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Tomasz Danek

Publications and source records attributed to Tomasz Danek.

7 recordsLinked to original sources
arXiv2021

On modelling bicycle power-meter measurements

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 techniques to maintain an agreement between modelled and measured power-meter values, which accounts for the associated uncertainties. The estimation of coefficients is performed for two typical scenarios of road cycling under windless conditions, along a course that is mainly flat as well as a course of near constant inclination. Also, we discuss relations between different combinations of two model parameters, where other quantities are constant, by the implicit function theorem. Using the obtained estimates of resistance coefficients for the two courses, we use the mathematical relations to make inferences on the model and physical conditions. Along with a discussion of results, we provide two appendices. In the first appendix, we illustrate the importance of instantaneous cadence measurements. In the second, we consider the model in constrained optimization using Lagrange multipliers.

physics.pop-ph↗
arXiv2020

On modelling bicycle power-meter measurements: Part II. Relations between rates of change of model quantities

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 imposed constraints.

physics.pop-ph↗
arXiv2020

Selecting velocity models using Bayesian Information Criterion

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 optimization. In the first step, we find the signal trajectory and, in the second step, we find media parameters by minimizing the misfit between the model and data.

physics.geo-ph↗
arXiv2018

Statistical and numerical considerations of Backus-average product approximation

In this paper, we examine the applicability of the approximation, $\overline{f\,g}\approx \overline f\,\overline g\,$, within Backus (1962) averaging. This approximation is a crucial step in the method proposed by Backus (1962), which is widely used in studying wave propagation in layered Hookean solids. According to this approximation, the average of the product of a rapidly varying function and a slowly varying function is approximately equal to the product of the averages of those two functions. Considering that the rapidly varying function represents the mechanical properties of layers, we express it as a step function. The slowly varying function is continuous, since it represents the components of the stress or strain tensors. In this paper, beyond the upper bound of the error for that approximation, which is formulated by Bos et al. (2017), we provide a statistical analysis of the approximation by allowing the function values to be sampled from general distributions. Even though, according to the upper bound, Backus (1962) averaging might not appear as a viable approach, we show that$-$for cases representative of physical scenarios modelled by such an averaging$-$the approximation is typically quite good. We identify the cases for which there can be a deterioration in its efficacy. In particular, we examine a special case for which the approximation results in spurious values. However, such a case$-$though physically realizable$-$is not likely to appear in seismology, where Backus (1962) averaging is commonly used. Yet, such values might occur in material sciences, in general, for which Backus (1962) averaging is also considered.

physics.geo-ph↗
arXiv2018

On Pareto Joint Inversion of guided waves

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 relations derived by Dalton et al. (2017). Numerical results are presented for three angular frequencies, 15 Hz, 60 Hz and 100 Hz, and for two, five and seven modes, respectively. Comparisons of the model parameters with the values inverted with error-free input indicate an accurate process. If, however, we introduce a 5% error to the input, the results become significantly less accurate, which indicates that the inverse operation, even though stable, is error-sensitive. Correlations between the inverted elasticity parameters indicate that the layer parameters are more sensitive to input errors than the halfspace parameters. In agreement with Dalton et al. (2017), the fundamental mode is mainly sensitive to the layer parameters whereas higher modes are sensitive to both the layer and halfspace properties; for the second mode, the results are more accurate for low frequencies.

physics.geo-ph↗
arXiv2016

On closest isotropic tensors and their norms

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 slowness-curve fit. We find that the isotropic tensors are similar to each other within the range of expected measurement errors.

physics.geo-ph↗