Variation of reflection point with offset

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Problem 4.11a

Equation (4.3a) for an offset geophone can be written


$ {\begin{aligned}(Vt)^{2}=(2h_{c})^{2}+(2s\cos \xi )^{2},\end{aligned}} $ (4.11a)

where $ 2s $ is the offset and $ h_{c} $ is the slant depth at the midpoint between the source $ S $ and receiver $ G $ (see Figure 4.11a). The point of reflection $ R(x_{1},z_{1}) $ is displaced updip the distance $ \Delta L $ from the zero-dip position $ P(x_{0},z_{0}) $. Show that the coordinates of a point $ x,z $ on the line $ IG $ must satisfy the relation


$ {\begin{aligned}(2s-x)/(s+h\ell )=z/hn=k,\end{aligned}} $ (4.11b)

where $ I $ is the image point, $ \ell ,n $ are the direction cosines of $ SI $, $ h $ is the slant depth at the source, and $ k $ is a parameter fixing the location of a point on $ IG $.

Solution

Referring to Figure 4.11a, $ (\ell ,n) $ are the direction cosines of $ SI $ where

$ {\begin{aligned}\ell =\sin \xi ,\quad n=\cos \xi ,\quad \xi =\tan ^{-1}(\ell /n).\end{aligned}} $

To get the coordinates of $ A(x,z) $, a point on $ IG $, we draw $ AB $ and $ IC $ perpendicular to $ TG $. Then, using the similar triangles $ ABG $ and $ ICG $, we have $ AB/BG=IC/CG $, that is,


$ {\begin{aligned}z/(2s-x)=2hn/2(s+h\ell ),\qquad \mathrm {so} \ z/hn=(2s-x)/(s+h\ell ),\end{aligned}} $ (4.11c)

$ x $ being the horizontal distance from $ S $. If we write

$ {\begin{aligned}k=z/hn=(2s-x)/(s+h\ell ),\end{aligned}} $

Figure 4.11a.  Displacement of reflection point for offset geophone.

we can vary $ k $ to get different points on $ IG $.

Problem 4.11b

Verify the following relations:


$ {\begin{aligned}x_{1}=x_{0}-s^{2}\ell n^{2}/h_{c},\quad z_{1}=z_{0}-s^{2}\ell ^{2}n/h_{c},\end{aligned}} $ (4.11d)


$ {\begin{aligned}{\hbox{and}}\qquad \qquad \Delta L=RP=-(s^{2}/2h_{c})\sin 2\xi .\end{aligned}} $ (4.11e)

Solution

To get $ R(x_{1},z_{1}) $, the point of intersection of $ IG $ and $ PT $, we first find the equation of $ PT $; the line $ PT $ has slope Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \tan\xi and passes through Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): T(-h/\ell,0) , so the equation is


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} z = x\tan \xi + h/n = (\ell/n)x + h/n = (\ell x h)/n. \end{align} (4.11f)

We now solve equations (4.11c) and (4.11f) as simultaneous equations. Eliminating Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} z = hn((2s-x_1)/(s+h\ell)) = (\ell x_1 + h)/n, \end{align}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \hbox{so} \qquad\qquad hn^2(2s - x_1) = (s + h\ell)(\ell x_1 + h). \end{align}

Using the equations Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): (\ell^2 + n^2) = 1 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h_c = (h + s\ell)  ; this reduces to


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} x_1 = h(s - 2s\ell^2 - h\ell)/h_c. \end{align} (4.11g)


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \hbox{Also} \qquad\qquad x_0 = s - h_c\ell = s - h - s\ell^2 = sn^2 - h\ell, \end{align} (4.11h)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \hbox{so} \qquad\qquad \Delta x = x_1 - x_0 &= [h(s-2s\ell^2-h\ell)-(sn^2 - h\ell)/h_c]/h_c \\ &= [hs - h\ell(h + 2s\ell)+(h+s\ell)(h\ell - sn^2)]/h_c \\ &=[hs - h\ell(h + 2s\ell)+h^2\ell-hsn^2 + hs\ell^2-s^2\ell n^2]/h_c \\ &=-s^2\ell n^2/h_c. \end{align}

From equation (4.11f) we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} z_1 &= (x_1\ell + h)/n = h\ell[(s - 2s\ell^2 - h\ell) + h_c h]/h_c n \\ &= [h\ell(2s - 2s\ell^2 - h\ell)+h^2]/h_c n. \end{align}

Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z_0 = h_c n ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \Delta z &= z_1 - z_0 = \left[h\ell(2s - 2s\ell^2 - h\ell) + h^2 - h_c^2n^2\right]/h_c n \\ &= [h\ell(2s - 2s\ell - h\ell) + h^2 - n^2(h^2 + 2hs\ell + s^2\ell^2)]/h_c n \\ &= [2hs\ell - 2hs\ell^3 + h^2(1-\ell^2-n^2) - 2hs\ell n - s^2\ell^2n^2]/h_c n\\ &= [2hs\ell(1-\ell^2-n)-s^2\ell^2n^2]/h_c n = -S^2\ell^2 n/h_c. \end{align}

We now have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \Delta L = [(\Delta x)^2 + (\Delta z)^2]^{1/2} = -s^2\ell n/h_c = -(s^2/2h_c)\sin 2\xi. \end{align}

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