Geodetics Preprint August 2000 Epoch-by-EpochÔ Positioning and Navigation

Paul J. de Jonge, Yehuda Bock, and Michael Bevis Geodetics, Inc. La Jolla, CA

BIOGRAPHY applications such as automated airborne landing systems, Paul de Jonge received his Ph.D. in geodesy at the Delft harbor navigation, automated highways, machine control, University of Technology in the Netherlands, and he and GPS/INS integration. Epoch-by-EpochÔ positioning currently works for Geodetics. Dr. Yehuda Bock is the also provides a useful architecture for static positioning, President and CEO of Geodetics. He is also a Research since relatively rare but highly discordant solutions (i.e. Geodesist and Senior Lecturer at the Scripps Institution of solution ‘outliers’ associated with extreme levels of Oceanography where he directs the Scripps Orbit and measurement noise) can be recognized and excluded from Permanent Array Center (SOPAC) and the California any position averaging. When single epoch solutions are Spatial Reference Center (CSRC). Dr. Michael Bevis is the averaged by a median filter, the scatter of the averaged Vice-President of Geodetics and a Professor of solutions is inversely proportional to the logarithm of Geophysics and Ge odesy at the University of Hawaii. averaging time.

ABSTRACT INTRODUCTION Geodetics, Inc. has developed a new class of Geodetics, Inc. has developed an innovative instantaneous, real-time GPS positioning algorithms based suite of software modules that implement Epoch-by- on dual frequency phase and pseudorange data, which we EpochÔ analysis of dual-frequency phase and range data call Epoch-by-EpochÔ positioning. The main advantage from two or more GPS receivers. The central feature of this of this approach over traditional real-time methods is that approach is that a high accuracy geodetic solution, based integer-cycle phase ambiguities are independently on instantaneous integer-cycle ambiguity resolution, is estimated for each and every observation epoch, allowing achieved for each measurement epoch using only the precise positions (1-2 cm horizontal and 5-10 cm vertical) observations collected at that epoch. Accordingly, each to be estimated given only a single epoch of data from two solution is independent of the solutions obtained for the or more receivers, in either baseline or network mode. Our previous and following epochs. Because Epoch-by- algorithms are extremely efficient – a single epoch on a EpochÔ analysis routinely resolves the complete set of single baseline can be processed in several milliseconds. double difference ambiguity parameters for each epoch, There is no need for the initialization period associated the technique delivers horizontal accuracies comparable to with conventional RTK algorithms. More importantly there those achieved in conventional kinematic and RTK is no need for reinitialization immediately following loss- processing. Our software can be used to process networks of-lock problems such as occur when a mobile GPS consisting of two or more GPS receivers, and allows receiver passes under a bridge or other obstruction. nearest-neighbor station spacings several times wider Accordingly, our approach minimizes the amount of time a than conventional RTK. receiver spends in a non-productive prepositioning mode Epoch-by-EpochÔ data processing is achieved of operations. Epoch-by-EpochÔ positioning has other by a core library of geodetic routines called the Rapid advantages, including a significantly extended range Network Analysis (RNA) library. All of the application relative to conventional RTK, and a better framework for software developed by Geodetics to date, and several quality control. The Epoch-by-EpochÔ approach can also applications currently under development, are built around be applied to the autonomous real-time determination of the proprietary RNA library. Our algorithms are extremely precise attitude and heading (about 0.05 degrees) of a efficient and allow the simultaneous, real-time network moving platform instrumented with 2-3 receivers and a positioning of at least 10 receivers sampling at a 1 Hz rigid multiple-antenna structure. A combination of this interval on a standard medium-scale PC workstation. A autonomous navigational capability with precise single epoch on a single baseline can be processed in only instantaneous (relative) positioning is ideal for many several milliseconds. Although Epoch-by-EpochÔ

Geodetics 2000 Geodetics Preprint August 2000 analysis is instantaneous, in the sense that it produces a CASE STUDY 1: VEHICLE TRACKING solution for each epoch or instant, without direct or Figures 1 and 2 show position and velocity indirect reference to data collected at any other time, it solutions obtained for a vehicle tracking test implemented provides a useful computational architecture for post- by Applanix Corporation on March 3, 2000 in Ontario, processing of multiple-epoch data batches as well as for Canada. The vehicle was a passenger van that traveled up real-time positioning. We illustrate some of these to 6 km away from a GPS base station at speeds of up to 70 applications in the following sections. km/hour. The GPS sampling interval was one second. All results shown in these figures are raw single-epoch KINEMATIC POSITIONING solutions. Figure 1 focuses on a subset of this test in Conventional RTK software partitions the integer which the vehicle made four circuits of the same C-shaped ambiguity problem into an initialization stage that typically section of road. We color code each of the four circuits so lasts 30-40 seconds. During this time phase ambiguities as to provide relative timing information in plots featuring are assessed using multiple epochs of data, and when the only spatial axes. This allows us to confirm that the two integer ambiguities are finally resolved with sufficient tracks apparent in the blow-up map (Fig. 1) correspond to confidence, initialization ends and actual positioning the two sides of the road, with both sides being traversed begins. Should a serious loss of satellite lock occur, as in all four circuits. The 3-D plot shows the first three visits might happen when a mobile GPS receiver passes under a to one end of the track, and one can clearly resolve the bridge, then once the hardware recovers and the GPS data manner in which the vehicle was turned around. Note that stream begins anew, it is necessary for the software to the vehicle arrives well-centered in the right lane, but reinitialize – that is, to return to the problem of resolving leaves the turn less well centered in the adjacent lane (at integer ambiguities as a preliminary to actual positioning. least for the first 40 meters). The total delay associated with loss of lock depends on The velocity solutions (Fig. 2) were obtained by the hardware as well as the software, since the receiver simple differencing of the position solutions, without any also has to reinitialize tracking after a loss of lock. The smoothing or editing. The velocity solutions are produced high performance, dual frequency GPS receivers available at a frequency of 1 Hz, but are offset by 0.5 seconds to civilians can recover from a very brief loss of lock in a relative to the position time series. Because they are second or two, but typically take 10-20 seconds to recover produced by differencing single-epoch positions, velocity from a sustained loss of lock. This hardware down time is time series are strongly sensitive to small errors in followed by additional down time associated with software position. We can detect an artifact in the vertical velocity initialization. Military dual frequency receivers, with direct (Vu) time series about 115 seconds into the test. access to the P code, can relock the satellites much more After completing the four circuits around the C- rapidly than non-classified receivers, with high shaped track, the vehicle stopped for 178 seconds. Both performance military units ‘cold starting’ in less than one the velocity solutions and the positions during this period second. are shown in the lower half of Figure 2. Note that the up Because Epoch-by-EpochÔ algorithms estimate coordinate has an order of magnitude more scatter than the integer ambiguities instantaneously, they completely does the north or east coordinate. In part this reflects the eliminate the software downtime associated with fact that we were estimating atmospheric parameters as initialization and reinitialization. Accordingly Geodetics’ part of each epoch analysis. A shorter static test took software cuts the total hardware plus software place at greater distance from the base station later in the initialization/reinitialization time to something very close to test, and the horizontal scatter associated with these the hardware initialization time. This leads to significantly solutions is shown in the lower left section of Figure 1. improved performance in civilian applications, and dramatically improved performance with military hardware. REAL-TIME ORIENTATION The second architectural advantage of Epoch-by- When three antennas are fixed to a rigid structure EpochÔ positioning is that a mobile receiver can be with spacings of order 1-2 meters, it is possible to positioned against a network of reference stations, and determine the orientation of the antenna array (i.e. its not just a single base station. This makes it easier to pitch, roll and heading angles) without reference to any implement wide-area RTK based on multiple reference external static base station. The solutions for the lengths stations. Using 3-5 reference stations will provide the of the inter-antenna vectors can be examined in real-time in infrastructure necessary to support RTK positioning order to monitor the integrity of the orientation throughout a major metropolitan region. Network-oriented determinations. Angular resolutions of about 0.05 degrees geodetic analysis also assists in system integrity are achieved using Geodetics’ Vector software, which is monitoring. based on RNA. Of course, when a static reference station is available, then it is also possible to solve for the

Geodetics 2000 Geodetics Preprint August 2000 position and velocity of the antenna array, thereby frequency solutions obtained by robust averaging in achieving a complete kinematic description of the system. successive (abutting but non-overlapping) time windows. The averaging is performed by computing the median of STATIC POSITIONING the north, east and up coordinates in each time window. For static positioning applications, such as Because these windows do not overlap, each of the surveying or deformation monitoring, it is advantageous averaged solutions is independent of its neighbors in the to collect and analyze multiple epochs of data, in order to same time series. mitigate positioning errors deriving from measurement A pair of semilog graphs show the relationship noise. Even in a multiple epoch framework, Epoch-by- between the scatter of the single epoch and multi-epoch EpochÔ positioning provides some significant solutions (obtained by averaging N single epoch advantages. The dominant source of error affecting dual solutions at intervals of (N-1)/2 minutes) and averaging frequency measurements is multipath noise. The level of time. Note that coordinate scatter varies in inverse multipath noise rises and falls with passing time, but is proportion to the logarithm of averaging time, as noted always present to some degree. The best means of previously by Bock et al. (2000). By averaging static suppressing multipath noise is to average the position solutions over longer periods of time, the precision of the solutions obtained over many epochs. The other class of resulting solution is improved but, since these solutions measurement error are high amplitude ‘glitches’ or outliers occur less frequently, this improvement comes at the cost associated with grossly problematical measurements, as of poorer temporal resolution. Of course it is possible to might occur, for example, when a receiver is subject to a set up an alarm system with a higher alarm threshold when burst of very high amplitude radio frequency interference. little or no averaging is taking place, and simultaneously While relatively rare in most settings, measurements consider one or more averaged time series associated with outliers can disrupt even position solutions obtained by more sensitive alarm levels. In this way one can “both averaging over multiple epochs. However, when solutions have one’s cake and eat it too”, and avoid the difficult are obtained on an instantaneous basis, it is relatively choice between detecting smaller levels of deformation simple to recognize badly discordant solutions, and simply and detecting larger motions more quickly. The lowest reject these solution outliers, and prevent them from frame in Figure 2 shows that monitoring a time series affecting the averaged position. Perhaps the simplest way produced by finding the median position of each station of averaging position coordinates over multiple epochs of every hour, should allow an alarm system to detect any data is by use of the median rather than the mean, as horizontal displacement greater than about 15 mm (about described by Bock et al. (2000). When the influence of 3s), without setting off significant numbers of false alarms. outliers is eliminated by robust averaging of the single Note that the ratio of vertical to horizontal scatter epoch solutions, the accuracy of the averaged solution in the single epoch solutions is much lower in this case tends to improve in proportion to the logarithm of total study than that seen in the vehicle tracking study. In large averaging time (Bock et al., 2000). Outliers can be part this is because atmospheric parameters were not identified and counted, and can be used as a means of estimated in this static analysis, though the low multipath monitoring the quality of the data being collected by a environment at the dam, and the use of choke ring CGPS network. antennas, may also have helped.

CASE STUDY 2 : DEFORMATION MONITORING ACKNOWLEDGMENTS The Metropolitan Water District of Southern We thank Applanix Corporation (Bruno Scherzinger) and California has constructed a continuous GPS network at the Metropolitan Water District of Southern California Diamond Va lley Lake reservoir to monitor the stability of (Michael Duffy and Cecilia Whitaker) for providing the its earthen dams. This network collects GPS observations data used in these case studies. with a sampling interval of 30 seconds. Figure 3 shows results we have obtained using Epoch-by-EpochÔ REFERENCE analysis for day 194 of this year (2000). These results were Bock, Y., R. Nikolaidis, P. J. de Jonge, and M. Bevis, 2000, obtained by solving for the geometry of the entire Instantaneous geodetic positioning at medium network, fixing just one station (the reference station distances with the Global Positioning System, WDR) to its prior coordinates. We show the raw single- Journal of Geophysical Research, in press. epoch results, and in several plots we superimpose lower

Geodetics 2000 Geodetics Preprint August 2000

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TURN-ABOUT # 2 (SEE MAP) U -0.4 STATIC VEHICLE TEST B (not in map) DISTANCE = 4.8 KM 40 SECS @1 Hz -0.6 0.83 -0.8 s = 3.3 mm N -1 0.82 -1.2

0.81 -1.4 760 TH - 4771 (METERS) s 750 E = 8.6 mm 5 0.8 740 0

NOR 0.12 0.13 0.14 0.15 730 -5 N 720 -10 EAST - 655 (METERS) -15 E 710 -20

Figure 1. Raw epoch-by-epoch solutions obtained using Geodetics RNA software from 1 Hertz dual- frequency GPS data collected during a vehicle tracking test. Data courtesy of Applanix Corporation. Geodetics 2000 Geodetics Preprint August 2000

VEHICLE TRACKING CONT'D VELOCITIES: North (Vn), East (Ve), Up (Vu), and total (V) 10 0 -10 Vn (m/s)

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V -5 700 720 740 760 780 800 820 840 860 time (seconds) 800 LOCATION HORIZONTAL SCATTER OF STATIC VERTICAL SCATTER about median position 600 TEST A 3 -140 th (m) NPTS = 178 NPTS = 178 nor 2 -145 400 -400 -200 0 1 east (m)

TH (CM) -150 Figure 2. Upper six frames show 0 NOR velocties, estimated by differencing UP (CM) D -155 the 1 Hertz position estimates, -1 during kinematic and static portions of the experiment. The two plots, right, -2 -160 show position repeatability when the -2 -1 0 1 0 10 20 30 vehicle was stopped. D EAST (CM) COUNT

Geodetics 2000 Geodetics Preprint August 2000

STATIC POSITIONING BASELINE: ESS - WDR Baseline length = 7.59 km 2565.80 2565.78

TH (M) 2565.76

NOR 2565.74 7145.02 7145.00

7144.98 EAST (M) 7144.96

-23.75 -23.80 -23.85 UP (M) -23.90 0 4 8 12 16 20 24 TIME (HRS) single epoch solns (ea. 30 secs) PRECISION VERSUS TEMPORAL RESOLUTION (AVERAGING TIME) 20-pt medians ( 9.5 min solns) 12 hourly WDNE 1.1 km 80-pt medians (39.5 min solns) 10 WDSE 1.7 km medians SADD 3.9 km 360-pt medians (3 hr solns) 8 ESS 7.6 km ontal

iz 6 30 single 4 epoch solns 20 2 RMS Hor Error(mm) HORIZONTAL 10 0 25 0 WDNE 1.1 km TH (MM) 20 WDSE 1.7 km SADD 3.9 km -10 tical 15 ESS 7.6 km er NOR V D -20 10

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Figure 3. Single-epoch and time-averaged positioning of the GPS deformation network at Diamond Valley Lake reservoir. This monitoring network incorporates Leica CRS-1000 GPS receivers with IGS-style choke rings. Atmospheric delays were not estimated in this analysis. Data courtesy of Metropolitan Water District of Southern California. Geodetics 2000