A Place where I can dump some ideas and questions

Since I started working on inversion problems, I've often heard many friends and co-workers say that the process of inversion is like a black box. There are many parameters that have to be tuned by user itself, making researchers confused if the model is reliable enough. 

I do think this problem is very important. However, I am still learning in this field. I am trying to figure it out on the basis of my limited knowledge. First, inversion is definitely a black box as long as the researchers are careful enough. It is built on strict mathematics and your input data. Practitioners can always check the data improvements (e.g. waveforms and traveltimes in seismic tomography) and realize why the model is updated in a certain way. Therefore, we should always ensure good data quality, even the dataset is very huge. This is the first step to make sure the robustness of inversion. 

Second, I also support the comment that the results of inversion can be largely affected by users because we can set up the parameters for regularization by ourselves. We can't avoid this problem due to the ill-posed property in the inversion. The thing we can do is testing diffrent combinations of the parameters (such as smoothing and damping) as many as possible and show these results to your audiences. However, the misfit comes from the misfit function you defined, which is not totally perfect, so smallest misfit doesn't mean that the model is the most realistic one. On the other hand, it's also impossible that you are able to say that the point you found is the global minimum. To solve this problem, incorporating known knowledge into your inversion becomes very important.

There are many common ways to add constraints to the inversion to ensure that model updates in a right direction, such as multi-stage strategy in Full-waveform inversion. We can also put some known data (e.g. seismic velocity from well logging data) to limit the change during the inversion. After the iterations, we can also evaluate the updated model based on some well-known structures and see whether the model has improved. For example, the shallow part of a plain area should reveal slow velocities. If it's not problematic in the big picture, and then we move on to the next stage.