# Flat 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 |

Consider the refraction wavefront and raypath geometry in Figure 3.4-11a associated with a single-layer nearsurface model. On top, we see a plot of first-breaks. For simplicity, consider a flat surface and flat refractor. For the head wave to form, and thus the refraction to occur, the requirement is that the overburden velocity *v _{w}* be smaller than the substratum velocity

*v*.

_{b}The traveltime profile depicts the first breaks seen on the shot record in Figure 3.4-11b. Note that to the left of the crossover offset *x _{c}* (also known as critical distance) are the first breaks associated with the the direct arrivals. Also note that to the right of offset

*x*are the first breaks associated with the refracted arrivals. From the refraction theory

_{c}^{[1]};

^{[2]}, the inverse of the slope of the line associated with the refracted wave arrivals is equal to the bedrock velocity

*v*. Also note that the inverse of the slope of the line associated with the direct wave arrivals is equal to the velocity of the weathering layer

_{b}*v*.

_{w}By picking the first breaks, the weathering and bedrock velocities, *v _{w}* and

*v*are estimated. By extending the line associated with the refracted arrivals to zero-offset, intercept time

_{b}*t*, the time at

_{i}*x*= 0, is estimated. From these three parameters, it is easy to show that depth to the bedrock

*z*is given by

_{w}

**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}}.}****(**)

We assume that *v _{b}* >

*v*. Derivation of this formula is left to Section C.5.

_{w}Alternatively, we can measure the critical distance corresponding to the change from the direct arrival to the refracted arrival on the traveltime plot and use it in computing the depth to the bedrock. Equation (**41a**), in terms of the critical distance *x _{c}*, takes the form

**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{1}{2}\sqrt{\frac{v_b-v_w}{v_b+v_w}}x_c.}****(**)

It may not be easy to measure the critical distance when depth to bedrock is small. In such cases, it is better to use the intercept time to compute the depth to the bedrock by way of equation (**41a**).

After computing *z _{w}*, the total static correction Δ

*τ*to the specified datum level can be applied by

_{D}

**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 \tau_D=-\frac{2z_w}{v_w}+\frac{2(E_D-E_S+z_w)}{v_b},}****(**)

where *E _{S}* is the surface elevation. If there is a difference between the elevations of shot and receiver stations, then an additional elevation correction using the bedrock velocity is required. Moreover, if the shots are located in boreholes, then the measured uphole time also must be incorporated into equation (

**42**). The estimated statics correction given by equation (

**42**) is an average value over a distance that can range from the critical distance to the spread length, depending on the number of traces used in estimating the bedrock velocity. Nevertheless, more than one shot-point is within a spread length. Therefore, an adequate definition of the near-surface model can be achieved and datum corrections can be computed for the entire profile.

## See also

- First breaks
- Field statics corrections
- Dipping 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