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

ð Þ¼ *n* � 1 *K*<sup>1</sup>

*Geodetic Sciences - Theory, Applications and Recent Developments*

ð Þ¼ *n* � 1 *K*<sup>0</sup>

1 *P <sup>T</sup>* <sup>þ</sup> *<sup>K</sup>*<sup>0</sup> 2 *e <sup>T</sup>* <sup>þ</sup> *<sup>K</sup>*<sup>3</sup>

Where *P* ¼ *Pd* þ *e*. This rewriting, was the first term is denominated as the hydrostatic component of the refractivity, was proposed by Davis et al. [7] and then has been widely accepted, but lead to a track of confusion in the literature between the meaning of "hydrostatic" and "dry". The word "hydrostatic" has specifically no meaning in Eq. (6), other than indicating that the total pressure is used instead of the partial pressure of the non-wet (dry) air, as in Eq. (5). The word "hydrostatic" has a precise meaning in numerical weather models [38], where it indicates that the equilibrium of an air column is a balance between the vertical pressure gradient and the buoyancy forces, neglecting convective processes [39] as a simplification of the Navier–Stokes primitive Equations [40]. This is also the assumption made in the Saastamoinen model of the atmosphere propagation delays [41], with the total pressure *P* at ground level taken as a parameter (and with also the assumption of an

To a good degree of approximation, the refractivity of air obeys a twofold

The terms *Nh*, *Hh* and *Nw*, *Hw* have, respectively, a value of 250 � <sup>10</sup>�<sup>6</sup>

<sup>128</sup> � <sup>10</sup>�<sup>6</sup> and 2.7 km for the location of our geodesy observatory in Tahiti (from the fit of radiosounding data over a typical year). The scale height *Hw* varies from 1.5 km to up to 8 km from place to place and according to a seasonal cycle [43]. For all practical GNSS purposes, one can consider that the water vapor is concentrated in the troposphere (from 8 km over the poles to 18 km at the Equator [44, 45], and that the atmosphere extends up to 100 km [46, 47]. The International Union of Telecommunications [48] recommends the use, for radio-link purposes, on a worldwide basis and for altitudes taken from sea level, of the formula (7), with

The prime integral (4) allows two things: 1/the computation of the path, 2/the

*L* ¼ ð *path*

The extra delay (in equivalent length) caused by the atmosphere is

*δL* ¼ ð *path*

*r* � *r*<sup>0</sup> *Hh* � �

, *Hh* = 7.35 km, the wet part being omitted (it is in fact included as a

þ *Nw* exp

*r* � *r*<sup>0</sup> *Hw* � �

*n ds* (8)

ð Þ *n* � 1 *ds* (9)

(7)

, 8.7 km,

and *K*<sup>3</sup> year after year [15, 36, 37].

atmosphere "at rest").

*Nh* = 315 � <sup>10</sup>�<sup>6</sup>

**128**

worldwide average in *Nh* and *Hh*).

computation of the time delay along the path as

exponential formula [42].

*n r*ð Þ¼ 1*:* þ *δnh* þ *δnw* ¼ 1*:* þ *Nh* exp

This formula can be easily rewritten as

*Pd <sup>T</sup>* <sup>þ</sup> *<sup>K</sup>*<sup>2</sup>

where *Pd* is the partial pressure of dry air in millibars,*T* is the temperature in Kelvin, *e* is the partial pressure of water vapor. *K*1, *K*<sup>2</sup> and *K*<sup>3</sup> are constants. The *Pd* term corresponds to the "dry" part of the refractivity, the *e* terms correspond to the "wet" part of the refractivity. Many authors have improved the coefficients *K*1, *K*<sup>2</sup>

*e <sup>T</sup>* <sup>þ</sup> *<sup>K</sup>*<sup>3</sup> *e*

*e*

*<sup>T</sup>*<sup>2</sup> (5)

*<sup>T</sup>*<sup>2</sup> (6)

By inserting Eq. (6) into Eq. (9) we get the separation of *δL* into additive "hydrostatic" *δLh* and "wet" *δLw* delays. The ratios of *δLh* and *δLw* with respect to the corresponding values taken along a vertical path are by definition (as in Eq. (1)) the hydrostatic (*mh*) and wet (*mw*) mapping functions that only depend on the elevation angle *e*<sup>0</sup> of the tangent of the bended ray at the receiver location.

Davis et al. [10] pushed the physical analysis of Eq. (9) a little bit further by introducing the notion of gradients. This notion is also based on the basic assumption of a main dependence of the refractivity with respect to height, with the refractivity in the neighborhood of the receiver written as

$$n = n\_V(r) + small \text{ lateral terms} \tag{10}$$

where *r* is taken along the local vertical of the receiver, and *nV* is the variation of *n* along the vertical of the observation site (the value of *n* at the receiver station is *n r*ð Þ¼ <sup>0</sup> *nV*ð Þ *r*<sup>0</sup> ). One can note that this writing violates, on a pure mathematical ground the dependence of *n* on only the geocentric radius, that was assumed for the computation of the path in Eq. (4) (i.e. no small lateral terms should be present). If we define a local frame with units vector ð Þ *x*^, ^*y* in the tangent plane perpendicular to the vertical direction of the station (usually defined by the North and East directions as in Eq. (1), we get, with also the assumption of a "flat Earth", the approximation

$$n\_{\boldsymbol{\alpha}}(\boldsymbol{r};\boldsymbol{\omega},\boldsymbol{\chi}) \simeq n\_{\boldsymbol{V}}(\boldsymbol{r}) + \left(\frac{\partial n}{\partial \boldsymbol{\alpha}}\right)\_{(\boldsymbol{r})} \boldsymbol{\varkappa} + \left(\frac{\partial n}{\partial \boldsymbol{\chi}}\right)\_{(\boldsymbol{r})} \boldsymbol{\jmath} \tag{11}$$

This is nothing else than a Taylor series, meaning that *x* and *y* are assumed to be small, and the subscript ð Þ*r* emphasizes that the partial derivatives of *n* are varying with the height *r* (i.e. they are not taken at *r* ¼ *r*0). For low elevation angles of the path, *x* and *y* are by no means "small", and can reach up to several hundreds of kilometers. We can define Eq. (11) as a "cylindrical" expansion of the refractivity.

If we insert this in Eq. (9), we get

$$\delta L(\varepsilon\_0) = \int\_{path} (n\_V - 1) \, d\mathfrak{s} + \int\_{path} \left(\frac{\partial n}{\partial \mathfrak{x}}\right)\_{(r)} \ge d\mathfrak{s} + \int\_{path} \left(\frac{\partial n}{\partial \mathfrak{y}}\right)\_{(r)} \mathcal{y} \, d\mathfrak{s} \tag{12}$$

If we now divide the first right term of Eq. (12) by

$$\delta L(e\_0) = \int\_{\text{vertical}} (n\_V - \mathbf{1}) \, ds \tag{13}$$

We get

$$\delta L(\mathfrak{e}\_0) = m(\mathfrak{e}\_0) \int\_{\text{vertical}} (n\_V - 1) \, d\mathfrak{s} + \int\_{\text{path}} \left( \frac{\partial n}{\partial \mathfrak{x}} \right)\_{(r)} \ge d\mathfrak{s} + \int\_{\text{path}} \left( \frac{\partial n}{\partial \mathfrak{y}} \right)\_{(r)} \mathfrak{y} \, d\mathfrak{s} \tag{14}$$

where *m e*ð Þ<sup>0</sup> <sup>≈</sup> <sup>1</sup> sin *<sup>e</sup>*<sup>0</sup> is by definition the mapping function. The value <sup>1</sup> sin *<sup>e</sup>*<sup>0</sup> is obtained by setting all the coefficients *a*, *b*,*c* … to 0 in Eq. (2).

By writing *<sup>R</sup>*<sup>2</sup> <sup>¼</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>y</sup>*2, *<sup>x</sup>* <sup>¼</sup> *<sup>R</sup>* cos *<sup>ϕ</sup>*, *<sup>y</sup>* <sup>¼</sup> *<sup>R</sup>* sin *<sup>ϕ</sup>*, and taking advantage of the fact that the path is nearly a straight line, as *n* is close to 1 at a 10�<sup>3</sup> level, we can write, for the two integrals involving the derivatives of *n*, *R* ¼ *rcotg e*ð Þ<sup>0</sup> and *ds* ¼ *dr* sin ð Þ *<sup>e</sup>*<sup>0</sup> . This is permissible, because physically these derivatives, as well as *<sup>x</sup>* and *<sup>y</sup>* are assumed to be small quantities. We obtain for the integral relative to the partial derivative *<sup>∂</sup><sup>n</sup> ∂x* � �

$$\int\_{path} \left(\frac{\partial n}{\partial \mathbf{x}}\right)\_{(r)} \mathbf{x} \, d\mathbf{s} = m(e\_0) \cot e\_0 \cos \phi \Big|\_{r=r\_0}^{r\_{\text{top}}} \left(\frac{\partial n}{\partial \mathbf{x}}\right)\_{(r)} r \, dr \tag{15}$$

All the tomography software treat, to obtain a tractable problem, the rays as straight

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

*<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

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 posi-

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

lines. This means that low-elevation slant delays cannot be considered.

sum *Ld*

*<sup>z</sup>* <sup>þ</sup> *Lw*

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

crude from a "meteorological" point-of-view.

**4. Beyond zenithal delays and gradients**

tioning and VLBI applications.

**131**

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> ∂y* � �.

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

$$\mathbf{G}\_{N}^{h} = \int\_{r=r\_{0}}^{r\_{\rm top}} \left(\frac{\partial n\_{h}}{\partial \mathbf{x}}\right)\_{(r)} r \, \mathbf{d}r,\\ \mathbf{G}\_{E}^{h} = \int\_{r=r\_{0}}^{r\_{\rm top}} \left(\frac{\partial n\_{h}}{\partial \mathbf{y}}\right)\_{(r)} r \, \mathbf{d}r \tag{16}$$

$$G\_N^w = \int\_{r=r\_0}^{r\_{np}} \left(\frac{\partial n\_w}{\partial \mathbf{x}}\right)\_{(r)} r \, \mathbf{d}\mathbf{r},\\G\_E^w = \int\_{r=r\_0}^{r\_{np}} \left(\frac{\partial n\_w}{\partial \mathbf{y}}\right)\_{(r)} r \, \mathbf{d}\mathbf{r} \tag{17}$$

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.
