Search NASA⌕ Search

SEARCH · Search NASA

Results for “Dipole approximation”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 91 records · Page 5

Efficient Implementation of the Invariant Imbedding T-Matrix Method and the Separation of Variables Method Applied to Large Nonspherical Inhomogeneous Particles

Three terms, ''Waterman's T-matrix method'', ''extended boundary condition method (EBCM)'', and ''null field method'', have been interchangeable in the literature to indicate a method based on surface integral equations to calculate the T-matrix. Unlike the previous method, the invariant imbedding method (IIM) calculates the T-matrix by the use of a volume integral equation. In addition, the standard separation of variables method (SOV) can be applied to compute the T-matrix of a sphere centered at the origin of the coordinate system and having a maximal radius such that the sphere remains inscribed within a nonspherical particle. This study explores the feasibility of a numerical combination of the IIM and the SOV, hereafter referred to as the IIMþSOV method, for computing the single-scattering properties of nonspherical dielectric particles, which are, in general, inhomogeneous. The IIMþSOV method is shown to be capable of solving light-scattering problems for large nonspherical particles where the standard EBCM fails to converge. The IIMþSOV method is flexible and applicable to inhomogeneous particles and aggregated nonspherical particles (overlapped circumscribed spheres) representing a challenge to the standard superposition T-matrix method. The IIMþSOV computational program, developed in this study, is validated against EBCM simulated spheroid and cylinder cases with excellent numerical agreement (up to four decimal places). In addition, solutions for cylinders with large aspect ratios, inhomogeneous particles, and two-particle systems are compared with results from discrete dipole approximation (DDA) computations, and comparisons with the improved geometric-optics method (IGOM) are found to be quite encouraging.

Invariant imbedding method↗

Absorption Efficiencies of Forsterite. I: DDA Explorations in Grain Shape and Size

We compute the absorption efficiency (Q(sub abs)) of forsterite using the discrete dipole approximation (DDA) in order to identify and describe what characteristics of crystal grain shape and size are important to the shape, peak location, and relative strength of spectral features in the 8 - 40 micron wavelength range. Using the DDSCAT code, we compute Q(sub abs) for non-spherical polyhedral grain shapes with a(sub eff) = 0.1 micron. The shape characteristics identified are: 1) elongation/reduction along one of three crystallographic axes; 2) asymmetry, such that all three crystallographic axes are of different lengths; and 3) the presence of crystalline faces that are not parallel to a specific crystallographic axis, e.g., non-rectangular prisms and (di)pyramids. Elongation/reduction dominates the locations and shapes of spectral features near 10, 11, 16, 23.5, 27, and 33.5 micron, while asymmetry and tips are secondary shape effects. Increasing grain sizes (0.1 - 1.0 micron) shifts the 10, 11 micron features systematically towards longer wavelengths and relative to the 11 micron feature increases the strengths and slightly broadens the longer wavelength features. Seven spectral shape classes are established for crystallographic a-, b-, and c-axes and include columnar and platelet shapes plus non-elongated or equant grain shapes. The spectral shape classes and the effects of grain size have practical application in identifying or excluding columnar, platelet or equant forsterite grain shapes in astrophysical environs. Identification of the shape characteristics of forsterite from 8 - 40 micron spectra provides a potential means to probe the temperatures at which forsterite formed.

circumstellar matter↗

Spectrally Consistent Scattering, Absorption, and Polarization Properties of Atmospheric Ice Crystals at Wavelengths from 0.2 to 100 um

A data library is developed containing the scattering, absorption, and polarization properties of ice particles in the spectral range from 0.2 to 100 microns. The properties are computed based on a combination of the Amsterdam discrete dipole approximation (ADDA), the T-matrix method, and the improved geometric optics method (IGOM). The electromagnetic edge effect is incorporated into the extinction and absorption efficiencies computed from the IGOM. A full set of single-scattering properties is provided by considering three-dimensional random orientations for 11 ice crystal habits: droxtals, prolate spheroids, oblate spheroids, solid and hollow columns, compact aggregates composed of eight solid columns, hexagonal plates, small spatial aggregates composed of 5 plates, large spatial aggregates composed of 10 plates, and solid and hollow bullet rosettes. The maximum dimension of each habit ranges from 2 to 10,000 microns in 189 discrete sizes. For each ice crystal habit, three surface roughness conditions (i.e., smooth, moderately roughened, and severely roughened) are considered to account for the surface texture of large particles in the IGOM applicable domain. The data library contains the extinction efficiency, single-scattering albedo, asymmetry parameter, six independent nonzero elements of the phase matrix (P11, P12, P22, P33, P43, and P44), particle projected area, and particle volume to provide the basic single-scattering properties for remote sensing applications and radiative transfer simulations involving ice clouds. Furthermore, a comparison of satellite observations and theoretical simulations for the polarization characteristics of ice clouds demonstrates that ice cloud optical models assuming severely roughened ice crystals significantly outperform their counterparts assuming smooth ice crystals.

Optical properties↗

Constraining Cometary Crystal Shapes from IR Spectral Features

A major challenge in deriving the silicate mineralogy of comets is ascertaining how the anisotropic nature of forsterite crystals affects the spectral features' wavelength, relative intensity, and asymmetry. Forsterite features are identified in cometary comae near 10, 11.05-11.2, 16, 19, 23.5, 27.5 and 33 microns [1-10], so accurate models for forsterite's absorption efficiency (Qabs) are a primary requirement to compute IR spectral energy distributions (SEDs, lambdaF lambda vs. lambda) and constrain the silicate mineralogy of comets. Forsterite is an anisotropic crystal, with three crystallographic axes with distinct indices of refraction for the a-, b-, and c-axis. The shape of a forsterite crystal significantly affects its spectral features [13-16]. We need models that account for crystal shape. The IR absorption efficiencies of forsterite are computed using the discrete dipole approximation (DDA) code DDSCAT [11,12]. Starting from a fiducial crystal shape of a cube, we systematically elongate/reduce one of the crystallographic axes. Also, we elongate/reduce one axis while the lengths of the other two axes are slightly asymmetric (0.8:1.2). The most significant grain shape characteristic that affects the crystalline spectral features is the relative lengths of the crystallographic axes. The second significant grain shape characteristic is breaking the symmetry of all three axes [17]. Synthetic spectral energy distributions using seven crystal shape classes [17] are fit to the observed SED of comet C/1995 O1 (Hale-Bopp). The Hale-Bopp crystalline residual better matches equant, b-platelets, c-platelets, and b-columns spectral shape classes, while a-platelets, a-columns and c-columns worsen the spectral fits. Forsterite condensation and partial evaporation experiments demonstrate that environmental temperature and grain shape are connected [18-20]. Thus, grain shape is a potential probe for protoplanetary disk temperatures where the cometary crystalline forsterite formed. The forsterite crystal shapes (equant, b-platelets, c-platelets, b-columns - excluding a- and c-columns) derived from our modeling [17] of comet Hale- Bopp, compared to laboratory synthesis experiments [18], suggests that these crystals are high temperature condensates. By observing and modeling the crystalline features in comet ISON, we may constrain forsterite crystal shape(s) and link to their formation temperature(s) and environment(s).

anisotropic crystal↗

Scattering and Diffraction of Electromagnetic Radiation: An Effective Probe to Material Structure

Scattered electromagnetic waves from material bodies of different forms contain, in an intricate way, precise information on the intrinsic, geometrical and physical properties of the objects. Scattering theories, ever deepening, aim to provide dependable interpretation and prediction to the complicated interaction of electromagnetic radiation with matter. There are well-established multiple-scattering formulations based on classical electromagnetic theories. An example is the Generalized Multi-particle Mie-solution (GMM), which has recently been extended to a special version ̶ the GMM-PA approach, applicable to finite periodic arrays consisting of a huge number (e.g., >>106) of identical scattering centers [1]. The framework of the GMM-PA is nearly complete. When the size of the constituent unit scatterers becomes considerably small in comparison with incident wavelength, an appropriate array of such small element volumes may well be a satisfactory representation of a material entity having an arbitrary structure. X-ray diffraction is a powerful characterization tool used in a variety of scientific and technical fields, including material science. A diffraction pattern is nothing more than the spatial distribution of scattered intensity, determined by the distribution of scattering matter by way of its Fourier transform [1]. Since all linear dimensions entered into Maxwell's equations are normalized by wavelength, an analogy exists between optical and X-ray diffraction patterns. A large set of optical diffraction patterns experimentally obtained can be found in the literature [e.g., 2,3]. Theoretical results from the GMM-PA have been scrutinized using a large collection of publically accessible, experimentally obtained Fraunhofer diffraction patterns. As far as characteristic structures of the patterns are concerned, theoretical and experimental results are in uniform agreement; no exception has been found so far. Closely connected with the spatial distribution of scattered intensities are cross sections, such as for extinction, scattering, absorption, and radiation pressure, as a critical type of key quantity addressed in most theoretical and experimental studies of radiative scattering. Cross sections predicted from different scattering theories are supposed to be in general agreement. For objects of irregular shape, the GMM-PA solutions can be compared with the highly flexible Discrete Dipole Approximation (DDA) [4,5] when dividing a target to no more than ~106 unit cells. Also, there are different ways to calculate the cross sections in the GMM-PA, providing an additional means to examine the accuracy of the numerical solutions and to unveil potential issues concerning the theoretical formulations and numerical aspects. To solve multiple scattering by an assembly of material volumes through classical theories such as the GMM-PA, the radiative properties of the component scatterers, the complex refractive index in particular, must be provided as input parameters. When using a PA to characterize a material body, this involves the use of an adequate theoretical tool, an effective medium theory, to connect Maxwell's phenomenogical theory with the atomistic theory of matter. In the atomic theory, one regards matter as composed of interacting particles (atoms and molecules) embedded in the vacuum [6]. However, the radiative properties of atomic-scaled particles are known to be substantially different from bulk materials. Intensive research efforts in the fields of cluster science and nanoscience attempt to bridge the gap between bulk and atom and to understand the transition from classical to quantum physics. The GMM-PA calculations, which place virtually no restriction on the component-particle size, might help to gain certain insight into the transition.

Xu, Yu-Lin↗

The Microwave Radiative Properties of Falling Snow Derived from Nonspherical Ice Particle Models. Part II: Initial Testing Using Radar, Radiometer and In Situ Observations

In this study, two different particle models describing the structure and electromagnetic properties of snow are developed and evaluated for potential use in satellite combined radar-radiometer precipitation estimation algorithms. In the first model, snow particles are assumed to be homogeneous ice-air spheres with single-scattering properties derived from Mie theory. In the second model, snow particles are created by simulating the self-collection of pristine ice crystals into aggregate particles of different sizes, using different numbers and habits of the collected component crystals. Single-scattering properties of the resulting nonspherical snow particles are determined using the discrete dipole approximation. The size-distribution-integrated scattering properties of the spherical and nonspherical snow particles are incorporated into a dual-wavelength radar profiling algorithm that is applied to 14- and 34-GHz observations of stratiform precipitation from the ER-2 aircraft-borne High-Altitude Imaging Wind and Rain Airborne Profiler (HIWRAP) radar. The retrieved ice precipitation profiles are then input to a forward radiative transfer calculation in an attempt to simulate coincident radiance observations from the Conical Scanning Millimeter-Wave Imaging Radiometer (CoSMIR). Much greater consistency between the simulated and observed CoSMIR radiances is obtained using estimated profiles that are based upon the nonspherical crystal/aggregate snow particle model. Despite this greater consistency, there remain some discrepancies between the higher moments of the HIWRAP-retrieved precipitation size distributions and in situ distributions derived from microphysics probe observations obtained from Citation aircraft underflights of the ER-2. These discrepancies can only be eliminated if a subset of lower-density crystal/aggregate snow particles is assumed in the radar algorithm and in the interpretation of the in situ data.

Ice particles; Radiances; Radiative transfer; Airc↗

Invariant Imbedded T-Matrix Method for Axial Symmetric Hydrometeors with Extreme Aspect Ratios

The single-scattering properties (SSPs) of hydrometeors are the fundamental quantities for physics-based precipitation retrievals. Thus, efficient computation of their electromagnetic scattering is of great value. Whereas the semi-analytical T-Matrix methods are likely the most efficient for nonspherical hydrometeors with axial symmetry, they are not suitable for arbitrarily shaped hydrometeors absent of any significant symmetry, for which volume integral methods such as those based on Discrete Dipole Approximation (DDA) are required. Currently the two leading T-matrix methods are the Extended Boundary Condition Method (EBCM) and the Invariant Imbedding T-matrix Method incorporating Lorentz-Mie Separation of Variables (IITM+SOV). EBCM is known to outperform IITM+SOV for hydrometeors with modest aspect ratios. However, in cases when aspect ratios become extreme, such as needle-like particles with large height to diameter values, EBCM fails to converge. Such hydrometeors with extreme aspect ratios are known to be present in solid precipitation and their SSPs are required to model the radiative responses accurately. In these cases, IITM+SOV is shown to converge. An efficient, parallelized C++ implementation for both EBCM and IITM+SOV has been developed to conduct a performance comparison between EBCM, IITM+SOV, and DDSCAT (a popular implementation of DDA). We present the comparison results and discuss details. Our intent is to release the combined ECBM IITM+SOV software to the community under an open source license.

Pelissier, Craig↗

Single-Scattering Properties of Melting Precipitation for GPM Passive Microwave and Radar Remote Sensing Applications

Over the past two decades, detailed computational simulations of the intricate three dimensional structures of ice-phase crystals and aggregates of those crystals have been developed. The microwave single-scattering properties of these simulated ice particles have been computed and used to improve quantitative estimates of snow rates and to better deRne the vertical structure of snow water contents in deep convective systems, as derived from satellite-borne passive microwave and/or radar remote sensing measurements. The same icephase particles have more recently been used as the starting point for simulations of melting precipitation using computational melting methods. In the current study, a heuristic melting method, as well as a physically-based melting procedure based on smoothed-particle hydrodynamics, are applied to ice particle models to describe the full evolution of the particles from dry snow to liquid drops. The discrete dipole approximation is utilized to calculate the single-scattering properties of the mixed-phase particles throughout the melting process. Then, the properties of the particles are “mapped” into simpliRed microphysical simulations of particle size spectra in the melting layers of stratiform, precipitating clouds. The bulk single-scattering properties of the melting layers and the sensitivity of their properties to modeling assumptions are explored, and the implications for combined radar-radiometer precipitation remote sensing from GPM are discussed.

William S Olson↗

Towards a Mass-Consistent Methodology for Realistic Melting Hydrometeor Retrieval

To address the acute challenge posed by the melting layer to accurate surface precipitation retrievals from space, we en-sure the compositional consistency in ice, liquid, and total masses of synthetic melting hydrometeors with a method of stochastic compensation. The method is applied to simulated melting hydrometeors prior to calculating their scattering properties using the discrete dipole approximation (DDA). We investigate the impact of this stochastic compensation to calculated scattering properties by contrasting it with a naïve approach and report our findings.

precipitation remote sensing↗

Evaluation of Higher-order Quadrature Schemes in Improving Computational Efficiency for Orientation-averaged Single-Scattering Properties of Nonspherical Ice Particles

We evaluate several high-order quadrature schemes for accuracy and efficacy in obtaining orientation-averaged single-scattering properties (SSPs). We use the highly efficient MIDAS to perform electromagnetic scattering calculations to evaluate the gain in efficiency from these schemes. MIDAS is shown to be superior to DDSCAT, a popular discrete dipole approximation (DDA) method. This study is motivated by the fact that quality physical precipitation retrievals rely on using accurate orientation-averaged SSPs derived from realistic hydrometeors as input to radiative transfer Models (RTMs). The DDA has been a popular choice for single-scattering calculations, due to its versatility with respect to target geometry. However, being iterative-solver-based (ISB), the most used DDA codes, e.g. DDSCAT and ADDA, must solve the scattering problem for each orientation of the target separately. As the size parameter and geometric anisotropy of the hydrometeor increase, the number of orientations needed to obtain accurate orientation-averages can increase drastically and so does the computation cost incurred by the ISB-DDA methods. MIDAS is a Direct-Solver-Based (DSB) code, its decomposition of the original large matrix with a high rank into multiple more manageable smaller matrices of lower ranks makes it much more computationally efficient while maintaining excellent accuracy. In addition, direct solvers consider all requested orientations at once, giving MIDAS further advantage over popular ISB-DDA methods. MIDAS, when combined with high-order quadrature for orientation averaging, can be greater than three orders of magnitude more efficient in obtaining RTM-ready SSPs of complex-shaped hydrometeors than existing ISB-DDA methods, with the native quadrature schemes they offer.

Ines Fenni↗

Sensitivity of Single-Scattering Properties to Precipitation Particle Meltwater Geometry for GPM Passive Microwave and Radar Remote Sensing Applications

The objective of the current work is to describe the microwave single-scattering properties of partially melted ice-phase precipitation particles using physically-based computational methods, taking into full account the varied geometries of the initially “dry” ice particles and the distributions of liquid water that develop during the melting process. Ultimately, the bulk properties of ensembles of these partially-melted particles will be included in “scattering tables” to support radar and combined radar-radiometer remote sensing of precipitation. A meshless Lagrangian melting procedure (snowMELT) based on smoothed-particle hydrodynamics is applied to both spherical and finely-structured ice particle models to describe the full evolution of the particles from dry ice particles to liquid drops. The melting of spherical ice particles using snowMELT is compared to an alternative continuum physics model to validate melt times and internal thermodynamics. The discrete dipole approximation is then utilized to calculate the single-scattering properties of different mixed-phase particles throughout the melting process. The sensitivities of particle single-scattering properties to thermodynamic and hydrodynamic assumptions in snowMELT are explored, and the implications for combined radar-radiometer precipitation remote sensing from GPM are discussed.

William S Olson↗

The Polarimetric Radar Scattering Properties of Oriented Aggregates

The scattering properties of aggregates are studied herein. Early aggregates (<7 monomers) of branched planar crystals and mature aggregates (up to 100 monomers) of columns are randomly generated with varying assumptions about the monomer attachment processes and the orientation behavior during collection. The resulting physical properties of the aggregates correspond well with prior in situ and retrieved sizes and shapes. Assumed azimuthally uniform orientations during collection and monomer pivoting upon attachment resulted in flatter and denser aggregates. The column aggregates had lower density and more spherical shapes than the branched planar crystal aggregates. The scattering properties were calculated using the discrete dipole approximation for a set of orientation angles and transformed to spectral coefficients representing modes of orientation angle variability. The zeroth- and second-order coefficients dominate this variability, with the zeroth-order coefficients representing the scattering properties for randomly oriented particles. The second-order coefficients for backscatter showed differences between horizontal and vertical polarization increasing with density, and these coefficients for specific differential phase increase with both mass and density. Similarly, coefficients for the copolar covariance decreased with density. Rapid changes in the contributions to the radar moments from the second-order coefficients from low to moderate density were observed, likely due to the increasing presence of horizontally aligned monomers in the aggregate structure. Differences in how differential reflectivity and correlation coefficient evolve with the orientation distribution parameters suggest that these measurements, along with specific differential phase and reflectivity, provide complementary information about aggregate sizes, shapes, and orientation distributions.

Cloud microphysics↗

Developing a Radar Signal Simulator for the Community Radiative Transfer Model

Active radar instruments provide vertically resolved clouds and precipitation measurements that cannot be provided by the passive instruments. These active measurements are not conventionally assimilated into the data assimilation systems because of the lack of fast forward radiative transfer models and also difficulties in the error modelling of the measurements. This paper describes the development, evaluation, and sensitivity analysis for a forward radar model implemented in the Community Radiative Transfer Model (CRTM). The scattering properties required by the forward model are provided by the hydrometeor lookup tables that were generated using the discrete dipole approximation. The model is able to calculate both the reflectivity and the attenuated reflectivity for any given radar instrument at any given zenith angles as long as CRTM instrument specific coefficients are available. The evaluation using CloudSat measurements shows a very good agreement between the simulations and measurements as long as the input profiles of hydrometeors are consistent with the measured reflectivity profiles. Major sources contributing to the differences between the measured and simulated reflectivities are input hydrometeor profiles, scattering lookup tables, lack of melting layer in the forward model, CRTM scattering solvers, and attenuation calculations. In addition to the forward model, both Tangent Linear and Adjoint of the model are also implemented and tested within CRTM. These components may be required by some data assimilation systems for the assimilation of radar measurements.

radar↗

A Comprehensive Forward Model for Spaceborne Radar Instruments

We present the development and validation of a comprehensive forward model designed to enhance remote sensing capabilities of spaceborne radar instruments. To overcome limitations in existing models, we integrated a Discrete Dipole Approximation (DDA) cloud scattering database into our Radiative Transfer Model (RTM), focusing on microwave frequencies. By simulating the optical properties of non-spherical frozen hydrometeors, the DDA technique effectively reduced discrepancies between simulated and observed values, surpassing traditional Mie tables. The evaluation of DDA lookup tables involved comparisons with a collocated dataset comprising short-term forecasts and satellite microwave data, providing evidence of their superiority. Additionally, we address the challenges of assimilating active radar measurements, which offer vertically resolved insights into clouds and precipitation. We explored the assimilation of spaceborne radar measurements in Numerical Weather Prediction (NWP) models by integrating a forward radar model, along with its adjoint and tangent linear, into the data assimilation system. Evaluation using CloudSat measurements demonstrated promising agreement between simulations and observations, particularly when the input hydrometeor profiles aligned with the measured reflectivity profiles, showcasing the potential of the developed forward radar model. Moreover, we discuss other challenges in radar measurement assimilation within NWP models, including potential observation errors and biases.

Isaac Moradi↗

Assimilation of Active MW and Radar Observations in the NWP Models

Passive instruments lack the ability to provide vertically resolved measurements of clouds and precipitation, which can be obtained by active radar instruments. However, incorporating these active measurements into data assimilation systems presents challenges due to the absence of fast forward radiative transfer models and difficulties in error modeling. This talk focuses on the development, evaluation, and sensitivity analysis of a forward radar model integrated into CRTM. The forward radar model utilizes scattering properties obtained from hydrometeor lookup tables generated using the discrete dipole approximation. By utilizing CRTM instrument-specific coefficients, the model can calculate both reflectivity and attenuated reflectivity for any given radar instrument and zenith angles. Evaluation using CloudSat measurements demonstrates a strong agreement between simulations and observations when the input profiles of hydrometeors align with the measured reflectivity profiles.

Isaac Moradi↗

Simulating Microwave and Radar Signals in Severe Weather Conditions

The development of radiative transfer simulators for radar and microwave signals, spanning a range of frequencies from 10 to 800 GHz, is paramount for enhancing weather forecasting accuracy, particularly for severe weather events. These tools facilitate the assimilation of microwave and radar observations into numerical weather prediction models, thereby improving accuracy of weather forecasts. Additionally, they directly simulate radar signals crucial for autonomous vehicle operation, with frequencies commonly used in radar systems such as 24, 74, 77, and 79 GHz. However, these frequencies are susceptible to weather phenomena like severe rain and snow, which can significantly impact vehicle safety and performance. These simulation tools, including radiative transfer simulators, serve a dual purpose. Firstly, they enhance forecasts for severe weather events, contributing to autonomous vehicle safety by providing early warnings and risk assessments. Secondly, they enable the direct simulation of radar signals in autonomous vehicle driving systems, allowing researchers and engineers to evaluate radar system performance under various weather conditions. In addition to the radar signal simulator, this abstract discusses the incorporation of advanced scattering properties developed using the discrete dipole approximation (DDA). The DDA technique enhances scattering calculations for frozen hydrometeors at microwave frequencies, thereby improving the accuracy of radar signal simulations and enabling more accurate assessments of radar system performance in adverse weather conditions. In summary, this abstract explores the development and utilization of comprehensive simulation tools, emphasizing their significance in simulating microwave and radar signals and improving weather forecasts. Special attention is given to the simulation of radar signals at critical frequencies for autonomous vehicle sensing and navigation, addressing challenges posed by severe weather phenomena and their effects on signal propagation and detection

Isaac Moradi↗

Higher-order ionospheric effects on the GPS observables and means of modeling them

Based on relaistic modeling of the electron density of the ionosphere and using a dipole moment approximation for the earth magnetic field, we are able to estimate the effect of the ionosphere on the Global Positioning Systems (GPS) signal for a ground user. The lowest-order (1/f(exp 2)) effect, which is of the order of .1 - 30 meters of zenith group delay, is subtracted out by forming a linear combination of the dual frequencies of the GPS signal. One is left with second- (1/f(exp 3)) and third-order (1/f(exp 4)) effects which are estimated typically to be approximately 0 - 2 cm, and approximately 0 - 2 mm at zenith respectively, depending on the time of day, time of year, the solar cycle and the relative geometry of the magnetic field and the line of sight. Given the total electron content along a line of sight, we derive an approximation to the second-order term which is accurate to approximately 90% within the magnetic dipole moment model; this approximation can be used to reduce the second-order term to the millimeter level, thus potentially improving precise positioning in space and on the ground. The induced group delay, or phase advance, due to second- and third-order effects are examined for two ground receivers located at equatorial and mid-latitude regions tracking several GPS satellites.

Bassiri, Sassan↗

Modeling the global positioning system signal propagation through the ionosphere

Based on realistic modeling of the electron density of the ionosphere and using a dipole moment approximation for the Earth's magnetic field, one is able to estimate the effect of the ionosphere on the Global Positioning System (GPS) signal for a ground user. The lowest order effect, which is on the order of 0.1-100 m of group delay, is subtracted out by forming a linear combination of the dual frequencies of the GPS signal. One is left with second- and third-order effects that are estimated typically to be approximately 0-2 cm and approximately 0-2 mm at zenith, respectively, depending on the geographical location, the time of day, the time of year, the solar cycle, and the relative geometry of the magnetic field and the line of sight. Given the total electron content along a line of sight, the authors derive an approximation to the second-order term which is accurate to approximately 90 percent within the magnetic dipole moment model; this approximation can be used to reduce the second-order term to the millimeter level, thus potentially improving precise positioning in space and on the ground. The induced group delay, or phase advance, due to second- and third-order effects is examined for two ground receivers located at equatorial and mid-latitude regions tracking several GPS satellites.

Bassiri, S.↗