Simultaneous Estimation of Submerged Floating Tunnel Displacement and Mooring Cable Tension through FIR Filter-Based Strain and Acceleration Fusion

. A submerged foating tunnel (SFT) is a tunnel structure that foats approximately 50 m below the water surface and is anchored to the seabed by mooring cables. It is a new alternative to conventional bridges and immersed tunnels. Tere have been ongoing eforts to construct SFTs worldwide, and the integrity of these SFTs needs to be monitored throughout their lifespan. Tis study simultaneously estimated the tunnel displacement and mooring cable tension force using acceleration and strain measurements to assess the integrity of an SFT. First, strain measurements were transformed to displacement using simplifed mode shapes and mode-scaling factors, which did not require the true mode shapes of the SFT. Te mode-scaling factors were automatically estimated using initial strain and acceleration measurements. Ten, the strain-based displacement was combined with the acceleration measurement using a fnite impulse response flter to improve the displacement estimation accuracy. In addition, the tension force of a mooring cable was estimated from the displacement at the connection between the tunnel and mooring cable. Te feasibility of the proposed technique was examined through a series of numerical simulations and laboratory tests on an 8 m-long aluminum SFT mock-up structure.


Introduction
A submerged foating tunnel (SFT) is a tunnel structure foating approximately 50 m below the water surface and anchored to the seabed by mooring cables, as shown in Figure 1. Te SFT is an attractive alternative to conventional bridges or immersed tunnels because its construction is not limited by the span length or depth of the ocean foor [1]. So far, no real-world SFT has been constructed, but there are ongoing eforts to build SFTs [2][3][4][5]. Te development of monitoring systems to continuously evaluate the integrity of these SFTs is necessary. In particular, the measurements of the tunnel displacement and mooring cable tension force are critical for the structural integrity assessment of an SFT.
Several techniques are available for direct displacement measurements, utilizing linear variable diferential transformers (LVDT) [6], real-time kinematic global navigation satellite systems (RTK-GNSS) [7], vision-based systems [8], and radar systems [9,10]. Indirect displacement estimation techniques using accelerometers, inclinometers, and strain sensors have been developed recently as well [11][12][13]. Another trend for structural displacement estimation involves combining multiple types of measurements to improve the displacement measurement accuracy and reliability. Examples include combining RTK-GNSS and accelerometer [14], strain gauge and accelerometer [15,16], vision camera and accelerometer [17,18], and millimeter-wave radar and accelerometer [19]. Obviously, GNSS, radar, and visionbased techniques cannot operate in underwater environments, and the fusion of strain gauges and accelerometers seems to be promising for SFT displacement monitoring.
In addition to displacement, monitoring of the mooring cable tension force is also essential because mooring cables are major load-carrying members. Several cable tension force estimation techniques have been proposed, utilizing load cells [20], strain gauges [21], accelerometers [22,23], electromagnetic (EM) sensors [24,25], and eddy current sensors [26]. However, all these discrete sensors must be physically placed on each cable, and such an installation can be time-consuming and labor-intensive. Although several attempts have been made to develop computer-vision techniques for noncontact tension force estimation [27,28], vision techniques cannot work in underwater environments.
Tis study simultaneously estimates the tunnel displacement and mooring cable tension force using acceleration and strain measurements. Te sensor confguration for the proposed technique is shown in Figure 1. Strain gauges are installed at the mid-span locations of each tunnel segment (x � x m,i , i � 1, · · · , N), and bidirectional accelerometers are installed at locations where the displacement needs to be estimated. Figure 2 shows the fowchart of the proposed technique. First, the unknown mode-scaling factors necessary for the strain-displacement transformation (derived in Section 2) are estimated using the initial strain and acceleration measurements (less than 1 min) (Section 3.1). Second, the strain measurements after the frst step are transformed to strain-based displacement using the derived transformation equation and estimated mode-scaling factors. Tird, a fnite impulse response (FIR) flter combines the strain-based displacement with acceleration measurements to improve displacement estimation accuracy (Section 3.2). Finally, the tension force of the mooring cable is estimated using the displacement estimated at the connection point of the mooring cable with the tunnel (Section 3.2.2). Te performance of the proposed technique is examined in Section 4 through a series of numerical simulations, and in Section 5, a laboratory test on an 8 m-long aluminum SFT mock-up structure is presented. Concluding remarks are provided in Section 6. Note that no real-world SFT construction has been realized; an SFT structure form, commonly used in previous studies [1][2][3] (as shown in Figure 1), is adopted in this study.
Tis study has the following contributions: (1) the displacement of an SFT is estimated by combining strain and acceleration measurements without knowing the true mode shapes of the SFT; (2) the tension force of a mooring cable is also estimated from the displacement at the connection point between the tunnel and the cable; and (3) the proposed displacement and tension force estimation technique are experimentally validated on an 8 m-long aluminum SFT structure.

Simplifed Model of SFTs.
Previous studies have shown that an SFT can be simplifed as a simple beam on an elastic foundation if the stifness ratio is less than 0.05 [29]: where K denotes the axial stifness of the mooring cables, D x is the distance between two adjacent mooring cables in the x direction, and EI denotes the bending stifness of the tunnel. When equation (1) is satisfed, the displacement mode shapes of a simply supported beam (SSB) (ψ j (x)), hereafter referred to as the simplifed displacement mode shapes, can approximate the true displacement mode shapes of an SFT (ψ j (x)). Here, subscript j denotes the j th mode. Tese simplifed displacement mode shapes have been commonly used to analyze the dynamic responses of an SFT under seismic, vehicle, and impact loadings [30][31][32].
In practice, it is difcult to measure the structural displacement, especially in underwater environments. To tackle this issue, the strain is measured and transformed to displacement using a strain-displacement transformation relationship, which requires the strain mode shapes. Figure 3 shows the diferences between the strain mode shapes of the SSB (φ j (x)) and the SFT (φ j (x)). However, if only the strain mode shapes at the mid-span locations are of interest (φ j (x) at x m,1 , x m,2 , . . ., x m,N ), φ j (x) can be easily approximated by scaling φ j (x) with a mode-scaling factor α j . Note that as the mode order increases, the diferences between φ j (x) and φ j (x) decrease, as well as the mode-scaling factor value ( Figure 3). Here, the strain mode shapes of the SFT are from one of the models investigated in numerical validation section. Figure 1 depicts an example of an SFT with strain gauges installed at N midspan locations along the tunnel (x � x m,i , i � 1, · · · , N). At the cross-section of each location, three strain gauges (S1-S3) are attached at three diferent points along the vertical (z) and horizontal (y) directions. Te measurements from strain gauges S1, S2, and S3 are denoted as ε 1 k , ε 2 k , and ε 3 k . Here, subscript k denotes the k th time step. Based on a mode superposition algorithm [16], the strain measurements ε 1 k Figure 1: A conceptual sketch of a submerged foating tunnel (SFT) anchored to the seabed by mooring cables and sensor confguration of the proposed technique.

Strain-Displacement Transformation.
Equation (4) involves N mode-scaling factors, and their estimation is discussed in Section 3.1. Similarly, the horizontal displacement (u y k ) can be estimated from the strain measurements ε 1 k and ε 3 k .  Figure 2: Flowchart of the proposed technique for simultaneous estimation of tunnel displacement and mooring cable tension using acceleration and strain measurements.
True strain mode shapes (φ j ) Amplitude First mode shape (j = 1) Second mode shape (j = 2) Third mode shape (j = 3) Mid-span locations Simplified strain mode shapes with scaling (α j φ j ) -Simplified strain mode shapes (φ j ) - Figure 3: Comparison of the true and simplifed strain mode shapes of an SFT.
Structural Control and Health Monitoring 3

Development of Displacement and Tension Force Estimation Technique for SFTs
Tis section proposes a simultaneous displacement and cable tension estimation technique for SFTs, using strain and acceleration measurements. As shown in Figure 2, the proposed technique consists of two steps: (1) automated mode-scaling factor estimation (Section 3.1) and (2) continuous displacement and mooring cable tension force estimation (Section 3.2.1). Note that the mode-scaling factor and displacement estimation steps are explained only for vertical displacement, but these steps can be easily extended to estimate the horizontal displacement as well.

Automated Mode-Scaling Factor
Estimation. An automated algorithm is proposed in this section to estimate the values of the unknown mode-scaling factors in equation (4), which are necessary for the strain-displacement transformation. Note that out of N unknown mode-scaling factors, only the frst P (<N) scaling factors are estimated, and the remaining high-order scaling factor values are set to 1. Tis approximation can be justifed on the basis of (1) the diference between the simplifed and true mode shapes and (2) the contribution of the higher modes to the structural displacement, where both decrease as the mode order increases ( Figure 3). It should be noted that because at least P equations are needed for the estimation of P mode-scaling factors, equation (4) is extended to include displacement estimation at P locations (x d,i , 1 ≤ i ≤ P):

Transformation Equation Decoupling
. First, equation (5) is decoupled into P independent equations before estimating the values of the mode-scaling factors. Te strain-based modal responses (q s j,k , 1 ≤ j ≤ P) are defned as follows: Te following equation is obtained by introducing equation (7) into equation (5): 4 Structural Control and Health Monitoring Equation (8) can be further rewritten as follows: where q u j,k denotes the j th modal response decomposed from u z k after removing the high-order ( > P) modal responses. Finally, P independent equations are obtained, and each of them includes only one mode-scaling factor.

Mode-Scaling Factor Estimation.
Assuming that accelerometers are installed at P locations and strain gauges are installed at N locations, a fowchart of the proposed modescaling factor estimation algorithm is presented in Figure 4. Note that the strain gauges must be installed at the mid-span locations of each tunnel segment, whereas the accelerometers can be installed at arbitrary locations along the tunnel. First, ∆ε 1,2 k (x m,i ), for i � 1, · · · , N, are computed from strain measurements using equation (3) and high-pass fltered. Subsequently, the high-pass fltered ∆ε 1,2 k (x m,i ) are decomposed into strain-based modal responses (q s i,k ) using equation (7). Te strain-based modal responses are divided into two parts: (1) high-order (j > P) and (2) low-order (j ≤ P).
Second, high-frequency displacements are estimated at P locations by double integrating and high-pass fltering acceleration measurements. Ten, acceleration-based loworder modal responses (q u j,k , 1 ≤ j ≤ P) are estimated from the acceleration-based high-frequency displacements and high-order(q s j,k , j > P)strain-based modal responses using equation (9).
Assuming that both the acceleration and strain are measured for a short time (less than 1 min), q u j,k and q s j,k are obtained at multiple time steps. Terefore, α z j can be estimated as the slope of the q s j,k vs. q u j,k plot using least-squares regression. Te purpose of high-pass fltering is to remove large low-frequency drifts in the acceleration-based displacements. Te cutof frequency of the flter should, therefore, be sufciently high to remove drifts and should be lower than the frst resonance frequency of the SFT. Note that an identical high-pass flter should be applied to both strain and acceleration measurements.

Continuous Displacement and Mooring Cable Tension
Force Estimation. After estimating the mode-scaling factors using the initial acceleration and strain measurements, the Structural Control and Health Monitoring displacement and mooring cable tension force can be continuously monitored.

FIR Filter-Based Displacement Estimation.
Displacement estimated from acceleration measurement using the double integration has a low-frequency drift caused by unknown initial conditions and measurement noises. A highpass flter can be used to remove the low-frequency drift [11], but it also removes the low-frequency structural displacement. Displacement estimated from strain measurement usually has low accuracy owing to relatively large noise levels in strain measurement [12,13]. An FIR flter was adopted in this study to estimate the SFT displacement by fusing strain and acceleration measurements, and the fnal displacement is estimated as a combination of low-frequency displacement from strain measurements and high-frequency displacement from acceleration measurements, which provides better accuracy than strain-based displacement. Figure 5 presents a fowchart of the proposed FIR flter-based technique, assuming that strain gauges are installed at N mid-span locations and an accelerometer is installed at the target location where displacement needs to be estimated.
First, multiple strain measurements are transformed into displacement at the target location using equation (4) and the estimated mode-scaling factors. Te strain-based displacement is then fltered using a low-pass flter (C L ) to obtain lowfrequency displacement, while the high-frequency displacement is obtained from double integration and the high-pass fltered acceleration measurements, where double integration and high-pass fltering are simultaneously performed by C H . Te cutof frequencies of the low-pass and high-pass flters are set to the frst resonance frequency of the SFT. More details on C L and C H can be found in the Appendix and the study by Ma et al. [16]. Te fnal displacement is obtained by combining the high-frequency displacement obtained from the acceleration measurement and the low-frequency displacement obtained from the strain measurements.

Mooring Cable Tension Force Estimation.
Te tension force of the mooring cable is estimated from both the horizontal and vertical displacements at the connection point between the tunnel and mooring cable. For the SFT shown in Figure 6, the following equation can be derived from the geometric relationship between the initial (t � 0) and current (t � k∆t) states of the SFT: where R denotes the outer radius of the tunnel and D y denotes the distance between two anchoring points along the y-direction. y 0 , z 0 denotes the initial position of the center of the tunnel cross-section. L l k and L r k denote the lengths of the left and right cables at the k th time step, respectively. L l k and L r k are related to the initial lengths (L l 0 and L r 0 ), axial stifness (K), and tension forces (T L k and T r k ) of the two cables, as follows: By combining equations (11) and (12), the tension forces can be estimated as follows: Note that the mooring cables are assumed to be straight without any sag and are always in tension because of the large initial tension.

Strain measurements at N locations
High-frequency strain differences at N locations Equation (7) Equation (3) & High-pass filtering Acceleration measurements at P locations High-frequency acceleration-based displacements at P locations Mode scaling factor estimation using least square regression Equation (9) Acceleration-based low-order modal responses (q u 1,k , q u 2,k , ..., q u p,k ) Double integration & High-pass filtering Strain-based high-order modal responses (q P+1 , q P+2 , ..., q N )     Figure 7(a) shows the model for an SFT, which consists of an 800 m-long tunnel and 54 m-long mooring cables spaced every 50 m along the longitudinal direction of the tunnel. Te tunnel was modeled in ABAQUS with the B31 beam element, and the mooring cables were modeled with the T3D2 truss element, similar to existing studies [31,32]. Te simply supported boundary condition, which has been widely used in previous studies [4,31,32], was adopted for the tunnel. Te diferent boundary conditions are investigated in Section 4.4.5. Strain gauges were installed at seven mid-span locations (i.e., 5, 9, 13, 15, 19, 23, and 27), and accelerometers were installed at locations 1-31, where the displacement needed to be estimated, as shown in Figure 7(a). Note that these thirty-one locations are either the connection points between the cable and tunnel or the mid-span locations. Additional parameters regarding the tunnel and mooring cables are shown in Figure 7(b), and the values reported in previous studies [31,33] are used here. Te gravity and buoyancy of the tunnel were 1.065 × 10 6 kN and 2.199 × 10 6 kN, respectively. Te remaining buoyancy of 1.134 × 10 6 kN was balanced by 30 mooring cables, resulting in 3.78 × 10 4 kN initial tension force for each mooring cable.

Model Description.
As shown in Figure 7(c), strain gauges were installed at four points on each cross-section at locations 5,9,13,15,19,23, and 27, for horizontal and vertical displacement estimations. Note that for the B31 beam element, only the strain responses at these four points are available in ABAQUS. Te tunnel had a tubular cross-section with an inner diameter of 18 m and a thickness of 1 m. One end of the mooring cable, inclined at 45°, was pin-connected to the tunnel and the other end to the seabed. Te distances from the tunnel to the seabed and the water surface were set to 38.5 m and 61.5 m, respectively, similar to the previous studies [33][34][35]. Te SFT model was simultaneously subjected to (1) wave and current loadings and (2) uniformly distributed loading with random amplitudes in both the horizontal and vertical directions, as shown in Figure 7(c). Te hydrodynamic environment was simulated using ABAQUS/Aqua, which has been commonly used in previous studies [36][37][38]. However, the real hydrodynamic environment may be more complex [39], and more research is needed to consider this.
Ten, the corresponding acceleration, strain, displacement responses, and cable tension forces were simulated for 100 s at a sampling rate of 100 Hz. Te simulated displacement and cable tension force were used as reference values to evaluate the estimation performance of the proposed technique. Te current velocity decreased linearly from 0.1 m/s at the water surface to 0 at the seabed. Te wave height and period were 11.7 m and 13 s, respectively. Te hydrodynamic environment was simulated using ABAQUS/Aqua.

Mode-Scaling Factor Estimation
Results. Strain gauges were installed at seven locations, and seven modes were extracted. Mode-scaling factors were estimated from the frst three modes (P � 3) using the 30 s acceleration responses recorded at three locations (i.e., 5, 15, and 27) and strain responses at all seven locations. Because the frst resonance frequency of the SFT was 0.35 Hz, the cutof frequency of the high-pass flter required for the mode-scaling factor estimation was set to 0.30 Hz. As shown in Figure 8(a), three diferent modal responses at 0.35 Hz, 0.49 Hz, and 0.96 Hz were decomposed from strain and acceleration measurements using equations (7) and (9), respectively.
Here, only the frst three modes were used for the displacement estimation because the contribution of the lower modes is more signifcant than that of the higher modes. Tat is, as the mode order increases, the magnitude of the modal responses decreases signifcantly. For example, the response amplitude of the third-order mode was only 1.4% of the frstorder mode response. Te corresponding scale-factor estimation results are shown in Figure 8(b). Apparent linear relationships were observed between the acceleration-and strain-based modal responses, with R 2 values greater than 0.97. Note that all results in this section were obtained in the vertical direction, and similar results obtained in the horizontal direction were omitted here owing to space limitations. Figure 9 shows a comparison of the frst three true strain mode shapes of the SFT model and the corresponding simplifed strain mode shapes. As mentioned in Section 2.1, the discrepancy decreased from the frst-mode shape to the third mode shape, and the diference was negligible for the third mode. Even for the frst-mode shape, good agreement was observed at mid-span locations (i.e., locations 5, 9, 13, 15, 19, 23, and 27) after scaling of the simplifed frst mode. Terefore, the use of simplifed strain mode shapes is acceptable if the strain responses at mid-span locations are used. For the SFT model used in this simulation, the stifness ratio defned in equation (1)

Displacements.
Te cutof frequency of the FIR flters was set to the frst resonance frequency of the SFT, i.e., 0.35 Hz. Te displacement at location 1 was estimated using the acceleration response at this location and the strain responses at locations 5,9,13,15,19,23, and 27. Tis procedure was repeated for all thirty one locations where accelerometers were placed to estimate the displacements at all locations. Figure 10 shows the displacements estimated at the middle of the tunnel (location 15) using the proposed and existing techniques [16]. Te existing technique frst transforms strain measurements to displacement using the assumed mode shapes and then simultaneously estimates the fnal displacement and a scaling factor, used to compensate for the discrepancy between the true and assumed mode shapes from the transformed displacement and acceleration measurement by integrating the FIR flter with a recursive least square (RLS) algorithm. Te displacements estimated using both techniques agreed closely with the reference displacements. Note that the existing technique computes a single scaling factor for all modes, but the scaling factor 8 Structural Control and Health Monitoring value varies with respect to the displacement estimation locations. In contrast, the proposed technique computes diferent scaling factors for each mode; however, each scaling factor value was constant at all locations. Te estimated displacements from 15 to 18 s were magnifed on the right-hand side of Figure 10 to clearly show the diferences between the estimated and reference displacements. Te displacement estimation performance was quantitatively evaluated using the root mean square error (RMSE) and normalized RMSE (NRMSE) as follows: where u r k and u k denote the reference and estimated displacements at the kth time step, respectively, and Q is the number of data points. Te proposed technique estimated displacements with NRMSE and RMSE values of less than 0.5% and 0.7 mm, respectively. Note that the NRMSE and RMSE values of the proposed technique were only one-ffth those of the existing technique.  (7) and (9)   Simplified strain mode shapes after scaling (α j φ j ) - Structural Control and Health Monitoring displacement estimation accuracy of the proposed technique was less sensitive to the displacement estimation location than that of the existing technique, and a similar level of performance was achieved at all locations. Note that the proposed technique estimated only three locationindependent mode-scaling factors, whereas the existing technique required thirty-one location-dependent scaling factors to estimate the displacement at thirty-one locations.

Tension Forces.
Te displacement estimated at the connection point between the tunnel and mooring cable was used to estimate the tension force of the corresponding mooring cable. Figure 12 shows the tension force estimated at the left mooring cable for location 16. Good agreement was observed between the estimated and reference tension forces. Figure 13 summarizes the NRMSE values of the tension forces estimated for all mooring cables. Te proposed technique showed less than 2% NRMSE and 60 kN RMSE for all mooring cables.

Measurement Noises.
Diferent levels of Gaussian white noise were added to the acceleration and strain responses to examine the efect of measurement noise on the displacement measurement accuracy. Because accelerometers typically have higher signal-to-noise ratios (SNRs) than strain gauges, the SNR of the acceleration responses was set to twice that of the strain responses. Figure 14 summarizes the displacement estimation performance under diferent measurement noise levels. Although the displacement estimation error increased with the measurement noise level, it was still less than 2% at most locations. Note that the NRMSE values were relatively large near the two fxed ends because of the relatively small displacements at these locations, particularly in the horizontal direction. However, the absolute displacement errors at these locations were very small, less than 1.5 mm.

Values of P and N.
First, the displacements were estimated by varying the N value from one to nine. Te corresponding RMSEs of the estimated displacements are shown in Figure 15. Signifcant RMSE reductions were observed when N increased from one to seven. However, further increases beyond seven did not improve the NRMSE value much. Te P value was set to one when N equaled one and to three in other cases. Next, the N value was fxed at seven, and the displacements were estimated by varying the P value. Te corresponding NRMSE values of the estimated displacements are shown in Figure 16. Increasing the P value from one to two resulted in a signifcant decrease in the NRMSE value. However, further increase of the P value did not reduce the NRMSE value much. Tis observation is consistent with Figure 8(a). As the mode order increases, the magnitude of the modal responses decreases signifcantly. Terefore, setting the values of the high-order scaling factors to one did not cause a signifcant diference.

Stifness Ratio.
Tree additional SFT models with diferent stifness ratios were simulated by changing the stifness of the mooring cables. Similar efects can be obtained by changing the distance between two adjacent mooring cables or the bending stifness of the tunnel. Note that the stifness ratios of all models still satisfy equation (1). Te displacements were estimated for the four SFT models following the procedures described in Section 4.1, except for the stifness ratios. Figure 18 compares the true and simplifed frst-mode shapes of the four SFTmodels. Apparently, the discrepancy increases as the stifness ratio increases, resulting in larger mode-scaling factors. For all models, the proposed technique estimated displacement with less than 2% NRMSE, as shown in Figure 19.

Boundary Condition.
By changing the boundary conditions of the tunnel from hinged at both ends to fxed at both ends, and fxed at one end and hinged at the other, two additional SFT models were simulated. Note that the realworld construction of an SFT has not yet been realized, and most existing studies have adopted either simply supported [4,31,32] or fxed-fxed [3,33,35,40] boundary conditions. Figure 20 summarizes the accuracy of displacement estimation. Because the simplifed mode shapes were obtained with simply supported boundary conditions, large discrepancies between the simplifed and true mode shapes appeared at locations near the two ends, i.e., locations 1-3 and 29-31 for the SFT model with fxed boundaries at both ends. Terefore, large NRMSE and RMSE values were observed at these six locations (1-3 and 29-31). For the remaining 25 locations, NRMSE values obtained using the proposed technique were below 0.8%. When the SFT model was fxed at one end and hinged at the other, extremely large errors were observed near the fxed end, i.e., at locations 1-3.

12
Structural Control and Health Monitoring Except for these three locations, displacements were still reliably estimated at the remaining 28 locations, with RMSE values less than 2.4%. Much larger errors were observed from the SFT model fxed at one end and hinged at the other owing to its nonsymmetric mode shapes. Structural Control and Health Monitoring of eight bolt-connected aluminum tubes. Note that the Young's modulus and density of the tube were 70 GPa and 2710 kg/m 3 , respectively. Each tube was 1 m long and had a uniform circular cross-section with an inner diameter of 13 cm and a thickness of 0.5 cm. Te tunnel was submerged into a 12 m long, 10 m wide, and 2 m deep ocean basin, as shown in Figure 21(a), and the tunnel was supported at its ends by a steel frame to simulate a hinge connection, as shown in Figure 21(b). Stainless steel wires of 0.5 mm diameter were used as mooring cables, and a total of 14 mooring cables were spaced equally at 1 m intervals along the longitudinal direction of the tunnel, as shown in Figure 21(b). For each mooring cable, one end was connected to the tunnel using metal ring buckles and the other end was connected to the pad eye welded to a steel frame. Te steel frame was fxed at the bottom of the 3D ocean basin using anchor bolts.

Experimental Validation
Five triaxial MEMS accelerometers, shown in Figure 21(c), were attached at locations 3-7, and 18 resistance-type strain gauges, shown in Figure 21(d), were attached at locations 1, 2, 4, 6, 8, and 9. Tree strain gauges were installed at each cross-section at locations 1, 2, 4, 6, 8, and 9 for bidirectional displacement estimation. Note that all strain gauges and accelerometers were installed on the external surface of the SFT mock-up structure and waterproofed. Fourteen load cells were adopted for all the mooring cables to measure the reference tension forces. All acceleration, strain, and load cell data were sampled at 100 Hz using National Instruments data acquisition devices. Table 1 lists the detailed specifcations of these sensors.
After sensor installation, water was poured into the 3D ocean basin to ensure that the SFT mock-up structure was immersed and foating. By adjusting the length of the mooring cables, the initial tension forces were set to approximately 150 N. Attempts were made to excite the test structure using diferent wave loadings, but no meaningful vibration was observed owing to the relatively high stifness of the SFT mock-up structure. Subsequently, the SFT mock-up structure was excited by manually pushing and releasing the tunnel along three diferent directions: (1) horizontal (y), (2) vertical (z), and (3)   acquisition channels, only the load cells of cables 1, 2, and 3 (as shown in Figure 21(b)) were used to measure the reference tension forces during the three excitations, whereas the other 11 load cells were used only to adjust the initial tension forces.

Reference Displacement Measurement Using Computer
Vision. Computer vision was adopted in this experiment to measure the reference (ground-truth) displacement. Because the tunnel was submerged during the experiment, it was difcult to directly measure its displacement. Instead, fve 20 cm-long rigid steel bars with two artifcial markers, as shown in Figure 22, were attached vertically to the top surface of the tunnel at locations 3-7. Because the tunnel may be subjected to torsion along the x axis, two markers were necessary. Five digital cameras (Sony DSC-WX800) were installed at stationary locations to individually track the movement of the markers at each location. Table 2 presents the detailed specifcations of the camera. In this test, each camera recorded the movement of two markers at each location with a 1920 × 1080 resolution and 119.88 Hz sampling rate. Te distance between the cameras and markers varied from 4 m to 7 m. Te water level was always below these markers during the experiments to ensure successful tracking of the markers. A commonly used templatematching algorithm [8] was adopted for marker tracking. Because the original displacement estimated from the vision measurement is in pixel units, a scale factor is required to convert the pixel unit to a length unit. By adjusting the focal length of each camera, a scale factor of 0.1 mm/pixel was achieved for all targets.
Te reference displacement was computed as the weighted average of the displacements of the two markers, as shown in Figure 22.
where L 1 and L 2 denote the vertical distances between each marker and tunnel cross-section center, respectively. u y 1 and u z 1 denote the horizontal and vertical displacements of Marker 1, respectively. u y 2 and u z 2 are defned in a similar manner. Because the strain and acceleration measurements were sampled at 100 Hz, the vision-based displacements, which were originally sampled at 119.88 Hz, were downsampled to 100 Hz. Figure 23 shows the frequency spectra of the acceleration measured at location 3 under an inclined excitation. Te contribution of the frst mode to the structural displacement was dominant in both the horizontal and vertical directions. Hence, the mode-scaling factors were estimated only for the frst mode (P � 1) in both directions. Te frst resonance frequencies of the SFTmock-up structure were observed at 5.16 Hz and 8.02 Hz in the vertical and horizontal directions, respectively. Te displacements were estimated at locations 3-7. Figure 24 shows the bidirectional displacements estimated at location 3 under an inclined excitation. Here, displacements were estimated from (1) strain measurement only, (2) the combination of strain and acceleration measurements using the existing technique, and (3) the combination of strain and acceleration measurements using the proposed technique. Signifcantly larger errors were observed for the  Figure 22: Vision-based reference displacement measurement using two artifcial markers at diferent heights of the rigid steel bar attached to the tunnel.  Figure 24. Note that the proposed technique computes diferent scaling factors for each mode; however, the scaling factor for a specifc mode is constant for diferent displacement estimation locations. In contrast, the existing technique computes a single scaling factor for all modes, and the scaling factor value varies at each displacement estimation location. However, this did not lead to a signifcant difference in the displacement estimation in this experiment. Because the contribution of the frst mode was signifcantly more dominant than that of any other mode, only a slight improvement was achieved using the proposed technique. Te displacement estimation performance at all fve locations is summarized in Table 3. Te overall RMSE and NRMSE of the strain-based displacement were 0.135 mm and 3.26%, respectively. Note that the estimation of strainbased displacement requires mode-scaling factors. Terefore, accelerometers were still required. Combining the acceleration measurements resulted in a signifcant improvement in displacement estimation accuracy, and around

Tension Forces.
Te displacements obtained at location 3 were used to estimate the tension force of cable 1, and those from location 5 were used to estimate the tension forces at cables 2 and 3. Figure 25 presents the tension force of cable 3 under inclined excitation. A good agreement between the estimated and reference tension forces was observed with 11.20 NRMSE and 3% NRMSE. Te tension forces estimated for all the cases are summarized in Figure 26. Te proposed technique successfully estimated the tension forces, and RMSE and NRMSE values were less than 12 N and 4.6%, respectively. It should be noted that the load cells installed on cables 1 and 2 leaked after horizontal excitation, resulting in the failure of the reference tension force measurement under inclined excitation. Terefore, the corresponding NRMSE values were not calculated.

Conclusions
Tis study proposes a simultaneous displacement and cable force estimation technique for SFTs using acceleration and strain measurements. Te main contribution of this study lies in the derivation of the strain-displacement transformation relationship for SFTs with simplifed mode shapes, automatic mode-scaling factor estimation, and displacement-based mooring cable tension force estimation. Te feasibility of the proposed technique was frst examined on a numerical SFT model by estimating the bidirectional displacements at thirty-one locations and tension forces at 30 mooring cables. Te NRMSE values of the displacements and tension forces estimated using the proposed technique were less than 1% and 2%, respectively. Experimental validation was conducted on an 8 m-long aluminum SFT mock-up structure. Te overall RMSE and NRMSE values of the estimated displacements were 0.082 mm and 1.98%, respectively. In addition, tension forces of three mooring cables were estimated with RMSE and NRMSE values less than 12 N and 4.6%, respectively. Te validation tests performed showed that the proposed technique can compensate for the discrepancy between the true and simplifed mode shapes at most locations, and then accurately estimate the SFT displacements. However, it may not work well near the two ends of the tunnel, particularly when the boundary conditions are not symmetric. In addition, in the laboratory test, the SFT mock-up structure was excited by manually pushing and releasing the tunnel due to its relatively large stifness. Further studies may be needed to examine the proposed technique under more realistic loading, such as vehicle and wave loading. In such a case, a longer SFT structure with less stifness than the one investigated in this study is required to generate meaningful vibration.