The normal vectors calculated from raw data of the MEMS sensor have inherited interferences and noises from different sources and thus should not be used directly to represent the position. Therefore, Kalman filter is used subsequently to estimate the state of the observations. In particular, this allows also to apply the concept of sensor fusion if more than one sensor is used e.g., to monitor inclination or acceleration.

Kalman filter goes back on 1960, when R. E. Kalman proposed a recursive solution to the discrete-data linear filtering problem [

16]. As one of the primary developers of the linear quadratic estimation (LQE), this method is named after his name. Kalman filter is a set of mathematical equations that uses a recursive means to estimate the underlying system state by minimizing the mean square error [

17]. Nowadays, Kalman filter and its extended version have been widely used in target tracking, navigation, and other relevant fields in data processing [

18,

19]. In the following a brief introduction of Kalman filter followed by his application to our problem of noisy positional data is given.

#### 4.1. A Brief Introduction of Kalman Filter

The first step of the Kalman filter is to build a model that represents the series of data. The state variable

x is addressed as an expression of discrete linear stochastic difference equation and the measurement value

z is described as a linear function of

x:

where:

A—State transition matrix relating the state at the previous time step k − 1 to that at the current step k;

u—Optional control input;

B—Control matrix that relates u to the state x;

w—Process noise vector, w(i) is normally distributed denoted by N(0, Q) and the covariance $cov\left[w\left(i\right),w\left(j\right)\right]=Q\cdot {\delta}_{ij}$, where ${\delta}_{ij}=\{\begin{array}{c}0\text{}if\text{}i\ne j\\ 1\text{}if\text{}i=j\end{array}$;

H—Meaurement transition matrix relating the state x to the measurement z;

v—Measurement noise vector, $v\left(i\right)$ is normally distributed denoted by $N\left(0,R\right)$ and the covariance $cov\left[v\left(i\right),v\left(j\right)\right]=R\cdot {\delta}_{ij}$, where ${\delta}_{ij}=\{\begin{array}{c}0\text{}if\text{}i\ne j\\ 1\text{}if\text{}i=j\end{array}$, and w and $v$ are statistically independent.

Once the Kalman filter model is established, the estimating process is applied by a predictor-corrector cycle. As is shown in

Figure 5, when the initial state value is established, the iteration can be immediately started. Specifically, time update equations and measurement update equations are respectively grouped that are responsible for the priori estimate for the next time step and the feedback of the priori estimate.

Table 2 provides the update equations from groups of time and measurement. A detailed explanation of Kalman filter can be found in [

17,

20]. There are several formulations Kalman filter update equations; in this paper, a specific form of Kalman gain

$K$ and a posteriori estimate error covariance

$P$ are selected on purpose of a concise expression of Kalman filter algorithms (

Table 2 and

Table 3) for both single and multi-sensor systems. More information can be found in [

21].

In a multi-sensor system, the main fusion methods include centralized and distributed means [

22], the sketches of which are illustrated in

Figure 6. Specifically, for centralized fusion methods, Willner, Chang [

23] has introduced three main linear Kalman filter algorithms for a multi-sensor system, namely parallel filter, sequentially filter and data compression filter, and all the three filters are equivalent and optimum. For distributed fusion methods, Carson has proposed a federated Kalman filter as a typical representation [

24,

25]. Generally speaking, the centralized fusion consumes intensive resources during calculating but assures a limited loss of information, while the distributed fusion calculates faster but has a less accuracy. In the following application, the centralized fusion method will be applied considering the relatively small amount of processed data at the same time.

#### 4.2. Application of Kalman Filter to Monitor Tunneling-Induced Ground Subsidence from Sensor Motion

A one-day field test of the X-SLEWS wireless sensor network was conducted from 10 a.m. to 5 p.m. on 27 January in 2013 at a tunneling construction site in Shanghai. The sensor nodes were deployed at the surface above the underground super highway tunnel at South Hongmei Road, while the shield machine would pass below the area at a depth of 41.65 m. Due to the subsidence related to the tunnel excavation, subsurface deformations was expected to monitor. Each node contains a 3-axis accelerometer and two orthogonal 1-axis inclinometers, and both of them are able to measure the tilting of the sensor with a settable frequency to the base station via a wireless network [

8]. The MEMS sensors of the sensor node are orientated on a plane defined as sensor plane, and we observe the change of the orientation of the normal vector and define it as the expected motion due to the tunnel driving.

Figure 7 shows the orientation of the MEMS sensor nodes in relation to the tunnel axis as well as the expected stages of motion caused by shield tunneling. When the excavation causes ground subsidence, the sensor node will firstly incline to the shield head, secondly restore to a certain extent with the expansion of the settlement area, and then tilt to the shield head again following the succeeding subsidence.

The data retrieved from two sensor nodes have been processed in this paper as an example. Firstly, the normal vectors over time have been derived as briefly described above and in detail in [

11]. To monitor the ground deformation, the change in inclination of the surface, respectively the sensor node, as depicted in

Figure 7, can be used as a measure and to visualize the changes over time. The normal vectors have been consequently projected into the longitudinal section of the tunnel, and the dip angle values have been hereby solved to express the vertical the best tilting of the sensor.

Figure 8 illustrates the derivation of the dip angle from the projected normal vector

${V}_{n\_prj}$ in the longitudinal section, and

Figure 9 shows the plots of dip angles of two sensor nodes located away from and on the track in the longitudinal section over the observing day. Particularly, Node 102 was deployed on the surface on the track of the tunnel for the purpose of an observation of ground subsidence, while Node 101 located 16 m away from the excavation center is used as a reference of Node 102.

The change of the inclination has been simplified as a uniform motion. Hereby, the movement model of the inclination of the sensor node can be defined as:

In which ${\theta}_{k}$ and ${\omega}_{k}$ are the tilt angle and the angular velocity at time $k$, $T$ is the measuring time interval, ${w}_{k}$ is the process noise vector at time $k$.

During the monitoring, the measurements were conducted every ten seconds, thus the measuring time interval

$T$ is taken as 10. Therefore, the Kalman filter model is built as:

where

${x}_{k}=\left[\begin{array}{c}{\theta}_{k}\\ {\omega}_{k}\end{array}\right]$,

$A=\left[\begin{array}{cc}1& 10\\ 0& 1\end{array}\right]$,

$\mathsf{\delta}=\left[\begin{array}{c}1\\ 1\end{array}\right]$,

$w$ is the process noise vector and it is normally distributed denoted by

$N\left(0,Q\right)$ and the covariance

$cov\left[w\left(i\right),w\left(j\right)\right]=Q\cdot {\delta}_{ij}$, where

${\delta}_{ij}=\{\begin{array}{c}0\text{}if\text{}i\ne j\\ 1\text{}if\text{}i=j\end{array}$.

Because the state transition matrix $A$ has a high reliability, the result will not be influenced much by $w$, hence we define a relatively small value of covariance as 10^{−7} and thus $Q=\left[\begin{array}{cc}0.0000001& 0\\ 0& 0.0000001\end{array}\right]$.

Since there are two types of sensors (accelerometers and an inclinometers) have been embedded in a sensor node, both of the measurements are going to be processed in the fusion center. Hereby the measurement value

$z$ known as the tilt angle at step

$k$ is defined as:

In which ${z}_{k}=\left[\begin{array}{c}{z}_{k,1}\\ {z}_{k,2}\end{array}\right]$, $H=\left[\begin{array}{c}{H}_{1}\\ {H}_{2}\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 1& 0\end{array}\right]$, ${v}_{k}=\left[\begin{array}{c}{v}_{k,1}\\ {v}_{k,2}\end{array}\right]$. Assuming the measurement noise vector $v$ is white noise with a covariance of 1, and thus ${R}_{k}=\left[\begin{array}{c}{R}_{k,1}\\ {R}_{k,2}\end{array}\right]=\left[\begin{array}{c}1\\ 1\end{array}\right]$.

The initial values of the model are defined as

${x}_{0}=\left[\begin{array}{c}0\\ 0\end{array}\right]$,

${P}_{0}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$. These values are substituted into Equation (21) to Equation (25) in

Table 3, following the Kalman filter cycle as shown in

Figure 5.

Before the application of Kalman filter, the values of dip angles of Nodes 101 and 102 were subtracted by their initial values individually, as there was an initial offset for each sensor due to the soldering. In this way the variation of dip angles of both sensors of Nodes 101 and 102 can be plotted, which are presented in

Figure 10 together with the filtered result. In general, the filtered dip angles show much lower variances compared to the values from the accelerometer and the inclinometer. A discussion and evaluation is given in the following.

#### 4.3. Evaluation and Discussion of the Filtering

As is shown in

Figure 10, vertical bars in blue and green indicate outliers caused by noise interferences from electrical fields and ground vibration when the tunnel boring machine and other construction measures approach, where the sensor node would be affected and thus would obtain inaccurate measurements. Regarding the accelerometer, the combination of the three 1-axis accelerometer has enabled a normalization in calculating the normal vector, but the relatively low sensitivity of 0.0637°/count has reduced the reliability of the deduced dip angle. As for the inclinometer, in spite of the high sensitivity of 0.00179°/count compared to the accelerometer, the output from the inclinometer could not be normalized because only two orthogonal axes were deployed with inclinometers. Therefore, to carry out a data filter to identify the outliers on the observed data before they are used for an assessment is necessary and valuable. In

Figure 10, the scale of the outliers for the inclinometer is evidently much bigger than that of the accelerometer for both nodes. Hence, it can be inferred that with a strong effect of noise interferences and ground vibration, the difference of sensitivity would not be the main factor of malfunction any more. Instead, the defect of normalization of the inclinometer has resulted in higher ratio of outliers and an unstable performance.

To explore more information hidden from the variations of dip angles, standard deviations of the filtered dip angles were calculated every hour for both sensor nodes, and amplified filtered variations of dip angles with sectional standard deviation values is presented in

Figure 11. The standard deviation values are mostly between 0.0172 and 0.0266, but two exceptions appear in the first two hours for Node 102 with a value of 0.0481 and 0.0373 respectively. On the other hand, Node 101 that was deployed 16 meters away performed regularly in the first two hours compared to its latter performance. Hereby, the abnormality of Node 102 indicates a disturbance of the data when the tunnel boring machine passed through the area underground.

Taking the second measuring hour for Node 102 as an example, a comparison was made between the filtered dip angles based on different algorithms of Kalman filter.

Figure 12 shows the plots of dip angle variations over time after distinct filtering. The blue line corresponds to the filtered dip angles based on measurements from the 3-axis accelerometer, while the green line represents the result corresponding to the two orthogonal 1-axis inclinometers. The red line shows centralized filtered dip angles on the basis of both the accelerometer and the inclinometer, and the average of two individual filtering (blue and green line) is also indicated with purple dot line. It can be observed that the centralized filtering is not always consistent with the curve of the average. The standard deviation of the inclinometer line is much higher than the accelerometer line, which indicates that the lack of z-axis of the inclinometer combination has led an instable performance. Nevertheless, considering its higher sensitivity, the variations calculated by inclinometers should still be taken into account, especially when the variations of dip angle are not notable. The standard deviation of the dip angle variations based on both sensor values is higher than that from the accelerometer and lower than that from the inclinometer, which have combined the advantages from both sensors and reduced to the instability to a certain level.

If the sensor node was deployed exactly on the ground above the tunneling center, it is not possible to evaluate the ground settlement by just calculating the tilt angle from only one single node located at the center along the longitudinal or the transversal direction. Nevertheless, when a series of sensor nodes are closely located on the ground from a distance away to the excavating center, a smooth subsidence map can be achieved.

Figure 13 presents a sketch showing the way of deploying sensor nodes in order to draw a settlement curve in the cross section. Simplifying the subsided ground surfaces between two adjacent sensor nodes are even, the absolute ground subsidence in the section can be calculated via the tangent function presented as a polyline. After that, a smooth subsidence curve can be obtained by interpolating the polyline (

Figure 13). Anyhow, it has to be considered that modern tunnel driving in urban areas and especially in the Shanghai areas is carried out with large experience and considering larger safety margins to reduce settlements. Therefore, ground displacements and tilt motion are always considerably low.

Finally, as Kalman filter is a recursive solution that can be continually applied to estimate the situation in the next time step and then be updated after obtaining the observation, the a priori estimate made by Kalman filter can be taken into account into a Building Information Model (BIM) to support the management and facility operation. Furthermore, it can be used as a kind of early warning systems to indicate changes in the ground response to the excavation.