Traveltime approximations and parameter estimation for orthorhombic media

Building anisotropy models is necessary for seismic modeling and imaging. However, anisotropy estimation is challenging due to the trade-off between inhomogeneity and anisotropy. Luckily, we can estimate the anisotropy parameters if we relate them analytically to traveltimes. Using perturbation theory, we have developed traveltime approximations for orthorhombic media as explicit functions of the anellipticity parameters [Formula: see text], [Formula: see text], and [Formula: see text] in inhomogeneous background media. The parameter [Formula: see text] is related to Tsvankin-Thomsen notation and ensures easier computation of traveltimes in the background model. Specifically, our expansion assumes an inhomogeneous ellipsoidal anisotropic background model, which can be obtained from well information and stacking velocity analysis. We have used the Shanks transform to enhance the accuracy of the formulas. A homogeneous medium simplification of the traveltime expansion provided a nonhyperbolic moveout description of the traveltime that was more accurate than other derived approximations. Moreover, the formulation provides a computationally efficient tool to solve the eikonal equation of an orthorhombic medium, without any constraints on the background model complexity. Although, the expansion is based on the factorized representation of the perturbation parameters, smooth variations of these parameters (represented as effective values) provides reasonable results. Thus, this formulation provides a mechanism to estimate the three effective parameters [Formula: see text], [Formula: see text], and [Formula: see text]. We have derived Dix-type formulas for orthorhombic medium to convert the effective parameters to their interval values.

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A new traveltime approximation for TI media

Geophysics ◽

10.1190/geo2011-0158.1 ◽

2012 ◽

Vol 77 (4) ◽

pp. C37-C42 ◽

Cited By ~ 28

Author(s):

Alexey Stovas ◽

Tariq Alkhalifah

Keyword(s):

Power Series ◽

Eikonal Equation ◽

Transversely Isotropic ◽

Transversely Isotropic Medium ◽

Efficient Tool ◽

Trade Off ◽

Background Medium ◽

The Common ◽

Anisotropy Parameters ◽

Ti Media

In a transversely isotropic (TI) medium, the trade-off between inhomogeneity and anisotropy can dramatically reduce our capability to estimate anisotropy parameters. By expanding the TI eikonal equation in power series in terms of the aneliptic parameter [Formula: see text], we derive an efficient tool to estimate (scan) for [Formula: see text] in a generally inhomogeneous, elliptically anisotropic background medium. For a homogeneous-tilted transversely isotropic medium, we obtain an analytic nonhyperbolic moveout equation that is accurate for large offsets. In the common case where we do not have well information and it is necessary to resolve the vertical velocity, the background medium can be assumed isotropic, and the traveltime equations becomes simpler. In all cases, the accuracy of this new TI traveltime equation exceeds previously published formulations and demonstrates how [Formula: see text] is better resolved at small offsets when the tilt is large.

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A complex point-source solution of the acoustic eikonal equation for Gaussian beams in transversely isotropic media

Geophysics ◽

10.1190/geo2019-0264.1 ◽

2020 ◽

Vol 85 (3) ◽

pp. T191-T207

Author(s):

Xingguo Huang ◽

Hui Sun ◽

Zhangqing Sun ◽

Nuno Vieira da Silva

Keyword(s):

Point Source ◽

Eikonal Equation ◽

Taylor Series Expansion ◽

Transversely Isotropic ◽

Complex Point ◽

Transversely Isotropic Media ◽

Isotropic Media ◽

Anisotropy Parameters ◽

Complex Point Source ◽

Complex Eikonal

The complex traveltime solutions of the complex eikonal equation are the basis of inhomogeneous plane-wave seismic imaging methods, such as Gaussian beam migration and tomography. We have developed analytic approximations for the complex traveltime in transversely isotropic media with a titled symmetry axis, which is defined by a Taylor series expansion over the anisotropy parameters. The formulation for the complex traveltime is developed using perturbation theory and the complex point-source method. The real part of the complex traveltime describes the wavefront, and the imaginary part of the complex traveltime describes the decay of the amplitude of waves away from the central ray. We derive the linearized ordinary differential equations for the coefficients of the Taylor-series expansion using perturbation theory. The analytical solutions for the complex traveltimes are determined by applying the complex point-source method to the background traveltime formula and subsequently obtaining the coefficients from the linearized ordinary differential equations. We investigate the influence of the anisotropy parameters and of the initial width of the ray tube on the accuracy of the computed traveltimes. The analytical formulas, as outlined, are efficient methods for the computation of complex traveltimes from the complex eikonal equation. In addition, those formulas are also effective methods for benchmarking approximated solutions.

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On parameterization in monoclinic media with a horizontal symmetry plane

Geophysics ◽

10.1190/geo2020-0301.1 ◽

2021 ◽

Vol 86 (1) ◽

pp. C37-C49

Author(s):

Alexey Stovas

Keyword(s):

Symmetry Plane ◽

Transversely Isotropic ◽

Azimuth Angle ◽

Background Model ◽

Phase Domain ◽

Horizontal Symmetry ◽

Anisotropy Parameters ◽

Converted Waves ◽

Anisotropy Model ◽

Sine Functions

I have derived accurate anisotropy parameters for a monoclinic anisotropy model with a horizontal symmetry plane based on normal moveout (NMO) ellipses for P-, S1-, S2-, and converted waves. The NMO velocity ellipse is also defined for all types of converted waves. The parameters are defined in the phase domain and compared with existing approximate monoclinic anisotropy parameters. These parameters are evaluated for two benchmark models consisting of two nonorthogonal fracture sets embedded into a transversely isotropic medium with a vertical symmetry axis. The dependence of monoclinic parameters on the azimuth angle between the fracture sets is analyzed. Being linearized with respect to fracture weaknesses, the monoclinic anisotropy parameters can be decomposed into sine functions of double and quartic azimuth angle between the fracture sets with the weights given by the stiffness coefficients of the background model. The discrimination between the fracture parameters computed from a given set of monoclinic parameters is dependent on the background model and controlled by the azimuth angle between the fracture sets.

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Implementation of a Bottom-Hole Assembly Program

Volume 2: Structures, Safety and Reliability; Petroleum Technology Symposium ◽

10.1115/omae2007-29467 ◽

2007 ◽

Author(s):

Kenneth Bhalla ◽

Lixin Gong ◽

George McKown

Keyword(s):

Computationally Efficient ◽

Efficient Tool ◽

Shock Absorbers ◽

Bottom Hole ◽

Size Number ◽

Weight On Bit ◽

Bottom Hole Assembly ◽

Parametric Studies ◽

Dip Angle ◽

Drill Collar

A state of the art windows graphical user interface (GUI) program has been developed to predict and design the bottom-hole assembly (BHA) performance for drilling. The techniques and algorithms developed in the program are based upon those developed by Lubinski and Williamson. The BHA program facilitates in conducting parametric studies, and in making field decisions for optimal performance. The input parameters may include: formation class, dip angle, hole size, drill collar size, number of stabilizers, stabilizer spacing. The program takes into consideration bit-formation characteristics and interaction, drill collar sizes, square collars, shock absorbers, MWD tools, reamer tools, directional tools, rotary steerable systems etc. The output may consist of hole curvature (build up or drop rate), hole angle, weight on bit and is presented in drilling semantics. Additionally, the program can perform mechanical analyses and solve for the bending moments and reactions forces. Moreover, the program has the capability to predict the wellpath using a drill ahead algorithm. The program consists of a mathematical model which makes assumptions of 2-D, static, constant hole curvature resulting in a robust computationally efficient tool that produces rapid reliable results in the field.

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A Simplified 3D Model of Whole Heart Electrical Activity and 12-Lead ECG Generation

Computational and Mathematical Methods in Medicine ◽

10.1155/2013/134208 ◽

2013 ◽

Vol 2013 ◽

pp. 1-10 ◽

Cited By ~ 20

Author(s):

Siniša Sovilj ◽

Ratko Magjarević ◽

Nigel H. Lovell ◽

Socrates Dokos

Keyword(s):

Electrical Activity ◽

Three Dimensional ◽

Model Complexity ◽

Bidomain Model ◽

Computationally Efficient ◽

Lead System ◽

Heart Electrical Activity ◽

Cell Tissue ◽

Whole Heart ◽

Electrocardiogram Ecg

We present a computationally efficient three-dimensional bidomain model of torso-embedded whole heart electrical activity, with spontaneous initiation of activation in the sinoatrial node, incorporating a specialized conduction system with heterogeneous action potential morphologies throughout the heart. The simplified geometry incorporates the whole heart as a volume source, with heart cavities, lungs, and torso as passive volume conductors. We placed four surface electrodes at the limbs of the torso: , , and and six electrodes on the chest to simulate the Einthoven, Goldberger-augmented and precordial leads of a standard 12-lead system. By placing additional seven electrodes at the appropriate torso positions, we were also able to calculate the vectorcardiogram of the Frank lead system. Themodel was able to simulate realistic electrocardiogram (ECG) morphologies for the 12 standard leads, orthogonal , , and leads, as well as the vectorcardiogram under normal and pathological heart states. Thus, simplified and easy replicable 3D cardiac bidomain model offers a compromise between computational load and model complexity and can be used as an investigative tool to adjust cell, tissue, and whole heart properties, such as setting ischemic lesions or regions of myocardial infarction, to readily investigate their effects on whole ECG morphology.

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Scanning anisotropy parameters in complex media

Geophysics ◽

10.1190/1.3553015 ◽

2011 ◽

Vol 76 (2) ◽

pp. U13-U22 ◽

Cited By ~ 61

Author(s):

Tariq Alkhalifah

Keyword(s):

Anisotropic Medium ◽

Homogeneous Medium ◽

Transversely Isotropic ◽

Transient Behavior ◽

Complex Media ◽

Linear Partial Differential Equations ◽

Anisotropy Parameters ◽

Axis Of Symmetry ◽

Ti Media ◽

Taylor’S Series

Parameter estimation in an inhomogeneous anisotropic medium offers many challenges; chief among them is the trade-off between inhomogeneity and anisotropy. It is especially hard to estimate the anisotropy anellipticity parameter η in complex media. Using perturbation theory and Taylor’s series, I have expanded the solutions of the anisotropic eikonal equation for transversely isotropic (TI) media with a vertical symmetry axis (VTI) in terms of the independent parameter η from a generally inhomogeneous elliptically anisotropic medium background. This new VTI traveltime solution is based on a set of precomputed perturbations extracted from solving linear partial differential equations. The traveltimes obtained from these equations serve as the coefficients of a Taylor-type expansion of the total traveltime in terms of η. Shanks transform is used to predict the transient behavior of the expansion and improve its accuracy using fewer terms. A homogeneous medium simplification of the expansion provides classical nonhyperbolic moveout descriptions of the traveltime that are more accurate than other recently derived approximations. In addition, this formulation provides a tool to scan for anisotropic parameters in a generally inhomogeneous medium background. A Marmousi test demonstrates the accuracy of this approximation. For a tilted axis of symmetry, the equations are still applicable with a slightly more complicated framework because the vertical velocity and δ are not readily available from the data.

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Azimuthally dependent anisotropic velocity model update

Geophysics ◽

10.1190/geo2013-0178.1 ◽

2014 ◽

Vol 79 (2) ◽

pp. C27-C53 ◽

Cited By ~ 21

Author(s):

Zvi Koren ◽

Igor Ravve

Keyword(s):

High Resolution ◽

Velocity Field ◽

Phase Angle ◽

Velocity Model ◽

Background Model ◽

Physical Parameters ◽

Model Parameters ◽

Effective Parameters ◽

Velocity Models

We consider a case where a 3D depth migration has been performed in the local angle domain (LAD) using rich-azimuth seismic data (e.g., conventional land surveys). The subsurface geologic model is characterized by considerable azimuthally anisotropic velocity variations. The background velocity field used for the migration can consist of azimuthally independent, e.g., vertical transverse isotropy, and/or azimuthally dependent (e.g., orthorhombic), velocity layers. The resulting 3D full-azimuth reflection angle gathers generated by the LAD migration represent in situ high-resolution amplitude preserved reflectivities associated with opening angles between incident and reflected slowness vectors in the specular directions. Residual moveouts (RMOs) automatically picked on these 3D image gathers along major horizons can indicate considerable residual periodic azimuthal variations. This situation is typical in depth imaging applied to unconventional shale plays, where the background velocity model doesn’t yet account for the aligned stress/fracture systems that exist in some of the target layers. We use the azimuthally dependent, phase-angle RMOs to update the interval parameters of the background model, accounting for the azimuthal anisotropy effect. Until now, this problem was mainly treated in the unmigrated time-offset domain, which is limited in describing the actual in situ changes of the velocity field with azimuths. The subsurface full-azimuth phase-angle domain RMOs provide better physical parameters to analyze the in situ azimuthal variations of the anisotropic media. Our method is grounded in a newly derived generalized Dix-based theory, where locally the background and updated models are assumed to be 1D anisotropic velocity models. At each lateral location, the orthorhombic axis [Formula: see text] points in the vertical direction across all layers, but the azimuthal orientations of the orthorhombic layers change from layer to layer. An effective model for such a layered structure (background or updated) is represented by a single layer with a vertical time identical to that of the whole package, effective fast and slow normal moveout (NMO) velocities, and an effective azimuthal orientation of the slow NMO velocity. Our approach begins with computation of these effective parameters for the background model and conversion of the high-resolution RMOs into a dense set of updated, effective, azimuthally dependent NMO velocities, which are then converted into three effective parameters of the updated model. Next, we apply a generalized Dix-based inversion approach to estimate the local NMO parameters for each updated layer. Finally, we convert the local parameters into interval azimuthally varying anisotropic model parameters (e.g., TTI, orthorhombic, or tilted orthorhombic) within each layer. The 1D Dix-based approach presented in this work should not be considered an alternative to more accurate 3D global inversion approaches, such as global anisotropic tomography. However, the proposed method can be effectively used for moderately laterally varying models, and some of the principal physical rules derived for the 1D model can be further used to improve the formulation and geophysical constraints applied to 3D global inversion methods.

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An Extended Maximum Likelihood Inference of Geographic Range Evolution by Dispersal, Local Extinction and Cladogenesis

10.1101/038695 ◽

2016 ◽

Cited By ~ 6

Author(s):

Champak Beeravolu Reddy ◽

Fabien Condamine

Keyword(s):

Local Extinction ◽

Rapid Expansion ◽

Model Complexity ◽

Published Data ◽

Computationally Efficient ◽

Likelihood Methods ◽

Origin And Evolution ◽

Statistical Framework ◽

Biological Realism

The origin and evolution of species ranges remains a central focus of historical biogeography and the advent of likelihood methods based on phylogenies has revolutionized the way in which range evolution has been studied. A decade ago, the first elements of what turned out to be a popular inference approach of ancestral ranges based on the processes of Dispersal, local Extinction and Cladogenesis (DEC) was proposed. The success of the DEC model lies in its use of a flexible statistical framework known as a Continuous Time Markov Chain and since, several conceptual and computational improvements have been proposed using this as a baseline approach. In the spirit of the original version of DEC, we introduce DEC eXtended (DECX) by accounting for rapid expansion and local extinction as possible anagenetic events on the phylogeny but without increasing model complexity (i.e. in the number of free parameters). Classical vicariance as a cladogenetic event is also incorporated by making use of temporally flexible constraints on the connectivity between any two given areas in accordance with the movement of landmasses and dispersal opportunity over time. DECX is built upon a previous implementation in C/C++ and can analyze phylogenies on the order of several thousand tips in a few minutes. We test our model extensively on Pseudo Observed Datasets and on well-curated and recently published data from various island clades and a worldwide phylogeny of Amphibians (3309 species). We also propose the very first implementation of the DEC model that can specifically account for trees with fossil tips (i.e. non-ultrametric) using the phylogeny of palpimanoid spiders as a case study. In this paper, we argue in favour of the proposed improvements, which have the advantage of being computationally efficient while toeing the line of increased biological realism.

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Characterization of fluid transport properties of reservoirs using induced microseismicity

Geophysics ◽

10.1190/1.1451597 ◽

2002 ◽

Vol 67 (1) ◽

pp. 212-220 ◽

Cited By ~ 192

Author(s):

Serge A. Shapiro ◽

Elmar Rothert ◽

Volker Rath ◽

Jan Rindschwentner

Keyword(s):

Large Scale ◽

Eikonal Equation ◽

Fluid Transport ◽

Heterogeneous Media ◽

Reservoir Characterization ◽

Homogeneous Medium ◽

Front Propagation ◽

Characteristic Size ◽

Permeability Tensor ◽

Heterogeneous Rock

We systematically describe an approach to estimate the large‐scale permeability of reservoirs using seismic emission (microseismicity) induced by fluid injection. We call this approach seismicity‐based reservoir characterization (SBRC). A simple variant of the approach is based on the hypothesis that the triggering front of hydraulically‐induced microseismicity propagates like a diffusive process (pore pressure relaxation) in an effective homogeneous anisotropic poroelastic fluid‐saturated medium. The permeability tensor of this effective medium is the permeability tensor upscaled to the characteristic size of the seismically active heterogeneous rock volume. We show that in a homogeneous medium the surface of the seismicity triggering front has the same form as the group‐velocity surface of thelow‐frequency anisotropic, second‐type Biots wave describing kinematic aspects of triggering‐front propagation in a way similar to the eikonal equation for seismic wavefronts. In the case of isotropic heterogeneous media, the inversion for the hydraulic properties of rocks follows from a direct application of this equation. In the case of an anisotropic heterogeneous medium, only the magnitude of a global effective permeability tensor can be mapped in a 3‐D spatial domain. We demonstrate the method on several field examples and also test the eikonal equation‐based inversion.

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Mapping moveout approximations in TI media

Geophysics ◽

10.1190/geo2013-0039.1 ◽

2014 ◽

Vol 79 (1) ◽

pp. C19-C26 ◽

Cited By ~ 7

Author(s):

Alexey Stovas ◽

Tariq Alkhalifah

Keyword(s):

Transversely Isotropic ◽

Vertical Axis ◽

Accurate Approximation ◽

Seismic Modeling ◽

Effective Parameters ◽

Transversely Isotropic Media ◽

Vertical Heterogeneity ◽

Isotropic Media ◽

Axis Of Symmetry ◽

Ti Media

Moveout approximations play a very important role in seismic modeling, inversion, and scanning for parameters in complex media. We developed a scheme to map one-way moveout approximations for transversely isotropic media with a vertical axis of symmetry (VTI), which is widely available, to the tilted case (TTI) by introducing the effective tilt angle. As a result, we obtained highly accurate TTI moveout equations analogous with their VTI counterparts. Our analysis showed that the most accurate approximation is obtained from the mapping of generalized approximation. The new moveout approximations allow for, as the examples demonstrate, accurate description of moveout in the TTI case even for vertical heterogeneity. The proposed moveout approximations can be easily used for inversion in a layered TTI medium because the parameters of these approximations explicitly depend on corresponding effective parameters in a layered VTI medium.

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