## 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.