Orthorhombic Model Research Articles

Uncertainty in orthorhombic model building: Analysis, mitigation, and validation

As seismic processing technologies have advanced, model definitions have grown correspondingly more complex, progressing from isotropy, to transverse isotropy, to the much more general orthorhombic anisotropy. This growth, while supported partly by continued improvements in acquisition techniques and survey geometries, has progressed at such a rate that current projects require solutions to a much greater relative number of unknowns than those in the near past. This added complexity creates ill-posed problems with huge model spaces, leading to a high degree of uncertainty in the final solutions. This uncertainty can be greatly mitigated through pragmatic use of a priori knowledge and intelligent data leverage and model regularization. Orthorhombic anisotropy has been well characterized both microscopically and macroscopically. This understanding allows an unprecedented level of constraint and confirmation for the model-building process by validating the directionality and strength of azimuthal velocity variations. Information within the data itself can also lead to a much more well-determined model-building result. The observed structure can be used to precondition inversion results to ensure geologic plausibility, constraining similar updates to similar events. Concurrently, nonparameterized residual moveout picking allows for complete freedom in describing gather events, yielding results which best resolve the various anisotropic parameters by accurately fitting the gather data and generating a high-resolution model. By properly combining the various constraints available in a well-designed processing project, the myriad uncertainties that arise when moving to an orthorhombic model space can be simplified and reduced, allowing for geologically reasonable solutions that fit data, yield valuable structural information, and enhance interpretation possibilities.

Perturbation‐based moveout approximations in anisotropic media

ABSTRACTThe moveout approximations play an important role in seismic data processing. The standard hyperbolic moveout approximation is based on an elliptical background model with two velocities: vertical and normal moveout. We propose a new set of moveout approximations based on a perturbation series in terms of anellipticity parameters using the alternative elliptical background model defined by vertical and horizontal velocities. We start with a transversely isotropic medium with a vertical symmetry axis. Then, we extend this approach to a homogeneous orthorhombic medium. To define the perturbation coefficients for a new background, we solve the eikonal equation with horizontal velocities in transversely isotropic medium with a vertical symmetry axis and orthorhombic media. To stabilise the perturbation series and improve the accuracy, the Shanks transform is applied for all the cases. We select different parameterisations for both velocities and anellipticity parameters for an orthorhombic model. From the comparison in traveltime error, the new moveout approximations result in better accuracy comparing with the standard perturbation‐based methods and other approximations.

Kinematic parameters of pure‐ and converted‐mode waves in elastic orthorhombic media

ABSTRACTI derive the kinematic properties of single‐mode P, S1, and S2 waves as well as converted PS1, PS2, and S1S2 waves in elastic orthorhombic media including vertical velocity, two normal moveout velocities defined in vertical symmetry planes, and three anelliptic parameters (two of them are defined in vertical symmetry plane and one parameter is the cross‐term one). I show that the azimuthal dependence of normal moveout velocity and anellipticity is different in phase and group domains. The effects on‐vertical‐axis singularity and on‐vertical‐axis triplication are considered for pure‐mode S1 and S2 waves and converted‐mode S1S2 waves. The conditions and properties of on‐vertical‐axis triplication are defined in terms of kinematic parameters. The results are illustrated in four homogeneous orthorhombic models and one multilayered orthorhombic model with no variation in azimuthal orientation for all the layers.

A new parameterization for waveform inversion in acoustic orthorhombic media

Orthorhombic anisotropic model inversion is extra challenging because of the multiple parameter nature of the inversion problem. The high number of parameters required to describe the medium exerts considerable trade-off and additional nonlinearity to a full-waveform inversion (FWI) application. Choosing a suitable set of parameters to describe the model and designing an effective inversion strategy can help in mitigating this problem. Using the Born approximation, which is the central ingredient of the FWI update process, we have derived radiation patterns for the different acoustic orthorhombic parameterizations. Analyzing the angular dependence of scattering (radiation patterns) of the parameters of different parameterizations starting with the often used Thomsen-Tsvankin parameterization, we have assessed the potential trade-off between the parameters and the resolution in describing the data and inverting for the parameters. The analysis led us to introduce new parameters [Formula: see text], [Formula: see text], and [Formula: see text], which have azimuthally dependent radiation patterns, but keep the scattering potential of the transversely isotropic parameters stationary with azimuth (azimuth independent). The novel parameters [Formula: see text], [Formula: see text], and [Formula: see text] are dimensionless and represent a measure of deviation between the vertical planes in orthorhombic anisotropy. Therefore, these deviation parameters offer a new parameterization style for an acoustic orthorhombic medium described by six parameters: three vertical transversely isotropic (VTI) parameters, two deviation parameters, and one parameter describing the anisotropy in the horizontal symmetry plane. The main feature of any parameterization based on the deviation parameters, is the azimuthal independency of the modeled data with respect to the VTI parameters, which allowed us to propose practical inversion strategies based on our experience with the VTI parameters. This feature of the new parameterization style holds for even the long-wavelength components of the model constrained by traveltimes.

Neutron diffraction study of (KxNa1−x)NbO3-based ceramics from low to high temperatures

A neutron diffraction study of (KxNa1−x)NbO3-based ceramics has been carried out from 5 K to high temperatures well above the Curie temperature. The diffraction data were analysed using Rietveld refinement. For pure KNN samples, especially at the Na-rich side of the phase diagram, the low-temperature structure of simple rhombohedral symmetry changes to a highly complex monoclinic structure at a higher temperature. Chemical analysis on the samples showed good agreement of the expected and actual compositions. Trigonal, monoclinic, orthorhombic, tetragonal and cubic phase models as well as two-phase mixtures are observed depending on the temperature of measurement. Space groupsR3c,P1m1,P11m,Amm2,P4mm, Pm\overline 3m and their combinations are used to refine the trigonal, monoclinic, orthorhombic, tetragonal, cubic and mixed phases, respectively. For the (K0.48Na0.48Li0.04)(Nb0.86Ta0.1Sb0.04)O3sample at temperatures between 5 and 300 K, the monoclinicP11mspace group gives the best refinement fit. For the (K0.17Na0.83)NbO3sample, a two-phase refinement using the trigonalR3cand monoclinicPmspace groups gave the best fit at 300 K, while at 5–150 K the trigonalR3cspace group gives the best fit. The understanding of the structure of these lead-free ceramics will help in the optimization of their piezoelectric properties.

Upscaling in orthorhombic media: Behavior of elastic parameters in heterogeneous fractured earth

A stack of horizontal homogeneous elastic arbitrary anisotropic layers in welded contact in the long-wavelength limit is equivalent to an elastic anisotropic homogeneous medium. Such a medium is characterized by an effective average description adhering to previously derived closed-form formalism. We have used this formalism to study three different inhomogeneous orthorhombic (ORT) models that could represent real geologic scenarios. We have determined that a stack of thin orthorhombic layers with arbitrary azimuths of vertical symmetry planes can be approximated by an effective orthorhombic medium. The most suitable approach for this is to minimize the misfit between the effective anisotropic medium, monoclinic in that case, and the desirable orthorhombic medium. The second model is an interbedding of VTI (transversely isotropic with a vertical symmetry axis) layers with the same layers containing vertical fractures (shales are intrinsically anisotropic and often fractured). We have derived a weak-anisotropy approximation for important P-wave processing parameters as a function of the relative amount of the fractured lithology. To accurately characterize fractures, inversion for the fracture parameters should use a priori information on the relative amount of a fractured medium. However, we have determined that the cracks’ fluid saturation can be estimated without prior knowledge of the relative amount of the fractured layer. We have used field well-log data to demonstrate how fractures can be included in the interval of interest during upscaling. Finally, the third model that we have considered is a useful representation of tilted orthorhombic medium in the case of two-way propagation of seismic waves through it. We have derived a weak anisotropy approximation for traveltime parameters of the reflected P-wave that propagates through a stack of thin beds of tilted orthorhombic symmetry. The tilt of symmetry planes in an orthorhombic medium significantly affects the kinematics of the reflected P-wave and should be properly accounted for to avoid mispositioning of geologic structures in seismic imaging.

P-wave slowness surface approximation for tilted orthorhombic media

We have developed an analytic and approximate formula for vertical slowness components of down- and upgoing plane P waves in 3D tilted orthorhombic media. A perturbation method and Shanks transform were used to derive the approximation for slowness surface of P waves in tilted orthorhombic media. We have also quantitatively described the validity range of the radial horizontal slowness components for the proposed formula. The validity range was affected by the strength of the anellipticity of an orthorhombic medium: the stronger the anellipticity, the smaller the validity range. Numerical examples determined that the proposed formula is accurate for tilted orthorhombic media with weak to strong anellipticity. We have also evaluated in detail the application of the proposed formula on calculating the P-wave intercept time in the [Formula: see text] domain for horizontally layered, tilted orthorhombic models. Our formula is useful for ray tracing, phase-shift migration, and [Formula: see text] domain intercept time approximation for tilted orthorhombic media.

Simulating propagation of decoupled elastic waves using low-rank approximate mixed-domain integral operators for anisotropic media

In elastic imaging, the extrapolated vector fields are decoupled into pure wave modes, such that the imaging condition produces interpretable images. Conventionally, mode decoupling in anisotropic media is costly because the operators involved are dependent on the velocity, and thus they are not stationary. We have developed an efficient pseudospectral approach to directly extrapolate the decoupled elastic waves using low-rank approximate mixed-domain integral operators on the basis of the elastic displacement wave equation. We have applied [Formula: see text]-space adjustment to the pseudospectral solution to allow for a relatively large extrapolation time step. The low-rank approximation was, thus, applied to the spectral operators that simultaneously extrapolate and decompose the elastic wavefields. Synthetic examples on transversely isotropic and orthorhombic models showed that our approach has the potential to efficiently and accurately simulate the propagations of the decoupled quasi-P and quasi-S modes as well as the total wavefields for elastic wave modeling, imaging, and inversion.

Experimental and numerical investigation of the thermal performance of a novel sintered-wick heat pipe

Thermal performance of a novel sintered wick heat pipe was investigated in this study. Two types of sintered wick heat pipes were fabricated and tested at different filling ratios of water, and their thermal resistances in different modes were compared. In the first type, wick was sintered annularly (conventional type), and in the other one (novel type of sintered wick) it was sintered only in one third of cross-section. Results showed that dry-out occurs at higher heat input by an increase in the filling ratio. Moreover, the best filling ratio is 20% for both heat pipes. Thermal resistances of the partly sintered wick heat pipe are approximately 28%, 17% and 47% lower than those of the annularly sintered one at 20% filling ratio in the vertical, horizontal and reverse-vertical modes, respectively. Gravity has a slight effect on partly sintered wick heat pipe performance in the horizontal mode. This novel type of sintered wick heat pipe has simpler structure, and its manufacturing is more affordable compared with the annularly sintered wick. Hence, the use of this type of novel heat pipe (partly sintered wick) rather than the conventional type (annularly sintered one) is recommended in most applications, especially in space conditions where the gravity is negligible. In addition, experimental results were compared with numerical ones, and it was shown that the Florez orthorhombic and Alexander models can provide reasonable predictions for the effective thermal conductivity of water-saturated sintered powder-metal wicks.

Azimuthally dependent kinematic properties of orthorhombic media

An orthorhombic velocity model covers anisotropy and vertical/horizontal heterogeneity in the uniform 3D framework. It is important to define the azimuthal dependence of kinematic parameters of this model in the phase and group domains. I have derived equations for azimuthal dependent anellipticity in orthorhombic media and a simple equation to link the group and phase azimuths at zero horizontal slowness projection (zero offset). I found that the orthorhombic media cannot be treated as azimuthally dependent transversely isotropic media. To define the effective kinematic properties from a multilayered orthorhombic medium is a nontrivial task, and I have developed an approximate least-squares solution for that problem. Numerical examples with the orthorhombic model and its special cases (a transversely isotropic model with vertical and horizontal symmetry axes and an elliptical isotropic model) were used to illustrate the effect of the individual orientation of symmetry planes on the effective kinematic properties.

Molecular Dynamics Simulation of Oxygen Ion Conduction in Orthorhombic Perovskite Ba-Doped LaInO3 Using Cubic and Orthorhombic Model.

The oxygen ion conduction behavior of an orthorhombic perovskite oxide Ba-doped LaInO3 was evaluated by molecular dynamics simulation. Comparative calculations for cubic and the orthorhombic models show that the orthorhombic model was in better agreement with the experimental results with respect to both ionic conductivity and activation energy. The results of the radial distribution function for O-O pair provided and explanation for why the ionic conductivity of cubic perovskite is higher than that of orthorhombic perovskite. The orthorhombic model has a higher peak intensity of O-O pair than the cubic model, it means that the orthorhombic model has a smaller number of oxygen-vacancy pairs in the lattice structure. Consequently, the ionic conductivity of orthorhombic perovskite is lower than that of cubic perovskite due to less conduction path.

Two-dimensional periodic arrangement of one-dimensional polymerized C60 evidenced by high-resolution cryo-transmission electron microscopy

We have observed an electron-beam (EB) irradiated C60 single crystal (SC) film, using high-resolution cryo-transmission electron microscopy (cryo-HRTEM) measured at a low temperature of 80 K, and obtained the lattice images of one-dimensional (1D) C60 polymer. The intermolecular distance between adjacent C60 molecules of the 1D polymer was obtained to be 0.92 nm, which is in good agreement with the previous result of 0.93 nm estimated from the electron diffraction (ED) patterns obtained at room temperature. Although the previous work indicated that the 1D C60 polymer is explained by a mixture of body-centered orthorhombic (BCO) and hexagonal-closed pack (HCP) models, the present cryo-HRTEM images and their fast Fourier-transformed patterns clearly exclude the BCO model, and demonstrate that the 1D C60 polymer structure can be explained only by the HCP-monoclinic model.

Family of Ti5O5 superstructures

Orthorhombic (space group Pmmm) and tetragonal (space group P422) models are proposed for the Ti5O5 (Ti5■O5□) family of nonstoichiometric titanium monoxide Ti x O z with double imperfection. The disorder-order (Ti x O z (space group Fm $$\overline 3 $$ m)-Ti5O5 (space group Pmmm) and Ti x O z (space group (Fm $$\overline 3 $$ m)-Ti5O5 (space group P422)) transition channels include five superstructure vectors, which belong to stars {k 10}, {k 4(1)}, and {k 4(2)} or {k 10}, {k 6(1)}, and {k 6(2)}, respectively. The functions of distribution of Ti and O atoms over the sites of the model Ti5O5 superstructures are calculated. The well-known cubic and monoclinic Ti5O5 superstructures are compared with the model orthorhombic and tetragonal superstructures. It is shown that, as temperature decreases, two physically allowable transformation sequences related to the Ti5O5 superstructure formation are possible in nonstoichiometric titanium monoxide Ti x O z .

The pressure effect on crystal structure of complex ferroelectrics Ba4Sm2Fe2Nb8O30 and Ba4Gd2Fe2Nb8O30

Abstract The crystal structure of complex oxide ferroelectrics Ba4Sm2Fe2Nb8O30 and Ba4Gd2Fe2Nb8O30 have been studied by means of energy-dispersive X-ray diffraction method at high pressure up to 4 GPa at room temperature. At ambient pressure, the crystal structure of both compounds is described by the tetragonal symmetry with space group P4/mbm. At pressure P∼1.8 GPa in Ba4Sm2Fe2Nb8O30 and at P∼2.2 GPa in Ba4Gd2Fe2Nb8O30 the structure phase transition have been observed. The orthorhombic structure model with Pba2 space group for high-pressure phase of those compounds have been proposed. The baric dependences of lattice parameters and unit cell volumes of Ba4Sm2Fe2Nb8O30 and Ba4Gd2Fe2Nb8O30 compounds have been obtained.

Effective orthorhombic anisotropic models for wavefield extrapolation

Wavefield extrapolation in orthorhombic anisotropic media incorporates complicated but realistic models, to reproduce wave propagation phenomena in the Earth's subsurface. Compared with the representations used for simpler symmetries, such as transversely isotropic or isotropic, orthorhombic models require an extended and more elaborated formulation that also involves more expensive computational processes. The acoustic assumption yields more efficient description of the orthorhombic wave equation that also provides a simplified representation for the orthorhombic dispersion relation. However, such representation is hampered by the sixth-order nature of the acoustic wave equation, as it also encompasses the contribution of shear waves. To reduce the computational cost of wavefield extrapolation in such media, I generate effective isotropic inhomogeneous models that are capable of reproducing the first-arrival kinematic aspects of the orthorhombic wavefield. First, in order to compute traveltimes in vertical orthorhombic media, I develop a stable, efficient and accurate algorithm based on the fast marching method. The derived orthorhombic acoustic dispersion relation, unlike the isotropic or transversely isotropic one, is represented by a sixth order polynomial equation that includes the fastest solution corresponding to outgoing P-waves in acoustic media. The effective velocity models are then computed by evaluating the traveltime gradients of the orthorhombic traveltime solution, which is done by explicitly solving the isotropic eikonal equation for the corresponding inhomogeneous isotropic velocity field. The inverted effective velocity fields are source dependent and produce equivalent first-arrival kinematic descriptions of wave propagation in orthorhombic media. I extrapolate wavefields in these isotropic effective velocity models using the more efficient isotropic operator, and the results compare well, especially kinematically, with those obtained from the more expensive anisotropic extrapolator.%%%%Wavefield extrapolation in orthorhombic anisotropic media incorporates complicated but realistic models, to reproduce wave propagation phenomena in the Earth's subsurface. Compared with the representations used for simpler symmetries, such as transversely isotropic or isotropic, orthorhombic models require an extended and more elaborated formulation that also involves more expensive computational processes. The acoustic assumption yields more efficient description of the orthorhombic wave equation that also provides a simplified representation for the orthorhombic dispersion relation. However, such representation is hampered by the sixth-order nature of the acoustic wave equation, as it also encompasses the contribution of shear waves. To reduce the computational cost of wavefield extrapolation in such media, I generate effective isotropic inhomogeneous models that are capable of reproducing the first-arrival kinematic aspects of the orthorhombic wavefield. First, in order to compute traveltimes in vertical…

An Orthorhombic Lattice Boltzmann Model for Pore-Scale Simulation of Fluid Flow in Porous Media

One application of the lattice Boltzmann equation (LBE) models is in combination with tomography to simulate pore-scale flow and transport processes in porous media. Most LBE models in the literature are based on cubic lattice, and if the voxels in a tomography image are not cubic or cannot be divided into cubes due to computational limitations, these models will lose most of their advantages. How to deal with such images is, hence, an interest in use of the LBE model to simulate pore-scale processes. In this paper, we present an orthorhombic LBE model based on the single-relaxation time approach with the relaxation parameter varying with lattice directions. The equilibrium distribution functions in the standard LBE model were modified to correct the anisotropy induced by the non-cubic lattice, and the calculations of the fluid density and momentum were also redefined in order to maintain the conservation of mass and momentum during the collision. We tested the model against analytical solution for fluid flow in a tube, and against the standard cubic-based LBE model for fluid flow in a duct with an island inside. The model was then applied to simulate fluid flow in a 3D image in attempts to analyse the errors if the voxels in the image are not cubic but are assumed to be cubic.

Crystal structure of Nd2NiO4.08

A neodymium nickel oxide of Nd2NiO4.08 has a very small orthorhombic strain at room temperature, 2(b − a) / (b + a) = 0.0014. An orthorhombic Pccn structure model of Nd2NiO4.08 is proposed for explaining the reason for the existence of the very small orthorhombic strain of Nd2NiO4.08. The orthorhombic Pccn structure model is drawn from the Fmmm structure model and the tetragonal P42/ncm structure model. Results of the Rietveld analyses using the Fmmm, the P42/ncm, and the Pccn structure models are compared. The Fmmm structure model can explain the reason for the existence of orthorhombic strain but cannot explain the presence of the forbidden reflections of the face-centered lattice in the X-ray powder diffraction pattern. On the contrary, the P42/ncm structure model can explain the presence of the forbidden reflections of the face-centered lattice in the diffraction pattern but cannot explain the presence of orthorhombic strain. The Pccn structure model can explain both the presence of the forbidden reflections of the face-centered lattice in the diffraction pattern and the presence of orthorhombic strain.

Estimation of elastic stiffness coefficients of an orthorhombic physical model using group velocity analysis on transmission data

We acquired 3C ultrasonic transmission seismograms, measured group velocities associated with the three quasi-body wave types, and determined density-normalized stiffness coefficients ([Formula: see text]) over an orthorhombic physical (laboratory) model. The estimation of [Formula: see text] is based on an approximate relationship between the nine orthorhombic [Formula: see text] and group velocities. Estimation of the anisotropic [Formula: see text] is usually done using phase velocities with well-known formulas expressing their theoretical dependence on [Formula: see text]. However, on time-domain seismograms, arrivals are observed traveling with group velocities. Group velocity measurements are found to be straightforward, reasonably accurate, and independent of the size of the transducers used. In contrast, the accuracy of phase velocities derived from the [Formula: see text] transform analysis was found to be very sensitive to small differences in picked arrival times and to transducer size. Theoretical phase and group velocities, calculated in a forward manner from the [Formula: see text] estimates, agreed with the originally measured phase and group velocities, respectively. This agreement confirms that it is valid to use easily measured group velocities with their approximate theoretical relationship to the [Formula: see text] to determine the full stiffness matrix. Compared to the phase-velocity procedure, the technique involving group velocities is much less prone to error due to time-picking uncertainties, and therefore is more suitable for analyzing physical model seismic data.

Tilted orthorhombic velocity model building and imaging of Zamzama gas field with full-azimuth land data

Many geologic plays show indications of azimuthal anisotropy caused by azimuthal stress differences and fractures, which need to be accounted for during velocity model building and depth imaging. To fully observe azimuthal anisotropy, multiple narrow-azimuth surveys or wide-azimuth or full-azimuth surveys are required. Modern land surveys usually are shot with full-azimuth coverage, which enables one to address azimuthal velocity changes if present. A reprocessing and imaging project has been conducted of a full-azimuth land survey over Pakistani Zamzama gas field where azimuthal anisotropy is observed. The major geologic structure in this field is a large anticline split by a prominent fault zone trending north-south. The prestack depth-imaging workflow first exhausted tilted-transverse-isotropy (TTI) assumptions and then attempted tilted orthorhombic velocity model building and imaging. The processing flow, velocity model-building methodology, and successful results of final imaging with tilted orthorhombic anisotropic assumptions can be demonstrated. Success is measured with seismic to well-tie analysis.

Structure analysis of a precipitate phase in an Ni-rich high-temperature NiTiHf shape memory alloy

Thermal aging of the high-temperature shape memory alloy 50.3Ni–29.7Ti–20Hf (at.%) introduces a novel precipitate phase that plays an important role in improving shape memory properties. The precipitate phase was investigated by conventional electron diffraction, high-resolution scanning transmission electron microscopy (STEM) and three-dimensional atom probe tomography. An unrelaxed orthorhombic atomic structural model is proposed based on these observations. This model was subsequently relaxed by ab initio calculations. As a result of the relaxation, atom shuffle displacements occur, which in turn yields improved agreement with the STEM images. The relaxed structure, which is termed the “H phase”, has also been verified to be thermodynamically stable at 0K.