# Ley de Snell

This page is a translated version of the page Snell’s law and the translation is 64% complete.
Other languages:
Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 2 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

Suppose we have two homogeneous isotropic media separated by a straight horizontal interface (Figure 10). We plot the depth axis y in the upward direction. Let ${\displaystyle n_{\rm {l}}{\rm {>}}n_{\rm {2}}}$. The low-velocity surface layer (layer 1, with constant slowness ${\displaystyle n_{\rm {l}}}$) is in the range ${\displaystyle {\rm {0<}}y{\rm {. The high-velocity deep layer (layer 2, with constant slowness ${\displaystyle n_{\rm {2}}}$) is in the range ${\displaystyle {\rm {y<}}y{\rm {<}}\infty }$. Ray EFG starts in a vertical direction. It strikes the horizontal interface at right angles. A particle on the ray speeds up when it passes from the low-velocity layer 1 into the high-velocity layer 2. The ray does not bend at the interface and travels in layer 2 in the same vertical direction. Ray ABC starts in a direction oblique to the interface. A particle on ray ABC speeds up when it passes from the low-velocity layer 1 into the high-velocity layer 2, just as the particle did for ray EFG. The particles on ray EFG and on ray ABC experience exactly the same contrast in velocity. However, ray ABC bends at the interface. Why?

The particle is like a little air-plane. The fuselage goes along the raypath. The airplane is steered by the wings, which extend at right angles to each side. The left wing enters the high-velocity layer first, whereas the right wing is still in the low-velocity layer. This swings the plane to the right, and the raypath bends at the interface. How much does it bend?

Ray vector AB in layer 1 has length ${\displaystyle n_{\rm {1}}}$ and makes angle ${\displaystyle {\theta }_{\rm {1}}}$ with the vertical. Ray vector BC in layer 2 has length ${\displaystyle n_{\rm {2}}}$ and makes angle ${\displaystyle {\theta }_{\rm {2}}}$ with the vertical. By Snell’s law, we have ${\displaystyle n_{1}\mathrm {sin} {\theta }_{\rm {l}}{\rm {=}}n_{\rm {2}}{\rm {\ sin\ }}{\theta }_{\rm {2}}}$. This equation says that the horizontal projections of the two ray vectors are the same. We expect this result because the slowness does not change in the horizontal direction. The slowness does E change at the interface in the vertical direction. The shallow medium has slowness ${\displaystyle n_{\rm {l}}}$, and the deep medium has a smaller slowness ${\displaystyle n_{\rm {2}}}$. Because the interface is level, it follows that slowness is constant along any horizontal line, so ${\displaystyle \partial n/\partial x{\rm {=0}}}$. However, slowness does change at the interface. The difference of the slowness at the interface is ${\displaystyle n_{\rm {2}}-n_{\rm {1}}}$, which is negative because ${\displaystyle n_{\rm {2}}{\rm {<}}n_{\rm {1}}}$. The gradient of slowness at the interface is proportional to ${\displaystyle \left(n_{\rm {2}}-n_{\rm {l}}\right)\mathbf {j} }$. This vector points straight into the low-velocity layer. The difference vector BD = BCAB must be proportional to the gradient of the slowness, and hence vector BD must point straight down. Thus, ${\displaystyle n_{\rm {1}}{\rm {\ cos\ }}{\theta }_{\rm {l}}{\rm {>}}n_{\rm {2}}{\rm {\ cos\ }}{\theta }_{\rm {2}}}$, which says that the raypath bends to the right at the interface. In this section, we have given a preview of the ray equation.

Figure 10.  Ray EFG, which is at right angles to the interface, does not bend at the interface. Ray ABC, which is skewed to the interface, does bend.

## Sigue leyendo

Sección previa Siguiente sección
La ecuación de Eikonal Ecuación del rayo
Capítulo previo Siguiente capítulo
Movimiento de ondas Visualización