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Study on accuracy of GPS for its application in SAR interferometry
Anti-spoofing
AS further alters the GPS signal by changing the characteristics of the P code by mixing it with a so-called W code resulting in the Y code. It is the latter that is modulated onto the carriers and is thus designed to prevent the ability of the receiver to make P code measurements. Many receiver manufacturers have already developed techniques to still make P code measurements with only a small addition in added noise (cross correlation techniques), Talbot (1992) or Ashjaee and Lorenz (1992).
Satellite Cloak Error
One billionth of a second (one nanosecond) of inaccuracy in a satellite clock results in about 30 centimeters (one foot) of error in measuring the distance to that satellite. For this reason, the satellites are equipped with very accurate (Cesium) atomic clocks. Even these very accurate clocks accumulate an error of 1 billionth of a second every three hours. We are trying to liniarize the pesudorange equations for the four satellites given as
with i = 4
Where Dti term represent the satellite clock corrections. GPS works on the principal that all of the satellites are synchronized with GPS master time. The pseudorange measurements must share a common time basis otherwise the position solution will be highly inaccurate. The satellite clock, however, are not in synchronization with one another. The satellite clock correction can be given by the equation
A further correction is made for single frequency users because the af0 clock offset term is based on dual frequency observations. af0, af1, af2, etc. are calculated separately.
which is the quantity measured on the ground
before satellite launch. Thus, V
GD could have a magnitude of 18 ns or larger for a given satellite. We consider here four psuedoranges because that is the minimum required to solve for the four unknowns of the four dimensional position and receiver clock offset. A taylor series is used to expand these equations about an estimated position (X,Y,Z,tB) with the higher order terms ignored. For the pseudorange measurement i
where (PRi - Dti) is the pseudorange based on the estimated
position. Thus, each pseudorange can be approximated as
The coefficients of ¶X, ¶Y, ¶Z are direction cosines. We can now write the linearized equations in matrix form
which can be written in the form
¶Y = H ¶ß
An iterative computation is carried out to
converge on the position solution. In general, ¶
Y is an m x 1 vector where m is the number of satellite in view. Accordingly H is an m x 4 matrix. One method of solving this equation is to take the generalized inverse of H
¶ß =
(HTH)-1HT ¶Y
The result can be used to converge on the solution within four to five iterations.
It is not necessary for the initial estimated position and the receiver clock offset to be accurate. The basic algorithm is as follows:
- Calculate the satellite positions
- Apply satellite clock corrections to the pseudorange measurements
- From initial position estimate Iterate
until convergence.
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