# Great geophysicists #2: Snellius

Willebrordus Snellius (1580–1626) was the Latin name that Dutchman Willebrord Snel van Royen, or plain Snell, used for all of his publications as a mathematician. He made great advances over his predessors by introducing trigonometric methods for measuring large distances across landcapes. He's regarded as the father of triangulation, which is widely applied in surveying, imaging, and event location. And he even tried to measure the radius of the earth.

His most famous contribution, Snell's law of refraction, has formed the basis of geometrical optics and is inherently ingrained in seismology. Snell's law is used to determine the direction of wave propagation through a refractive interface:

$\large \dpi{120} \frac{\sin \theta_1}{\sin \theta_2} = \frac{v_1}{v_2} = \frac{n_2}{n_1}$

How much waves bend depends on the ratio of velocities between the two media v2/ v1. You will notice that the right hand side of this equation is where Ibn Sahl left off.

Other mathematicians before him, Ibn Sahl for instance, were aware that light rays refracted when they entered media of different velocities, but Snellius was the first to describe this problem using trigonometry. He made his discovery in 1621, when he was 41 years old, but it was never published in his lifetime. René Descartes, the inventor of the cartesian coordinate and analytical geometry, published this law of refraction 16 years after Snell's death, as Descarte's law of refraction. But Snell was eventually widely attributed with the discovery in 1703 when Christiaan Huygens published Snell's results in his Dioptrica to explain, among other things why successive wavefronts travel in parallel.

Oleg Alexandrov via Wikipedia

In a classic analogy, a 'fast' region is the beach, a ' slow' region is the water, and the fastest way for a rescuer on the beach to get to a drowning person in the water is to run, then swim, along a path that follows Snell's law. The path a ray will take upon entering a media is the one that minimzes the travel time through that media (see Fermat's principle). Notice too that there are no arrows indicating the direction of ray propagation: whether the ray enters from above or below, the refraction behaviour is the same.

In seismology, Snell's law is used to describe how seismic waves bend and turn in accordance with contrasting velocities in the subsurface which is the foundation of surveying, image focusing, and event detection. It appears in ray-tracing, ray-parameterization, offset to angle estimations (used in AVO), anisotropy problems, velocity modeling, and traveltime tomography. Snellius, we salute you!