1,045 research outputs found
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 . This is in
contrast to parameters and , 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 ,
and
. Moreover, we
consider a new parameter, , indicating the anisotropy of the
equivalent medium, and we show its range and properties. Subsequently, we
compare to the Thomsen parameters, emphasizing its usefulness as a
supportive parameter showing the anisotropy of the equivalent medium or as an
alternative parameter to . We argue with certain Berryman et al.
considerations regarding the properties of the anisotropy parameters
and . 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 . Both methods are based on
detecting variations of the Lam\'e parameter, , 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 and ---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 and \citet{Thomsen} parameters and . We
exhibit , and in a form of cross-plots and
distributions that are relevant to chosen range of and . We
repeat that process for various ranges of Lam\'e parameters. Additionally, to
support the MC simulations, we consider several numerical examples of growing
, by using scale factors. As a result of the thorough analysis of the
relations among , and , 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
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