Theory of GNSS Radio Occultation

Abstract

In this chapter, a brief history of the radio occultation remote sensing technique is introduced. The physical principles of GNSS radio occultation (RO) technique are discussed, and the detailed GNSS RO processing steps are presented finally

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1 Introduction

The word “occult” is original from Latin word “occultus” referring to “hide”. The usage of “occultation” in science field starts in astronomy. An occultation refers to an event that occurs when one celestial body is occulted (or hidden) by another celestial body of interest that passes between it and the observer. During an occultation event, the signal from the occulted object travel through the limb (or edge) of the middle object could be distorted due to the existence of medium, such as atmosphere at the limb. The change of the signal can then be used to infer the atmospheric structure of the celestial body of interest. This is called the occultation technique. Various signal sources at different wavelength have been used to apply the occultation remote sensing technique. For example, by looking at the stars that is setting or rising behind a celestial body of interest is called stellar occultation. The way in which the star’s signals bend and change relative to the observer can be used to characterize the atmospheres and/or ring systems in the solar system or beyond.

1.1 Radio Occultation in Planetary Sciences

The radio occultation (RO) technique, which uses the radio signals, emerged at the dawn of interplanetary space exploration in the 1960s. It was pioneered by two independent research groups at Stanford University and the NASA’s Jet Propulsion Laboratory (JPL), for determining the size and atmospheric composition of distant planets. The first planetary radio occultation profile were collected in 1965, when the Mariner-4 spacecraft flied by Mars and sent the radio signals through its atmosphere and ionosphere and intercepted at the Earth (Fjeldbo and Eshleman 1965; Kliore et al. 1965). As seen in Fig. 5.1, by precisely recording the change of the radio signals scanning through the planetary atmosphere, the vertical structure of the atmosphere can be precisely measured. Since then, the radio occultation technique has been widely used to probed the planetary atmosphere and ionosphere of nearly every planets in the solar system: e.g., Venus, Mars and Mercury (e.g., Kliore et al. 1967; Fjeldbo and Eshleman 1968; Howard et al. 1974), the gas giants Jupiter, Saturn, Uranus and Neptune (e.g., Fjeldbo et al. 1975; Kliore et al. 1980; Lindal et al. 1987; Tyler et al. 1989). In addition, the RO technique has also been used to study the moons of the outer planet, such as Jupiter’s satellites: Io, Europa, Ganymede and Callisto (Kliore et al. 1975, 1997), Saturn’s satellite Titan (Lindal et al. 1983) and Neptune’s satellite Triton (Tyler et al. 1989), as well as the planetary ring structure (e.g., Marouf and Tyler 1986).

Fig. 5.1

Schematic plot of planetary occultation geometry with radio signal ray paths (solid line) connecting a flyby spacecraft (trajectory in dashed line) with a distant receiver or transmitter on the Earth

Other than the US interplanetary missions, the former USSR also launched multiple planetary missions and applied RO technique for sensing the planetary atmosphere and ionosphere (e.g., Kolosov et al. 1976; Ivanov et al. 1979). Table 5.1 shows a list of celestial bodies in the solar system that have been investigated by the radio occultation technique in the past half century. The RO technique continues to thrive and provide highly valuable atmospheric measurements that contribute to an improved understanding of structure, circulation, dynamics, and transport in the planetary atmosphere.

Table 5.1 Celestial bodies investigated by the radio occultation technique except the Earth

1.2 GNSS Radio Occultation in Earth Sciences

The application of the radio occultation technique to probe the Earth’s atmosphere was first suggested in the mid-1960s (Fishbach 1965; Kliore 1969; Lusignan et al. 1969). Several more studies of satellite configurations were carried out by a group of Russian scientists later (Kalashnikov and Yakovlev 1978; Gurvich and Krasil'nikova 1987; Eliseev and Yakovlev 1989; Yakovlev et al. 1995).

A few pioneering RO experiments from satellite-to-satellite tracking link demonstrated the feasibility and the great benefit of the technique for the Earth’s atmosphere research. The first radio occultation experiment to sense the Earth’s atmosphere was carried out within the framework of the historical (joint US-Soviet) Apollo-Soyuz mission using the ATS-6 (Applications Technology Satellite-6) geostationary satellite in July 1975 (Rangaswamy 1976). Another RO experiment based on the radio link between ATS-6 and GEOS-3 (Geodynamics Experimental Ocean Satellite-3) on a circular polar orbit was documented by Liu (1978). Yakovlev et al. (1995) presented preliminary results of radio occultation experiments, which were performed by using the orbital station MIR and a retranslating geostationary satellite.

Note that the RO observation requires a pair of a radio source and a suitable receiver off the Earth. To monitor the state of the atmosphere will require a sufficient number of daily RO soundings to fulfill the meteorological tasks. However, the high cost of implementing a large number of new transmitters and receivers in the space to achieve the sampling goal makes the technique not attractable by then.

In the 1980s, the emergence of the Global Navigation Satellite System (GNSS) constellation that provides reliable and low-cost radio signal sources, enabled the RO application for Earth’s atmospheric sounding. The US Global Positioning System (GPS) and the Russian Globalnaya Navigatsionnaya Sputnikovaya Sistema (GLONASS) became the first two operational GNSS in the world. The RO concept to profile Earth’s atmosphere using the GNSS radio signal was first proposed in 1980s (e.g., Gurvich and Krasil'nikova 1987; Yunck 1988). Figure 5.2 shows a schematic view of a setting occultation as a Low-Earth-Orbit (LEO) receiver satellite tracks the GPS transmitter while it sets behind the Earth. The dashed straight line is the line-of-sight without the presence of the medium. Whereas, the bended black line represents the GPS signal ray path travels through the ionosphere and neutral atmosphere before reaching the LEO receiver.

Fig. 5.2

Schematic plot of GPS radio occultation geometry

On 3 April 1995, the proof-of-concept GPS/MET (for GPS Meteorological experiment) led by the University Corporation for Atmospheric Research (UCAR) was launched and became the first GPS RO satellite mission to probe the Earth’s atmosphere (Feng et al. 1995; Ware et al. 1996; Rocken et al. 1997). The RO measurement precisely reveals the fine vertical structure of temperature across the tropopause inversion as indicated in the closely coincident radiosonde sounding and the weather model profile. The temperature retrievals agreed well with co-located analyses and radiosondes to within 1 K between ∼5 and 25 km.

The GPS/MET RO receiver aboard the small research satellite MicroLab-1 continued to collect over a hundred RO soundings per day until the end of the mission in mid 1997. The remarkably successful mission demonstrated the capability of GPS RO technique to provide global coverage, accurate and high-vertical resolution soundings of the Earth’s atmosphere in all-weather conditions. The breakthrough success of the mission paves the road for developing a worldwide constellation of LEO satellites, which shall operationally provides a dense global observation of fundamental atmospheric variables. The RO observations will provide a great complement to the conventional nadir-viewing satellite sounders and the sparse in-situ measurements from radiosondes and aircrafts.

Following the pioneering example of GPS/MET, the same type of GPS RO receiver were launched into the space aboard two small international flight projects in 1999 and one US Air Force (USAF) satellite mission in 2001, i.e., Denmark’s Ørsted mission, designed primarily for magnetic field mapping, South Africa’s Sunsat, a student-built satellite carrying a high-resolution imager, and USAF’s IOX for ionosphere sensing (Yunck et al. 2000).

The GPS/MET type of basic receiver built by NASA’s Jet Propulsion Lab (JPL) is a dual-frequency Turbo Rogue GPS receiver that produces only a small number of RO soundings (∼125) each day. Many of the soundings failed in the lower troposphere (∼3–5 km above surface), in particular over the moist regions in tropics and mid-latitudes. Nevertheless, such unique GPS/MET measurements provided not only pioneering atmospheric science but also enormous insight into the behavior of RO signals, which ultimately benefited and led to the new design for the next generation occultation receivers.

In 2000, the JPL-built second-generation occultation receivers (known as the “BlackJack”) were launched on two international satellite missions. One is on the German satellite CHAMP (CHAllenging Minisatellite Payload) for magnetometry and gravity mapping (Wickert et al. 2001). Another is on the Argentina spacecraft SAC-C (Satellite de Aplicaciones Cientificas-C) carrying a multispectral imager and magnetometer (Hajj et al. 2004). SAC-C became the first RO mission to carry occultation antennas in both the fore and aft velocity directions, and thus was the first to observe rising occultations. One unique feature of SAC-C is that all flight software can be modified and reloaded after launch. Both satellite missions provide much stable daily RO soundings measurements as compare to earlier RO missions. For several years, CHAMP and especially SAC-C have served as developmental test beds for GPS sounding. In 2001, the BlackJack receivers were placed on one of the twin spacecraft GRACE-A (Gravity Recovery and Climate Experiment-A). However, the RO occultation measurements were not activated until 2006 to avoid the potential disturbance to the major gravity field measurement mission (Wickert et al. 2009).

On 15 April 2006, the joint Taiwan-US six RO satellite constellation, FORMOSAT-3/COSMIC (Formosa Satellite mission #3/Constellation Observing System for Meteorology, Ionosphere, and Climate), were successfully launched. Each COSMIC satellite is equipped with a newer generation GPS RO receiver (called IGOR, Integrated GPS and Occultation Receiver). The RO receiver preserves a high degree of the Blackjack heritage and the hardware architecture. In addition it offers open-loop tracking capability that is highly desirable for moist lower troposphere soundings where the BlackJack close-loop receivers from earlier missions encounter problems. COSMIC was the first constellation of satellites dedicated primarily to RO and delivering 1,500–2,500 daily RO soundings in near-real-time to operational weather centers around the world shortly after its launch (Anthes et al. 2008). On the same horizon, the MetOp-A, the Europe’s first polar orbiting satellite for operational meteorology, was launched on October 19, 2006. The satellite payload includes a GPS RO receiver (GRAS – Global Navigation Satellite System Receiver for Atmospheric Sounding), which was independently developed by the European Space Agency (ESA) (Luntama et al. 2008).

Following the success of these missions, a series of RO missions were launched on many international satellite payloads. For example, the UC C/NOFS, the German TerraSAR-X and its sister satellite TanDEM-X; OceanSat2 and Megha Tropiques (a joint India-France mission); the SAC-D/Aquarius from Argentina; and a follow-on MetOp-B from ESA. Moreover, a series of single-receiver RO missions have been planned to be launched in the near future such as Chinese FY-3C equipped with GNOS (GNNS Radio Occultation Sounder); ROHP-PAZ from Spain; KOMPSAT-5 from South Korea and EQUARS from Brazil.

In general, a single GPS RO receiver satellite can recover ∼500 rising and setting RO profiles each day, distributed almost uniformly around the globe; a large constellation would recover many thousands of profiles, which could have a profound impact on both long term climatological studies and short term weather predictions.

The follow-on mission of COSMIC-II are planned to have 12 satellites with 6 satellites on low-inclination (equatorial) orbits and another 6 satellites on high inclination orbits to produce uniform global sounding coverage. The next generation RO receiver (e.g., TriG developed by JPL) will be capable of tracking the GPS, GLONASS and Galileo satellites at the same time and will significantly increase the sounding densities. The likely over 10,000 daily profiles will provide extremely valuable atmospheric observations that are essential for mesoscale weather forecasting, such as hurricane/typhoon, thunderstorms etc. The GNSS RO also attracts strong interests from the private sectors. The CICERO (Community Initiative for Cellular Earth Remote Observation) was form to seek private funds for launching many micro-satellites in Low-Earth Orbit and providing dense RO soundings.

All these RO missions provide essential global atmospheric measurements with high vertical resolution and significantly benefit the weather and climate research communities. A comprehensive list of the past and current RO missions as well as many on plan is summarized in the Table 5.2.

Table 5.2 Radio occultation technique for the Earth’s atmosphere

2 Principle of GNSS Radio Occultation

As the LEO satellite equipped with a GNSS receiver orbits around the Earth, an occultation event occurs (Fig. 5.2) when the received navigation signal from a setting (rising) GNSS satellite scan through successively deeper (higher) layers of the Earth’s atmosphere until the GNSS signals descend below the Earth surface (rise above the atmosphere). The GNSS signal is bent or delayed before arriving at the LEO due to the Earth’s atmosphere.

Strictly speaking, the propagation of the GNSS signal through the atmosphere obeys Maxwell’s equation in which the propagation medium (e.g., the Earth’s atmosphere) is characterized by a three-dimensional spatial distribution of a complex and dispersive refractive index. The GNSS radio occultation technique takes advantage of the extremely precise phase and amplitude measurement of the GNSS navigation signals that pass through the Earth’s atmosphere to provide accurate retrieval of the vertical profiles of refractive index of the atmosphere. Consequently, the atmospheric properties such as air density, temperature, pressure, and humidity can be inferred (Kursinski et al. 1997; Rocken et al. 1997).

2.1 Atmospheric Refraction

Before the introduction of the RO retrieval process, we first explore how the atmospheric properties influence the refractive index n.

The refractive index (or index of refraction) n in a medium is defined as the ratio between the speed of light in a vacuum and the speed of light in the medium. In the neutral atmosphere, n is very close to unity, such that it is conveniently expressed in terms of refractivity defined as N = (n − 1) × 10⁶. The refractivity at GPS frequencies contains contributions from four major components, i.e., the dry neutral atmosphere (N ^dry), water vapor (N ^vapor), free electrons in the ionosphere (N ^iono), and particulates (primarily liquid water and ice water content, N ^scatt) through the following relationship (Kursinski et al. 1997; Hajj et al. 2002):

$$\begin{array}{llllllll} N &=77.6\frac{P}{T}+3.73\times {10}^5\frac{P_w}{T^2}-\left(40.3\times {10}^7\frac{n_e}{f^2}+O\left(\frac{1}{f^3}\right)\right)\\ & \quad +\left(1.4{W}_{\it liquid}+0.6{W}_{\it ice}\right) \end{array}$$

(5.1)

$$ ={N}^{\it dry}+{N}^{\it vapour}+{N}^{\it iono}+{N}^{\it scatt} $$

(5.2)

where P and P _w are total pressure and water vapor partial pressure in hectopascal (hPa), T is temperature in Kelvin (K), n _e is electron number density per cubic meter, f is signal frequency in Hertz (Hz), and W _liquid and W _ice are referred to liquid water content and ice water content in gram per cubic meter, respectively.

Dry refractivity is proportional to molecular number density and is dominant below 60–90 km. The dry refractivity term is due to the polarizability of molecules in the atmosphere, i.e., the ability of an incident electric field to induce an electric dipole in the molecules. The moist refractivity term is due primarily to the large permanent dipole moment of water vapor and becomes significant in the lower troposphere, especially in the tropics and subtropics (Kursinski et al. 2000). The ionospheric term in Eq. (5.1) includes a first-order approximation (1/f ²) to the Appleton-Hartree equation (Papas 1965), which is mainly due to free electrons in the ionosphere and becomes important above 60–90 km. The second-order term (1/f ³) is generally neglected (e.g., Kursinski et al. 1997). The scattering term given in Eq. (5.1) is due to liquid water droplets and ice crystals suspended in the atmosphere. For realistic suspensions of water or ice, the scattering term is small in comparison with the other terms and is therefore neglected in most RO applications.

2.2 Geometric Optics Approximation

At GNSS frequencies, it is convenient to assume that the refractive index is real (i.e., zero absorption). For simplicity, we can further assume that the signals are monochromatic, which is largely valid in the dry atmosphere. Because the wavelengths of the GNSS radio signals are generally small compared to the characteristic scale of the atmospheric problem, the geometric optics (or ray optics) concept can be applied to describe the GPS radio occultation measurements. The signals (light waves) propagate in a direction orthogonal to the geometrical wavefronts defined as the surface on which the signal phase is constant (i.e., stationary phase) (Born and Wolf 1980). Lines representing these signal propagating trajectories are called ray paths (e.g., black solid curve in Fig. 5.2).

In the geometric optics (GO) approximation to the propagation of electromagnetic radiation, the path of a ray passing through a region of varying refractive index is determined globally by Fermat’s principle of least time and locally by Snell’s law. Therefore, the differential equation of the ray path can be described by the Eikonal equation (Born and Wolf 1980) as

$$ \frac{d}{ ds}\left(n\frac{d\overrightarrow{r}}{ ds}\right)=\overrightarrow{\nabla}n, $$

(5.3)

where $\overrightarrow{r}$ is the position vector of a typical point on a ray, $\overrightarrow{\nabla}n$ is the gradient of the refractive index n, and ds is an incremental length along the ray path (i.e., $d\overrightarrow{r}=\overrightarrow{s} ds$).

2.3 Spherically Symmetric Atmosphere Assumption

Consider the variation of the vector $\overrightarrow{r}\times \left(n\overrightarrow{s}\right)$ along the ray. We have

$$ \frac{d}{ ds}\left(\overrightarrow{r}\times n\overrightarrow{s}\right)=\frac{d\overrightarrow{r}}{ ds}\times n\overrightarrow{s}+\overrightarrow{r}\times \frac{d}{ ds}\left(n\overrightarrow{s}\right)\!, $$

(5.4)

Since $d\overrightarrow{r}=\overrightarrow{s} ds$, the first term on the right vanishes. The second term, on account of Eq. (5.3), can be rewritten as:

$$ \frac{d}{ ds}\left(\overrightarrow{r}\times n\overrightarrow{s}\right)=\overrightarrow{r}\times \overrightarrow{\nabla}(n), $$

(5.5)

Equation (5.5) shows that only the non-radial portion of the refractive index gradient contributes to changes in $\overrightarrow{r}\times \left(n\overrightarrow{s}\right)$. Now, let’s consider rays in a medium with spherical symmetry, i.e., where the refractive index only varies on the radial direction. This is a simple approximation of the earth’s atmosphere, when the curvature of the earth is taken into account.

$$ n=n\left(\overrightarrow{r}\right) $$

(5.6)

Therefore, $\overrightarrow{\nabla}n=\frac{\overrightarrow{r}}{r}\frac{ dn}{ dr},$ i.e., the refractive index gradient is only in the radial direction. The second term on the right-hand side of Eq. (5.4) also vanishes. Hence $\overrightarrow{r}\times \left(n\overrightarrow{s}\right)= {\it const}.$ This relation implies that all the rays are plane curves and along each ray,

$$ nr \sin \phi =a, $$

(5.7)

where ϕ is the angle between the position vector $\overrightarrow{r}$ and the tangent of the ray path (see Fig. 5.3), and the constant a in a spherically symmetric atmosphere is called impact parameter and is also known as Bouguer’s rule, which represents Snell’s law in a spherically symmetric medium.

Fig. 5.3

A ray path (thick line) in a medium with spherical symmetry that satisfies the Bouguer’s formula

2.4 Bending Angle and Refractive Index

The accumulated change in the ray path direction along a ray path is defined as the bending angle. According to Eq. (5.3), the rate of change in ray path tangential direction is given as

$$ \frac{d\overrightarrow{s}}{ ds}=\frac{1}{n}\left(\overrightarrow{\nabla}n-\overrightarrow{s}\frac{ dn}{ ds}\right)=\frac{1}{n}\left({\overrightarrow{\nabla}}_{\perp \overrightarrow{s}}n\right), $$

(5.8)

Thus, the bending is only due to the refractive index gradient that is orthogonal to the ray path tangent direction $\overrightarrow{s}$, i.e., the projection of $\overrightarrow{\nabla}n$ into the plane perpendicular to the ray direction $\overrightarrow{s}$ (i.e., ${\overrightarrow{\nabla}}_{\perp \overrightarrow{s}}n$). Now we can define a local coordinate system where x is orthogonal to r and lies in the plane defined by s and r, and y is in the direction orthogonal to the r-x plane (Kursinski et al. 2000). The bending angle increment along the ray path can be written as

$$ d\alpha =\frac{\left|d\overrightarrow{s}\right|}{\left|\overrightarrow{s}\right|}=\frac{ ds}{n}{\left[{\left(\frac{\partial n}{\partial r} \sin \phi +\frac{\partial n}{\partial x} \cos \phi \right)}^2+{\left(\frac{\partial n}{\partial y}\right)}^2\right]}^{1/2}. $$

(5.9)

The largest gradients of refractivity are generally found in the lower level of the atmosphere near the tangent point (ϕ ≈ 90°, cosϕ ≈ 0) along a ray path. Since the magnitude of horizontal gradient is generally much smaller than those of vertical gradients in the Earth’s atmosphere, the bending of a ray path is largely caused by the refractivity gradient in radial direction. But it is worth noting that the greatest horizontal gradient contribution will come from the gradient in y direction (i.e., perpendicular to the ray tangent direction).

The variation of n along a limb path in the Earth’s atmosphere is dominated by the vertical density gradient so that, to the first order, we can assume the gradient of n is directed radially and the local refractive index field is spherically symmetrical, i.e., n = n(r). Combining Eq. (5.9) with the Bouguer’s rule in Eq. (5.7), the bending angle increment along the ray path can be simplified as

$$ d\alpha =-\frac{d \ln (n)}{ dr}\frac{a}{\sqrt{n^2{r}^2-{a}^2}} dr. $$

(5.10)

Since the refractivity generally decreases at higher altitudes, to allow the bending angle to be positive values, a negative sign is added. The total bending angle thus becomes

$$ \alpha (a)=2{\displaystyle {\int}_{r_t}^{\infty } d\alpha =-2a{\displaystyle {\int}_{r_t}^{\infty}\frac{d \ln (n)}{ dr}\frac{ dr}{\sqrt{n^2{r}^2-{a}^2}}}} $$

(5.11)

where r is distance from the center of curvature of a ray path and the integral is over the portion of the atmosphere above r _t , the radius of the tangent point (i.e., the point on the ray path that is closest to the Earth’s center). By introducing the integration variable x = nr, Eq. (5.11) can be modified as

$$ \alpha (a)=-2a{\displaystyle {\int}_a^{\infty}\frac{d \ln (n)}{ dx}\frac{ dx}{\sqrt{x^2-{a}^2}}} $$

(5.12)

Equation (5.12) provides the forward calculation of bending angle α given the refractive index profile n(r). By inverting the equation through the Abelian transformation, the n(r) can be expressed as a function of α and a (Fjeldbo et al. 1971):

$$ n(r)= {\it exp}\left[\frac{1}{\pi }{\displaystyle {\int}_a^{\infty}\frac{\alpha (x) dx}{\sqrt{x^2-{a}^2}}}\right]. $$

(5.13)

Given impact parameter a and the refractive index n, the radius r at each tangent point can be derived according to Bouguer’s formula:

$$ r=\frac{a}{n(r)}. $$

(5.14)

Note that Eq. (5.13) embeds the assumption of local spherically symmetric atmosphere, i.e., the refractive index only varies along radius direction. However, the ellipsoidal shape of the Earth (with an equatorial radius roughly 20 km larger than its polar radius) and horizontal gradients in atmospheric structure produce non-spherical symmetry in the refractive index field. Moreover, the ray paths for a given occultation do not necessarily scan the atmosphere vertically nor are they coplanar. Therefore, measurement of α(a) will be affected by the tangential refractivity gradients and occultation geometry, and Eq. (5.13) could introduce systematic errors into the retrieved refractive index profile n(r). As shown in Fig. 5.4, First-order errors due to the ellipticity of the Earth can be eliminated by selecting a center and radius of curvature (r′) appropriate to the latitude and orientation of the occultation near the tangent point, where most of the bending is accumulated (Kursinski et al. 1997; Hajj et al. 2002).

Fig. 5.4

GPS radio occultation geometry. The bending angle, impact parameter and curvature radius are denoted by α, a and r′, respectively

3 GNSS Radio Occultation Processing

Based on the physical principle of the GNSS RO sensing technique described in previous section, the GNSS RO retrieval process generally consists of three major steps. Firstly, the GNSS signal phase and amplitude records are calibrated to derive the precise occultation geometry (e.g., positions and velocities of GNSS/LEO satellites) and the excess phase delay due to the atmosphere (Sect. 5.3.1). Secondly, the bending angle and the refractivity profiles are retrieved based on the physical principle of GNSS RO technique. Thirdly, the geophysical parameters in the neutral atmosphere and ionosphere can be retrieved, which will be described in Chaps. 6 and 7, respectively.

3.1 Calibrating and Extracting GNSS RO Observables

As the GNSS signal goes through the Earth’s atmosphere, the signal is bent and delayed due to the media. The difference between the observed phase delay and the geometric phase of the occultation link is referred as the excess atmospheric delay, i.e., a combination of the phase delay caused by the neutral atmosphere and ionosphere. By precisely measuring the excess phase delay and its differential form, or call Doppler, the bending angle and therefore the vertical structure of atmospheric refractive index can be derived. In following sections, we will focus on retrieving the excess atmospheric delay from the RO observations through the calibration process.

The basic observables at a LEO receiver for an occultation event are the occulting GPS signal amplitude and phase measurements. Specifically for a GPS occultation, the phase observations consist of L1 (C/A) and L2 (P2) phase measurements between the LEO receiver satellite and the occulting GPS satellite. These raw phase measurements L can be modeled (in dimension of distance) as (e.g., Hajj and Romans 1998; Hajj et al. 2002):

$$ {L}_k^{TR}=-\frac{c}{f_k}{\phi}_k^{TR}={\rho}^{TR}+{\Delta}_k^{TR}+{C}^T+{C}^R+{\varepsilon}_k $$

(5.15)

$$ {\Delta}_k^{TR}={\Delta}_k^{neutral}+{\Delta}_k^{iono}={\Delta}_k^{neutral}+\left(d\frac{ {\it TEC}_k^{TR}}{f_k^2}+\mathrm{O}\left({f}^2\right)\right) $$

(5.16)

with

• ϕ ^TR _k the recorded phase in cycles for the signal propagated from transmitter (T) to receiver (R);

• c the speed of light in vacuum; k = 1 or 2 for L1 and L2, respectively;

• ρ ^TR the geometric range (distance) between the transmitter and the receiver;

• Δ ^TR _k the total excess delay due to neutral atmosphere (Δ ^neutral _k ) and ionosphere (Δ ^iono _k );

• C ^T,  C ^R time dependent clock errors of the transmitter and the receiver, respectively;

• ε _k measurement noise due to the receiver’s thermal noise and local multipath;

• d a constant;

• TEC ^TR _k the integrated electron density along the raypath, and

• O(f ²) the higher order ionospheric terms (order 1/f ³ or higher).

Subscript k in Eqs. (5.15 and 5.16) implies the dependency on the frequency. The ionosphere is dispersive media whereas the neutral atmosphere is non-dispersive at radio frequencies. However, since the electromagnetic signal has to travel through the ionosphere before and after it reaches the lower neutral atmosphere, the L1 and L2 signals received at a given time also sense slightly different parts of the neutral atmosphere. This is why the neutral atmospheric delay term Δ ^neutral _k remains frequency dependent (Hajj et al. 2002).

Here we assume the phase errors caused by the transmitter and receiver antennas’ relative orientation as well as the phase center variations have been modeled and removed. Also, a constant bias (additive constant) corresponding to a large integer number (constant) of cycles during an occultation is ignored, since the derivative of the phase but not the absolute phase delay is of interest for an occultation measurement. It is worth noting that the high order ionospheric terms O(f ²) (Bassiri and Hajj 1993) is normally small and can be ignored, but it becomes a dominant error term at high altitudes (>40–60 km) during solar-maximum day-time conditions (Kursinski et al. 1997).

In Eq. (5.15), the dominant term on the right-hand-side is the geometrical range ρ between the transmitter and receiver. Whereas the excess atmospheric phase delay accumulated in the GPS L1 and L2 phase measurements in the neutral atmosphere (Δ ^neutral _k ) and ionosphere (Δ ^iono _k ) is of primary interest. By differentiating the excess phase delay, the excess Doppler due to the atmosphere can be derived, which is the fundamental building blocks for retrieving the vertical profile of bending angle and refractive index of the atmosphere.

The process of extracting the excess phase delay due to the atmosphere is generally referred as the calibration processes, which consists of two major steps. Firstly, Precision-Orbit-Determination (POD) is performed to derive the precise orbits (positions and velocities) for both the transmitter (e.g., GPS) and LEO RO receiver satellites on the occultation link. This process will allow removal of the dominant geometrical range term (ρ) in Eq. (5.15) between the transmitting and receiving satellites. Secondly, through the differential technique with simple linear combination, the clock errors of both GPS transmitter and LEO receiver can be removed. Consequently, the sum of the neutral and the ionospheric delays is isolated (up to a constant). The details of the calibration process is presented in the following two sessions.

3.1.1 Precision Orbit Determination (POD) Method

In order to derive useful atmospheric profiles from radio occultations, the velocities of both GPS and LEO satellites need to be estimated with high accuracy within sub-mm/s level, as it is directly related to the excess Doppler due to the atmosphere. Given the fast movement of GPS (∼3.8 km/s) and LEO (∼7 km/s) satellites, this is a daunting challenge. The way to achieve highly accurate orbit information is referred as Precision Orbit Determination (POD) process.

The objective of POD is to obtain an accurate orbit (position and velocity vectors) that accounts for the dynamical environment in which the motion occurs, including all relevant forces affecting the satellite’s motion. In practice, each LEO satellite is generally equipped with at least two upward-looking antennas, with one to track a high-elevation GPS satellite for POD purpose and another to track a reference GPS satellite. In addition one or two downward-looking (forward or rear-looking along satellite trajectory) are to the limb of the Earth’s atmosphere for occultation measurements (Fig. 5.5). For example, each COSMIC spacecraft is equipped with four antennas, with two upward-looking antennas and two limb viewing occultation antennas.

Fig. 5.5

Data links used for various differencing technique

Since the beginning of the space age the POD technique has been used in the geodesy research community as a means to improve or even determine geodetic models. As a major legacy of GPS geodesy, the International GPS Service for Geodesy (IGS) was formed in 1992, to oversee the deployment and operation of a permanent global reference ground network to provide precise GPS orbits and reference data to geodesists (Mueller and Beutler 1992). These reference data are critical for the occultation missions.

For each occultation, the POD process consists of the orbit determination for the occulting and reference GPS transmitters and for the LEO receiver. The POD of GPS involves processing the ground reference stations data to estimate the high-rate clock offsets and precise orbits for the GPS satellites. Once the ground-based processing is completed, the space-based LEO POD processing is executed to solve for the LEO orbits and clock offsets with several processing options, which includes the dynamic approach, kinematic or geometric approach, and the reduced-dynamic approach (e.g., Rim and Schutz 2002).

The dynamic orbit determination approach (Tapley 1973) directly solves for the equation of the motion and thus requires precise models of the forces acting on the satellite. This can be achieved by accurate modeling various forces on the satellite, including the gravitational forces (e.g., gravitational effects of sun, moon and planets, tides and relativistic effects), non-gravitational forces (e.g., atmospheric drag, solar and Earth radiation pressure and thermal radiation). Some other unmodeled forces will generally need to be estimated. Dynamic model errors are the limiting factor for this technique, which, however, can be reduced by the continuous, global, and high precision GPS tracking data.

Alternatively, the kinematic approach doesn’t require modeling the orbit dynamics except for possible interpolation between solution points for the satellite (i.e., relies purely upon observation data), and the orbit solution is referenced to the phase center of the on-board GPS antenna instead of the satellite’s center of mass. However, kinematic solutions are more sensitive to geometrical factors, such as the direction of the GPS satellites and the GPS orbit accuracy, and they require the resolution of phase ambiguities, which are not always available.

The reduced-dynamic approach (Wu et al. 1987), on the other hand, uses both kinematic and dynamic information and optimally weighs their relative strength by solving for local geometric position corrections using a process noise model to absorb dynamic model errors.

There are various software packages for POD process in the radio occultation research community. For example, the GFZ group (Geoforschungszentrum, German Research Center for Geosciences) use the EPOS-OC; NASA Jet Propulsion Laboratory (JPL) utilize the GIPSY-OASIS (GNSS-Inferred Positioning System and Orbit Analysis) software and apply the reduced-dynamic strategy; The UCAR COSMIC group use the Bernese GPS data processing package to solve for satellite orbits with a reduced-dynamic approach (e.g., Ho et al. 2009).

3.1.2 Differencing Technique to Remove Clock Errors

Once the effect of satellite motion is removed, the GPS and LEO satellite clock errors need to be eliminated through the differencing technique to derive the atmospheric excess phase of the occultation link.

In the ideal case, when both GPS and LEO satellite clocks are sufficiently stable and require no calibrations, i.e., the LEO and GPS clock offset are zero or known, the total excess phase delay due to the atmosphere can be directly resolved from Eq. (5.15) after removing the satellite geometric term. This is referred as zero-difference (or un-difference), i.e., no differencing is needed. Beyerle et al. (2005) demonstrate the zero-difference processing can produce highly accurate excess phase data by applying prior estimated LEO and GPS clocks in GRACE-B occultation measurements. Note that the twin GRACE (Gravity Recovery and Climate Experiment AɖB) spacecrafts are equipped with an ultra-stable-oscillator (USO), which allows highly accurate clock measurements without need for clock calibration.

However, for most of the other GPS occultation missions, the less stable LEO receiver’s clocks generally require calibration. Sometimes the GPS clocks are also need to be calibrated. A commonly known calibration process is the differencing technique. Differential GPS/GNSS (DGPS/DGNSS) is a technique for reducing the error in GPS-derived positions by using additional data from a reference GNSS receiver at a known position. The technique was originally developed by GPS geodesists to significantly improve the precision of ground GPS measurement, in the presence of large errors due to the selective availability (SA) process. The SA was introduced by the U.S. Department of Defense to degrade the performance of GPS. The intentional, time varying errors of up to 100 m were intentionally added to the L1 publicly available navigation signals to destabilize GPS signals for “unauthorized” users. This significantly limited the usage of the civilian GPS application that requires much higher precision of measurements without a costly classified receiver. In the early and mid 1980s, the pioneering work of GPS geodesists leads to a DGPS technique. Generally, the DGPS involves determining the combined effects of navigation message ephemeris and satellite clock errors (including the effects of propagation) at a reference station and transmitting corrections, in real time, to a user’s receiver. Given a reference receiver with known exact position, the vector displacement or differential, between the known position and the position the reference receiver get from the GPS satellites, can be calculated. This differential or correction can then be applied to achieve higher accuracy of positioning for the GPS receiver of interest that collects the exact same data from the same GPS satellites at the same time. The DGPS technique eliminates selective availability and other clock errors and allows civilian receivers with broad-beam antennas to achiever millimeter precision in phase measurements (e.g., Wu 1984). The widespread usage of differential GPS services as well as the new technologies to deny GPS service to potential adversaries on a regional basis by the US military eventually leads to the termination of the SA service in 2 May, 2000.

The similar technique can be applied for deriving high-precision LEO satellite obits. The two most widely applied procedures are called double-differencing and single-differencing, and they differ in how the effect of the GPS satellite oscillator fluctuations are removed from the LEO phase data (Kursinski et al. 1997; Wickert et al. 2002; Hajj et al. 2002; Beyerle et al. 2005). Figure 5.5 shows a diagram that illustrates differencing geometry.

The single-differencing procedure requires the LEO receiver viewing simultaneously an occulting transmitter (GPS_occ) and a non-occulting reference transmitter (GPS_ref) during the occultation (Fig. 5.5). The occultation link (LEO-GPS_occ) and reference link (LEO-GPS_ref) data are differenced to remove the effect of the receiver clock errors, and the solved-for high-rate GPS clock offsets are interpolated based on the IGS final clock products to remove the effects of the transmitter clock errors (Wickert et al. 2002; Schreiner et al. 2010).

However, in some situations, especially before the deactivation of Selective Availability (S/A), the GPS clock can be sufficiently unstable and need to be calibrated. In such case, the double-differencing procedure is needed to remove both the GPS and LEO clock errors (Hajj et al. 2002; Wickert et al. 2002; Schreiner et al. 2010). Therefore, other than the two data links between LEO and the occulting/reference GPS transmitters, two additional data links between a ground reference station (GS_ref, precisely known position) and the same occulting/reference transmitters need to be measured (Fig. 5.5). The similar single-differencing technique needs to be applied twice to eliminate the clock errors of both the GPS transmitters and LEO receiver.

One significant disadvantage of double-difference processing is its susceptibility to availability of ground fiducial network station data (Galas et al. 2001; Wickert et al. 2001) as well as error sources including multipath, residual atmospheric and ionospheric noise, data interpolation, and thermal noise. Since deactivation of Selective Availability (S/A), the GPS clock errors are reduced by orders of magnitude. Without S/A GPS clocks are sufficiently stable, therefore, double differencing can be replaced by the single difference technique to eliminate the need for concurrent high-rate ground station observations (Wickert et al. 2002).

After the POD and differencing process, the L1 and L2 atmospheric excess phases can be derived along with the precise orbits (positions and velocities) of both the LEO and occulting GPS satellites. The further derivation of the atmospheric properties based on these occultation measurements will be explored in the next section.

3.2 Bending Angle Retrieval

After the phase delays are calibrated to remove special and general relativistic effects and to remove the GPS and LEO clock errors, a time series of excess phase delay at both GNSS frequencies (e.g., L1 and L2 for GPS) are derived. Then the atmospheric bending in the ionosphere and neutral atmosphere can be inferred.

In the ionosphere and upper part of the neutral atmosphere, the radio signals can be assumed to be monochromatic (e.g., single-tone or single-ray), i.e., only one ray connects the transmitter and receiver at one instant. The geometric optics (or ray optics) concept can be applied to describe the radio signal propagation, as the diffraction effect can be neglected. The computation of bending angles is thus straightforward as they are unambiguously related to the instantaneous frequency of the received signal.

In the lower troposphere, however, radio signals become non-monochromatic (e.g., multiple-tone) and may have a very complex structure due to multipath effects caused mainly by water vapor structures (Gorbunov and Gurvich 1998; Sokolovskiy 2001). The atmospheric multipath occurs when sharp vertical variations in atmospheric refractivity structure create multiple, simultaneous signal paths connecting the transmitter and receiver through the atmosphere. In the multipath regions, the bending angles cannot be derived directly from the instantaneous frequency of the measured signal because the instantaneous frequency will be related, not to a single ray, but to two or more rays.

Generally, it is practical to split the bending angle retrieval into two altitude ranges. Above the lower troposphere where atmospheric multipath is not significant, the bending angles at both frequencies are derived based on geometric optics from the differential form of the excess phase (or excess Doppler) after appropriate noise filtering (Vorob'ev and Krasil'nikova 1994; Kursinski et al. 1997; Hajj et al. 2002). In the lower troposphere, where atmospheric multipath cannot be neglected, the radio-holographic (wave optics) techniques are needed to accurately reconstruct the bending angle from the phase and amplitude measurements (e.g., Gorbunov 2002; Jensen et al. 2003).

3.2.1 Geometric Optics Method

The sum of the extra phase delay due to the ionosphere and neutral atmosphere is determined after the calibration process. The differential form of the excess phase delay, also called extra Doppler shift can be derived. Based on geometric optics assumption, the accumulation of atmospheric bending along a ray path can be measured as an extra Doppler shift relative to that expected for a straight-line (in vacuum) signal path (Vorob'ev and Krasil'nikova 1994; Kursinski et al. 1997; Hajj et al. 2002). By using the geometry and notation in Fig. 5.4, the extra Doppler shift f _D in the GPS transmitter frequency f _T at the receiver can be expressed as the projection of the satellite velocities on the ray paths (Kursinski et al. 1997), such as

$$ \begin{array}{llllllllllll}{f}_D\hfill & =\dfrac{f_T}{c}\left({\overrightarrow{V}}_T\cdot {\overrightarrow{k}}_T+{\overrightarrow{V}}_R\cdot {\overrightarrow{k}}_R\right),\hfill \\ {}\hfill & =-\dfrac{f_T}{c}\left({V}_T^r \cos {\phi}_T+{V}_T^{\theta } \sin {\phi}_T+{V}_R^r \cos {\phi}_R+{V}_T^{\theta } \sin {\phi}_R\right),\hfill \end{array} $$

(5.17)

where ${\overrightarrow{V}}_R$ and ${\overrightarrow{V}}_T$ are velocity vectors for receiver and transmitter, and ${\overrightarrow{k}}_R$ and ${\overrightarrow{k}}_T$ are unit vectors representing the direction of the ray path at the receiver and transmitter, and c is the velocity of light in vacuum. The superscripts (r and θ) represent the radial and tangential components of the velocity vectors, and ϕ _T and ϕ _R are the angels between the ray path and the satellite position vectors at the transmitter and receiver (Fig. 5.4).

According to Bouguer’s rule in Eq. (5.7) and the geometry of Fig. 5.4, we have:

$$ a=n\left({r}_T\right){r}_T \sin {\phi}_T=n\left({r}_R\right){r}_R\; \sin {\phi}_R=n\left({r}_t\right){r}_t. $$

(5.18)

As the transmitter and the receiver are at reasonably high altitudes, we can simply setting the refractive index equals to unity at both transmitter and receiver, thus we have

$$ a={r}_T \sin {\phi}_T={r}_R \sin {\phi}_R. $$

(5.19)

The total bending angle α according to the geometry in Fig. 5.4 follows

$$ \alpha ={\phi}_T+{\phi}_R+\theta -\pi, $$

(5.20)

where, r _T , r _R , and r _t are the distance of the transmitter, receiver and the tangent point from the center of curvature, and θ is the open angle between the transmitter and receiver vectors.

Given the precise measurements of the occultation geometry (transmitter and receiver position and velocity vectors) and extra Doppler shift, the bending angle can be derived iteratively from Eqs. (5.17, 5.18, 5.19, and 5.20) under the assumption of local spherically symmetric atmosphere.

3.2.2 Radio-Holographic (RH) Method

In the lower troposphere, the multipath problem due to the water vapor gradients could result in large error in the geometric optics bending retrieval. Also the vertical resolution of GO retrieval is limited by the size of the Fresnel zone (Kursinski et al. 1997). In order to tackle the multipath problem and improve the vertical resolution of the bending angle retrieval, it is necessary to use the radio-holographic (RH) methods, which is based on the analysis of the records of complex radio signals, or radio holograms.

There are many RH methods proposed for processing radio occultation signals in multipath regions: (1) back-propagation (BP) (also referred to as diffraction correction), which propagates signals backward in the vacuum toward a plane located in a single-ray area (Hinson et al. 1997; Gorbunov and Gurvich 1998); (2) radio-optics, which analyzes the local spatial spectra of the measured complex wave field through Fourier analysis (Lindal et al. 1987; Pavelyev 1998; Hocke et al. 1999; Sokolovskiy 2001); (3) Fresnel diffraction theory (Marouf et al. 1986; Mortensen and Høeg 1998); (4) canonical transform (CT) (Gorbunov 2002); (5) full-spectrum-inversion (FSI) (Jensen et al. 2003) and (6) phase matching method (Jensen et al. 2004).

The widely used CT method consists of using Fourier Integral Operators (FIO) to find directly the dependence of the bending angle on the impact parameter for each physical ray in multi-path conditions. The CT method requires that the electromagnetic field is back-propagated to a straight line before the canonical transformation can be applied to transform the coordinate to impact parameter representation (Gorbunov 2002). The back-propagation from the observation trajectory to the auxiliary straight-line trajectory results in the most computational time in CT processing. The Fourier transform method has theoretical limitation that requires circular satellite orbits. The approximation developed by Jensen et al. (2003) in FSI method expands the Fourier method to be applied for near-circular/realistic satellite orbits. The FSI method utilizes the relation between the derivative of the phase of a physical ray on the instantaneous frequency in the full Fourier spectrum of the RO signal and the time of intersection of the physical ray with LEO satellite orbit. Similar to FSI method, the phase-matching method is based on the synthetic aperture concept and the method of the stationary phase and thus preserves the high vertical resolution properties. Moreover, it is a more general method that is directly applicable for noncircular/realistic orbits and eliminates the intermediate step of propagation of complex electromagnetic field to circular orbits (e.g., for FSI) or straight line (e.g., for CT). However the phase-matching method comes with higher computational cost, as it can’t apply Fast-Fourier-Transform (FFT).

All the RH methods are based on stationary electromagnetic theory. Other than BP method, the other RH methods provide the bending angle as a single-valued function of impact parameter in the multi-path region. The BP method, however, may result in muli-valued function in particular in the presence of ducting or super-refraction (Gorbunov 2002).

In terms of resolution and ability to handle multipath, the most efficient radio-holographic methods are currently the Fourier operator based methods such as canonical transform and FSI. Both methods are widely adapted by many GNSS occultation data processing centers. For GPS and close to circular LEO orbit, the FSI method is an optimal method (most accurate and fast) for bending angle retrieval in the mult-path condition.

3.3 Ionosphere Retrieval

After the calibration process, the sum of the neutral and the ionospheric delays is isolated (up to a constant). When the tangent point of the occultation link is in the ionosphere, the excess phase delay due to the neutral atmosphere is negligible. There are two types of processing technique for extracting ionospheric delay along the ray path (Hajj and Romans 1998; Schreiner et al. 1999): (1) single-frequency approach, i.e., deriving the ionospheric delay at each carrier frequency (e.g., L1 and L2) separately; (2) a dual-frequency approach that directly isolates the ionospheric delay through linear combination by assuming L1 and L2 signals travel along the same ray path in the ionosphere.

The first approach results in less noisy determination of ionospheric delay but requires the a-priori calibration processes (Sect. 5.3.1) to remove the orbit and clock errors. The second approach is inherently simpler by eliminating the calibration process, as the orbit and clocks errors cancel out when forming the L1 and L2 linear combination. However, the simplicity in the dual-frequency approach is at the cost of lower precision due to the noise added by L2. Also it assumes L1 and L2 signals travel along the same ray path in the ionosphere, which could be violated and result in extra errors in the presence of significant bending in the ionosphere (Hajj and Romans 1998).

By using the single-frequency approach, the bending angle at each signal frequency can be retrieved from the ionospheric delay or Doppler. Following the Abel transform in Eq. (5.13), the vertical profile of refractive index can be derived. Note that the Abel integral requires knowledge of bending all the way up to the top of the ionosphere. GPS is above most of the ionosphere, however, the LEO receiver satellite are generally located inside the ionosphere. Therefore the bending due to the ionosphere above LEO altitude might not be neglected and needs to be modeled (e.g., Hajj and Romans 1998).

The electron density profile n _e (in per cubic meter) can then be derived through the following relation (Hajj and Romans 1998):

$$ {n}_e(r)=\left[1-n(r)\right]\cdot {f}^2/(40.3). $$

(5.21)

In the ionosphere, the total electron content (TEC in electrons numbers per square meters) along a ray is related to electron density n _e , refractive index n and the excess phase S (in meters) by

$$ TEC={\displaystyle \int {n}_e dl=-\frac{f^2}{40.3\times {10}^{16}}{\displaystyle \int \left(n-1\right) dl=-\frac{f^2S}{40.3}}}. $$

(5.22)

Due to the dispersive characteristics of the ionosphere, the L1 and L2 signals propagate on slightly different paths and thus result in slightly different TECs. TEC may be calculated from excess ionospheric phase delay (S ₁ and S ₂ at two carrier frequencies) after removing the orbit and clock errors through calibration process.

While the bending angle is rather small in the F2 layer of the ionosphere, a straight-line propagation assumption (i.e., the path difference is negligible) can be applied, such that the TEC is related to the calibrated phase ΔS = (S ₁ − S ₂) as

$$ TEC=\frac{\Delta S}{40.3}\frac{f_1^2{f}_2^2}{f_1^2-{f}_2^2}, $$

(5.23)

where f ₁ and f ₂ are the GPS L1 and L2 carrier frequencies, respectively. The phase difference (ΔS) cancels out the orbit and clock errors automatically and eliminates the complicated calibration process. The dual-frequency approach greatly reduces the amount of data processing and allows electron density profile n _e to be computed on orbit and disseminated in near real time that is essential for space weather forecast.

Note that the total TEC is the integral of electron density from the GPS transmitter to LEO receiver. With the assumption of spherical symmetric ionosphere, the ionosphere contribution above the LEO can be removed. The TEC along the section of a ray below the LEO, $\overline{ TEC}$ (r₀), is a function of electron density (Schreiner et al. 1999)

$$ \overline{ TEC}\left({r}_0\right)=2{\displaystyle {\int}_{r_0}^{r_{LEO}}\frac{n_e(r)\cdot r}{\sqrt{r^2-{r}_0^2}}} dr. $$

(5.24)

Therefore, the electron density profile can be reconstructed similar to the Abel transform through the following:

$$ {n}_e(r)=-\frac{1}{\pi }{\displaystyle {\int}_{r_0}^{r_{LEO}}\frac{d\left(\overline{ TEC}\right)}{d{r}_0}\frac{d{r}_0}{\sqrt{r_0^2-{r}^2}}}. $$

(5.25)

The inversion of the electron density n _e (r) is based on the assumption of local spherical symmetry of the electron density in a large region (a few thousand kilometers in radius) around the ray path tangent points. This assumption may be violated due to the presence of large horizontal gradients in electron density, in particular below the F-layer (sometimes giving large negative or positive electron density). At the same time the geographical location of the ray path tangent points at the top and at the bottom of a profile may differ by several hundred kilometers (horizontal smear). Therefore, retrieved electron density profiles should generally not be interpreted as actual vertical profiles, but rather as a mapping of both vertical and horizontal ionospheric structure into a 1D profile, given particular occultation geometry.

3.4 Neutral Atmosphere Retrieval

In the neutral atmosphere (generally below 70 km), the inversion process includes removing the ionospheric effects, deriving the neutral atmosphere bending angle, retrieving refractivity, and further inferring the geophysical parameters.

3.4.1 Ionospheric Calibration on Bending

When the tangent points descend into the neutral atmosphere, the bending angles at both GPS frequencies include the bending contribution from both the ionosphere and neutral atmosphere. In the most general situation, an ionospheric correction is needed in order to derive the neutral atmospheric bending. The process to separate the neutral atmospheric bending (or ionosphere-free bending) by removing the ionosphere induced bending is referred as ionospheric calibration.

Because of the dispersive nature of the ionosphere, the L1 and L2 signals travel along slightly different paths and have slightly different bending. The separation of the two signal paths in the ionosphere near the tangent point varies from 100 to 5 km, depending on the tangent height and location of the occultation, the local time and the solar conditions (e.g., Hajj and Romans 1998; Hajj et al. 2002).

Generally, a first-order ionospheric correction is accomplished through a simple linear combination of the L1 and L2 bending angle profiles following a procedure first suggested by Vorob'ev and Krasil'nikova (1994), such that

$$ {\alpha}_{neutral}(a)=\frac{f_{L1}^2}{f_{L1}^2-{f}_{L2}^2}{\alpha}_{L1}(a)-\frac{f_{L2}^2}{f_{L1}^2-{f}_{L2}^2}{\alpha}_{L2}(a), $$

(5.26)

where α _L1 and α _L2, the bending at two GPS frequencies f _L1 and f _L2, are interpolated to the same level of impact parameter (a).

Note that the L2 phase measurements are usually noisier and less accurate than L1 due to lower frequency (i.e., more prone to the ionospheric scintillation and delay) and lower SNR (signal to noise ratio) (e.g., Hajj et al. 2002). To reduce the effect of larger noise on L2, a slightly modified linear combination is used (Rocken et al. 1997):

$$ {\alpha}_{neutral}(a)={\alpha}_{L1}(a)+\frac{f_{L2}^2}{f_{L1}^2-{f}_{L2}^2}\left[{\overline{\alpha}}_{L1}(a)-{\overline{\alpha}}_{L2}(a)\right], $$

(5.27)

where ${\overline{\alpha}}_{L1}(a)$ and ${\overline{\alpha}}_{L2}(a)$ are obtained from L1 and L2 excess phases subjected to filtering with larger smoothing window (e.g., longer intervals) than that used for α _L2(a). This results in overall error reduction by reducing the effects of L2 noise, however, at the expense of a certain increase of uncalibrated ionospheric effects.

Deeper into the lower atmosphere (e.g., below ∼10 km), the L2 signal becomes too weak for robust tracking due to the low L2 signal power, incomplete knowledge of P-code modulation as well as large atmospheric defocusing effects. In such case, the ionospheric correction term (i.e., ${\overline{\alpha}}_{L1}(a)-{\overline{\alpha}}_{L2}(a)$), is linear extrapolated from higher altitudes downward to the surface for continuing ionospheric correction in the lower altitudes with the absence of L2 measurement. Also note that the ionospheric calibration should not be applied above a certain height, when the neutral atmosphere signature on the occulted signal is comparable to residual ionospheric effects or the receiver’s thermal noise. This tends to occur at the altitude of ∼50–90 km, depending on the ionospheric conditions (Hajj et al. 2002).

The ionospheric calibration process described above effectively removes the first order ionospheric term (1/f ²) in Eq. (5.1). Higher order contributions constitute the major source of error during day-time solar maximum at high altitudes (e.g., Kursinski et al. 1997) will require further calibration (e.g., Bassiri and Hajj 1993).

3.4.2 Refractivity Retrieval from Abel Inversion

Under the assumption of spherically symmetric atmosphere, the refractive index profile can be derived from the neutral atmospheric bending through the Abel transform in Eq. (5.13). As the upper limit of the Abel integral requires knowledge of the bending angle as a function of impact parameter up to the top of the atmosphere. However, the estimated bending is reasonable accurate only up to a certain upper height (e.g., ∼50–90 km dependent on the ionospheric condition as discussed in previous section). Therefore, the a-priori (or background) bending angle is needed to extend the observational neutral bending angle at higher altitudes. This a-priori bending is often referred as the upper boundary condition for Abel inversion. In practice, the a-priori bending can be derived from weather or climate models (e.g., Hedin 1991) or from a simple model (e.g., exponential extrapolation of bending upward from a certain altitudes (Hajj et al. 2002)). The uncertainty in the a-priori bending model could introduce errors in the refractivity retrieval from Abel integral. To reduce the effects of error propagation downward from the upper stratosphere, the optimization technique that mixes the observational neutral atmospheric bending with the a-priori model can be used (e.g., Lohmann 2005).

After the refractive index as a function of impact parameter n(a) at the tangent point, is derived from the Abel inversion, the tangent point radius is obtained from (5.14), i.e., r = a/n. The radius in turn is converted into height above an ellipsoidal fit to the mean sea-level geoid.

3.4.3 Quality Control

In the GNSS RO data processing centers, various quality control methods are applied and they are used at different processing stages. For example, in the early processing stage, the quality of the measured signal SNR, excess phase and excess Doppler are examined; whereas in the later stages, the bending angle and/or refractivity profiles are compared with either climatology or weather model analysis or reanalysis products. Various criteria can be applied to ensure the quality of RO retrieval dataset for suited research or operational usage (e.g., Ho et al. 2009).

After going through the calibration and retrieval processes, the bending angle and refractivity profiles can be derived from occultation measurements. Further process will lead to the derivation of the atmospheric properties in the neutral atmosphere and ionosphere, which will be elaborated in the following Chaps. 6 and 7, respectively.