Dipping refractor

**Seismic Data Analysis**
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

Figure 3.4-11 (a) Geometry for refracted arrivals. Here, v_w = weathering velocity, v_b = bedrock velocity, z_w = depth to the refractor equivalent to the base of the weathering layer, θ_c = critical angle, and x_c = crossover distance. The direct wave arrival has a slope equal to 1/v_w and the refracted wave arrival has a slope equal to 1/v_b. (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

t^{-}=t_{i}^{-}+{\frac {x}{v_{b}^{-}}}

(43a)

and

t^{+}=t_{i}^{+}+{\frac {x}{v_{b}^{+}}}.

(43b)

The inverse slopes are given by

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

(44a)

and

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

(44b)

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

\sin \theta _{c}={\frac {v_{w}}{v_{b}}}.

(44c)

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

t_{i}^{-}={\frac {2z_{wS}\cos \theta _{c}\cos \varphi }{v_{w}}}

(45a)

and

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

\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 v_b using the expression

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

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).