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Underground displacement monitoring is an effective method to explore deep into rock and soil masses for execution of subsurface displacement measurements. It is not only an important means of geological hazards prediction and forecasting, but also a forefront, hot and sophisticated subject in current geological disaster monitoring. In previous research, the authors had designed a novel electromagnetic underground horizontal displacement sensor (called the H-type sensor) by combining basic electromagnetic induction principles with modern sensing techniques and established a mutual voltage measurement theoretical model called the Equation-based Equivalent Loop Approach (EELA). Based on that work, this paper presents an underground displacement inversion approach named “EELA forward modeling-approximate inversion method”. Combining the EELA forward simulation approach with the approximate optimization inversion theory, it can deduce the underground horizontal displacement through parameter inversion of the H-type sensor. Comprehensive and comparative studies have been conducted between the experimentally measured and theoretically inversed values of horizontal displacement under counterpart conditions. The results show when the measured horizontal displacements are in the 0–100 mm range, the horizontal displacement inversion discrepancy is generally tested to be less than 3 mm under varied tilt angles and initial axial distances conditions, which indicates that our proposed parameter inversion method can predict underground horizontal displacement measurements effectively and robustly for the H-type sensor and the technique is applicable for practical geo-engineering applications.

For people throughout the world, geological hazards and disasters are always a potentially catastrophic and costly risk in terms of human lives, property and ecosystem destruction. In the past 100 years, large and super large geological disasters have frequently occurred worldwide [

Underground displacement monitoring can measure the subsurface layered displacement deep inside the studied rock and soil masses. It can effectively determine the deformation mode of geological hazards and geotechnical engineering, locate the position and depth of sliding surfaces, and evaluate the deformation ranges and changing dynamics, so it can provide more comprehensive and reliable information for deformation mechanics analysis, stability/safety assessment, hazard prediction and forecast, prevention and mitigation projects design.

However, due to the invisibility, complexity and terribleness of underground monitoring objects and conditions, underground displacement monitoring technology has suffered from slow development. Up to now there exist only a few practicable underground displacement monitoring instruments such as inclinometers, settlement gauges, extensometers and TDRs [_{i}_{0}), mutual inductance voltage measurement (_{o}

The geometrical parameters of the proposed H-type sensor are listed in

For the H-type underground displacement sensor, these two sensing units should be vertically pre-buried into the drilling hole and tightly backfilled to make them synchronously deform with the surrounding rock and soil masses before working. During the working process, the lower and the upper sensing units function as the signal excitation unit and signal receiving unit, respectively, and are called Solenoid I and II accordingly. To adapt to the sensor's sensing properties [_{0} (_{0} between them remains unchanged, so the H-type underground displacement sensor mainly serves for projects as translational sliding, side slope, foundation pit engineering, highway and railway roadbed measurements which mainly require underground displacement monitoring in the horizontal direction.

According to the electromagnetic induction law, if a sine voltage _{i}_{o}_{0} = 0) as shown in _{0} = 0) as shown in _{0}) between Solenoid I and II in real-time, so the sensor can reversely derive the relative horizontal displacement Δ_{o}_{0}) under the given geometrical parameters for these two solenoids (e.g., diameter

Centered by the H-type underground displacement sensor and combined with the design idea of a linear sensor array, we have further proposed a novel distributed underground displacement measuring apparatus. As

Meanwhile, after comprehensive research on various factors and parameters affecting the sensing characteristics of the H-type sensor, a mutual inductance voltage measurement model called the Equation-based Equivalent Loop Approach (EELA) was put forward [

In our previous work [

as our previous studies have proven, firstly, it's more efficient to study the horizontal displacement parameter (Δ_{o}_{o}_{i}

after neglecting the helicity and ununiformity of the circular coils wound on Solenoids I and II, each solenoid may be replaced by two “equivalent current loops”, whose diameter and position are determined according to the following rules: the magnetic field generated by these two loops is nearly equal to that generated by the original Solenoid I and II under the counterpart magnetic potential. That is, the mutual inductance between Solenoids can be equivalent as:
_{13}, _{14}, _{23}, and _{24} are the mutual inductances between two equivalent loops 1 and 3, 1 and 4, 2 and 3, 2 and 4, respectively.

full applying related electromagnetic field theory and equations to deduce the analytic expressions of mutual inductance _{13}, _{14}, _{23} and _{24}.

For example, when Solenoid I and II are arranged in the parallel-axial state as depicted in _{ij}_{i}_{ij}_{ij}_{ij}_{ij}_{ij}_{ij}_{i}_{j}_{2}_{n}_{ij}_{ij}_{0} = 4π × 10^{−7}

It deserves emphasis, however, that the monitoring objects of H-type sensors are invisible and complex subsurface rock and soil masses with highly nonlinear characteristics, and the sensor's output is not directly a measured underground displacement and tilt angle but a mutual inductance voltage that lacks a clear physical meaning. Moreover, the specialized EELA mutual inductance voltage measuring model is quite abstract and complex. Hence, another key research topic in underground displacement measurement is how to use our proposed EELA model as the theoretical basis of underground horizontal displacement measurement to establish an efficient and practical underground displacement parameter inversion algorithm, which can directly convert the real-time output of mutual inductance voltage into the H-type sensor's measuring parameters-the relative horizontal displacement and tilt angle at different subsurface depth within the monitored mass.

For these, this paper proposes an underground displacement inversion method based on the EELA forward modeling and approximate optimization inversion theory. It utilizes the above introduced semi-analysis EELA model as the forward model to generate the reference signals of the mutual inductance voltage, which is then used as entry data of the inversion system, together with the measured data of mutual inductance voltage and axial tilt angle of the H-type sensor. Through further application of a comprehensive optimization algorithm, the inversion system can finally realize the underground horizontal displacement parameter inversion for the H-type sensor with fairly high prediction precision.

The parameter inversion method, also called the reverse analysis method in system identification theory, is characterized by the establishment of a reasonable mathematical model to simulate the unknown practical system, and by implemention of an iterative process to modify the model parameter gradually so as to minimize or optimize the error between the model output and the actual output of system in a certain sense [

Output data of the unknown system must be collected as accurately as possible;

Mathematical model to simulate the unknown system must be reasonable and effective;

Parameter adjustment algorithm must be as quick and efficient as possible.

According to the differences between calculation methods, the parameter inversion method can be divided into two categories: the analytical method and the numerical method. The analytical inversion method is mainly used for the parameter inversion with simple geometry and boundary conditions and is generally not suitable for parameter inversion under the complex geotechnical engineering conditions. By comparison, the numerical inversion method has better universality. According to the different back analysis process, it can be classified into three types: the backward solution method, the direct/forward inversion method, and the mapping method. Among them, the backward solution method is established on the basis of inverse matrix calculations, so this method is not universal and only suitable for linear inversion problems. The basic idea of the direct/forward inversion method is converting the parameter inverse process into a certain objective function optimization problems. That is, it first establishes an object function, then directly takes on the forward analysis process and executes the iteration process of least error function to modify and optimize step by step the trial value of the unknown function, thus the direct inversion method is mainly featured by a quite large calculation burden, a wide scope of back-analysis adaptation, and the capability of executing parameter inversions for all kinds of linear and nonlinear problems. At present, the parameter inversion method has mainly developed towards two directions: one is some intelligent parameter inversion methods put forward for the purpose of improving the inversion theoretical study depth, such as the genetic algorithm, artificial neural network algorithm, ant colony algorithm, and simulated annealing algorithm [

Starting from the practical problems of underground displacement monitoring, this paper utilizes the above mentioned EELA theoretical model as the mathematical model of parameter inversion, and combines it with the optimization inversion method to solve the inverse problems of underground displacement for the H-type underground displacement sensor.

Firstly, we introduce briefly the optimization inversion method. Generally speaking, it is a direct parameter back analysis approach based on the optimal control principle, whose basic ideas are as follows [

If we denote a point in _{1}, _{2}, …, _{s}_{i}_{i}_{i}

Methods to solve this kind of problems are called optimal control methods. Basically, they implement various iterative methods for concrete solving: first to set a group of initial estimates for the parameters to invert, then use the iteratively obtained values to modify the initial value gradually, so as to continuously decrease the value of the objective function

One key point in the parameter inversion process is to establish a reasonable and accurate mathematical model to simulate the unknown system. For the H-type underground displacement sensor, thanks to our previous theoretical researches, a mutual inductance voltage measuring theoretical model with relatively high calculation accuracy and suitable for hardware implementation has been established, namely the EELA model, to describe the complex relationship among the H-type sensor's measuring underground horizontal displacement and sliding angle, the sensor's output mutual inductance voltage, and the geometry and shape parameters of sensor units. Therefore, to meet the practical demands of underground displacement monitoring, this paper uses the EELA model as the parameter inversion mathematical model and calls it the forward simulation model. Then, we input the initial model parameters and trial estimate values of inversion parameters into the EELA forward model, and run this model's calculation program to obtain the output sequence of simulated mutual inductance voltage (

Acquisition of mutual inductance voltage data from the H-type sensor. The data may come from two measurable ways: one is the

Execution of the EELA forward simulation process, that is, input of the initial model parameters and horizontal displacement estimate values into the execution program of the EELA forward simulation model to generate the corresponding mutual inductance voltage simulated/theoretical values. The initial model parameters include the shape parameters (e.g., length _{0} and tilt angle _{0}).

Execution of the module parameter adjustment process. That is, compare the simulated values of mutual inductance voltage based on the EELA forward simulation in Step 2 with the measured values of mutual inductance voltage in Step 1 so as to gradually adjust the relative horizontal displacements that are input into the EELA model through the module parameter adjustment algorithm, until the simulated and the measured mutual inductance voltage can meet the required accuracy or reach the minimum discrepancy between them. The final iterative result is then output as the horizontal displacement inversion result.

To examine and evaluate the above proposed underground displacement parameter inversion method, we will conduct a series of experiments of underground displacement parameters inversion for the H-type underground displacement sensor in this section.

The H-type sensor's output includes the mutual inductance voltage _{o}_{0} between two sensor units (Solenoid I and II), and the parameter to invert is the relative horizontal displacement value between them, namely the sensor's measuring underground horizontal displacement at some given underground depth. The related parameter inversion experiment setup and assessment scheme will first be briefly introduced.

Experiments were performed on the electromagnetic underground displacement measurement testing platform, which is established at China Jiliang University's Geological Disaster Monitoring Equipment Research Center and has been detailed in our previous paper [_{0} between Solenoid I and II, and the sensor's output of mutual inductance voltage _{o}

The horizontal displacement inversion experiments and evaluation are set up as follows:

Before the experiment, the initial axial distance _{0} and axial tilt angle _{0} for the H-type sensor are set as fixed values, and the initial central distance _{0} is fixed to 0. During the experiment, we change point-by-point the sensor's relative horizontal displacement _{i}_{oi}_{o}_{o}_{1}, _{o}_{2}, …, _{om}_{1} = {_{1}, _{2}, …, _{m}

As the initial model parameters of the H-type sensor, the shape parameters (length _{0}, central distance _{0} and tilt angle _{0}, are all required to be set as the same values of step (a)) are input into EELA forward simulation model. At the same time, a trial estimated sequence of horizontal displacement _{1}, _{2}, …, _{m}

Implementation of the proposed EELA forward simulation-optimization inversion algorithm on every point (_{oi}_{i}_{1}, _{2}, …, _{m}

Comparison of the degree of fitting between the measured horizontal displacement series in step (a) and the inversed horizontal displacement series in step (c), so as to verify the adaptability of the above proposed parameters inversion method for the H-type sensor.

_{0} (0°, 5°, …, 30°) when specifying the axial distance _{0} at 115 mm. _{0} and _{0} into the calculation program of EELA forward simulation model together with the sensor's geometrical parameters (diameter

_{0} to 0°, 5°, …, and 30°, respectively.

After a detailed analysis on the above table and figures, it can be concluded that: during the process of varying _{0} from 0° to 30°, there only exist small overall deviations between the modeling inversed and experimentally measured horizontal displacement, and good curve fitness has displayed between them. It can be seen that when _{0} is varied from 0° to 30°, the relative and absolute mean inversion deviations of horizontal displacement are ranged between −1.54–1.06 mm and 0.32–2.50 mm, respectively. These experimental results initially show that it is reliable and effective to applying the proposed “forward simulation-optimization inversion method” on the underground horizontal displacement inversion for the H-type sensor with quite acceptable prediction accuracy and stability.

To analyze the influence of measurement noise for parameter inversion, the comparisons between _{0} is 30°, there exists relatively big measurement deviation between the measured and modeled voltage, the absolute average inversion deviation reaches 2.50 mm. This infers that to achieve the good underground displacement parameter identification, the following basic conditions are required: small measurement noise, reliable parameter inversion algorithm and accurate forward modeling.

As can be seen from _{0} are equal to 25° and 30°, there are relatively obvious deviations between the experimentally measured and EELA simulated mutual inductance voltage. Here we give a brief explanation. The mutual inductance voltage graph shown in _{0}. Before measurement of each mutual inductance voltage curve, it is required that the initial central distance _{0} between two sensing units be adjusted to 0 mm. However, along with the increase of _{0} it's hard to guarantee adjustment of _{0} to be precise zero, thus causing some voltage measurement error to parameter Δ

On the other hand, as _{0} is large, both the measured and simulated mutual inductance voltage are not decreasing monotonically as the horizontal displacement Δ

All these have overcome the problem of non-convergence or serious inversion distortion during the inversion process of parameter Δ_{0} (such as 110 mm, 115 mm, …, 135 mm) and made comparisons with the counterpart experimentally measured values. These inversion and comparison results show that during the entire process of _{0} varied from 110 mm to 135 mm (varied by 5 mm intervals), the horizontal displacement inversed values always show quite good fitness with the measured values. More specifically, both the relative and absolute inversion deviation averages are generally controlled within 3 mm when the variation range of measured horizontal displacements is set as 0–100 mm. It further testifies the effectiveness and accuracy of the proposed parameter inversion method to deduce the measuring underground displacement parameter for the H-type sensor.

To provide a direct and close-up comparison to the inversion results when _{0} is set as 115 mm, _{0} is set as 125 mm while _{0} is changed from 0° till 30° with 5° change intervals. Similarly, it can been found that when _{0} is varied from 0° to 30°, the relative and absolute mean inversion deviations of horizontal displacement are changed between −2.87–2.24 mm and 0.59–3.02 mm, respectively. So it proves again that it is quite reliable and accurate to apply the proposed “EELA forward model-optimization inversion method” to derive the measuring underground horizontal displacement for the H-type sensor.

Sudden geological disasters cause great harm. They not only seriously threaten the safety of human life, but can also greatly damage the development of human society, economic development and the resource environment. Displacement variation is a direct reflection of the movement and deformation characteristics of geological disaster masses, so underground displacement monitoring is one of the important methods and bases for geological disaster prediction and forecasting.

In our previous studies we designed a simple and novel electromagnetic underground displacement sensor, namely the H-type sensor. It can convert the measuring underground horizontal displacement and tilt angle to the variations of mutual inductance and inclination measuring voltage. Meanwhile, we proposed a quite accurate and efficient mutual inductance voltage measuring model, namely, the EELA model, to describe the functional relationship among the H-type sensor's output of mutual inductance voltage, the measuring parameters of underground horizontal displacement and tilt angle, and its geometry and shape parameters.

Based on the above research, this paper has presented an underground horizontal displacement parameter inversion approach called the EELA forward simulation- optimization inversion method, which has the following features: first, it applies the EELA-based mutual inductance voltage measuring model as the H-type sensor's forward simulation model to generate the simulated signal of mutual inductance voltage. Second, both the simulated signal and measured signal of mutual inductance voltage, together with the sensor measured tilt angle and initial model parameters, are input into the parameter inversion system and applied with the comprehensive optimization inversion algorithm, to realize the inversion of underground displacement parameter for the H-type sensor.

A series of comparative studies between the inversed and the measured values of horizontal displacement when both the tilt angles _{0} and initial axial distances _{0} are changed have been conducted. The study results show that the inversion deviation is stable and less than 3 mm when the measured horizontal displacements are varied in 0–100 mm range under different _{0} and _{0} conditions, so it is verified that the proposed EELA forward simulation-optimization inversion method is effective for deducing the underground horizontal displacement for the H-type sensor with quite good inversion accuracy and stability.

This work is funded by the National Natural Science Foundation of China (NSFC) of the Special Fund for Basic Research on Scientific Instruments under Grant 61027005, the NSFC General Project under Grant No. 51074146 and 41376111, the National Science and Technology Support Plan of China under Grant No. 2012BAK10B05-3, and Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ13F010003.

Schematic diagram of the H-type electromagnetic underground displacement sensor. (_{0}.

Schematic diagram of the distributed underground displacement measuring apparatus made up of a series of electromagnetic underground displacement sensors.

Schematic diagram of displacement parameter inversion for the H-type sensor.

Schematic diagram of the experimental setup.

Measured mutual inductance voltage of H-type sensor (_{0} = 115 mm).

Forward simulated mutual inductance voltage of H-type sensor (_{0} = 115 mm).

Horizontal displacement inverse results of H-type sensor (_{0} = 0°, _{0} = 115 mm).

Inversed horizontal displacement of H-type sensor (_{0} = 5°, _{0} = 115 mm).

Horizontal displacement inverse results of H-type sensor (_{0} = 10°, _{0} = 115 mm).

Horizontal displacement inverse results of H-type sensor (_{0} = 15°, _{0} = 115 mm).

Horizontal displacement inverse results of H-type sensor (_{0} = 20°, _{0} = 115 mm).

Horizontal displacement inverse results of H-type sensor (_{0} = 25°, _{0} = 115 mm).

Horizontal displacement inverse results of H-type sensor (_{0} = 30°, _{0} = 115 mm).

Geometrical parameters of Solenoids I and II for the proposed H-type sensor.

Diameter ( |
mm | 70 | |

Length ( |
mm | 75 | |

Axial distance (_{0}) |
mm | 115 | |

Coil turns ( |
mm | 400 | wound by 3 layers |

Comparison of the inversed and the measured values of horizontal displacement (_{0} = 115 mm).

0 | −1.54 | 1.59 |

5 | −0.59 | 0.90 |

10 | 1.06 | 1.09 |

15 | −0.24 | 0.32 |

20 | 0.62 | 0.87 |

25 | 0.84 | 1.72 |

30 | 0.60 | 2.50 |

Comparison of the inversed and the measured values of horizontal displacement (_{0} = 125 mm).

0 | −2.87 | 2.87 | |

5 | −2.20 | 2.20 | |

10 | 0.335 | 0.59 | |

15 | −0.652 | 0.65 | |

20 | 0.805 | 1.29 | |

25 | 1.52 | 2.08 | |

30 | 2.24 | 3.02 |