## 2 Literature Review
### Radio Occultation Concept  
The RO method, elucidated by various researchers [4; 16; 17], precisely measures how radio waves emitted by GPS satellites bend due to refractive index gradients before reaching LEO satellites [18]. This technique, pivotal in understanding planetary atmospheres within our solar system [19], involves measuring the bending angle (\(\alpha\)) of radio waves as they intersect the atmosphere, providing data on pressure, temperature, Water Vapor Pressure (WVP), and refractivity [20]. In ionized mediums, this angle helps determine Total Electronic Content (TEC) [21]. The methodology utilizes a RADAR transmitter on a spacecraft beyond the target planet's atmosphere and a receiver on Earth's surface. As the spacecraft passes behind the planet's limb, electromagnetic rays traverse atmospheric layers, revealing insights into the vertical structure of the planet's ionosphere and neutral atmosphere [22]. Adaptation of the RO technique to study Earth's atmosphere was enabled by the U.S. Global Positioning System (GPS) [23], later augmented by the GLONASS, GALILEO, and BEIDOU systems [24]. Pyxis, a notable GPS-RO sensor, stands out for its compact size and capability to receive signals from major GNSS constellations--GPS, GLONASS, GALILEO, and BEIDOU. It provides over ten times the data of existing sensors, probing even the lowest atmospheric layers where severe weather events occur [25]. Figure 1 illustrates the RO geometry for a single event involving LEO satellites, such as PlanetiQ, with symbols defined as follows: r\({}_{\text{G}}\), r\({}_{\text{L}}\) are GNSS and PlanetiQ satellite positions; V\({}_{\text{G}}\), V\({}_{\text{L}}\) are their velocities; \(\theta_{\text{L}}\) and \(\theta_{\text{G}}\) are angles between Earth's center and the satellites' ray directions; a is the impact parameter, and \(\alpha\) is the signal bending angle. In the context of the neutral atmosphere, the refractivity (N), a dimensionless parameter represented as N = 10\({}^{6}\)(n - 1), where n denotes the refractive index, can be formally expressed in relation to atmospheric conditions. Refractivity is intricately linked to atmospheric pressure (P) measured in hectopascals, temperature (T) in kelvins, and water vapor partial pressure (P\({}_{\text{w}}\)) in hectopascals [27]. \[N=77.6\frac{P}{T}+3.73\times 10^{5}\frac{P_{w}}{T^{2}} \tag{1}\]  
In the framework of a locally spherically symmetric atmosphere, a ray adheres to [PERSON]'s law, implying that the impact parameter in Equation (2) remains constant for a specific ray within the geometric optics approximation [28]. \[a=rn(r)sin(\theta_{L}) \tag{2}\]  
The total refractive bending angle \(\alpha\), as a function of rt (i.e., the radius of the ray at the tangent point), is given by [19]. in Equation (3), which can be further simplified to Equation (4) given that a(r) is a monotonic function and using the substitution x = n(r)r [27]:  
\[\alpha(a)=-2a\int_{r_{t}}^{\infty}\frac{1}{n}\frac{dn}{dr}\frac{dr}{\sqrt{\left( nr\right)^{2}-a^{2}}} \tag{3}\]  
\[\alpha(a)=-2a\int_{r_{t}}^{\infty}\frac{1}{n}\frac{dn}{dx}\frac{dx}{\sqrt{x^{2 }-a^{2}}} \tag{4}\]  
In the existence of a ducting layer, the impact parameter a exhibits a non-monotonic behavior concerning the radial distance r within and immediately below the ducting layer, as demonstrated by [29]. Additionally, the calculation of a requires an alternative method, specifically not adhering to Equation (4). To address this, a more suitable approach, described in detail by [29], is employed for solving Equation (3) in the presence of the ducting layer. Once the bending angle profile is provided, the refractive index function n'(r)  
Figure 1: Occultation event geometry, defining important location and angular variables of an RO event. Adapted from [26]. is determined by inverting Equation (5) using the Abel inversion technique, as outlined by [19]. From refractivity, pressure, temperature, and WVP, refractivity can be calculated.