**1. Introduction**

The effect of the Earth atmosphere on the propagation of light was noticed just after the invention of the telescope by Galileo Galilei, and tables of atmospheric refraction (bending of ray lights) were already available in the XVII century. After the advent of VLBI observations in the fifties and the launch of the first Earth satellite in the sixties, the modeling of the time delays caused by the neutral atmosphere became a necessity.

The current mathematical structure of the modeling of the propagation time delays, used in almost all GNSS software is given by [1].

$$\begin{split} \delta L(e\_0) &= m\_h(e\_0) \left[ L\_x^h + \text{cotg}(e\_0) (G\_N^h \cos \left( \mathcal{Q} \right) + G\_E^h \sin \left( \mathcal{Q} \right) \right] \\ &+ m\_w(e\_0) \left[ L\_x^w + \text{cotg}(e\_0) \left( G\_N^w \cos \left( \mathcal{Q} \right) + G\_E^w \sin \left( \mathcal{Q} \right) \right) \right] \end{split} \tag{1}$$

where *δL* is the slant (extra) delay with respect to propagation in vacuum along the bended ray, *L<sup>h</sup> <sup>z</sup>* and, *Lw <sup>z</sup>* are the hydrostatic and wet zenith delays, *e*<sup>0</sup> and ∅ are the satellite elevation and azimuth angles as seen from the station, respectively; *G<sup>h</sup> N* and *G<sup>w</sup> <sup>N</sup>*, *G<sup>h</sup> <sup>E</sup>* and *G<sup>w</sup> <sup>E</sup>* are the north and east components of the hydrostatic and wet delays gradients; *mh* and *mw* are the hydrostatic and wet mapping functions. Eq. (1) is used in precise GNSS processing software through the modeling of the phase signal [2–4].

The mapping functions "map" the so-called slant neutral atmosphere (extra) delay *δL* (i.e. the delay along the bended ray from the observer to the emitter) to two "zenithal delays", named hydrostatic delay (very often improperly called "dry" delay) *Lh <sup>z</sup>* and wet delay *Lw <sup>z</sup>* , essentially caused by the water vapor. The mapping functions are usually written [5] in the form of continuous fractions, that were introduced by Marini [6, 7] and normalized by Herring [8] in the form

$$m(e\_0) = \frac{\mathbf{1} + \frac{a}{1 + \frac{b}{1 + \epsilon}}}{\sin\left(e\_0\right) + \frac{a}{\sin\left(e\_0\right) + \frac{b}{\sin\left(e\_0\right) + \epsilon}}}\tag{2}$$

Other (simplified) forms of the mapping functions can be found in the literature [9], but the mainstream form is always Eq. (2). The gradients themselves, noted by the upper case letter *G* in Eq. (1) were introduced to compensate azimuthal anisotropic effects [10–12].

For the last thirty years, the improvements on these formulas mainly focused on better and better determinations of the coefficients *a, b, c*, by comparisons of these formulas with ray tracing. The literature acknowledges as "best" models the VMF1 and VMF3 families, with some seasonally adjusted coefficients constrained from ray-tracing results with respect to Numerical Weather Models (NWM) [13, 14].

The role of the water vapor in the neutral delay is important, as it can be up to 20% (about 45 cm of the zenithal delay *L<sup>w</sup> <sup>z</sup>* ), with respect to the total zenithal delay *Lh <sup>z</sup>* <sup>þ</sup> *Lw <sup>z</sup>* (about 2.3 m). The other gases, including carbon dioxide, have a negligible role in the neutral delay [9, 15], thus cannot be detected through GNSS processing.

Water is present in its three phases on Earth atmosphere, hydrosphere and continents: solid, liquid and water vapor, with important latent heats between phases. Water vapor in the atmosphere has large sources (evaporation, evapotranspiration) and sinks (rain, snow). Water vapor is also the most important greenhouse gas (beyond carbon dioxide) and the driver of cloud coverage. To describe the water cycle [16] is therefore of the uttermost importance, as evidenced by the so-called Energy Balance models [17, 18] that can be written

$$\mathbf{C}\_{1}\mathbf{S}\_{0}(\mathbf{1}-a) = \mathbf{C}\_{2}\frac{dT}{dt} + \mathbf{C}\_{3}T^{4}(\mathbf{1}-\beta) \tag{3}$$

mainly lidars [24], photometers [25], and water vapor radiometers [26]. The only source providing *in situ* meteorological data are the radiosondes [27], launched twice per day in a limited amount of worldwide sites. Many studies have been devoted to the causal relationship between water vapor and rain [28, 29], including

*The mapping functions mh (blue) and mw (red) plotted against each other for a typical GNSS station in Beijing (latitude: 39.6086*° *N, longitude: 115.8922*° *E), winter time on January 16th, 2012, with the VMF1 model [13], parameterized by inputting data from the ECMWF numerical weather model EAR-40 [31].*

It is therefore important to separate the water vapor modeling coefficients *Lw*

easier said than done, as the functions *mh*ð Þ *e*<sup>0</sup> and *mw*ð Þ *e*<sup>0</sup> in Eq. (1) have almost the

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

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

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

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],

*<sup>z</sup>* , *G<sup>h</sup>*

*<sup>N</sup>* and *G<sup>h</sup>*

*n r*ð Þ *r* cos eð Þ¼ *n r*ð Þ<sup>0</sup> *r*<sup>0</sup> cosð Þ *e*<sup>0</sup> (4)

*<sup>E</sup>* from the hydrostatic coefficients *Lh*

same dependence on the elevation angle *e*<sup>0</sup> (see **Figure 1**).

geocentric radius *r* of the refractivity *n*), the prime integral relation

derived for laboratory conditions (air perfectly mixed), as

*z* ,

*<sup>E</sup>* in Eq. (1). But this is

extreme events [30].

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

**and gradients**

at the receiver location.

**127**

*G<sup>w</sup>*

**Figure 1.**

*<sup>N</sup>* and *G<sup>h</sup>*

Where *S*<sup>0</sup> is the solar constant (1360 W/m2 ),*T* is the mean temperature on the Earth surface in Kelvin, *t* is the time. *C*1, *C*<sup>2</sup> and *C*<sup>3</sup> are constants.

The coefficients *α* and *β* are albedos, respectively in the visible and infrared wavelengths, both mainly driven by the water vapor contents of the Earth atmosphere [19]. The coefficients *α* reflects the cloud coverage, typically today at the 30% level, and the coefficient *β* is an infrared albedo, keeping our planet warm at around 15 °C. Without the greenhouse gases, our planet will be at a freezing mean temperature of �18 °C. They have antagonist effects, an increase of *α* means a cooling of Earth surface, and an increase of *β* means a warming, with a lot of intricacies between the positive and negative feedbacks related to the water vapor cycle of the climate models [20]. The ultimate goal of global long-term climate models [21] is to predict which effect will prevail (this is the *dT/dt* term in the right side of Eq. (3)).

The study [22] highlights the difficulty of measuring atmospheric water vapor with sufficient spatial and temporal resolution, and with sufficient accuracy, to provide observational constraints. GNSS processing is not the only source of water vapor data in the atmosphere. Remote sensing by satellites is the main provider [23], but the resolution of their data sets is limited by the distance between the satellites and the Earth and their orbital cycles. Besides, satellites are expensive. GNSS receivers, even precise ones, are a lot cheaper, and can provide long-term time series with high temporal resolution. Other ground-based instruments are

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

#### **Figure 1.**

delay) *Lh*

*Lh <sup>z</sup>* <sup>þ</sup> *Lw*

side of Eq. (3)).

**126**

tropic effects [10–12].

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

20% (about 45 cm of the zenithal delay *L<sup>w</sup>*

*<sup>z</sup>* , essentially caused by the water vapor. The mapping

(2)

functions are usually written [5] in the form of continuous fractions, that were

<sup>1</sup> <sup>þ</sup> *<sup>a</sup>* <sup>1</sup><sup>þ</sup> *<sup>b</sup>* 1þ*c*

> sin ð Þþ *<sup>e</sup>*<sup>0</sup> *<sup>b</sup>* sin *<sup>e</sup>*ð Þ <sup>0</sup> <sup>þ</sup>*<sup>c</sup>*

*<sup>z</sup>* ), with respect to the total zenithal delay

*dt* <sup>þ</sup> *<sup>C</sup>*3*T*<sup>4</sup>ð Þ <sup>1</sup> � *<sup>β</sup>* (3)

),*T* is the mean temperature on the

sin ð Þþ *<sup>e</sup>*<sup>0</sup> *<sup>a</sup>*

Other (simplified) forms of the mapping functions can be found in the literature [9], but the mainstream form is always Eq. (2). The gradients themselves, noted by the upper case letter *G* in Eq. (1) were introduced to compensate azimuthal aniso-

For the last thirty years, the improvements on these formulas mainly focused on better and better determinations of the coefficients *a, b, c*, by comparisons of these formulas with ray tracing. The literature acknowledges as "best" models the VMF1 and VMF3 families, with some seasonally adjusted coefficients constrained from ray-tracing results with respect to Numerical Weather Models (NWM) [13, 14]. The role of the water vapor in the neutral delay is important, as it can be up to

*<sup>z</sup>* (about 2.3 m). The other gases, including carbon dioxide, have a negligible role in the neutral delay [9, 15], thus cannot be detected through GNSS processing. Water is present in its three phases on Earth atmosphere, hydrosphere and continents: solid, liquid and water vapor, with important latent heats between phases. Water vapor in the atmosphere has large sources (evaporation, evapotranspiration) and sinks (rain, snow). Water vapor is also the most important greenhouse gas (beyond carbon dioxide) and the driver of cloud coverage. To describe the water cycle [16] is therefore of the uttermost importance, as evidenced by the

*dT*

The coefficients *α* and *β* are albedos, respectively in the visible and infrared wavelengths, both mainly driven by the water vapor contents of the Earth atmosphere [19]. The coefficients *α* reflects the cloud coverage, typically today at the 30% level, and the coefficient *β* is an infrared albedo, keeping our planet warm at around 15 °C. Without the greenhouse gases, our planet will be at a freezing mean temperature of �18 °C. They have antagonist effects, an increase of *α* means a cooling of Earth surface, and an increase of *β* means a warming, with a lot of intricacies between the positive and negative feedbacks related to the water vapor cycle of the climate models [20]. The ultimate goal of global long-term climate models [21] is to predict which effect will prevail (this is the *dT/dt* term in the right

The study [22] highlights the difficulty of measuring atmospheric water vapor with sufficient spatial and temporal resolution, and with sufficient accuracy, to provide observational constraints. GNSS processing is not the only source of water vapor data in the atmosphere. Remote sensing by satellites is the main provider [23], but the resolution of their data sets is limited by the distance between the satellites and the Earth and their orbital cycles. Besides, satellites are expensive. GNSS receivers, even precise ones, are a lot cheaper, and can provide long-term time series with high temporal resolution. Other ground-based instruments are

introduced by Marini [6, 7] and normalized by Herring [8] in the form

*m e*ð Þ¼ <sup>0</sup>

*Geodetic Sciences - Theory, Applications and Recent Developments*

so-called Energy Balance models [17, 18] that can be written

Where *S*<sup>0</sup> is the solar constant (1360 W/m2

*C*1*S*0ð Þ¼ 1 � *α C*<sup>2</sup>

Earth surface in Kelvin, *t* is the time. *C*1, *C*<sup>2</sup> and *C*<sup>3</sup> are constants.

*The mapping functions mh (blue) and mw (red) plotted against each other for a typical GNSS station in Beijing (latitude: 39.6086*° *N, longitude: 115.8922*° *E), winter time on January 16th, 2012, with the VMF1 model [13], parameterized by inputting data from the ECMWF numerical weather model EAR-40 [31].*

mainly lidars [24], photometers [25], and water vapor radiometers [26]. The only source providing *in situ* meteorological data are the radiosondes [27], launched twice per day in a limited amount of worldwide sites. Many studies have been devoted to the causal relationship between water vapor and rain [28, 29], including extreme events [30].

It is therefore important to separate the water vapor modeling coefficients *Lw z* , *G<sup>w</sup> <sup>N</sup>* and *G<sup>h</sup> <sup>E</sup>* from the hydrostatic coefficients *Lh <sup>z</sup>* , *G<sup>h</sup> <sup>N</sup>* and *G<sup>h</sup> <sup>E</sup>* in Eq. (1). But this is easier said than done, as the functions *mh*ð Þ *e*<sup>0</sup> and *mw*ð Þ *e*<sup>0</sup> in Eq. (1) have almost the same dependence on the elevation angle *e*<sup>0</sup> (see **Figure 1**).
