MODIFIED GPS-OTF ALGORITHMS FOR BRIDGE MONITORING: APPLICATION TO THE PIERRE-LAPORTE SUSPENSION BRIDGE IN QUEBEC CITY

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1 MODIFIED GPS-OTF ALGORITHMS FOR BRIDGE MOITORIG: APPLICATIO TO THE PIERRE-LAPORTE SUSPESIO BRIDGE I QUEBEC CIT Rock Santerre and Luc Lamoureux Centre de Recherche en Géomatique Université Laval Québec, Canada G1K 7P4 Abstract Algorithms for On-The-Fly ambiguity resolution have been modified for the deformation monitoring of a suspension bridge. Instantaneous relative positioning of a deformation network at a precision of about ±5 mm horizontally and ±1 mm vertically has been achieved using GPS L1 phase observations. Particular attention has been paid to the modeling of relative tropospheric delay and to the phase center calibration between antennas of different types. Moving averages on station coordinates have also been applied to reduce multipath effects. The Pierre-Laporte bridge is a 6-lane, 14-m-long suspension bridge which crosses the St. Lawrence river in Québec City. Three 48-hour GPS sessions have been conducted during the months of July and October 1996 and February For each session, 5 GPS receivers were observing at a data rate of 2 seconds. Daily variation in the vertical position of the antenna located at the deck center shows clear correlation with respect to temperature and vehicle loading. Transverse movement of the deck center has been monitored and correlated with (transverse) wind speed. Seasonal variation in temperature caused a vertical displacement of the deck center, a contraction of the towers as well as a displacement of the towers towards the river banks. Modified OTF algorithms In this study, a suspension bridge is seen as a kinematic deforming structure but with limited movement amplitude. The OTF (On-The-Fly) algorithms used to resolve GPS phase ambiguity, even if the receiver is in motion, have been modified for the particularity of a suspension bridge. The algorithms have been designed to work with single frequency receivers. The steps of the evaluation and the validation of L1 phase ambiguities and least-squares solutions are schematically presented in Figure 1 and described in the next paragraphs. To reduce the number of ambiguity combinations, good a priori coordinates must be available. Unlike dam deformation GPS surveys, one cannot use the coordinates determined from a previous session because the deformation of a suspension bridge (even the towers) can easily exceed 1 cm (half of a L1 wavelength) between 2 sessions. The coordinates obtained from filtered code solutions or from double difference float solutions, as used in pure kinematic mode, give too many ambiguity sets to be tested. A solution is to start with a static-like processing of a short data span. The time span must be long enough to provide a positioning precision of ±5 cm (one quarter of a L1 wavelength), but within this time period the amplitude of the movement of the bridge

2 should not exceed 5 cm. For bridge towers this criteria is always respected, but attention Start End A: A priori coordinates Test 5: on cycle slips ext epoch? E: Keep and display the L.S.A. solution B: A priori ambiguity Test 4: on ratio of variance factors Test 1: on ambiguity search range ext ambiguity set? ext satellite? must be paid to the deck center movement when there are rapid fluctuations of temperature, wind speed or traffic jam during this static session. For the other epochs, the coordinates of the previous epoch are used as a priori coordinates. With the a priori coordinates, double difference ambiguity is calculated, for each pair of satellites, starting from the first epoch of the observation session, with the following equation: Φ ρ dtrop = (1) λ If the decimal part of equation (1) departs by more than.25 cycle from the closest integer value (this can be due to multipath or observation noise) two integer ambiguity values associated to this pair of satellites will be tested. If not, only the closest integer value will be retained. ote that the ambiguities are not parameters to be estimated in the least-squares adjustment (LSA). In fact, different potential ambiguity combinations are tried and tested to isolate the correct ambiguity set among all possible candidates. Least-squares adjustment is performed with all ambiguity combinations and 3 criteria are used to identify the correct ambiguity set: - the correction to the a priori coordinates must be smaller than 1 cm (in absolute value); - the residuals must be smaller than 1 cm (in absolute value); - the ratio of the second smallest and the smallest a posteriori variance factors must be larger than 2. A test is also done to detect potential cycle slips between epochs. In this step, Doppler data at epochs i and i+1 are used to predict phase variation between epochs i and i+1, as follows: Di + Di+ 1 ( Φi+ 1 Φi) ( Ti+ 1 Ti) λ (2) 2 If the difference (in absolute value) is larger than one wavelength, the observation at epoch i+1 is flagged, a new ambiguity value is calculated with equation (1), and the validation steps are again checked. Equation (2) is very efficient when the data rate is high and the receiver is static or moving slowly between the observation epochs. To avoid mathematical correlation in double difference observations, the Hatch approach [Hatch, 199] has been used. With this approach the observation weight matrix is kept diagonal and the residuals are associated to each satellite, instead of each pair of satellites. This last feature is very important for a detailed analysis of remaining unmodeled errors. At every epoch (at each 2 seconds in our case), four unknown parameters are solved in a least-squares adjustment. C: L.S.A. for each ambiguity set Test 2: on L.S.A. correction terms Test 3: on individual residuals D: Store L.S.A. solution(s) Figure 1: Data flow for ambiguity resolution and LSA solution.

3 GPS error sources and modeling The main GPS errors to deal with are relative tropospheric delay, multipath and antenna phase center variation. The effect of other errors such as ionospheric refraction, ephemerides and Selective Availability (SA) can be assumed negligible for baselines as short as 1 km. The readers are referred to [Santerre, 1991] for a detailed analysis of GPS error propagation in GPS network. Tropospheric refraction: Because some baselines of the monitoring network have height differences of about 6 m, attention has been paid to the modeling of relative tropospheric delay. Meteorological data collected at the level of the bridge deck were extrapolated in altitude for the GPS stations located on the 2 bridge towers. The height extrapolation algorithms for temperature, pressure and relative humidity are described in [Rothacher et al., 1986] and used as input for the Hopfield s tropospheric model. Multipath: To avoid multipath as much as possible, antennas have been setup above any reflective objects at the reference stations and on the top of the 2 towers. At the deck center, the antenna was installed on a 4-m-high beam to avoid multipath (and obstructions) from the vehicles. However, the towers and suspension cables could be a source of multipath. A method to filter (short period) multipath effect is the use of the moving average of instantaneous coordinate results. The moving average window must be carefully selected, i.e., wide enough to reduce multipath effect but not too wide to avoid the loss of information about real bridge movement. Unfortunately, no chokering antennas were available to us during the observation sessions. Antenna phase center variation: Three types of antennas have been used, namely Ashtech Z-XII and LD-XII and ovatel 51. The relative phase centers of the antennas have been calibrated using a 1-m calibration beam. This aluminum beam, with precisely known length, was setup and leveled on a geodetic point and oriented (with an optical device) towards the direction of another geodetic point [Bourassa, 1994]. The comparison of the known baseline components of the calibration beam with the baseline components obtained from GPS observation sessions allows to determine relative phase center variation between pairs of GPS antennas. Four 24-hour observation sessions have been conducted for the relative phase center calibration of the 5 GPS antennas. The relative phase center offset never exceeded 2 mm in the 3 baseline components. These measured offsets are within the manufacturers technical specifications and are not significant, considering the precision associated to the calibration process. Accordingly, no further correction has been applied for the phase center variation. Before surveying the Pierre-Laporte bridge, the methodology has been tested on known baselines with a device which mimics realistic bridge displacements. The length of the baselines and the height differences were typical of the bridge monitoring network. For independent and precise comparisons, the baseline height differences have been determined by geometrical leveling (relative geoid undulations were also taken into account). This analysis showed that instantaneous GPS relative positioning has typically a precision of about ±5 mm, horizontally and about ±1 cm, vertically, when PDOP factor does not exceed a value of 6. Deformation monitoring network Figure 2 shows the stations of the deformation monitoring network. The baseline length (D) and the height difference ( h) between the stations are given in Table 1. Two reference stations (RI1 and RI2) have been setup in bedrock, on the north river bank close to the bridge. Stations TO and TOS are located on the top of the orth and South towers, respectively. Station TACE is located on the deck center of the bridge.

4 TO a TACE RI1 b d TOS RI2 c e f 1 m UL25 Figure 2: Deformation monitoring network of the Pierre-Laporte bridge. Table 1: Baseline length and height difference. Baseline a b c d e f D (km) h (m) To verify the stability of the 2 reference stations, they have been connected, by GPS observations and precise geometric leveling, to station UL25 located on the Laval University campus. This concrete pillar, located 4 km away from the bridge, is part of the GPS calibration network in Quebec City area, established by the Geodetic Surveys of Canada. Between the 3 sessions, the coordinates of stations RI1 and RI2, relative to UL25, did not change significantly. So, the same set of coordinates for the reference stations was used for all 3 sessions. The Pierre-Laporte bridge is a 6- lane, 14-m-long suspension bridge which crosses the St. Lawrence river in Québec City. The center span is 67-m-long supported by 2 11-m- towers. The bridge structure, of high a total weight of 18, tons, is suspended at 2 main cables of 62 cm in diameter. Such an engineering structure is affected by strong constraints caused by winds, traffic loading and temperature variations. Analysis of the movement of station TACE (deck center) Figure 3 illustrates the vertical displacement of the deck center (station TACE) and the temperature as a function of time for the July 1996 session. The epochs where the temperature is the highest roughly correspond to the lowest vertical positions of the deck center. A variation of temperature of 8 C has caused a vertical displacement of about 12 cm. Thermal dilatation of the suspension cables explains this movement of the deck center. The vibrations caused by vehicle traffic on the bridge are more apparent in Figure 4 which is an enlargement of Figure 3, between 13: and 14:3 (EDT). Moreover, the influence of vehicle loading is clearly detected in Figure 4. In fact, one can see the impact of a car accident leading to the entire blockage of 3 lanes of the bridge, for about 3 minutes. The center deck of the bridge, half-full with stopped vehicles, has been bent by about 4 cm during this time period. Figure 5 shows the transverse displacement of station TACE caused by the transverse wind speed. The wind speed was recorded on the bridge deck with an anemometer. The high correlation between these 2 quantities are clearly visible. For example, a transverse wind speed of 2 km/h (from the East direction) has caused a transverse displacement of the deck center of about 5 cm, in the West direction. In Figures 3 to 5, the gray line, superimposed on the raw displacement results, is the moving average applied onto the coordinates to reduce the impact of observation noise and multipath. The length of the moving average window is 6 seconds, in Figures 3 and 4, and 12 seconds in Figure 5.

5 Vertical displacement (m) Eastern Daylight Time 16: 22: 4: 1: 16: 22: 4: Temperature ( o C) Elapsed Time (hours) Figure 3: Vertical displacement of station TACE as a function of temperature (July 1996)..5 Eastern Daylight Time 13:15 13:3 13:45 14: 14:15 Vertical displacement (m) min. 4 cm -.5 3: 3:15 3:3 3:45 4: 4:15 4:3 Elapsed Time ( hours) Figure 4: Vertical displacement of station TACE due to traffic loading during a car accident. Transverse displacement (m) East West Eastern Daylight Time 17: 23: 5: 11: 17: 23: 5: Transverse wind speed (km/h) Elapsed Time (hours) Figure 5: Transverse displacement of station TACE as a function of wind speed (October 1996).

6 Analysis of the seasonal displacements of the bridge (towers and deck center) Table 2: Seasonal variations of the average coordinates of the bridge stations. variations between July and February sessions L: 4. cm station TO T: -1. cm V: -5.5 cm L: -3.8 cm station TOS T: -.6 cm V: -6.1 cm L:.2 cm station TACE T: -4.5 cm V: 45.5 cm temperature -25 C transverse 21 km/h wind speed L: longitudinal, T: transverse, V: vertical Table 2 presents the seasonal variations (between July 1996 and February 1997) of the average coordinates for each station. The analysis of these results shows that the 2 towers moved away from each other of 8 cm along the longitudinal axis of the bridge. The vertical variation of -6 cm at stations TO and TOS is a consequence of the thermal contraction of the 11-m-high towers. In fact, between the 2 sessions the average temperatures dropped by 25 C. The deck center (station TACE) rose of 46 cm between the 2 seasons. The thermal contraction of the suspension cables explains this movement. Finally, the average transverse displacement of station TACE (5 cm towards the West direction) is caused by the variation of the average transverse wind speed of 21 km/h between the 2 sessions. Conclusions and further research A methodology, algorithms and software have been developed for bridge deformation monitoring in post-processing. The precision of instantaneous relative positioning is about ±5 mm horizontally and ±1 mm vertically, when PDOP values are smaller than 6. The analysis of the displacements shows: i) a transverse movement of the deck center due to a transverse wind speed of about: -2 mm / 1 km/h; ii) a vertical displacement of the deck center with temperature variation of about: -2 cm / 1 C; iii) a vertical variation of the deck center correlated with traffic flow; iv) an elongation of the tower with respect to temperature variation of about: 2 mm / 1 C; and v) a displacement of the towers towards the river banks of about: -1.5 mm / 1 C. Other GPS sessions (with a higher GPS data rate) for studying the high frequency movements of the bridge are planned. Provisions are made to add more receivers with chokering antennas. The methodology will also be implemented for real-time deformation monitoring and applied to other engineering structures such as towers, skyscrapers and dams. Acknowledgments: We would like to thank the Quebec Transportation Department for their helps during the GPS campaigns, and the atural Sciences and Engineering Research Council of Canada and the Laval University Faculty of Forestry and Geomatics for their financial supports. References Bourassa, M. (1994). Étude des effets de la variation des centres de phase des antennes GPS. Mémoire de maîtrise, Département des sciences géomatiques, Université Laval, Québec, 19 p. Hatch, R. (199). Instantaneous Ambiguity Resolution. IAG Symposium o. 17, Kinematic Systems in Geodesy, Surveying, and Remote Sensing, Banff, Alberta, Canada, pp Rothacher, M. et al. (1986). The Swiss 1985 GPS campaign. Proceedings of the Fourth International Geodetic Symposium on Satellite Positioning, Austin, Texas, pp Santerre, R. (1991). Impact of GPS satellite sky distribution. Manuscripta Geodaetica, 16, pp