RF signals are delayed by a small amount as they travel through the Ionosphere, a thin layer of charged particles high up in the atmosphere. The delay is proportional to the path length of the signal through the ionosphere and also changes with frequency in a know manner. If a signal comes straight through the ionosphere the path length will be shorter than one that comes through at an acute angle. Because the effect of the ionosphere is well know for different frequencies of RF signals it is possible to directly calculate the ionospheric delay if measurements can be made at two well seperated frequencies, again this is not usually the case for commercial GPS receivers which utilise only a single frequency. The single frequency delay calculation was originaly based on a model by Klobuchar, but since the early days of WAAS it has been replaced sometimes by an interpolation method based on the delay at the 4 corners of a square containing the ionospheric point where the RF signal passes through.
$$T_{iono} = F[5\times 10^{-9} + AMP (1 - {X^2 \over {2}} + {X^4 \over {24}})]$$ |X| < 1.57 $$T_{iono} = F ( 5 \times 10^{-9} )$$ |X| >= 1.57 Where \(T_{iono}\) is referenced to the L1 frequency.
The Obliquity Factor (F) in the model is given by:
$$F = 1 + 16(0.53 - E)^3$$
Where E = Elevation angle between the receiver and satellite positions in semicircles.
The vertical delay amplitude (AMP) in the model is given by:
$$AMP = \alpha_n\phi^n_m, AMP >= 0$$
$$AMP = 0, AMP < 0$$
where \(\alpha_n\) are the satellite transmitted coefficients of a cubic equation representing the amplitude of the vertical delay with n = 0-3.
\(\phi_m\) is the Geomagnetic latitude of the earth projection of the ionospheric intersection point. The mean ionospheric height is assumed to be 350 km.
\(= \phi_i + 0.064 cos (\lambda_i - 1.617)\) semicircles
\(\phi_i\) Geodetic latitude of the earth projection of the ionospheric intersection point.
\( = \phi_u + \psi cos A\) semicircles
for \(|\phi_i| <= 0.416\) semicircles
if \(\phi_i > 0.416\), then \(\phi_i = +0.416 \)
if \(\phi_i < 0.416\), then \(\phi_i = -0.416 \)
\(\phi_u\) Receiver WGS84 geodetic latitude in semicircles.
\(\psi\) Earth's central angle between receiver position and Earth projection of the ionospheric intersection point.
\({0.0137 \over{E + 0.11 }} - 0.022\) semicircles.
A Azimuth angle between the rceiver and the satellite, measured clockwise positive from True North in semicircles.
\(\lambda_i\) Geodetic longitude of the earth projection of the ionospheric intersection point.
\(= \lambda_u + {\phi sin A\over{cos \psi_i}}\) semicircles.
\(\lambda_u\) Receiver WGS84 geodetic longitude in semicircles.
The Phase (X) in the model is given by:
$$ X = {2\pi(t - 50400)\over{PER}} $$
where t = Local time \(= (4.32\times 10^4)\lambda_i + GPS time\) seconds
for \(0 \le t \le 86400 \) seconds
if \( t \ge 86400 \) seconds, subract 86400 seconds.
if \( t \lt 0 \) seconds, add 86400 seconds.
GPS time = Receiver- computed system time
PER = Period of the model = \(\{ \sum_{n=0}^3 \beta_n \phi_m^n, PER \ge 72000 \}\) seconds
\(\beta \) = The satellite-transmitted coefficients of a cubic equation representing the period of the model with n = 0, 1, 2 and 3.
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