34 GPS Error and Biases

Prof. Masood Ahsan Siddiqui

   Outline

• GPS error and Bias introduction
• GPS Ephemeris Errors
• Selective availability
• Satellite and Receiver Clock
• Errors Multipath Error
• Antenna phase center variation
• Receiver measurement noise Ionospheric dela
• Tropospheric delay
• Satellite geometry measures
• User equivalent range error

    Introduction

GPS pseudo range and carrier phase measurements are both affected by several types of random errors and biases (systematic errors). These errors may be classified as those originating at the satellites, those originating at the receiver, and those that are due to signal propagation (atmospheric refraction). Figure 1 shows the various errors and biases.

The errors originating at the satellites include ephemeris, or orbital errors, satellite clock errors, and the effect of selective availability. The later was intentionally implemented by the U.S. department of defense to degrade the autonomous GPS accuracy for security reasons. It was, however, terminated at midnight (eastern daylight time) on May 1, 2000. The errors originating at the receiver include receiver clock errors, multipath error, receiver noise, and antenna phase center variations. The signal propagation errors include the delay of the GPS signal as it passes through the ionospheric and tropospheric layers of the atmosphere. In fact, it is only in a vacuum (free space) that the GPS signal travels, or propagates, at the speed of light.

In addition to the effect of these errors, the accuracy of the computed GPS position is also affected by the geometric locations of the GPS satellites as seen by the receiver. The more spread out the satellites are in the sky, the better the obtained accuracy.

Fig.1: GPS Error

http://www.globalspec.com/reference/66277/203279/chapter-4-gps-errors-and-biases.

GPS Ephemeris Errors

Satellite positions as a function of time, which are included in the broadcast satellite navigation message, are predicted from previous GPS observations at the ground control stations. Typically, overlapping 4-hour GPS data spans are used by the operational control system to predict fresh satellite orbital elements for each 1-hour period. As might be expected, modeling the forces acting on the GPS satellites will not be perfect, which causes some errors in the estimated satellite positions, known as ephemeris errors. Nominally, an ephemeris error is usually in the order of 2m to 5m, and can reach up to 50 m under selective availability. According to, the range error due to the combined effect of the ephemeris and the Satellite clock errors is of the order of 2.3m. An ephemeris error for a particular satellite is identical to all GPS users worldwide. However, as different users see the same satellite at different view angles, the effect of the ephemeris error on the range measurement, and consequently on the computed position, is different. This means that combining (differencing) the measurements of two receivers simultaneously tracking a particular satellite cannot totally remove the ephemeris error. Users of short separations, however, will have an almost identical range error due to the ephemeris error, which can essentially be removed through differencing the observations. For relative positioning, the following rule of thumb gives a rough estimate of the effect of the ephemeris error on the baseline solution.

Some applications, such as studies of the crustal dynamics of the earth, require more precise ephemeris data than the broadcast ephemeris. To support these applications, several institutions [e.g., the International GPS Service for Geodynamics (IGS), the U.S. National Geodetic Survey (NGS), and Geomatics Canada have developed post mission precise orbital service. Precise ephemeris data is based on GPS data collected at a global GPS network coordinated by the IGS. At the present time, precise ephemeris data is available to users with some delay, which varies from 12 hours for the IGS ultra rapid orbit to about 12 days for the most precise IGS precise orbit. The corresponding accuracies for the two precise orbits are in the order of a few decimeters to 1 decimeter, respectively. Users can down load the precise ephemeris data free of charge from the IGS center

Selective Availability

GPS was originally designed so that real-time autonomous positioning and navigation with the civilian C/A code receivers would be less precise than military P-code receivers. Surprisingly, the obtained accuracy was almost the same from both receivers. To ensure national security, the U.S. department of defense implemented the so called selective availability (SA) on Block II GPS satellites to deny accurate real time autonomous positioning to unauthorized users. SA was officially activated on March 1990.

SA introduces two types of errors. The first one, called delta error, results from dithering the satellite clock, and is common to all users worldwide.

The second one, called epsilon error, is an additional slowly varying orbital error. With SA turned on, nominal horizontal and vertical errors can be up to 100m and 156 m, respectively, at the 95% probability level. Figure 2 shows how the horizontal position of a stationary GPS receiver varies over time, mainly as a result of the effect of SA. Like the range error due to ephemeris error, the range error due to epsilon error is almost identical between users of short separations. Therefore, using differential GPS (DGPS; see Chapter S) would overcome the effect of the epsilon error. In fact, DGPS provides better accuracy than the standalone P-code receiver due to the elimination or the reduction of the common errors, including SA.

Following extensive studies, the U.S. government discontinued SA on May 1, 2000, resulting in a much improved autonomous GPS accuracy. With the SA turned off, the nominal autonomous GPS horizontal and vertical accuracies would be in the order of 22m and 33m.

Fig.2: GPS accuracy error before and after deactivation of selective availability

Source: http://gps.sref.info/course/4g.html

Satellite and Receiver Clock Errors

The clocks in the satellite are very accurate (to about 3 nanoseconds), they do sometimes drift slightly and cause small errors, affecting the accuracy of the position. The US department of defense monitors the satellite clocks using the Control Segment and can correct any drift that is found. Each GPS Block II and Block IIA satellite contains four atomic clocks, two cesium and two rubidium. The newer generation Block IIR satellites carry rubidium clocks only. One of the onboard clocks, primarily a cesium for Block II and IIA, is selected to provide the frequency and the timing requirements for generating the GPS signals. The others are backups. The GPS satellite clocks, although highly accurate, are not perfect. Their stability is about 1 to 2 parts in 1013 over a period of one day. This means that the satellite clock error is about 8.64 to 17.28 ns per day. The corresponding range error is 2.59m to 5.18m, which can be easily calculated by multiplying the clock error by the speed of light. Cesium clocks tend to behave better over a longer period of time compared with rubidium clocks. In fact, the stability of the cesium clocks over a period of 10 days or more improves to several parts in 1014. The performance of the satellite clocks is monitored by the ground control system. The amount of drift is calculated and transmitted as a part of the navigation message in the form of three coefficients of a second-degree polynomial. Satellite clock errors cause additional errors to the GPS measurements. These errors are common to all users observing the same satellite and can be removed through differencing between the receivers. Applying the satellite clock correction in the navigation message can also correct the satellite clock errors. This, however, leaves an error of the order of several nanoseconds, which translates to a range error of a few meters (one nanosecond error is equivalent to a range error of about 30 cm) .GPS receivers, in contrast, use inexpensive crystal clocks, which are much less accurate than the satellite clocks . As such, the receiver clock error is much larger than that of the GPS satellite clock. It can, however, be removed through differencing between the satellites or it can be treated as an additional unknown parameter in the estimation process. Precise external clocks (usually cesium or rubidium) are used in some applications instead of the internal receiver clock. Although the external atomic clocks have superior performance compared with the internal receiver clocks, they cost between a few thousand dollars for the rubidium clocks to about $ 20,000 for the cesium clocks.

Fig.3: Clock Error in GPS

Source: http://www.soi.wide.ad.jp/class/20050026/slides/01/64.html

Multipath Error

    Multipath error is one of the predominant error sources in all GPS applications. Particularly the multipath error has to be precisely estimated in the Global Navigation Satellite Systems (GNSSs) as it is the major error source (2 – 4 m) that limits the GPS receiver’s performance. Whenever, a signal is transmitted from a GPS satellite it follows a “multiple” number of propagation “paths” on its way to receiving antenna. These multiple signal paths are due to the fact that the signal gets reflected back to the antenna off surrounding objects, including the earth’s surface. The GPS receiver tracks both the direct and reflected signal components. The radio wave transmitted from a satellite radiates in all directions, these radio waves including reflected waves that are reflected off due to various obstacles, diffracted waves, scattering waves, and the direct wave from the satellite to GPS receiver (fig.4) In this case, since the path lengths of the direct, reflected, diffracted, and scattering waves are different, the time each takes to reach the GPS receiver will be different. In addition the phase of the incoming wave varies because of reflections. As a result, the receiver receives a superposition consisting of several waves having different phase and times of arrival. The generic name of a radio wave in which the time of arrival is retarded in comparison with this direct wave is called a delayed wave. Then, the reception environment characterized by a superposition of delayed waves is called a multipath propagation environment. In a multipath propagation environment, the received signal is sometimes intensified. This phenomenon is called multi-path fading and the signal level of the received wave changes from moment to moment.

Multipath is a major error source for both the carrier-phase and pseudo range measurements. It occurs when the GPS signal arrives at the receiver antenna through different paths. These paths can be the direct line of sight signal and reflected signals from objects surrounding the receiver antenna. Multipath distorts the original signal through interference with the reflected signals at the GPS antenna. It affects both the carrier-phase and pseudo range measurements; however, its size is much larger in the pseudo range measurements. The size of the carrier-phase multipath can reach a maximum value of a quarter of a cycle (about 4.8 cm for the L1 carrier phase). The pseudo range multipath can theoretically reach several tens of meters for the C/A-code measurements. However, with new advances in receiver technology, actual pseudo range multipath is reduced dramatically. Examples of such technologies are the Strobe correlated (Ashtech, Inc.) and the MEDLL (NovAtel, Inc.). With these multipath mitigation techniques, the pseudo range multipath error is reduced to several meters, even in a highly reflective environment. Under the same environment, the presence of multipath errors can be verified using a day-to-day correlation of the estimated residuals. This is because the satellite-reflector-antenna geometry repeats every sidereal day. However, multipath errors in the undifferenced pseudo range measurements can be identified if dual-frequency observations are available. A good general multipath model is still not available, mainly because of the variant satellite reflector antenna geometry. There are, however several options to reduce the effect of multipath. The option is to select an observation site with no reflecting objects in the vicinity of the receiver antenna. Another option to reduce the effect of multipath is to use a chock ring antenna (a chock ring device is a ground plane that has several concentric metal hoops, which attenuate the reflected signals). As the GPS signal is right handed circularly polarized while the reflected signal is left handed, reducing the effect of multipath may also be achieved by using an antenna with a matching polarization to the GPS. The disadvantage of this option, however, is that the polarization of the multipath signal becomes right handed again if it twice reflected.

Fig.4.Multipath Error

Source: http://file.scirp.org/Html/7-8501064_31523.htm

Antenna phase center variation

A GPS antenna receives the incoming satellite signal and then converts its energy into an electric current, which can be handled by the GPS receiver. The point at which the GPS signal is received is called the antenna phase center. Generally, the antenna phase center does not coincide with the physical (geometrical) center of the antenna. It varies depending on the elevation and the azimuth of the GPS satellite as well as the intensity of the observed signal. As a result, additional range error can be expected. The size of the error caused by the antenna phase center variation depends on the antenna type, and is typically in the order of a few centimeters. It is, however, difficult to model the antenna phase center variation and, therefore, care has to be taken when selecting the antenna type. For short baselines with the same types of antennas at each end, the phase center error can be canceled if the antennas are oriented in the same direction. Mixing different types of antennas or using different orientations will not cancel the error. Due to its small size, this error is neglected in most of the practical GPS applications. It should be pointed out that phase center errors could be different on L1 and L2 carrier phase observations. This can affect the accuracy of the ionosphere free linear combination, particularly when observing short baselines. As mentioned before, for short baselines, the errors are highly correlated over distance and cancel sufficiently through differencing. Therefore, using a single frequency might be more appropriate for short baselines in the static mode.

Receiver measurement noise

The receiver measurement noise results from the limitations of the receiver’s electronics system. A good GPS system should have a minimum noise level. Generally, a GPS receiver performs a self test when the user turns it on. However, for high-cost precise GPS systems, it might be important for the user to perform the system evaluation. Two tests can be performed for evaluating a GPS receiver (system) a. zero baseline and b. short baseline tests.

A zero baseline test is used to evaluate the receiver performance. The test involves using one antenna/preamplifier followed by a signal splitter that feeds two or more GPS receivers (see Figure.5). Several receiver problems such as inter channel biases and cycle slips can be detected with this test. As one antenna is used, the baseline solution should be zero. In other words, any nonzero value is attributed to the receiver noise. Although the zero baseline test provides useful information on the receiver performance, it does not provide any information on the antenna/preamplifier noise. The contribution of the receiver measurement noise to the range error will depend very much on the quality of the GPS receiver. Typical average value for range error due to the receiver measurement noise is of the order of 0.6m.

To evaluate the actual field performance of a GPS system, it is necessary to include the antenna/preamplifier noise component. This can be done using short baselines of a few meters apart, observed on two consecutive days (see Figure.5). In this case, the double difference residuals of one day would contain the system noise and the multipath effect. All other errors would cancel sufficiently. As the multipath signature repeats every sidereal day, differencing the double difference residuals between the two consecutive days eliminates the effect of multipath and leaves only the system noise.

Fig.5: Zero baseline test for evaluating the performance of a GPS receiver

Source:http://what-when-how.com/gps/gps-errors-and-biases-part-2/

Ionospheric delay

At the uppermost part of the earth’s atmosphere, ultraviolet and X-ray radiations coming from the sun interact with the gas molecules and atoms. These interactions result in gas ionization: a large number of free “negatively charged” electrons and “positively charged” atoms and molecules. Such a region of the atmosphere where gas ionization takes place is called the ionosphere. It extends from an altitude of approximately S0 km to about 1,000 km or even more (see Figure.6). In fact, the upper limit of the ionospheric region is not clearly defined. The electron density within the ionospheric region is not constant; it changes with altitude. As such, the ionospheric region is divided into sub regions, or layers, according to the electron density. These layers are named D (S0-90 km), E (90-140 km), F1 (140-210 km), and F2 (210-1,000 km), respectively, with F2 usually being the layer of maximum electron density. The altitude and thickness of those layers vary with time, as a result of the changes in the sun’s radiation and the Earth’s magnetic field. For example, the F1 layer disappears during the night and is more pronounced in the summer than in the winter. The question that may arise is: How would the ionosphere affect the GPS measurements? The ionosphere is a dispersive medium, which means it bends the GPS radio signal and changes its speed as it passes through the various ionospheric layers to reach a GPS receiver. Bending the GPS signal path causes a negligible range error, particularly if the satellite elevation angle is greater than 5°. It is the change in the propagation speed that causes a significant range error, and therefore should be accounted for. The ionosphere speeds up the propagation of the carrier phase beyond the speed of light, while it slows down the PRN code (and the navigation message) by the same amount. That is, the receiver-satellite distance will be too short if measured by the carrier phase and too long if measured by the code, compaired with the actual distance. The ionospheric delay is proportional to the number of free electrons along the GPS signal path, called the total electron content (TEC). TEC, however, depends on a number of factors: (1) the time of day (electron density level reaches a daily maximum in early afternoon and a minimum around midnight at local time); (2) the time of year (electron density levels are higher in winter than in summer);(3) the 11-year solar cycle (electron density levels reach a maximum value approximately every 11 years, which corresponds to a peak in the solar flare activities known as the solar cycle peak-in 2001we are currently around the peak of solar cycle number 23), and (4) the geographic location (electron density levels are minimum in mid latitude regions and highly irregular in polar and equatorial regions). As the ionosphere is a dispersive medium, it causes a delay that is frequency dependent. The lower the frequency, the greater the delay; that is, the L2 ionospheric delay is greater than that of L1. Generally, ionospheric delay is of the order of 5m to 1Sm, but can reach over 150m under extreme solar activities, at midday, and near the horizon. This discussion shows that the electron density level in the ionosphere varies with time and location. It is, however, highly correlated over relatively short distances, and therefore differencing the GPS observations between users of short separation can remove the major part of the ionospheric delay. Taking advantage of the ionosphere’s dispersive nature, the ionospheric delay can be determined with a high degree of accuracy by combining the P-code pseudo range measurements on both L1 and L2. Unfortunately, however, the P-code is accessible by authorized users only. With the addition of a second C/A-code on L2 as part of the modernization program, this limitation will be removed. The L1 and L2 carrier-phase measurements may be combined in a similar fashion to determine the variation in the ionospheric delay, not the absolute value. Users with dual- frequency receivers can combine the L1 and L2 carrier phase measurements to generate the ionosphere-free linear combination to remove the ionospheric delay. The disadvantages of the ionosphere-free linear combination, however, are: (1) it has a relatively higher observation noise, and (2) it does not preserve the integer nature of the ambiguity parameters. As such, the ionosphere-free linear combination is not recommended for short baselines. Single-frequency users cannot take advantage of the dispersive nature of the ionosphere. They can, however, use one of the empirical ionospheric models to correct up to 60% of the delay. The most widely used model is the Klobuchar model, whose coefficients are trans- mitted as part of the navigation message. Another solution for users with single-frequency GPS receivers is to use corrections from regional networks. Such corrections can be received in real time through communication links.

Figure.6: Short baseline test for evaluating the performance of a GPS system

Source:http://what-when-how.com/gps/gps-errors-and-biases-part-2/

    Tropospheric delay

The troposphere is the electrically neutral atmospheric region that extends up to about 50 km from the surface of the earth. The troposphere is a nondispersive medium for radio frequencies below 1S GHz. As a result, it delays the GPS carriers and codes identically.

That is, the measured satellite to receiver range will be longer than the actual geometric range, which means that a distance between two receivers will be longer than the actual distance. Unlike the ionospheric delay, the tropospheric delay cannot be removed by combining the L1 and the L2 observations. This is mainly because the tropospheric delay is frequency independent. The tropospheric delay depends on the temperature, pressure, and humidity along the signal path through the troposphere. Signals from satellites at low elevation angles travel a longer path through the troposphere than those at higher elevation angles. Therefore, the tropospheric delay is minimized at the user’s zenith and maximized near the horizon. Tropospheric delay results in values of about 2.3m at zenith (satellite directly overhead), about 9.3m for a 15°elevation angle, and about 20-28m for a 5°elevation angle. Tropospheric delay may be broken into two components, dry and wet. The dry component represents about 90% of the delay and can be predicted to a high degree of accuracy using mathematical models. The wet component of the tropospheric delay depends on the water vapor along the GPS signal path. Unlike the dry component, the wet component is not easy to predict. Several mathematical models use surface meteorological measurements (atmospheric pressure, temperature, and partial water vapor pressure) to compute the wet component. Unfortunately, however, the wet component is weakly correlated with surface meteorological data, which limits its prediction accuracy. It was found that using default meteorological data (1,010 mb for atmospheric pressure, 20°C for temperature, and 50% for relative humidity) gives satisfactory results in most cases.

Satellite geometry measures

The various types of errors and biases discussed earlier directly affect the accuracy of the computed GPS position. Proper modeling of those errors and biases and/or appropriate combinations of the GPS observables will improve the positioning accuracy. However, these are not the only factors that affect the resulting GPS accuracy. The satellite geometry, which represents the geometric locations of the GPS satellites as seen by the receiver, plays a very important role in the total positioning accuracy. The better the satellite geometry strength, the better the obtained positioning accuracy. As such, the overall positioning accuracy of GPS is measured by the combined effect of the unmodeled measurement errors and the effect of the satellite geometry. Good satellite geometry is obtained when the satellites are spread out in the sky. In general, the more spread out the satellites are in the sky, the better the satellite geometry, and vice versa. Figure shows a simple graphical explanation of the satellite geometry effect using two satellites assuming a two-dimensional (2-D) case. In such a case, the receiver will be located at the intersection of two arcs of circles; each has a radius equal to the receiver-satellite distance and a center at the satellite itself. Because of the measurement errors, the measured receiver-satellite distance will not be exact and an uncertainty region on both sides of the estimated distance will be present. Combining the measurements from the two satellites, it can be seen that the receiver will in fact be located somewhere within the uncertainty area, the hatched area. It is known from statistics that, for a certain probability level, if the size of the uncertainty area is small, the computed receiver’s position will be precise. As shown in Figure.6(a), if the two satellites are far apart (i.e., spread out), the size of the uncertainty area will be small, resulting in good satellite geometry. Similarly, if the two satellites are close to each other Figure.6(b), the size of the uncertainty area will be large, resulting in poor satellite geometry. The satellite geometry effect can be measured by a single dimensionless number called the dilution of precision (DOP). The lower the value of the DOP numbers, the better the geometric strength, and vice versa. The DOP number is computed based on the relative receiver-satellite geometry at any instance, that is, it requires the availability of both the receiver and the satellite coordinates. Approximate values for the coordinates are generally sufficient though, which means that the DOP value can be determined without making any measurements. As a result of the relative motion of the satellites and the receiver(s), the value of the DOP will change over time. The changes in the DOP value, however, will generally be slow except in the following two cases: (1) a satellite is rising or falling as seen by the user’s receiver, and (2) there is an obstruction between the receiver and the satellite (e.g., when passing under a bridge).

     In practice, various DOP forms are used, depending on the user’s need. For example, for the general GPS positioning purposes, a user may be interested in examining the effect of the satellite geometry on the quality of the resulting three-dimensional (3-D) position (latitude, longitude, and height). This could be done by examining the value of the position dilution of precision (PDOP). In other words, PDOP represents the contribution of the satellite geometry to the 3-D positioning accuracy. PDOP can be bro- ken into two components: horizontal dilution of precision (HDOP) and vertical dilution of precision (VDOP). The former represents the satellite geometry effect on the horizontal component of the positioning accuracy, while the latter represents the satellite geometry effect on the vertical component of the positioning accuracy. Because a GPS user can track only those satellites above the horizon, VDOP will always be larger than HDOP. As a result, the GPS height solution is expected to be less precise than the horizontal solution. The VDOP value could be improved by supplementing GPS with other sensors, for example, the pseudolites Other commonly used DOP forms include the time dilution of precision (TDOP) and the geometric dilution of precision (GDOP). GDOP represents the combined effect of the PDOP and the TDOP. To ensure high-precision GPS positioning, it is recommended that a suitable observation time be selected to obtain the highest possible accuracy. A PDOP of five or less is usually recommended. In fact, the actual PDOP value is usually much less than five, with a typical average value in the neighborhood of two. Most GPS software packages have the ability to predict the satellite geometry based on the user’s approximate location and the approximate satellite locations obtained from a recent almanac file for the GPS constellation. The almanac file is obtained as part of the navigation message, and can be downloaded free of charge over the Internet (e.g., from the U.S. Coast Guard Navigation Center.

https://www.slideshare.net/maneeb/errors-and-biases-in-gps

User equivalent range error

It has been shown that the GPS positioning accuracy is measured by the combined effect of the unmodeled measurement errors and the effect of the satellite geometry. The unmodeled measurement errors will certainly be different from one satellite to another, mainly because of the various view angles. In addition, the ranging errors for the various satellites will have a certain degree of similarity. To rigorously determine the expected GPS positioning accuracy, we may apply an estimation technique such as the least squares method .The least squares method estimates the user’s position (location) as well as its covariance matrix. The latter tells us how well the user’s position is determined. In fact, the covariance matrix reflects the combined effect of the measurement errors and the satellite geometry.

A more simplified way of examining the GPS positioning accuracy may be achieved through the introduction of the user equivalent range error. Assuming that the measurement errors for all the satellites are identical and independent, then a quantity known as the UERE may be defined as the root sum square of the various errors and biases. Multiplying the UERE by the appropriate DOP value produces the expected precision of the GPS positioning at the one sigma (1-a) level. To obtain the precision at the 2-a level, sometimes referred to as approximately 95% of the time, we multiply the results by a factor of two. For example, assuming that the UERE is 8m for the standalone GPS receiver, and taking a typical value of HDOP as 1.5, then the 95% positional accuracy will be 8 x1.5×2 = 24m.

Fig.7: User equivalent range error

Source:http://nptel.ac.in/courses/105104100/lectureB_11/B_11_2sat_geo.htm