Course on Inverse Problems Albert Tarantola Lesson VI: a) General - - PowerPoint PPT Presentation

Princeton University

Department of Geosciences

Course on Inverse Problems

Albert Tarantola

Lesson VI: a) General Formulation of the Inverse Problem

Inverse Problems

In a typical inverse problem, one has: model parameters M = {m1, m2, . . . , mp} ,

• bservable parameters O = {o1, o2, . . . , oq} ,

a relation oi = oi(m1, m2, . . . , mp) predicting the outcome of the possible observations. For short, M → O = ϕ(M) .

The three basic elements of a typical inverse problem are:

• some a priori information on the model parameters, rep-

resented by a volumetric probability fprior(M) defined

• ver M ,
• some experimental information obtained on the observ-

able parameters, represented by a volumetric probability gobs(O) defined over O ,

• the ‘forward modeling’ relation M → O = ϕ(M) that

we have just seen. This leads to the posterior information represented by fpost(M) = 1 ν fprior(M) gobs( ϕ(M) ) , where ν is a normalization constant.

One also has a posterior information gpost(O) on the observ- able parameters gpost = ϕ( fpost ) . The explicit formula for gpost(O) is complicated, but we ob- tain samples of gpost(O) by mapping samples of fpost(M) via ϕ .

Stanford University

School of Earth Sciences

Course on Inverse Problems

Albert Tarantola

Lesson VI: b) Explicit Plotting of the Posterior Volumetric Probability

Example I: Explicitly plotting the poste- rior probability density

⇒ mathematica notebook

Example II: Explicitly plotting the poste- rior probability density

⇒ mathematica notebook