## 3 Results
### Three-Satellite Constellation
#### 3.1.1 Representation of Precipitation at RO Observation Times  
Superimposing the RO events simulated above on actual coincident precipitation data allows for a physically realistic characterization of the number of cloud structures associated with heavy precipitation that will be sampled per year, such that the constellation will be able to provide RO soundings inside and nearby heavy precipitation almost simultaneously. Following the detectability threshold proposed by [PERSON] et. al. [35], a precipitation detection threshold of \(\Delta\phi\,>\) 1.5 mm is set. This detection threshold has been confirmed with ROHP data as a conservative threshold for rays with a tangent point height of \(\sim\)2 km [25; 37]. In [PERSON] et. al. [37], detectability was defined as vertical averages of \(\Delta\phi\), but since IMERG is providing only surface precipitation, for the purpose of this study, a single ray close to the surface is assumed to represent the precipitation conditions at the time of the RO. In the following discussion, the definition of \"outside precipitation\" is better expressed as \"undetected precipitation\" (i.e., when \(\Delta\phi\) < 1.5 mm) for the purposes of this study. This is because one is not able to completely rule out the possibility that there is precipitation present when the \(\Delta\phi\) that is induced is below the instrument detectability threshold [37]. For the computation of the full end-to-end contributions to \(\Delta\phi\) along each ray path, [36] carried out scattering computations using the T-matrix approach for liquid- and solid-phase precipitation. Since IMERG only provides surface precipitation, to calculate the precipitation-induced \(\Delta\Phi\) in a computationally efficient fashion, a simplified power-law equation that relates the rain rate (\(R\)) to the specific differential phase shift (\(K_{DP}\)) at the GNSS L-band frequency (1.575 GHz) was derived [44] and is as follows:  
\[K_{DP}=cR^{b} \tag{1}\]  
where \(c\) = 0.00868 and \(b\) = 1.218, \(R\) is in mm hr\({}^{-1}\) and \(K_{DP}\) is in units of deg km\({}^{-1}\). The rain rates provided by IMERG are interpolated into the representative RO ray, which is simulated by a straight line of 300 km (i.e., 150 km on either side of the tangent point in the azimuthal direction between the GNSS satellite and the RO receiver). \(K_{DP}\) is integrated along the ray path to obtain \(\Delta\phi\). Figure 7: **(a)** Histogram of the distance (km) between the tangent point locations of the RO acquired with the first and second satellite of the train constellation (blue), and with the first and third satellites (orange). **(b)** Same as left panel, but for the time difference (seconds) between the RO tangent point locations. To use a single straight-line ray is a simplification of the actual ray paths that result as the receiving and transmitting satellites occult each other. In reality, the set of RO rays do not create a perfect vertical plane while descending deeper into the atmosphere, but rather a slant collection of rays that follow the geometry created by the relative motion of the receiver with respect to the transmitter. In Figure 8, an RO event captured by three satellites is shown in Figure 8a, where two of the observations intersect precipitation, while the third is sounding outside of the precipitation structure. A realistic representation of the RO ray projection onto the Earth surface is shown in Figure 8a (colored collection of rays), in addition to the simplified 300 km ray for each RO (black dashed line). The realistic rays are generated using actual ray-tracing from the retrievals of a real RO observation that occurred in a different location and time, and have been transported here for illustration purposes. Only the portion of these rays that lie below 6 km are shown, which are the portions that are more likely to be affected by precipitation.