Dipping refractor

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Seismic Data Analysis
Seismic-data-analysis.jpg
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, 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


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}} (43a)

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}.} (43b)

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)}} (44a)

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)},} (44b)

where φ is the refractor dip and θc 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}.} (44c)

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}} (45a)

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}.} (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 — 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].} (46a)

Then, we compute the bedrock velocity vb 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}}.} (46b)

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}}.} (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).


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}}.} (41a)

See also

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Dipping refractor
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