Editing Inverse problem (section)

==== Inverse problems in the wave equations ====

The goal is to recover the wave-speeds (P and S waves) and the density distributions from [[seismogram]]s. Such inverse problems are of prime interest in seismology and [[exploration geophysics]].
We can basically consider two mathematical models:
* The [[Wave equation|acoustic wave equation]] (in which S waves are ignored when the space dimensions are 2 or 3)
* The [[Linear elasticity|elastodynamics equation]] in which the P and S wave velocities can be derived from the [[Lamé parameters]] and the density.
These basic [[Hyperbolic partial differential equation|hyperbolic equations]] can be upgraded by incorporating [[attenuation]], [[anisotropy]], ...

The solution of the inverse problem in the 1D wave equation has been the object of many studies. It is one of the very few non-linear inverse problems for which we can prove the uniqueness of the solution.<ref name="sciencedirect.com"/> The analysis of the stability of the solution was another challenge.<ref name="ReferenceA">{{cite journal |last1=Bamberger |first1=Alain |last2=Chavent |first2=Guy |last3=Lailly |first3=Patrick |title=About the stability of the inverse problem in the 1D wave equation, application to the interpretation of the seismic profiles |journal=Journal of Applied Mathematics and Optimization |date=1979 |volume=5 |pages=1–47 |doi=10.1007/bf01442542 |s2cid=122428594 }}</ref> Practical applications, using the least-squares approach, were developed.<ref name="ReferenceA"/><ref>{{cite journal |last1=Macé |first1=Danièle |last2=Lailly |first2=Patrick |title=Solution of the VSP one dimensional inverse problem |journal=Geophysical Prospecting |date=1986 |volume=34 |issue=7 |pages=1002–1021 |osti=6901651 |doi=10.1111/j.1365-2478.1986.tb00510.x |bibcode=1986GeopP..34.1002M }}</ref>
Extension to 2D or 3D problems and to the elastodynamics equations was attempted since the 80's but turned out to be very difficult ! This problem often referred to as Full Waveform Inversion (FWI), is not yet completely solved: among the main difficulties are the existence of non-Gaussian noise into the seismograms, cycle-skipping issues (also known as phase ambiguity), and the chaotic behavior of the data misfit function.<ref>{{cite journal |last1=Virieux |first1=Jean |last2=Operto |first2=Stéphane |title=An overview of full-waveform inversion in exploration geophysics |journal= Geophysics|date=2009 |volume=74 |issue=6 |pages=WCC1–WCC26 |url=https://www.researchgate.net/publication/228078264 |doi=10.1190/1.3238367}}</ref> Some authors have investigated the possibility of reformulating the inverse problem so as to make the objective function less chaotic than the data misfit function.<ref name="ReferenceB">{{cite journal |last1=Clément |first1=François |last2=Chavent |first2=Guy |last3=Gomez |first3=Suzana |title=Migration-based traveltime waveform inversion of 2-D simple structures: A synthetic example |journal= Geophysics |date=2001 |volume=66 |issue=3 |pages=845–860|doi=10.1190/1.1444974 |bibcode=2001Geop...66..845C }}</ref><ref name="ReferenceC">{{cite journal |last1=Symes |first1=William |last2=Carrazone |first2=Jim |title=Velocity inversion by Differential semblance optimization |journal= Geophysics |date=1991 |volume=56 |issue=5 |pages=654–663 |doi=10.1190/1.1443082 |bibcode=1991Geop...56..654S }}</ref>