# Dipping refractor

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

Figure 3.4-11  (a) Geometry for refracted arrivals. Here, vw = weathering velocity, vb = bedrock velocity, zw = depth to the refractor equivalent to the base of the weathering layer, θc = critical angle, and xc = crossover distance. The direct wave arrival has a slope equal to 1/vw and the refracted wave arrival has a slope equal to 1/vb. (b) A shot record that exhibits the direct wave and the refracted wave depicted in (a). (c) Geometry for a dipping refractor with forward traveltime profile associated with the direct wave and refracted wave arrivals, and (d) with both forward and reverse traveltime profiles. See text for details.

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

 ${\displaystyle t^{-}=t_{i}^{-}+{\frac {x}{v_{b}^{-}}}}$ (43a)

and

 ${\displaystyle t^{+}=t_{i}^{+}+{\frac {x}{v_{b}^{+}}}.}$ (43b)

The inverse slopes are given by

 ${\displaystyle v_{b}^{-}={\frac {v_{w}}{\sin(\theta _{c}+\varphi )}}}$ (44a)

and

 ${\displaystyle v_{b}^{+}={\frac {v_{w}}{\sin(\theta _{c}-\varphi )}},}$ (44b)

where φ is the refractor dip and θc is the critical angle of refraction given by

 ${\displaystyle \sin \theta _{c}={\frac {v_{w}}{v_{b}}}.}$ (44c)

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

 ${\displaystyle t_{i}^{-}={\frac {2z_{wS}\cos \theta _{c}\cos \varphi }{v_{w}}}}$ (45a)

and

 ${\displaystyle t_{i}^{+}={\frac {2z_{wR}\cos \theta _{c}\cos \varphi }{v_{w}}}.}$ (45b)

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 — ${\displaystyle {v_{w}},\ {v_{b}^{-}},\ {\text{and }}{v_{b}^{+}}.}$ These measurements are then inserted into the expression

 ${\displaystyle \varphi ={\frac {1}{2}}\left[\sin ^{-1}{\frac {v_{w}}{v_{b}^{-}}}-\sin ^{-1}{\frac {v_{w}}{v_{b}^{+}}}\right].}$ (46a)

Then, we compute the bedrock velocity vb using the expression

 ${\displaystyle v_{b}={\frac {2\cos \varphi }{\begin{pmatrix}{\frac {1}{v_{b}^{-}}}+{\frac {1}{v_{b}^{+}}}\end{pmatrix}}}.}$ (46b)

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

 ${\displaystyle z_{w}={\frac {v_{b}v_{w}t_{i}^{-}}{2\cos \varphi {\sqrt {v_{b}^{2}-v_{w}^{2}}}}}.}$ (46c)

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).

 ${\displaystyle z_{w}={\frac {v_{b}v_{w}t_{i}}{2{\sqrt {v_{b}^{2}-v_{w}^{2}}}}}.}$ (41a)