4.3 Electromagnetic signal propagation

4.3.1 Solar-Terrestrial framework

• The Sun: sources of energy, photosphere, cromosphere and corona.
• The interplanetary magnetic field
• The Earth magnetic field and its coupling with the interplanetary one: solar events and geomagnetic storms.
• Differents parts of the atmosphere: ionized (plasmasphere and ionosphere) and neutral (troposphere).

4.3.2 Group and Phase velocities

Let's consider two different electromagnetic plane waves in the propagation direction:

Then:

In the resulting superposition, we have the phase velocity for the carrier phase and the group velocity , the velocity of the transmitted information:

And from this expression we can relate the corresponding refraction indices and in terms of the frequency dependences:

 

4.3.2 Electromagnetic waves

The electromagnetic (EM) waves are well described by the Maxwell equations, with the relationships between electric and magnetic field, jointly with the "constitutive relationships" (12) expressing the medium response to the EM waves.

 

The Maxwell equations in integral form are:

 

And in differential form:

 

And from 14 in the vacuum with , , the equation of the EM wave with propagation velocity can be obtained:

 

4.3.3 Some plasma features

• The Ionosphere is a weak ionized gas with the Earth magnetic field superimposed (magnetoplasma).
• The more determinant charged particles, regarding to the interaction with the EM waves, are the free electrons, due to its smaller mass.
• We can assume than at a given scale, there are the same number of ions (molecules without some electrons) for instance with mass M and charge -e and free electrons (mass m and charge e).
• There are two kind of typical movements:
  1. The electrons (mainly) oscillate in the neighbourhood of the ions.
  2. The electrons (and ions) move around the magnetic field lines in a spiral path.

Let's quantify the movement 1: let's consider a displacement of all the electrons a distance x, generating an electrostatic field like a plane condensator:

And the movement equation is then harmonic:

with associated frequencies for electrons and ions respectively:

with frequencies respectively related as:

For the second kind of movement, due to the Earth magnetic field, we have a typical spiral path due to the Lorentz force:

where is the Larmor frequency, which are related, between ions and electrons as:

762 1.0

47.7 4.1

25.4 5.7 909

23.8 5.9 879

Finally to say that the answer of the plasma to the simultaneous and is being the velocity of the colisionless guide center:

4.3.4 Plasma colisions

A simplified model for the electrons-(neutral) atom colision is:

1. Mean free length:
2. Equidistribution energy theorem:

From here, the corresponding colision frequency can be approximated as:

whith typical values of s

4.3.5 Debye length

The organized movement scale in the plasma must be smaller than the electric shielding/screening length free electrons-ions: the Debye length (  1cm).

An approximated expression is:

4.3.6 Ionospheric delay of the GPS code and phase

We are going to consider only the effect due to the free electrons, due to its smaller mass. Let's consider the electric field as a plane wave:

and the movement equation for the free electron (we neglect the magnetic field):

One solution for the velocity is:

From these equations, we can get:

The resulting current density is:

 

begin the ionospheric free electron density. Now applying the third Maxwell equation in 14:

and assuming an isotropic medium (the vacuum):

and the equation 16, we get:

This equation can be rewritten as:

being

From here, the phase refraction index can be written as:

 

being in electrons/m and f in Hz.

From equation 11 the group refraction index is

 

And finally the ionospheric delay produced in the code, , will be:

 

being and the position vector of the receiver and satellite j. Substituting equation 18 in 19 and taking into account 17 we reach to the following relationships:

 

where is the Slant Total Electron Content (STEC) along the raypath between the given receiver and the transmitter j.

From here, the ionosphere positive/negative delay for the GPS code/phase can be written for both frequencies and as:

And from here it can be eliminated the ionosphere term in the "ionospheric-free" combinations PC, LC or to isolate the ionospheric dependence for mapping purposes with the "ionospheric" combinations PC, LI:

where the last expressions for and is for the STEC in units of electrons/m and the delay in meters, including also the instrumental biases for the transmitter and receiver.

4.3.7 Tropospheric delay of GPS signals

• The water vapour, one of the components of the Earth atmosphere, has several special features:
  • Its distribution is correlated with clouds and rains.
  • The latent heat of the water associated to the transition from liquid to vapour is large, then it contain an important part of the atmospheric intern energy.
  • It play also an important role in the atmospheric chemistry.
• As a consequence, the knowledge of its spatial distribution is one of the main error sources in the weather forecasting.
• In order to measure the water vapour spatial distribution:
  • Radiosondes, onboard balloons (vertical profiles of temperature ( 0.2 C), humidity ( 4%) and pressure.
  • Microwave radiometers, looking to the bright temperature in certain emision/absortion line ("sky from ground" and "ground from sky" respectively, depending on the bacground temperature).
  • GPS receivers, the tropospheric delays, converted to Integrated Water Vapour (IWV) appears as a geodetic product in the high precision positioning.
• The neutral atmosphere (troposphere) produces a non-dispersive delay to the code and carrier phase, that has two physical origins:
  1. The EM interaction with the dipolar magnetic momentum: this is the wet delay (typical vertical values are  35cm, 10% of the total troposperic delay).
  2. The EM interaction with the remaining air molecules, mainly the , with produces the hydrostatic (or dry) delay (  2.3 m, 90% of the total).
• The hydrostatic delay is the largest one but can be very well predicted with models ( is homogenously mixed at global scale). The reciprocal is valid for the wet delay.

Let's develop an easy approach (the Hopfield model) in order to quantify the zenit tropospheric delay :

where n is the refraction index and N is the refractivity (the same for both code and phase in an non-dispersive medium).

To get a model for the dependence of N with the height h, we are going to assume that:

• The temperature gradient is constant
• Hydrostatic equilibrium
• The refractivity N is proportional to the mass density .

where M is the molar mass, P is the pressure, T is the temperature, R is the Rydberg constant and h is the height.

Applying the hydrostatic law for the pressure

and the hypothesis of temperature gradient constant,

then,

.

From these last relationships, we can integrate:

And:

 

being the root height, and the refractivity in h=0, that for the hidrostatic (dry) and wet components are respectively , :

where e is the water vapour pressure.

The Hopfield model is an empirical realization of our model 21 ( 6.71 C/km):

To relax a bit the hypothesis, we will allow a certain dependence of with the temperature:

Then, we can get the tropospheric vertical delays:

with:

And finally, in order to relate the vertical delay with the slant delay, we have to apply an oblicuity factor usually named mapping function M, that depends on the elevation angle E:

It can be approximated for high elevation with the cosecant funcion, corresponding to the plane-parallel approximation of the atmosphere:

Some mapping functions used, for the hydrostatic and wet delay, are:

If we want to avoid the tropospheric modelling with GPS, we need to have a close reference GPS receiver and to perform differential GPS (DGPS).