During its five years of operation as of 2017, the Sample Analysis at Mars (SAM) Tunable Laser Spectrometer (TLS) on board the Curiosity rover has detected six methane spikes above a low background abundance in Gale crater. The methane spikes are likely sourced by nearby emission from the surface. Here we use inverse Lagrangian modeling techniques to identify upstream emission regions on the Martian surface for these methane spikes at unprecedented spatial resolutions. Inside Gale crater, the northwestern crater floor casts the strongest influence on the detections. Outside Gale crater, the upstream regions extend towards the north. The contrasting results from two consecutive TLS methane measurements point to an active emission site to the west and the southwest of the Curiosity rover on the northwestern crater floor. The observed spike magnitude and frequency also favor emission sites on the northwestern crater floor, unless there are fast methane removal mechanisms at work, or either the TLS methane spikes or the Trace Gas Orbiter (TGO) non-detections can not be trusted.