*4.2. Data Collection*

On 31 December 2020, the experimental measurements of dynamic displacement monitoring were carried out on the platform located in the south square of Beijing University of Civil Engineering and Architecture (BUCEA). The equipment used in the tests includes the shaking table, a GNSS deformation monitoring system independently developed by BUCEA, the ADIS16505 IMU (inertial measurement unit) produced by Analog Devices, Inc. (containing an accelerometer and gyroscope, but only the accelerometer data were used in this test) and laptops. The param of GNSS and accelerometer are shown in the following Table 1.

Two sets of equipment, the GNSS and accelerometer, were attached to the shaking table using a metal plate. The connecting line between the centers of MIMU and GNSS antenna was directed toward north–south. Since the accelerometer was located toward the northern GNSS antennas at the distance of 10 cm using a northeast down system, the installation direction of the X-axis would point to the north. The layout of monitoring systems and monitoring points are shown in Figure 8.


**Table 1.** Principal specifications of GNSS and accelerometer.

**Figure 8.** Instrument setup for deformation monitoring systems: (**a**) the layout of monitoring systems; (**b**) the monitoring point.

### *4.3. Comparison of PSD, HHT, and VMD–HHT*

To illustrate the advantages of the VMD–HHT model in the time domain and energy domain compared with PSD and HHT, three groups of tests were designed with time sequence amplitudes of 10 mm, 30 mm, and 50 mm. A GNSS receiver using a 5 Hz sampling rate and an accelerometer using a 100 Hz sampling rate were used to obtain the three original data sets for comparison.

The shaking table tests were conducted for 10 min (each test lasted approximately 3 min). The test site had an open view of the sky, thus avoiding multipath effects. Additionally, during the experiments, there was no wind to prevent wind-induced frequency noise. Figure 9 gives the frequency extract from three vibration events using the method of PSD; as seen from Figure 9, the frequency response peaked at 1 Hz is consistent with the vibration frequency of the shaking table. However, the time of the three groups of tests with amplitudes of 10 mm, 30 mm, and 50 mm is unknown from Figure 9 due to the frequency of all tests are 1 Hz.

HHT and VMD–HHT methods can separate each event in the time dimension and reflect the intensity of each event (Figure 10). It can be seen that the frequency range is about 1 Hz, and the intensity of three events is gradually strengthening, corresponding to amplitudes at 10 mm, 30 mm, and 50 mm. Comparing Figures 10 and 11 show that the HHT spectrum is noisier than the VMD–HHT spectrum. We can also observe that the intensity displayed in the HHT spectrum is smaller than the VMD–HHT spectrum, and the latter is more clearly detected. However, the frequency values at the beginning and end of each signal also reach higher values, especially the signal with amplitude at 50 mm. It is probably caused by endpoint effect. Overall, VMD–HHT can extract the characteristics of time, frequency, and energy more reliably than HHT.

**Figure 10.** Time–frequency–energy extracted by HHT.

**Figure 11.** Time–frequency–energy extracted by VMD–HHT.

### *4.4. Reconstruction of Dynamic Displacement Using FDIA Based on the VMD–HHT Model and GPS*

To assess the effectiveness and accuracy of FDIA based on the VMD–HHT model, displacements measured by GNSS are given for comparison. Firstly, the VMD approach was applied to remove residual and high-frequency noise from accelerometer data. Then, FDIA was used to obtain vibration displacements based on the cleaned data proposed by VMD. Figure 12 shows the displacement comparison of GNSS and FDIA; as seen from Figure 12, there is a high correlation between GNSS and FDIA. It is evident that the vibration amplitude is about 10 mm, and frequency is about 1 Hz, which is consistent with the set param. It can be concluded that VMD–HHT-based FDIA has successfully calculated the vibration displacements.

**Figure 12.** Comparative results of vibration displacements of GNSS and FDIA.

### **5. Trial Analysis of CB4A Offshore Oil Platform**

Located at 38◦839 north latitude and 118◦5038 east longitude, the CB4A offshore oil platform lies in the Yellow River Delta along the coast of the Bohai Sea. It is a fixed offshore oil platform in Shengli Oilfield at Dongying City with a total area of 700 m<sup>2</sup> (see in Figure 13). The sea area near the platform is windy all year round, affected by the southeast Pacific monsoon and the northwest monsoon of the Asian continent. More specifically, as frequently affected by the monsoon of the Asian continent, the offshore oil platform also suffers from heavy wind all year round. In addition, being close to the maritime transportation hub, the platform is under heavy traffic. As a result, ship collisions occur from time to time, and thus, the monitoring of offshore oil platforms becomes exceptionally urgent.

**Figure 13.** The view of the offshore oil platform.
