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The Mexican Migration Project (MMP) is a study that includes samples of undocumented Mexican immigrants to the United States after their return to Mexico. Of particular interest are the departure and return dates of a sampled migrant's most recent sojourn in the United States, and the total number of such journeys undertaken by that migrant household, for these data enable the construction of data-driven undocumented immigration models. However, such data are subject to an extreme physical bias, for to be included in such a sample, a migrant must have returned to Mexico by the time of the survey, excluding those undocumented immigrants still in the US. In our analysis, we account for this bias by jointly modeling trip timing and duration to produce the likelihood of observing the data in such a "snapshot" sample. Our analysis characterizes undocumented migration flows including single visit migrants, repeat visitors, and "retirement" from circular migration. Starting with 1987, we apply our models to 30 annual random snapshot surveys of returned undocumented Mexican migrants accounting for undocumented Mexican migration from 1980-2016. Contrary to published estimates based on these same data, our results imply migrants remain in the US much longer than previously estimated based on analysis that ignored the physical snapshot bias. Scaling to population quantities, we produce lower bounds on the total number of undocumented immigrants that are much larger than conventional estimates based on US-based census-linked surveys, and broadly consistent with the estimates reported by Fazel-Zarandi, Feinstein and Kaplan (2018).