Civil and Environmental Engineering and Geodetic Science This file can be found on the course web page: Where also GPS reference links are provided Part III POINT POSITIONING DIFFERENTIAL GPS GS609

Civil and Environmental Engineering and Geodetic Science GPS Positioning (point positioning with pseudoranges) tt signal transmitted signal received range,  = c  t    

Civil and Environmental Engineering and Geodetic Science

Point Positioning with Pseudoranges Assume that ionospheric effect is removed from the equation by applying the model provided by the navigation message, or it is simply neglected Assume that tropospheric effect is removed from the equation by estimating the dry+wet effect based on the tropospheric model (e.g., by Saastamoinen, Goad and Goodman, Chao, Lanyi) Satellite clock correction is also applied based on the navigation message Multipath and interchannel bias are neglected The resulting equation : corrected observable 

Civil and Environmental Engineering and Geodetic Science Point Positioning with Pseudoranges Linearized observation equation Geometric distance obtained from known satellite coordinates (broadcast ephemeris) and approximated station coordinates Objective: drive (“observed – computed” term) to zero by iterating the solution from the sufficient number of satellites (see next slide)

Civil and Environmental Engineering and Geodetic Science Point Positioning with Pseudoranges Minimum of four independent observations to four satellites k, l, m, n is needed to solve for station i coordinates and the receiver clock correction Iterations: reset station coordinates, compute better approximation of the geometric range Solve again until left hand side of the above system is driven to zero

Civil and Environmental Engineering and Geodetic Science In the case of multiple epochs of observation (or more than 4 satellites)  Least Squares Adjustment problem! Number of unknowns: 3 coordinates + n receiver clock error terms, each corresponding to a separate epoch of observation 1 to n

Civil and Environmental Engineering and Geodetic Science Point Positioning with Pseudoranges Minimum of three independent observations to three satellites k, l, m is needed to solve for station i coordinates when the receiver clock error is neglected Iterations: reset station coordinates, compute better approximation of the geometric range Solve again until left hand side of the above system is driven to zero

Civil and Environmental Engineering and Geodetic Science If point is occupied for a longer period of time  receiver clock error will vary in time, thus multiple estimates are needed New clock correction is estimated at every epoch for total of n epochs Multiple satellites are observed at every epoch (can vary from epoch to epoch) Superscripts 1,2,…,n denote epochs; thus rows in the above system represent a single epoch (all m satellites observed at the epoch) in the form of eq. (1) two slides back [c] is a column of c with the number of rows equal the number of satellites, m, observed at the given epoch

Civil and Environmental Engineering and Geodetic Science In the case of multiple epochs of observation (or more than 4 satellites)  adjustment problem! Number of unknowns: 3 coordinates + n receiver clock error terms, each corresponding to a separate epoch of observation 1 to n Rewrite eq. (2) using matrix notation Y is a vector of “observed – computed” A is a design matrix of partial derivatives is a vector of unknowns is a white noise vector Where is the observation standard deviation, uniform for all measurements

Civil and Environmental Engineering and Geodetic Science B j =[1 1 1 … 1] T where the number of 1 equal to the number of satellites (1,…,m) observed at epoch j (j=1,…,n) Rearranging terms in eq. (2) leads to a simplified form of a design matrix A, and subsequently to a normal matrix easy to handle by Gaussian elimination

Civil and Environmental Engineering and Geodetic Science Where is a vector of unknown station coordinates [X i Y i Z i ] and matrices A i (size (m,3)) are of the form of (1a), written for m satellites (ranges) observed at the epoch y j is a m-element vector of the form where j is the epoch between 1 and n Final system of normal equations following from eq. (3) Rewrite eq. (3) in the following form: VERIFY ! Where m is number of observations at one epoch

Civil and Environmental Engineering and Geodetic Science Dilution of Precision Accuracy of GPS positioning depends on: the accuracy of the range observables the geometric configuration of the satellites used (design matrix) the relation between the measurement error and the positioning error:  pos = DOP  obs DOP is called dilution of precision for 3D positioning, PDOP (position dilution of precision), is defined as a square root of a sum of the diagonal elements of the normal matrix (A T A) -1 (corresponding to x, y and z unknowns)

Civil and Environmental Engineering and Geodetic Science Dilution of Precision PDOP is interpreted as the reciprocal value of the volume of tetrahedron that is formed from the satellite and user positions Receiver Good PDOP Bad PDOP Position error  p =  r PDOP, where  r is the observation error (or standard deviation)

Civil and Environmental Engineering and Geodetic Science Dilution of Precision The observation error (or standard deviation) denoted as  r or  obs is the number that best describes the quality of the pseudorange (or phase) observation, thus is is about 0.2 – 1.0 m for P-code range and reaches a few meters for the C/A-code pseudorange. Thus, DOP is a geometric factor that amplifies the single range observation error to show the factual positioning accuracy obtained from multiple observations It is very important to use the right numbers for  r to properly describe the factual quality of your measurements. However, most of the time, these values are pre-defined within the GPS processing software (remember that Geomatics Office never asked you about the observation error (or standard deviation)) and user has no way to manipulate that. This values are derived as average for a particular class of receivers (and it works well for most applications!)

Civil and Environmental Engineering and Geodetic Science Dilution of Precision DOP concept is of most interest to navigation. If a four channel receiver is used, the best four-satellite configuration will be used automatically based on the lowest DOP (however, most of modern receivers have more than 4 channels) This is also an important issue for differential GPS, as both stations must use the same satellites (actually with the current full constellation the common observability is not a problematic issue, even for very long baselines) DOP is not that crucial for surveying results, where multiple (redundant) satellites are used, and where the Least Squares Adjustment is used to arrive at the most optimal solution However, DOP is very important in the surveying planning and control (especially for kinematic and fast static modes), where the best observability window can be selected based on the highest number of satellites and the best geometry (lowest DOP); check the Quick Plan option under Utilities menu in Geomatics Office

Civil and Environmental Engineering and Geodetic Science Differential GPS (DGPS) DGPS is applied in geodesy and surveying (for the highest accuracy, cm-level) as well as in GIS-type of data collection (sub meter or less accuracy required) Data collected simultaneously by two stations (one with known location) can be processed in a differential mode, by differing respective observables from both stations The user can set up his own base (reference) station for DGPS or use differential services provided by, for example, Coast Guard, which provides differential correction to reduce the pseudorange error in the user’s observable

Civil and Environmental Engineering and Geodetic Science By differencing observables with respect to simultaneously tracking receivers, satellites and time epochs, a significant reduction of errors affecting the observables due to: satellite and receiver clock biases, atmospheric as well as SA effects (for short baselines), inter-channel biases is achieved DGPS: Objectives and Benefits

Civil and Environmental Engineering and Geodetic Science Differential GPS Selective Availability (SA), if it is on Satellite clock and orbit errors Atmospheric effects (for short baselines) Using data from two receivers observing the same satellite simultaneously removes (or significantly decreases) common errors, including: Base station with known location Unknown position Single difference mode

Civil and Environmental Engineering and Geodetic Science Differential GPS Receiver clock errors Atmospheric effects (ionosphere, troposphere) Receiver interchannel bias Using two satellites in the differencing process, further removes common errors such as: Base station with known location Unknown position Double difference mode

Civil and Environmental Engineering and Geodetic Science

Consider two stations i and j observing L1 pseudorange to the same two GPS satellites k and l:

Civil and Environmental Engineering and Geodetic Science Consider two stations i and j observing L1 phase range to the same two GPS satellites k and l:

Civil and Environmental Engineering and Geodetic Science Let’s consider differential pseudoranging first The single-differenced (SD) measurement is obtained by differencing two observable of the satellite k, tracked simultaneously by two stations i and j: It significantly reduces the atmospheric errors and removes the satellite clock and orbital errors; differential effects are still there (like iono, tropo and multipath, and the difference between the clock errors between the receivers) In the actual data processing the differential tropospheric and multipath errors are neglected, while remaining ionospheric, differential clock error, and interchannel biases might be estimated (if possible)

Civil and Environmental Engineering and Geodetic Science DGPS in Geodesy and Surveying single-differenced The single-differenced measurement is obtained by differencing two observables of the satellite k, tracked simultaneously by two stations i and j: Non-integer ambiguity !

Civil and Environmental Engineering and Geodetic Science DGPS Concept, cont. By differencing one-way observable from two receivers, i and j, observing two satellites, k and l, or simply by differencing two single differences to satellites k and l, one arrives at the double-differenced (DD) measurement: In the actual data processing the differential tropospheric, ionospheric and multipath errors are neglected; the only unknowns are the station coordinates Double difference Two single differences

Civil and Environmental Engineering and Geodetic Science Differential Phase Observations Double difference Two single differences Single difference ambiguity

Civil and Environmental Engineering and Geodetic Science DGPS in Geodesy and Surveying double-differenced By differencing one-way observable from two receivers, i and j, observing two satellites, k and l, or simply by differencing two single differences to satellites k and l, one arrives at the double-differenced measurement:

Civil and Environmental Engineering and Geodetic Science Differential Phase Observations Double differenced (DD) mode is the most popular for phase data processing In DD the unknowns are station coordinates and the integer ambiguities In DD the differential atmospheric and multipath effects are very small and are neglected The achievable accuracy is cm-level for short baselines (below km); for longer distances, DD ionospheric-free combination is used (see the future notes for reference!) Single differencing is also used, however, the problem there is non-integer ambiguity term (see previous slide), which does not provide such strong constraints into the solution as the integer ambiguity for DD

Civil and Environmental Engineering and Geodetic Science Triple Difference Observable Differencing two double differences, separated by the time interval dt provides triple-differenced measurement, that in case of phase observables effectively cancels the phase ambiguity biases, N 1 and N 2 In both equations the differential effects are neglected and the station coordinates are the only unknowns

Civil and Environmental Engineering and Geodetic Science Note: Observed phases (in cycles) are converted to so-called phase ranges (in meters) by multiplying the raw phase by the respective wavelength of L1 or L2 signals  Thus, the units in the above equations are meters!  Positioning with phase ranges is much more accurate as compared to pseudoranges, but more complicated since integer ambiguities (such as DD ambiguities) must be fixed before the positioning can be achieved  Triple difference (TD) equation does not contain ambiguities, but its noise level is much higher as compared to SD or DD, so it is not recommended if the highest accuracy is expected

Civil and Environmental Engineering and Geodetic Science St. 1 St. 2 2 (base) Positioning with phase observations: A Concept

Civil and Environmental Engineering and Geodetic Science Positioning with phase observations: A Concept Three double difference (based on four satellites) is a minimum to do DGPS with phase ranges after ambiguities have been fixed to their integer values Minimum of five simultaneously observed satellites is needed to resolve ambiguities Thus, ambiguities must be resolved first, then positioning step can be performed Ambiguities stay fixed and unchanged until cycle slip (CS) happens

Civil and Environmental Engineering and Geodetic Science Covariance Matrix for Phase Combination Four single differencesThree double differences Where A is a differencing operator matrix

Civil and Environmental Engineering and Geodetic Science SD differencing operator

Civil and Environmental Engineering and Geodetic Science DD differencing operator Thus DD covariance matrix is a full matrix for one epoch For several epochs – it will be a block diagonal matrix

Civil and Environmental Engineering and Geodetic Science We have talked so far about single, double and triple differences of GPS observable (predominantly phase), which are nothing else but linear combinations of direct measurements. These linear combinations become very handy in removal (or at least significant reduction) of various error sources and nuisance parameters, making positioning process rather simple (at least for short baselines). Keep in mind that the covariance matrix becomes more complicated, but that is a small price to pay for a limited number of unknowns offered in double differencing! There are, however, even more “advanced” linear combinations whose specific objectives would be to further eliminate some errors that might still be present in differential form in the, for example, double difference equation, and to simplify (or enable) certain actions – such as ambiguity resolution (we know that ambiguities must be resolved before we can do positioning with GPS phase observations). So, let’s take a look at some of the most useful linear combinations (you can create any combination you like, the point is to make it in a smart way so that it would make your life easier!

Civil and Environmental Engineering and Geodetic Science Useful linear combinations Created usually from double-differenced phase observations Ion-free combination based on L1 and L2 observable eliminates ionospheric effects (actually, the first order only) Ion-only combination based on L1 and L2 observable, (useful for cycle slip tracking) eliminates all effects except for the ionosphere, thus can be used to estimate the ionospheric effect Widelane – its long wavelength of 86.2 cm supports ambiguity resolution; based on L1 and L2 observable

Civil and Environmental Engineering and Geodetic Science Ionosphere-free combination ionosphere-free phase measurement similarly, ionosphere-free pseudorange can be obtained The conditions applied are that sum of ionospheric effects on both frequencies multiplied by constants to be determined must be zero; second condition is for example that sum of the constants is 1, or one constant is set to 1 (verify!)

Civil and Environmental Engineering and Geodetic Science Ionosphere-free combination Take the ionospheric terms on L1 and L2 and assume that they meet the following conditions (where  1 and  2 are the “to be determined” coefficients defining the iono-free combination: However, we only considered the 1 st order ionospheric term here!

Civil and Environmental Engineering and Geodetic Science Estimated ionospheric group delay for GPS signal (see the table) The first order effects are most significant In the phase/range equation we use only 1 st order ionospheric terms Thus the iono-free combination is in fact only ion 1 st order iono-free L1L1 L2L2 Residual Range Error First Order: 1/f m26.7 m0.0 Second Order: 1/f 3 ~ 1.6 cm~ 3.3 cm~ -1.1 cm Third Order: 1/f 4 ~ 0.86 mm~ 2.4 mm~ mm Calibrated 1/f 3 Term Based on a Thin Layer Ionospheric Model ~ 1-2 mm The phase advance can be obtained from the above table by multiplying each number by -1, -0.5 and -1/3 for the 1/f 2, 1/f 3 and 1/f 4 term, respectively

Civil and Environmental Engineering and Geodetic Science Integration of the refractive index renders the measured range, and the ionospheric terms for range and phase (see earlier notes) Denoting the 1 st and 2 nd order iono term as follows (after the integration, in cycles; a and be are constants): We can now consider forming so-called iono-free combination phase equation, but including the second order iono term (see the enclosed hand- out) Based on the L1 and L2 frequencies, and assuming the proposed third GPS frequency called L5, we can form two iono-free combinations, and combine them further to derive a 2 nd order ion-free linear combination (future!)

Civil and Environmental Engineering and Geodetic Science Notice that the two 1 st order iono-free combinations and used here, were derived under the assumption that  1 was set to 1, as opposed to our condition used earlier that  1 +  2 =1 (see also the handout) We can now derive the 2 nd order ionospheric term as follows (by using the above ion-free combinations for the ionospheirc terms only, including the 2 nd order, as shown on the slide above): Now, the general form of the 2 nd order iono-free combination is as follows: Where the inospheric terms above are used to estimate the n 1 and n 2 under the assumption that the final iono term in the linear combination will disappear 2 nd order ion-free combination

Civil and Environmental Engineering and Geodetic Science Assuming n 1 =1 and using the conditions above we can write: And finally arrive at the n2 value for this combination: 2 nd order ion-free combination thus Represents the 2 nd order ionosphere-free linear combination (future!) Notice the non-integer ambiguity!

Civil and Environmental Engineering and Geodetic Science Other useful linear combinations widelanewidelane where is in cycles the corresponding wavelength ionospheric-only ionospheric-only (geometry-free) combination is obtained by differencing two phase ranges [m] belonging to the frequencies L 1 and L 2 [meter] Non-integer ambiguity!

Civil and Environmental Engineering and Geodetic Science Widelane Difference between phase observable on L1 and L2 (in cycles) Widelane in [m] Widelane wavelength

Civil and Environmental Engineering and Geodetic Science Phase observable, although very accurate, must have an initial integer ambiguity resolved before it can be used for positioning. Any time we loose lock to the satellite or so called cycle slip happens, we need to re-establish the ambiguity value before we can continue with positioning! What is a cycle slip and what do we do to fix it? The ambiguity resolution algorithm is coming soon!

Civil and Environmental Engineering and Geodetic Science Cycle Slips Sudden jump in the carrier phase observable by an integer number of cycles All observations after CS are shifted by the same integer amount Due to signal blockage (trees, buildings, bridges) Receiver malfunction (due to severe ionospheric distortion, multipath or high dynamics that pushes the signal beyond the receiver’s bandwidth) Interference Jamming (intentional interference) Consequently, the new ambiguities must be found

Civil and Environmental Engineering and Geodetic Science time Initial ambiguity Phase observations with cycle slip at epoch t 0 t0t0 New ambiguity Cycle slips must be found and fixed before we can use the data (at the given epoch and beyond) for positioning

Civil and Environmental Engineering and Geodetic Science Cycle Slip Detection and Fixing Use ionosphere-only combination under normal conditions, ionosphere changes smoothly with time, so any abrupt changes in ionosphere-only combination indicates cycle slip Single, double or triple difference residuals can be tested Phase and range combination can also be used, however, this will not detect small cycle slips due to large noise on pseudorange Receivers try to resolve CS using extrapolation, flag the data with possible cycle slips

Civil and Environmental Engineering and Geodetic Science Cycle Slip Detection and Fixing Cycle slips can be located by comparing either one of the listed quantities between two consecutive epochs (jump occurs) Also, a time series of the testing quantity can be examined (1 st, 2 nd, 3 rd and 4 th differences of the series of testing quantity) To find the correct size of CS the curve fit to the testing quantity is performed before and after CS Shift between the curves indicates the cycle slip amount Kalman filter prediction can also be used (predicted value – observed value indicates the size of CS) The testing quantities are then corrected by adding the size of CS to all the subsequent quantities

Civil and Environmental Engineering and Geodetic Science

Cycle Slip Detection and Fixing, Final Solution A good method is to carry out a triple difference solution first Since only one TD is affected it can be treated as a blunder, and a least squares solution can still be obtained The residuals of converged TD solution indicate the size of cycle slips Before using DD for the final solution, DD should be corrected for CS Least squares solution with non-integer ambiguities (float solution) Fix ambiguities Final Least squares solution with integer ambiguities (constraints)

Civil and Environmental Engineering and Geodetic Science Differential Global Positioning System (DGPS) is a method of providing differential corrections to a GPS receiver in order to improve the accuracy of the navigation solution. DGPS corrections originate from a reference station at a known location. The receivers in these reference stations can estimate errors in the GPS observable because, unlike the general population of GPS receivers, they have an accurate knowledge of their position. As a result of applying DGPS corrections, the horizontal accuracy of the system can be improved from m (100m under SA), 95% of the time, to better than 1m (95% of the time). Differential GPS (DGPS) Services

Civil and Environmental Engineering and Geodetic Science Differential GPS (DGPS) Services There exists a reference station with a known location that can determine the range corrections (due to atmospheric, orbital and clock errors), and transmit them to the users equipped with proper radio modem. The DGPS reference station transmits pseudorange correction information for each satellite in view on a separate radio frequency carrier in real time. DGPS is normally limited to about 100 km separation between stations. Improves positioning with ranges by 100 times (to sub-meter level)

Civil and Environmental Engineering and Geodetic Science Some DGPS Services Starfix II OMNI-STAR (John E. Chance & Assoc, Inc.) U.S. Coast Guard Federal Aviation Administration GLOBAL SURVEYOR™ II NATIONAL, Natural Resources Canada Differential Global Positioning System (DGPS) Service, AMSA, Australia

Civil and Environmental Engineering and Geodetic Science Wide Area Differential GPS (WADGPS) Differential GPS operation over a wider region that employs a set of monitor stations spread out geographically, with a central control or monitor station. WADGPS uses geostationary satellites to transmit the corrections in real time (5-10 sec delay). For example: OMNISTAR, Differential Corrections Inc., WAAS (FAA-developed Wide Area Augmentation System)

Civil and Environmental Engineering and Geodetic Science Atmospheric layer A Schematic of the WAAS

Civil and Environmental Engineering and Geodetic Science The WAAS improves the accuracy, integrity, and availability of the basic GPS signals A WAAS-capable receiver can give you a position accuracy of better than three meters, 95 percent of the time This system should allow GPS to be used as a primary means of navigation for enroute travel and non-precision approaches in the U.S., as well as for Category I approaches to selected airports throughout the nation The wide area of coverage for this system includes the entire United States and some outlying areas such as Canada and Mexico. The Wide Area Augmentation System is currently under development and test prior to FAA certification for safety-of-flight applications. WAAS

Civil and Environmental Engineering and Geodetic Science Total correction estimation is accomplished by the use of one or more GPS Base Stations that measure the errors in GPS pseudo-ranges to all satellites in view, and generate corrections Subsequently, the corrections are sent to the users Thus, real-time DGPS always involves some type of wireless transmission system (one-way, i.e., the user does not send any info back) VHF systems for short ranges (FM Broadcast) low frequency transmitters for medium ranges (Beacons) geostationary satellites (OmniSTAR) for coverage of entire continents. So, we know how to communicate with DGPS (or WADGPS) services, but how does the system generate the actual corrections, and how do they get customized for the user’s location? WADGPS: operational aspects

Civil and Environmental Engineering and Geodetic Science A GPS base station tracks all GPS satellites that are in view at its location. Given the precise surveyed location of the base station antenna, and the location in space of all GPS satellites at any time from the ephemeris data (navigation message broadcast from all GPS satellites), an expected (or “theoretical”)range to each satellite can be computed for any time The difference between that computed range and the measured range is the range error If that information can quickly be transmitted to other nearby users, they can use those values as corrections to their own measured GPS ranges to the same satellites (DGPS) In case of WADGPS, the local base stations send their corrections to the master station that is responsible for the communication via the geostationary satellite Thus, the satellite would receive and disseminate a set of corrections coming from all the WADGPS network base stations WADGPS: operational aspects

Civil and Environmental Engineering and Geodetic Science How does the user get customized/optimized correction? For example, OmniSTAR user sets receive these packets of data from the satellite transponder (an exact duplicate of the data as it was generated at each base station) Next, the atmospheric errors must be corrected. Every base station automatically corrects for atmospheric errors at its location, because it is a part of the overall range error; but the user is likely to be not at any of those locations, so the corrections are not optimized for the user. Also, the OmniSTAR system has no information about each individual's location So, if these corrections are to be automatically optimized for each user's location, then it must be done in each user's Omnistar. WADGPS: operational aspects

Civil and Environmental Engineering and Geodetic Science For this reason, each OmniSTAR user set must be given an approximation of its location (from the GPS receiver being a part of OmniSTAR set) Given that information, the OmniSTAR user set can use a Model to compute and remove most of the atmospheric correction contained in satellite range corrections from each Base Station message, and substitute a correction for its own location. After the OmniSTAR processor has taken care of the atmospheric corrections, it then uses its location - versus the eleven base station locations, in an inverse distance-weighted least-squares solution. The output of that least-squares calculation is a synthesized Correction Message that is optimized for the user's location. This technique of optimizing the corrections for each user's location is called the Virtual Base Station Solution WADGPS: operational aspects

Civil and Environmental Engineering and Geodetic Science All WADGPS systems generate range and range rate correction The range correction is an absolute value, in meters, for a given satellite at a given time of day. The range-rate term is the rate that correction is changing, in meters per second. That allows GPS users to continue to use the "correction, plus the rate-of-change" for some period of time while waiting for a new message. In practice, OmniSTAR would allow about 12 seconds in the "age of correction" before the error from that term would cause a one-meter position error. OmniSTAR transmits a new correction message every two and one/half seconds, so even if an occasional message is missed, the user's "age of data" is still well below 12 seconds. WADGPS: operational aspects

Civil and Environmental Engineering and Geodetic Science

OmniSTAR's unique "Virtual Base Station" technology generates corrections optimized for the user's location. OmniSTAR receivers output both high quality RTCM-SC104 (Radio Technical Commission for Maritime Services) Version 2 corrections and differentially corrected Lat/Long in NMEA format (National Marine Electronics Association).

Civil and Environmental Engineering and Geodetic Science

OmniSTAR receiver

Civil and Environmental Engineering and Geodetic Science Radio Modems