An efficient method for inferring Manning’s n coefficients using water surface elevation data was presented in Sraj et al. (Ocean Modell 83:82–97 2014a) focusing on a test case based on data collected during the Tōhoku earthquake and tsunami. Polynomial chaos (PC) expansions were used to build an inexpensive surrogate for the numerical model GeoClaw, which were then used to perform a sensitivity analysis in addition to the inversion. In this paper, a new analysis is performed with the goal of inferring the fault slip distribution of the Tōhoku earthquake using a similar problem setup. The same approach to constructing the PC surrogate did not lead to a converging expansion; however, an alternative approach based on basis pursuit denoising was found to be suitable. Our result shows that the fault slip distribution can be inferred using water surface elevation data whereas the inferred values minimize the error between observations and the numerical model. The numerical approach and the resulting inversion are presented in this work.