**2. Basic assumptions at the core of the definition of mapping functions and gradients**

Mapping functions, as they were introduced by Marini [6, 32] are based on the assumption of a totally layered atmosphere. This means that the refractivity *n* is only a function of height (the exact meaning of the word height is related to the definition of geoid). The ray equation of radio waves (including light) obeys, in the spherical approximation and again for a totally layered atmosphere (dependence on geocentric radius *r* of the refractivity *n*), the prime integral relation

$$n(r)\,r\cos\left(\mathbf{e}\right) = n(r\_0)r\_0\cos\left(e\_0\right)\tag{4}$$

where *r* is the geocentric radius, *r*<sup>0</sup> is the geocentric radius at the receiver location, *n r*ð Þ is the refractivity at geocentric radius *r*, *e* is the angle between the tangent to the bended ray and the local horizon (the plane perpendicular to the direction of *r* at height *r*). *e*<sup>0</sup> is the elevation angle of the tangent of the bended ray at the receiver location.

The details of the computation of the ray path can be found in [6, 33, 34]. The refractivity of the atmosphere is a function of pressure, temperature and water vapor contents. A formula widely used is the Smith and Weintraub formula [35], derived for laboratory conditions (air perfectly mixed), as

$$\mathbf{u}(n-1) = K\_1 \frac{P\_d}{T} + K\_2 \frac{e}{T} + K\_3 \frac{e}{T^2} \tag{5}$$

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

where *r* is taken along the local vertical of the receiver, and *nV* is the variation of *n*

*∂n ∂x* � �

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.

> *∂n ∂x* � �

ð Þ*r*

*∂n ∂x* � �

sin *<sup>e</sup>*<sup>0</sup> is by definition the mapping function. The value <sup>1</sup>

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

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

ð Þ*r*

*x ds* þ

ð *path*

*x ds* þ

ð *path*

ð Þ*r x* þ

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* ¼ *nV*ð Þþ *r small lateral terms* (10)

*∂n ∂y* � �

ð Þ*r*

*∂n ∂y* � �

ð Þ*r*

ð Þ *nV* � 1 *ds* (13)

*∂n ∂y* � �

ð Þ*r*

*y* (11)

*y ds* (12)

*y ds* (14)

sin *<sup>e</sup>*<sup>0</sup> is

refractivity in the neighborhood of the receiver written as

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

*n r*ð Þ ; *x*, *y* ≃ *nV*ð Þþ *r*

ð Þ *nV* � 1 *ds* þ

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

*δL e*ð Þ¼ <sup>0</sup>

ð Þ *nV* � 1 *ds* þ

obtained by setting all the coefficients *a*, *b*,*c* … to 0 in Eq. (2).

ð *path*

ð

ð *path*

*vertical*

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

ð *path*

ð

*vertical*

*δL e*ð Þ¼ <sup>0</sup>

We get

*dr*

**129**

derivative *<sup>∂</sup><sup>n</sup>*

*δL e*ð Þ¼ <sup>0</sup> *m e*ð Þ<sup>0</sup>

where *m e*ð Þ<sup>0</sup> <sup>≈</sup> <sup>1</sup>

*∂x* � �

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

This formula can be easily rewritten as

$$\mathbf{r}(n-1) = K\_1' \frac{P}{T} + K\_2' \frac{e}{T} + K\_3 \frac{e}{T^2} \tag{6}$$

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 atmosphere "at rest").

To a good degree of approximation, the refractivity of air obeys a twofold exponential formula [42].

$$n(r) = 1. + \delta n\_h + \delta n\_w = 1. + N\_h \exp\left(\frac{r - r\_0}{H\_h}\right) + N\_w \exp\left(\frac{r - r\_0}{H\_w}\right) \tag{7}$$

The terms *Nh*, *Hh* and *Nw*, *Hw* have, respectively, a value of 250 � <sup>10</sup>�<sup>6</sup> , 8.7 km, <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 *Nh* = 315 � <sup>10</sup>�<sup>6</sup> , *Hh* = 7.35 km, the wet part being omitted (it is in fact included as a worldwide average in *Nh* and *Hh*).

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

$$L = \int\_{path} n \, ds \tag{8}$$

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

$$
\delta L = \int\_{path} (n - 1) \, ds \tag{9}
$$
