Editing Inverse problem (section)

===== Diffraction tomography =====
Diffraction tomography is a classical linear inverse problem in exploration seismology: the amplitude recorded at one time for a given source-receiver pair is the sum of contributions arising from points such that the sum of the distances, measured in traveltimes, from the source and the receiver, respectively, is equal to the corresponding recording time. In 3D the parameter is not integrated along lines but over surfaces. Should the propagation velocity be constant, such points are distributed on an ellipsoid. The inverse problems consists in retrieving the distribution of diffracting points from the seismograms recorded along the survey, the velocity distribution being known.  A direct solution has been originally proposed by [http://amath.colorado.edu/~beylkin/papers/BEYLKI-1983a.pdf Beylkin] and Lambaré et al.:<ref>{{cite journal |last1=Lambaré |first1=Gilles |last2=Virieux |first2=Jean |last3=Madariaga |first3=Raul |last4=Jin |first4=Side |title=Iterative asymptotic inversion in the acoustic approximation |journal=  Geophysics |date=1992 |volume=57 |issue=9 |pages=1138–1154 |doi=10.1190/1.1443328 |bibcode=1992Geop...57.1138L |s2cid=55836067 }}</ref> these works were the starting points of approaches known as amplitude preserved migration (see Beylkin<ref>{{cite journal |last1=Beylkin |first1=Gregory |title=The inversion problem and applications of The generalized Radon transform |journal=Communications on Pure and Applied Mathematics |date=1984 |volume=XXXVII |issue=5 |pages=579–599 |url=http://amath.colorado.edu/faculty/beylkin/papers/BEYLKI-1984.pdf |doi=10.1002/cpa.3160370503}}</ref><ref>{{cite journal |last1=Beylkin |first1=Gregory |title=Imaging of discontinuities in the inverse scaterring problem by inversion of a causal generalized Radon transform |journal=J. Math. Phys. |date=1985 |volume=26 |issue=1 |pages=99–108|doi=10.1063/1.526755 |bibcode=1985JMP....26...99B }}</ref> and Bleistein<ref>{{cite journal |last1=Bleistein |first1=Norman |title=On the imaging of reflectors in the earth |journal=Geophysics |date=1987 |volume=52 |issue=7 |pages=931–942 |doi=10.1190/1.1442363 |bibcode=1987Geop...52..931B |s2cid=5095133 }}</ref>).  Should geometrical optics techniques (i.e. [https://www.encyclopediaofmath.org/index.php/Ray_method rays])  be used for the solving the wave equation, these methods turn out to be closely related to the so-called least-squares migration methods<ref>{{cite journal |last1=Nemeth |first1=Tamas |last2=Wu |first2=Chengjun |last3=Schuster |first3=Gerard |title=Least-squares migration of incomplete reflection data |journal = Geophysics |date=1999 |volume=64 |issue=1 | pages=208–221 | url=https://csim.kaust.edu.sa/web/FWI&LSM_papers/LSM/Geophysics1999Nemeth.pdf |doi=10.1190/1.1444517 |bibcode=1999Geop...64..208N }}</ref> derived from the least-squares approach (see Lailly,<ref name='Proceedings of the international conference on "Inverse Scattering, theory and applications", Tulsa, Oklahoma'>{{cite book |last1=Lailly |first1=Patrick |title=The seismic inverse problem as a sequence of before stack migrations |date=1983 |publisher=SIAM |location=Philadelphia |isbn=0-89871-190-8 |pages=206–220}}</ref> Tarantola<ref>{{Cite journal | doi=10.1190/1.1441754| title=Inversion of seismic reflection data in the acoustic approximation| journal=Geophysics| volume=49| issue=8| pages=1259–1266| year=1984| last1=Tarantola| first1=Albert | bibcode=1984Geop...49.1259T| s2cid=7596552}}</ref>).