversion. For this reason, though the inverse dynamic deconvolution procedures are old algorithms, they could be useful today for solving inverse scattering problems arising in exploration geophysics and various fields. [ABSTRACT FROM AUTHOR]

ABSTRACT: This paper introduces a novel method of modelling acoustic and elastic wave propagation in inhomogeneous media with sharp variations of physical properties based on the recently developed grid‐characteristic method which considers different types of waves generated in inhomogeneous linear‐elastic media (e.g., longitudinal, transverse, Stoneley, Rayleigh, scattered PP‐, SS‐waves, and converted PS‐ and SP‐waves). In the framework of this method, the problem of solving acoustic or elastic wave equations is reduced to the interpolation of the solutions, determined at earlier time, thus avoiding a direct solution of the large systems of linear equations required by the FD or FE methods. We apply the grid‐characteristic method to compare wave phenomena computed using the acoustic and elastic wave equations in geological medium containing a hydrocarbon reservoir or a fracture zone. The results of this study demonstrate that the developed algorithm can be used as an effective technique for modelling wave phenomena in the models containing hydrocarbon reservoir and/or the fracture zones, which are important targets of seismic exploration. [ABSTRACT FROM AUTHOR]

In this paper, we study seismic responses to an earthquake source in horizontally stratified transversely isotropic porous media with a vertical axis of symmetry that is, transversely isotropic porous media. We present a semi‐analytical method to simulate the three‐dimensional time–space–domain seismic wavefields. We obtain the solution in the frequency–wavenumber domain by the global matrix method and transform it to the time–space domain by the discrete wavenumber method and fast Fourier transform. Our method incorporates the moment tensor source so that it allows simulating seismic responses to an earthquake. We validate our method by degenerating the transversely isotropic porous media into the isotropic porous media and transversely isotropic solid media, respectively, and compare the results with the corresponding reference solutions. We then investigate the characteristics of seismic wavefields in the transversely isotropic porous media by considering an explosive source, a shear‐wave source and a double‐couple source. The results reveal interesting observations in the transversely isotropic porous media versus the isotropic porous media. For example, the qSV wave can have a response to the pore pressure during its propagation in a transversely isotropic porous medium while the shear‐wave does not generate any disturbance of the pore pressure. Our method is semi‐analytical and computationally efficient and it can be used for rapid evaluation of the seismic responses in transversely isotropic porous media. [ABSTRACT FROM AUTHOR]

Least squares migration uses the assumption that, if we have an operator that can create data from a reflectivity function, the optimal image will predict the actual recorded data with minimum square error. For this assumption to be true, it is also required that: (a) the prediction operator must be error‐free, (b) model elements not seen by the operator should be constrained by other means and (c) data weakly predicted by the operator should make limited contribution to the solution. Under these conditions, least squares migration has the advantage over simple migration of being able to remove interference between different model components. Least squares migration does that by de‐convolving or inverting the so‐called Hessian operator. The Hessian is the cascade of forward modelling and migration; for each image point, it computes the effects of interference from other image points (point‐spread function) given the actual recording geometry and the subsurface velocity model. Because the Hessian contains illumination information (along its diagonal), and information about the model cross‐correlation produced by non‐orthogonality of basis functions, its inversion produces illumination compensation and increases resolution. In addition, sampling deficiencies in the recording geometry map to the Hessian (both diagonal and non‐diagonal elements), so least squares migration has the potential to remove sampling artefacts as well. These (illumination compensation, resolution and mitigating recording deficiencies) are the three main goals of least squares migration, although the first one can be achieved by cheaper techniques. To invert the Hessian, least squares migration relies on the residual errors during iterations. Iterative algorithms, like conjugate gradient and others, use the residuals to calculate the direction and amplitudes (gradient and step size) of the necessary corrections to the reflectivity function or model. Failure of conditions (a), (b) or (c) leads the inversion to calculate incorrect model updates, which translate to noise in the final image. In this paper, we will discuss these conditions for Kirchhoff migration and reverse time migration. [ABSTRACT FROM AUTHOR]

In an acoustic transversely isotropic medium, there are two waves that propagate. One is the P‐wave and another one is the S‐wave (also known as S‐wave artefact). This paper is devoted to analyse the S‐wave in two‐dimensional acoustic transversely isotropic media with a tilted symmetry axis. We derive the S‐wave slowness surface and traveltime function in a homogeneous acoustic transversely isotropic medium with a tilted symmetry axis. The S‐wave traveltime approximations in acoustic transversely isotropic media with a tilted symmetry axis can be mapped from the counterparts for acoustic transversely isotropic media with a vertical symmetry axis. We consider a layered two‐dimensional acoustic transversely isotropic medium with a tilted symmetry axis to analyse the S‐wave moveout. We also illustrate the behaviour of the moveout for reflected S‐wave and converted waves. [ABSTRACT FROM AUTHOR]

In this paper, we deduced the corresponding first‐order velocity–stress equation for curvilinear coordinates from the first‐order velocity–stress equation based on the modified Biot/squirt model for a two‐dimensional two‐phase medium. The equations are then numerically solved by an optimized high‐order non‐staggered finite difference scheme, that is, the dispersion relation preserving/optimization MacCormack scheme. To implement undulating free‐surface topography, we derive an analytical relationship between the derivatives of the particle velocity components and use the compact finite‐difference scheme plus a traction‐image method. In the undulating free surface and the undulating subsurface interface of two‐phase medium, the complex reflected wave and transmitted wave can be clearly recognized in the numerical simulation results. The simulation results show that the curvilinear‐grid finite‐difference method, which uses a body‐conforming grid to describe the undulating surface, can accurately reduce the numerical scattering effect of seismic wave propagation caused by the use of ladder‐shaped grid to fit the surfaces when undulating topography is present in a two‐phase isotropic medium. [ABSTRACT FROM AUTHOR]

Orthorhombic models are often used in the seismic industry nowadays to describe azimuthal and polar anisotropy and reasonably realistic in capturing the features of the earth interior. It is challenging to handle so many model parameters in the seismic data processing. In order to reduce the number of the parameters for P wave, the acoustic orthorhombic medium is proposed by setting all on‐axis S wave velocities to zero. However, due to the coupled behaviour for P and S waves in the orthorhombic model, the 'S wave artefacts' are still remained in the acoustic orthorhombic model, which kinematics needs to be defined and analysed. In this paper, we analyse the behaviour of S wave in acoustic orthorhombic media. By analysis of the slowness surface in acoustic orthorhombic media, we define the S waves (or S wave artefacts) that are more complicated in shape comparing to the one propagating in an acoustic transversely isotropic medium with a vertical symmetry axis. The kinematic properties of these waves are defined and analysed in both phase and group domain. The caustics, amplitude and the multi‐layered case for S wave in acoustic orthorhombic model are also discussed. It is shown that there are two waves propagating in this acoustic orthorhombic medium. One of these waves is similar to the one propagating in acoustic vertical symmetry axis media, whereas another one has a very complicated shape consisting of two crossing surfaces. [ABSTRACT FROM AUTHOR]

Self‐potential is a passive geophysical method that can be applied in a straightforward manner with minimum requirements in the field. Nonetheless, interpretation of self‐potential data is particularly challenging due to the inherited non‐uniqueness present in all potential methods. Incorporating information regarding the target of interest can facilitate interpretation and increase the reliability of the final output. In the current paper, a novel method for detecting multiple sheet‐like targets is presented. A numerical framework is initially described that simulates sheet‐like bodies in an arbitrary 2D resistivity distribution. A scattered field formulation based on finite differences is employed that allows the edges of the sheet to be independent of the grid geometry. A novel analytical solution for two‐layered models is derived and subsequently used to validate the accuracy of the proposed numerical scheme. Lastly, a hybrid optimization is proposed that couples linear least‐squares with particle‐swarm optimization in order to effectively locate the edges of multiple sheet‐like bodies. Through numerical and real data, it is proven that the hybrid optimization overcomes local minimal that occurs in complex resistivity distributions and converges substantially faster compared to traditional particle‐swarm optimization. [ABSTRACT FROM AUTHOR]

Mesh‐free discretization, flexibly distributing nodes without computationally expensive meshing process, is able to deal with staircase problem, oversampling and undersampling problems and saves plenty of nodes through distributing nodes suitably with respect to irregular boundaries and model parameters. However, the time‐domain mesh‐free discretization usually exhibits poorer stability than that in regular grid discretization. In order to reach unconditional stability and easy implementation in parallel computing, we develop the frequency‐domain finite‐difference method in a mesh‐free discretization, incorporated with two perfectly matched layer boundary conditions. Furthermore, to maintain the flexibility of mesh‐free discretization, the nodes are still irregularly distributed in the absorbing zone, which complicates the situation of artificial boundary reflections. In this paper, we implement frequency‐domain acoustic wave modelling in a mesh‐free system. First, we present the perfectly matched layer boundary condition to suppress spurious reflections. Moreover, we develop the complex frequency shifted–perfectly matched layer boundary condition to improve the attenuation of grazing waves. In addition, we employ the radial‐basis‐function‐generated finite difference method in the mesh‐free discretization to calculate spatial derivatives. The numerical experiment on a rectangle homogeneous model shows the effectiveness of the perfectly matched layer boundary condition and the complex frequency shifted–perfectly matched layer boundary condition, and the latter one is better than the former one when absorbing large angle incident waves. The experiment on the Marmousi model suggests that the complex frequency shifted–perfectly matched layer boundary condition works well for complicated models. [ABSTRACT FROM AUTHOR]

Seismic anisotropy has an important influence on seismic data processing and interpretation. Although the frequency‐domain seismic wavefield simulation has a problem of solving the large scale linear sparse matrix due to the computational limitations, it has some advantages over the time‐domain seismic wavefield simulation including efficient inversion using only a limited number of frequency components and easy implementation of multiple sources. To accurately simulate seismic wave propagation in the frequency domain, we also need to choose the absorbing boundary conditions to absorb artificial reflections from edges of the model as we do in the time domain. Compared with the classical boundary conditions including the perfectly matched layer and complex frequency‐shifted perfectly matched layer, the complex frequency‐shifted multi‐axial perfectly matched layer has been proven to effectively suppress the unwanted reflections at grazing incidence and solve the instability problem in the time‐domain seismic numerical modelling in anisotropic elastic media. In this paper, we propose to extend the complex frequency‐shifted multi‐axial perfectly matched layer absorbing boundary condition to the frequency‐domain seismic wavefield simulation in anisotropic elastic media. To test the validity of our proposed algorithm, we compare the results (snapshots and seismograms) of the frequency‐domain seismic wavefield simulation with those of the time‐domain modelling. The model studies indicate that the complex frequency‐shifted multi‐axial perfectly matched layer absorbing boundary condition is stable in the frequency‐domain seismic wavefield simulation in anisotropic media, and provides better absorbing performance than the complex frequency‐shifted perfectly matched layer boundary condition. [ABSTRACT FROM AUTHOR]

Understanding how physical properties and seismic signatures of present day rocks are related to ancient geological processes is important for enhanced reservoir characterization. In this paper, we have studied this relationship for the Kobbe Formation sandstone in the Barents Sea. These rocks show anomalous low shear velocities and high VP/VS ratios, which does not agree well with conventional rock physics models for moderately to well consolidated sandstones. These sandstones have been buried relatively deeply and subsequently uplifted 1–2 km. We compared well log data of the Kobbe sandstone with velocity–depth trends modelled by integrating basin modelling principles and rock physics. We found that more accurate velocity predictions were obtained when first honouring mechanical and chemical compaction during burial, followed by generation of micro‐cracks during uplift. We suspect that these micro‐cracks are formed as overburden is eroded, leading to changes in the subsurface stress‐field. Moreover, the Kobbe Formation is typically heterogeneous and characterized by structural clays and mica that can reduce the rigidity of grain contacts. By accounting for depositional and burial history, our velocity predictions become more consistent with geophysical observables. Our approach yields more robust velocity predictions, which are important in prospect risking and net erosion estimates. [ABSTRACT FROM AUTHOR]

Obtaining an accurate image of the subsurface still remains a great challenge for the seismic method. Migration algorithms aim mainly on positioning seismic events in complex geological contexts. Multiple reflections are typically not accounted for in this process, which can lead to the emergence of artefacts. In Marchenko imaging, we retrieve the complete up‐ and downgoing wavefields in the subsurface to construct an image without such artefacts. The quality of this image depends on the type of imaging condition that is applied. In this paper, we propose an imaging condition that is based on stabilized unidimensional deconvolution. This condition is computationally much cheaper than multidimensional deconvolution, which has been proposed for Marchenko imaging earlier. Two specific approaches are considered. In the first approach, we use the full up‐ and downgoing wavefields for deconvolution. Although this leads to balanced and relatively accurate amplitudes, the crosstalk is not completely removed. The second approach is to incorporate the initial focussing function in the deconvolution process, in such a way that the retrieval of crosstalk is avoided. We compare images with the results of the classical cross‐correlation imaging condition, which we apply to reverse‐time migrated wavefields and to the up‐ and downgoing wavefields that are retrieved by the Marchenko method. [ABSTRACT FROM AUTHOR]

ABSTRACT Vertical seismic profiling data can provide a high-resolution reservoir image because the receivers are located close to the reservoir and, hence, the wavefield encounters less distortion in the overburden compared to surface seismic data. However, conventional migration of vertical seismic profiling data using only upgoing primary wavefields often suffers from poor illumination, imaging artefacts and low image reliability, especially at image locations away from the well. We propose full-wavefield migration to image vertical seismic profiling data using the primaries and all orders of surface multiples and internal multiples. The downgoing internal or surface multiples are treated as an additional source of illumination. In full-wavefield migration, we aim to estimate a high-resolution reflectivity image of the subsurface with this extended illumination. The algorithm is recursive in depth like conventional wavefield extrapolation-based migration, however, it incorporates the non-linear transmission and scattering effects at each depth level. Further, full-wavefield migration is posed as a constrained least-squares inversion problem that could be solved using a conjugate gradient method. We propose an iterative full-wavefield forward modelling engine as the backbone of this inversion scheme. The parameter used in the modelling is subsurface reflectivity. Full-wavefield modelling iteratively incorporates the non-linearity of the wavefield due to multiple scattering, where every iteration utilizes one higher order of scattered wavefields to estimate the subsurface reflectivity. In addition, the constrained inversion helps in reducing the extrapolation artefacts and provides a high-resolution image of the reservoir. In this paper, we discuss the concept of full-wavefield migration and demonstrate its potential as an imaging tool for vertical seismic profiling data using synthetic examples. [ABSTRACT FROM AUTHOR]

ABSTRACT: Acoustic transversely isotropic models are widely used in seismic exploration for P‐wave processing and analysis. In isotropic acoustic media only P‐wave can propagate, while in an acoustic transversely isotropic medium both P and S waves propagate. In this paper, we focus on kinematic properties of S‐wave in acoustic transversely isotropic media. We define new parameters better suited for S‐wave kinematics analysis. We also establish the travel time and relative geometrical spreading equations and analyse their properties. To illustrate the behaviour of the S‐wave in multi‐layered acoustic transversely isotropic media, we define the Dix‐type equations that are different from the ones widely used for the P‐wave propagation. [ABSTRACT FROM AUTHOR]

ABSTRACT: Borehole guided waves that are excited by explosive sources outside of the borehole are important for interpreting borehole seismic surveys and for rock property inversion workflows. Borehole seismograms are typically modelled using numerical methods of wave propagation. In order to benchmark such numerical algorithms and partially to interpret the results of modelling, an analytical methodology is presented here to compute synthetic seismograms. The specific setup is a wavefield emanating from a monopole point source embedded within a homogeneous elastic medium that interacts with a fluid‐filled borehole and a free surface. The methodology assumes that the wavelength of the seismic signal is much larger than the borehole radius. In this paper, it is supposed that there is no poroelastic coupling between the formation and the borehole. The total wavefield solution consists of P, PP, and PS body waves; the surface Rayleigh wave; and the low‐frequency guided Stoneley wave (often referred as the tube wave) within the borehole. In its turn, the tube wave consists of the partial responses generated by the incident P‐wave and the reflected PP and PS body waves at the borehole mouth and by the Rayleigh wave, as well as the Stoneley wave eigenmode. The Mach tube wave, which is a conic tube wave, additionally appears in the Mach cone in a slow formation with the tube‐wave velocity greater than the shear one. The conditions of appearance of the Mach wave in a slow formation are formulated. It is shown that the amplitude of the Mach tube wave strongly depends on Poisson's ratio of the slow surrounding formation. The amplitude of the Mach tube wave exponentially decreases when the source depth grows for weakly compressible elastic media with Poisson's ratio close to 0.5 (i.e., saturated clays and saturated clay soils). Asymptotic expressions are also provided to compute the wavefield amplitudes for different combinations of source depth and source‐well offset. These expressions allow an approximate solution of the wavefield to be computed much faster (within several seconds) than directly computing the implicit integrals arising from the analytical formulation. [ABSTRACT FROM AUTHOR]

ABSTRACT The electromagnetic response of a horizontal electric dipole transmitter in the presence of a conductive, layered earth is important in a number of geophysical applications, ranging from controlled-source audio-frequency magnetotellurics to borehole geophysics to marine electromagnetics. The problem has been thoroughly studied for more than a century, starting from a dipole resting on the surface of a half-space and subsequently advancing all the way to a transmitter buried within a stack of anisotropic layers. The solution is still relevant today. For example, it is useful for one-dimensional modelling and interpretation, as well as to provide background fields for two- and three-dimensional modelling methods such as integral equation or primary-secondary field formulations. This tutorial borrows elements from the many texts and papers on the topic and combines them into what we believe is a helpful guide to performing layered earth electromagnetic field calculations. It is not intended to replace any of the existing work on the subject. However, we have found that this combination of elements is particularly effective in teaching electromagnetic theory and providing a basis for algorithmic development. Readers will be able to calculate electric and magnetic fields at any point in or above the earth, produced by a transmitter at any location. As an illustrative example, we calculate the fields of a dipole buried in a multi-layered anisotropic earth to demonstrate how the theory that developed in this tutorial can be implemented in practice; we then use the example to examine the diffusion of volume charge density within anisotropic media-a rarely visualised process. The algorithm is internally validated by comparing the response of many thin layers with alternating high and low conductivity values to the theoretically equivalent (yet algorithmically simpler) anisotropic solution, as well as externally validated against an independent algorithm. [ABSTRACT FROM AUTHOR]

ABSTRACT: Potential field data such as geoid and gravity anomalies are globally available and offer valuable information about the Earth's lithosphere especially in areas where seismic data coverage is sparse. For instance, non‐linear inversion of Bouguer anomalies could be used to estimate the crustal structures including variations of the crustal density and of the depth of the crust–mantle boundary, that is, Moho. However, due to non‐linearity of this inverse problem, classical inversion methods would fail whenever there is no reliable initial model. Swarm intelligence algorithms, such as particle swarm optimisation, are a promising alternative to classical inversion methods because the quality of their solutions does not depend on the initial model; they do not use the derivatives of the objective function, hence allowing the use of L1 norm; and finally, they are global search methods, meaning, the problem could be non‐convex. In this paper, quantum‐behaved particle swarm, a probabilistic swarm intelligence‐like algorithm, is used to solve the non‐linear gravity inverse problem. The method is first successfully tested on a realistic synthetic crustal model with a linear vertical density gradient and lateral density and depth variations at the base of crust in the presence of white Gaussian noise. Then, it is applied to the EIGEN 6c4, a combined global gravity model, to estimate the depth to the base of the crust and the mean density contrast between the crust and the upper‐mantle lithosphere in the Eurasia–Arabia continental collision zone along a 400 km profile crossing the Zagros Mountains (Iran). The results agree well with previously published works including both seismic and potential field studies. [ABSTRACT FROM AUTHOR]

In this paper, we investigate production induced microseismicity based on modelling material failure from coupled fluid-flow and geomechanical simulation. The model is a graben style reservoir characterized by two normal faults subdividing a sandstone reservoir into three compartments. The results are analysed in terms of spatial and temporal variations in distribution of material failure. We observe that material failure and hence potentially microseismicity is sensitive to not only fault movement but also fluid movement across faults. For sealing faults, failure is confined to the volume in and around the well compartment, with shear failure localized along the boundaries of the compartment and shear-enhanced compaction failure widespread throughout the reservoir compartment. For non-sealing faults, failure is observed within and surrounding all three reservoir compartments as well as a significant distribution located near the surface of the overburden. All shear-enhanced compaction failures are localized within the reservoir compartments. Fault movement leads to an increase in shear-enhanced compaction events within the reservoir as well as shear events located within the side-burden adjacent to the fault. We also evaluate the associated moment tensor mechanisms to estimate the pseudo scalar seismic moment of failure based on the assumption that failure is not aseismic. The shear-enhanced compaction events display a relatively normal and tight pseudo scalar seismic moment distribution centred about 106 Pa, whereas the shear events have pseudo scalar seismic moments that vary over three orders of magnitude. Overall, the results from the study indicate that it may be possible to identify compartment boundaries based on the results of microseismic monitoring. [ABSTRACT FROM AUTHOR]

ABSTRACT Mechanical compaction or loss of porosity due to increase in effective stress is a fundamental geological process that governs many of the rock elastic and transport parameters, all of great importance in exploring and developing subsurface reservoirs. The ability to model the compaction process enables us to improve our understanding of the seismic signature of the basin and better relate the geology of deposition to current porosity, velocity, pore pressure, and other mechanical parameters that depend on the state of compaction of the sediment. In this paper, a set of mathematical equations that can be used to model the plastic deformation associated with primary and secondary loading curves is presented. Compaction laws are posed in terms of natural strain increment formulation often used in plasticity theory to model large deformation. Laboratory and field estimates of constitutive plastic deformation relations for sand-shale mixtures are used in a numerical model that generates estimates of porosity under various pore pressures, shale content, and loading scenarios. These estimates can be used in a variety of settings to predict various basin and reservoir properties associated with different loading conditions and/or sedimentation processes. [ABSTRACT FROM AUTHOR]

Elastic interactions between pores and cracks reflect how they are organized or spatially distributed in porous rocks. The principle goal of this paper is to understand and characterize the effect of elastic interactions on the effective elastic properties. We perform finite element modelling to quantitatively study how the spatial arrangement of inclusions affects stress distribution and the resulting overall elasticity. It is found that the stress field can be significantly altered by elastic interactions. Compared with a non-interacting situation, stress shielding considerably stiffens the effective media, while stress amplification appreciably reduces the effective elasticity. We also demonstrate that the T-matrix approach, which takes into account the ellipsoid distribution of pores or cracks, can successfully characterize the competing effects between stress shielding and stress amplification. Numerical results suggest that, when the concentrations of cracks increase beyond the dilute limit, the single parameter crack density is not sufficient to characterize the contribution of the cracks to the effective elasticity. In order to obtain more reliable and accurate predictions for the effective elastic responses and seismic anisotropies, the spatial distribution of pores and cracks should be included. Additionally, such elastic interaction effects are also dependent on both the pore shapes and the fluid infill. [ABSTRACT FROM AUTHOR]

This paper studies aspects that influence the de-ghosting performance of marine multi-level sources based on a modified Johnson model. The normalized squared error between actual signature and its corresponding ghost-free signature is introduced to evaluate the multi-level source design. The results show that optimum depth combinations and volume combinations exist in the design. However, there is also some flexibility in the volume combination which makes it possible to optimize with respect to residual bubble oscillation. By considering both operational aspects and performance, we propose that three or four levels in a multi-level source are reasonable. Compared to a horizontal source, a multi-level source can be designed to reduce the notch effect, strengthen the down-going energy and improve the energy transmission directivity. Studies of the influence of depth and firing time deviations indicate that a multi-level source is more stable than a normal horizontal source in an operational environment. [ABSTRACT FROM AUTHOR]

ABSTRACT For an accurate interpretation of seismic data, multiple-free data are of great value. Removing surface multiples and interbed multiples proves to be challenging in many cases. The nowadays widely used method of Surface-Related Multiple Elimination (SRME) has lately been redefined as a full-waveform inversion process, resulting in the method of Estimation of Primaries by Sparse Inversion (EPSI). The new method is shown to be more accurate than the former method in several situations, because it estimates primaries such that they, together with their multiples, explain the input data. Its main advantage is that the minimum energy assumption in traditional multiple subtraction is avoided. The SRME methodology has been extended to the case of internal multiples by several authors, however, the involved subtraction of predicted multiples is probably even more challenging than for the surface-multiple case. Therefore, in this paper the EPSI method is generalized to remove both surface and interbed multiples. As in previous implementations of internal multiple removal based on data-driven convolution, the newly proposed scheme requires some knowledge about the subsurface: the data should be divided into (macro) layers and appropriate time windows must be selected. The method is tested on two 2D synthetic datasets to prove its viability. Furthermore, application to a 2D field dataset showed improved accuracy compared to conventional prediction and subtraction. [ABSTRACT FROM AUTHOR]

ABSTRACT In this paper we present a case history of seismic reservoir characterization where we estimate the probability of facies from seismic data and simulate a set of reservoir models honouring seismically-derived probabilistic information. In appraisal and development phases, seismic data have a key role in reservoir characterization and static reservoir modelling, as in most of the cases seismic data are the only information available far away from the wells. However seismic data do not provide any direct measurements of reservoir properties, which have then to be estimated as a solution of a joint inverse problem. For this reason, we show the application of a complete workflow for static reservoir modelling where seismic data are integrated to derive probability volumes of facies and reservoir properties to condition reservoir geostatistical simulations. The studied case is a clastic reservoir in the Barents Sea, where a complete data set of well logs from five wells and a set of partial-stacked seismic data are available. The multi-property workflow is based on seismic inversion, petrophysics and rock physics modelling. In particular, log-facies are defined on the basis of sedimentological information, petrophysical properties and also their elastic response. The link between petrophysical and elastic attributes is preserved by introducing a rock-physics model in the inversion methodology. Finally, the uncertainty in the reservoir model is represented by multiple geostatistical realizations. The main result of this workflow is a set of facies realizations and associated rock properties that honour, within a fixed tolerance, seismic and well log data and assess the uncertainty associated with reservoir modelling. [ABSTRACT FROM AUTHOR]

ABSTRACT Seismic wave propagation in transversely isotropic (TI) media is commonly described by a set of coupled partial differential equations, derived from the acoustic approximation. These equations produce pure P-wave responses in elliptically anisotropic media but generate undesired shear-wave components for more general TI anisotropy. Furthermore, these equations suffer from instabilities when the anisotropy parameter ε is less than δ. One solution to both problems is to use pure acoustic anisotropic wave equations, which can produce pure P-waves without any shear-wave contaminations in both elliptical and anelliptical TI media. In this paper, we propose a new pure acoustic transversely isotropic wave equation, which can be conveniently solved using the pseudospectral method. Like most other pure acoustic anisotropic wave equations, our equation involves complicated pseudo-differential operators in space which are difficult to handle using the finite difference method. The advantage of our equation is that all of its model parameters are separable from the spatial differential and pseudo-differential operators; therefore, the pseudospectral method can be directly applied. We use phase velocity analysis to show that our equation, expressed in a summation form, can be properly truncated to achieve the desired accuracy according to anisotropy strength. This flexibility allows us to save computational time by choosing the right number of summation terms for a given model. We use numerical examples to demonstrate that this new pure acoustic wave equation can produce highly accurate results, completely free from shear-wave artefacts. This equation can be straightforwardly generalized to tilted TI media. [ABSTRACT FROM AUTHOR]

ABSTRACT In this paper we present a workflow for microseismic feasibility studies that allows a thorough assessment of the probability of success of the monitoring project to be made. The workflow includes the following elements: assessment of the value of information to establish the business case; identification of hardware options to investigate deployment options; survey of analogue cases to confirm the do-ability of the project; modelling of event location errors and detectability to establish the most favourable array geometry; generation of full waveform synthetics to anticipate undesired seismic features. A study comprising all these elements provides the reference frame for discussions with service companies, limiting misunderstandings and avoiding missed opportunities. [ABSTRACT FROM AUTHOR]

ABSTRACT In this paper, we derive analytical expressions for one-way and two-way kinematical parameters in elliptical tilted transverse isotropy media. We show that the homogeneous elliptical tilted transverse isotropy models result in hyperbolic moveout with a reflection point sideslip x0, which can be considered as an additional traveltime parameter for one-way wave propagation. For homogeneous elliptical tilted transverse isotropy models we show that the inversion of one-way traveltime parameters suffers from the ambiguity for large tilts. It is shown that the accuracy of the inversion is sensitive to the error in x0. We also derive and invert the traveltime parameters for a vertically heterogeneous elliptical tilted transverse isotropy model with a tilt gradually changing with depth. The a priori knowledge of parameter δ is very important for inversion. The wrong choise of this parameter results in significant errors in inverted model parameters. [ABSTRACT FROM AUTHOR]

ABSTRACT Nowadays, geostatistics is commonly applied for numerous gridding or modelling tasks. However, it is still under used and unknown for classical geophysical applications. This paper highlights the main geostatistical methods relevant for geophysical issues, for instance to improve the quality of seismic data such as velocity cubes or interpreted horizons. These methods are then illustrated through four examples. The first example, based on a gravity survey presents how a geostatistical interpolation can be used to filter out a global trend, in order to better define real anomalies. In the second case study, dedicated to refraction surveying, geostatistical filtering is used to filter out acquisition artefacts and identify the main geological structures. The third one is an example of porosity being integrated geostatistically with a seismic acoustic impedance map. The last example deals with geostatistical time to depth conversion; the interest of performing geostatistical simulations is finally discussed. [ABSTRACT FROM AUTHOR]

This paper tests the ability of various rock physics models to predict seismic velocities in shallow unconsolidated sands by comparing the estimates to P and S sonic logs collected in a shallow sand layer and ultrasonic laboratory data of an unconsolidated sand sample. The model fits are also evaluated with respect to the conventional model for unconsolidated sand. Our main approach is to use Hertz-Mindlin and Walton contact theories, assuming different weight fractions of smooth and rough contact behaviours, to predict the elastic properties of the high porosity point. Using either the Hertz-Mindlin or Walton theories with rough contact behaviour to define the high porosity endpoint gives an over-prediction of the velocities. The P-velocity is overpredicted by a factor of ∼1.5 and the S-velocity by a factor of ∼1.8 for highly porous gas-sand. The degree of misprediction decreases with increasing water saturation and porosity.Using the Hertz-Mindlin theory with smooth contact behaviour or weighted Walton models gives a better fit to the data, although the data are best described using the Walton smooth model. To predict the properties at the lower porosities, the choice of bounding model attached to the Walton Smooth model controls the degree of fit to the data, where the Reuss bound best captures the porosity variations of dry and wet sands in this case since they are caused by depositional differences. The empirical models based on lab experiments on unconsolidated sand also fit the velocity data measured by sonic logs in situ, which gives improved confidence in using lab-derived results. [ABSTRACT FROM AUTHOR]

This paper discusses the asymptotic behaviour of the electromagnetic fields received on the sea-bed (target response), as well as the fields distributed inside a thin resistive target, generated by a horizontal electric dipole above the sea-bed in marine controlled-source electromagnetics for hydrocarbon exploration. It is found that the guided wave supported by a thin resistive target can be expressed as a single-mode exponential function. A simple closed-form expression is derived to relate the single-mode wavenumber of the guided wave to the model parameters: the resistivity and thickness of the target layer, the sea-bed resistivity and the frequency. When the air-wave is removed, the guided wave is dominant among the fields received on the sea-bed at far offset. Hence the wavenumber of the guided wave can be calculated from the fields measured on the sea-bed. The closed-form expression can then be used to invert the target property from the calculated wavenumber and hence, can be considered as a hydrocarbon indicator. [ABSTRACT FROM AUTHOR]

Anisotropy in subsurface geological models is primarily caused by two factors: sedimentation in shale/sand layers and fractures. The sedimentation factor is mainly modelled by vertical transverse isotropy (VTI), whereas the fractures are modelled by a horizontal transversely isotropic medium (HTI). In this paper we study hyperbolic and non-hyperbolic normal reflection moveout for a package of HTI/VTI layers, considering arbitrary azimuthal orientation of the symmetry axis at each HTI layer. We consider a local 1D medium, whose properties change vertically, with flat interfaces between the layers. In this case, the horizontal slowness is preserved; thus, the azimuth of the phase velocity is the same for all layers of the package. In general, however, the azimuth of the ray velocity differs from the azimuth of the phase velocity. The ray azimuth depends on the layer properties and may be different for each layer. In this case, the use of the Dix equation requires projection of the moveout velocity of each layer on the phase plane. We derive an accurate equation for hyperbolic and high-order terms of the normal moveout, relating the traveltime to the surface offset, or alternatively, to the subsurface reflection angle. We relate the azimuth of the surface offset to its magnitude (or to the reflection angle), considering short and long offsets. We compare the derived approximations with analytical ray tracing. [ABSTRACT FROM AUTHOR]

Well mineralogy can be estimated from probabilistic, direct and machine learning models; however, all these models have limitations. The maximum number of components in probabilistic models is restricted to the number of logs plus one. Direct models require the precise composition of minerals. Machine learning models demand unbiased databases, a challenge as the samples are collected in reservoir intervals. These limitations impact the evaluation for the Santos Basin pre‐salt rocks due to the complexity of facies and magnesian clays. This work proposes creating a hybrid model through the combination of probabilistic and machine learning models. First, mineral fractions of calcite, dolomite, quartz, k‐feldspar, detrital clay, plagioclase and pyroxene are estimated by the algorithm XGBoost trained using rock samples. Then, a probabilistic model reconstructs the well logs and machine learning estimations through the seven minerals mentioned plus magnesian clays, pyrite, barite and fluids. The difference between the real and reconstructed responses is minimized, weighted by the curves' uncertainties. The hybrid model is used to estimate the mineralogy of three wells drilled in the Santos Basin, honouring the mineralogy of the rock samples collected in these wells and improving the quantification of dolomite, pyroxene and magnesian clay. Among the advances introduced, the following stand out: The use of machine learning estimates and well logs improved the quantification of magnesian clay; the machine learning estimates regularized the probabilistic model, generating more coherent results; the uncertainties of the machine learning algorithms dealt with database bias. The hybrid model mitigated limitations related to database bias without the costs associated with collecting more samples. [ABSTRACT FROM AUTHOR]

Airborne radiometrics (radiometric surveying) has been used in many applications, such as geological mapping, environmental studies, mineral exploration and lithology mapping. The standard processing of gamma‐ray spectrometry data provides good results when the acquisition conditions such as the flight and sensor orientation are constant. In practice, abrupt changes in flight height when flying in mountainous terrain are common, and standard processing neglecting this factor can lead to erroneous interpretation. The primary cause is the fact that the successive corrections applied to radiometric data in standard processing do not consider the effective sampled area of a survey (i.e. the field of view), which can have significant and variable overlaps between adjacent samples due to changing observation height. For this reason, standard height and sensitivity corrections may lead to incorrect estimates of the concentrations of the radioelements on the ground. To ameliorate this problem, we have developed a two‐dimensional inversion‐based processing method that incorporates the aircraft height, replaces the sensitivity correction and accounts for a positivity condition using a logarithmic barrier. Some minerals associated with potassium, uranium and thorium allow the mapping of hydrothermal alteration, especially potassium which is an important constituent of hydrothermal fluids. We demonstrate the new methods by comparing the standard processing results with the inversion‐based processing of airborne gamma‐ray spectrometry data in the gold deposit‐rich area of Mara Rosa Magmatic Arc, Brazil. The inversion‐based processing results enhance the anomalies and suppress the interpolation artefacts as well as increase the signal‐to‐noise ratio. The application of airborne radiometric transformed maps also improved the footprint knowledge around Cu–Au mineralization and highlighted potential areas for further studies. [ABSTRACT FROM AUTHOR]

Due to high computational costs and memory requirements, it is necessary to use coarse grids for least‐squares reverse time migration when processing large‐scale field data. According to the dispersion relation, strong numerical dispersion will occur in least‐squares reverse time migration images when the migration velocity of the shallow strata is low, reducing the resolution of the migration results. To overcome this shortcoming, we propose new least‐squares reverse time migration implementation method with a modified flux‐corrected transport technique. This algorithm involves two main steps: obtain an appropriate calibration parameter for flux‐corrected transport through forward modelling and perform least‐squares reverse time migration using flux‐corrected transport. We remove the non‐linear processing from conventional flux‐corrected transport algorithm often used in computational fluid dynamics based on the assumption that the seismic wavefield is relatively continuous and smooth without strong ripples, which makes it possible to apply flux‐corrected transport to least‐squares reverse time migration without increasing too much computation complexity. To suppress the numerical dispersion more effectively, the diffusion fluxes along the diagonal direction are added to flux‐corrected transport. Numerical tests on synthetic data and field data illustrate that compared to the conventional least‐squares reverse time migration method, the proposed method produces images with fewer migration artefacts, more accurate amplitudes and faster convergence speed on the coarse grids. [ABSTRACT FROM AUTHOR]

Well logs are geophysical measurements of rock properties acquired continuously along a borehole. Because of the physics underlying their operation, borehole logging instruments perform local spatial averages of rock properties in the vicinity of the borehole, yielding well logs that are often affected by tool design, layer boundaries, borehole environmental conditions and mud‐filtrate invasion. Such environmental effects are ubiquitous and can lead to errors in petrophysical–property estimations from well logs if not accounted for in the interpretation. Separate well‐log inversion mitigates environmental effects by matching well logs with their numerical simulations. The latter simulations rely on specific assumptions about the relative geometry of the borehole and the rocks penetrated by the well. For vertical wells penetrating horizontal layers, it is common to assume piecewise‐constant layer properties that we collectively refer to as the earth model; some of these properties have a direct relationship to well logs (e.g. resistivity, gamma ray, density and acoustic slowness). Well‐log inversion is traditionally approached with deterministic methods such as the Levenberg–Marquardt algorithm to minimize the differences between well logs and their numerical simulations, often without accounting for the relationship between measurement noise and model uncertainty. Bayesian inversion, on the other hand, yields the posterior probability distribution of estimated properties and intrinsically quantifies their uncertainty. However, Bayesian Well‐log inversion usually involves the implementation of Markov chain Monte Carlo sampling, which requires a prohibitive number of forward simulations and is therefore not suitable for rapid petrophysical and/or elastic and mechanical evaluations of rocks. We introduce an efficient Bayesian Well‐log inversion method using a gradient‐based Markov chain Monte Carlo method. Gradient‐based Markov chain Monte Carlo is a group of relatively new Markov chain Monte Carlo algorithms that combine gradient updates with Hessian‐based sampling. gradient‐based Markov chain Monte Carlo draws samples efficiently from the posterior probability distribution using the gradient information to guide samples towards high‐probability regions and the Hessian to approximate the posterior probability distribution locally. We verify the gradient‐based Markov chain Monte Carlo inversion method with the inversion of synthetic well logs, including gamma ray, resistivity, density, neutron porosity, photoelectric factor and compressional/shear‐wave slowness. Results show that gradient‐based Markov chain Monte Carlo decreases the computational cost by more than 90% compared to conventional Markov chain Monte Carlo sampling. Next, gradient‐based Markov chain Monte Carlo inversion is applied to a field example from the Central North Sea, where conventional interpretation methods yield shale concentration, sandstone porosity and water saturation with errors up to 19.2%, 14.9% and 48.8%, respectively. Gradient‐based Markov chain Monte Carlo inversion products satisfactorily reproduce the available measurements with only a modest increase in computational cost compared to deterministic inversion approaches. [ABSTRACT FROM AUTHOR]

Characterizing the expressions of seismic waves in elastic anisotropic media depends on multiparameters. To reduce the complexity, decomposing the P‐mode wave from elastic seismic data is an effective way to describe the considerably accurate kinematics with fewer parameters. The acoustic approximation for transversely isotropic media is widely used to obtain P‐mode wave by setting the axial S‐wave phase velocity to zero. However, the separated pure P‐wave of this approach is coupled with undesired S‐wave in anisotropic media called S‐wave artefacts. To eliminate the S‐wave artefacts in acoustic waves for anisotropic media, we set the vertical S‐wave phase velocity as a function related to propagation directions. Then, we derive a pure P‐wave equation in transversely isotropic media with a horizontal symmetry axis by introducing the expression of vertical S‐wave phase velocity. The differential form of new expression for pure P‐wave is reduced to second‐order by inserting the expression of S‐wave phase velocity as an auxiliary operator. The results of numerical simulation examples by finite difference illustrate the stability and accuracy of the derived pure P‐wave equation. [ABSTRACT FROM AUTHOR]

Calculating the electric potential for 3‐D resistivity inversion algorithms can be time consuming depending on the structure of the mesh. There have been generally two approaches to generating finite‐element meshes. One approach uses a structured rectangular mesh with hexahedral elements on a rectangular model grid. The distribution of model cells can be designed to follow known boundaries, and directional roughness constraints can be easily imposed. A 1‐D wavelet transform that takes advantage of the regular arrangement of the model cells can also be used to reduce the computer time and memory required to solve the smoothness‐constrained least‐squares equation. However, the structured rectangular mesh uses an unnecessarily fine mesh in parts of the model that are far away from the electrodes where the potential changes gradually. A second approach uses an unstructured mesh with tetrahedral elements created automatically by a mesh generation program with finer elements nearer the electrodes and coarser elements in the more remote regions. This generates a mesh with a much smaller number of nodes. The disadvantage is that an irregular model grid is normally used. We examine an alternative approach that combines structured and unstructured meshes. We employ a regular model grid with a finer mesh near the surface and a coarser mesh in deeper regions using a combination of hexahedral and tetrahedral elements. The semi‐structured mesh reduces the calculation time by more than three times compared with a structured mesh. An adaptive semi‐structured mesh that also uses a coarser mesh for model cells near the surface if they are more than one unit electrode spacing from the nearest electrode was also developed for surveys with non‐uniform data coverage. For the Bonsall Leys field survey, which used a capacitively coupled mobile system and collected a data set with nearly a million electrode positions, the adaptive mesh reduces the calculation time by about 80%. The calculation time can be further reduced by about 93% when it is combined with a mesh segmentation method. [ABSTRACT FROM AUTHOR]

Shear wave velocity information is valuable in many aspects of seismic exploration and characterization of reservoirs. However, shear wave logs are not always available in the interval of interest due to cost and time‐saving purposes. In this study, we present a tailored supervised learning approach to estimate shear wave velocity from well‐log measurements in the Lower Cretaceous succession of the Valdemar and Boje fields in the Danish North Sea. Our objective is to investigate the performance of four supervised learning regression models (linear, random forest, support vector and multi‐layer perceptron). A limited well‐log data set from six wells is used for training and testing the supervised learning models. A set of well data containing normalized gamma ray, compressional wave velocity, neutron porosity and medium resistivity logs gave reasonable shear wave velocity estimates in the test wells with root‐mean‐square error scores within the range of other published studies. Based on limited input data and complex geology, the multi‐layer perceptron was the most successful model in predicting the reservoir sections of the test wells. However, all models lacked stability in the overburden zones. Lastly, re‐training the multi‐layer perceptron on the six wells to predict missing shear wave velocity in a nearby well showed promising results for further reservoir characterization. The obtained results can yield useful input into, for example, seismic pre‐stack inversion, amplitude versus offset analysis and rock physics analysis. [ABSTRACT FROM AUTHOR]

Most often, fast induced polarization in frozen rocks appears in the transient electromagnetic data at early (less than a few hundreds of microseconds) times as a nonmonotone voltage response. Usually, such transients are interpreted in terms of Pelton's resistivity/conductivity model. The relaxation time τIS found in this way determines the decay rate of transient voltage response to a current step. Conversion of τIS into the relaxation time τVS, determining the decay rate of the current response to a voltage step, gives τVS values typical for dielectric relaxation of ice at temperatures of about several degrees below zero. This result, along with the fact that both fast induced polarization and dielectric polarization of ice are described by the Debye relaxation model, suggests that fast induced polarization in frozen rocks is controlled by the dielectric polarization of ice. As a possible mechanism for such a control, the effect of ice polarization on the surface conductivity of frozen rocks is considered. [ABSTRACT FROM AUTHOR]

Inversion of seismic data using information from horizontal wells is often hampered by cumulative well‐location errors. These errors can have a significant influence on the final subsurface model derived from the data. To achieve a proper data integration and arrive at correct uncertainty estimates, we formulate the problem in a fully probabilistic framework and present a numerical approach for improving subsurface imaging using uncertain well‐log data and their uncertain locations as well as uncertain seismic data. The result is improved model error quantification in the seismic inversion process. [ABSTRACT FROM AUTHOR]

Frequency‐domain finite‐difference modelling is widely used in exploration geophysics. However, it involves solving a large system of linear algebraic equations, which may cause huge computational costs, especially for large‐scale models. To address this issue, we have proposed an improved 25‐point finite‐difference scheme for wavefield modelling of two‐dimensional frequency‐domain elastic wave equations. The biggest difference between the improved 25‐point scheme with other 25‐point schemes is the finite‐difference formulas for spatial derivatives. The proposed 25‐point scheme applies to equal and unequal grid intervals, and its optimization coefficients depend on the grid‐spacing ratio and Poisson's ratio. The dispersion analysis indicates that within the phase velocity errors of 1% and 2%, the improved 25‐point scheme only requires approximately 2.3 and 2.2 grid points per shear wavelength, which has greater accuracy than the existing schemes. To eliminate artificial boundary reflections, we apply the perfectly matched layer boundary conditions to the model edges. Several numerical examples are presented to prove the validity of the improved 25‐point scheme. [ABSTRACT FROM AUTHOR]

Three‐dimensional geoelectrical surveys are widely used to map the subsurface in areas with complex geology. Field survey methods using the offset pole–dipole and dipole–dipole arrays have been proposed to map large areas efficiently. However, it has been found that negative apparent resistivity values are sometimes encountered in areas with large resistivity contrasts, particularly when large offsets between the current electrodes and potential dipoles are used. The vector array configuration that makes two measurements of the electric field at each station with two potential dipoles at right angles avoids this problem. The combined potential measured at the two dipoles is independent of the orientation the current electrodes to the receiver station. The amplitude of the vector apparent resistivity value is always positive. A commonly used configuration is to lay out the current and potential electrodes along alternating parallel lines. We examine the use of different offsets between the current electrodes and potential dipoles to improve the survey spatial resolution. A new inversion method using both the amplitude and direction of the vector array potentials is described to improve the resolution. Results from the inversion of a synthetic model data set and a field survey over a deep‐seated landslide are shown. [ABSTRACT FROM AUTHOR]

ABSTRACT In this paper, we derive analytical expressions for one-way and two-way kinematical parameters in elliptical tilted transverse isotropy media. We show that the homogeneous elliptical tilted transverse isotropy models result in hyperbolic moveout with a reflection point sideslip x0, which can be considered as an additional traveltime parameter for one-way wave propagation. For homogeneous elliptical tilted transverse isotropy models we show that the inversion of one-way traveltime parameters suffers from the ambiguity for large tilts. It is shown that the accuracy of the inversion is sensitive to the error in x0. We also derive and invert the traveltime parameters for a vertically heterogeneous elliptical tilted transverse isotropy model with a tilt gradually changing with depth. The a priori knowledge of parameter δ is very important for inversion. The wrong choise of this parameter results in significant errors in inverted model parameters. [ABSTRACT FROM AUTHOR]

The state‐of‐the‐art joint migration inversion faces the so‐called amplitude‐versus‐offset challenge, due to adopting over‐simplified one‐way propagation, reflection and transmission operators to avoid over‐parameterization in the inversion process. To overcome this challenge, we apply joint migration inversion to horizontally layered media (or 1.5‐dimensional media) and parameterize the solution space via density and velocity models. In this scenario, one‐way propagation, reflection and transmission operators required by the joint migration inversion process can be analytically and correctly derived from the subsurface models, so the amplitude‐versus‐offset challenge is successfully overcome. We introduce a new concept, which is named 'inverse propagation', into our 1.5‐dimensional amplitude‐versus‐offset joint migration inversion. It can correctly reconstruct subsurface wavefields by using a surface‐recorded receiver wavefield with all the influence of transmission, reflection and multiples accounted for. A synthetic example is used to demonstrate the correctness of the inverse propagation. This work is the foundation to further develop the 1.5‐dimensional amplitude‐versus‐offset joint migration inversion technology. [ABSTRACT FROM AUTHOR]

Seismic wave imaging in complex media requires an accurate wavefield simulation method that can accurately describethe wave propagation in realistic media. Reverse time‐depth migration is the preferred method for seismic wave imaging in complex media. Although it is relatively expensive, its imaging accuracy is usually better than migrations based on the ray method. Migration of primary reflection data requires a wave propagation simulation method that can accurately describe primary reflected/scattered wave energy and incorporate anisotropy. Accordingly, we propose the simulation of wave propagation in tilted transversely isotropic media using a 15° one‐way wave equation in a ray‐centred coordinate system, combining the flexibility of ray theory and the accuracy of wave theory. We use this equation to describe the propagation of body waves in a single ray tube, a 'beam'. The wavefield along the beam, guided by its central raypath, has an angle limit defined only by the ray angle; therefore, wave propagation in complex and steeply dipping media can be simulated with a 15° one‐way wave equation. Numerical experiments show that the simulation results for beam propagation using the 15° equation in the ray‐centred coordinate system have good accuracy. For prestack depth migration in tilted transversely isotropic media, we built a beam imaging method using this propagator, and this migration method yielded accurate images with greater efficiency than reverse time‐depth migration. [ABSTRACT FROM AUTHOR]