Navigation

Content Actions

Download module PDF
Add to ...
Add the module to:
- My Favorites
- A lens
- An external social bookmarking service
- My FavoritesLogin required (What is 'My Favorites'?)
  'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
- A lensLogin required (What is a lens?)
  Definition of a lens
  Lenses
  A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.
  What is in a lens?
  Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.
  Who can create a lens?
  Any individual Connexions member, a community, or a respected organization.
- External bookmarks
E-mail the author
Print this Web page

Lenses

Definition of a lens

Lenses

A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual Connexions member, a community, or a respected organization.

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.

Rice University ELEC 301 Projects
This module is included inLens: Rice University ELEC 301 Project Lens
By: Rice University ELEC 301As a part of collection:"ELEC 301 Projects Fall 2004"
Click the "Rice University ELEC 301 Projects" link to see all content affiliated with them.

Related material

Collections using this module

Recently Viewed: This feature requires Javascript to be enabled.

Mathematical Foundations

Module by: Wing-kei Yu

Summary: The mathematical basis for why seismic imaging works.

Sound waves travel at definite velocities characteristic of the propagation medium. Reflection occurs when the waves hit a change in medium and experience a change in velocity. Fermat’s principle of least time states that a wave, in going between two points S and R, must traverse a path length that is stationary with respect to variations of that path. While this principle was formulated for light beams, it also holds for other waves.

Fermat’s principle can be understood from a phasor perspective. Waves that traverse paths close to that of a maximum or minimum (stationary) path will arrive by routes that only differ slightly in path length. Hence, they will arrive from S to R nearly in-phase, and they will add constructively. Waves taking other paths far away from the stationary one will arrive mainly out of phase with each other and cancel.

For an ellipse, the sum of the distances from the two focal points to a point on the ellipse is a constant. Thus the paths SQR for all points Q on an ellipse are stationary. If a pulse is fired from the source S and arrives at receiver R a time t later, we can then place the source and receive at the focal points of an ellipse. The sum of the distances from each focal point to a point on the ellipse is given by:

r 1 + r 2 =vt

r 1 r 2 v t

(1)

where v is the speed of sound in the given medium. The ellipse represents the locus of all possible image points for one source/receiver pair.

Figure 1: Source and Receiver diagram demonstrating the locus of possible reflection points.

This way, we can draw such an ellipse for each source-receiver pair. Each image exhibits elliptical wavefronts. The intensity of an oscillatory integral is given by:

Iω=∫dxaxⅇⅈwψx

I ω x a x w ψ x

(2)

where a(x) is the amplitude function, ω is the frequency, and φ(x) is the phase of the signal. Because we are sending a pulse, we can make a large ω approximation and look at the case where . For a small section dx, the high number of oscillations due to large ω causes the integral to equal zero except where . This point is a point of stationary phase, and occurs where the wavefront is. Therefore, after summing up all the ellipses generated by each source-receiver pair, only the outline of the reflecting surface is visible.

Below are images showing the summing of increasing numbers of ellipses. The reflecting surface is mostly horizontal, with a mountain barely visible on the right.

Figure 2