Figure 1 - uploaded by Hiroyuki Nagahama
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Indicatrix of Finsler metric (solid curve; mZ2.1) and wavefront of ray velocity for peridotite (dashed curve) at a certain depth (the crystal b-axis is parallel to the direction of depth, z -axis), where aZ8.85 km s K1 and bZ7.60 km s K1 .
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The seismic ray theory in anisotropic inhomogeneous media is studied based on non-Euclidean geometry called Finsler geometry.
For a two-dimensional ray path, the seismic wavefront in anisotropic media can be geometrically expressed by Finslerian parameters.
By using elasticity constants of a real rock, the Finslerian parameters are estimated from a...
Context in source publication
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... the elastic constants (4.2) into the function of the phase velocity (3.13), the wavefront of peridotite is obtained from (4.3) and (4.4) (figure 1). In figure 1, the crystal b-axis is parallel to direction of depth, z -axis. Then, velocity of the slow qP-wave is directed to the olivine b-axis. ...
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... A geometric curve that can assume a concavity is the deformed superellipse, which has been employed to describe prostate shapes and for segmentation in TRUS images by Gong et al. [12] with very good results. Superellipses have found applications in several other fields [49][50][51][52][53][54][55][56][57][58]), and the following equation defines them: ...
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... In the past decade these curves and transformations have been used successfully to model of a wide variety of natural shapes, from plant leaves and tree rings to starfish and avian eggs [5]. Further applications of Finsler geometry in the natural sciences involving superellipses or Gielis curves, are found in forest ecology [6,7], seismic ray paths in anisotropic media [8], and the spreading of wildfires [9]. This has inspired various researchers to study various geometrical transformations, including inversions [10], whereby the inversion occurs with Lamé-Gielis curves as inversion circles. ...
While there are well-known synthetic methods in the literature to find the image of a point under circular inversion in l2−normed geometry (Euclidean geometry), there is no similar synthetic method in Minkowski geometry, also known as the geometry of finite-dimensional Banach spaces. In this study, we have succeeded in giving a synthetic construction for the circular inversion in l1−normed spaces, which is one of the most fundamental examples of Minkowski geometry. Moreover, this synthetic construction has been given using the Euclidean circle, independently of the l1−norm.
... This quantum interaction would become manifest in a dispersion relation which is close to that given by GR at low energies, but exhibits larger deviations at higher energies. Commonly studied example of MDRs include models derived from doubly or deformed special relativity (DSR) [9][10][11], more generally deformed relativistic kinematics [12][13][14][15], non-commutative spacetime geometries [16][17][18][19], string theory [20,21], loop quantum gravity [22][23][24][25], the Standard-Model Extension (SME) [26], as well as the propagation of fields through media [27][28][29][30][31][32][33][34]. ...
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... This ecological metric was first mentioned in [42] and it is worth noting that this is also directly related to allometric laws because of the exponential function in the metric. Inspired by Lamé curves, the same metric has been used in the study of seismic wave propagation [43], whereby the indicatrix expresses the anisotropy of the shape of the seismic wavefront. ...
Our scientific and technological worldviews are largely dominated by the concepts of entropy and complexity. Originating in 19th-century thermodynamics, the concept of entropy merged with information in the last century, leading to definitions of entropy and complexity by Kolmogorov, Shannon and others. In its simplest form, this worldview is an application of the normal rules of arithmetic. In this worldview, when tossing a coin, a million heads or tails in a row is theoretically possible, but impossible in practice and in real life. On this basis, the impossible (in the binary case, the outermost entries of Pascal's triangle xn and yn for large values of n) can be safely neglected, and one can concentrate fully on what is common and what conforms to the law of large numbers, in fields ranging from physics to sociology and everything in between. However, in recent decades it has been shown that what is most improbable tends to be the rule in nature. Indeed, if one combines the outermost entries xn and yn with the normal rules of arithmetic, either addition or multiplication, one obtains Lamé curves and power laws respectively. In this article, some of these correspondences are highlighted, leading to a double conclusion. First, Gabriel Lamé's geometric footprint in mathematics and the sciences is enormous. Second, conic sections are at the core once more. Whereas mathematics so far has been exclusively the language of patterns in the sciences, the door is opened for mathematics to also become the language of the individual. The probabilistic worldview and Lamé's footprint can be seen as dual methods. In this context, it is to be expected that the notions of information, complexity, simplicity and redundancy benefit from this different viewpoint.
... However, it is observed that the Earth exhibits more complicated and especially anisotropic behavior with respect to the sound speed [11,20,51]. In the anisotropic case, seismic rays propagate along geodesics of a Finsler norm [4,58] and Riemannian geometry is not enough to describe the most general types of anisotropies. The boundary rigidity problem is already a difficult non-linear inverse problem in the Riemannian case and anisotropies complicate things even more. ...
... Our main theorem implies boundary rigidity up to first order for a conformal family of spherically symmetric reversible Finsler norms satisfying the Herglotz condition. In terms of elasticity, if we have a conformal family of stiffness tensors (a family of factorized anisotropic inhomogeneous media [9,58]) such that the induced family of Finsler norms give the same distances between boundary points and satisfy the assumptions of Theorem 1.1, then the stiffness tensors are equal up to first order (see Sect. 5.3). ...
... One can also see from the geodesic equations that the Herglotz condition forbids circles being geodesics. An example of a spherically symmetric reversible Finsler norm satisfying (6) is the Finsler norm (see also [58]) ...
We show that the geodesic ray transform is injective on scalar functions on spherically symmetric reversible Finsler manifolds where the Finsler norm satisfies a Herglotz condition. We use angular Fourier series to reduce the injectivity problem to the invertibility of generalized Abel transforms and by Taylor expansions of geodesics we show that these Abel transforms are injective. Our result has applications in linearized boundary rigidity problem on Finsler manifolds and especially in linearized elastic travel time tomography.
... [3][4][5]12]) and Huygens' principle (see e.g. [8,14,15,20]), with realworld applications in seismic theory [1,21], sound waves [13] or wildfire modeling [7,11,16,19], among others. In the latter, it becomes a general version of Zermelo's navigation problem, which seeks the fastest trajectory between two points for a moving object in a medium which, in turn, may also be moving with respect to the observer (see e.g. ...
We study the variational problem of finding the fastest path between two points that belong to different anisotropic media, each with a prescribed speed profile and a common interface. The optimal curves are Finsler geodesics that are refracted — broken — as they pass through the interface, due to the discontinuity of their velocities. This “breaking” must satisfy a specific condition in terms of the Finsler metrics defined by the speed profiles, thus establishing the generalized Snell’s law. In the same way, optimal paths bouncing off the interface — without crossing into the second domain — provide the generalized law of reflection. The classical Snell’s and reflection laws are recovered in this setting when the velocities are isotropic. If one considers a wave that propagates in all directions from a given ignition point, the trajectories that globally minimize the traveltime generate the wavefront at each instant of time. We study in detail the global properties of such wavefronts in the Euclidean plane with anisotropic speed profiles. Like the individual rays, they break when they encounter the discontinuity interface. But they are also broken due to the formation of cut loci — stemming from the self-intersection of the wavefronts — which typically appear when they approach a high-speed profile domain from a low-speed profile.
... Roughly speaking, conic Zermelo pseudo-metrics model the anisotropic mechanics of a system submitted to the influence of some vector field, as is obviously the case for Zermelo's navigation problem. Other applications include the spread of forest fires [44,37], the propagation of sound in a wind [29], the path of a seismic ray [66], a model for a graphene sheet when an electric field is externally applied [21], the deviation of a charged particle through an electromagnetic field [18], analogue models [28,24,38], and general Finslerian spacetimes [11,40,34]. ...
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... Historically, the …rst to use geometry based on Finslerian line elements to describe physical interactions was Randers [168], in his search for a uni…ed geometric description of gravity and electromagnetism. Later on, Finsler geometry (and, to some extent, its generalizations, Lagrange and generalized Lagrange geometry, introduced by R. Miron and his collaborators, [146]- [148]) were used in physics in various other contexts, for example: the geometric description of …elds in media, [8], [47], [56], [106], [134], [158], [171], [222], the study of non-local Lorentz invariant extensions of fundamental physics, [185], [186], [178], [108]- [110], [36]- [38], and, …nally, the search for an improved description of gravity [81], [124], [145], [172], [163], [93], [94], which might explain dark matter or dark energy geometrically, [90], [112], [130], [138], [139], [204]. ...
The thesis was submitted to Transilvania University of Brasov.
Its full text can be found at:
https://unitbv.ro/documente/cercetare/doctorat-postdoctorat/abilitare/teze-de-abilitare/voicu-nicoleta/05-Voicu-teza-abilitare-ENG.pdf
... Since the 1990's we have many, many examples from biology and other fields in the natural sciences. GT are used in the geometry of wave propagation in seismic phenomena[83]. Meanwhile, GT have already won their place in the history of geometry and mathematics[84]. ...
A uniform description of natural shapes and phenomena is an important goal in science. Such description should check some basic principles, related to 1) the complexity of the model, 2) how well its fits real objects, phenomena and data, and 3) a direct connection with optimization principles and the calculus of variations. In this article, we present nine principles, three for each group, and we compare some models with a claim to universality. It is also shown that Gielis Transformations and power laws have a common origin in conic sections.
... [3,4,5,12]) and Huygens' principle (see e.g. [8,14,15,20]), with real-world applications in seismic theory [1,21], sound waves [13] or wildfire modeling [7,11,16,19], among others. In the latter, it becomes a general version of Zermelo's navigation problem, which seeks the fastest trajectory between two points for a moving object in a medium which, in turn, may also be moving with respect to the observer (see e.g. ...
We study the variational problem of finding the fastest path between two points that belong to different anisotropic media, each with a prescribed speed profile and a common interface. The optimal curves are Finsler geodesics that are refracted -- broken -- as they pass through the interface, due to the discontinuity of their velocities. This "breaking" must satisfy a specific condition in terms of the Finsler metrics defined by the speed profiles, thus establishing the generalized Snell's law. In the same way, optimal paths bouncing off the interface -- without crossing into the second domain -- provide the generalized law of reflection. The classical Snell's and reflection laws are recovered in this setting when the velocities are isotropic. If one considers a wave that propagates in all directions from a given ignition point, the trajectories that globally minimize the traveltime generate the wavefront at each instant of time. We study in detail the global properties of such wavefronts in the Euclidean plane with anisotropic speed profiles. Like the individual rays, they break when they encounter the discontinuity interface. But they are also broken due to the formation of cut loci -- stemming from the self-intersection of the wavefronts -- which typically appear when they approach a high-speed profile domain from a low-speed profile.
