Author Topic: Global Navigation Satellite Systems -- tutorial (Read 28024 times)

Neil Obstat · « **Reply #195 on:** September 29, 2017, 07:03:03 PM »

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In Depth -- doppler effect

[There is another short video here but it's not worth mentioning.]

Note that the doppler effect will naturally occur as with any other waveform emanating from a moving object. This effect is noticeable as the distortion of sound from a passing car or train when you are standing still. A train whistle or car horn has a higher pitch when approaching than when it is receding.

When the satellite is approaching the receiver, the GPS signal wavelengths will be compressed, resulting in shorter wavelengths. The receiver compares the actual wavelength received with the expected wavelength to help verify that it is indeed observing the correct satellite (one that is approaching or receding). The receiver takes into account the doppler effect in estimating the range from the GPS radio wave signals.

Neil Obstat · « **Reply #196 on:** September 29, 2017, 07:10:54 PM »

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3h. Post-processing of GPS location data

The fixing of ambiguities can be done in the “post-processing” of the GPS data, typically after they are downloaded onto a computer and processed with GPS data processing software, such as the National Geodetic Survey’s Online Position User Service, or OPUS. Post-processing is used to reduce all of the individual GPS observations to one robust and accurate estimate of the position being measured. OPUS post-processing is achieved using NOAA’s Continuously Operating Reference Stations, known as CORS. We will explore the CORS network in more detail in Unit 4.

As a result of post-processing using OPUS, a report is generated giving not only the estimated coordinates for the mark, but some data quality and error estimates. They include: 1) the fraction of observations used to compute an accurate position, 2) the fraction of ambiguities that were successfully fixed, and 3) an overall estimate of error, known as residual mean square error.

Because there are errors (offsets and biases, etc.) at both the satellite end and the receiver end, estimating the ambiguities (called ambiguity “fixing”) is usually done in conjunction with other GPS data coming from GPS reference stations or other simultaneous GPS observations. In Unit 4 we will explore how a process called “double-differencing” uses the data from a second location to cancel out errors.

Once we have solved for (or fixed) the integer ambiguity, we can use it along with the phase measurement to compute the entire range using this formula:

At the initial epoch, ρ0 = λ × (φo + N)

Where:
ρ = range
λ = wavelength
φ = phase measurement
N = integer ambiguity

At epoch 1, ρ1 = λ × (Counted Cycles + N) + (λ × φ1)

Note that high precision GPS signal transmission and processing involves dual frequencies, L1 and L2. The two wavelengths can be combined to create new signals of larger or smaller wavelengths which can also aid in resolving the ambiguity.

For additional information on how wavelengths can be combined in GPS signal processing, please see APPENDIX 3: The role of combining wavelengths in signal processing.

Neil Obstat · « **Reply #197 on:** September 29, 2017, 07:18:39 PM »

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3i. Summary

In this unit, we explored how the physical properties of radio waves can be used to provide a more precise estimate of distance than is possible through the timing of signal transmission and reception. Although the wavelengths provide the potential for very precise determination of distances (down to the millimeter level), we discussed the inherent problems with counting the wavelengths that spanned the large distance between the satellite and the receiver. We looked at how the initial count is referred to as the “ambiguity,” and how tracking each satellite across the sky eventually allows us to get a better and better estimate of this initial ambiguity, and all the subsequent distances between the satellite and receiver. We also briefly examined the role of “post-processing” of GPS data to estimate the ambiguities (ambiguity “fixing”), which results in an estimation of the position of the receiver antenna with higher accuracy than was possible by just using the pseudorange.

Neil Obstat · « **Reply #198 on:** September 29, 2017, 07:21:50 PM »

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For unit 3 you only get 4 questions.
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3j. Review questions

Question 1 of 4

To use the radio wave from a GPS satellite for precise positioning, what information is needed to compute distance? (Choose all that apply.)
a) Frequency of the signal
b) Time offset and PRN code
c) Range code
d) Counted whole cycles
e) Phase measurement

Question 2 of 4

Assume the wavelength of the L1 signal is 0.19 m/cycle. How many whole L1 cycles would you count between you and the satellite if the distance were known to be 20,150,000 m? (Choose the best answer.)
a) 106,052,631
b) 82,521,298
c) 1,059,747
d) 825,212
e) 42

Question 3 of 4

What would the partial cycle (phase measurement) of the above computation be (in cycles and in cm)? (Choose the best answer.)
a) 0.579 cycles, 11 cm
b) 0.202, 3.8 cm
c) 0.5 cycles, 12.2 cm
d) 0.985 cycles, 24 cm

Question 4 of 4

If a GPS carrier frequency is 1GHz, what is the expected positional accuracy based on phase-measurement error? The speed of light/radio signal is 299,792,458 m/s. (Choose the best answer.)
a) 3 m
b) 0.3 m
c) 0.03 m
d) 0.003 m

Neil Obstat · « **Reply #199 on:** September 30, 2017, 06:39:04 PM »

Quote from: Neil Obstat on September 29, 2017, 06:16:22 PM

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The tutorial supplies this answer to the question above. But they made one mistake, which then results in several errors.
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Can you find the errors or the one mistake? (Hint: They're off by more than "several centimeters.")
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[quoted from the tutorial, 3e. Computation of GPS wavelengths, bottom of page, "Question"]:
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20,135,196.834 m ÷ 0.19 m/wavelength = 105,974,720.179 wavelengths [Nor did they spell out that the 0.179 is the rational portion of this quantity of wavelengths]
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If we could count just the instantaneous number of cycles at this one epoch, we would have counted 105,974,720 wavelengths - the integer portion of the solution.
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Clearly, we’d be missing a fraction of the next cycle (shown as the extended part of the curve in the figure above). In the example, we’d be off by 0.179 cycles. At 0.19 m/wavelength, this error would be:
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0.179 m x 0.19 cm = 0.034 m = 3.4 cm.
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So we’d be off by several centimeters.

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(I'm keeping my text in blue font so you can tell it's mine.)
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In this sample question's answer above, they used the wrong distance to the satellite. I'm not sure where they got the wrong distance, maybe from another problem somewhere. The question they asked was as follows:
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Question
Assume the satellite is 20,166,318.727 meters away from the GPS antenna. How many wavelengths of the L1 signal would make up this distance?
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NOTICE, the satellite is 20,166,318.727 meters away, whereas in the answer they have 20,135,196.834 m, which is off by 31,121.893 meters. That's over 31 kilometers, and it amounts to an error of 1.54%. They're worried about a fraction of one cycle but this mistake makes them off 163,800 cycles!!
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The correct answer would say:
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~~20,135,196.834~~ 20,166,318.727 m ÷ 0.19 m/wavelength = ~~105,974,720~~ 106,138,519.616 wavelengths
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If we could count just the instantaneous number of cycles at this one epoch, we would have counted ~~105,974,720~~ 106,138,519 wavelengths - the integer portion of the solution.

Clearly, we’d be missing a fraction of the next cycle (shown as the extended part of the curve in the figure above). In the example, we’d be off by ~~0.179~~ 0.616 cycles. At 0.19 m/wavelength, this error would be:
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~~0.179~~ 0.616 m x 0.19 cm = ~~0.034~~ 0.117 m = ~~3.4~~ 11.7 cm.
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So we’d be off by ~~several~~ a dozen centimeters.
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This correction goes to show how one mistake can have a chain reaction of effects. Since the GPS system runs calculations by the thousands, in a fraction of a second, there has to be a way of catching errors before they get out of hand, otherwise they could snowball and wreck the whole project.

Neil Obstat · « **Reply #200 on:** September 30, 2017, 08:32:58 PM »

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Next Unit!!
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Foundations of Global Navigation Satellite Systems (GNSS)
Reducing errors in GNSS positioning

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Curiously, we just went through a question-and-answer example where the tutorial made one error which turned into additional errors. So error reduction is a pretty important principle with these satellite systems.
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Neil Obstat · « **Reply #201 on:** September 30, 2017, 08:38:06 PM »

4. Unit description and objectives

In this unit you will learn how multiple sources of error enter into the distance equations that affect our estimated distances between the GNSS satellites and the GNSS receivers on Earth. After completing this unit you should be able to:

1. Describe different potential error sources in GNSS operation and how they are mitigated:
- ---GPS satellite and receiver clock errors
- ---Ionospheric delay
- ---Satellite orbital errors
- ---Tropospheric error
- ---Multipath errors
2. Explain how increased observation time increases the accuracy of GNSS-derived position.
3. Explain how the mathematical process of of double differencing is used to increase accuracy and reduce errors.
4. Explain the role of continuous GNSS active-relative positioning systems such as NOAA’s CORS.

Neil Obstat · « **Reply #202 on:** September 30, 2017, 08:46:30 PM »

4a. Post-processing of GPS data

We have presented the basic principles involved in determining the range between GPS satellites in space and the GPS receiver, including pseudorange estimation and GPS radio signal processing. Unfortunately, due to errors in the satellite ranging system, these ranges alone will not provide us with a sufficiently accurate position for precise surveying applications. Clock offsets, atmospheric disturbances, reflected waves—anything that affects our ability to measure time precisely or that affects how the radio signal travels from the satellite to the receiver— will lead to errors in the estimate of our position.

Luckily, computational methods can be employed to remove many of these sources of error. Many of these methods may be applied after the initial observation. For example, satellite clock offsets are computed and made available as part of GPS satellite orbital products available online. This information can be used in the precise positioning computations conducted after the GPS observations have been made, using post-processing software.

Post-processing
Post-processing, in GPS terms, means computing accurate positions based on data made available after the GPS observations have been made. Post-processing allows us to remove sources of error, such as satellite clock offsets, errors in the GPS satellite orbits, and atmospheric effects.

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Neil Obstat · « **Reply #203 on:** September 30, 2017, 09:14:51 PM »

4b. Understanding the sources of error

Since precise time synchronization is so important—recall that the receiver must determine the exact satellite position at a given time using the ephemeris—clock offsets at both the satellite and the receiver are found to be the largest potential source of error in satellite positioning as shown in this figure. Additional variables affecting the magnitude of the error include atmospheric effects, multipath, and receiver noise, among others. Also, one should consider the potential errors in the broadcast ephemerides; these can be mitigated by post-processing the data with the more accurate “precise ephemerides.” These are available a couple of weeks after the date of the GPS observation (more on this later).
The following equation shows the relationship between some of the major sources of error.

Known values:

c = Speed of light (constant, given)

δS = Satellite clock offset (value obtained from the broadcast ephemeris)

P = The pseudorange

Unknown values:

ρ = The true range

εorb = Satellite orbital errors

δR = Receiver clock offset

εion = Ionosphere delay

εtrop = Tropospheric delay

εmp = Multipath

εP = Receiver noise

We can express the pseudorange, P, as a function of variables, some of which are known, and others which are unknown (and have to be resolved in our computations).

We know the speed of light, which is a constant (299,792,458m/s).
The satellite clock offset from true GPS time is provided within the broadcast ephemeris (or alternatively from the precise orbit files available online from the International GNSS Service), so it is known.
The pseudorange is also computed, so it is considered “known.”

What remains unknown is the true range, ρ, as well as additional sources of error.

Next, let’s look more closely at five main sources of error and how they can be addressed:
GPS satellite and receiver clock errors,
ionospheric effect,
satellite orbital errors,
tropospheric error, and
multipath errors.
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Neil Obstat · « **Reply #204 on:** September 30, 2017, 09:22:47 PM »

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GPS satellite and receiver clock errors

Recall that the largest potential source of error in GPS signal processing comes from clock errors at the satellite and the receiver. Satellites contain atomic clocks which are extremely accurate, however they still experience some drift as well as relativistic errors. These errors are monitored and corrected by the ground-based observing systems.

The other major source of error is directly related to the clock within the receiver. The quartz oscillator is quite inexpensive compared to the atomic clocks onboard the GPS satellites, and is relatively stable. However, since the oscillator is used to reproduce the GPS satellite’s PRN code as well as compute the GPS signal phase measurement, any error will have multiple and potentially compounding effects.

We discussed earlier that synchronization between the satellite clock and the receiver clock is critical to the determination of the pseudorange, and that the receiver clock offset can be estimated by including the information from a fourth satellite in our trilateration solution. So we approximate the clock bias in our solution for the pseudorange, although the bias is not entirely removed.

Neil Obstat · « **Reply #205 on:** September 30, 2017, 09:35:34 PM »

Ionospheric effect

Atmospheric effects on the transmission of the GPS signal have the potential to be the second largest source of error in GPS positioning, next to clock offsets. GPS signals travel at the speed of light as long as the signals are in the vacuum of space. However, as the signals traverse our atmosphere, they are altered through refraction and diffraction. This is especially the case with the ionosphere, lying between 80 and 600 km above Earth’s surface. The ionosphere contains numerous free electrons that can affect the transmission of radio waves. The higher the total electron count (TEC) per square meter, the greater the potential interference. Solar storms can be a leading cause of high TEC.
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www.swpc.noaa.gov/news/g3-strong-geomagnetic-storm-levels-reached-early-cme-arrival
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Text from bottom of screenshot above:

Quote

G3 (STRONG) GEOMAGNETIC STORM LEVELS REACHED WITH EARLY CME ARRIVAL
published: Sunday, August 16, 2015 16:23 UTC
The CME associated with a filament eruption on August 12, 2015 arrived a little earlier than expected. An interplanetary shock was observed by the ACE spacecraft at approximately 0745 UTC (3:45 am ET) on August 15th. G3 (Strong) Geomagnetic Storm levels were reached by 1143 UTC (7:43 am ET), however, activity levels appear to be decreasing slightly as the initial phase of the impact passes. G2 (Moderate) Geomagnetic Storm conditions are still in progress as CME effects continue. Stay tuned for updates!

NOAA’s Space Weather Prediction Center (http://www.swpc.noaa.gov/communities/global-positioning-system-gps-community-dashboard) provides geomagnetic storm warnings, alerts, and watches to specifically address potential impacts to GPS observations on Earth.

Fortunately, different signal frequencies are affected slightly differently by the ionosphere. The following formula shows how ionospheric delay is inversely related to the frequency of the signal squared.

𝑣 = ionospheric delay (m)

𝑐 = speed of light (m/s)

𝑓 = frequency of the GPS signal (Hz)

TEC = the quantity of free electrons per square meter

Higher frequencies are affected much less than lower frequencies (note the squared term in the denominator). A main strength of the dual-frequency GPS receiver is that we can use the difference between how the two frequencies are attenuated to estimate (and remove) most of the ionospheric delay.

For additional information on how L1 and L2 frequencies are combined to produce the "ionosphere-free" combination, please see APPENDIX 4: Ionosphere-free linear combination of L1 and L2.

Neil Obstat · « **Reply #206 on:** September 30, 2017, 09:41:40 PM »

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Satellite orbital errors

Variations in the actual path of satellites from expected/modeled orbits can be large enough to be significant in GPS calculations. These orbital paths are affected by a number of variables, most notably the variations in Earth’s gravitational field. (See Gravity for Geodesy for more information.)

Other phenomena such as the satellite passing into and out of the sun’s illumination can alter the orbits slightly. Fortunately, satellite path variations are one of the easiest sources of error to correct in your GPS positioning.

Satellite orbital errors are corrected through post-processing GPS data based on the published “precise” satellite orbits. These orbits are used to replace the less-accurate modeled broadcast orbits.

Truth is Eternal · « **Reply #207 on:** September 30, 2017, 09:43:25 PM »

NEIL OBSTAT, YOU HAVE A PROBLEM.

Neil Obstat · « **Reply #208 on:** September 30, 2017, 09:45:35 PM »

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Linked site has following modulus available:
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https://www.meted.ucar.edu/training_module.php?id=1164&tab=04#.WdBVdPlSyQM
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Define the basic properties of gravitation.
Explain the properties of gravitation near a large mass such as a planet.
Explain the relationship between gravitation and gravity.
Identify density, altitude, and latitude as the three key influencing factors on gravity.
Explain how gravity’s magnitude varies with latitude.
Show how gravity’s magnitude decreases with altitude.
Give examples of high and low density features of Earth and the resulting changes in gravity.
Describe the impact of mass movement on gravity.
Identify the signals that gravimeters measure.

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Neil Obstat · « **Reply #209 on:** September 30, 2017, 09:54:47 PM »

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Tropospheric error

The troposphere can also cause a delay in the GPS satellite signal. The troposphere is generally thought of as Earth’s atmosphere between the surface and about 50 km. Unlike the ionosphere, the effect of the troposphere is independent of frequency so it is not so easily removed.

It is the temporal and spatial variations in water vapor content within the troposphere that can cause unexplained error. The net outcome of the tropospheric effect is a delay such that the apparent range between the satellite and the receiver is longer than it truly is. The tropospheric effect depends on the amount of atmosphere that the GPS signal has to cross. It is less from a satellite directly overhead compared to one at the horizon.

The atmospheric errors (ionosphere and troposphere) are not homogeneous over space and time. However, two GPS receivers located relatively close to each other (say, within 100 km) will likely share similar atmospheric errors. This fact leads to an elegant way to reduce atmospheric (and other) errors called “double differencing,” (See “4c Removing error via double differencing”.) [4c is two more sections down the list so it's coming up very soon.]

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Author Topic: Global Navigation Satellite Systems -- tutorial (Read 28024 times)

Neil Obstat

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Neil Obstat

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Neil Obstat

Re: Global Navigation Satellite Systems -- tutorial

Neil Obstat

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Neil Obstat

Re: Global Navigation Satellite Systems -- tutorial

Neil Obstat

Re: Global Navigation Satellite Systems -- tutorial

Neil Obstat

Re: Global Navigation Satellite Systems -- tutorial

Neil Obstat

Re: Global Navigation Satellite Systems -- tutorial

Neil Obstat

Re: Global Navigation Satellite Systems -- tutorial

Neil Obstat

Re: Global Navigation Satellite Systems -- tutorial

Neil Obstat

Re: Global Navigation Satellite Systems -- tutorial

Neil Obstat

Re: Global Navigation Satellite Systems -- tutorial

Truth is Eternal

Re: Global Navigation Satellite Systems -- tutorial

Neil Obstat

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Neil Obstat

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