1 Introduction

A large part of bridges in Germany and Europe was built in 1950s–1970s. They are approaching a critical age and experiencing permanently increasing operational loads, while their condition is continuously deteriorating. Under such conditions, the highway administrations try to develop and establish cost-effective and safety-conscious bridge management systems. An existing bridge evaluation system in Germany prescribes regular inspections by trained specialists and allows the measurements within a SHM, if necessary. However, the goals and the extent of monitoring, suitable monitoring techniques and an integration of the SHM into the established safety evaluation procedures for bridges are currently rather a subject of development than a standard. The main reasons for a slow implementation of SHM in civil engineering are usually quite large dimensions, a very long service life and unique properties of each civil structure. The entire spectrum of monitoring methods can be conditionally subdivided into three groups that focus on static behavior, e.g., [1], real-time dynamic response of structures [2] as well as regular non-destructive testing of structural components in use, e.g., [3]. An overview of a wide variety of methods can be found in [4, 5].

The present work focuses on the static behavior of a highway bridge due to temperature effects. In this field, satellite-based synthetic aperture radar interferometry deformation measurements are considered to offer synergies to conventional measurement methods in the context of SHM. First of all, they are conducted remotely and are theoretically applicable to a set of structures in the entire area of interest at the same time. Satellite-based InSAR allows for the measurement of displacements on ground with millimetric precision and makes use of phase differences between two or more Synthetic Aperture Radar (SAR) satellite images [6]. Precise long-term monitoring is facilitated by Multi-Temporal InSAR stacking techniques, like the Persistent Scatterer Interferometry (PSI) [7, 8] or others e.g., [9,10,11]. These techniques are widely used for infrastructure and facility monitoring purposes like dams [12,13,14] or tailing site facilities in the mining sector [15]. Recently, with increasing relevance, the discussion on InSAR usability for bridge monitoring is evolving [16,17,18,19,20,21,22,23,24,25].

Technically, InSAR requires stable signal backscatter conditions during the observation period. Thus, no or only a small number of measurement points are found in areas with dense vegetation (forest) while the point density is high in urban areas with many man-made features. A density of 100,000 measurement points per square kilometer is typically exceeded using high-resolution satellite imagery. This high level of detail in combination with a large spatial coverage (depending on the satellite image footprint) provides added value to a broad range of applications. Another satellite-based measurement technique based on the global navigation satellite system GNSS shows equally high potential for monitoring static and even dynamic behavior of structures, like in [26].

Temperature-induced deformations of bridges are generally recognized to be a potential source of structural damage and a reason for monitoring [27]. In this context, the goal of the present study was to test the applicability of the InSAR technique for deformation monitoring of a large, curved highway bridge and to compare the obtained results with alternative measurement data of classical geodesy surveying and with the finite element simulation. Such a comparison of various measurement and simulation techniques promises an important insight into the accuracy, efficiency and limitations of the InSAR technique in the context of SHM.

2 Highway bridge

The skeleton of the Wehretal bridge (Fig. 1) as a part of the federal highway A44 was completed in 2019, but it was still not opened to traffic during the project time. This highway bridge is a prestressed concrete structure with a total length of approximately 670 m and a monolithic deck laying on a series of column pairs (Fig. 2). The structure bridges a gap between two tunnels over the valley of the stream Wehre. A special feature of the bridge is a relatively large curvature of the longitudinal axis with a constant radius of approximately 480 m. The bridge consists of two independent substructures, which correspond to two opposite traffic directions. The present study is dealing with the southern or inner part of the bridge that has the smaller radius of curvature.

Fig. 1
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View of the Wehretal bridge (left) and its geographic location (right)

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Fig. 2
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Geometry and support scheme of the bridge including pillar positions and support degrees-of-freedom

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The main load bearing component of the bridge is a 15-span continuous deck as a double T-beam (Fig. 3). This substructure also includes two abutments at each end of the bridge and 14 pairs of pillars arranged along two concentric support axes with constant radii of curvature of 482.2 m and 489.9 m, respectively. The transverse axes for each pillar pair are numbered consecutively, giving together with both abutments a total amount of 16 support positions, as shown in Fig. 2. The pillars are straight-lined members of reinforced concrete with a variable cross-section and a variable length between 6.85 and 17.35 m. The pillars are monolithically connected to the deck at support positions 4 to 7 and equipped with sliding bearings at all other supports which enable a longitudinal and transverse displacement of the deck with a certain friction. The details of the support configuration are given in Fig. 2.

Fig. 3
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Exemplary cross-section of the bridge deck at a position with sliding bearing (left) and monolithic joint (right)

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The 15 spans of the bridge have variable lengths from 29.2 to 67.5 m, as shown in Fig. 2. The cross-section of the deck (Fig. 3) includes two longitudinal girders with a constant height of 1.47 m and a slab with a constant width of 15.6 m. Only in the support area at positions 5 and 6, the girder cross-section has a haunch with the height of 3.27 m, as can be seen in Fig. 4b. Both girders are post-tensioned over the entire length of the bridge. Additionally, the deck is equipped with two sound protection walls with a height of 2 to 4 m.

Fig. 4
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Finite element model of the bridge: a isometric view of the finite element model, b view of the rendered finite element model at positions 4–6 with main constructional elements and corresponding finite element types, c view of the non-rendered finite element model at positions 4–6

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Due to its length and curvature, the bridge is sensitive to environmental temperature variations. It is designed for a maximum deformation up to 100 cm in the longitudinal direction and up to 5 cm in the transverse direction. The normal behavior of the deck and the pillars requires a flawless operation of the sliding bearings. Otherwise, an unscheduled loading of the deck and the pillars can be expected. Therefore, the 24 bearings can be identified as critical components of the bridge construction.

3 Monitoring goals

The main goal of the present SHM is to monitor static deformations of the bridge induced by the variation of environmental temperature and to evaluate the deformations with respect to critical states. In this context, thermal deformations were measured by two alternative methods: the satellite-based InSAR stacking technique and a traditional geodetic technique on site by means of a total station. In addition, the structural simulation by use of the finite element method should provide the overall picture of the bridge behavior under temperature variation and should serve as a basis to compare the independent measurement results. The outcomes of this work shall support the realization of efficient and reliable bridge monitoring systems on a larger scale in the future. In addition, vibration monitoring and dynamic response analysis are subject of the ongoing investigation, therefore, they are not considered in the present contribution.

4 Finite element simulation

4.1 Finite element model

The linear finite element model of the Wehretal bridge was developed using the ANSYS software (version 2020 R2). Figure 4a shows the whole model in isometric projection, Fig. 4b represents a fragment of the rendered model including real cross-sections of the structural elements and a denomination of the used elements. Figure 4c shows the same fragment of the non-rendered model. The whole model with a total of 300,000 degrees-of-freedom is quite large to be presented in one figure with all details. Therefore, the corresponding details on the main structural elements are given below.

Both abutments and the foundations of all pillars were assumed to be rigid, fixed bodies and, therefore, not modeled explicitly. The pillars were modeled by beam elements, taking into account their individual length and variable cross-section. The deck as a double T-beam was divided into two longitudinal girders and a roadway slab above them. The girders were modeled with beam elements considering their variable cross-sections in the vicinity of support positions 5 and 6. The roadway slab was modeled by shell elements with different thicknesses according to the true cross-sections. The prestressing tendons were modeled using truss elements with a stiff bond to the surrounding concrete. The sound protection walls and the pavement were considered as dead load.

A special attention was paid to the modeling of the sliding bearings since they influence the overall behavior of the bridge under temperature variations significantly. Each bearing includes the absolute displacement restrictions and the relative displacements between the sub- and superstructure, according to Fig. 2. The modeling of the sliding bearings is shown in Fig. 5. The friction that is present in the bearings was modeled using contact elements governed by the Coulomb friction model. The coefficient of friction \(\mu\) can be set separately for each of the 24 bearings in the FE model. Since the real coefficient of friction is generally unknown, certain assumptions were made in course of numerical simulations regarding \(\mu\) for each support position which is discussed in more detail in Sect. 7.1.

Fig. 5
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Modeling of sliding bearings using contact elements with friction μ and coupling of the vertical degrees-of-freedom: the bearing on the north side allows for the longitudinal displacement and no transverse; the bearing on the south side allows for both longitudinal and transverse displacement

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4.2 Simulation of temperature deformations

The bridge deformation under a combination of the dead load, prestressing and a unified temperature increment \(\Delta T\) for all parts of the structure was simulated by use of the linear elastic model. Since the real temperature distribution over the cross-section and also along the structure is unknown, the assumption of the same temperature increment \(\Delta T=30 ~ \mathrm{K}\) was utilized to study the exemplary deformation behavior. The coefficient of friction was set to \(\mu =0.05\) for all bearings. The longitudinal displacements \({u}_{\varphi }\) and transverse ones \({u}_{r}\) at all support positions 1 to 16 are calculated for the pillar heads and the deck separately. The simulation results shown in Fig. 6 allow for direct visualization of relative displacements \(\Delta u\) between the deck and the pillars.

Fig. 6
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Displacements of the deck and the pillar heads at all support positions under the combination of dead load, prestressing and total temperature increment \(\Delta T=30 ~ \mathrm{K}\); the unified coefficient of friction is set to \(\mu =0.05\): a longitudinal direction, b transverse direction

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The longitudinal displacements of the deck represented by the two girders under a constant temperature increment show a nearly linear evolution along the longitudinal axis that matches well the theoretical behavior. The so-called neutral point (NP) with the zero longitudinal displacement lies between the support positions 5 and 6. It is worth remembering that the deck and the pillars are fixed to each other at positions 4 to 7. Therefore, the pillar heads at these positions exhibit the same displacement as those of the girders. At every other position a relative displacement \({\Delta u}_{\varphi ,i}\) between the girder and the pillar head can be directly extracted from the diagram in Fig. 6a. The absolute displacements of the different pillar heads vary due to their different lengths and bending stiffnesses. The maximum transverse displacements of the deck are observed between the support positions 9 and 11 (Fig. 6b). At that, a relative displacement between the outer girder and the corresponding pillars is prevented by a special construction element (guide rail), while the inner girder can glide over the sliding bearing in transverse direction. This behavior can be directly seen as a difference \({\Delta u}_{r ,i}\) for supports 1–3 and 8–16 in Fig. 6b.

Since no temperature data were available directly on the bridge construction, the temperature measurements from the closest weather station of the German weather service (Deutscher Wetterdienst, DWD) were used. This weather station is located at about 12 km air distance from the bridge. The measurements refer to the air temperature at a height of 2.50 m over ground and take place every 10 min. Figure 7 shows the corresponding air temperature measured during the entire observation period of the present study, as well as the corresponding observation periods from the satellite and geodetic measurements. Both the seasonal and the daily temperature fluctuations can be observed directly. Temperature changes over the season can reach magnitudes of up to \(50 ~ {\text{K}}\). The short-term temperature difference within a few days between day and night is sometimes up to \(30 ~ {\text{K}}\). For the present study, the temperature difference of \(30 ~ {\text{K}}\) is considered representative to study the thermal deformation of the bridge.

Fig. 7
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Temperature measurements taken from the closest weather station for the entire observation period (source: Deutscher Wetterdienst DWD)

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5 InSAR measurements

5.1 Measurement technique

Spaceborne SAR sensors are actively transmitting high-energetic pulses with a frequency from the microwave region of the electromagnetic spectrum in a side-looking geometry (Line-of-Sight, LOS, Fig. 8a and b) and receiving the echoes of the backscattering signals. By revisiting a given sensor position within a determined time interval, the difference in the distance measurements (Fig. 8c), in particular the comparison of the phase information acquired during the two overflights, allows for detection and monitoring of displacements at the earth’s surface with a sensitivity in the millimeter range. A prerequisite is precise orbit information of the satellite. Displacement time series can be derived from a set of SAR images acquired in the same viewing geometry and the usage of an InSAR stacking technique. [24] provides well described introductory details on InSAR theory and accuracy. For the InSAR PSI analysis of the Wehretal bridge, the technique was used, which is implemented in the commercial SARscape software package (Version 5.6.1) provided by SARMAP. S.A.

Fig. 8
figure 8

© DLR e.V. 2019–2021 and Airbus Defence and Space 2021

a Schemes for descending orbit geometry and viewing direction, b satellite viewing geometry and looking angle, c interferometric distance measurement principle, d mean amplitude image of the descending TerraSAR-X images showing the Wehretal bridge

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The correlation of the backscattered signals between pixels of different image acquisitions has a decisive influence on a successful InSAR analysis. It is expressed as a coherence parameter and ranges theoretically from 0 to 1, i.e., from total decorrelation to complete correlation. Several independent sources of signal decorrelation effects contribute to the coherence value, amongst others temporal decorrelation [28]. Especially for monitoring purposes, temporal signal decorrelation, i.e., the change of the physical backscattering characteristics, can be crucial. In natural terrains, seasonal variations causing phenological evolution and soil moisture change affect signal correlation in time, and in urban areas with many artificial reflectors, construction work is a main source of signal decorrelation. The coherence values of considered measurement points are used as quality measure and thresholding is typically applied for point selection purposes.

The PSI method (simplified workflow is presented in Fig. 9) is described in detail in [7]. Initially, a set of SAR satellite images is co-registered to one reference image, ideally temporally centered in the data stack to reduce temporal signal decorrelation effects. Thus, all images are connected to the reference image. A reference image at the very beginning of the data stack, for example, may contain signal reflectors that are not available at the end of the monitoring period. The corresponding interferograms are computed by pixelwise multiplication of the complex signal from one image with the complex conjugate of the other. The resulting interferometric phase, computed for each pixel, represents the phase difference between the two image acquisitions and the respective acquisition dates.

Fig. 9
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Simplified PSI processing workflow

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The PSI method applied in this study identifies measurement points, which are temporally persistent scatterers (PS). PS points are distinct radar reflectors visible in the SAR satellite image with a stable radar reflection characteristic in all satellite scenes of the input image stack and during the complete monitoring period. The identification of PS points is based on amplitude and phase stability analyses conducted in a selection process. The interferometric phase of the PS is driven by the actual displacement, but also by topographic impact, atmospheric runtime delays, satellite orbit inaccuracies and other “noise” sources. With focusing on surface displacements, all other phase contributing factors need to be separated and removed. For the removal of the topographic phase contribution, for example, a Digital Elevation Model (DEM) can be used by converting the elevation information it contains into phase values and thus subtracting it. In this study, the high-resolution and high-quality WorldDEM™ [29] from Airbus was fused with local LiDAR measurement data of the Wehretal bridge that was collected from AllTerra in October 2019. This combination updates the WorldDEM™ dataset with the elevation information of the newly built bridge under consideration.

Several impacts, like the orientation of the bridge relatively to the sensor viewing geometry, ground material and design of the bridge, as well as wavelength and spatial resolution of the SAR-sensor influence the number of PSs, which are detectable. The quality of the analysis is driven, amongst other factors, by the number and the spatial distribution of PSs on the bridge.

With data acquired by the satellite from one viewing direction only, the measured one-dimensional (1D) LOS movements comprise vertical as well as horizontal (east–west) motion components. Horizontal motion components in North–South direction cannot be measured due to the polar orbit design (Fig. 8a) of the SAR satellites and consequently a very low measurement sensitivity to those motion components. A decomposition into the two-dimensional (2D) components — vertical and east–west — is possible by using datasets from two different viewing geometries and a suitable amount of PS (overlapping and large number) for the combined gridding.

5.2 Data and measurement results

In the present study, high-resolution spotlight (HS) images of the TerraSAR-X (TSX) satellite mission with precise science orbit information were used for the InSAR time series measurements. They were acquired from ascending and descending orbits starting in March 2019 with acquisition intervals of 22 days and 11 days, respectively. TSX operates at X-band (3.1 cm signal wavelength). HS images cover an area of about 10 km \(\times\) 5 km (range \(\times\) azimuth) with a spatial resolution of approximately 1 m \(\times\) 1 m. To ensure a high number of PS, an appropriate monitoring period for InSAR analysis must be selected. Following impacts were evaluated to find the most suitable set-up:

  • First, the bridge was under construction during the study period. Surface construction went on and sound protection walls were erected at both sides of the bridge substructures. As the detection of PS requires a stable signal in time (resulting in high coherence), and preliminary analyses identified the sound protection walls as highly potential source for measurement points, their completion by end of October 2019 was awaited. In contrast, the road surface on the bridge does not contain PS as its smooth surface results in a specular reflection of the radar signal. Consequently, almost no signal is backscattered towards the SAR antenna;

  • Second, snowfall in winter 2021 caused changes in signal characteristics for four images in a row (mid of January until beginning of March 2021) resulting in low radar signal coherence. The exclusion of the respective images from the InSAR analysis caused a temporal data gap of 55 days during the observation period;

  • Third, the applied PSI algorithm uses a linear temporal model for motion derivation. Challenges occur in case of significant nonlinear movements in time by the PS. Simulation of deformation with respect to the temperature change show strong deformations in longitudinal direction, especially for the eastern part of the bridge (Fig. 6a). Based on these simulations, it has been decided to run the InSAR analysis not on the complete observation interval (November 2019 until January 2021), but rather to split the data set into two, considering an approximately linear temperature trend during these sub-intervals.

As a consequence, only the data of the descending orbit with a shorter acquisition sampling interval—the larger and denser satellite data stack—allowed for a reliable InSAR analysis. The mean amplitude image subset of the Wehretal bridge of the descending data stack is presented in Fig. 8d.

Two sub-periods of observation were selected and independently processed: a winter–summer period including the satellite images from 27th November 2019 until 15th July 2020 and a summer–winter period covering the data from 12th June 2020 until 7th January 2021 (Table 1, Fig. 7). The temporal overlap between both results forms a prerequisite of the PSI technique to use a minimum number of input images. The LOS movement for each sub-interval has been derived for those PS exceeding a temporal coherence threshold of 0.75. Higher values are being better suited for the displacement estimation [30].

Table 1 Acquisition dates of the TSX HS images used for InSAR analysis
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The mean LOS velocity (in mm/year) for the 2 periods is presented color-coded in Fig. 10 and overlays the mean amplitude image of the respective period of analysis. Obviously, the eastern part of the bridge is underrepresented regarding measuring points. All PSs in this section of the bridge have a temporal coherence smaller than 0.65, and hence, are below the chosen threshold and are not shown. For the western part of the bridge, mainly oriented in east–west direction, measurement points were detected predominantly between the supports 4 and 8. The coloring in Fig. 10 represents the seasonal effect of expansion and contraction with a stable section represented by green color. At that, blue color shows motion towards the SAR-sensor and red color shows motion away from the sensor. Besides the bridge under investigation, the InSAR results also show subsidence at an area close to the south-eastern end of the bridge.

Fig. 10
figure 10

© DLR e.V. 2019–2021 and Airbus Defence and Space 2021

Line-of-Sight movement velocity (mm/year) for the winter–summer period (27/11/2019–15/07/2020) and for the summer–winter period (12/06/2020–07/01/2021) based on TerraSAR-X High-resolution Spotlight images. Background images are the corresponding mean amplitude images. Traffic directions are indicated with arrows. Color range is ± 25 mm/year,

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To compare the measurements of the various monitoring techniques and the FE simulation, the derived 1D-LOS displacement values of each PS belonging to the bridge structure were transformed first into horizontal direction and finally into bridge longitudinal direction. The transformation was done under the assumption that no vertical movement component occurs. Although the bridge design and the FE simulations suggest predominantly horizontal motions, potential settlements from ground compaction might be an error source.

According to the acquisition geometry of the radar satellite, there is nearly no sensitivity for motions in north–south direction. Since the bridge-to-SAR-sensor orientation changes continuously in longitudinal direction (curved structure), the eastern part of the bridge runs approximately north–south. The longitudinal movements for this section would not have been accurately derivable with InSAR even if the PS would have high temporal coherence values.

6 Traditional geodetic monitoring

Complementary to the satellite-based InSAR measurements, classical geodetic measurements of point displacements on the bridge by use of an automated total station were executed as well. A typical accuracy of the displacement measurements is declared to be of about 1–2 mm. This widely used technique in civil engineering is considered as a reference for comparison, although its accuracy is also subject to several influence factors like weather and sight conditions, etc. A general description of the geodetic surveying technique is given, for example, in the handbook [31] and in [32].

6.1 Monitoring system

The geodetic monitoring system was developed for measuring static and thermal deformations. Several reflection prisms were put on the structure and served as measurement points. They were installed in each transverse axis between the positions 6 and 16, except number 15. An automated Trimble S9 total station with an autonomous energy supply was placed in the field within a sight distance from the southern side of the bridge (Fig. 11a and b). The reflection prisms were installed at three positions in each transverse axis: pillar foot, pillar head and girder (Fig. 11c), excluding the abutment at axis 16 with only two prisms (girder and abutment head). The total number of measurement points amounts to 29. The system is able to measure all three displacements (x,y,z) for each reflector point. Thus, the calculation of the absolute and relative spatial displacements, for example, between girder and pillar head is possible. The entire observation covers the period from December 9th, 2020 to February 28th, 2022. At that, the system provided measurement data at time intervals of around 45 min. It is worth noting that the measured displacement values are not absolute but relative to the time instant of the operation start of the monitoring system and to the corresponding deformation state of the bridge at this time.

Fig. 11
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Arrangement of the geodetic monitoring system: a total station, b bridge ground plan with positions of the total station and the observation points (reflectors), c typical position of reflectors in each transverse axis of the bridge

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6.2 Measurement results

As mentioned above, the measurement data contain three spatial displacement values \({u}_{x},{u}_{y},{u}_{z}\) at 29 measurement locations every 45 min. Some of these measurement sequences were used for calibration purposes. For example, the pillar feet experience negligible displacements under normal conditions in general and, thus, can be considered as conditionally stationary. Transverse and vertical displacements of the pillars and girders are also minimal, compared to those in the longitudinal direction. Therefore, they are irrelevant for the present study. The focus was set on the longitudinal deformation of the bridge due to temperature fluctuations. In that context, the longitudinal displacements \({u}_{\varphi }\) at the pillar heads and the corresponding girders (Fig. 11c) were monitored and evaluated.

Figure 12 shows the longitudinal displacements of the girder at positions 6, 13, and 16 over a period of 55 days with the corresponding temperature measurements from the weather station. The displacement values refer to the initial state of deformation at the measurement start on December 9th, 2020. The daily oscillations of displacements can be clearly seen in all three graphs as well as their correlation with the temperature. Solely, the longitudinal displacements at position 6 show quite small oscillations, since the girders at this position are fixed to the pillars. The displacement is thus limited by the bending stiffness of the pillar.

Fig. 12
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Exemplary sequence of measured longitudinal displacements uφ of the girder in relation to the measured environmental temperature

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Figure 13 shows the temperature \(T\) and the longitudinal displacement \({u}_{\varphi ,16}\) over a period of 24 h. The already mentioned correlation between the measured displacement and the temperature can be clearly seen, as well as a certain time delay of approximately 3 to 4 h between the characteristic points of the two graphs. Based on this observation, it can be concluded that the deformation of the bridge depends not only on the instantaneous temperature, but rather on the temperature history of the past hours. The same effect can be observed with respect to the seasonal temperature history. For instance, the deformation state of the bridge at the same environmental temperature differs during the summer or winter observation periods.

Fig. 13
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Monitored longitudinal girder displacements uφ at position 16 and temperature measurements T during the day of 17/06/21

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7 Comparison of measurement results

The present study offers a rare and important opportunity to compare various monitoring techniques on a real bridge with each other and with a computer simulation. It provides a useful insight into the efficiency and limitations of the considered approaches and their applicability to SHM. Total station monitoring is an established standard measurement technique in civil engineering. Therefore, the corresponding measurement data is considered rather as a reference although it is also subject to uncertainties due to weather and sight conditions. In comparison, satellite-based InSAR measurements have not yet become an established standard for bridge monitoring purposes, although the method has been successfully applied in several types of civil structure monitoring. A critical point is the question whether the processing and interpretation of the measurements require some pre-knowledge about the individual structure under observation. The answer to this question will necessarily influence the chances for a “blind” monitoring of multiple structures in a whole region or along highways. Another open question is about the value of the InSAR data for SHM and its potential combinations with other monitoring techniques. Last but not least is also the question on the accuracy and uncertainty of the measurement data or its independent validation. Therefore, the present study includes comprehensive finite element simulations of structural behavior under temperature fluctuations to provide a pre-knowledge and a basis for comparison of alternative measurement techniques. Evidently, any finite element model cannot reflect all realistic conditions and is also subject to model uncertainties. The quantification of the involved uncertainties is, however, out of the scope of the present article. The focus is put rather on the qualitative comparison of the techniques and their results.

7.1 Difficulties, assumptions and solutions

There are several objective conditions that make a comparison of the applied measurement techniques challenging, among others:

  • strongly different sampling rates and time scales for the measurements;

  • generally different measurement points for different techniques;

  • different sensitivity to spatial orientation of the structure and its environment.

Specific properties of the structure itself like the unknown coefficient of friction \(\mu\) in the bearings or the temperature distribution over the cross-section and along the bridge due to sunshine are additional difficulties regarding the interpretation of the obtained results.

7.1.1 Sampling rate

The InSAR measurements were processed with a sampling rate of 11 days at a specific time of day of 5:35 am at the location of the Wehretal bridge. Thus, the intra-day fluctuation of the temperature induced deformation of the bridge cannot be observed. Instead, InSAR is sensitive to seasonal deformation changes. The total station measurements are conducted in the present case every 45 min. That is enough to catch the daily and the seasonal variations at specified points. But even the selection of the daytime for the seasonal comparison is a matter of compromise since the time instants of both techniques do not match. In the present case, the total station measurement at 6:00 am was selected for comparison. No additional interpolation of the total station results between two time instants has been done.

7.1.2 Measurement points

The measurement points on the structure for the total station are selected for the monitoring purposes as shown in Fig. 11b and c. The measurement points of the InSAR technique (PS) cannot be selected in advance, except by installing so-called corner reflectors that are adjusted towards the satellite’s viewing geometry. In general, such reflectors are usually installed in cases where no measurement points are expected to be found or at specific critical locations of interest. Independently from this, a PS represents a strong and dominant point scatterer within a resolution cell that has stable radar signal reflection towards the sensor over time. This reflection can come from different elements of the bridge itself or from objects on and around the bridge that are in view of the sensor. Obviously, the top surfaces of the deck and the sound protection walls could be potential sources of reflection. Thus, the girder and the pillar deformation measured by the total station underneath the deck cannot be caught by the InSAR technique. However, due to the monolithic connection of the slab and the girder, the corresponding longitudinal deformation of them both can be considered as identical. The element joints within the sound protection walls are designed in such a way that the wall can follow the deformation of the deck. Thus, the radar reflections from the walls can also be considered to be identical to those of the girder. All detected PS with a temporal coherence threshold larger than 0.75 at the bridge are involved into comparison.

7.1.3 Bridge and displacement orientation

The total station is able to determine all three spatial displacements of any measurement point. The InSAR technique provides the distance change between the satellite and the object along the LOS. For a better interpretation of the InSAR results, it is of advantage when the spatial displacements are decomposed into the longitudinal component \({u}_{\varphi }\) and the transversal component \({u}_{r}\) related to the bridge axes at any point, which was also done for the present study.

However, as the InSAR technique is constrained by the right-looking viewing geometry and the near-polar orbiting flight direction of the satellite, there is nearly no sensitivity for north–south directed movements as compared to east–west and vertical movements. The bridge longitudinal axis changes its geographic orientation continuously, as can be seen in Fig. 2. The best results with InSAR can then be achieved for the spans between the supports 1 to 7 (approximately east–west direction). However, the deck is monolithically connected to the pillars between supports 4 to 7 and the thermal deformations over there are generally quite small. The displacements between supports 8 and 16 are expected to be significantly larger, but the bridge orientation there turns to the north–south direction. The longitudinal displacement cannot be detected at this part of the bridge using InSAR due to the mentioned low north–south sensitivity. Unfortunately, just this part of the bridge could be best monitored by the total station due to sight conditions in the field close to the bridge. The experience in the present study shows that such specific difficulties should be considered in advance.

7.1.4 Temperature distribution

Because of different exposure with respect to the sun (sunshine and shadow areas), various components of the bridge construction will have different temperatures. This fact influences the real deformation. Since no temperature sensors were installed on the bridge, the same temperature was assumed for all components, that is taken to be equal to the temperature measured by the weather station. Similar difficulties can always be expected to appear in monitoring, as the amount of the temperature sensors will usually be unable to satisfy all specific demands. It is expected that the influence of the environmental temperature is larger than that of the local surface temperature variation due to the sunshine.

7.1.5 Friction of the bearings

The unknown value of the coefficient of friction that must be assumed within in the FE model has a significant impact on the results of the numerical simulation. It is therefore another uncertain parameter regarding the comparison between the measured and the simulated temperature deformation of the structure.

Interestingly, the geodetic monitoring system enables to estimate the true value of the friction coefficient μ from the measured displacement of every pillar head. It was observed that each pillar head experiences a certain maximum value of deflection under increasing temperature. Any further temperature increase does not affect this value since the maximum friction force is already achieved. This friction force and thus the coefficient of friction μ can be then calculated from the measured displacement and the known bending stiffness of the pillar as inverse mathematical problem. In the present study, the real coefficient of friction was determined to be \(\mu \approx 0.012\) for all bearings of the inner longitudinal axis. This experimentally determined value agrees very well with the information given by design codes for bearings of this type. It is therefore assumed within the FE model. The bearings of the outer longitudinal axis possess a guide rail that prevents relative displacements between the pillar heads and the girder in transverse direction. Since there are no measurements available at the outer longitudinal axis, the value of \(\mu =0.07\) given by the mentioned design codes was taken. This combination of \(\mu =0.012\) and \(\mu =0.07\) for the inner and outer longitudinal axis, respectively, is then used for all subsequent finite element simulations.

7.1.6 Observation period

Logistical difficulties in on-site instrumentation during the COVID-19 pandemic resulted in the geodetic measurement data being only shortly overlapping with the InSAR satellite measurements. The jointly covered observation period is very small (Fig. 7). A direct comparison of results for the same time is quite limited. Therefore, the following approach was chosen to get a meaningful indirect comparison of measurements and simulations. Based on the derived InSAR results for the two selected investigation periods 11/2019–07/2020 (winter–summer) and 06/2020–01/2021 (summer–winter), three representative intervals were determined for each of them as listed in Table 2. The first interval is equal to the entire observation period. The second one represents the maximum absolute value of temperature fluctuation \(\Delta T\) and the third one was arbitrarily selected. The last column in Table 2 shows the difference of the temperature value between the start and the end date of the corresponding interval.

Table 2 Selection of characteristic observation intervals within the InSAR measurements
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For each of the 6 periods listed in Table 2, suitable intervals within the geodetic measurements were selected for comparison. These selected intervals exhibit the same temperature fluctuation \(\Delta T\) as the respective InSAR intervals and also have a minimum duration of 2 weeks to exclude daily fluctuation from this consideration. Since multiple geodetic observation intervals fulfill these conditions, they were all included and processed separately. The corresponding displacement fluctuation values of each of these intervals were used to determine the mean value, which serves as a reference for comparison with the InSAR measurements. Finally, the numeric simulation was performed for each of the six characteristic values of \(\Delta T\) using the described FE model.

Despite all challenges mentioned above, the presented approach allows to reasonably compare the results of two alternative measurement techniques and the numerical simulations. The main idea behind is based on the fact that thermal deformations of the deck evolve linearly along the longitudinal axis of the bridge, as can be seen from the two lines in Fig. 6a. Thus, a linear regression of all measured displacements of the superstructure at all positions in each data set should deliver a straight line over the lengths of the bridge similar to those of Fig. 6a. Ideally, the slope of the respective regression lines should be the same for the alternative measurements and the numerical simulation. This idea is implemented below.

7.2 Results

In fact, the comparison of numerical simulations for a certain temperature increment \(\Delta T\) with the linear regression curves for the InSAR and the geodetic measurement data provides a meaningful result, as shown in Figs. 14 and 15.

Fig. 14
figure 14

Comparison of the geodetic measurements, the InSAR measurements and the simulations of the longitudinal displacements of the superstructure along the longitudinal axis for the observation period winter–summer, number 2 according to Table 2

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Fig. 15
figure 15

Comparison of the geodetic measurements, the InSAR measurements and the simulations of the longitudinal displacements of the superstructure along the longitudinal axis for the observation period summer–winter, number 2 according to Table 2

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Both cases depicted therein correspond to the maximum temperature fluctuation of \(\Delta T=\pm ~ 20.2 ~ {\text{K}}\) for the respective intervals 2 according to Table 2. A good correlation between the three results can be observed. The regression line for the total station measurements lies quite close to the line obtained by simulation. The regression line of the InSAR data shows a slightly different slope that is still close to that of the geodetic measurements and simulation. The available measurement points of the InSAR data set are shown in both figures to illustrate their scatter and limitation to the spans between supports 4 and 8. These point clouds, especially in Fig. 15, exhibit a nonlinear evolution of the displacements measured by InSAR along the bridge. Regardless of the reason for this effect, every subset of the selected points can produce a linear regression line with a certain deviation. Nevertheless, the obtained results show a quite good qualitative agreement of three independent assessment approaches.

The individual results can be compared quantitatively by means of the slope \(S\) of the regression line and the location of the neutral point marking the zero longitudinal displacement as shown in Fig. 6a. This position corresponds to the zero crossing \({x}_{0}\) of the regression lines and is recalculated in meters as the distance from the left end of the bridge (arch length). Taking the simulation result as reference, a relative error \(E\) for the measurements was then calculated for the slope S and the neutral point position \({x}_{0}\) as follows:

$$E=\frac{\left| {{\text{y}}}_{{\text{FEM}}}-{{\text{y}}}_{{\text{MEAS}}} \right|}{\left| {{\text{y}}}_{{\text{FEM}}} \right|} \cdot 100 ~ \%$$
(1)

with \({{\text{y}}}_{{\text{FEM}}}\): \(S\) or \({x}_{0}\) respectively from the finite element simulation, \({{\text{y}}}_{{\text{MEAS}}}\): \(S\) or \({x}_{0}\) respectively from the corresponding measurement.

The obtained results for all observation intervals listed in Table 2 are summarized in Tables 3 and 4.

Table 3 Comparison results for the winter–summer observation period
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Table 4 Comparison results for the summer–winter observation period
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8 Discussion and analysis

Although the numerical results are taken as reference values for the relative errors in Eq. (1) as well as in Tables 3 and 4, they should not contain necessarily the true values. They are also subject to several uncertainties. Nevertheless, the trends reflected in Tables 3 and 4 can be considered as realistic. They serve as a baseline for the first critical analysis. This paper describes a qualitative and quantitative comparison of two completely different measurement approaches with a numerical solution concerning the thermal deformation of the bridge. The obtained results show a good agreement regarding the longitudinal displacement of the bridge deck. In addition, the results in Tables 3 and 4 allow to draw essential conclusions summarized below:

  • the slope of the regression lines for the two observation intervals No. 2 within the winter–summer and the summer–winter periods with the same absolute temperature fluctuation of \(\pm ~ 20.2 ~ {\text{K}}\) are different for each of the three compared approaches. It means that the absolute temperature change \(\Delta T\) is not the only factor that influences the overall thermal deformation;

  • the smaller the temperature difference and the longer the observation period, the larger is the relative error and, consequently, the lower is the accuracy. This is reflected by the intervals No. 1 in both observation periods. Perhaps, the reason is a superposition of multiple effects with uncertainties for small absolute values;

  • the slope S of the regression lines should correlate with the coefficient of thermal expansion and the tensile stiffness of the deck according to the mechanical beam model. Therefore, this slope should be a nearly constant value along the bridge. Such simple mechanical models could be helpful for plausibility checks of monitoring systems;

  • some relative error levels in Tables 3 and 4 reach high values of 30–40 %. Other error values are below 15 % or even below 10 %, which is very good for practical purposes in view of numerous difficulties and assumptions described above. We see the potential for even improved values in future scenarios;

  • the total station measurements provide sometimes better and sometimes worse results compared to the InSAR measurements. Although the absolute accuracy of the total station measurements should be better than that of the InSAR measurements for each individual measurement position, both techniques deliver comparable results with respect to the slope S and the NP position;

  • the determination of the NP position from the monitoring seems to be a useful result that allows to check and to prove the design assumptions based solely on the simulation. There is a difference in \({x}_{0}\) between the simulation and the measurement of up to 20 %. Such results are unknown from the literature so far.

Some additional remarks for the discussion of usability of InSAR measurements: In the context of this study mid-resolution (5 m \(\times\) 20 m) Sentinel-1 data was investigated in addition. These data sets are available free of charge and cover large regions within a single satellite scene (potentially advantageous for monitoring of several bridges in one region). However, the spatial resolution is considered as too poor for the given scenario to provide meaningful PS detection. Therefore, very high-resolution data is generally recommended for any application in the context of SHM.

9 Conclusions and recommendations

A comparison of alternative measurement techniques for SHM is a challenge due to different sampling rates, measurement points and other specific features. Nevertheless, a suitable approach has been developed and implemented in the present study for the InSAR and total station measurements, providing an important insight and new results. Several specific challenges during the application of alternative techniques have been identified and systematically analyzed. Some of them could and should be eliminated in advance by means of a thorough planning and implementation. Some others must be always taken into account.

The InSAR measurements exhibit a strong sensitivity with respect to the spatial orientation of the structure and the flight direction of the satellite. In the present case, the InSAR deformation measurements covering the north–south oriented part of the curved bridge are significantly less accurate than those covering the west–east oriented part. In general, InSAR measurement of displacement vectors that are affine with the flight direction is technically restricted. This is a limiting factor for a regional “blind” monitoring of multiple structures. Here, a combined implementation of measurement techniques can overcome the limitations of the individual ones, for example the usage of total station measurements at north–south oriented structures. On the other hand, the InSAR approach has proven its advantage of temporal data availability when local restrictions or risks prevent appropriate on-site measurements.

The present study shows that a pre-knowledge about the structure can be very important for a reliable interpretation of the InSAR measurements. In addition, a pre-knowledge allows for the selection of optimum observation scenarios given by the existing SAR sensors. Thus, the “blind” monitoring is generally not recommended. A numerical model of the structure is usually essential for plausibility checks and the correct interpretation of measurement results. In combination with the measurement data flow, it can form a core of a digital twin for SHM. The typical sampling rate of today’s SAR satellite systems (several days) makes the monitoring of daily or even intra-day deformations impossible. Thus, the InSAR technique is suitable for monitoring the seasonal elastic deformation and long-term inelastic deformations due to damage or settlements. It is shown, that InSAR provides meaningful insights for parameters relevant for SHM, particularly for the neutral point position \({x}_{0}\) or the elongation gradient along the bridge \(S\). Based on the authors’ experience from the present study, the following recommendation could be formulated:

  • InSAR measurement techniques can be a reasonable supplement for SHM in combination with alternative techniques on site, provided there is an intersection between the structural parts observed by alternative techniques for cross-checking;

  • InSAR measurement techniques should be applied taking into account a pre-knowledge of the geometry model, structural orientation and specific features of construction to optimize the accuracy and efficiency. Blind monitoring is not recommended;

  • traditional measurement techniques like geodetic surveying on site could be quite expensive for large structures due to a significant amount of required sensors and equipment or inapplicable for hidden structural parts due to vegetation or neighboring constructions. The InSAR technique as a supplement could be very helpful to reduce costs or to monitor hidden constructions from the top;

  • the sampling rate of the InSAR measurements (days or weeks) is insufficient to cover the daily fluctuation of structural deformations. It is recommended to use alternative techniques on site and apply the methods of artificial intelligence to extend the available data sets onto the time intervals and structural locations that are missing or not covered by the monitoring system. That could be the matter of further research;

  • it is strongly recommended to use numerical models to correctly interpret measurement results for any measurement technique in use;

  • alternative measurement techniques, for example based on the GNSS technology, could be in addition applied on site taking into account the cost–benefit relation and the visibility of the structure to multiple satellites. It could increase the reliability and the accuracy of the whole monitoring system;

  • when observing temperature induced deformation, it is also recommended to install several temperature sensors directly on the structure to cover the temperature difference across the cross-section and along the bridge. This would provide a more precise picture of the actual temperature distribution, which should result in a better agreement between simulated and measured deformation states;

In the authors’ opinion, the above recommendations should be taken into account in future developments of SHM systems. Corresponding technical innovations, both in the area of satellite-based and laser-based displacement measurements, can be integrated into the presented approach without serious difficulties.