The sequential reconstruction of source waveforms under a sparsity constraint is considered from a Bayesian perspective. 

Let the wave field which is observed by a sensor array be caused by a spatially-sparse set of sources.
A spatially weighted Laplace-like prior is assumed for the source field and the corresponding weighted LASSO cost function is derived.  After the weighted LASSO solution has been calculated as the maximum a posteriori estimate at the current time step, the posterior distribution of the source amplitudes is analytically approximated.  This results in a sequential update for the LASSO weights. 

The  method is evaluated numerically using a uniform linear array.