Editing Inverse problem (section)

==== Tools to overcome the first difficulty ====
The first difficulty reflects a crucial problem: Our observations do not contain enough information and additional data are required. Additional data can come from physical '''prior information''' on the parameter values, on their spatial distribution or, more generally, on their mutual dependence. It can also come from other experiments: For instance, we may think of integrating data recorded by gravimeters and seismographs for a better estimation of densities. 
The integration of this additional information is basically a problem of [[statistics]]. This discipline is the one that can answer the question: How to mix quantities of different nature? We will be more precise in the section "Bayesian approach" below.

Concerning distributed parameters, prior information about their spatial distribution often consists of information about some derivatives of these distributed parameters. Also, it is common practice, although somewhat artificial, to look for the "simplest" model that reasonably matches the data. This is usually achieved by [[Penalty method|penalizing]] the [[Lp space|<math>L^1</math> norm]] of the gradient (or the [[total variation]]) of the parameters (this approach is also referred to as the maximization of the entropy). One can also make the model simple through a parametrization that introduces degrees of freedom only when necessary.

Additional information may also be integrated through inequality constraints on the model parameters or some functions of them. Such constraints are important to avoid unrealistic values for the parameters (negative values for instance). In this case, the space spanned by model parameters will no longer be a vector space but a '''subset of admissible models''' denoted by <math>P_\text{adm}</math> in the sequel.