A Theoretical Model to Predict Both Horizontal Displacement and Vertical Displacement for Electromagnetic Induction-Based Deep Displacement Sensors

Deep displacement observation is one basic means of landslide dynamic study and early warning monitoring and a key part of engineering geological investigation. In our previous work, we proposed a novel electromagnetic induction-based deep displacement sensor (I-type) to predict deep horizontal displacement and a theoretical model called equation-based equivalent loop approach (EELA) to describe its sensing characters. However in many landslide and related geological engineering cases, both horizontal displacement and vertical displacement vary apparently and dynamically so both may require monitoring. In this study, a II-type deep displacement sensor is designed by revising our I-type sensor to simultaneously monitor the deep horizontal displacement and vertical displacement variations at different depths within a sliding mass. Meanwhile, a new theoretical modeling called the numerical integration-based equivalent loop approach (NIELA) has been proposed to quantitatively depict II-type sensors' mutual inductance properties with respect to predicted horizontal displacements and vertical displacements. After detailed examinations and comparative studies between measured mutual inductance voltage, NIELA-based mutual inductance and EELA-based mutual inductance, NIELA has verified to be an effective and quite accurate analytic model for characterization of II-type sensors. The NIELA model is widely applicable for II-type sensors' monitoring on all kinds of landslides and other related geohazards with satisfactory estimation accuracy and calculation efficiency.