# Effect of horizontal velocity gradient

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

Title | Problems in Exploration Seismology and their Solutions |

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

Chapter | 10 |

Pages | 367 - 414 |

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

ISBN | ISBN 9781560801153 |

Store | SEG Online Store |

## Problem

In Figure 10.14a, **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_{1} =2.00\ {\rm km/s}}**
, **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} =4.00\ {\rm km/s}}**
, the horizon 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 20^{\circ}}**
, and the diffracting 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 P}**
is at a vertical depth of 2.00 km, the dipping horizon being midway 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 P}**
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 P'}**
(at the surface). Compare the diffraction curve with what would have been observed on a CMP section if **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_{1} =V_{2} =3.00\ {\rm km/s}}**
.

### Solution

The curve for the diffracting point below the dipping interface can be solved analytically or graphically by ray tracing. The crest of the diffraction is located beneath where the angle of approach to the surface 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 90^{\circ}}**
; such a raypath is called an *image ray*. If we denote the angle between a ray 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 P}**
and the vertical as **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}**
, then Snell’s law at the interface gives for the image ray the equation

**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} \sin \left(\theta +20^{\circ} \right)=\left(V_{2} /V_{1} \right)\sin 20^{\circ} =2\sin 20^{\circ} =0.684,\\ {\rm or} \qquad\qquad\qquad\qquad \theta ={\sin}^{-1} 0.684-20^{0}=23^{\circ}. \end{align} }**

The fact that an image ray approaches the surface vertically is employed in depth migration (see problem 10.16) to accommodate horizontal changes in velocity.

The diffraction curve (solid curve, Figure 10.14c) is not symmetrical and the crest is displaced about 500 m updip from

The diffraction curve for the constant velocity case (dashed curve, Figure 10.14c) can be calculated using the equations

**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} \tan \theta =x/z=x/2.00,\\ t=2z/(V\cos \theta)=4.00/(3.00\cos \theta)=1.33/\cos \theta, \end{align} }**

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 \theta}**
is the angle of approach at the surface.

## Continue reading

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Unconformities | Stratigraphic interpretation book |

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Data processing | Refraction methods |

## Also in this chapter

- Improvement due to amplitude preservation
- Deducing fault geometry from well data
- Structural style
- Faulting
- Mapping faults using a grid of lines
- Fault and stratigraphic interpretation
- Interpretation of salt uplift
- Determining the nature of flow structures
- Mapping irregularly spaced data
- Evidences of thickening and thinning
- Recognition of a reef
- Seismic sequence boundaries
- Unconformities
- Effect of horizontal velocity gradient
- Stratigraphic interpretation book
- Interpretation of a depth-migrated section
- Hydrocarbon indicators
- Waveshapes as hydrocarbon accumulation thickens