# Dipping refractor

Series | Investigations in Geophysics |
---|---|

Author | Öz Yilmaz |

DOI | http://dx.doi.org/10.1190/1.9781560801580 |

ISBN | ISBN 978-1-56080-094-1 |

Store | SEG Online Store |

When the refractor is dipping, it turns out that the inverse slope of the refracted arrival is no longer equal to the bedrock velocity (Figure 3.4-11c). An extra parameter — the dip of the refractor, needs to be estimated (Section C.6). This requires reverse profiling as illustrated in Figure 3.4-11d. We have the refracted arrival in the forward direction and the refracted arrival in the reverse direction obtained by interchanging the shots with receivers. The traveltimes for the refracted arrivals of the forward and reverse profiles are expressed 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 t^{-}=t^{-}_i+\frac{x}{v^{-}_b}}****(**)

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 t^{+}=t^{+}_i+\frac{x}{v^{+}_b}.}****(**)

The inverse slopes are given by

**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^{-}_b=\frac{v_w}{\sin (\theta_c+\varphi)}}****(**)

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 v^{+}_b=\frac{v_w}{\sin (\theta_c-\varphi)},}****(**)

where *φ* is the refractor dip and *θ _{c}* is the critical angle of refraction given by

**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 \sin \theta_c=\frac{v_w}{v_b}.}****(**)

Finally, the intercept times are given by the following relations:

**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^{-}_i=\frac{2z_{wS}\cos \theta_c\cos\varphi}{v_w}}****(**)

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 t^{+}_i=\frac{2z_{wR}\cos \theta_c\cos\varphi}{v_w}.}****(**)

Derivation of the relations (3-44a,b) and (3-45a,b) are left to Section C.6.

To estimate the thickness of the near-surface layer, first we compute the refractor dip *φ* from the slope measurements — **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_{w}},\ {v_{b}^{-}},\ \text{and }{v_{b}^{+}}.}**
These measurements are then inserted into the expression

**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 \varphi=\frac{1}{2}\left[\sin^{-1}\frac{v_w}{v^{-}_b}-\sin^{-1}\frac{v_w}{v^{+}_b}\right].}****(**)

Then, we compute the bedrock velocity *v _{b}* using the expression

**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_b=\frac{2\cos\varphi}{\begin{pmatrix}\frac{1}{v^{-}_b}+\frac{1}{v^{+}_b}\end{pmatrix}}.}****(**)

Finally, we compute the depth to the bedrock at shot/receiver stations

**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 z_w=\frac{v_bv_wt^{-}_i}{2\cos\varphi\sqrt{v^2_b-v^2_w}}.}****(**)

Again, equations (**46a**,**46c**) reduces to equation (**41a**).

Keep in mind that, whether it is the flat refractor (equation **41a**) or dipping refractor case (equation **46c**), the depth to bedrock estimation at a shot-receiver station requires the knowledge of weathering velocity, bedrock velocity and intercept time. In the case of a flat refractor, these can be measured directly from shot profiles; whereas, in the case of a dipping refractor, they can be computed by way of equations (**46a**, **46b**, **46c**).

**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 z_w=\frac{v_bv_wt_i}{2\sqrt{v^2_b-v^2_w}}.}****(**)

## See also

- First breaks
- Field statics corrections
- Flat refractor
- The plus-minus method
- The generalized reciprocal method
- The least-squares method
- Processing sequence for statics corrections
- Model experiments
- Field data examples
- Exercises
- Topics in moveout and statics corrections