On inhomogeneity parameters for Backus average

In this paper, we discuss five parameters that indicate the inhomogeneity of a stack of parallel isotropic layers. We show that, in certain situations, they provide further insight into the intrinsic inhomogeneity of a Backus medium, as compared to the Thomsen parameters. Additionally, we show that the Backus average of isotropic layers is isotropic if and only if $\gamma=0$. This is in contrast to parameters $\delta$ and $\epsilon$, whose zero values do not imply isotropy.Comment: 11 pages, 4 figures, 4 table

On relationship between trigonal and cubic symmetry classes of an elasticity tensor

In the literature, there is an ambiguity in defining the relationship between trigonal and cubic symmetry classes of an elasticity tensor. We discuss the issue by examining the eigensystems and symmetry groups of trigonal and cubic tensors. Additionally, we present numerical examples indicating that the sole verification of the eigenvalues can lead to confusion in the identification of the elastic symmetry.Comment: 11 pages, 1 figur

On conditions for long-wave equivalent medium to be isotropic and on analysis of parameters indicating anisotropy of equivalent TI medium

In this paper, we consider a long-wave equivalent medium to a finely parallel-layered inhomogeneous medium, obtained using the Backus average. Following the work of Postma and Backus, we show explicitly the derivations of the conditions to obtain the equivalent isotropic medium. We demonstrate that there cannot exist a transversely isotropic (TI) equivalent medium with the coefficients $c^{\overline{\rm TI}}_{1212} \neq c^{\overline{\rm TI}}_{2323}$, $c^{\overline{\rm TI}}_{1111} = c^{\overline{\rm TI}}_{3333}$ and $c^{\overline{\rm TI}}_{1122} = c^{\overline{\rm TI}}_{1133}$. Moreover, we consider a new parameter, $\varphi$, indicating the anisotropy of the equivalent medium, and we show its range and properties. Subsequently, we compare $\varphi$ to the Thomsen parameters, emphasizing its usefulness as a supportive parameter showing the anisotropy of the equivalent medium or as an alternative parameter to $\delta$. We argue with certain Berryman et al. considerations regarding the properties of the anisotropy parameters $\epsilon$ and $\delta$. Additionally, we show an alternative way---to the one mentioned by Berryman et al.---of indicating changing fluid content in layered Earth.Comment: 10 page

On constraints imposed on a transversely isotropic elasticity tensor

We discuss several physical constraints imposed on elasticity parameters of a transversely isotropic (TI) tensor. There are three types of restrictions we investigate; a fundamental one of stability conditions, and two additional ones, commonly considered in seismology. The first commonly considered restriction comes from an assumption of a wave with a greater speed in the horizontal than vertical direction. The second constitute the assumption that quasi-P wave is faster than quasi-S waves. We show several numerical examples to examine how these restrictions affect a TI tensor with known values of certain elasticity constants that could be acquired from the vertical or horizontal measurements.Comment: 6 pages, 4 figure

Uniform linear bound in Chevalley's lemma

We obtain a uniform linear bound for the Chevalley function at a point in the source of an analytic mapping that is regular in the sense of Gabrielov. There is a version of Chevalley's lemma also along a fibre, or at a point of the image of a proper analytic mapping. We get a uniform linear bound for the Chevalley function for a closed Nash (or formally Nash) subanalytic set.Comment: 12 page

On problematic case of product approximation in Backus average

Elastic anisotropy might be a combined effect of the intrinsic anisotropy and the anisotropy induced by thin-layering. The Backus average, a useful mathematical tool, allows us to describe such an effect quantitatively. The results are meaningful only if the underlying physical assumptions are obeyed, such as static equilibrium of the material. We focus on the only mathematical assumption of the Backus average, namely, product approximation. It states that the average of the product of a varying function with nearly-constant function is approximately equal to the product of the averages of those functions. We discuss particular, problematic case for which the aforementioned assumption is inaccurate. Further, we focus on the seismological context. We examine numerically if the inaccuracy affects the wave propagation in a homogenous medium -- obtained using the Backus average -- equivalent to thin layers. We take into consideration various material symmetries, including orthotropic, cubic, and others. We show that the problematic case of product approximation is strictly related to the negative Poisson's ratio of constituent layers. Therefore, we discuss the laboratory and well-log cases in which such a ratio has been noticed. Upon thorough literature review, it occurs that examples of so-called auxetic materials (media that have negative Poisson's ratio) are not extremely rare exceptions as thought previously. The investigation and derivation of Poisson's ratio for materials exhibiting symmetry classes up to monoclinic become a significant part of this paper. Except for the main objectives, we also show that the averaging of cubic layers results in an equivalent medium with tetragonal (not cubic) symmetry. Additionally, we present concise formulations of stability conditions for low symmetry classes, such as trigonal, orthotropic, and monoclinic.Comment: 35 pages, 5 figures, 4 table

Effective elasticity of a medium with many parallel fractures

We consider an alternative way of obtaining the effective elastic properties of a cracked medium. Similarly, to the popular linear-slip model, we assume flat, parallel fractures, and long wavelengths. However, we do not treat fractures as weakness planes of displacement discontinuity. In contrast to the classical models, we represent fractures by a thin layer embedded in the background medium. In other words, we follow the Schoenberg-Douma matrix formalism for Backus averaging, but we relax their assumptions of infinite weakness and marginal thickness of a layer so that it does not correspond to the linear-slip plane. To represent the properties of a fracture, we need a fourth order elasticity tensor and a thickness parameter. The effective tensor becomes more complicated, but it may describe a higher concentration of parallel cracks more accurately. Apart from the derivations of the effective elasticity tensors, we perform numerical experiments in which we compare the performance of our approach with a linear-slip model in the context of highly fractured media. Our model becomes pertinent if filled-in cracks occupy more than one percent of the effective medium.Comment: 23 pages, 3 figure

On anisotropy parameters and fluid detection in equivalent TI medium

We consider a long-wave transversely isotropic (TI) medium equivalent to a series of finely parallel-layered isotropic layers, obtained using the \citet{Backus} average. In such a TI equivalent medium, we verify the \citet{Berrymanetal} method of indicating fluids and the author's method \citep{Adamus}, using anisotropy parameter $\varphi$. Both methods are based on detecting variations of the Lam\'e parameter, $\lambda$, in a series of thin isotropic layers, and we treat these variations as potential change of the fluid content. To verify these methods, we use Monte Carlo (MC) simulations; for certain range of Lam\'e parameters $\lambda$ and $\mu$---relevant to particular type of rocks---we generate numerous combinations of these parameters in thin layers and, after the averaging process, we obtain their TI media counterparts. Subsequently, for each of the aforementioned media, we compute $\varphi$ and \citet{Thomsen} parameters $\epsilon$ and $\delta$. We exhibit $\varphi$, $\epsilon$ and $\delta$ in a form of cross-plots and distributions that are relevant to chosen range of $\lambda$ and $\mu$. We repeat that process for various ranges of Lam\'e parameters. Additionally, to support the MC simulations, we consider several numerical examples of growing $\lambda$, by using scale factors. As a result of the thorough analysis of the relations among $\varphi$, $\epsilon$ and $\delta$, we find eleven fluid detectors that compose a new fluid detection method. Based on these detectors, we show the quantified pattern of indicating change of the fluid content.Comment: 32 pages, 123 figure

PP-wave reflection coefficient for vertically cracked media: Single set of aligned cracks

The main goal of this paper is to analyse the influence of cracks on the azimuthal variations of amplitude. We restrict our investigation to a single set of vertical, circular, and flat cavities aligned along a horizontal axis. Such cracks are embedded in either isotropic surroundings or transversely isotropic background with a vertical symmetry axis. We employ the effective medium theory to obtain either transversely-isotropic material with a horizontal symmetry axis or an orthotropic medium, respectively. To consider the amplitudes, we focus on a Vavrycuk-Psencik approximation of the PP-wave reflection coefficient. We assume that cracks are situated in one of the halfspaces being in welded contact. Azimuthal variations depend on the background stiffnesses, incidence angle, and crack density parameter. Upon analytical analysis, we indicate which factors (such as background's saturation) cause the reflection coefficient to have maximum absolute value in the direction parallel or perpendicular to cracks. We discuss the irregular cases, where such extreme values appear in the other than the aforementioned directions. Due to the support of numerical simulations, we propose graphic patterns of two-dimensional amplitude variations with azimuth. The patterns consist of a series of shapes that change with the increasing value of the crack density parameter. Schemes appear to differ depending on the incidence angle and the saturation. Finally, we extract these shapes that are characteristic of gas-bearing rocks. They may be treated as gas indicators. We support the findings and verify our patterns using real values of stiffnesses extracted from the sedimentary rocks' samples.Comment: 41 pages, 9 figures, 8 tables, Matlab cod

Realistic Simulation of the MAPS Response

The MAPS technology is considered as a possible choice for the ILC Vertex Detector. Test results of MIMOSA-5 sensors indicate that the pixel multiplicity and the single point resolution depend significantly on the incident particle angle. We propose a simple model describing charge distribution in the detector, which can be used for detailed simulation of the Vertex Detector response. Good agreement with beam test data is obtained. A new class for Track Detailed Simulation (TDS) has been developed and implemented in the EUTelescope software framework.Comment: 3 pages, 3 figures, to appear in the proceedings of International Linear Collider Workshop (LCWS08 and ILC08), Chicago, Illinois, 16-20 Nov 200