# Michael A. SlawinskiMemorial University of Newfoundland · Department of Earth Sciences

Michael A. Slawinski

Ph.D. Theoretical seismology

## About

111

Publications

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Introduction

I strive to enhance quantitative descriptions of physical phenomena by formulating mathematical models of the Earth, for seismology, and of power, for bicycle science. In view of recent enhancements in data acquisition and computational techniques, there is more information in seismic data than can be extracted within standard theories. Similarly, in view of recent developments of bicycle power meters, there is information, whose interpretation requires further theoretical understanding.

Additional affiliations

January 2009 - present

September 2008 - January 2009

September 2005 - present

Education

September 1992 - March 1996

September 1986 - August 1988

September 1982 - August 1985

## Publications

Publications (111)

We model the power applied by a cyclist on a velodrome -- for individual time trials -- taking into account its straights, circular arcs, and connecting transition curves. The dissipative forces to be overcome by power are air resistance, rolling resistance, lateral friction and drivetrain resistance. Also, the power is used to increase the mechani...

We model the instantaneous power applied by a cyclist on a velodrome -- for individual pursuits and other individual time trials -- taking into account its straights, circular arcs, and connecting transition curves. The forces opposing the motion are air resistance, rolling resistance, lateral friction and drivetrain resistance. We examine the cons...

We present a strategy for selecting the values of model parameters by comparing walkaway Vertical Seismic Profiling data with a multilayered model in the context of Bayesian Information Criterion. We consider -wave traveltimes and assume elliptical polar velocity dependence. A model with different propagation speeds, depending on the angle of propa...

We combine power-meter measurements with GPS measurements to study the model that accounts for the use of power by a cyclist. The model takes into account the change in elevation and speed along with adverse effects of air, rolling and drivetrain resistance. The focus is on estimating the resistance coefficients using numerical optimization techniq...

We present a strategy for selecting the values of elasticity parameters by comparing walk-away vertical seismic profiling data with a multilayered model in the context of Bayesian Information Criterion. We consider $P$-wave traveltimes and assume elliptical velocity dependence. The Bayesian Information Criterion approach requires two steps of optim...

We prove the equivalence—under rotations of distinct terms—of different forms of a determinantal equation that appears in the studies of wave propagation in Hookean solids, in the context of the Christoffel equations. To do so, we prove a general proposition that is not limited to \({{\mathbb {R}}}^3\), nor is it limited to the elasticity tensor wi...

We model the instantaneous power on a velodrome for individual pursuits, taking into account its straights, circular arcs, and connecting transition curves. The forces opposing the motion are air resistance, rolling resistance, lateral friction and drivetrain resistance. We examine the constant-cadence and constant-power cases, and discuss their re...

For a constant power output, the mean ascent speed (VAM) increases monotonically with the slope. This property constitutes a physical background upon which various strategies for the VAM maximization can be examined in the context of the maximum sustainable power as a function of both the gear ratio and cadence.

For a moving bicycle, the power meters respond to the propulsion of the centre of mass of the bicycle-cyclist system. Hence, an accurate modelling of power measurements|on a velodrome|requires a distinction between the trajectory of the wheels and the trajectory of the centre of mass. We formulate and examine an individual-pursuit model that takes...

Power-meter measurements together with GPS measurements are used to study the model that accounts for the use of power by a cyclist. The focus is on estimating the coefficients of the air, rolling and drivetrain resistance, uncertainties of these estimates, as well as relations between them. Expressions used in the main text are derived in the appe...

Power-meter measurements are used to study a model that accounts for the use of power by a cyclist. The focus is on relations between rates of change of model quantities, such as power and speed, both in the context of partial derivatives, where other quantities are constant, and Lagrange multipliers, where other quantities vary to maintain the imp...

We show that, in general, the translational average over a spatial variable---discussed by Backus \cite{backus}, and referred to as the equivalent-medium average---and the rotational average over a symmetry group at a point---discussed by Gazis et al. \cite{gazis}, and referred to as the effective-medium average---do not commute. However, they do c...

The only restriction on the values of the elasticity parameters is the stability condition. Within this condition, we examine Christoffel equation for nondetached qP slowness surfaces in transversely isotropic media. If the qP slowness surface is detached, each root of the solubility condition corresponds to a distinct smooth wavefront. If the qP s...

As shown by Backus (J Geophys Res 67(11):4427–4440, 1962), the average of a stack of isotropic layers results in a transversely isotropic medium. Herein, we consider a stack of layers consisting of randomly oriented anisotropic elasticity tensors, which—one might reasonably expect—would result in an isotropic medium. However, we show—by means of a...

We illustrate properties of guided waves in terms of a superposition of body waves. In particular, we consider the Love and SH waves. Body-wave propagation at postcritical angles--required for a total reflection--results in the speed of the Love wave being between the speeds of the SH waves in the layer and in the halfspace. A finite wavelength of...

The only restriction on the values of the elasticity parameters is the stability condition. Within this condition, we examine Christoffel equation for nondetached $qP$ slowness surfaces in transversely isotropic media. If the $qP$ slowness surface is detached, each root of the solubility condition corresponds to a distinct smooth wavefront. If the...

The purpose of this paper is to prove the equivalence$-$under rotations of distinct terms$-$of different forms of a determinantal equation that appears in the studies of wave propagation in Hookean solids, in the context of the Christoffel equations. To do so, we prove a general proposition that is not limited to ${\mathbb R}^3$, nor is it limited...

We derive a general expression for the deformation-gradient tensor by invoking the standard definition of a gradient of a vector field in curvilinear coordinates. This expression shows the connection between the standard definition of a gradient of a vector field and the deformation gradient tensor in continuum mechanics. We illustrate its applicat...

In this paper, we examine the applicability of the approximation, fg‾≈f‾g‾\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\overline{f g}\approx \overline{f}\,\overline{g...

As shown by Backus (1962), the average of a stack of isotropic layers results in a transversely isotropic medium. Herein, we consider a stack of layers consisting of a randomly oriented anisotropic elasticity tensor, which-one might expect-would result in an isotropic medium. However, we show-by means of a fundamental symmetry of the Backus average...

We use the Pareto Joint Inversion, together with the Particle Swarm Optimization, to invert the Love and quasi-Rayleigh surface-wave speeds, obtained from dispersion curves, in order to infer the elasticity parameters, mass densities and layer thickness of the model for which these curves are generated. For both waves, we use the dispersion relatio...

In general, the Backus average of an inhomogeneous stack of isotropic layers is a transversely isotropic medium. Herein, we examine a relation between this inhomogeneity and the strength of resulting anisotropy, and show that, in general, they are proportional to one another. There is an important case, however, in which the Backus average of isotr...

We examine the Backus average of a stack of isotropic layers overlying an isotropic halfspace to examine its applicability for the quasi-Rayleigh and Love wave dispersion curves, both of which apply to the same model. We compare these curves to values obtained for the stack of discrete layers using the propagator matrix. The Backus average is appli...

It is common to assume that a Hookean solid is isotropic. For a generally anisotropic elasticity tensor, it is possible to obtain its isotropic counterparts. Such a counterpart is obtained in accordance with a given norm. Herein, we examine the effect of three norms: the Frobenius 36-component norm, the Frobenius 21-component norm and the operator...

This remarkable collaboration between a mathematical physicist and a science philosopher concerns foundational and conceptual issues in seismology. Their aim is to present mathematical, physical and philosophical topics in a clear and concise manner. They provide an extensive philosophical discussion of the methods of science and show how seismolog...

In this paper, following the Backus (1962) approach, we examine expressions
for elasticity parameters of a homogeneous generally anisotropic medium that is
long-wave-equivalent to a stack of thin generally anisotropic layers. These
expressions reduce to the results of Backus (1962) for the case of isotropic
and transversely isotropic layers. In ove...

We examine the sensitivity of the Love and the quasi-Rayleigh waves to model parameters. Both waves are guided waves that propagate in the same model of an elastic layer above an elastic halfspace. We study their dispersion curves without any simplifying assumptions, beyond the standard approach of elasticity theory in isotropic media. We examine t...

"The author dedicates this book to readers who are concerned with finding out the status of concepts, statements and hypotheses, and with clarifying and rearranging them in a logical order. It is thus not intended to teach tools and techniques of the trade, but to discuss the foundations on which seismology — and in a larger sense, the theory of wa...

Dalton and Slawinski (2016) show that, in general, the Backus (1962) average and the Gazis et al. (1963) average do not commute. Herein, we examine the extent of this noncommutativity. We illustrate numerically that the extent of noncommutativity is a function of the strength of anisotropy. The averages nearly commute in the case of weak anisotropy...

We examine two types of guided waves: the Love and the quasi-Rayleigh waves. Both waves propagate in the same model of an elastic isotropic layer above an elastic isotropic halfspace. From their dispersion relations, we calculate their speeds as functions of the elasticity parameters, mass densities, frequency and layer thickness. We examine the se...

An anisotropic elasticity tensor can be approximated by the closest tensor belonging to a higher symmetry class. The closeness of tensors depends on the choice of a criterion. We compare the closest isotropic tensors obtained using four approaches: the Frobenius 36-component norm, the Frobenius 21-component norm, the operator norm and the L2 slowne...

We show that the Backus (1962) equivalent-medium average, which is an average
over a spatial variable, and the Gazis et al. (1963) effective-medium average,
which is an average over a symmetry group, do not commute, in general. They
commute in special cases, which we exemplify.

We postulate that validity of the Backus (1962) average, whose weights are
layer thicknesses, is limited to waves whose incidence is nearly vertical. The
accuracy of this average decreases with the increase of the source-receiver
offset. However, if the weighting is adjusted by the distance travelled by a
signal in each layer, such a modified avera...

We examine the Backus averaging method - which, in general, allows one to represent a series of parallel layers by a transversely isotropic medium-using a repetitive shale-sandstone model. To examine this method in the context of experimental data, we perturb the model with random errors, in particular, the values of layer thicknesses and elasticit...

In the physical realm, an elasticity tensor that is computed based on measured numerical quantities with resulting numerical errors does not belong to any symmetry class for two reasons: (1) the presence of errors, and more intrinsically, (2) the fact that the symmetry classes in question are properties of Hookean solids, which are mathematical obj...

A b s t r a c t A generally anisotropic elasticity tensor can be related to its closest counterparts in various symmetry classes. We refer to these counterparts as effective tensors in these classes. In finding effective tensors, we do not assume a priori orientations of their symmetry planes and axes. Knowledge of orientations of Hookean solids al...

"I thought I knew everything I needed to know about characteristics and their role in the solution of partial differential equations. After reading this book, I discovered that I was wrong. The authors draw us along on an incredible mathematical journey by starting from simple examples and moving on to progressively more sophisticated examples. Thu...

We introduce the effective elasticity tensor of a chosen material-symmetry class to represent a measured generally anisotropic elasticity tensor by minimizing the weighted Frobenius distance from the given tensor to its symmetric counterpart, where the weights are determined by the experimental errors. The resulting effective tensor is the highest-...

The present book-which is the third, signi cantly revised edition of the textbook originally published by
Elsevier Science-emphasizes the interdependence of mathematical formulation and physical meaning in the
description of seismic phenomena. Herein, we use aspects of continuum mechanics, wave theory and ray
theory to explain phenomena resulting f...

Geophysics—similarly to astrophysics—relies on remote sensing.
Inferring material properties of the Earth’s interior is akin to inferring
the composition of a distant star. In both cases, scientists rely
on matching theoretical predictions or explanations with observations.
Notably, obtaining a sample of a material from the interior of
our planet m...

A generally anisotropic elasticity tensor, which might be obtained from physical measurements, can be approximated by a tensor belonging to a particular material-symmetry class; we refer to such a tensor as the effective tensor. The effective tensor is the closest to the generally anisotropic tensor among the tensors of that symmetry class. The con...

This textbook - incorporated with many illuminating examples and exercises - is aimed at graduate students of physical sciences and engineering. The purpose is to provide a background of physics and underlying mathematics for the concept of rays, filling the gap between mathematics and physics textbooks for a coherent treatment of all topics. The a...

We consider the problem of representing a generally anisotropic elasticity tensor, which might be obtained from physical measurements,
by a tensor belonging to a chosen material symmetry class, so-called ‘effective tensor'. Following previous works on the subject,
we define this effective tensor as the solution of a global optimization problem for...

In the conclusion of their seminal paper Dahlen et al.(Geophys. J. Int., 141, 157–174, 2000) state that their results extend ray theory to finite-frequency waves. In view of the importance of that paper, which led to developments in theoretical and computational seismology, we feel that it is important to clarify certain mathematical statements the...

Bayesian interpretation of a three-parameter model of anisotropy and inhomogeneity for a VSP data set obtained in the Western Canada Basin indicates that introduction of the anisotropy parameter is justified by the data to a high grade of evidence. The acquisition geometry is a key factor in the resolving power of the data, and hence in deciding on...

We propose a numerical ray-tracing algorithm for isotropic media discretized by finite elements. This algorithm, which is based on tetrahedral elements, combines the flexibility of finite-element modelling with the calculation of rays through a three-dimensional (3D) speed field, given a direction of a ray at its initial point. We implement this al...

In the seminal paper by F. A: Dahlen et al. [“Fréchet kernels for finite-frequency traveltimes. I: Theory”, Geophys. J. Int. 141, No. 1, 157–174 (2000; doi:10.1046/j.1365-246X.2000.00070.x)], the authors formulated an important expression as a first-order estimate of the signal-traveltime delay for seismic studies. The authors left out a term in a...

We present a method to identify the symmetry class of an elasticity tensor whose components are given with respect to an arbitrarily oriented coordinate system. The method is based on the concept of distance in the space of tensors, and relies on the monoclinic or transversely isotropic distance function. Since the orientation of a monoclinic or tr...

We formulate a method of representing a generally anisotropic elasticity tensor by an elasticity tensor exhibiting a material symmetry: an effective tensor. The method for choosing the effective tensor is based on examining the features of the plot of the monoclinic-distance function of a given tensor, choosing an appropriate symmetry class, and th...

We prove that the symmetry group of an elasticity tensor is equal to the symmetry group of the corresponding Christoffel matrix. Comment: This note completes the argument of Bóna, A., Bucataru, I., Slawinski, M.A. (2007) Material symmetries versus wavefront symmetries. The Quarterly Journal of Mechanics and Applied Mathematics 60(2), 73-84

The present book — which is the second, and significantly extended, edition of the textbook originally published by Elsevier Science — emphasizes the interdependence of mathematical formulation and physical meaning in the description of seismic phenomena. Herein, we use aspects of continuum mechanics, wave theory and ray theory to explain phenomena...

Ray-centred coordinates are used in investigating seismic theory and in computing its results. It is commonly stated that
they are local in nature. In this paper, we discuss the region of invalidity of such coordinates. Notably, the existence of
such regions limits the generality of theoretical conclusions based on a proof that makes use of these c...

In the seminal paper by Dahlen et al. the authors formulate an important expression as a first-order estimate of traveltime delay. The authors left out a term which would at first glance seem nontrivial, on the basis that their intention was to derive the Fr\'echet derivative linking the observed delay to the model perturbation (Nolet 2009, pers. c...

We derive the characteristic equations, and the so-called Christoffel equation, for the vector elastodynamic equations in
terms of both hypersurfaces of nonuniqueness and as wavefronts based on a physical definition. We follow Courant and Hilbert
in defining a wavefront as a surface for which a solution may be zero on one side but nonzero on the ot...

We consider the problem of obtaining the orientation and elasticity parameters of an effective tensor of particular sym-metry that corresponds to measurable traveltime and polar-ization quantities. These quantities — the wavefront-slow-ness and polarization vectors — are used in the Christoffel equation, a characteristic equation of the elastodynam...

We have proved that the innermost wavefront-slowness sheet of a Hookean solid is convex, whether or not it is detached from the other sheets. This theorem is valid for the generally anisotropic case, and it is an extension of theorems whose proofs require the detachment of the innermost sheet. Although the Hookean solids that represent most materia...

We consider the problem of obtaining the effective orthotropic tensor that corresponds to a given generally anisotropic one;
herein, by ‘effective’, we mean the closest in the sense of the Frobenius norm, without a priori assuming the orientation of the orthotropic tensor. It is difficult to find the absolute minimum of the distance function
since...

We propose a new ray-tracing method based on the concept of simulated annealing. Using this method, we find rays between fixed sources and receivers that render traveltime globally minimal.With our method, we are able to construct rays and their associated traveltimes to satisfactory precision in complex media. Furthermore, our algorithm can be mod...

Using orthogonal projections, we investigate distance of a given elasticity tensor to classes of elasticity tensors exhibiting particular material symmetries. These projections depend on the orientation of the elasticity tensor, hence the distance is obtained as the minimization of corresponding expressions with respect to the action of the orthogo...

We consider the problem of finding the transversely isotropic elasticity tensor closest to a given elasticity tensor with respect to the Frobenius norm. A similar problem was considered by other authors and solved analytically assuming a fixed orientation of the natural coordinate system of the transversely isotropic tensor. In this paper we formul...

We construct an eighteen-dimensional orbifold that is in a one-to-one correspondence with the space of SO (3)-orbits of elasticity tensors. This allows us to obtain a local parametrization of SO (3)-orbits of elasticity tensors by six SO (6)-invariant and twelve SO (3)-invariant parameters. This process unravels the structure of the space of the or...

We construct a method for finding the elasticity parameters of an anisotropic homogeneous medium using only ray velocities and corresponding polarizations. We use a linear relation between the ray velocities and wavefront slownesses, which depends on the corresponding polarizations. Notably, this linear relation circumvents the need to use explicit...

We compare two methods that determine the components of the elasticity tensor describing a Hookean solid: a method based on the wavefront-slowness and polarization measurements, and a method based on the ray-velocity and polarization measurements. The first method assumes a lateral homogeneity of the medium; the second one assumes its complete homo...

In this presentation, we discuss the one-to-one relation between the elasticity parameters and the traveltime and polarization of a propagating signal in the context of the measurement errors. The one-to-one relationship between seismic measurements and a model postulated in the realm of the constitutive equation of an elastic continuum provides th...

We formulate coordinate-free conditions for identifying all the symmetry classes of the elasticity tensor and prove that these conditions are both necessary and sufficient. Also, we construct a natural coordinate system of this tensor without the a priory knowledge of the symmetry axes.

We show that, in general, wavefronts are more symmetric than the medium in which they propagate. This means that we cannot determine the symmetries of the medium based solely on the symmetries of the wavefronts. However, we show that we can determine the symmetries of the medium from the symmetries of wavefronts and polarizations together.

We formulate coordinate-free conditions for identifying all the symmetry classes of the elasticity tensor and prove that these
conditions are both necessary and sufficient. Also, we construct a natural coordinate system of this tensor without the a
priory knowledge of the symmetry axes.

We study wave propagation in anisotropic inhomogeneous media. Specifically, we formulate and analytically solve the ray-tracing equations for the factorized model with wavefront velocity increasing linearly with depth and depending elliptically on direction. We obtain explicit expressions for traveltime, wavefront (phase) angle, and ray (group) vel...

We prove that in elastic anisotropic inhomogeneous media, rays and wavefronts are orthogonal to each other with respect to the metric induced by the phase-velocity function. The standard orthogonality of rays and wavefronts in elastic isotropic inhomogeneous media is a special case of this formulation.