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Next: V. GPS observations and Up: IV. Physical fundamentals Previous: 4.2 Time

4.3 Electromagnetic signal propagation

4.3.1 Solar-Terrestrial framework

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



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:

Ion [Hz] [Hz] [m] [m s ]

0.02
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
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:

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).


next up previous
Next: V. GPS observations and Up: IV. Physical fundamentals Previous: 4.2 Time

Manuel Hernandez Pajares
Thu Jun 4 14:25:37 GMT 1998