## 1 Introduction  
Ground surface deformation monitoring during urban tunnel excavations is a common means for ensuring construction quality and safety, including the safety of people and equipment inside the tunnel and structures located on and near the ground surface. Conventional ground surface deformation monitoring involves the use of total stations, levels, inclinometers, and other devices that need to be deployed on the ground surface. However, it is inconvenient to utilize these conventional surveying techniques in open water areas, such as rivers and lakes. During the past three decades, numerous underwater transportation tunnels have been built in large cities in China. Underwater tunnel excavations often carry additional risks compared to ordinary tunnel excavations and could suffer severe consequences. Usually, shallow tunnels beneath rivers are driven in \"soft\" sediments comprising of silt, clay, sand, and gravel. Surrounding soil disturbances and tunnel deformations may lead to cracks and water infiltration, thus, threatening the safety of tunnel construction and operation. Accurate, continuous, long-term, and near real-time riverbed deformation monitoring has become a challenging topic for both researchers and engineers in urban tunneling. There is an urgent need for developing new techniques and methods for riverbed deformation monitoring. Global positional system (GPS) techniques, global navigation satellite systems (GNSS) in general, have been frequently applied for studying dynamic behaviors of engineering structures during the past two decades [1; 2; 3; 4; 5; 6; 7; 8]. Recently, GPS techniques have been applied for monitoring long-term stability and static deformation of engineering structures [9; 10; 11]. The deformation information is usually used to determine the state of the structure's health. Thus, the process of implementing GPS for detecting and monitoring the stability and deformation of engineering structures is a component of structural health monitoring (SHM), which focuses on assessing the health status of infrastructure using multiple techniques. SHM has become a cross-disciplinary research field encompassing fields such as civil engineering, surveying, remote sensing (ground-based, drone-based, satellite-based), mechanical engineering, structural dynamics, signal processing, computational hardware and software, data telemetry, smart sensors, machine learning and pattern recognition, as well as other fields yet to be defined [12; 13; 14; 15]. The principal advantage of using satellite positioning techniques for SHM, as opposed to classical surveying techniques, is that they provide continuous and real-time (or near real-time) measurements in three directions. There have been a large number of studies on GPS-based SHM systems over the last two decades, mostly utilizing kinematic GPS positioning techniques with a sampling rate of one sample per second (1 Hz) or higher to measure rapid movements of an GPS antenna, which indicate oscillations or kinematic deformations of engineering structures. Kinematic GPS has become a powerful tool for SHM. However, kinematic GPS techniques often fail to detect minor and gradual movement (deformation) of structures. This study focuses on the detection and monitoring of minor and gradual movements using static GPS positioning. The term \"detection\" primarily focuses on identifying abrupt positional-changes, which often requires near real-time positioning updates and decision making. The term \"monitoring\" focuses on slow and gradual positional-changes, such as settlement, uplift, tilt, and rotation of engineering structures. The distinction between the terms \"abrupt displacement\" and \"gradual displacement\" is somewhat nebulous because a field event always happens during a time period. For example, a positional shift continued for several hours may appear as an abrupt or instant step in daily-displacement time series; however, it may appear as a slope in hourly-displacement time series.