Seismic inversion based on L1-norm misfit function and total variation regularization

Abstract

To solve the inverse problems when outliers exist in the seismic data and discontinuities such as layer boundaries need to be clearly delineated and merge the low frequency information to the inverted parameters. L1-norm misfit function, total variation regularization, a priori information constraints, method of Lagrange multipliers, and iteratively re-weighted least squares. Integrating the L1-norm misfit function, total variation regularization and a priori information constraints via the method of Lagrange multipliers, we create the objective function of seismic inversion to solve the inverse problems that outliers exist in the seismic data and discontinuities such as layer boundaries need to be clearly delineated. In addition, the priori information constraints ensure the inverted parameters have low frequency components. The proposed inversion method is successfully tested on noisy synthetic seismic data with outliers and real seismic data. If there are a small number of outliers in the seismic data, we need to do the seismic inversion in a way that minimizes their effect on the estimated parameters. However, the L2-norm misfit function is highly susceptible to even small numbers of inconsistent seismic observations. As an alternative to L2-norm, one can consider the solution that minimizes the L1-norm misfit function (L1MF) which will be more outlier-resistant, or robust, than the L2-norm solution. Of course, there are some alternative techniques to find the favorable regularization parameters. A set of good regularization parameters is the key of the seismic inversion process.