Journal of Geophysical Research: Atmospheres

Development of a curved ray tracing method for modeling of phase paths from GPS radio occultation: A two-dimensional study

Authors


Abstract

[1] A two-dimensional curved ray tracer (CRT) is developed to study the propagation path of radio signals across a heterogeneous planetary atmosphere. The method, designed to achieve improvements in both computational efficiency and accuracy over conventional straight-line methods, takes rays' first-order bending into account to better describe curved raypaths in the stratified atmosphere. CRT is then used to simulate the phase path from GPS radio occultation (RO). The merit of the ray tracing approach in GPS RO is explicit consideration of horizontal variation in the atmosphere, which may lead to a sizable error but is disregarded in traditional retrieval schemes. In addition, direct modeling of the phase path takes advantage of simple error characteristics in the measurement. With provision of ionospheric and neutral atmospheric refractive indices, in this effort, rays are traced along the full range of GPS-low Earth orbiting (LEO) radio links just as the measurements are made in real life. Here, ray shooting is employed to realize the observed radio links with controlled accuracy. CRT largely reproduces the very measured characteristics of GPS signals. When compared, the measured and simulated phases show remarkable agreement. The cross validation between CRT and GPS RO has confirmed not only the strength of CRT but also the high accuracy of GPS RO measurements. The primary motivation for this study is enabling effective quality control for GPS RO data, overcoming a complicated error structure in the high-level data. CRT has also shown a great deal of potential for improved utilization of GPS RO data for geophysical research.

1. Introduction

[2] In recent years, measurements from the Global Positioning System (GPS) radio occultation (RO) technique have increasingly been considered a valuable source of information for monitoring terrestrial atmosphere and space weather. After the successful campaign of the proof-of-concept GPS Meteorology (GPS/MET) mission from 1995 to 1997 [Ware et al., 1996; Rocken et al., 1997], several missions such as Ørsted, Stellenbosch UNiversity SATellite (SUNSAT), CHAllenging Minisatellite Payload (CHAMP) [Reigber et al., 2004], Satélite de Aplicaciones Cientificas-C (SAC-C) [Colomb et al., 2004], and Gravity Recovery and Climate Experiment (GRACE) [Tapley et al., 2004] followed. More recently the Constellation Observing System for Meteorology, Ionosphere and Climate (COSMIC) was launched in April 2006, and it is providing an unprecedented amount of data from the six-satellite network [Anthes et al., 2008]. These experiments have confirmed the unique strengths of the GPS RO technique, which attracts a wide spectrum of applications [Kursinski et al., 1996; Hajj et al., 2002; Wickert et al., 2004; Kuo et al., 2004].

[3] The primary observable quantity in the GPS RO technique is precise measurement of the carrier phase typically available at two GPS L-band frequencies, f1 = 1.57542 GHz (L1) and f2 = 1.2276 GHz (L2), recorded by a GPS receiver on board a low Earth orbiting (LEO) satellite tracking a setting or rising GPS satellite. The measured phase varies essentially with atmospheric conditions, relative orbital motion between transmitter and receiver, and satellite clock drift errors that include relativistic effect. The satellite clock errors can be corrected with the aid of auxiliary links (i.e., differencing techniques which make use of additional ground stations or satellites), whereas phase variations due to the orbital motion are estimated with the knowledge of precise positions and velocities of the satellites. Minor phase residual consists of receiver thermal noise, hardware delays, and cycle slips. When all other terms are suitably assessed, the phase variation that relates to characteristics of the Earth's atmosphere, the so-called excess phase, can be isolated. Thanks to the time-varying geometry between a pair of GPS and LEO satellites, the collected measurements during an event of occultation can be used to infer the spatial structure of the atmosphere at the time.

[4] During the past few decades, considerable efforts have been made to assess ray bending angles in the atmosphere from phase (or along with amplitude) measurements. These include methods of geometric optics [Kursinski et al., 1997], back propagation [Gorbunov et al., 2000], sliding spectrum [Sokolovskiy, 2001], canonical transform [Gorbunov, 2002], and full spectrum inversion [Jensen et al., 2003]. These methods differ in their time-frequency representation of received signals. Meanwhile, the atmosphere that embodies the time-frequency content is heterogeneous. It is thus natural that inversion efforts, recovery of atmospheric structure from the measured time series, seek to take the horizontal variation in the atmosphere into account. Nonetheless, the problem is generally complicated and it is not easy to work out a solution. If the atmosphere were spherically symmetric as a special case, however, the reconstruction would be reduced to a one-dimensional problem that is considerably easier to deal with. For this reason, the aforementioned methods share the premise that the atmospheric refractive index is spherically symmetric. Typically, the estimated profile of the bending angle is then converted to a profile of the refractive index via an Abel transform. Afterward, other higher-level products (e.g., temperature, pressure, water vapor pressure, and electron density) could also be derived from the refractive index. In doing so, however, the horizontal gradient in the atmosphere would be deserted.

[5] Phase measurement has been widely proven to be precise and accurate for research into the terrestrial atmosphere. The good quality of phase measurement is also well assessed by navigational and positioning studies. However, retrieved parameters from phase measurement are subject to various errors. For instance, the bending angle, derived under the assumption of spherical symmetry, is unable to explain any indication of atmospheric asymmetries in the recorded phase, which is thus considered to be a sort of error or noise. The resulting “error” is then carried forward through the data processing chain. Beginning with these errors in the phase and bending angle, others (some random, others systematic) might be introduced to higher-level parameters at several points as the data processing proceeds. Mathematical operations inside the data stream transform one variable's error to another's, and propagate them to the immediate vicinity, so that the error structure becomes increasingly complicated for higher-level parameters. Some of the operations (e.g., low-pass filtering and Abel transform) which are nonlinear in nature make it even more cumbersome to properly characterize the retrieval errors.

[6] The key motivation for this study is to improve quality control for GPS RO data. As implied above, ensuring effective quality control for higher-level data is challenging, since their error structure is complicated. For example, it may be the case that a “bad” temperature profile, obtained from a phase measurement dubious in quality, by chance shows fairly good agreement with collocated data sets. Probably, the misleading agreement is due to a deceptive interplay between the errors in both RO and verifying data. It is also possible that the error propagation in the data stream makes a distinguished phase error dispersed and turns it into a temperature error as moderate as the error in the verifying data can easily mask. Nonetheless, the measurement of phase has much simpler error characteristics. Therefore, one could identify the measurement errors with an elevated assurance and possibly correct some of those occurring first, so the rectified errors do not remain in the data chain afterward. We anticipate retrievals of improved quality when such correction is possible. Even if such a correction is impractical, at least the error detection itself is informative for the quality assurance of individual occultations: the data screening at this stage is so effective that one can obtain a quality-controlled data set while discarding a lesser number of useful occultations. This reduces the risk that good retrievals are erroneously rejected, so one could rescue a valuable piece of information that compares poorly with verifying data but indeed is of good quality.

[7] In identifying possible flaws in measured phase, a realistic simulation of the measurement is quite useful. The forward modeling of wave propagation, calculating a raypath for a known atmospheric medium, is a well-understood problem. In this study, we first establish realistic environments in which measurements are likely to be made in real life, by combining various sources of information. This includes information on the ionosphere and plasmasphere as well as the neutral atmosphere. We then simulate the phase measurement using an advanced ray tracing method, and we next compare the modeled phase with the measured one. Ray tracing (RT) is computationally more efficient than solving the wave equation in complex structures, for instance, in a moving source of signals such as a GPS satellite.

[8] There have been some RT studies for GPS RO that simulate bending angles for the neutral atmosphere [e.g., Zou et al., 2000; Healy, 2001; Liu et al., 2001; Zou et al., 2004; Poli and Joiner, 2004]. Without any explicit treatment of the ionosphere, the previous studies commonly circumvented ionospheric bending by using the well-known “ionosphere-corrected” combination of L1 and L2 bending angles. The frequency-weighted linear combination eliminates the first-order ionospheric effect, which is in proportion to carrier frequency. This combination works perfectly when the ionosphere does not cause rays to bend and the two (L1 and L2) raypaths coincide. In reality, however, neither occurs. As will be discussed later, explicit consideration for the ionosphere yields more accurate neutral atmospheric bending angles, especially for high altitudes where the ionospheric bending is significant. This also reduces the error due to the split between L1 and L2 rays. Meanwhile, one of the major obstacles in modeling the phase measurement with RT is computational cost. The factors responsible for the high cost are as follows: first, long GPS-LEO distances along which rays travel, requiring that ray equations be integrated; second, demands on extracting and storing all necessary information needed to describe time-varying observing geometry that changes from one epoch to another in three full dimensions; and, last, multiple iterative end-to-end tracing of a ray for each epoch to realize the observed GPS-LEO link, so-called ray shooting (RS). Earlier studies are based on the traditional RT that approximates a short raypath as straight line. In general, the refractive index either in the neutral atmosphere or in the ionosphere is strongly stratified. Consequently, a ray seldom travels along a straight line. Accordingly, the straight-line ray tracing (SLRT) has to divide a segment of a raypath into smaller pieces in order to achieve an acceptable level of accuracy as the ray bends more and more. The situation is no different even for a climatological atmosphere that does not contain any small-scale structures, but still has a strong vertical gradient in the refractive index. Devoting our attention to this fact, we developed a curved ray tracing (CRT) method that well represents rays' first-order bending. This allows one to use a longer step size for the same error tolerance, leading to remarkable cost reduction.

[9] Most previous RT studies assumed that the exact location of a ray's tangent point (the closest approach to the Earth), where the RT was initiated, was known. In fact, the position is a mere estimate provided by conventional RO processing methods in connection with the bending angles derived under the assumption of spherical symmetry. Unless the atmosphere is spherically symmetric, therefore, a ray initiated from the a priori tangent point does not end up at actual GPS and LEO positions. This compels a repeated adjustment of the tangent point until the ray eventually hits the two satellites at the same time, or at least until the observed satellite-to-satellite angle seen from the Earth's center is reproduced. Otherwise, the simulated bending angle lacks a basis in reality. Even after a successful RS, furthermore, the merit of simulation still varies depending on the availability or reality of a model ionosphere. In the meantime, each iteration in the RS requires a complete RT along the whole path between the two satellites. The RS terminates its iteration when a candidate path approaches the two satellites concurrently, closer than the specified error tolerance. Given that the RS has to be applied individually to each epoch of an occultation (with 50 Hz epoch rate, for instance, 3000–6000 epochs are available for each occultation that lasts 1–2 min), an effective RS leads directly to a great deal of cost reduction. As this is irrelevant to the effectiveness of RT, separate efforts are made in this study to accelerate the RS.

[10] The uniqueness and strength of our approach, which we believe can overcome most of the aforementioned problems, are summarized as follows: (1) distribution of electron density in the ionosphere and plasmasphere is explicitly modeled and used to describe the ionospheric medium of interest; additionally, the neutral atmosphere above the top of a numerical weather prediction (NWP) model and up to a high enough altitude (e.g., 200 km) is presented with a realistic empirical model; (2) a novel, effective CRT, which has the potential that characterizes accurate ray parameters along the entire radio link from GPS to LEO satellites across horizontally inhomogeneous atmospheric media, is developed; (3) an operative RS is sought to accomplish the observed radio links; and (4) “raw” phase measurements, instead of bending angles, are directly simulated.

[11] The outline of the paper is as follows. We describe the CRT in section 2. In section 3 the data sources and numerical procedures used to simulate refractive indices are presented; then the RS employed to fulfill the observed radio links is described. Key findings from our simulations are explained through a few occultations in section 4. A summary and outlook are given in section 5.

2. Curved Ray Tracing

[12] The theoretical basis for ray tracing is geometric optics. For basic concepts and equations, readers are referred to a book by Born and Wolf [1964]. In terms of geometric optics, the propagation of radio waves in an atmospheric medium can be described by the eikonal equation

equation image

where S is the eikonal and n is the refractive index. The major approximation that leads to the eikonal equation is the asymptotic high-frequency (short-wavelength) limit of Maxwell's equations, which implies that the medium contains only perturbations larger than the wavelengths of GPS signals (approximately 19 cm for L1 and 24 cm for L2). The physical significance of the eikonal is that surfaces of constant S correspond to phase fronts of the propagating wave, while rays are defined as orthogonal trajectories to the phase fronts. For a wave emanating from a single point source, eikonal solvers completely determine the three-dimensional structure of the phase fronts. For a moving source like the transmitter on GPS satellites, however, the repetitive construction of phase fronts conditional on each location of the source makes their application impractical.

[13] The eikonal equation can be transformed into the ray equations, so one can calculate the raypath of interest directly without having to construct the wavefronts. A commonly used form of the ray equations is a set of first-order differential equations:

equation image
equation image

where r is the position vector, ds is the differential displacement along the raypath, and t is the tangent unit vector of the raypath. This first-order system can be solved with standard numerical techniques such as the Runge-Kutta method. The required inputs for RT are the initial position r0, initial tangent unit vector t0, step size Δs, and refractive indices and their spatial gradients along the computed raypath.

[14] Taking advantage of the atmospheric structure that is spherically symmetric to a good extent, the above ray equations can be expressed more conveniently in polar coordinates (r,θ):

equation image
equation image
equation image

where r and θ are the radius and polar angle (or angular coordinate) at an arbitrary point on the raypath, respectively, and ψ is the local zenith angle of the raypath. We define θ as the clockwise rotation from the positive y axis and ψ as the clockwise measure of the tangent vector from the local zenith. Note that equations (3a)(3c) are equivalent with equations (2a) and (2b), except for the coordinate transform involved, and so are their integrations.

[15] In the first-order Euler's method, RT is carried out in a piecewise-linear way, meaning that a neighboring new position on a raypath is on the line collinear with the tangent vector at previous location. When planetary-scale propagation in the atmosphere is considered, rays are bent in order to traverse the stratified atmosphere as quickly as possible (Fermat's principle). This demands that a path segment has to be divided into smaller integration steps to a degree that straight lines can approximate the actual curved path within an acceptable accuracy. This is an inherent problem of the straight-line approach that arises from approximating a curve as a set of line segments. Higher-order schemes may reduce the numerical error by trading off some computational costs. In most practical problems, however, refractive index and their spatial gradient are not readily available in analytical forms. Instead, these local values have to be evaluated from computational grids that are limited in resolution. The relevant interpolation results in nonnegligible errors and so makes it less attractive to use the higher-order schemes. Another problem is that the path length undertaken by connected line segments following a curve is always shorter than actual arc length, leading to systematically underestimated phase paths.

[16] In the differential geometry of curves, the Frenet-Serret formulas [Born and Wolf, 1964] provide methods to describe smooth curves in space. In the two-dimensional case, the formulas are

equation image
equation image

where κ is the curvature whose magnitude is ∥t′∥. The curvature indicates the deviance of the curve from a straight line. The normal unit vector u is defined as

equation image

which is perpendicular to the tangent unit vector t and points to the curvature center. When applied to RT, the Frenet-Serret formulas offer an intuitive geometric description of a raypath. Particularly, equation (4a) corresponds to (3a), while the right-hand side of (3a) relates the curvature to refractive index and the ray's direction in polar coordinates.

[17] RT starts off as point-to-point computation and is carried out in sequence by evaluating state parameters (i.e., r, θ, and ψ) at each point along the computed raypath. Additionally, a curve that connects two neighboring points has to be suitably parameterized to provide the path length between the points. In order to choose the parameterization that best suits a specific problem, one may deliberate over the structure and complexity of the medium that has to be dealt with, and the scale of interest in the medium. We have chosen the osculating circle, which has radius R =|κ|−1 and whose center lies along a line collinear with u. The osculating circle is simple and well appropriate, we believe, for large-scale terrestrial studies. On the basis of the parameterization and thanks to the relation between equations (3a) and (4a), a ray between P1 and P2 (whose position vectors are denoted by r1 and r2, respectively) can be traced in a piecewise-circular manner, as illustrated in Figure 1. The key idea here is that a new (ending) position of path P2 lies on the osculating circle defined at current (starting) position P1. Therefore, one can determine r2 by rotating r1rc (subscript c denotes the ray's curvature center) by an angle ɛ about the binormal unit vector, b = t × u, at the curvature center. The procedure starts with computing κ from the spatial gradient of the refractive index and the ray's direction (i.e., right-hand side of 3a). The sign of κ determines the correct u between two unit vectors normal to t in the osculating plane:

equation image
Figure 1.

Geometry of the piecewise-circular ray tracing for a segment of the raypath that connects P1 and P2, in the polar coordinates whose center is the curvature center of the Earth's ellipsoid O. Position vectors of P1 and P2 are denoted by r1 and r2, respectively. Here the curve is parameterized with an arc of the osculating circle, defined by the local Frenet-Serret system (t,u,b) at P1, with radius Rc and center Oc. The new endpoint P2 is determined by rotating r1rc by an angle ɛ about the binormal unit vector, b = t × u, at the curvature center. The upper left inset depicts the coordinate system involved and also ray parameters (r,θ,ψ) in the coordinates.

[18] For right-propagating rays with the geometry shown in Figure 1, a positive (negative) curvature represents the raypath as concave (convex) when viewed from the Earth's center. The position vector of ray's curvature center rc is given by

equation image

where Rc is the curvature radius. The rotation angle ɛ determines the step size ds, which in turn controls the accuracy of the integration. Here ds can be point-by-point adaptive along an intact transmitter-receiver path to achieve optimal efficiency in computation. To do so, rigorous theoretical analysis might be required; however, this is beyond the scope of this study. Given that κ is pertinent to a local complexity of the medium, the region of large curvature (small radius) may require a smaller step size. On the basis of that idea, in this study ɛ is kept fixed in size throughout the integration, so ds changes in proportion to the curvature radius. Being closely relevant to ɛ, as long as no backward propagation of the wave occurs, the polar angle θ also serves well as the independent variable for the integration; however, ɛ is preferred in this study as it enables ds to be adaptive to κ. For RTs in the polar coordinates, on the other hand, it is customary to integrate with respect to r, the radius from the Earth's center. However, using r is troublesome, especially near the tangent point, since r may not vary monotonically during the integration. It is also difficult to control the step size for the reason that a ray's direction can make a sizable difference in ds at the point.

[19] The relation among ray parameters is now described. The radius of position vector r2 can be written as

equation image

where

equation image

[20] The arc length ds between P1 and P2 is given by

equation image

and the chord length is computed as

equation image

[21] Using equation (3a), the local zenith angle at P2 is estimated as ψ2 = ψ1 + κds + θ1θ2. The optical path for the segment is approximated as dΦ = equation imageds, where equation image is the refractive index evaluated at the middle point of the segment. At this point, all parameters that describe the position and direction of the ray are known at P2 and thus the integration proceeds to the next segment. Once RT for the whole path that connects a transmitter (Tx) and its receiver (Rx) is completed, the excess phase path L is computed as

equation image

where Φ and ρ are the total optical path and the geometric distance between the transmitter and the receiver, respectively.

3. Data and Numerical Procedure

3.1. GPS Radio Occultation Data

[22] The GPS RO data used in this study are produced by the COSMIC Data Analysis and Archive Center (CDAAC) at the University Corporation for Atmospheric Research (UCAR). Here the GPS RO processing algorithm described by Schreiner et al. (preprint, 2002) is applied (data as well as more information on processing are available online at http://www.cosmic.ucar.edu). The procedure includes acquisition of raw LEO data, GPS orbit information, and ground station data; precise orbit determination; and isolation of phase measurement. The orbit of LEO is determined through a postprocessing mode in which the precise orbits of GPS satellites and information on ground stations published by the International Global Navigation Satellite System Service (IGS) are utilized. This provides an accurate solution to phase measurements and accordingly builds a condition favorable to the error assessment of our forward modeling by reducing the uncertainty between errors of measurement and the model. Description of preprocessing methods used by other research groups can be found elsewhere [e.g., Kursinski et al., 1997; Wickert et al., 2002; Hajj et al., 2004].

[23] Although our discussion in this paper is limited to a small number of examples, we have simulated 23,918 CHAMP and 19,309 SAC-C occultations over a 4 month period, May–August 2002. This was needed to ensure the stability and performance of CRT. Whenever possible, all L1 and L2 phase measurements of 50 Hz epoch rate are simulated. This necessitates RTs for about 10,000 epochs on average for each occultation. Time-varying positions of GPS and LEO satellites are also used in order to fulfill the measured GPS-LEO radio link by means of an RS. Although possible in the framework of RT [see Kursinski et al., 1997], no attempt is made to simulate amplitude measurements. Instead, we resort to measured signal-to-noise ratio (SNR) in order to assure the data quality of individual occultations.

[24] For carrier frequency fi, measured phase path Φi can be modeled in distance units as

equation image

where ρ is the range between GPS and LEO satellites, η is the extra delay due to the neutral atmosphere, and I is the ionospheric group delay pertinent to total electron content (TEC) along the raypath. The term bi represents the measurement bias that is involved with the unknown number of cycles or turns of phase in the initial measurement when a receiver locks onto the signal carrier of a GPS satellite. This quantity, often called initial ambiguity, is constant during an occultation event unless any cycle slip occurs. The term ɛi represents phase residuals such as receiver thermal noise, local multipath errors, and clock errors. The residuals are assumed to be time-independent, zero-mean Gaussian errors. We regard this term as measurement error (or noise) in the phase path. Readers are referred to works by Leick [1995], Teunissen and Kleusberg [1998], Hofmann-Wellenhof [2001], and Hajj et al. [2002] for in-depth description of the GPS observation equation. Subtracting the range from the phase path yields the excess phase path: Li = Φiρ.

3.2. Refractive Index

[25] RT requires knowledge of the refractive index along a raypath as an input parameter. The refractivity N can be approximated to first order by

equation image

where n is the refractive index, T is the temperature in K, P is the (total) pressure in hPa, Pw the is water vapor pressure in hPa, ne the is electron number per cubic meter, and f is the GPS carrier frequency in Hz. The coefficients are k1 = 77.6 hPa K−1, k2 = 3.73 × 105 K2 hPa−1, and k3 = 4.03 × 107m3 s−2. Here the neutral atmospheric part is modeled according to Smith and Weintraub [1953] and the ionospheric contribution is the first-order expansion of the Appleton-Hartree formula [Budden, 1985]. In the following, we describe the data sources used to model the refractive index for the neutral atmosphere and the ionosphere and plasmasphere, respectively.

3.2.1. Neutral Atmosphere

[26] The 45-year reanalysis (ERA40) of the European Center for Medium-Range Weather Forecasts (ECMWF) has been used to model the refractive index for the neutral atmosphere. The ERA40 project [Uppala et al., 2005] is a global atmospheric analysis of many conventional observations and satellite data streams for the period of September 1957 to August 2002. The analyses were produced four times daily (at 0000, 0600, 1200, and 1800 UT) for the entire period by a three-dimensional variational data assimilation that was used operationally at ECMWF from January 1996 to November 1997. The data resolution is T159 spectral truncation in the horizontal (about 125 km grid spacing) and 60 levels in the vertical, with variables represented up to a pressure of 0.1 hPa (about 65 km). The neutral atmosphere above the top of the ERA40 and up to 200 km is extended with the empirical model of the U.S. Naval Research Laboratory (NRL), MSIS, which stands for Mass Spectrometer and Incoherent Scatter Radar (the two primary data sources for development of earlier versions of the model [Hedin, 1991]). Recently, total mass density from satellite accelerometers and orbit determination has been added to the model, which now extends from the ground through the exosphere [Picone et al., 2002]. This model provides estimates of the temperature, composition, and total mass density derived from multi-instrument measurements for given geophysical location and time. Historical values of 10.7 cm solar radio flux (F10.7) and geomagnetic amplitude index (Ap) are given as inputs. While combining the MSIS with the ERA40 analyses, a 5 km zone is used to ensure a smooth transition. The linear transition, performed with respect to the height, ends at the top of the ERA40. Total mass density in the transition layer and above is then hydrostatically adjusted to ensure consistency with the combined thermal structure. In doing so, any possible change in the air composition above the turbopause is not considered.

3.2.2. Ionosphere and Plasmasphere

[27] The International Reference Ionosphere (IRI) is used to provide the refractive index for the ionosphere in terms of electron density. It provides climatological (for given location and the time of year) electron density, electron temperature, ion temperature, and ion composition for magnetically quiet conditions in the 50–2000 km range. The major data sources are the worldwide network of ionosondes, incoherent scatter radars, International Satellites for Ionospheric Studies (ISIS), and Alouette topside sounders, and in situ instruments on several satellites and rockets [Bilitza, 2001]. We have used a recent release that includes the International Reference Geomagnetic Field (IGRF) model of the International Association of Geomagnetism and Aeronomy (IAGA) for the magnetic coordinates and the Time Empirical Ionospheric Correction Model (so-called STORM) of the Space Environment Center (SEC) for storm-time updating of the F2 layer peak density [Fuller-Rowell et al., 2001].

[28] Electron density in the plasmasphere is obtained from the Russian Standard Model of Ionosphere (SMI) [Chasovitin et al., 1998]. SMI is an empirical model based on whistler and satellite observations. The model presents global analytical profiles of electron density, smoothly fitted to the IRI electron density profile at a specified altitude and extended toward the plasmapause, up to 36,000 km. In the model, the shape of the IRI topside electron density profile is improved using ISIS 1, ISIS 2, and intercosmos (IK) 19 satellite inputs [Gulyaeva, 2003]. We forced the SMI to fit the IRI-produced peak height (hmF2) and peak electron density (NmF2), so the electron density is determined by the IRI below the hmF2 and by the SMI above. Although electron density is smaller in the plasmasphere than in the ionosphere, GPS signals travel a much longer distance through the plasmasphere. This results in a considerable contribution of the plasmasphere to the TEC perceived at the receiver. A previous study shows that the plasmaspheric contribution could comprise about 10% of TEC in a GPS signal during daytime and 30%–35% at night (J. A. Klobuchar, P. H. Doherty, G. J. Bailey, and K. Davies, Limitations in determining absolute total electron content from dual-frequency GPS group delay measurements, paper presented at the International Beacon Satellite Symposium, Aberystwyth, U. K., 11–15 July 1994). Therefore, the inclusion of plasmaspheric density provides a complete description of the refractive index along a GPS-LEO link. Observed historical values such as sunspot number (Rz) and geomagnetic amplitude index (Ap) are used for the required input parameters.

3.3. Coordinate System and Computational Grids

[29] In section 2, CRT is described in a two-dimensional geometry. Nonetheless, the method is general enough to be used for three-dimensional applications. The only component not considered in this study for this purpose is the rays' out-of-plane bending. The three-dimensional application, however, incurs significant amounts of additional resources since it requires computing and holding refractive indices in each epoch's occultation plane. Moreover, the ray's torsion as well as curvature has to be considered. Our main purpose in this study is developing a fast ray tracer appropriate for real-time applications, and thus only two-dimensional use is considered here.

[30] Shown in Figure 2 is the ray coordinate system (RCS) used in this study: two-dimensional (polar) coordinates whose origin is the local curvature center of the Earth's ellipsoid, defined by the World Geodetic System 1984 (WGS-84). It is commonly used for conventional GPS RO processing (e.g., geometric optics) to correct the planetary-scale spherical asymmetry caused by the Earth's oblateness [Syndergaard, 1998]. Although RT does not have to be performed at these particular coordinates, RCS is worthwhile for some practical reasons. During an early stage of our development, the fast and well-known geometric optics provided a reliable reference for the quantities derived from the ray tracer, e.g., the bending angle, the impact parameter (also known as refraction constant), and the refractivity at tangent point. Since the two methods were performed in the same RCS, it was possible to compare their results directly on a epoch-by-epoch basis. This enabled many quick and convenient tests. Additionally, using the RCS is advantageous to a faster convergence of RS iteration, since the structure of the refractive index seen in the coordinates is closer to a spherical symmetry.

Figure 2.

Geometry involved with the ray coordinate system. The system's center is moved from the Earth's center to the local curvature center of the Earth's ellipsoid, to partially correct the planetary-scale spherical asymmetry in the atmosphere due to the Earth's oblateness. The local spheroid is defined at the point of tangency (TS) where the straight-line LEO-GPS is tangential with the ellipsoid; the x axis is parallel to the LEO-GPS and the y axis is perpendicular to the vector pointing TS from the curvature center; the polar angle is the clockwise measure from the y axis.

[31] The local curvature of the ellipsoid, held fixed during an occultation event, is calculated at the point of tangency between the ellipsoidal surface and the straight line connecting a paired transmitter and receiver. The moment at which the straight line makes the closest approach to the point of tangency is referred to as the reference epoch; when the ray's bending is considered, the actual tangent height for the reference epoch is about 12 km. The RCS is defined in the reference plane, the occultation plane at the reference epoch: the x axis is parallel to the straight line, while the y axis is perpendicular to the surface of the ellipsoid. As shown in Figure 1, the polar angle θ is measured clockwise from the y axis. The direction of the x axis is chosen by placing the receiver (LEO) at the left side (x < 0) and the transmitter (GPS) at the right side (x > 0) of the y axis, for the sake of convenience.

[32] Orbital motion of the satellites makes the occultation plane slide horizontally, so strictly speaking, each epoch has its own occultation plane. As collected, there are almost-parallel occultation planes as numerous as the number of epochs that belong to an observed occultation event. In order to perform RT for the occulation, assuming all relevant rays reside in the single reference plane, instantaneous positions of GPS and LEO at each epoch have to be projected onto the reference plane (in many places in this section, GPS and LEO refer to their corresponding satellites, respectively). In doing so, GPS is projected first and then LEO follows. We start off with GPS since it moves slowly, as a consequence of the distant orbit. This results in a smaller error in the projection. Here not only the orbital radii but also the central angle between the two satellites is preserved, so orbital motion of the satellites is properly portrayed. The temporal side slip of the occultation plane, rather than the ray's out-of-plane bending, is a major error source for a two-dimensional RT. The error increases with out-of-plane displacement of the raypath. For GPS RO, however, the displacement may be acceptably small since a GPS radio occultation event lasts no longer than 3 min for the neutral atmosphere. Poli and Joiner [2004] show that the drift of tangent point, which includes components both parallel and perpendicular to the raypath, causes error up to 1.2% in their simulated bending angles. In their study, the largest error appears at high altitudes above 30 km, which is the most distant location from their reference (lowermost) tangent height. The error may be reduced in our study, where a higher (12 km) reference tangent height is used and along-path drift is well accounted for. For the sake of simplicity, the error pertinent to the side slip is disregarded in this study.

[33] After placing the satellites in the RCS, computational grids, merely big enough to hold the satellites inside, are established. For a typical neutral atmospheric occultation, the angular span of the grids is about 100°. The grids consist of 181 equally spaced columns in the angular direction, providing a grid spacing comparable to the horizontal resolution of the NWP data used in this study. The angular spacing of the grid is about 0.55°, which corresponds to 60 km in the great-circle distance. The maximum radial extent is slightly larger than the GPS orbit. We used 600 fixed vertical layers with a variable resolution: 1 m near the Earth's surface, 50 m around the tropopause, 0.5 km at the stratopause, and stretched to 1000 km at the height of the GPS. Although the depth of layers is chosen rather arbitrarily, it is intended to consider the vertically varying resolutions of GPS measurements and the model refractive index. The quasi-exponential vertical variation of refractivity in the neutral atmosphere is also taken into account. The vertical grids are initially designed for an inverse scheme of RT that makes every endeavor to recover small-scale features in the atmosphere from available GPS measurements. For a forward modeling, such a fine grid, compared with the native resolution of model refractive index, is unnecessary. However, it does not negatively impact the results either, so the resolution is retained.

[34] After the grids are configured, refractive index is assigned to each grid point. Being available for a given time and location, the NWP data and empirical models are combined to provide a refractive index for the computational grids. As we know the geophysical location of each grid point, the procedure is straightforward. The parameters interpolated into the grids are temperature, pressure, amount of water vapor, and electron density. Given that these are not required for a forward RT, we instead keep separate refractive indices for L1 and L2 frequencies or, rather, their cubic-spline coefficients for each radial column of the grids. Otherwise, the frequency-dependent refractive index has to be recalculated whenever needed. From the cubic-spline coefficients, the refractive index and its vertical gradient at any height in a column can be easily computed. This conveniently provides a continuous gradient of refractive index even for irregular vertical grids. For angular direction, however, a simple linear interpolation is applied.

3.4. Ray Shooting

[35] Beginning with launching a ray at one satellite, CRT is carried out step by step as described in section 2 until the ray passes through the other satellite. The size of each integration step is adjusted according to the ray's local curvature; hence a smaller size is used in the vicinity of the tangent point, where considerable bending occurs. While the step size near ground level is about 0.1° minimally, it is as big as 2° above the height of LEO. In our numerical tests the model phase was rather insensitive to the step size. This is partly due to the limited resolution of the model refractive index used in this study and to some extent may indicate the strength of CRT, which is designed to allow a longer step size than SLRT does.

[36] In the meantime, the known position of GPS and LEO satellites imposes another constraint on the RT. In general, the aptness of a launch angle for the boundary value problem is unknown until the traced ray passes through the destination. Under the circumstances, one may shoot as many rays as needed until a ray hits the target. The ray shooting iteratively updates the launch angle at a departing satellite in order to reduce the distance between the targeted satellite and the arrival point of the ray closest to the target. An effective shooting algorithm as well as a good initial guess for the launch angle trims the number of iterations needed, leading to a direct cost reduction.

[37] Our RT starts from an LEO and ends at the corresponding GPS, which is opposite to the actual direction of signal transmission. This choice is made to speed up RS, as will be explained in the following. When the impact parameter (= nr sin ψ) is slightly perturbed, local zenith angles of the ray at GPS ψG and at LEO ψL are related to each other approximately as follows:

equation image

where rG and rL are orbital radii of GPS and LEO and DG and DL are distances to the tangent point from the GPS and the LEO, respectively. For a typical neutral atmospheric CHAMP occultation, rGrL−1 ≈ 3.9 and DGDL−1 ≈ 10–15. In other words, ΔψG makes bigger changes in the impact parameter (or the ray's arrival position) by a factor of 10–15, when compared with a ΔψL value of the same size. Because of this fact, a tinkering of ψG, when the ray is traced from GPS, often results in an erratic effect on the arrival position at the LEO side. This can be compared to tuning a very sensitive analog radio dial. Meanwhile, RT is subject to a numerical error that builds up as the RT goes on along the lengthy GPS-LEO path. The accumulated error adds unpredictable variation to the ray's ending position and eventually makes the RS iteration more difficult to converge. Being irrelevant to the direction of RT, furthermore, the error is adverse to the RT conducted from GPS to LEO. In practice, the change in ψG between two consecutive iterations can be diminutive, and this tends to cause numerical difficulties, especially when a small tolerance for the positional error at the LEO side (e.g., 1 mm) is pursued. Nonetheless, dequation imageG/L, where equation imageG is the arrival point of the ray at the GPS side, behaves less wildly with iteration. Consequently, tracing a ray from LEO to GPS, updating ψL iteratively, eases the shooting. For a spherically symmetric atmosphere, RT can be initiated from an estimated tangent point. In this case, one-sided tracing, from the tangent point to one of the satellites, is sufficient. For an asymmetric atmosphere, however, the RS has to deal with two control parameters, the location of the tangent point and the refractive index, in order to lower positional errors at both satellites. This significantly slows down the convergence of iteration and thus has little practical value.

[38] For the RS iteration, the Newton-Raphson method is employed with 1 mm tolerance for the position error at the GPS side. Over epochs, CRT is conducted sequentially in the top-to-bottom direction. For the first few epochs at the very top of an occultation, the geometric launch angle, the ray's zenith angle with respect to the straight line connecting GPS and LEO, serves as a good initial guess because the ray's bending at the high altitudes is negligible. When more than two epochs are successfully traced, a prediction based on the results obtained from previous epochs provides an excellent initial guess for the next epoch. We use a weighted least squares error algorithm that minimizes the root-mean-square error in the predicted and actual values. In the method, increased weighting is given to more recent epochs. Thanks to the high epoch rate of GPS measurements and also the nature of the atmospheric medium (which is largely continuous and strongly stratified), the prediction for the launch angle is straightforward and also reliable. In a large-scale view, a GPS-LEO raypath is very close to a straight line since the ray's bending is a few degrees at most. Therefore, the prediction-based shooting is quite effective as long as rays are traced from LEO to GPS direction.

[39] So far, we have described the RS used in this study in terms of launch angle, because doing so is conceptually easier to understand. For a practical implementation, the method suffers from the temporal change of occultation geometry. Accordingly, our prediction provides a guess as to impact parameter at LEO for the next epoch, because it varies more linearly over time. Since the position of LEO and the refractive index at the location are fixed, however, the launch angle directly relates to the impact parameter.

[40] The RS used in this study implies that only a single ray arrives at the receiver at a time. It is also assumed that the ray has been transmitted from the position of transmitter at the moment of reception. However, multiple rays, transmitted concurrently but through different paths or transmitted at different times, can arrive at the receiver simultaneously. In this case, the RS iteration typically oscillates among possible solutions. The atmospheric multipath is beyond the scope of this study, and thus has not been considered.

4. Results

[41] The distribution of modeled electron density for the CHAMP occultation that took place at 1121 UT 17 May 2002 (hereafter CH1) is shown in Figure 3, and it varies considerably in both horizontal (angular) and vertical (radial) directions. As a result, the GPS side differs significantly from the LEO side in the intensity of F2 peaks on the raypath. This leads to unequal contribution by TEC to the phase path from each side. Surely, the assumption of spherical symmetry for the electron density in this case is unsatisfactory. Figure 4 compares measured L1 and L2 excess phases with their model counterparts for the same occultation. It should be mentioned that the measured and simulated phases are matched up epoch by epoch according to temporal sequence in the measurements. Here GPS RO does not measure the tangent height at each epoch. Strictly speaking, therefore, the comparison is valid only in the time domain. In order to offer a geophysical description, however, the result is depicted in relation to the tangent height. Specifically, at each epoch the measurement is assumed to possess the same tangent height as its model counterpart. In the comparison shown in Figure 4, a large and systematic difference (for which the measurements are mainly responsible) appears throughout the height range. Once the bias is corrected, good agreement is established in both frequencies, including the bumps at 2–3 km (not shown).

Figure 3.

Distribution of modeled electron density in the occultation plane (color shaded) for CH1. Shown for the electron density is log10dion, where dion is electron number density (m−3). A simulated radio link for a pair of GPS and LEO satellites (black solid line) and its projection on the Earth's surface (pink solid line) are also shown.

Figure 4.

Measured (solid) and modeled (dashed) profiles of excess phase path (m) for CH1: (left) L1 and (right) L2. Measurement biases for L1 and L2 are denoted by b1 and b2, respectively.

[42] Figure 5 shows the trace of model refractivity along the raypath that has its tangent point just above the ground. The along-path refractivity is about 310 at the tangent point and decreases rapidly with height. In the ionosphere, the LEO-side refractivity reaches −75 at the peak height of electron density. The LEO-side trace differs from the GPS-side trace, especially in the lower troposphere (below 5 km) and in the ionosphere (above 250 km) as a consequence of the spherical asymmetry in the model atmosphere. One of the key parameters that can best describe the propagation of radio waves in the atmosphere is the ray's curvature, which is determined by both the gradient of refractive index along the raypath (Figure 5) and the direction of the raypath. The maximum curvature appears at around 2 km at the GPS side, which is about 0.28 times that of the Earth's curvature (Figure 6). This means that the ray's curvature radius is 3.6 times larger than the Earth's radius. Above the F2 peak, electron density locally decreases with height, leading to a negative curvature. Similar features appear for D and E layers as well. The ray bends toward outer space at the height of its negative curvature. Even though the ionospheric refractivity itself at the F2 peak is quite large in size, as shown in Figure 5, the weak vertical gradient there limits magnitude of the curvature; however, the ionospheric curvature is still comparable to the neutral atmospheric curvature at 30 km.

Figure 5.

Trace of refractivity along a simulated raypath for CH1. LEO side (solid line) and GPS side (dashed line) are shown with respect to the tangent point.

Figure 6.

The same as Figure 5, except for the trace of ray curvature. Shown here is the curvature (m−1) multiplied by the local curvature radius of the Earth (m). Positive (negative) curvature stands for a raypath bending toward the Earth (outer space).

[43] CRT works best if the ray's curvature in an integration step is constant. As shown in Figure 6, the variation in the curvature along the raypath is indeed moderate. There is no particular concern with regard to the bump at 2 km, since the raypath there is mostly horizontally oriented. So to speak, the thickness of integration steps of a fixed size decreases rapidly as the ray approaches the tangent point. In addition, in view of the fact that a smaller step size is to be used where the ray's curvature is large, the curvature changes little within a step. The slow variation in the curvature supports the strength of CRT over the straight-line (κ = 0) method. The large positive curvature below 30 km explains why SLRT needs a considerably smaller step size in order to fit the actual raypath as accurately as CRT can. The effectiveness of CRT is boosted in the region in which the ray's curvature is larger, leading to a considerable cost reduction.

[44] In spherically symmetric media, a ray conserves its impact parameter. The trace of the impact parameter along a raypath thus shows the degree to which the particular medium deviates from being spherically symmetric. Along the same raypath as in Figures 5 and 6, rapid changes in the impact parameter appear as big as 80 m near the tangent point (Figure 7). Disregarding spherical asymmetries in the atmosphere, therefore, conventional methods for GPS RO possess this much uncertainty in their impact parameter. The change of 80 m in the impact parameter corresponds to an approximate 4% deviation in refractivity. This yet again draws attention to caring about spherical asymmetries in GPS RO technique.

Figure 7.

Trace of impact parameter along a simulated raypath for CH1. Shown here is the deviation of impact parameter (m) from the initial value at LEO versus polar angle (degrees) at LEO side (solid line) and GPS side (dashed line), with respect to the tangent point where the polar angle is zero.

[45] One of the practical problems in processing currently available dual-frequency GPS data is a lower SNR at the L2 channel. This makes the L2 data unusable below a certain height even if L1 data remain useful. As an example, the CHAMP occultation that took place at 19.1°N, 122.44°E, 0503 UT 28 May 2002 (hereafter CH2), is shown in Figure 8. Here we show Ld(≡ L1L2) that is largely in proportion to the TEC along the raypath (often called STEC). While the model is continuous all the way down to near the Earth's surface, the measurement shows an abrupt change at 5 km. This is due to L2 data, which are discarded after the discontinuity (hereafter the point of L2 cutoff is referred to as CL2). If a ray has its tangent point at an ionospheric altitude, STEC is large. On the contrary, when the ray penetrates deep into the neutral atmosphere and travels a shorter distance in the ionosphere, the phase advance due to the ionosphere decreases. In addition, all the rays that have their tangent heights lower than the bottom of the ionosphere cover the same extent in the ionosphere. In this case, the STEC is largely determined by the change in the occulting geometry: namely, the angle between raypath and local horizon at ionospheric heights. Consequently, the horizontal structure in the ionosphere becomes less important. This allows one to estimate the truncated part of Ld from the measurements available at higher altitudes when there are no immense ionospheric irregularities. Therefore, one can remove the ionospheric influence in the L1 phase below the height of CL2. We proceed to a regression for the measured Ld over a 10 s window that ends at CL2, and the subsequent regression equation is then used to evaluate the Ld below CL2. The transition from the measurement to the regression curve starts from the point in the window where the measured Ld gets noisy. Once the correction is made and the measurement bias in Ld is taken into account, the measured L1 usually differs by no more than 1 m from the model, even in the lower troposphere.

Figure 8.

Profiles of L1-L2 excess phase path (m) for CH2. The solid line indicates the original measurement; the dashed line indicates the measurement after L2 correction; the dotted line indicates the model.

[46] The ionosphere-corrected phase Lc lessens the ionospheric effect:

equation image

where ηc is the neutral atmospheric contribution to phase; bc and ɛc denote measurement bias and noise in Lc, respectively. Nevertheless, this combination does not completely eliminate the ionospheric influence, because L1 and L2 raypaths are not coincident. In the meantime, RT is capable of simulating the ionosphere-free phase Lf, when applied to a model of purely neutral atmosphere (i.e., the case that the ionosphere is not accounted for model refractive index). The two neutral atmospheric phases, Lc and Lf, differ from each other to some extent since they are obtained from scanning different parts of the neutral atmosphere as a consequence of ionospheric bending. For the real atmosphere, it is impossible to deduce Lf from measured L1 and L2 phases; instead, Lc is treated as a surrogate for Lf. For the model, however, one can simulate Lc with the aid of an ionospheric model. In that case the modeled Lc, rather than the modeled Lf, compares better with the measured Lc, since the ionospheric bending has been taken into account in modeling the Lc. If the large-scale structure in the model of electron density is realistic, furthermore, the split between the L1 and L2 paths will also be closely reproduced. This makes the modeled and measured Lc values even closer.

[47] The measured and modeled Lc values for CH2 are presented in Figure 9a. The systematic difference in the mean indicates the associated measurement bias bc. After a correction for the bias, they show a remarkable agreement below 90 km, except for the measurement noise (Figure 9b). In order to estimate bc, the center of a moving window with a width of about 20 km is made to slide over a 50–70 km range, where both ionospheric and neutral atmospheric effects are relatively small and hence significant mismodelings for both delays can be avoided. It should be mentioned that the model-based bias correction is vulnerable to a model's own bias if the window is placed in the troposphere. However, the neutral atmospheric delay in the 50–70 km range is so small that the model's bias in Lc is of little concern. For each position of the moving window, a linear least squares fitting for measurement minus model in Lc is carried out. Among all positions, the one that achieves the best fit is considered to be a candidate. The regression at the selected position is rejected if the cumulative residual for the fitting exceeds a threshold. This happens in cases where the measurement is exceptionally noisy or the model refractive index is problematic. Once the candidate is accepted, bc is estimated from the regression model at the chosen center of the moving window. The moving window holds a fixed number of epochs inside, over 500 in most cases. As the window slides, the instantaneous least squares estimation can be conveniently updated by replacing the last trailing epoch with a new leading one and taking the change due to swapping of the two epochs into account. It should be noted that, in principle, GPS RO measurements alone could determine the measurement bias; however, the technique, known as ambiguity resolution, usually requires additional information about the group delay (i.e., range) (e.g., see a review article by Teunissen and Verhagen [2008] for a comprehensive review on this topic) and is not considered in the present study. Although Lc is commonly treated as a proxy of Lf, they are quite different from each other, as shown in Figure 9b. In Lc, however, model and measurement show a noteworthy agreement. This demonstrates that ionospheric modeling is complementary to neutral atmospheric simulation, and vice versa.

Figure 9.

Profiles of neutral atmospheric excess phase path (m) for CH2: ionosphere-corrected (Lc) measurement and model and ionosphere-free (Lf) model. In Figure 9a, bc designates the measurement bias in Lc. The measured Lc in Figure 9b is bias corrected, while that in Figure 9a shows the original (bias uncorrected).

[48] An example that leaves bc undetermined is cycle slips in received GPS signals. The cycle slip is a sudden jump of cycles in the measured carrier phase, due to a temporary loss of lock in the tracking loop of a GPS receiver. Being momentary, intermittent, and at times correctable, the error is distinguished from other “random” noises. Well-developed data preprocessing may detect and repair most cycle slips beforehand, although occasionally large-amplitude measurement noises embedded in a rapidly varying signal hide them from detection. Figure 10 shows some occultations for which our preprocessing failed in the resolution. Here is shown the measurement residual, i.e., measurement minus model. The cycle slips are identifiable by abrupt changes, discontinuous and stepwise, in the Lc difference. L2 phases are more susceptible to cycle slips due to their lower SNR, and a slip of one cycle in the frequency causes an error of about 38 cm in Lc, as can be inferred from equation (16); indeed, leftward shifts of the residuals with decreasing height indicate that the cycle slips occur in L2 phases. The examples shown in Figure 10 contain slips of half-cycle on which Hajj et al. [2002] discussed some causes and possible corrections. It should be emphasized that CRT makes the detection of the hidden cycle slips much easier since it provides a model Lc, which eliminates a likely temporal variation from the measurements and is also noise free. Therefore, the cycle slips are more discernible in the residuals than in the measurements themselves. For a short period (e.g., over a few consecutive epochs), the error combining measurement noise and model error is smaller than a half-cycle in Lc, especially for high altitudes above the tropopause, so the RT elucidates the location and size of cycle slips.

Figure 10.

Hidden cycle slips revealed by ray tracing. Measurement residuals in Lc are shown. The slips, denoted by squares, are identifiable by discontinuous and stepwise change. Leftward shifts of the residuals with decreasing height indicate that they occur in L2 phases; the slips are half-cycles, about 19 cm in distance unit.

[49] Once bc is determined, the measured Lc can be compared with the corresponding model on an epoch-by-epoch basis. The residuals in Lc for CH2, with and without the Ld correction below the CL2, are shown in Figure 11. In forming the Lc combination, noises in L1 and L2 measurements are also combined and amplified. As a result, the measured Lc is noisier than either L1 or L2 measurement. For the Lc in which Ld is corrected below CL2, a recursive low-pass, fourth-order Butterworth filter [Rabiner and Gold, 1975], is applied to suppress the Ld noise. In doing so, the filter is carried out twice, from head to tail and the reverse, in the time domain so as to remove the phase shift involved with the recursive operation. The primary purpose of the filtering is anti-aliasing, which makes the comparison less harsh when the measurements are statistically compared with their model counterparts for a large number of occultations. It also makes the Ld at both sides across the CL2 more consistent, since the low-order regression used to estimate the truncated part of Ld acts as a low-pass filter. Going back to the comparison, the filtering explains the Lc noise above the CL2. The comparison shown in Figure 11 demonstrates well that CRT can simulate the measured Lc very closely, down to the lowest height. Considering the fact that the measured Lc can be as big as 1 km near ground level, the agreement in Lc is remarkable.

Figure 11.

Measurement residuals in Lc (m) for CH2: (solid line) original measurement and (dashed line) L2-corrected measurement, respectively.

5. Summary and Outlook

[50] The measurements from the GPS RO technique offer numerous attractions in various geophysical research areas. However, shortcomings of current data usage hinder exploitation of the technique's full potential, especially in studies that make use of higher-level parameters such as soundings of temperature, water vapor content, and pressure. The main problem in using these parameters is their complicated error characteristics, which hamper effective quality control. This stems from nonlinear propagation of phase error into subsequent data streams, incidence of new errors in each processing step, and subsequent interactions between the errors. For conventional methods that rely on the assumption of a spherically symmetric atmosphere, the disregard of horizontal variation is one of the major error sources.

[51] In an effort to better deal with this problem, we look into the benefits of quality control for phase measurement. This requires a reliable reference that can enlighten flaws in the raw measurement. To do so, we have developed a new ray tracing method, CRT, and employed it to directly simulate the phase measurement. The novelty of the method ensues from the piecewise-circular approach that approximates a segment of a raypath with an arc of the osculating circle. The method is designed to achieve improvements in both computational efficiency and accuracy over the traditional straight-line approach, SLRT, taking advantage of the atmospheric structure that is strongly stratified in the large scale. We provide realistic refractive indices for CRT, combining various sources of information including empirical models for the ionosphere (IRI) and plasmasphere (SMI), the ERA40 global analyses, and the MSIS for the neutral atmosphere in high altitudes. The refractive indices cover the entire radio link between GPS and LEO satellites. Given that satellites' precise locations are known, a ray shooting algorithm is used to realize the observed radio link with controlled accuracy. We endeavor to closely describe the complete procedure that leads to a model counterpart of phase measurement.

[52] Key features in the phase measurement are elucidated with the aid of CRT and are intently described through two CHAMP occultations. The altitude-dependent significance of horizontal variation in the atmosphere is demonstrated with model traces of refractivity, ray curvature, and impact parameter along a raypath. Comparing the simulation with corresponding measurements, not only individual L1 and L2 phases but also the ionosphere-corrected phases show good agreement. The outstanding agreement between measured and modeled Lc proves that realistic ionospheric modeling is useful for an accurate neutral atmospheric simulation and vice versa. It is concluded that the direct simulation of phase is advantageous to the quality control of GPS RO data, thanks to the simple error structure of the measurement. The effectiveness is attested to by the detection of cycle slips, which is difficult otherwise. In this study, we have intervalidated CRT with GPS RO measurements; the results clearly indicate their great performance and accuracy.

[53] In order to describe CRT in great detail but to keep the discussion concise, we have used a relatively small number of occultations for our analysis. One of our extended studies has further confirmed the findings presented in this study, and the result is currently in preparation for a separate publication. Although we have restricted our analysis in this study to the simulated phase, the CRT has shown great potential for a thorough characterization of the radio link. Many aspects pertinent to atmospheric multipath, ground reflection, and amplitude variation due to atmospheric defocusing are observed in the simulations. These require separate efforts, but may broaden our understanding especially of the measured amplitude. Our approach is expandable to the ground-based GPS technique, in which a receiver is placed on the Earth's surface, without any major modifications. One more possible step forward in that direction is a seamless integration of the two observing systems (space- and ground-based GPS techniques) in the context of ray tracing. Although CRT is developed for GPS RO, it may also suit other limb-viewing observing systems.

Acknowledgments

[54] This material is based upon work supported by the National Science Foundation (NSF) Division of Arctic Sciences award ARC-0733058, the NSF Division of Atmospheric and Geospace Sciences CSA-0939962, under Cooperative Agreement AGS-0918398, and a subaward from The Ohio State University under the National Aeronautics and Space Administration award NNX08AN57G. One of the authors is supported by the Global Partnership Program (GPP) of the Korea Foundation for International Cooperation of Science and Technology (KICOS), sponsored by the Korean Ministry of Science and Technology (MOST). Special thanks are given to the UCAR/CDAAC team for useful discussions and data provision. We thank GFZ for data release of CHAMP and JPL for SAC-C.

Ancillary