# Difference between revisions of "Reflection/transmission coefficients at small angles and magnitude"

(added page) |
(→Continue reading: updated) |
||

(2 intermediate revisions by the same user not shown) | |||

Line 14: | Line 14: | ||

| isbn = ISBN 9781560801153 | | isbn = ISBN 9781560801153 | ||

}} | }} | ||

− | == Problem 3. | + | == Problem 3.9 == |

'''Show that, when angles in the Zoeppritz equations (3.2e,f,h,i) are small (so that squares and products are negligible), equations (3.6a) and (3.6b) for reflection and transmission at normal incidence are still valid, and that the reflected and transmitted S-waves are given by''' | '''Show that, when angles in the Zoeppritz equations (3.2e,f,h,i) are small (so that squares and products are negligible), equations (3.6a) and (3.6b) for reflection and transmission at normal incidence are still valid, and that the reflected and transmitted S-waves are given by''' | ||

Line 120: | Line 120: | ||

|- | |- | ||

| align="center" | [[Amplitude/energy of reflections and multiples]] | | align="center" | [[Amplitude/energy of reflections and multiples]] | ||

− | | align="center" | [[ | + | | align="center" | [[Magnitude]] |

|- | |- | ||

! style="background: #426580; color: white;" | Previous chapter | ! style="background: #426580; color: white;" | Previous chapter | ||

Line 141: | Line 141: | ||

* [[Reflection and transmission coefficients]] | * [[Reflection and transmission coefficients]] | ||

* [[Amplitude/energy of reflections and multiples]] | * [[Amplitude/energy of reflections and multiples]] | ||

+ | * [[Magnitude]] | ||

* [[AVO versus AVA and effect of velocity gradient]] | * [[AVO versus AVA and effect of velocity gradient]] | ||

* [[Variation of reflectivity with angle (AVA)]] | * [[Variation of reflectivity with angle (AVA)]] |

## Latest revision as of 10:18, 25 February 2019

Series | Geophysical References Series |
---|---|

Title | Problems in Exploration Seismology and their Solutions |

Author | Lloyd P. Geldart and Robert E. Sheriff |

Chapter | 3 |

Pages | 47 - 77 |

DOI | http://dx.doi.org/10.1190/1.9781560801733 |

ISBN | ISBN 9781560801153 |

Store | SEG Online Store |

## Problem 3.9

**Show that, when angles in the Zoeppritz equations (3.2e,f,h,i) are small (so that squares and products are negligible), equations (3.6a) and (3.6b) for reflection and transmission at normal incidence are still valid, and that the reflected and transmitted S-waves are given by**

**where** , -

### Solution

When the angle of incidence is small, and and the same is true for . In this case Snell’s law and the Zoeppritz equations (3.2e,f,h,i) become

In matrix notation, the Zoeppritz equations are now

**(**)

To get the amplitude ratios and , we solve this equation either by inverting the left-hand matrix [see Sheriff and Geldart, 1995, equation (15.20)] or by using Cramer’s rule (see Wylie, 1966, 453). Using the latter method, and neglecting squares and products of the angles, we first calculate the value of det(), the determinant of the matrix in equation (3.9a). We shall expand by elements in the first row [see Sheriff and Geldart, 1995, equation (15.2)]; when we do this we see that the 2nd and 4th determinants in the expansion are multiplied by and , respectively, and since we are neglecting products and squares of angles, angles inside these two determinants have been replaced with zeros. The expansion about the first row becomes

Next we calculate the values of and , , 2, where is with column 1 replaced with the elements of the right-hand matrix in equation (3.9a), etc. (see Cramer’s rule in Sheriff and Geldart, 1995, problem 15.2j). Expanding about the first row and setting the angles in the 2nd and 4th determinant equal to zero as before, the expansion becomes

The second and fourth determinants are zero, so

Dividing by , we get

which is the same as equation (3.6a). Similarly, we find that

where , . Note that

Also, when

so *q* and

## Continue reading

Previous section | Next section |
---|---|

Amplitude/energy of reflections and multiples | Magnitude |

Previous chapter | Next chapter |

Theory of Seismic Waves | Geometry of seismic waves |

## Also in this chapter

- General form of Snell’s law
- Reflection/refraction at a solid/solid interface and displacement of a free surface
- Reflection/refraction at a liquid/solid interface
- Zoeppritz’s equations for incident SV- and SH-waves
- Reinforcement depth in marine recording
- Complex coefficient of reflection
- Reflection and transmission coefficients
- Amplitude/energy of reflections and multiples
- Magnitude
- AVO versus AVA and effect of velocity gradient
- Variation of reflectivity with angle (AVA)