Variation of reflection point with offset

Problems in Exploration Seismology and their Solutions

Series	Geophysical References Series
Title	Problems in Exploration Seismology and their Solutions
Author	Lloyd P. Geldart and Robert E. Sheriff
Chapter	4
Pages	79 - 140
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	SEG Online Store

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 $\tan \xi$ and passes through $T(-h/\ell ,0)$, so the equation is

${\begin{aligned}z=x\tan \xi +h/n=(\ell /n)x+h/n=(\ell xh)/n.\end{aligned}}$

(4.11f)

We now solve equations (4.11c) and (4.11f) as simultaneous equations. Eliminating $z$ gives

${\begin{aligned}z=hn((2s-x_{1})/(s+h\ell ))=(\ell x_{1}+h)/n,\end{aligned}}$

${\begin{aligned}{\hbox{so}}\qquad \qquad hn^{2}(2s-x_{1})=(s+h\ell )(\ell x_{1}+h).\end{aligned}}$

Using the equations $(\ell ^{2}+n^{2})=1$, $h_{c}=(h+s\ell )$ ; this reduces to

${\begin{aligned}x_{1}=h(s-2s\ell ^{2}-h\ell )/h_{c}.\end{aligned}}$

(4.11g)

${\begin{aligned}{\hbox{Also}}\qquad \qquad x_{0}=s-h_{c}\ell =s-h-s\ell ^{2}=sn^{2}-h\ell ,\end{aligned}}$

(4.11h)

${\begin{aligned}{\hbox{so}}\qquad \qquad \Delta x=x_{1}-x_{0}&=[h(s-2s\ell ^{2}-h\ell )-(sn^{2}-h\ell )/h_{c}]/h_{c}\\&=[hs-h\ell (h+2s\ell )+(h+s\ell )(h\ell -sn^{2})]/h_{c}\\&=[hs-h\ell (h+2s\ell )+h^{2}\ell -hsn^{2}+hs\ell ^{2}-s^{2}\ell n^{2}]/h_{c}\\&=-s^{2}\ell n^{2}/h_{c}.\end{aligned}}$

From equation (4.11f) we get

${\begin{aligned}z_{1}&=(x_{1}\ell +h)/n=h\ell [(s-2s\ell ^{2}-h\ell )+h_{c}h]/h_{c}n\\&=[h\ell (2s-2s\ell ^{2}-h\ell )+h^{2}]/h_{c}n.\end{aligned}}$

Since $z_{0}=h_{c}n$,

${\begin{aligned}\Delta z&=z_{1}-z_{0}=\left[h\ell (2s-2s\ell ^{2}-h\ell )+h^{2}-h_{c}^{2}n^{2}\right]/h_{c}n\\&=[h\ell (2s-2s\ell -h\ell )+h^{2}-n^{2}(h^{2}+2hs\ell +s^{2}\ell ^{2})]/h_{c}n\\&=[2hs\ell -2hs\ell ^{3}+h^{2}(1-\ell ^{2}-n^{2})-2hs\ell n-s^{2}\ell ^{2}n^{2}]/h_{c}n\\&=[2hs\ell (1-\ell ^{2}-n)-s^{2}\ell ^{2}n^{2}]/h_{c}n=-S^{2}\ell ^{2}n/h_{c}.\end{aligned}}$

We now have

${\begin{aligned}\Delta L=[(\Delta x)^{2}+(\Delta z)^{2}]^{1/2}=-s^{2}\ell n/h_{c}=-(s^{2}/2h_{c})\sin 2\xi .\end{aligned}}$

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Table of Contents (book)

Also in this chapter

• Accuracy of normal-moveout calculations
• Dip, cross-dip, and angle of approach
• Relationship for a dipping bed
• Reflector dip in terms of traveltimes squared
• Second approximation for dip moveout
• Calculation of reflector depths and dips
• Plotting raypaths for primary and multiple reflections
• Effect of migration on plotted reflector locations
• Resolution of cross-dip
• Cross-dip
• Variation of reflection point with offset
• Functional fits for velocity-depth data
• Relation between average and rms velocities
• Vertical depth calculations using velocity functions
• Depth and dip calculations using velocity functions
• Weathering corrections and dip/depth calculations
• Using a velocity function linear with depth
• Head waves (refractions) and effect of hidden layer
• Interpretation of sonobuoy data
• Diving waves
• Linear increase in velocity above a refractor
• Time-distance curves for various situations
• Locating the bottom of a borehole
• Two-layer refraction problem

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