# Difference between revisions of "Refractions and refraction multiples"

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The traveltime curves are plotted as curves (a) in Figure 6.16b. | The traveltime curves are plotted as curves (a) in Figure 6.16b. | ||

− | [[file:Ch06_fig6-16b.png|thumb|{{figure number|6.16b.}} Traveltime curves. Letters denote curves for respective parts (a), (b), (c).]] | + | [[file:Ch06_fig6-16b.png|thumb|center|{{figure number|6.16b.}} Traveltime curves. Letters denote curves for respective parts (a), (b), (c).]] |

== Problem 6.16b == | == Problem 6.16b == |

## Revision as of 16:27, 8 November 2019

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

Title | Problems in Exploration Seismology and their Solutions |

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

Chapter | 6 |

Pages | 181 - 220 |

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

ISBN | ISBN 9781560801153 |

Store | SEG Online Store |

## Contents

## Problem 6.16a

Determine the traveltime curve for the refraction **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathit{SMNPQR}}**
and the refraction multiple **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathit{SMNTUWPQR}}**
in Figure 6.16a.

### Solution

We assume that the velocities are known to three significant figures. Then, using equation (3.1a),

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \theta _{2} ={\rm sin}^{-1} (2.00/4.20)=28.4^{0} ; \theta _{1} ={\rm sin}^{-1} (2.80/4.20)=41.8^{\circ} \end{align} }**

The traveltimes can be obtained either graphically or by calculation. Calculating, we get for the refraction traveltimes **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t_{R}}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} t_{R} = 2SM/2.80+2MN/2.00+NP/4.20\\ = 2\times 0.75/2.80 {\rm \; cos\; }41.8^{\circ} +2\times 3.25/(2.00{\rm \; cos\; }28.4^{\circ} )\\ +(x-2\times 0.75{\rm \; tan\; }41.8^{\circ} -2\times 3.25{\rm \; tan\; }28.3^{\circ} )/4.20\\ = x/4.20+2\times 0.75{\rm \; cos\; }41.8^{\circ} /2.80+2\times 3.25{\rm \; cos\; }28.4^{\circ} /2.00\\ = x/4.20+3.26. \end{align} }**

The critical distance (see equation (6.15a) is

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} x^{'} =2(0.75{\rm \; tan\; }41.8^{\circ} +3.25{\rm \; tan\; }28.4^{\circ} )=4.86\ {\rm km}. \end{align} }**

The traveltime curve for *SMNTUWPQR* is parallel to that for *SMNPQR* and displaced toward longer time by the amount **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta t}**
where

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \Delta t=2TU/2.00-TW/4.20\\ =2\times 3.25/(2.00{\rm \; cos\; }28.4^{\circ} )-2\times 3.25({\rm \; tan\; }28.4^{\circ} )/4.20=2.86\ {\rm s}. \end{align} }**

The critical distance for *SMNTUWPQR* is increased to

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} x^{'} =2\times 0.75{\rm \; tan\; }41.8^{\circ} +4\times 3.25{\rm \; tan\; }28.4^{\circ} =8.37\ {\rm km}. \end{align} }**

The traveltime curves are plotted as curves (a) in Figure 6.16b.

## Problem 6.16b

Determine the traveltime curves when both refractor and reflector dip **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 8^{\circ}}**
down to the left, the depths shown in Figure 6.16a now being the slant distances from **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S}**
to the interfaces.

### Solution

A combined graphical and calculated solution probably provides the easiest solution although Adachi’s method (see problem 11.5) could be used to give greater precision if the data accuracy warranted. A large-scale graph was used to achieve better accuracy; Figure 6.16c is a reduced-scale replica. The traveltime curves are shown in Figure 6.16b labeled (b).

The critical distance for the refraction is **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle SR_{1} =4.38}**
km, and

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} t_{R_{1} } =(1.00+0.18)/2.80+2\times 3.71/2.00=4.13\ {\rm s}. \end{align} }**

The **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_{2}}**
-layer outcrops at **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R_{2} =5.39}**
km,

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} t_{R_{2} } =1.00/2.80+2\times 3.71/2.00+1.12/4.20=4.33\ {\rm s}. \end{align} }**

The headwave has a different slope to the right of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R_{2}}**
. To plot the curve in this zone, we use point **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R_{4}}**
at the offset **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x=8.79\ {\rm km}}**
. Then,

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} t_{R_{4}} = 1.00/2.80+(3.71+3.21)/2.00+4.71/4.20=4.94\ {\rm s}, \end{align} }**

and the headwave curve is a straight line joining the traveltimes at the points **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R_{2}}**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R_{4}}**
.

We have two types of reflected refractions: a typical path for the first type is **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle SMNP_{1}U_{2}WP_{4} R_{4}}**
, the reflection occurring at the shallow dipping interface, The second type, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle SMNP_5 R_{6} P_{7} R_{7}}**
, involves reflection at the surface. The first type exists between **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R_{3}}**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R_{4}}**
, and the curve is parallel to the head-wave curve to the right of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R_{3}}**
. The second type exists to the right of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R_{3}}**
. For the first type, we find the coordinates of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R_{5}}**
:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} x=6.80 \ \hbox {km},\\ t_{R_{5} } =1.00/2.80+(2\times 3.71+3.32+3.12)/2.00+1.12/4.20=7.55\ {\rm s}. \end{align} }**

For the second type we find coordinates of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R_{7}}**
:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} x=9.15\ {\rm km},\\ t_{\rm R_{7}} =1.00/2.80+(3.71+3.30+2.97+2.78)/2.00+3.85/4.20=7.65\ {\rm s}. \end{align} }**

## Problem 6.16c

What happens when the reflector dips **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 3^{\circ}}**
to the left and the refractor **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 5^{\circ}}**
to the left?

### Solution

A summary of the detailed graphical solution is as follows. The traveltime curves are shown in Figure 6.16b labeled (c). The various angles in Figure 6.16d are

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \theta _{c} =28.4^{\circ} \;,\; \theta _{2} =(28.4^{\circ} +5^{\circ} -3^{\circ} )\; =30.4^{\circ},\\ \theta _{1} ={\rm sin}^{-1} [(2.80/2.00) {\rm \; sin\; }30.4^{\circ} ] =45.1^{\circ},\\ \alpha _{1} = {\rm angle\ of\ approach}\ =45.1^{\circ} +3^{\circ} =48.1^{\circ},\\ \theta _{2}^{'} =(28.4^{\circ} -5^{\circ} +3^{\circ} )\; =26.4^{\circ},\\ \theta _{1}^{'} ={\rm sin}^{-1} [(2.80/2.00) {\rm \; sin\; }38.5^{\circ} ] =38.5^{\circ},\\ \alpha _{1^{'} } =38.5^{\circ} -3^{\circ} \; =35.5^{\circ} \end{align} }**

The refraction curve is a straight line though **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R_{5}}**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R_{7}}**
:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} R_{5} : x^{'} =4.64\ {\rm km} = {\rm critical\ distance},\\ t_{R_{5} } =(1.09+0.44)/2.80+(3.65+3.50)/2.00=4.20\ {\rm km};\\ R_{7} : x=8.15\ {\rm km},\\ t_{R_{7}} =(1.09+0.44)/2.80+(3.64+3.35)/2.00+3.80/4.20\\ =4.95\ {\rm s}. \end{align} }**

The incident angle at **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle W}**
is **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta _{3} =24.4^{\circ}}**
, which is less than the critical angle, so that no refraction will be generated there, only a reflection. However, the refraction that starts at **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle N}**
will give rise to upgoing rays which will be reflected, giving a reflected refraction, such as the ray that ends at **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R_{8}}**
. The traveltime curve is parallel to the refraction curve and exists beyond **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R_{6}}**
whose coordinates are

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} x=7.79\ {\rm km},\\ t_{R_{6} } =(1.09+0.48)/2.80+(3.64+3.50+3.42+3.36)/2.00=7.52\ {\rm s}. \end{align} }**

## Continue reading

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Reflections/diffractions from refractor terminations | Destructive and constructive interference for a wedge |

Previous chapter | Next chapter |

Geometry of seismic waves | Characteristics of seismic events |

## Also in this chapter

- Characteristics of different types of events and noise
- Horizontal resolution
- Reflection and refraction laws and Fermat’s principle
- Effect of reflector curvature on a plane wave
- Diffraction traveltime curves
- Amplitude variation with offset for seafloor multiples
- Ghost amplitude and energy
- Directivity of a source plus its ghost
- Directivity of a harmonic source plus ghost
- Differential moveout between primary and multiple
- Suppressing multiples by NMO differences
- Distinguishing horizontal/vertical discontinuities
- Identification of events
- Traveltime curves for various events
- Reflections/diffractions from refractor terminations
- Refractions and refraction multiples
- Destructive and constructive interference for a wedge
- Dependence of resolvable limit on frequency
- Vertical resolution
- Causes of high-frequency losses
- Ricker wavelet relations
- Improvement of signal/noise ratio by stacking