Rigid Bridges Health Dynamic Monitoring Using 100 Hz GPS Single-Frequency and Accelerometers

This article presents the modal frequency recordings of a rigid bridge, monitored by the GPS receivers (Global Positioning System) with a data recording rate of 100 Hz and accelerometers. The GPS data processing was performed through the double-difference phase, using the adjusted interferometry technique (i.e. phase residue method—PRM  ). In the method, the double-difference phase of the carrier L1 is realized by using two satellites only, one was positioned at the zenith of the structure and the other satellite was positioned near the horizon. The results of the parametric adjustment of the PRM observations were finalized through software Interferometry, mathematical algorithm were applied and compared with the accelerometer. The comparison served to validate the use of GPS as a fast and reliable instrument for the preliminary monitoring of the dynamic behavior of the bridge, road artworks which are common in several countries, especially in the Brazilian road network. The data time series from the GPS and accelerometers were processed using the Wavelet. The detection of frequencies means that the combination of 100 Hz GPS receivers and the PRM allows detecting vibrations up to 5 mm. It presented significant results which were never obtained by the Fourier

Furthermore, due to the number of rigid bridges e.g. small and medium-sized reinforced concrete bridges, on which the Brazilian road network is notably constituted [5], it was necessary to continue the development of the previous method. Other reasons were to empower the modified methodology to meet monitoring requirements of rigid structures and to allow the engineer to have another tool and technique in hand, for the ease of installation, improvement in measurement speed and reliability in the results.

Methodology
The methodology of this work is divided into three different stages: 1) data collection using the PRM technique, 2) observable processing and 3) spectral analysis of GPS data.
The methodology presented in this paper is based on interferometer principle.
It uses the L1 carrier phase that needs to be collected from two GPS satellites via  [14].
To register the modal frequency of the structure, for example, it is necessary that one satellite should be close to zenith and another to be close to the horizon (reference satellite) ( Figure 1). In the processing of the double-difference (DD) phase, the lowest satellite is the reference satellite, allowing collection of residuals from the highest satellite, which is also called the zenith satellite. Adopting this configuration, there will be a greater contribution of double difference phase in the final data processing results. It will be due to the changes in the phase, i.e. the signal of the zenith's satellite in relation to the reference satellite, which hardly detects any movement of the antenna.
The DDPR is generated by atmosphere scintillation, satellite and receiver electronics noise, multipath, antennas phase center pattern, satellites orbital dynamics and any antenna movement ranging from millimeters to some centimeters. Fortunately, in this case, the double differenced time variations of each one of these effects are quite distinguishable, presenting a different frequency spectrum. The electronics and atmosphere presents a rapid random behavior as a function of the equivalent phase detector noise bandwidth; multipath in most scenarios is a slow time varying function; antennas phase center pattern and satellites dynamics can be neglected for a short time observations and, finally, the antennas movement contribution will depend on inner product between the movement vector and unit vector to the satellites direction. The movement behavior mostly is a periodic time function [1].
The residuals, from the parametric-adjusted double-difference, incorporate all phase position deviations calculated during the observation. These phase deviations are due to electronic receiver noise, multipath, small dynamic antenna movements and other error sources. By converting the residuals in the frequency domain, via Morlet Continuous Wavelet Transform, it is possible to see the different behaviors of the receiver phase noise, multipath, and periodic oscillations of bridge's span allowing the distinction between them, because these interferences are located at low frequency ranges. An important activity for obtaining the phase residuals from the raw data is to verify the data quality, by looking for cycle slips and missing epochs. This can be achieved by observing the data continuity of the chosen satellites. The presence of some sporadic epoch and any cycle slips that stays within the noise level will not compromise the results.
The L1 double difference phase observable is given by Leick (2004): The first term of the right side of Expression (2) The term S comprises of the stationary components and N is the random phase noise. The topocentric distance, between satellite p and receiver k as a function of time is given by the expression: Same expressions are for the other three distances. The double difference among topocentric distances, in a short baseline as a function of time can be represented by a polynomial function: In the Expression (6) the coefficient 0 a can be added to the steady satiate components S. The time behavior of observed double-difference can be fitted to the polynomial function. The residuals contain the time-dependent phase disturbances and random phase noise. Expression (7) gives the phase residuals, The polynomial fit can be done by parametric minimum least square method.
With this approximation, it is possible to obtain the phase residual directly from the raw data, independent of a regular data processing program, to be analyzed in the frequency domain by the CWT, as mentioned previously.
Millimeter or unstable periodic oscillations caused by movements of a large structure are difficult to separate from the random noise, which results in degrading the precision of the measurement. One-way to improve the signal to noise ratio is the use of auto-correlation technique. Autocorrelation enhances periodic functions and lessens random values. The autocorrelation of data of n samples is converted to a half time sample because the delay can only be shifted by half the original sample. Expression (8) presents the applied autocorrelation function with the delay (t) ranging from 0 to n/2.
Stage 2: processing of data by Interferometry  software.
For the data processing, the authors had difficulties to find scientific or commercial software that allowed applying the proposed methodology exclusively. Several softwares were tested. However, none was applicable for two main reasons, 1) the selection of reference satellite in the post-processing of GPS observations was not allowed by any software because it is not a commercial purpose and 2) the software does not process observations collected at 100 Hz. Since, GPS receivers were from the manufacturer called JAVAD, the authors chose to propose a scientific partnership with the manufacturer. The company accepted the invitation to develop software that would perform post-processing with the permission of selecting the reference satellite. After two years of combined efforts, sharing ideas, suggestions and testing, the authors of this work and the manufacturer's developer team were able to finalize the "Interferometry software"-used only by authors. It was developed for the sole purpose of meeting the demands of this research. The "Interferometry package", despite being available for free, can be found within the Justin  trading software platform of Javad manufacturer in versions 2016 onwards (Justin v.2.123.161.2). Additionally, it is possible to use open source software such as RTKLIB that can be modified to replace the post-processing reference satellite. In addition, 100 Hz data can also be processed (http://www.rtklib.com/).
In order to use the software, it is necessary that the user proceed in the same manner as the conventional software. It is essential to create project, assign coordinate system, import data, select antenna, set and assign coordinates of the reference station, view quality of GPS observables, assign cutting angle, and observation rate, among others. Nevertheless, the software simply processes and adjusts data by parametric mode only with the proposed methodology. The user has to use the commercial version of the software, if he attempts to use conventional post-processing to get adjusted coordinates or a geodetic traverse.   The analyses for detecting the frequency due to the small dynamic vibration were done by applying the Continuous Wavelet Transform algorithm-C-WT-with the Morlet Wavelet [16]. The selection of the best mother wavelet is not a simple task. Usually, there are more than a couple of alternatives [17]. This research aims to study the frequency in the time domain from GPS data that have the contribution of electronic noise and multipath. It is observed that Morlet Wavelet is most efficient at identifying the signs of the frequencies expected due to a signal with the amplitude variation in peak to peak up to 5 mm in the low frequency region. The first study developed by the authors, using CWT was published in 2009 [2]. Similar to that, a particular wavelet, Morlet is used, and is defined by Equation (9)  steps dt is defined as the convolution of ( ) f t with the complex combination of the scaled and normalized mother wavelet, see Equation (10): where ( ) , j k W t represents the similarity between Wavelet function and the analyzed time seriess ( ) f t , i.e., the higher the value of ( ) , j k W t , the greater the similarity between the analyzed function and mother wavelet function which modulates the signal analyzed.
The idea behind the CWT is to apply the wavelet as a band pass filter to the time seriess. It is important to note that the authors did several tests with the Fourier Transform (FT) and their variations, in order to analyze the random data originated by the bridge even though FT is not an appropriate method for analyzing random vibrations [2]. The reason that prompted the authors to use Wavelet was that FT represents spectral responses through peaks only, i.e., detected frequency (The peak is recorded), or undetected, the peak with amplitude close to the multipath spectral response threshold [3]. However, it is not indicative in what time or period of time, one or other frequency were recorded. This means that only one Wavelet has scaling (expansion/compression) and transla-

Bridge Characteristics
The bridge is located on the Jaguari River, on a portion of the Federal Highway  They have a width of 40 cm and a total height of 2.80 m.

Instrumentation Layout of the Structure
The instrumentation used: 1) a pair of GPS JAVAD Sigma receivers, with 100 Hz data rate, 2) choke ring antennas, model RegAnt_DD_E and 3) two K-Beam  accelerometers, AC10g and AC2g.

Selection of Jaguari Bridge
The Jaguari bridge has been monitored since 2011 by the team from the infra-  progress of this study can be observed in Araújo Neto et al. [21], Larocca et al. [3], Oliveira et al. [4].

Data Collected at Jaguari Bridge
The field study was focused on the central span of the Jaguari concrete bridge. It started at 9:10 am in July 2016 and a pair of GPS receivers with a 100 Hz recording rate was used. One of the receivers was installed on the bridge whereas the base receiver remained in the geodetic frame. In the same period, two axial accelerometers of high sensitivity were installed. Although the measurements were made by both accelerometers, data from the AC2g accelerometer will be presented in this work, considering that the AC10g accelerometer served as a safety measure, in case of failure of AC2g accelerometer.
Accelerometers AC2g, AC10g and radio data were installed on the parapet of the Jaguari bridge near the GPS antenna in a way that they did not interfere with the operation of the GPS antenna (Figure 4(B)). The data of the accelerometers was measured in the bands of ±2 g, ±10 g respectively, and the recording was performed on a Laptop with radio signal reception, at recording rate of 100 Hz.  in software were observed as well. Precise ephemeris was not necessary due to the short length of the base vector [19].
In order to test the proposed method, the observations were conducted under a controlled condition. The weather conditions were favorable, with an average temperature of 22˚C, wind speed of 18 km/h and 71% humidity. The traffic conditions were normal without having any restriction at traffic lanes. The software Interferometry v.1.0 was used for the post-processing, it allows to select the ref- erence and measurer satellites, as suggested by the proposed methodology. The residues used to apply CWT were from "measure" satellite PRN14.

Bridge's Frequency Due Vibration from GPS Receiver 100 Hz
In Figure 7, the graphical information of the time seriess, obtained through the residuals resulting from the adjustment of observations of the double phase difference L1, is shown. Although the observation time is around 12-minutes, wavelet graphs will be displayed in 5-minute intervals (30,000 observations).
This proposed interval will present sufficient information to represent the modal frequency of the monitored bridge and can be clearly represented on a scalogram ( Figure 8) [3]. Experience has shown that a graph with a data rate of 100 Hz masks the visualization of results without the zoom tool. In addition, the authors assert that the information contained in the remaining intervals maintains the pattern of the spectral response presented, without compromising the interpretation. Figure 7 shows the data of the residuals without any type of bandpass filter, through which it is possible to visualize the response peaks of the structure. However, it is not possible to identify the estimated frequency.
For a better representation of the modal frequency, it was essential to apply a filter (mathematical model) that presented the real response characteristic of the structure, with easy visualization and comprehension. Hence, it was necessary to    structure as well as moving objects, such as vehicles and people, would be potential sources of multipath. It is assumed that the other errors and interferences inherent to the GPS system are: tropospheric and ionospheric delay errors, orbit errors and clock errors, which were eliminated or significantly reduced with DD.

Bridge's Frequency Due Vibration from Axial Accelerometer
To match with GPS recording rate, the accelerometer data rate was down sampled at the 100 Hz level to be compatible with GPS data, even though the As shown in Figure 10, it is possible to identify the dominant frequency in the entire measured ranging from 4 Hz up to 8 Hz, which corresponds to the vertical vibration mode. It is worth mentioning that although the modal frequency of the structure presents a single value, by the spectral analysis of the signals obtained in the field, it is possible to verify a variable performance in the interval between 4 and 8 Hz. A magnification of the first minute of the image generated by the interval of 5-minute allows to clarify the spectral response of the structure due to the excitations caused by vehicular traffic on the Jaguari bridge and it means that it was possible to detected the vibrations close to 5 mm of rigid span that has 30 meters longer.
In Figure 10, the intensity of the spectral response of the vehicle or vehicle group is also observed, keeping in mind that the bridge has two lanes of traffic and two or more vehicles are constantly in transit. By the magnitude recorded in the range, bounded by the contour between 4 to 8 Hz, it is possible to verify the intensity of the spectral response in the intervals of each influence independently. This characteristic was also clearly observed in the presented methodology.
In this study, the spectral response of the concrete bridge was conferred, measured by the accelerometer, similarly, the results found by GPS are presented. In order to facilitate the comparison of the monitoring results obtained by the different forms presented in this work, all figures from CWT and the accelerometer data are generated from Morlet Continuous Wavelet Transform to identify modal-frequency from Jaguari Bridge. Moreover, in Figure 10, the level Figure 10. CWT of the Acceleration spectrum (frequency) from accelerometer with 5-min interval. The 5% statistical significance level of sine wave detection is shown as a red thick contour. Positioning of significant signal information is limited by the thick contour with 5% of significance and 95% confidence. The abscissa axis represents the number of observations (for each 0.01 s) and the left vertical axis represents the value of the frequency in hertz and the right vertical axis represents the energy intensity scale in which the frequency is displayed in the area of confidence

Conclusions
In this work, the results of the tests carried out at Jaguari Concrete Bridge showed that it is possible to detect and monitor modal frequency and structure behavior using GPS receivers, combined with Continuous Wavelet Transform algorithm (CWT) under PRM. Regarding the representativeness of the data measured by the GPS receiver, it is observed that the CWT is more functional in relation to the application of the Fourier transform, particularity in representing the frequencies existing in the time series in the same interval of time. It is concluded that bridge characteristics found by CWT would hardly be seen by the frequency graphs generated by the Fourier transform.
The values from double-difference phase residuals exactly correlate with the theoretical results of the frequencies presented by the axial accelerometer (conventional instrument). Both measurements were performed simultaneously for the first time for this structure and settled for a value between 4 and 8 Hz, which are bridge's natural frequency and its harmonics due to the traffic. Furthermore has its limits, when it is applied to meet engineering needs, multipath is still one of the major degradation sources for this system.
The use of modern GPS receivers with a high recording rate as 100 Hz, does not only limit the device's ability to record structure information, but also improves the quality of GPS observations with the help of the modern hardware and software used, as well as with sampling rate similar to the main accelerometers available in the market. GPS receivers also help in recording the acceleration of the structure, with greater quantity of samples per unit time of a signal, without error of "aliasing", according to Nyquist's Theorem.
In view of the results obtained, this research validates the capability of GPS At no time, the research presented here is based on the replacement of conventional instrumentation used in engineering structures. Rather, it enhances pioneer the potential in aggregate with existing tools. That is, making the GPS 100 Hz as the first instrument to be used in the monitoring of rigid bridges, in any climate or unfavorable condition, without the need for any calibration, with the only difference of allowing global positioning in a milli-meter manner.