Editing Inverse problem (section)

==== Travel-time tomography ====

Realizing how difficult is the inverse problem in the wave equation, seismologists investigated a simplified approach making use of geometrical optics. In particular they aimed at inverting for the propagation velocity distribution, knowing the arrival times of wave-fronts observed on seismograms. These wave-fronts can be associated with direct arrivals or with reflections associated with reflectors whose geometry is to be determined, jointly with the velocity distribution.

The arrival time distribution <math>{\tau}(x)</math> (<math>x</math> is a point in physical space) of a wave-front issued from a point source, satisfies the [[Eikonal equation]]:
<math display="block">\|\nabla \tau (x)\| =  s(x),</math>
where <math>s(x)</math> denotes the [[Slowness (seismology)|slowness]] (reciprocal of the velocity) distribution. The presence of <math>\| \cdot \| </math> makes this equation nonlinear. It is classically solved by shooting [[Ray tracing (physics)|rays]] (trajectories about which the arrival time is stationary) from the point source.

This problem is tomography like: the measured arrival times are the integral along the ray-path of the slowness. But this tomography like problem is nonlinear, mainly because the unknown ray-path geometry depends upon the velocity (or slowness) distribution. In spite of its nonlinear character, travel-time tomography turned out to be very effective for determining the propagation velocity in the Earth or in the subsurface, the latter aspect being a key element for seismic imaging, in particular using methods mentioned in Section "Diffraction tomography".