Using a velocity function linear with depth

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 4.17a

When the velocity increases linearly with depth according to the relation

$ {\begin{aligned}V=V_{0}+az,\end{aligned}} $

(4.17a)

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/":): a being constant, show that

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/pa)(\cos i_0 - \cos i), \end{align}

(4.17b)

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} t = (1/a)\ln \left(\frac{\tan i/2}{\tan i_0/2}\right), \end{align}

(4.17c)

where 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/":): V = velocity, 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/":): V_0 = velocity at depth 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 , $ x= $ source-geophone distance, 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/":): p = (\sin i)/V= raypath parameter, 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= arrival time, and 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/":): i= angle of incidence.

Background

When the velocity is a function of depth only, as in equation (4.17a), expressions for the offset 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/":): x (see problem 4.1) and traveltime 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 can be found by dividing the medium into horizontal layers, each of infinitesimal thickness (Figure 4.17a) and then integrating. We 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 x_n = \Delta z_n \tan i_n, \end{align}

$ {\begin{aligned}\Delta t_{n}=\Delta z_{n}/(V_{n}\cos i_{n}).\end{aligned}} $

Also, using equation (3.1a), we 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} \sin i_n/V_n = \sin i_0/V_0 = p. \end{align}

In the limit we get the following integrals for 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/":): x and 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 :

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 = \int_0^z \tan i\ \mathrm{d}z, \quad t = \int_0^z \mathrm{d}z/(V\cos i). \end{align}

(4.17d)

When the velocity increases monotonically with depth, a ray must eventually return to the surface (see Figure 4.20a). For horizontal velocity layering the raypaths are symmetrical about the deepest point.

Figure 4.17a.  Raypaths where 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/":): V=V(z) .

Solution

In equation (4.17d) we substitute 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/":): u = pV = \sin i , 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/":): {\rm d}z = {\rm d}V/a = {\rm d}u/pa from equation (4.17a), also 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 {i} = u/(1 - u^2)^{1/2} . Thus, 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} x &= \frac{1}{pa}\int_{u_0}^{u} \frac{d{\rm d}u}{[1-u^2]^{1/2}} = \frac{1}{2pa}\int_{u_0}^{u}\dfrac{{\rm d}(u^2)}{(1-u^2)^{1/2}} = \frac{1}{pa}(1-u^2)^{1/2}\left|\right._{u}^{u_0} \\ &= (1/pa)(\cos i_0 - \cos i); \\ t &= \frac{1}{a}\int_{u_0}^{u}\frac{{\rm d}u}{u(1-u^2)^{1/2}} = \frac{1}{a}\ln\left[\frac{u}{1 + (1-u^2)^{1/2}}\right]\left|\right._{u_0}^{u} \\ &= \frac{1}{a}\ln\left[\left(\frac{\sin i}{\sin i_0}\right)\left(\frac{1+\cos i_0}{1+\cos i}\right)\right] = \frac{1}{a}\ln \left(\frac{\tan i/2}{\tan i_0/2}\right). \end{align}

Problem 4.17b

Show that the angle of incidence 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/":): i and the depth 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 can be written

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} i &= 2 \tan^{-1}(e^{at}\tan i_0/2), \end{align}

(4.17e)

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 &= (\sin i - \sin i_0)/pa. \end{align}

(4.17f)

Solution

From equation (4.17c) 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} e^{at} = \left(\frac{\tan i/2}{\tan i_0/2}\right), \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{hence} \qquad\qquad i = 2 \tan^{-1}(e^{at}\tan i_0/2). \end{align}

Solving equation (4.17a) for 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 = (V - V_0)/a = (\sin i - \sin i_0)/pa. \end{align}

Problem 4.17c

Given the velocity function $ V=1.60+0.600\ z $ km/s, find the depth and offset of the point of reflection and the reflector dip when 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_0 = 4.420 s and 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/":): \Delta t/\Delta x = 0.155 s/km. What interpretation would you give the result?

Solution

First, we note that 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 in the preceding equations is one-way time, so we take 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_0 = 2.210 s, 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/":): V_0 = 1.60 km/s, 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/":): a = 0.600 . To 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/":): p , we find the angle of approach 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/":): i_0 [see equation (4.2d)] for 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/":): \Delta t/\Delta x = 0.155 s/km:

$ {\begin{aligned}i_{0}&=\sin ^{-1}(1.60\times 0.155)=14.4^{\circ },\\i&=2\tan ^{-1}[e^{at}\tan(i_{o}/2)]=2\tan ^{-1}(e^{0.600\times 2.21}\tan 7.2^{\circ })=50.9^{\circ }.\end{aligned}} $

The dip of the reflector 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/":): \xi equals 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/":): i , 50.9Failed 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/":): ^{\circ} in this case, because, in order for the ray to return to the source, it must have been incident on the reflector at right angles.

Because of the unusually steep dip, this event might be from a fault plane or it might represent a steeply dipping bed in an area of highly folded sediments.

Problem 4.17d

If the ray continued without reflection, when and where would it emerge? What moveout would be observed at the recording point? Calculate the maximum depth of penetration.

Solution

[Equations (4.20a,b,c) could be used here instead of the following.]

Because the path is symmetrical about the midpoint, the angle of approach at the receiver equals 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/":): (\pi - i_0) = (180^{\circ}-14.4^{\circ}) = 165.6^{\circ} .

From equations (4.17b,c) we 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} x &= (1/pa)[\cos i_0 - \cos \ (\pi - i_0)] = 2(V_0/a)\cot i_0 \\ &= (2\times 1.60/0.600)\cot 14.4^{\circ} = 20.8\ \mathrm{km}, \\ t &= 2(1/0.600) \ln \left(\frac{\tan(\pi/2-i_0/2)}{\tan i_0/2}\right) = 3.33 \ \ln \ (\cot^2(i_o/2))] \\ &=6.66 \ \ln \ (\cot 7.2^{\circ}) = 6.66 \ \mathrm\ {In} \ (\tan 82.8^{\circ}) = 13.8\ \mathrm{s}. \end{align}

Because of symmetry, the moveout on emergence would be 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/":): -0.155 s/km. To find the maximum depth of penetration 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_m , we use the fact that the ray is traveling horizontally at the depth 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_m , that 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/":): i = \pi/2 . From part (c) we have $ i_{0}=14.4^{\circ } $, so 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/":): p = \sin i_0/V_0 = \sin 14.4^{\circ}/1.60 = 0.155 . Equation (4.17f) now 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 = h_m = (\sin 90^\circ - \sin 14.4^\circ)/(0.155 \times 0.600) = 8.08\ \mathrm{km}. \end{align}

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Weathering corrections and dip/depth calculations	Head waves (refractions) and effect of hidden layer
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Partitioning at an interface	Seismic velocity
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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