Effect of horizontal velocity gradient

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Problem

In Figure 10.14a, 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_{1} =2.00\ {\rm km/s}} , 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_{2} =4.00\ {\rm km/s}} , the horizon dips 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 20^{\circ}} , and the diffracting point 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 P} is at a vertical depth of 2.00 km, the dipping horizon being midway between 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 P} 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 P'} (at the surface). Compare the diffraction curve with what would have been observed on a CMP section if 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_{1} =V_{2} =3.00\ {\rm km/s}} .

Figure 10.14a.  Geometry of problem.

Solution

The curve for the diffracting point below the dipping interface can be solved analytically or graphically by ray tracing. The crest of the diffraction is located beneath where the angle of approach to the surface is 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 90^{\circ}} ; such a raypath is called an image ray. If we denote the angle between a ray at 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 P} and the vertical 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 \theta} , then Snell’s law at the interface gives for the image ray the equation


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 \begin{align} \sin \left(\theta +20^{\circ} \right)=\left(V_{2} /V_{1} \right)\sin 20^{\circ} =2\sin 20^{\circ} =0.684,\\ {\rm or} \qquad\qquad\qquad\qquad \theta ={\sin}^{-1} 0.684-20^{0}=23^{\circ}. \end{align} }

The fact that an image ray approaches the surface vertically is employed in depth migration (see problem 10.16) to accommodate horizontal changes in velocity.

The diffraction curve (solid curve, Figure 10.14c) is not symmetrical and the crest is displaced about 500 m updip from 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 P} . The right limb is nearly flat because the increased traveltimes at the high velocity is almost compensated for shorter travel distance at the low velocity.

Figure 10.14b.  Ray tracing for the diffracting point 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 P} .
Figure 10.14c.  Diffraction curves. Solid curve is 2-layer case, dashed is constant velocity case.

The diffraction curve for the constant velocity case (dashed curve, Figure 10.14c) can be calculated using the equations

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 \begin{align} \tan \theta =x/z=x/2.00,\\ t=2z/(V\cos \theta)=4.00/(3.00\cos \theta)=1.33/\cos \theta, \end{align} }

where 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 \theta} is the angle of approach at the surface.

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