# Water Pipeline Leakage Detection Based on Coherent \(\varphi\)-OTDR and Deep Learning Technology
## 2 Sensing Principle of \(\varphi\)-OTDR  
Optical pulses propagate through optical fibers, and due to the uneven refractive index distribution of fibers, Rayleigh scattering occurs [17]. Some of the scattered light will be back-guided towards the input end of the fiber and will be detected with a delay proportional to the distance of the diffusing point, which is called optical time-domain reflection. There is a relationship between the length \(d\) of the optical fiber link through which the incident light passes and the time delay \(\tau\) between the incident light and the scattered light:  
\[d=\frac{c}{2n}\tau \tag{1}\]  
Here, \(c\) represents the speed of light in a vacuum and \(n\) denotes the effective refractive index of the fundamental mode of SMF. The system's positioning accuracy, denoted as \(z\), primarily depends on the pulse width, represented by \(T\); then, the spatial resolution \(z\) can be written as follows:  
\[z=\frac{Tc}{2n} \tag{2}\]  
When the refractive index of fiber section at the corresponding location undergoes slight changes, an initial determination of the signal position can be achieved. However, within the optical fiber, the power of Rayleigh scattering light is weak, making it challenging to work over long-distance fiber links. To enhance the sensitivity of scattered light detection, coherent detection is commonly employed. The optical time-domain reflectometer (OTDR) technology based on coherent detection is referred to as a Coherent Optical Time-Domain Reflectometer (COTDR) [18; 19]. The schematic diagram of a COTDR is shown in Figure 1.  
The laser beam generated by the narrow-linewidth laser (NLL) is divided into two paths through a fiber coupler. The upper path serves as the probing light while the other acts as the local oscillator light. The probing light passes through an acousto-optic modulator (AOM), where it is modulated into optical pulses. These pulses are injected into the optical fiber through a circulator, resulting in Rayleigh scattering along the fiber under test (FUT). The scattered light then enters the coupler and interferes with the local oscillator light. The interference results in an optical intensity proportional to the local oscillator's light amplitude. Coherent detection effectively amplifies the scattered light. In Figure 1, the local oscillator is denoted as \(E_{L}\) and the scattered light is denoted as \(E_{S}\), and according to [PERSON]'s equations, we can derive the following:  
\[E_{L}=E_{0}{}^{i\omega_{0}t} \tag{3}\]  
Here, \(E_{0}\) represents the amplitude of the local oscillator, \(\omega_{0}\) represents the angular frequency of the local oscillator, and \(t\) represents the detection time. \[E_{s}=rE_{1}e^{-\alpha z}e^{i(\omega_{s}t+2\beta z+\varphi_{s})} \tag{4}\]  
Here, \(r\) represents the Rayleigh scattering coefficient, \(E_{1}\) represents the incident light amplitude, \(\alpha\) represents the fiber attenuation coefficient, \(z\) represents the location of the Rayleigh scattering point, \(\omega_{s}\) represents the angular frequency of the detection light, \(\beta\) represents the propagation constant of the propagating mode at the operating wavelength, \(t\) represents the detection time, and \(\varphi_{s}\) represents the Rayleigh scattering phase shift. The intensity of the light after the interference between the scattered light and the local oscillator is denoted as \(I\).