**Acknowledgements**

The only case where the hypothesis of a small *ϵ<sup>w</sup>* can be violated occurs during

increases. The warm inversion layer then acts as a cap and stops atmospheric mixing [93] causing a large deviation of the refractivity with respect to the exponential

The end-product for the meteorology community of the inversion of Eq. (19) cannot only be the set of *Hw* and *λ<sup>n</sup>* coefficients, that are too difficult to handle. We propose, in addition, to give the results in the form of records over a grid the resolution of which is in agreement with the maximum degree of the expansion in Eq. (19), with respect to a suitable ellipsoid (like WGS84), and with these fields: *Observation Time, Latitude, Longitude, Geometrical height, total refractivity, wet*

The refractivity fields can then be converted, if needed, to water vapor levels according to Eq. (6) with suitable temperature profiles over the troposphere and/or feed high resolution NWM taking natively into account turbulent/convective processes [94]. Xia et al. [95] tried to derive the refractivity field from slant delays by substituting Eq. (6) into Eq. (9), but the underlying hypothesis is an atmosphere at rest, in a similar fashion of the neutral delay model of Saastamoinen [41, 96]. Is the approach developed in this article directly implementable in GNSS software, as a replacement of the usual approach of Eq. (1)? The response is a careful yes [69]. Strictly speaking, a mapping function defines, from the point of view of differential geometry, a time-evolving coordinate chart that is a non-orthogonal system of coordinates made of the refracted elevation at ground level, the length along the bended ray, and the azimuth. We think that such an implementation in GNSS software implies at least the use of a constant (i.e., not evolving with time) system of coordinates (i.e., a constant mapping function), that therefore must be computed with respect to some standard model of the atmosphere, carefully designed and normalized [97]. For this purpose, it should be noted that the variation of the propagation delay caused by the bending is of second order with respect

Finally, the modeling of the wet refractivity field through an expansion series in

We discussed in this brief paper the pros and cons of the standard approach mapping functions + gradients to model the neutral delays of the atmosphere, and more specifically the wet delays caused by the presence of water vapor in the troposphere. If this standard approach is almost perfect for people doing positioning, deformation and VLBI studies, as they see the neutral delays as "noise", it is not so well adapted to people looking at these delays as signals to study atmospheric processes. In particular, the standard definition of gradients is too crude, and does not permit to have access to the horizontal turbulence/convection scales, that are key parameters to model these processes in high resolution NWM models. We therefore propose an alternative way to model the wet tropospheric delays, through a representation of the wet refractivity field as a perturbation over an exponential decay with altitude with a locally adjusted scale height and a time/space series expansion over a suitable basis of orthogonal functions. Our approach is computa-

tionally expensive, and maybe not suited for real-time applications, but its

time and space (Eq. (19)) can be also used to model tropospheric delays, in a correlated way, between uplink and downlink signals to planetary space crafts, where the uplink and downlink separation in time can reach tens of minutes or even

to the integration of the refractivity along the path [98].

inversion episodes, where atmospheric temperature increases when altitude

*Geodetic Sciences - Theory, Applications and Recent Developments*

decay.

*refractivity.*

hours [99].

**134**

**5. Conclusion**

This research was funded by a DAR grant from the French Space Agency (CNES) to the Geodesy Observatory of Tahiti. Feng Peng did this research in the framework of his Ph.D., with a 14-months stay at the Observatory of Tahiti funded through the Cai Yuanpei program.
