Reference Ray Research Articles

Improving GNSS-IR Sea Surface Height Accuracy Based on a New Ionospheric Stratified Elevation Angle Correction Model

Approximately 71% of the Earth’s surface is covered by vast oceans. With the exacerbation of global climate change, high-precision monitoring of sea surface height variations is of vital importance for constructing global ocean gravity fields and preventing natural disasters in the marine system. Global Navigation Satellite System Interferometry Reflectometry (GNSS-IR) sea surface altimetry is a method of inferring sea surface height based on the signal-to-noise ratio of satellite signals. It enables the retrieval of sea surface height variations with high precision. However, navigation satellite signals are influenced by the ionosphere during propagation, leading to deviations in the measured values of satellite elevation angles from their true values, which significantly affects the accuracy of GNSS-IR sea surface altimetry. Based on this, the contents of this paper are as follows: Firstly, a new ionospheric stratified elevation angle correction model (ISEACM) was developed by integrating the International Reference Ionosphere Model (IRI) and ray tracing methods. This model aims to improve the accuracy of GNSS-IR sea surface altimetry by correcting the ionospheric refraction effects on satellite elevation angles. Secondly, four GNSS stations (TAR0, PTLD, GOM1, and TPW2) were selected globally, and the corrected sea surface height values obtained using ISEACM were compared with observed values from tide gauge stations. The calculated average Root Mean Square Error (RMSE) and Pearson Correlation Coefficient (PCC) were 0.20 m and 0.83, respectively, indicating the effectiveness of ISEACM in sea surface height retrieval. Thirdly, a comparative analysis was conducted between sea surface height retrieval before and after correction using ISEACM. The optimal RMSE and PCC values with tide gauge station observations were 0.15 m and 0.90, respectively, representing a 20.00% improvement in RMSE and a 4.00% improvement in correlation coefficient compared to traditional GNSS-IR retrieval heights. These experimental results demonstrate that correction with ISEACM can effectively enhance the precision of GNSS-IR sea surface altimetry, which is crucial for accurate sea surface height measurements.

A novel approach for incidence angle modifier calculation of arbitrarily oriented linear Fresnel collectors: Theory, simulations and case studies

The easiest possibility to assess the combined opto-energy efficiency of Linear Fresnel Collectors (LFC) is to refer to the Incidence Angle Modifier (IAM) concept: for this kind of solar collector, IAM calculation is often performed according to the so-called factorization method. Unfortunately, the factored IAM fails in providing reliable results of the real LFC yield. In this investigation, hourly 3D ray tracing simulations have been carried out on two case-study plants located in Morocco and Italy: reference ray tracing calculated IAMs have been used to develop simple correlations for inferring the instantaneous IAM of arbitrarily aligned LFCs. Instantaneous IAM formulas resulted to be able to provide yield predictions with a Root Mean Square Error equal to 5.57 % of the yearly average IAM, much lower than the factorization method (19.42 %). Subsequently, a reduced 3-day dataset of ray tracing values has been considered to calibrate the regression constants, resulting in still high accuracy for orientations not exceeding 30°. Finally, the analysis of available power at the receiver allowed to further assess the robustness of the proposed approach regardless of the plant orientation angles.

Tracing real-valued reference rays in anisotropic viscoelastic media

Tracing real-valued reference rays in anisotropic viscoelastic media

Error correction algorithms in non-null aspheric testing next generation sequencing data

Non-null tests are more versatile than null tests and could give fast, generic tests with sufficient performance. The non-null testing methods are able to measure aspheric surfaces with substantial deviations from a reference sphere. The ray will return along a different path and interfere with a separate reference ray in a non-null arrangement. The retrace error occurs because of the significant disparity between the aspheric and the reference. To correct retracing errors and replicate aspheric fabrication imperfections, complex systems modelling and simulations are required. We propose a ray-tracing-based technique for simulating the non-null interferometer that estimates the optical path differences by tracking lasers across the reference and testing paths. An error correction method is used by ray trace analysis to correct the retracing errors in non-null tests of aspheric surfaces. Experiments were conducted to demonstrate the effectiveness of this strategy. The result of the study shows that the proposed system successfully and effectively rectifies the retracing error compared to other practical findings and computer simulation methods. It is more versatile, faster, and provides adequate aspheric assessment performance.

Primary aberration optimization for double-plane symmetric beam shaping systems using a pair of curved reference surfaces.

The geometric aberration of centered refracting double-plane symmetric optical systems (DPSOS) is investigated. For DPSOS with different defocus values in the tangential plane and the sagittal plane (astigmatic wavefront), a pair of curved reference surfaces which vanishes the quadratic terms of the optical path difference (OPD) between a general ray and a reference ray are deduced. With the curved reference surfaces, the primary (fourth-order) wave aberration function for DPSOS is calculated and analyzed, which can be used for beam shaping designs with astigmatic input wavefront, such as slab lasers and semiconductor lasers. Further, the proposed curved reference surfaces can be applied to analyze the aberrations of general DPSOS.

Weak-anisotropy approximation of P-wave geometric spreading in horizontally layered anisotropic media of arbitrary symmetry: Tilted transversely isotropic specification

Understanding the role of geometric spreading and estimating its effects on seismic wave propagation play an important role in several techniques used in seismic exploration. The spreading can be estimated through dynamic ray tracing or determined from reflection traveltime derivatives. In the latter case, derivatives of nonhyperbolic moveout approximations are often used. We offer an alternative approach based on the weak-anisotropy approximation. The resulting formula is applicable to P-waves reflected from the bottom of a stack of horizontal layers, in which each layer can be of arbitrary anisotropy. At an arbitrary surface point, the formula depends, in each layer, on the thickness of the layer, on the P-wave reference velocity used for the construction of reference rays, and on nine P-wave weak-anisotropy (WA) parameters specifying the layer anisotropy. Along an arbitrary surface profile, the number of WA parameters reduces to five parameters related to the profile. WA parameters represent an alternative to the elastic moduli and, as such, can be used for the description of any anisotropy. The relative error of the approximate formula for a multilayered structure consisting of layers of anisotropy between 8% and 20% is, at most, 10%. For models including layers of anisotropy stronger than 20%, the relative errors may reach, locally, even 30%. For any offset, relative errors remain under a finite limit, which varies with the anisotropy strength.

Normal and five-fingered hand: comparative X-ray morphometry in the post-natal age.

Five-fingered hand (5-FH) with completely developed phalanges is a rare phenotype observed so far only in humans and characterised by three phalanges of the 1st ray. A long-lasting, debated question is if the missing element of the normal hand 1st ray is the metacarpal or the phalanx. In this study, comparative X-rays morphometry of long bones in normal and 5-FH is carried out with the aim to face this question through homology analysis of long bone segments in the transverse and longitudinal line of normal hand and 5-FH. In the normal hand X-rays (n =20) and in a 5-FH X-rays series (n = 9) the relative length of each segment on the ray total length and the index of growth rate (IGR) were assessed. The calculation of the first parameter in normal hand bi-phalangeal thumb was carried out on the 3rd ray total length in the same hand. The parameters of relative length and the proximal/distal growth rate asymmetry in the post-natal period (assessed through the IGR) confirmed in 5-FH the homology of all the five segment on the transverse line. In the normal control hand, the relative length assessment methodology was biased by the missing segment of the thumb, therefore, the reference to the 3rd ray total length in the same hand (instead of the 1st), allowed the homology analysis of the thumb metacarpal and 1st phalanx with the lateral segments (2nd-5th ray) of the same hand. The 5-FH analysis was used to choose the more appropriate reference ray for the normal hand group. The comparative analysis of relative lengths and IGRs in the two groups suggested homology of the (anatomical) 1st metacarpal with the 2nd-5th proximal phalanges in the same hand and that of the (anatomical) 1st proximal phalanx with the 2nd-5th mid phalanges. These data suggest that the missing segment of the normal hand thumb is the metacarpal.

Higher-order Hamilton–Jacobi perturbation theory for anisotropic heterogeneous media: transformation between Cartesian and ray-centred coordinates

SUMMARY Within the field of seismic modelling in anisotropic media, dynamic ray tracing is a powerful technique for computation of amplitude and phase properties of the high-frequency Green’s function. Dynamic ray tracing is based on solving a system of Hamilton–Jacobi perturbation equations, which may be expressed in different 3-D coordinate systems. We consider two particular coordinate systems; a Cartesian coordinate system with a fixed origin and a curvilinear ray-centred coordinate system associated with a reference ray. For each system we form the corresponding 6-D phase spaces, which encapsulate six degrees of freedom in the variation of position and momentum. The formulation of (conventional) dynamic ray tracing in ray-centred coordinates is based on specific knowledge of the first-order transformation between Cartesian and ray-centred phase-space perturbations. Such transformation can also be used for defining initial conditions for dynamic ray tracing in Cartesian coordinates and for obtaining the coefficients involved in two-point traveltime extrapolation. As a step towards extending dynamic ray tracing in ray-centred coordinates to higher orders we establish detailed information about the higher-order properties of the transformation between the Cartesian and ray-centred phase-space perturbations. By numerical examples, we (1) visualize the validity limits of the ray-centred coordinate system, (2) demonstrate the transformation of higher-order derivatives of traveltime from Cartesian to ray-centred coordinates and (3) address the stability of function value and derivatives of volumetric parameters in a higher-order representation of the subsurface model.

Efficient seismic ray tracing based on the shortest path method

SUMMARYLimiting the number of valid nodes around a reference ray can greatly reduce the calculation time of the shortest-path method (SPM). The calculation is executed by increasing the number of cells and/or nodes in the target area, step by step, until the rays converge. The ray obtained in the previous step is used as a reference ray, and the initial reference rays are given by the ordinary SPM. The Dijkstra algorithm and binary heap sorting method are used, as in the ordinary SPM. As the cell and node numbers increase, the calculation time for the modified SPM (mSPM) is reduced compared to that for the ordinary SPM. In the 3-D (100 × 100 × 100 km3) checkerboard velocity pattern model, the relative calculation time becomes two to four orders of magnitude smaller. The calculation time for the mSPM itself is approximately proportional to E log2(V), where E is the edge (ray path segment) number and V is the vertex (node) number, as seen in the heap sorting algorithm. The mean traveltime and ray path differences between the mSPM and pseudo-bending method (PBM) are small, less than 0.005 s and around 0.6 km, respectively, and slightly larger than those with respect to the ordinary SPM. The total differences from the exact solution are estimated to be less than 0.01 s and 1.0 km, which are sufficiently small for traveltime tomography. The traveltime and ray path can be improved by utilizing iterative calculations, shifting of the starting point and more neighbour nodes. The ray path obtained by the mSPM can be a local minimum, according to velocity models. The relative mean traveltime and ray path differences between adjacent cells and nodes generally show trends similar to those of the traveltime and ray path differences from the PBM. Hence, these relative differences can help reveal the behaviour of the differences from the PBM.

Aberration analysis of offset cylindrical lens using differential algebra method

This study introduces a differential algebra (DA) method to the high-order aberration analysis of a multi-offset cylindrical lens. The axial field functions (AFF) of harmonic field components for a cylindrical lens are interpolated into an analytic expression using a Gaussian wavelet, which can be transferred into DA arguments and can be operated as per DA methodology. The tracing of one reference ray can enable the high-order aberration coefficients of the multi-offset cylindrical lens to be obtained. As an example, a monochromator (MC) with a multi-offset cylindrical lens is computed, and up to third rank aberration coefficients are obtained using the developed DA method. The software has been developed and tested extensively.

Exact equations to measure highly aberrated wavefronts with the Hartmann test.

An exact vector expression for the deformations of a wavefront from any chosen reference surface, as a function of the directions of the real and reference rays, is deduced. It can be used with slope measuring test methods, such as Hartmann or Ronchi tests, but the need for a spherical reference is removed. We present simulated and experimental results to show the feasibility of this proposal.

Flux of radiation from pointlike sources in general relativity

In the paper we study propagation of light in general relativity through a spacetime filled with cold plasma with infinite conductivity. We use the geometric optics based on Synge's approach. As the main result we provide a formula for calculation of spectral flux of radiation emitted from a pointlike source. The formula employs connecting vectors that are obtained by integrating the ray deviation equation along the reference ray connecting the source and observation event. As a byproduct we formulate Etherington's reciprocity theorem with the inclusion of plasma, interrelating angular size distance and luminosity distance. We also discuss the Liouville theorem and its formulation in terms of connecting vectors.

P-wave reflection-moveout approximation for horizontally layered media of arbitrary moderate anisotropy

We present an approximate nonhyperbolic P-wave moveout formula applicable to horizontally layered media of moderate or weak anisotropy of arbitrary symmetry and orientation. Anisotropy symmetry and its orientation may differ from layer to layer. Instead of commonly used Taylor-series expansion of the square of the reflection traveltime in terms of the square of the offset, we use the weak-anisotropy approximation, in which the square of the reflection traveltime is expanded in terms of weak-anisotropy (WA) parameters. The resulting formula is simple, and it provides a transparent relation between the traveltimes and WA parameters. Along an arbitrarily chosen single surface profile, it depends, in each layer, on the thickness of the layer, on the reference P-wave velocity used for the construction of reference rays, and on three WA parameters specified in the Cartesian coordinate system related to the profile. In each layer, these three “profile” WA parameters depend on “local” WA parameters specifying anisotropy of a given layer in a local coordinate system and on directional cosines specifying the orientation of the local coordinate system with respect to the profile one. The number of local P-wave WA parameters may vary from three for transverse isotropy or six for orthorhombic symmetry to nine for triclinic symmetry. Our tests of the accuracy indicate that the maximum relative traveltime errors do not exceed 0.5% or 2.5% for weak or moderate P-wave anisotropy, respectively.

Effect of black hole–plasma system on light beams

In the paper we discuss propagation of light around Kerr black hole surrounded by non-magnetized cold plasma with infinite conductivity. For that purpose, we use equations for propagation of light rays obtained within Synge's approach in the approximation of geometrical optics. We derive equation of deviation of a ray propagating close to the reference ray, which is a generalization of the well-known Jacobi equation, and use it to calculate the modification of angular distribution of stars observed close to a black hole surrounded by plasma, compared to the uniform star distribution that would be seen without black hole or plasma. We place the observer in the equatorial plane of the Kerr black hole and try various choices of plasma distributions described by mathematically simple formulae. Key features of star distribution on a local sky near the black hole are identified and the influence of plasma on them is discussed.

Reflection moveout approximation for a P-SV wave in a moderately anisotropic homogeneous vertical transverse isotropic layer

A description of the subsurface is incomplete without the use of S-waves. Use of converted waves is one way to involve S-waves. We have developed and tested an approximate formula for the reflection moveout of a wave converted at a horizontal reflector underlying a homogeneous transversely isotropic layer with the vertical axis of symmetry. For its derivation, we use the weak-anisotropy approximation; i.e., we expand the square of the reflection traveltime in terms of weak-anisotropy (WA) parameters. Traveltimes are calculated along reference rays of converted reflected waves in a reference isotropic medium. This requires the determination of the point of reflection (the conversion point) of the reference ray, at which the conversion occurs. This can be done either by a numerical solution of a quartic equation or by using a simple approximate solution. Presented tests indicate that the accuracy of the proposed moveout formula is comparable with the accuracy of formulas derived in a weak-anisotropy approximation for pure-mode reflected waves. Specifically, the tests indicate that the maximum relative traveltime errors are well below 1% for models with P- and SV-wave anisotropy of approximately 10% and less than 2% for models with P- and SV-wave anisotropy of 25% and 12%, respectively. For isotropic media, the use of the conversion point obtained by numerical solution of the quartic equation yields exact results. The approximate moveout formula is used for the derivation of approximate expressions for the two-way zero-offset traveltime, the normal moveout velocity and the quartic term of the Taylor series expansion of the squared traveltime.

Numerical description of mode coupled waves in inhomogeneous magnetized plasmas

Geometrical optics (GO) ray tracing has been widely used for a description of electron cyclotron resonance waves in inhomogeneous magnetized fusion plasmas. However, this reduced approach is not correct in sufficient low density plasmas with a sheared magnetic field, where mode coupling between two electromagnetic-like cold plasma modes can occur. Here, we extend a ray tracing method based on the first-order theory of extended geometrical optics (XGO), which captures mode coupled complex amplitude between O and X mode along the ray trajectory. In our approach, reference ray is calculated with ray equation to satisfy the lowest-order part of XGO theory and an evolution of complex amplitude profile along the reference ray is calculated with partial differential equation derived from first-order terms. Calculation results performed by extended ray tracing are in good agreement with 1D full wave analysis. By introducing second-order terms into our numerical approach, diffraction will be treated.

Comparisons of moveout approximations and inversions for P- and S-waves in acoustic transversely isotropic media with a vertical symmetry axis

Acoustic transversely isotropic medium with a vertical symmetry axis (acoustic VTI) is a very practical anisotropic model for P-wave kinematic analysis. However, apart from P-wave, there is also one additional wave that propagates in this medium, widely known as S-wave artifact. This S-wave possesses bizarre properties and is thoroughly different from the conventional shear waves in elastic VTI media. This paper is devoted to the analysis of S-wave artifact moveout approximations and the Dix-type inversion in acoustic VTI media. We consider four different traveltime approximations originally developed for P-wave and apply them for S-wave moveout. The accuracies of P- and S-waves approximations are numerically compared for both homogeneous and multi-layered acoustic VTI models. With an additional reference ray, the generalized moveout approximation performs the best for S-wave traveltime for a large range of offsets. From the effective parameters estimated for P- and S-waves moveout approximations, we also estimate the interval layer parameters from corresponding Dix-type equations developed for P- and S-waves, respectively. The accuracy and feasibility of the Dix-type inversions are demonstrated in a multi-layered model test.

Generalized non‐hyperbolic approximation for qP‐wave relative geometrical spreading in a layered transversely isotropic medium with a vertical symmetry axis

ABSTRACTCompensation for geometrical spreading along the ray‐path is important in amplitude variation with offset analysis especially for not strongly attenuative media since it contributes to the seismic amplitude preservation. The P‐wave geometrical spreading factor is described by a non‐hyperbolic moveout approximation using the traveltime parameters that can be estimated from the velocity analysis. We extend the P‐wave relative geometrical spreading approximation from the rational form to the generalized non‐hyperbolic form in a transversely isotropic medium with a vertical symmetry axis. The acoustic approximation is used to reduce the number of parameters. The proposed generalized non‐hyperbolic approximation is developed with parameters defined by two rays: vertical and a reference rays. For numerical examples, we consider two choices for parameter selection by using two specific orientations for reference ray. We observe from the numerical tests that the proposed generalized non‐hyperbolic approximation gives more accurate results in both homogeneous and multi‐layered models than the rational counterpart.

Comparison of the anisotropic-ray-theory rays and anisotropic common S-wave rays with the SH and SV reference rays in a velocity model with a split intersection singularity

We describe the behaviour of the anisotropic–ray–theory S–wave rays in a velocity model with a split intersection singularity. The anisotropic–ray–theory S–wave rays crossing the split intersection singularity are smoothly but very sharply bent. While the initial–value rays can be safely traced by solving Hamilton’s equations of rays, it is often impossible to determine the coefficients of the equations of geodesic deviation (paraxial ray equations, dynamic ray tracing equations) and to solve them numerically. As a result, we often know neither the matrix of geometrical spreading, nor the phase shift due to caustics. We demonstrate the abrupt changes of the geometrical spreading and wavefront curvature of the fast anisotropic–ray–theory S wave. We also demonstrate the formation of caustics and wavefront triplication of the slow anisotropic–ray–theory S wave.Since the actual S waves propagate approximately along the SH and SV reference rays in this velocity model, we compare the anisotropic–ray–theory S–wave rays with the SH and SV reference rays. Since the coupling ray theory is usually calculated along the anisotropic common S–wave rays, we also compare the anisotropic common S–wave rays with the SH and SV reference rays.

Minimal size of the focal spot when focusing a field in a waveguide using a vertical array

The paper considers the problem of using a vertical array to focus a monochromatic field in the vicinity of a given point of an underwater sound channel. The solution to the problem is discussed, which has been obtained with a limitation on the initial grazing angles of the emitted waves. The limitation is introduced to increase the stability of the solution to the influence of random perturbations of the sound velocity field. Then, the radiated field represents a wave beam propagating along a reference ray that connects the center of the array and the focusing point. In the geometric optics approximation, analytic estimates of the smallest possible dimensions of the focal spot are obtained. A simplified version of the formula is obtained for the vertical scale of the spot convenient for rough express estimates and comparison of the focusing properties of waveguides with different sound velocity profiles.