Analytical Parameterization Research Articles

Optical beam spread in seawater

We study the spatial and angular distribution of light in a narrow stationary beam propagating in a medium like seawater. For a power-law parametrization of the seawater phase function, using radiative transfer analysis, we derive analytical expressions that determine the radiance distribution in the medium. Both the case of intermediate source–receiver distances, where absorption only begins to affect spreading of the beam, and the asymptotic regime of beam propagation are examined. For the main characteristics of the radiance distribution (the total power, the variance of the angular and radial distributions), scaling relations involving the inherent optical parameters of seawater are found. To validate the expressions obtained we carry out Monte Carlo simulation for commonly adopted seawater scattering models (of Petzold and Morel et al) and their analytical parameterizations. For optical parameters typical to seawater, the predicted analytical laws are in excellent agreement with the results of the Monte Carlo computations.

Analytic Parameterization of Longwave Optical Properties of Bulk Vegetation Layer Permitting Non‐Zero Leaf Reflectivity and Its Implementation in CLM5

AbstractFor modern land surface models (LSMs) representing a singular bulk vegetation layer, the longwave optical properties (i.e., emissivity, reflectivity, and transmittivity) of vegetation layer are derived with a simplified constraint of assuming zero leaf reflectivity. This constraint is necessary, for instance, to the Beer–Lambert (B–L) law to establish a relationship between the optical properties and leaf area index. However, the simplified constraint leads to an overestimation of land surface emissivity in the vegetated regions. In this study, we introduce a new scheme considering realistic leaf reflectivity values rather than assuming zero. This new scheme is based on the relationship derived from the B–L law, but it is statistically augmented to consider the effects of leaf reflections. It is designed to emulate a multi‐vegetation‐layer numerical model known as the Norman model, which is capable of numerical calculations of multi‐reflections among leaves. This new method consists of only a couple of simple equations; despite its simplicity, it very closely mimics the Norman model; The discrepancy of the results between the new method and the Norman model is less than measurement uncertainties for any combination of input parameters. When the new scheme is implemented in the Community Land Model version 5 (CLM5), the land surface emissivity values are simulated much more consistently with global measurements, resulting in significant alterations of land surface energy budget. The enhanced realism through our new scheme is poised to contribute to more accurate numerical weather and climate simulations.

Analytical parameterization of Bragg curves for proton beams in muscle, bone, and polymethylmethacrylate.

Proton dose calculation in media other than water may be of interest for either research purposes or clinical practice. Current study aims to quantify the required parameters for analytical proton dosimetry in muscle, bone, and PMMA. Required analytical dosimetry parameters were extracted from ICRU-49 report and Janni study. Geant4 Toolkit was also used for Bragg curve simulation inside the investigated media at different proton energies. Calculated and simulated dosimetry data were compared using gamma analysis. Simulated and calculated Bragg curves are consistent, a fact that confirms the validity of reported parameters for analytical proton dosimetry inside considered media. Furthermore, derived analytical parameters for these media are different from those of water. Listed parameters can be reliably utilized for analytical proton dosimetry inside muscle, bone, and PMMA. Furthermore, accurate proton dosimetry inside each medium demands dedicated analytical parameters and one is not allowed to use the water coefficients for non-water media.

Monte-carlo simulation of the effective lunar aperture for detection of ultra-high energy neutrinos with LOFAR

Ultra-high-energy (UHE) cosmic neutrinos interacting with the Moon’s regolith generate particle showers that emit Askaryan radiation. This radiation can be observed from the Earth using ground-based radio telescopes like LOFAR. We simulate the effective detection aperture for UHE neutrinos hitting the Moon. Under the same assumptions, results from this work are in good agreement with previous analytic parameterizations and Monte Carlo codes. The dependence of the effective detection aperture on the observing parameters, such as observing frequency and minimum detection threshold, and lunar characteristics like surface topography have been studied. Using a Monte Carlo simulation, we find that the detectable neutrino energy threshold is lowered when we include a realistic treatment of the inelasticity, transmission coefficient, and surface roughness. Lunar surface roughness at large scales enhances the total aperture for higher observation frequencies (ν≥1GHz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u \\ge 1~\ extrm{GHz}$$\\end{document}) but has no significant effect on the LOFAR aperture. However, roughness at scales small compared to the wavelength reduces the aperture at all frequencies.

Parameterization of polydisperse aerosol optical properties during hygroscopic growth

Aerosol water uptake and the related influence on its optical properties owing to the hygroscopic growth of polydisperse aerosols have a significant effect on air quality and climate. This study develops an analytical parameterization for a single hygroscopic optical parameter and the scattering enhancement factor for polydisperse aerosol sizes. Polydisperse lognormal size distributions for geometric mean diameters of 0.05–1 μm and geometric standard deviations of 1.3–2.5 are considered with real refractive indices between 1.35-1.6. The analytical expression obtained for the optical parameter is compared to Mie theory and reasonable agreements within the size range investigated in this study are observed. Further, our results show that the analytical expression could express polydispersed aerosol optical properties as a function of geometric mean diameter and geometric standard deviation of a lognormal distribution and the refractive index. The obtained analytical expression can be efficiently used for 3D models with a reduced computational burden.

Improved flux-surface parameterization through constrained nonlinear optimization

Parameterization of magnetic flux-surfaces is often used for magnetohydrodynamic stability analysis and microturbulence modeling in tokamaks. Shape parameters for such local parameterization of a (numerical) equilibrium are traditionally computed analytically using geometrically derived quantities. However, often the shape is approximated by the average of values for different sections of the flux-surface contour or a truncated series, which does not guarantee an optimal fit. Here, instead nonlinear least squares optimization is used to compute these parameters, with a weighted sum of squared error cost function that is robust to outliers. This method results in a lower total absolute error for both the parameterization of the flux-surface contour and the poloidal magnetic field density than current methods for several parameterizations based on the well-known “Miller geometry.” Furthermore, rapid convergence of shape parameters is achieved, no approximate geometric measurements of the contour are needed, and the method is applicable to any analytical shape parameterization. Validation with local, linear gyrokinetic simulations using these optimized shape parameters showed reduced root mean square errors in both the growth rate and frequency spectra when compared with simulations based on numerical equilibria. In particular, the popular Turnbull–Miller parameterization benefits from this approach, extending its usability closer toward the last-closed flux-surface for cases with minor up-down asymmetry.

The isovector density-dependence of the M3Y nucleon-nucleon interactions and related soft and hard equations of state

Towards a precise equation of state (EOS) of nuclear matter (NM) based on a realistic interaction, we use non-relativistic Brueckner-Hartree-Fock (BHF) scheme to deduce the isovector (IV) density dependence (DD) of M3Y-Paris and Reid nucleon-nucleon interactions, via comparison with previous BHF calculations of nucleon optical potential. The single particle rearrangement correction is taken into account. With analytical parameterization of the isoscalar DD, the IV DD is parameterized (IVF1) for a broad range of EOSs, from soft EOS of saturation incompressibility K0 = 150 MeV to stiff one with K0 = 300 MeV. These Paris (Reid) EOSs indicate symmetry energy coefficient of 30.06±0.02 MeV (31.76±0.06 MeV), with corresponding density-slope sticks around L = 49.7 MeV (51.9 MeV). We found that the IVF1 DD indicates larger (smaller) saturation binding-energy (density) of asymmetric NM, and more high-density repulsion increasing with density and isospin asymmetry, as well as higher pressure, than those based on the IVF0 DD that scaled in terms of the isoscalar DD. Also, the IVF1 confirms the mostly indicated hard symmetry energy at supra-saturation density, on the contrary of the IVF0 that indicates soft one. Increasing the stiffness of the EOS increases the NM energy, though it decreases the symmetry energy. The more neutron-rich NM gets less bound, over less bound-density range. Only the fine ranges of Paris (K0 = 240 - 250 MeV) and Reid (K0 = 230 - 240 MeV) EOSs maintain simultaneous consistency with earlier experimental constraints and accurate calculations on NM energy, symmetry energy, and pressure.

Geometry of the Ovoids: Avian Eggs and Similar Asymmetric Forms

Despite the longstanding interest in the shapes of the eggs the available parametric descriptions in the modern literature are given only via purely empirical formulas without any clear relationships with their measurable parameters. Here we present geometrical models of the eggs based on Perseus spirics and Cassinian ovals which were known since the ancient time but their analytical parameterization was also absent in the meantime. Such ones have been found recently and the present work is based on the idea to use spirics or Cassinians as geometrical models of the eggs shapes. New explicit formulas for the volumes, surface areas and the curvatures of the avian eggs have been derived from the first principles and these have been compared with the available experimental data.

Geometry of the Ovoids: Reptilian Eggs and Similar Symmetric Forms

Despite the longstanding interest in the shapes of the eggs since the ancient time till nowadays, the available parametric descriptions in the modern literature are given only via purely empirical formulas without any clear relationships with their measurable physical parameters. Here we present a geometrical model of the eggs based on Perseus spirics which were known as well since the ancient time but their analytical parameterizations were absent in the meantime. Such parameterizations have been found recently and the present work is based on the idea to use the spirics as a geometrical model of the egg's shapes. Explicit formulas for the volume, surface area and the curvatures of the eggs are derived from the first principles and these have been compared with the available empirical formulas and experimental data.

Assessing spatially heterogeneous scale representation with applied digital soil mapping

Topographic data are increasingly important to environmental models as fine-scale resolution, wide coverage data sets become available. Scale is an important consideration for predictive model quality. Recent advances in multiscale terrain analysis led to scaling techniques that allow the scale at which a topographic parameter is represented to vary spatially. This research compared predictive soil model performance across feature sets generated with different scaling strategies; including multiple heterogeneous strategies, common feature selection algorithms applied to homogeneously scaled data, and unscaled data. Model performance was assessed for accuracy and uncertainty. The results showed that unscaled data performed worse in all circumstances compared to multiscale feature sets. Overall, heterogeneous and homogeneous feature sets did not differ substantially in accuracy, prediction uncertainty, or error. However, one scaling strategy exploited the flexibility of heterogeneous scaling to consistently perform better than other feature sets for most soil properties in terms of accuracy, and consistently ranked among the least uncertain and least error prone (up to a 0.080 increase in accuracy with a corresponding 0.017 decrease in prediction uncertainty and 0.011 decrease in error relative to the second best method, in the case of the proportion of clay modelled at 5–15 cm depth). This was achieved by decoupling the definition of process scales from analytical parameterization, allowing the optimization to occur within broadly defined process scales. This research demonstrates how to exploit heterogeneous scaling of topographic attributes to improve model performance.

The Karcher mean of equilateral triangles

In this paper we present a class of Riemannian equilateral triangles ΔABC of the Cartan-Hadamard Riemannian manifold of N×N positive definite matrices of determinant 1 whose Karcher mean of vertices is of the formΛ(A,B,C)=A+B+Cdet⁡(A+B+C)N. The obtained class of triples (A,B,C) forms an analytic manifold with an analytic parameterization over SL(N,C)±×R. The determinantal identity det⁡(A+B+C)=det⁡(A−1+B−1+C−1) and an extended arithmetic-Karcher-harmonic mean inequalities is derived for such triples.

Parameterization of a Novel Nonlinear Estimator for Uncertain SISO Systems with Noise Scenario

Dynamic observers are commonly used in feedback loops to estimate the system’s states from available control inputs and measured outputs. The presence of measurement noise degrades the performance of the observer and consequently degrades the performance of the controlled system. This paper presents a novel nonlinear higher-order extended state observer (NHOESO) for efficient state and disturbance estimation in presence of measurement noise for nonlinear single-input–single-output systems. The proposed nonlinear function allows a fast reconstruction of the system’s states and is robust against uncertainties and measurement noise. An analytical parameterization technique is proposed to parameterize the coefficients of the proposed nonlinear higher-order extended state observer in the case of measurement noise in the output signal. Several scenarios are simulated to demonstrate the effectiveness of the proposed observer.

Analytical model of a short DC glow discharge in the presence of significant radial ambipolar diffusion losses

We present a simple analytical model of a short direct-current glow discharge (without positive column) that is applicable for the case of narrow discharge tubes when radial losses of charged particles due to ambipolar diffusion significantly influence discharge properties. The model is based on the analytical parameterization of the non-local ionization produced by fast electrons, which allows obtaining an exact solution of the ambipolar diffusion equation with radial particle losses written in the τ-approximation. Analysis of the spatial distribution of electron density in the near-cathode plasma regions of a discharge allow obtaining an explicit expression for the position of electric field reversal in the negative glow that is independent of electron temperature and interelectrode distance. The latter fact makes the model viable for description of near-cathode plasma regions in the general case of a discharge with a positive column. The procedure for obtaining quantitative estimates and interpretation of experimental data on discharge properties using the proposed model is presented and discussed in detail.

Flushing the Lake Littoral Region: The Interaction of Differential Cooling and Mild Winds

AbstractThe interaction of a uniform cooling rate at the lake surface with sloping bathymetry efficiently drives cross‐shore water exchanges between the shallow littoral and deep interior regions. The faster cooling rate of the shallows results in the formation of density‐driven currents, known as thermal siphons, that flow downslope until they intrude horizontally at the base of the surface mixed layer. Existing parameterizations of the resulting buoyancy‐driven cross‐shore transport assume calm wind conditions, which are rarely observed in lakes and thereby restrict their applicability. Here, we examine how moderate winds (≲5 m s−1) affect this convective cross‐shore transport. We derive simple analytical solutions that we further test against realistic three‐dimensional numerical hydrodynamic simulations of an enclosed stratified basin subject to uniform and steady surface cooling rate and cross‐shore winds. We show cross‐shore winds modify the convective circulation, stopping or even reversing it in the upwind littoral region and enhancing the cross‐shore exchange in the downwind region. The analytical parameterization satisfactorily predicted the magnitude of the simulated offshore unit‐width discharges in the upwind and downwind littoral regions. Our scaling expands the previous formulation to a regime where both wind and buoyancy forces drive cross‐shore discharges of similar magnitude. This range is defined by the non‐dimensional Monin‐Obukhov length scale, χMO: 0.1 ≲ χMO ≲ 0.5. The information needed to evaluate the scaling formula can be readily obtained from a traditional set of in situ observations.

MARMOT: magnetism, anisotropy, and more, using the relativistic disordered local moment picture at finite temperature

We present MARMOT, a hybrid Python/FORTRAN implementation of the disordered local moment picture within multiple scattering density-functional theory. MARMOT takes atom-centred, scalar-relativistic potentials and constructs an effective medium (within the coherent potential approximation) to describe the disordered magnetic moment orientations at finite temperature. By solving the single-site scattering problem fully relativistically, spin–orbit effects are included, allowing the magnetocrystalline anisotropy to be calculated. Magnetic transition temperatures, spin and orbital moments, the density-of-states, and analytical parameterizations of the magnetic potential energy surface can also be calculated. Here, we describe the theory and practical implementation of MARMOT, and demonstrate its use by calculating Curie temperatures, magnetizations and anisotropies of bcc Fe, GdFe2 and YCo5.

Analytical Closed-form Parameterizations for Bounded Rationality of either of Overconfidence or Underconfidence

Analytical Closed-form Parameterizations for Bounded Rationality of either of Overconfidence or Underconfidence

Explicit Solution of the Focus Locus Problem for the Harmonic Oscillator Orbits in the Plane

Dynamical orbits of the harmonic oscillator potential in the plane are ellipses which depend on a real parameter. Some time ago in this journal it has been proven by purely geometrical methods that the locus of the focuses of these ellipses are Cassinian ovals. Here we present several explicit analytic parameterizations of these remarkable curves. Nominally, their forms depend on the magnitude of the initial distance from the center of attraction and the magnitude of the initial velocity. We have found a few parameterizations in which the roles of the size and shapes can be clearly distinguished.

Energy spectra of secondaries in proton-proton interactions

We compare the predictions of aafrag for the spectra of secondary photons, neutrinos, electrons, and positrons produced in proton-proton collisions to those of the parametrizations of Kamae et al., Kelner et al. and Kafexhiu et al. We find that the differences in the normalization of the photon energy spectra reach 20%--50% at intermediate values of the transferred energy fraction $x$, growing up to a factor of two for $x\ensuremath{\rightarrow}1$, while the differences in the neutrino spectra are even larger. We argue that LHCf results on the forward production of photons and neutral pions favor the use of the qgsjet-II-04m model on which aafrag is based. The differences in the normalization have important implications in the context of multimessenger astronomy, in particular, for the prediction of neutrino fluxes, based on gamma-ray flux measurements, or regarding the inference of the cosmic ray spectrum, based on gamma-ray data. We note also that the positron-electron ratio from hadronic interactions increases with energy toward the cutoff, an effect which is missed using the average electron-positron spectrum from Kelner et al. Finally, we describe the publicly available python package aafragpy, which provides the secondary spectra of photons, neutrinos, electrons, and positrons. This package complements the aafrag results for protons with energies above 4 GeV with previous analytical parameterizations of particle spectra for lower energy protons.

Bjorken variable and scale dependence of quark transport coefficient in semi-inclusive lepton production of hadron off nuclei

Nuclear modification of hadron production in deep inelastic lepton-nucleus scattering can be applied to study the parton propagation mechanism in cold nuclear matter. By means of the analytic parameterization of quenching weight based on BDMPS formalism with the target nuclear geometry effect, the leading-order computations for hadron multiplicity ratios are performed with comparison to the HERMES charged pions production data on the quarks hadronization occurring outside the nucleus. The relation is discovered between quark transport coefficient and the measurable kinematic variables in deep inelastic scattering. Four models are proposed on the quark transport coefficient. The constant model, the power-law model and the double power-law model can be ruled out because of the experimental fact that the transverse momentum broadening increases as a function of the photon virtuality $$Q^{2}$$ . The quark transport coefficient is determined as a function of the Bjorken variable x and scale $$Q^2$$ . The trend of quark transport coefficient in respect of Bjorken variable x and scale $$Q^2$$ is qualitatively in partial agreement with HERMES experimental data on transverse momentum broadening. It is hoped that our effort is conducive to understanding of jet quenching phenomenon in relativistic heavy ion collisions.

Magnetic diffusion and interaction effects on ultrahigh energy cosmic rays: Protons and nuclei

The flux of ultrahigh energy cosmic rays reaching the Earth is affected by the interactions with the cosmic radiation backgrounds as well as with the magnetic fields that are present along their trajectories. We combine the SimProp cosmic ray propagation code with a routine that allows to account for the average effects of a turbulent magnetic field on the direction of propagation of the particles. We compute in this way the modification of the spectrum which is due to the magnetic horizon effect, both for primary nuclei as well as for the secondary nuclei resulting from the photo-disintegration of the primary ones. We also provide analytic parameterizations of the attenuation effects, as a function of the magnetic field parameters and of the density of cosmic ray sources, which make it possible to obtain the expected spectra in the presence of the magnetic fields from the spectra that would be obtained in the absence of magnetic fields. The discrete nature of the distribution of sources with finite density also affects the spectrum of cosmic rays at the highest energies where the flux is suppressed due to the interactions with the radiation backgrounds, and parameterizations of these effects are obtained.