## 1 Introduction  
Natural gas is a kind of high-quality, efficient, clean energy and raw material. Since the 1970s, the development of natural gas in the world has been accelerated, and the research on gas exploitation, transportation, and storage has received increasing attention [1; 2; 3; 4; 5; 6; 7]. Nowadays, natural gas pipelines account for almost half of the total length of pipelines in the world. Natural gas is poisonous and explosive. Therefore, it is important that natural gas pipelines are maintained in a safe manner. Leakages are the major causes of in-service natural gas pipeline accidents. In practice, natural gas pipeline leakages can be divided into two types--sudden and rapid leakages with a high leakage rate, and slow leakages with a low leakage rate, also known as seepage. Sudden leakages are mostly caused by third-party earth work, heavy vehicle rolling, geological changes, and impacts [8; 9]. There is extensive literature on sudden leakage detection [10; 11; 12; 13], such as the commonly used negative pressure wave (NPW) method [14; 15; 16] and the fiber-optic-based method [17; 18]. The principle of NPW is that when leaks develop in a natural gas pipeline, the gas density near the leak point will decrease rapidly. This phenomenon results in a negative pressure wave which propagates through the gas from the leak point. The NPW method often employs pressure sensors or other types of sensors that can detect wave propagation. Thanks to the recent development in structural health monitoring [19; 20; 21; 22; 23; 24; 25; 26] and damage detection [27; 28; 29; 30; 31; 32; 33], many new types of sensors, such as piezoceramic transducers [34; 35; 36; 37; 38; 39] and fiber optic sensors [40; 41; 42; 43], have been developed for civil infrastructure. Because of their high bandwidth and low cost, piezoceramic transducers have been applied to pipeline monitoring [44; 45; 46]. On the other hand, because of their distributive nature and small size, fiber optic sensors are often reported for pipeline monitoring [47; 48; 49]. In the NPW method, these sensors are installed upstream and downstream of a pipeline to detect negative pressure waves [50; 51; 52; 53; 54]. The signal captured by these pressure sensors can be processed to determine whether there is a gas leak. Fiber-optic-based methods usually use distributed fiber optical sensors to detect leakages in natural gas pipelines; a distributed fiber optical sensor installed along the pipeline can detect the temperature variation or vibrations caused by leakage. The commonality of these two methods is the utilization of sensors. This leads to a potential problem that if the sensors are not accurate enough, it is highly possible to miss the alarms for small leakages. Slow small leakages (seepage) usually occur in weak links of the pipelines, such as welding seams, flanges, or valves [55; 56; 57], and they are difficult to detect due to their concealment. In addition, pipelines are subject to corrosion [58; 59; 60], erosion [61], and cracking [62; 63; 64], which may also cause small leakages at the initial stage. Currently, there are only limited studies in this area [65; 66]. At present, the effective method for small leakages is based on the real-time model method. Compared with other leakage detection methods, the real-time model method can provide all the characteristics of pipeline flow and fluid properties. It also considers the impact of physical properties of pipelines. In addition, the real-time model considers the influence of system noise and measurement noise and has advantages in small leakage detection. However, it also has some disadvantages, such as complex computation and slow convergence. The Kalman filter is a kind of filtering algorithm that estimates the required states through algorithms from the observation quantity related to the extracted signal, however, it can only be used for linear systems. The Kalman filter that can estimate states for a nonlinear system is called the extended Kalman filter (EKF) [67].