# Preliminary Results on Tropospheric ZTD Estimation by Smartphone
## 2 Materials and Methods  
### ZTD Estimation from GNSS  
It is widely understood that GNSS was initially designed to determine the position, expressed by cartesian coordinates (X, Y, Z) of a receiver by means of pseudo-range or carrier-phase measurements. Both these approaches are affected by biases, primarily produced by the Earth atmosphere, which increases the real distance between satellite and the receiver, and by instrumental (both receiver and satellites) and site-specific causes: receiver and satellites clock errors, multipath, etc. The basic GNSS carrier-phase observable, in units of length, can be written as follows [36]:  
\[\varphi_{\mathrm{R}}^{\mathrm{S}}(\mathrm{t})=\ \ \mathrm{g}_{\mathrm{R}}^{ \mathrm{S}}(\mathrm{t})+\mathrm{c}\cdot\left(\delta_{\mathrm{R}}(\mathrm{t})- \delta^{\mathrm{S}}(\mathrm{t})\right)\ +\ \lambda\mathrm{N}_{\mathrm{R}}^{\mathrm{S}}(\mathrm{t})-\mathrm{I}_{\mathrm{R}}^{ \mathrm{S}}(\mathrm{t})+\ \mathrm{T}_{\mathrm{R}}^{\mathrm{S}}(\mathrm{t})+\mathrm{MP}(\mathrm{t})+\epsilon \tag{1}\]where \(\lambda\) is the wavelength of the carrier-phase, \(\rho_{\rm R}^{\rm 6}\) is the geometric satellite-receiver distance (obtained by the satellite and the receiver coordinates), \(\epsilon\) is the speed of light, \(\delta_{\rm S}\) and \(\delta^{\rm R}\) are the satellite and receiver clock errors, respectively, \(\rm N_{R}^{\rm S}\) is the unknown initial phase ambiguity, \(\rm I_{R}^{\rm S}\) is the ionospheric (slant) delay, \(\rm T_{R}^{\rm S}\) is the tropospheric (slant) delay, MP is the multipath effect, and \(\epsilon\) contains all the remaining unmodelled errors. Most of the terms in Equation (1) can be neglected by means of appropriate precautions during the installation of the instrumentation and expedients when the GNSS receiver is operative. For example, \(\rho_{\rm R}^{\rm S}\) estimation can be improved through accurately known satellite and receiver positions, \(\delta_{\rm S}\) and \(\delta^{\rm R}\) can be eliminated using the double differences strategy or can be modelled in PPP, \(\rm N_{R}^{\rm S}\) is resolved in the initialization phase, and \(\rm I_{R}^{\rm S}\) can be eliminated using the iono-free dual frequencies combination for double-frequency receivers, or it can be modelled for single-frequency receivers. The multipath effect MP can be mitigated by choosing an antenna position that is far from reflecting objects, installing special antennas (e.g., choke-ring antennas), masking low elevation signals, or by using techniques and models designed to recognize the reflected signals and discard them [37]. The presence of the atmosphere introduces effects on the signal crossing that influence the antenna positioning. At the same time, these effects may be a starting point to monitor the troposphere itself, representing a useful knowledge base for meteorological purposes. Since the early 1990s, GNSS has also been used for meteorological purposes [38] because of its high accuracy, all-weather operation capability, high temporal resolution, and compatibility with other observation systems, e.g., radiosondes, water vapor radiometers, and lately, environmental satellites. GNSS meteorology exploits the delay of electromagnetic GNSS signal transmission in the atmosphere due to water vapor, dry gases, hydrometeors, and other particulates [39; 40]. As already stated, GNSS observations are affected by different sources of bias, one of which is caused by tropospheric refractions during the crossing of the atmosphere by the electromagnetic GNSS signal. The tropospheric effect is frequency-independent, and it cannot be reduced, but it produces a bias in each satellite-receiver observation that can be related to the so-called ZTD. The estimation of ZTD helps to enhance the positioning precision and represents a contribution to meteorological studies. ZTD can be estimated by GNSS CORSs network adjustments to correct the computed delay, thus exploiting a tropospheric model that can be adopted in atmospheric conditions that are not standard.