## 1. Introduction

The effective monitoring of underground displacement is an important means to prevent and mitigate such major geological hazards as landslide, collapse, debris flow and subsidence [

1,

2,

3]. It is also a practical engineering tool to evaluate quality and risk for a wide range of hydraulic and geotechnical projects, including roads, railways, tunnels, dams, mining sites, and so on [

4,

5,

6,

7]. It can go deeply into the studied rock and soil mass to detect and measure the underground layered displacements and deformation quantity. For various kinds of geotechnical mass, underground displacement monitoring can effectively locate the sliding surfaces, determine the deformation mode, evaluate the deformation ranges and predict the deformation trend. Therefore, it can provide more objective and detailed information for deformation mechanics analysis, stability/safety assessment, hazard prediction/forecasting and prevention and mitigation project design [

8,

9,

10]. However, due to the extremely variable and complex properties of monitored underground geotechnical mass, such as spatial invisibility, temporal randomness, environmental terribleness and geological heterogeneity, underground displacement monitoring technology was slowly developed. Most of the existing underground displacement monitoring methods, including inclinometers, settlement gauges, extensometers and TDR (time domain reflectometry), suffer from certain limitations, including low accuracy, high cost, narrow range, poor adaptability, low automation, poor long-term durability and susceptibility to electromagnetic interference [

11,

12,

13,

14,

15]. In recent years, optical fiber sensing technologies have been rapidly developed for structural health monitoring of geotechnical infrastructures, such as bridges, dams and tunnels [

16,

17,

18], and many exploratory research projects and experiments have been actively conducted for their application on the subsurface displacement monitoring of geotechnical mass [

19,

20]. Compared to the conventional underground displacement instrumentation, optical fiber sensors have specific merits, including tiny size, light weight, high precision, immunity to electromagnetic interference and resistance to corrosion. In the geotechnical domain, Ho [

21] designed an FBG (fiber Bragg grating)-segmented deflectometer to measure the relative deflection segment by segment within the geotechnical mass concerned. Pei [

22] developed an FBG-based in-place inclinometer to monitor the underground displacement of the Weijiagou Landslide in China. Zhu [

23] developed a surface-adhered FBG sensing bar to monitor the internal displacements of a model gravity dam. Optical fiber sensor techniques have revealed application potential to underground displacement monitoring. However, some challenges still exist before their full practical application, including the fiber fragility, limited measuring range, low embedding survival rate and cross sensitivity of strain and temperature [

24].

Thanks to our previous research [

25,

26], an electromagnetic underground displacement three-dimensional (3D) measuring sensor had been advocated on the basis of a unique array structure design of integrated sensing units and integration of various magnetoelectric effects, including electromagnetic induction, the Hall effect and the magnetoresistance (MR) effect. It was abbreviated as the electromagnetic underground displacement 3D sensor.

As

Figure 1 shows, the proposed underground displacement 3D sensor is composed of a number of cylindrical electromagnetic sensing units and an information central processing unit. Each sensing unit has an identical structure, whose outer wall is an air-cored solenoid and inner wall is embedded with the integrated sensing PCB (printed circuit board). All sensing units are serially connected through the power lines, communication lines and signal lines and linked as a chain with high elastic connecting materials after epoxy resin coverage. The monitoring data of sensing units can be further transmitted to the remote computer through the GPRS (general packet radio service) wireless network for processing, display and prediction, thereby realizing a real-time measuring and monitoring of underground 3D displacements.

**Figure 1.**
Schematic diagram of the electromagnetic underground 3D measuring sensor. GPRS, general packet radio service.

**Figure 1.**
Schematic diagram of the electromagnetic underground 3D measuring sensor. GPRS, general packet radio service.

During the actual installation process, all of the sensing units are vertically buried into a drilling hole of equidistance and backfilled tightly. The coordinates of the top (surface) sensing unit are determined by GPS or mapping standard point coordinates. Along with the sliding and deformation of surrounding rock and soil mass, the relative displacement and inclination may occur between any two adjacent sensing units. According to the electromagnetic induction and Hall effects, the output values of mutual inductance U_{o} and Hall voltage U_{H} between two adjacent units will be changed accordingly. Meanwhile, the sensor’s built-in integrated tilt measuring module can measure the relative tilt angle θ_{0} between them. By implementing a certain measurement algorithm, the simultaneous variations of U_{o}, U_{H} and θ_{0} can be converted to the measuring values of relative horizontal displacement, vertical displacement and tilt angle between two corresponding adjacent sensing units. During the whole measuring process, under the centralized control of the information processing unit, the relative horizontal displacements, vertical displacements and tilt angles between two adjacent units can be successively measured from bottom to top. Therefore, the sensor can realize an automatic, real-time and complete monitoring and measuring of underground horizontal displacements, vertical displacements and tilt angles from the surface to different depths (till to the bedrock) within the detected underground mass.

Compared to the existing underground displacement monitoring instrumentations, our proposed underground displacement 3D measuring sensor has such sensing properties as follows

- (1)
Simple structure, convenient manufacturing and relatively low cost.

- (2)
Relatively convenient installation and good portability thanks to its measuring chain assembly.

- (3)
Big measuring range, quite high measuring and positing accuracy.

- (4)
Good consistency between the measured and the actual deformation of geotechnical mass, thanks to the flexible structure of the sensing unit measuring array, where each sensing unit can freely move and deform accompanying the movement of the surrounding geotechnical mass.

- (5)
Three-dimensional measurement of the geotechnical underground displacement, rather than one-dimensional measurement by most of the existing instruments.

For the proposed sensor, in order to achieve an automatic and accurate measurement toward the underground 3D displacements, not only high quality design and manufacturing are required, but also an in-depth theoretical research on the underground displacement sensing mechanism and measuring property is urgently needed. Among them, how to establish some precise and efficient underground displacement parameter measuring models and parameter inversion methods is of vital importance.

In our previous works, thanks to comprehensive studies on various factors and parameters affecting the proposed sensor’s sensing characteristics, two underground displacement measuring models with such virtues as sound estimation accuracy, high computation efficiency and easy hardware implementation were advocated. They are the NIELA (numerical integration-based equivalent loop approach) mutual inductance voltage measuring model [

26] and EMC-NI (equivalent magnetic charge-numerical integration model) Hall voltage measuring model [

27]. These two models can quite accurately evaluate the complicated relationship among the proposed 3D sensor’s varied output of mutual inductance voltage (

U_{o}) and Hall voltage (

U_{H}), respectively, the measuring parameters—The relative horizontal displacement, vertical displacement and tilt angle between any two adjacent sensing units—And the shape, geometry and material property parameters of the sensing units. The measuring parameters can represent the relative horizontal displacement, vertical displacement and tilt angle at different underground depths, namely the different buried depths of sensing units within the monitored geological mass.

However, the monitoring objects of the underground displacement 3D sensor are mainly the complex, variable and invisible subsurface rock and soil mass with complicated nonlinear characteristics. The sensor’s output are not directly measuring underground displacements and sliding angles, but the mutual inductance voltage and Hall voltage, short of physical meaning. Moreover, these two mentioned measuring models are quite abstract and complex. Therefore, a focal point should be taken when conducting the fundamental research of underground displacement measurement theory and methods, namely the study of the underground displacement parameter inversion approach for the proposed sensor. This implies how to make full use of the previously presented NIELA and EMC-NI models as an underground displacement measuring theoretical basis to establish some reliable and efficient underground displacement inversion approaches, to convert the real-time output of mutual inductance voltage and Hall voltage into the sensor’s measuring parameters—Underground horizontal displacement, vertical displacement and tilt angle at different depths within rock and soil mass—Directly.

In this paper, an innovative displacement joint inversion method coupling innovative forward modeling and approximate optimization inversion has been proposed. It applies the semi-analysis NIELA and EMC-NI models as the join forward models to generate the reference signals of mutual inductance voltage and Hall voltage simultaneously, which combined with the measured signals and some related parameters, are input into the proposed inversion system. Through further execution of the joint optimization algorithm, the inversion system can finally realize joint inversions of the measuring underground horizontal displacement and vertical displacement parameters for the proposed 3D sensor with fairly high prediction precision and efficiency.

## 2. Sensor Working Principle and Theoretical Modeling

As shown in

Figure 2, our designed electromagnetic underground displacement 3D measuring sensor is mainly worked on the following principle: driven by the movement of surrounding rock and soil mass, a relative horizontal displacement Δ

X, vertical displacement Δ

Z and tilt angle

θ_{0} might synchronously occur between any two adjacent sensing units (referred to as Sensing Units I and II). Due to the electromagnetic induction and Hall effect, the mutual inductance voltage

U_{o} and Hall voltage

U_{H} generated between Sensing Units I and II will be varied simultaneously. Meanwhile, the relative axially tilt angle

θ_{0} between them can be automatically measured by the built-in tilt measuring integrated module. Therefore, by virtue of the establishment of underground displacement measurement relationship models and GPS-based ground coordinate measurement, the proposed sensor can realize sequentially from bottom to top the measurement of Δ

X, Δ

Z and

θ_{0} at different depths within the monitored mass and convert them into the underground 3D deformation coordinates consistent with the GPS space coordinates. It should be stated that the proposed underground displacement measurement relationship models can effectively describe the complex relationship among the variations of

U_{o},

U_{H} and

θ_{0} between any two adjacent sensing units, the measuring parameters (relative horizontal displacement Δ

X, vertical displacement Δ

Z and tilt angle

θ_{0} at corresponding underground depths) and the geometry and property parameters of the sensing units. The geometry and property parameters mainly include the length, diameter and winding coil turns of the sensing units, the shape, size and magnetization characteristics of the permanent magnet, the sensing characteristics of the Hall sensor and the I/O relationship of the Hall voltage measuring circuitry.

**Figure 2.**
Principle diagram of the proposed underground displacement 3D sensor (only two adjacent sensing units are given). (**a**) Initial geometrical arrangement; (**b**) appearances of relative horizontal displacement ΔX, vertical displacement ΔZ and tilt angle θ_{0}.

**Figure 2.**
Principle diagram of the proposed underground displacement 3D sensor (only two adjacent sensing units are given). (**a**) Initial geometrical arrangement; (**b**) appearances of relative horizontal displacement ΔX, vertical displacement ΔZ and tilt angle θ_{0}.

Each sensing unit has the same structure. As shown in

Figure 3, its outer wall is an air-cored cylindrical solenoid, and the inner wall is wedged with several integrated sensing PCB combining such function modules as the permanent magnet, Hall sensor, SCM (single-chip microcomputer), sine voltage generation, mutual inductance voltage measurement, Hall voltage measurement, tilt angle measurement, A/D conversion and RS485 communication.

**Figure 3.**
Sensing unit component diagram.

**Figure 3.**
Sensing unit component diagram.

In consideration of various factors affecting the sensing characteristics and after in-depth theoretical study of the proposed 3D sensor, we had established two quite efficient and accurate semi-analytic underground displacement measuring theoretical models: the NIELA-based mutual inductance voltage measuring model and the EMC-NI based Hall voltage measuring model. These two models have been detailed in our previously published papers [

26,

27]. Here, we only make a brief introduction.

The NIELA (numerical integration-based equivalent loop approach) mutual inductance voltage measuring model was established by technical fusion of the electromagnetic field theoretical study, equivalent loop modeling on solenoid and the numerical integration approach. As

Figure 2 shows, for the proposed 3D sensor, it can qualitatively characterize the functional relationship among the varied outputs of mutual inductance voltage

U_{o} between any two adjacent sensing units, the measuring parameters between them and the geometrical and property parameters of sensing units. Among them, the measuring parameters include the relative horizontal displacement Δ

X, relative vertical displacement Δ

Z and relative tilt angle

θ_{0}. The geometrical and property parameters mainly include the solenoids’ length

h, diameter

d, number of windings

w and the initial vertical distance

Z_{0} and horizontal distance

X_{0} between them. Generally speaking, NIELA is essentially a semi-analytic calculation approach with quite high accuracy.

Through the comprehensive application of Hall sensing mechanism analysis, 3D spatial distribution modeling to the magnetic field of the permanent magnet and the multidimensional numerical calculation method, the EMC-NI (equivalent magnetic charge-numerical integration approach) was proposed and served as the proposed sensor’s Hall voltage measuring model. It provides a quite precise and efficient description of the functional relationship among the sensor’s Hall voltage output U_{H} between Sensing Units I and II, the measuring parameters ΔX, ΔZ and θ_{0} between them and the geometrical and sensing properties of the sensing units and permanent magnet.

## 3. Underground Displacement Parameter Joint Inversion Method

The built-in circuitry modules in the proposed 3D sensor cannot directly output the measuring underground horizontal displacement and vertical displacement, but the relatively abstract physical quantity: the mutual inductance voltage and Hall voltage. Therefore, another important research content of underground displacement measurement is how to make further usage of the above theoretical modeling results to work out some quite efficient and practical underground displacement parameter inversion approaches. The purpose is to inversely deduce the measuring underground horizontal and vertical displacement parameters at different underground depths (the axial tilt angle θ_{0} can be directly measured by the sensor circuitry) from the synchronous output variables of mutual inductance voltage and Hall voltage between Sensing Units I and II.

Generally speaking, the parameter inversion method is an effective way to inversely predict one or more initial parameters (such as speed, displacement, initial stress or geometric parameters) through minimizing the differences between the monitoring data and the modeling calculation results [

28]. It is characterized by intensively studying some field measurable physical parameters (e.g., load, stress and strain) capable of describing the system behaviors and establishing some reasonable mathematical or physical back-analysis models to simulate the unknown practical system. Then, it implements the effective parameter adjustment process to modify the model parameter gradually, so as to minimize the error (difference) between the inversion calculation results and the system actual measurements [

29,

30]. An effective parameter inverse procedure at least consists of three elements [

31]:

- (1)
Data collected from the unknown system should be as precise and timely as possible;

- (2)
The calculation model to describe the unknown system should be sensible and representative;

- (3)
The parameter adjustment algorithm must be efficient and convergent.

Presently, the parameter inversion methods are mainly developed along two ways: (1) some evolutionary inversion methods are developed mainly aimed at raising the inversion theoretical depth or improving the inversion scheme, such as genetic algorithm, artificial neural network, ant colony algorithm, evolution algorithm and simulated annealing [

32,

33,

34,

35]; and (2) some practical and simple inversion approaches are put forward mainly targeted at solving some practical geo-engineering problems [

36,

37].

According to the practical engineering requirements of underground displacement monitoring, this paper proposes a simple and efficient inversion approach of underground displacement parameters with quite high prediction accuracy and efficiency. It is called the “joint forward simulation-optimization inversion method”. It is aimed at solving the simultaneous and direct measurement problem of underground horizontal, vertical displacement and tilt angle for the proposed 3D sensor.

As mentioned, it is very important to establish some representative and accurate calculation models to simulate the unknown system. For our proposed sensor, by virtue of previous studies, two high efficiency and approximate underground displacement measuring theoretical models have been established, namely the NIELA and EMC-NI models, to describe the complex relationships among the sensor’s output of mutual inductance voltage and Hall voltage, respectively, its measuring parameters—Underground horizontal displacement, vertical displacement and tilt angle—And the sensor’s geometry and sensing property parameters. Thereby, according to the practical requirement of underground displacement monitoring and measurement, firstly, this paper combines the NIELA model with the EMC-NI model to establish the parameter inversion mathematical models, which are called the joint forward simulation models of underground displacement. Secondly, the initial model parameters and trial estimates of inversing parameters are input into the NIELA and EMC-NI forward simulation models simultaneously, and after a run of the execution program of forward simulation, output sequences of the simulated (theoretical) values of the mutual inductance voltage and Hall voltage are obtained simultaneously. This process is called the “joint forward simulation process”. Based on these, integration of NIELA and EMC-NI forward simulation models with the combined optimization inversion algorithm makes up our proposing joint parameter inversion method, which can realize a simultaneous inversion of underground horizontal displacement and vertical displacement for the proposed 3D sensor.

Here, we give a brief introduction to the optimization inversion procedure. It is a direct parameter inverse analysis approach following the following optimal control principle:

Let a point in

l-dimensional space be depicted as

h = (

p_{1},

p_{2}, …,

p_{l}); then, the point collection satisfying

m_{i} <

p_{i} <

n_{i} constitutes a specific domain in the

l-dimensional space and may be marked as

p. If the value of target function

$J(\stackrel{-}{p})$ is set as the standard to measure whether {

p} accords with the actual requirement, the parameter inversion problem converts to an optimization search problem, namely to solve {

p} when satisfying:

The solutions to such kinds of problems are called the optimal control methods, which usually resort to the iterative process for solution. First, set a group of initial estimate values for the parameters to be inversed (predicted). Next, repeatedly execute the iterative process and revise these parameters according to the iteration results or feedbacks step by step, so that the value of objective function J (such as the minimum error function) can be gradually decreased under certain restrictive conditions. Then, the optimized values are tested with the specified convergence criteria. Once the convergence conditions are satisfied, the optimization process ends, and the final iteration results are output as parameter inversion values.

**Figure 4.**
Schematic diagram of the displacement parameter joint inversion method. NIELA, numerical integration-based equivalent loop approach; EMC-NI, equivalent magnetic charge-numerical integration.

**Figure 4.**
Schematic diagram of the displacement parameter joint inversion method. NIELA, numerical integration-based equivalent loop approach; EMC-NI, equivalent magnetic charge-numerical integration.

Figure 4 shows a schematic diagram of the “joint forward simulation-optimization inversion method” for the proposed 3D sensor. The inversion process mainly includes three steps:

- (1)
Data acquisition of mutual inductance voltage U_{o}, Hall voltage U_{H} and tilt angle θ_{0} for the proposed sensor. The data may come from two ways: Usually, it is the on-site measuring results when the sensor has been buried into the monitored rock and soil masses for months or years. Sometimes, it can be the experimental measurement values for sensor testing and research purposes, where the values of relative displacement and tilt angle (ΔX, ΔZ, θ_{0}) between two adjacent sensing units are artificially varied, so as to measure and record the synchronous variations of U_{o} and U_{H}. In this paper, the latter way is adopted.

- (2)
Execution of the NIELA and EMC-NI joint forward simulation process. The initial model parameters and the initial estimate values of relative horizontal displacement and vertical displacement are fed into the execution programs of the forward simulation models. Therefore, the simulated values of mutual inductance voltage and Hall voltage can be generated. The initial model parameters include the geometrical and property parameters of the solenoids and permanent magnets.

- (3)
Execution of the joint optimization inverse process. Make a combined comparison between the simulated values of the mutual inductance voltage and Hall voltage obtained in Step (2) with the corresponding measured values in Step (1) and gradually adjust the relative horizontal displacement and vertical displacement values by the joint parameters modification algorithm. This process continues until not only the fitness between the simulated and the measured mutual inductance voltage can meet the required accuracy (or gets the minimum discrepancy under certain conditions), but also the fitness between the simulated and measured Hall voltage achieves optimality under specified constrains. The final iterative results are then output as the joint inversion results of horizontal displacement ΔX and vertical displacement ΔZ for the proposed sensor. They can be treated as the measuring values of relative horizontal displacement and vertical displacement at some underground depth where the corresponding sensing units are buried. The iterative values of ΔX and ΔZ combined with the measured value of relative tilt angle θ_{0} constitute the complete monitoring results at the corresponding subsurface depth within the monitored geological mass for the proposed 3D sensor.