In this assignment you will work with relations pertaining to ionospheric delay, and perform ionosphere-free position fixes from dual-frequency data.
In all cases where appropriate turn in Matlab (or equivalent) code and supporting functions with your results. Please comment liberally and write your code for clarity.
Problem 1. Single Frequency Estimate of Ionospheric Obliquity Factor.
If the ionosphere is assumed to be an infinitesimally thin shell at altitude h, a simple model for the ionospheric delay is that it is a function of the zenith delay and 1/cos(d'), where d' is the angle that the line of sight vector from the receiver to the satellite makes with the zenith vector at the ionospheric pierce point. d' is called the declination angle which is 0 at zenith and p/2 at the horizon. 1/cos(d') is called the obliquity factor (= OF) of the ionosphere. That is
DI (d') = zenith delay . OF(d')
Starting with OF(d') = 1/cos(d'), derive an equivalent expression for OF in terms of the declination angle, d, between the line of sight vector and the zenith vector at the receiver location based on a spherical Earth with radius R and ionospheric shell height h. That is:
DI (d) = zenith delay . OF(d)
For the model as described, what is OF(d)? Assume as an approximation that the line of sight path that the signal travels is not bent by the ionosphere. Note d and d' are not the same angle due to the curvature of the ionospheric shell at constant height above the (curved) Earth surface. To solve this problem, you should draw the picture and then apply some geometric relations to get OF(d) as a function of d (not d').
Problem 2. Comparison of Single Frequency and Dual Frequency Navigation Solutions.
a). The observation files assigned to you for homework 1 have been replaced with headers that contain the estimated position by the observer. Use the initial xyz WGS-84 position estimates in the header files as “truth” to make a 4x1 plot of xyz and 3-axis error residuals. (3-axis is the rss of x,y,z errors). Report the mean and standard deviation of each of these (4) variables in a table labeled “Single Frequency Code Position Accuracy.” (Aside: How does this compare to what you reported in Homework 1?)
b). All the RINEX data files contain C/A and P2 pseudoranges based on dual frequency measurements. The C/A measurements are made at L1 frequency and the P2 measurements are made at L2 frequency. Using the ionosphere-free position estimation technique discussed in class, create a new function (starting with your original function, do not reinvent the wheel) that forms an ionosphere-free dual frequency estimate of position. Make a 4x1 plot dual-frequency xyz and 3-axis error residuals and put the original single frequency error residuals on the same plot (same scale). Report the mean and standard deviations of each of these (4) variables in a table labeled “Dual Frequency Code Position Accuracy.”
c). Discuss and compare the results from parts a and b.
In your discussion address the following questions:
- Which measurement system is more accurate?
What is the average accuracy improvement? Is it higher or lower than
you expected? Why might that be the case?
- Is the “improved” system always more accurate,
or just some of the time? Why or why not?
- Which measurement is more noisy? Is that
what you expected? Why might that be the case?
Problem 3. Determination of Ionospheric Delay for a Satellite Measurement Set
Pick 1 "good satellite" in your observation data file that you have been given. Ideally, this satellite should be continuously visible and possess high signal strength (i.e. elevation) during the entire data set. State which satellite observation you are using and comment on the expected quality of the data versus time. For this satellite measurement set:
a). Form the iono-free pseudorange and the iono-free carrier phase measurements (start the carrier arbitrarily at 0). Plot and compare the measurements on the same scale (meters). If you wish, remove a linear slope from the data in the plot so that approximate clock drift does not dominate the plot (this should allow you to see the structure of the raw measurements in more detail).
b). Using both the iono-free PR and carrier phase measurements, make a single best plot of estimated iono-delay. Explain in your results how you obtained this curve. Plot the iono-free pseudorange on the same plot for comparison. Correlate your data to what you know about the satellite's pass (e.g. its point of maximum and minimum elevation).