Accurate kinematic models are fundamental to enhance our knowledge of the seismic cycle as well as to improve surface ground motion prediction. However, the solution of the ill-posed kinematic inverse problem is non-unique (e.g., Cohee & Beroza, 1994; Wald & Heaton, 1994; Cotton & Campillo, 1995 and Minson et al., 2013) and, according to current acquisition systems surrounding active faults, this problem is highly underdetermined, in spite of its rather simple formulation as a linear inverse problem. Non-linear formulations of the problem, based on model reduction strategies, alleviate the underdetermination of the problem. However, non-linear formulations imply drastic assumptions on the rupture history and they complicate the use of linear algebra tools to assess the uncertainties of results. Regardless of the assumed inverse formulation, the incorporation of physical constrains and prior information into the inverse problem is necessary to provide more robust and plausible solutions. In this work (Sanchez-Reyes et al. 2018), we present a new hierarchical linear time domain kinematic source inversion method able to assimilate data traces through evolutive time windows. This progressive approach, both on the data and model spaces, does require mild assumptions based on prior knowledge or preconditioning strategies on the slip rate local gradient estimations. Contrary to similar approaches (Fan et al., 2014), this strategy benefits from the sparsity and causality of the seismic rupture while still ensuring the positivity of the solution. While standard regularization terms are used for stabilizing the inversion, strategies based on parameter reduction leading to a non-linear relationship between the source history and the observed seismograms are avoided. Rise time, rupture velocity and other attributes can be extracted later on from the slip-rate inversion we perform. . Satisfactory results are obtained on synthetic benchmarks proposed by the Source Inversion Validation project (Mai et al. 2016) and for the 2016 M$_w$7.0 Kumamoto mainshock. Our specific formulation combined with simple prior information, as well as numerical results obtained so far, yields interesting perspectives for a quasi-real-time implementation and to ease the uncertainty quantification of such ill-conditioned problem.