**3. Physical meaning of zenithal delays and gradients**

The modeling of the extra-delays caused by the atmosphere by the combination of mapping functions and gradients of Eq. (1) has proved very effective since Davis introduced his formula 30 years ago [49–51]. But what is the real meaning of effective?

We have to remember that this model was primarily introduced to model atmospheric delays in VLBI, then to improve positioning estimates from GNSS data, and it is now battle-proven for these two applications. But another application, being known today as GNSS meteorology, emerged during the nineties, first with the modeling of the integrated water vapor contents along the vertical of the GNSS receiver (i.e. no gradients), known as "precipitable water" (or PW), that used the *Lw <sup>z</sup>* zenithal delay converted to PW through a multiplicative constant, known as the Π constant introduced by Bevis et al. [52]. Because the wet and dry mapping functions cannot be separated, for any practical purposes, in Eq. (1), the separation between the sum *Ld <sup>z</sup>* <sup>þ</sup> *Lw <sup>z</sup>* and *Lw <sup>z</sup>* must be done by introducing an "external hydrostatic estimate" *L<sup>h</sup> <sup>z</sup>* , the model of choice being the so-called Saastamoinen model [41]. By its own inception, a PW time series is relative to a particular GNSS station, and does not provide any information about the lateral gradients of the water vapor contents of the atmosphere for this site. But a dense network of GNSS receivers do. An even more powerful way to grasp the 3D and even 4D (with the inclusion of time) variations of the water vapor contents of the atmosphere is the tomography, first promoted by [1, 53, 54]. In the approach of tomography, Eq. (1) is just seen as an intermediate tool, the data inputted in the tomography software being the reconstructed *δLw* (the "wet" part of Eq. (1)). The tomography approach needs a dense network of GNSS receivers over a limited area, and take advantage of a multiple crossing paths between the receivers and the satellites of the GNSS constellations to invert the intrinsically ill-posed correspondence between the *δLw* and the 3D atmospheric water vapor refractivity field over the area.

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

ð *path*

*∂n ∂x* � �

"hydrostatic" and "wet" gradients of Eq. (1) as

ð*rtop r*¼*r*<sup>0</sup>

ð*rtop r*¼*r*<sup>0</sup>

*∂nh ∂x* � �

*∂nw ∂x* � �

**3. Physical meaning of zenithal delays and gradients**

ð Þ*r*

ð Þ*r*

*Gh N* ¼

*G<sup>w</sup> <sup>N</sup>* ¼

effective?

*Lw*

**130**

between the sum *Ld*

hydrostatic estimate" *L<sup>h</sup>*

*<sup>z</sup>* <sup>þ</sup> *Lw*

*<sup>z</sup>* and *Lw*

ð Þ*r*

*Geodetic Sciences - Theory, Applications and Recent Developments*

*x ds* ¼ *m e*ð Þ<sup>0</sup> cot*e*<sup>0</sup> cos *ϕ*

where *rtop* is the top of the atmosphere with respect to the geocentric radius (around 100 km), and a similar expression in sin *ϕ* for the partial derivative *<sup>∂</sup><sup>n</sup>*

*r* dr, *G<sup>h</sup>*

*r* dr, *G<sup>w</sup> <sup>E</sup>* ¼

The significations of the gradients are therefore the integration, along the altitude, weighted by the altitude, of the North and East directional derivatives of the "hydrostatic" and "wet" parts of the refractivity, evaluated along the vertical of the receiver location. It is in fact an integration along the geometrical line-of-sight.

The modeling of the extra-delays caused by the atmosphere by the combination of mapping functions and gradients of Eq. (1) has proved very effective since Davis introduced his formula 30 years ago [49–51]. But what is the real meaning of

We have to remember that this model was primarily introduced to model atmospheric delays in VLBI, then to improve positioning estimates from GNSS data, and it is now battle-proven for these two applications. But another application, being known today as GNSS meteorology, emerged during the nineties, first with the modeling of the integrated water vapor contents along the vertical of the GNSS receiver (i.e. no gradients), known as "precipitable water" (or PW), that used the

*<sup>z</sup>* zenithal delay converted to PW through a multiplicative constant, known as the

model [41]. By its own inception, a PW time series is relative to a particular GNSS station, and does not provide any information about the lateral gradients of the water vapor contents of the atmosphere for this site. But a dense network of GNSS receivers do. An even more powerful way to grasp the 3D and even 4D (with the inclusion of time) variations of the water vapor contents of the atmosphere is the tomography, first promoted by [1, 53, 54]. In the approach of tomography, Eq. (1) is just seen as an intermediate tool, the data inputted in the tomography software being the reconstructed *δLw* (the "wet" part of Eq. (1)). The tomography approach needs a dense network of GNSS receivers over a limited area, and take advantage of a multiple crossing paths between the receivers and the satellites of the GNSS constellations to invert the intrinsically ill-posed correspondence between the *δLw* and the 3D atmospheric water vapor refractivity field over the area.

*<sup>z</sup>* must be done by introducing an "external

*<sup>z</sup>* , the model of choice being the so-called Saastamoinen

Π constant introduced by Bevis et al. [52]. Because the wet and dry mapping functions cannot be separated, for any practical purposes, in Eq. (1), the separation

*E* ¼

ð*rtop r*¼*r*<sup>0</sup>

ð*rtop r*¼*r*<sup>0</sup>

*∂nh ∂y* � �

*∂nw ∂y* � �

ð Þ*r*

ð Þ*r*

The precise details of the mathematical machinery linking Eq. (11) to Eq. (1) can be found in Davis et al. [10]. The important fact, from a physical point-of-view is that, if we split the refractivity into a "hydrostatic" and a "wet" part, we get the

ð*rtop r*¼*r*<sup>0</sup>

*∂n ∂x* � �

ð Þ*r*

*r dr* (15)

*r* dr (16)

*r* dr (17)

*∂y* � � . All the tomography software treat, to obtain a tractable problem, the rays as straight lines. This means that low-elevation slant delays cannot be considered.

Some authors [51, 55, 56] tried to assess the physical meaning of tropospheric gradients, but their effort were limited to qualitative assessments and correlations studies. Up to our knowledge [57], nobody is using gradients as data to constraint operational NWMs, albeit efforts having made to extract gradients from NWM numerical simulations [14] or make comparisons with NWMs outputs [58], or even to propose the use of slant delays for such a use [59]. The only GNSS data products that are currently inputted (assimilated) in NWMs are total zenithal delays (i.e. the sum *Ld <sup>z</sup>* <sup>þ</sup> *Lw <sup>z</sup>* ), as for example in the latest Météo-France AROME model [60].

This is clearly sending the message that the meteorology community does not yet consider gradients as a usable data set. We think that the main reason for this is the underlying assumption of the cylindrical Taylor's expansion [Eq. (11)], at the basis of the notion of gradients, where a strict separation between vertical variations and lateral variations is assumed, and supposed valid over all the troposphere (at least as seen from the receiver location). This assumption is closely related to the hydrostatic assumption, itself closely linked to the highly non-linear Navier–Stokes equations, which admit as solutions a combination of laminar and turbulent/convective flows. At scales larger than a few tens of kilometers, the atmospheric flows are mostly horizontal [61]. This corresponds to the highest resolution available for typical MNW models, built around the hydrostatic assumption [62]. The atmospheric turbulence [63] itself is organized as "vortices", or eddies, with scales varying over several orders of magnitude, from a few meters to several hundreds of kilometers [64, 65]. A combination of laminar and turbulence is also possible, and it is known as "frozen flow", where "frozen turbulence" is carried by laminar flow [66]. This is illustrated for the layman by clouds driven by the wind. Atmospheric turbulence/convection is modeled through statistical tools, the structure functions [67], that obeys an exponential decay with altitude (i.e. turbulence is "higher" in the boundary layer) [68]. The definition of gradients by Davis et al. [10] is simply too crude from a "meteorological" point-of-view.
