This study quantifies methane emissions in clean areas of Western China using TROPOMI XCH4 observations and a mass conserving method, while explicitly considering observational uncertainty. All resulting physically unrealistic emissions values are successfully identified and filtered using a mixture of a threshold-based approach and a stochastic approach focused on the impacts of the pixel-by-pixel and day-by-day XCH4 uncertainty. A resulting minimum emission retrieval value of 0.5-1.8 μg/m²/s is obtained, which is lower than existing minimum thresholds using TROPOMI. Additionally, our results demonstrate that there is a 5% emissions bias due to the non-linearity of the gradient term when the XCH4 contains a 10% error. This method does not randomly remove negative values and retain positive values, therefore providing robust and reliable quantification of emissions under conditions of data uncertainty.