**4. Beyond zenithal delays and gradients**

Therefore, what can be the future of the modeling of neutral delays in GNSS meteorology? Applications in GNSS positioning and VLBI clearly show that Eq. (1) is sufficient for these applications, because what is of interest to these users are the integrated delays, not directly the variations of refractivity in the atmosphere. Eq. (1) is sufficient by itself to model these slant (extra) delays, as evidenced by tomography applications and the statistical analysis of these delays [69]. The zenithal total delays have proven to have a physical meaning, as they are related to the modeling of PW through an a priori model of the "dry" atmosphere and a proportional correspondence to zenithal wet delays. They are also feeding the current medium resolution NWM models. The gradients themselves are more questionable. They are merely *ad'hoc*, empirical corrections introduced for positioning and VLBI applications.

Can the definition of gradients be improved? From a physical point-of-view, we do not think so. The main assumption to derive the delay gradients in Davis et al. formula (Eqs. (16) and (17)) is an integration, along the line-of-sight receiversatellite, of the gradients of the refractivity. Even with a better "geometrical definition" of the gradients, taking into account the curvature of Earth, the bending of the rays, etc. … , the main problem is that a line-of-sight station-satellite usually cross – and average– many eddies. According to [70], the shape and size of the eddies

depend on the altitude. Close to the ground (0–2 km of altitude), the eddies are assumed to be small and not far from isotropic, while the irregularities at higher altitudes are highly anisotropic, i.e., the eddies become more flattened laterally. Along the vertical, the refractivity variation is mainly dominated by an exponential decay [71], but this is not the case along the horizontal direction. Besides, the repartition of the lines-of-sight in the sky can be scarce or uneven. For example, the GPS constellation, the most used one because of the high quality of its orbit modeling, offer quasi-repeating repeating tracks where only a few satellites (4 to 12) are visible from a particular location (**Figure 2**). This means that only a few lines-ofsight can be used at any time, and that there is, from a practitioner point of view, not enough data to constraint a better representation of the slant delays than the six-parameters Eq. (1).

this scale height is related to the rate at which the PW decorrelates with horizontal

*<sup>ε</sup>n*ð Þ¼ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>*, *<sup>t</sup>* <sup>X</sup>

renormalized according to the Gram-Schmidt scheme [80].

*geometrical path*

low-elevation rays cannot be taken into account.

ð

framework of radar tomography [87, 91, 92].

*with altitude and pushed by the wind [62, 63].*

*δLw*ð Þ¼ *e*<sup>0</sup> *Nw*

**Figure 3.**

**133**

3D (or 4D if the time is present) series expansion

*Beyond Mapping Functions and Gradients DOI: http://dx.doi.org/10.5772/intechopen.96982*

On the contrary of Davis et al. [10], we fully represent the term *εw*ð Þ *x*, *y*, *z*, *t* as a

*λn*Φ*n*ð Þ *x*, *y*, *z*, *t* (19)

ð Þ <sup>1</sup>*:* <sup>þ</sup> *<sup>ε</sup>w*ð Þ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>*, *<sup>t</sup> ds* (20)

*n*

where the Φ*n*ð Þ *x*, *y*, *z*, *t* are a set of suitably chosen orthogonal functions in the atmospheric lens comprised between the local horizon of the station and the local tropopause. The *λ<sup>n</sup>* are the coefficients of the expansion. If the shape of the tropopause boundary is known [78], the Φ*<sup>n</sup>* functions can be defined as empirical orthogonal functions (EOF) [79] or as a pre-defined set of orthogonal functions

A preliminary attempt with a small data set was made by [81] with the assump-

*r* � *r*<sup>0</sup> *Hw* � �

This integral relationship is averaging the wet refractivity field along the linesof-sight (fan-beam tomography [87, 88]), and the inversion in terms of *λ<sup>n</sup>* coefficients must be regularized. By construction, the *ε<sup>w</sup>* correction must be small, so we

tion of a constant altitude tropopause (see **Figure 3**), where the Φ*<sup>n</sup>* orthogonal functions are a subset of Zernike functions [82]. The line-of-sight are assumed to be straight-lines to obtain tractable equations, as it is the case for tomography [83, 84]

and the statistical analysis of the slant delays [85, 86]. This implies that

The integral relation to be solved with respect to *ϵ<sup>w</sup>* is therefore

exp

can use a truncated Singular Value decomposition (the EOF approach) or a Tikhonov approach [89] to enforce this smallness with respect to 1. The use of a priori refractivity values along the vertical for sites collocated with radiosoundings can also be envisaged [90] (in preparation). The Tikhonov approach, and its ability to model local variations of the refractivity field has been investigated in the

*The geometry of the inversion of the wet delays, with the representation of eddies in the troposphere, flattened*

separation.

Hopefully, Augmented Constellations and Low-Earth-Orbits constellations (LEO) will become soon a reality [72–74], thanks to the ever-decreasing size and costs of satellites, as well as the availability of miniaturized atomic clocks [75]. LEO constellations are particularly interesting for GNSS meteorology, as their satellites will cross the sky in a few minutes instead of hours, with a boost by one order of magnitude, or even two, of the available line-of-sight geometries. Our proposal to keep the separation of the refractivity into a "hydrostatic" and "wet" part, with the "hydrostatic" slant part evaluated separately from proven models like the Saastamoinen [41] model and subtracted from the total slant delay, then to represent the wet refractivity field based on a mean exponential decay of the wet refractivity as

$$
\delta n\_w(r) = N\_w \exp\left(\frac{r - r\_0}{H\_w}\right) (\mathbf{1}. + e\_w(\mathbf{x}, \mathbf{y}, z, t)) \tag{18}
$$

where the *ϵ<sup>w</sup>* terms represent the departure of the wet refractivity field from the exponential local decay law and *x*, *y*, *z*, *t* are local coordinates with respect to a frame linked with the local GNSS receiver and *t* is time. As the wet scale height can vary by a factor of four, it must be provided from external sources (for example from the ECMWF-ERA series of climate models, see [76]). An estimate of *Hw* can also be determined from the slant wet delays themselves, but only if a reliable estimate of the wet refractivity is available, as the integral over the geometrical path between the GNSS satellite and the receiver is proportional to *NwHw* for a pure exponential decay of the wet refractivity. Empirical relations also exist between the ground value of the refractivity and scale height for example [77], but they are probably out-of-date. *Hw* is by itself a very important parameter, as [71] demonstrated that

#### **Figure 2.**

*The sky-tracks (in elevation and azimuth) of the GPS satellites (one color per satellite) visible from the THTI station (latitude: 17.5769*° *S, longitude: 149.6063*° *W), in the wet season on January 10th, 2018.*

depend on the altitude. Close to the ground (0–2 km of altitude), the eddies are assumed to be small and not far from isotropic, while the irregularities at higher altitudes are highly anisotropic, i.e., the eddies become more flattened laterally. Along the vertical, the refractivity variation is mainly dominated by an exponential decay [71], but this is not the case along the horizontal direction. Besides, the repartition of the lines-of-sight in the sky can be scarce or uneven. For example, the GPS constellation, the most used one because of the high quality of its orbit modeling, offer quasi-repeating repeating tracks where only a few satellites (4 to 12) are visible from a particular location (**Figure 2**). This means that only a few lines-ofsight can be used at any time, and that there is, from a practitioner point of view, not enough data to constraint a better representation of the slant delays than the

*Geodetic Sciences - Theory, Applications and Recent Developments*

Hopefully, Augmented Constellations and Low-Earth-Orbits constellations (LEO) will become soon a reality [72–74], thanks to the ever-decreasing size and costs of satellites, as well as the availability of miniaturized atomic clocks [75]. LEO constellations are particularly interesting for GNSS meteorology, as their satellites will cross the sky in a few minutes instead of hours, with a boost by one order of magnitude, or even two, of the available line-of-sight geometries. Our proposal to keep the separation of the refractivity into a "hydrostatic" and "wet" part, with the

> *r* � *r*<sup>0</sup> *Hw*

*The sky-tracks (in elevation and azimuth) of the GPS satellites (one color per satellite) visible from the THTI*

*station (latitude: 17.5769*° *S, longitude: 149.6063*° *W), in the wet season on January 10th, 2018.*

where the *ϵ<sup>w</sup>* terms represent the departure of the wet refractivity field from the exponential local decay law and *x*, *y*, *z*, *t* are local coordinates with respect to a frame linked with the local GNSS receiver and *t* is time. As the wet scale height can vary by a factor of four, it must be provided from external sources (for example from the ECMWF-ERA series of climate models, see [76]). An estimate of *Hw* can also be determined from the slant wet delays themselves, but only if a reliable estimate of the wet refractivity is available, as the integral over the geometrical path between the GNSS satellite and the receiver is proportional to *NwHw* for a pure exponential decay of the wet refractivity. Empirical relations also exist between the ground value of the refractivity and scale height for example [77], but they are probably out-of-date. *Hw* is by itself a very important parameter, as [71] demonstrated that

ð Þ 1*:* þ *εw*ð Þ *x*, *y*, *z*, *t* (18)

"hydrostatic" slant part evaluated separately from proven models like the Saastamoinen [41] model and subtracted from the total slant delay, then to represent the wet refractivity field based on a mean exponential decay of the wet

*δnw*ð Þ¼ *r Nw* exp

six-parameters Eq. (1).

refractivity as

**Figure 2.**

**132**

this scale height is related to the rate at which the PW decorrelates with horizontal separation.

On the contrary of Davis et al. [10], we fully represent the term *εw*ð Þ *x*, *y*, *z*, *t* as a 3D (or 4D if the time is present) series expansion

$$\varepsilon\_n(\mathbf{x}, \mathbf{y}, z, t) = \sum\_{n} \lambda\_n \Phi\_n(\mathbf{x}, \mathbf{y}, z, t) \tag{19}$$

where the Φ*n*ð Þ *x*, *y*, *z*, *t* are a set of suitably chosen orthogonal functions in the atmospheric lens comprised between the local horizon of the station and the local tropopause. The *λ<sup>n</sup>* are the coefficients of the expansion. If the shape of the tropopause boundary is known [78], the Φ*<sup>n</sup>* functions can be defined as empirical orthogonal functions (EOF) [79] or as a pre-defined set of orthogonal functions renormalized according to the Gram-Schmidt scheme [80].

A preliminary attempt with a small data set was made by [81] with the assumption of a constant altitude tropopause (see **Figure 3**), where the Φ*<sup>n</sup>* orthogonal functions are a subset of Zernike functions [82]. The line-of-sight are assumed to be straight-lines to obtain tractable equations, as it is the case for tomography [83, 84] and the statistical analysis of the slant delays [85, 86]. This implies that low-elevation rays cannot be taken into account.

The integral relation to be solved with respect to *ϵ<sup>w</sup>* is therefore

$$\delta L\_w(e\_0) = N\_w \begin{cases} \text{[geometric]} & \exp\left(\frac{r - r\_0}{H\_w}\right) (\mathbf{1} + \epsilon\_w(\mathbf{x}, y, z, t)) \,\text{ds} \\ \text{path} & \end{cases} \tag{20}$$

This integral relationship is averaging the wet refractivity field along the linesof-sight (fan-beam tomography [87, 88]), and the inversion in terms of *λ<sup>n</sup>* coefficients must be regularized. By construction, the *ε<sup>w</sup>* correction must be small, so we can use a truncated Singular Value decomposition (the EOF approach) or a Tikhonov approach [89] to enforce this smallness with respect to 1. The use of a priori refractivity values along the vertical for sites collocated with radiosoundings can also be envisaged [90] (in preparation). The Tikhonov approach, and its ability to model local variations of the refractivity field has been investigated in the framework of radar tomography [87, 91, 92].

#### **Figure 3.**

*The geometry of the inversion of the wet delays, with the representation of eddies in the troposphere, flattened with altitude and pushed by the wind [62, 63].*

The only case where the hypothesis of a small *ϵ<sup>w</sup>* can be violated occurs during inversion episodes, where atmospheric temperature increases when altitude 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 decay.

end-product are records of the total and wet refractivity values with high-resolution

needs of future numerical weather models [38], the emerging field of the modeling of atmospheric rivers [100, 101] and besides does not require the additional step

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

in time (minute-scale) and distance (sub km-scale), in accordance with the

of water vapor tomography, with lower cost, better mobility and simpler

operation [102].

**Acknowledgements**

**Conflict of interest**

**Author details**

**135**

Jean-Pierre Barriot<sup>1</sup>

through the Cai Yuanpei program.

*Beyond Mapping Functions and Gradients DOI: http://dx.doi.org/10.5772/intechopen.96982*

The authors declare no conflict of interest.

\* and Peng Feng<sup>2</sup>

2 LIESMARS State Key Laboratory, Wuhan University, China

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: jean-pierre.barriot@upf.pf

1 University of French Polynesia, French Polynesia

provided the original work is properly cited.

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 refractivity.*

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

Finally, the modeling of the wet refractivity field through an expansion series in 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 hours [99].
