## 1 Introduction
### Introduction of GPS Multipath Error and Its Mitigation  
Multipath effect is caused when direct signals are received with reflected signals at an antenna as shown in Figure 1. In theory, the maximum magnitude of carrier phase multipath error is a quarter of the signal wavelength. It affects positioning accuracy and availability (due to incorrect or failed ambiguity resolution) in high precision GPS positioning applications. Many methods have been developed to detect and mitigate GPS multipath errors. [PERSON] et al. [4; 5] propose using Support Vector Machine (SVM) to estimate multipath errors with inputs of satellite elevation and azimuth by taking advantage of sidereal repeatability of GPS multipath signals. There are many other GPS multipath mitigation techniques based on the sidereal repeatability of GPS multipath signals [6; 7; 8; 9]. Wavelet decomposition is also widely used to mitigate multipath errors [10; 11; 12]. These methods can only be applied to static antennas (such as at permanent International GNSS Service (IGS) stations) in unchanged environments, where the multipath repeats every sidereal day.  
Multipath errors can be mitigated using stochastic models based on Carrier-to-Noise Ratio (CNR) or Signal-to-Noise Ratio (SNR) [13; 14]. Simple stochastic models are based on the satellite elevation angle (E) [13; 15]. For example, assuming the variance of the carrier phase error to be:  
\[\sigma^{2}=1/\sin^{2}\mathrm{E}. \tag{1}\]  
Some improved stochastic models are based on Carrier-to-Noise Ratio (CNR) or Signal-to-Noise Ratio (SNR). e.g., SIMGA-\(\varepsilon\) model [15] uses CNR to estimate carrier phase variances \(\sigma^{2}\) using the equation:  
\[\sigma^{2}=\mathrm{C_{i}}\cdot 10^{-\frac{\mathrm{CNR}}{\mathrm{I}0}}, \tag{2}\]  
where \(\mathrm{C_{i}}\) denotes a factor consisting of the carrier loop noise bandwidth and a conversion term from cycle2 to millimetres2, which includes the \(\mathrm{L_{i}}\) wavelength. CNR is the measured CNR from the GPS data. Some receiver and antenna designs can mitigate multipath effects. Special hardware designs such as vision correlator [16] and Multipath Estimating Delay-Lock-Loop [17] have been used by some receiver manufacturers to mitigate multipath error. [PERSON] and [PERSON] [18] use multiple closely spaced antennas to model multipath error based on the multipath phase differences of antenna pairs.  
Figure 1: Geometry of multipath signals, a reflector, and an antenna (adapted from [13]).  
Most of the research above is based on static GPS data; this is because it is difficult to precisely assess the carrier phase multipath errors of kinematic data. We present a new convolutional neural network based multipath detection method for both static and kinematic data. With the development of a rotating arm rig by the authors, the multipath error of kinematic data can be assessed in the measurement domain and used to validate the detection method. In addition, a stochastic model based on the detection result is used so that the performance of the new method in the position domain can be compared with a simple elevation model and SIGMA-\(\epsilon\) model. The proposed methods can be used in carrier phase based kinematic positioning, including Real-Time Kinematic (RTK) applications for improving kinematic positioning performance in difficult environments.