Aiming at fabricating a false positioning solution towards the victim user, spoofing attacks are nowadays challenging the security of Global Navigation Satellite Systems (GNSS). Due to the relative motion between the spoofer and victim user, the potential inconsistency involved in the measurement of Doppler shift could be an efficient basis to reveal the attacks. Although several researches on anti-spoofing techniques via measured Doppler have been developed, weaknesses in terms of implementation complexity and strict assumptions such as single-antenna spoofing still remain. To this end, Doppler Residual Monitoring (DRM) is proposed and investigated in this paper as a novel anti-spoofing scheme. Firstly, caused by the presence of the spoofing attacks, abnormal bias within the measured Doppler shifts is demonstrated based on the modeling of GNSS spoofing. Due to the difficulty in directly observing the Doppler bias concealed in the counterfeit signal, Doppler positioning technique is exploited to reveal this part of the variation in an indirect way. In particular, Doppler residual provided by Doppler positioning equations is proposed and the relationship between Doppler bias and Doppler residual is illustrated. More specifically, Sum of Squares of Errors (SSE), which denotes the sum of squares of Doppler residual, is utilized as the test statistics for spoofing detection. To further distinguish all the counterfeit signals, a random traversal technique is investigated in this paper. Moreover, Doppler Shift Difference (DSD) is proposed to match the spoofing signals with its transmitting antenna based on the discrimination results, which could be an essential basis for the anti-spoofing techniques towards multi-antenna spoofing scenario. Also provided is the analysis of the theoretical performance along with the simulation results. Finally, the validity of our method is demonstrated through field tests with two spoofing scenarios, where the spoofing attacks are respectively achieved by one antenna and two antennas. Without external facilities assistance, our DRM-based scheme could be a significant complement to the GNSS anti-spoofing techniques as a novel spoofing discrimination method.