To stabilize the inversion, I use a form of Tikhonov regularization (). This makes the objective function:
39#39 is the regularization operator, and the parameter 40#40 is a Lagrange multiplier that allows us to determine how strong our regularization will be. Note that I have also added the weighting function 41#41.By using an iterative approach, we can create 41#41 in such a way as to allow us to ``expand'' the data space so that the ``new'' events we generate in the model space with the regularization can be propagated to points outside of the true survey geometry, essentially allowing us to keep this energy in the iterative inversion scheme. 41#41 can also allow us to include the Jacobian () so that the amplitudes of the events are correct.
This objective function can be written in terms of the more intuitive fitting goals:
The first fitting goal is the ``data fitting goal,'' meaning that it is responsible for making a model that is consistent with the data. The second fitting goal is the ``model styling goal,'' meaning that it allows us to impose some idea of what the model should look like using the regularization operator 39#39. If we iterate until convergence, solving both fitting goals at the same time, the data fitting goal will prevent the regularization from creating anything in the model that conflicts with the recorded data. The regularization of the model will prevent the noise in the null space of the inversion from blowing up, helps clean up artifacts, and helps to fill in the shadow zones. Unfortunately, 39#39 is generally a poorly conditioned matrix, thereby making the iterative inversion converge slowly.