Blondeau weathering corrections

**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	8
Pages	253 - 294
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	SEG Online Store

Problem

The Blondeau method of making weathering corrections (Musgrave and Bratton, 1967, 231−246) is useful in areas where, because of appreciable compaction within the low-velocity layer, the velocity is given approximately by the equation

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} V=az^{1/n}; \end{align}

(8.21a)

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 and $$ n $$ being constants, 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/":): n>+1.

The Blondeau method starts with a curve of first breaks versus offset [Figure 8.21a(ii)] plotted on log-log graph paper; the curve is approximately a straight line with 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/":): B=\left(1-1/n\right) (see Musgrave and Bratton, 1967, 244).

To remove the effect of a surface layer of thickness 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_{m} , we find $$ F $$ , a tabulated function of 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/":): B . Then 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=Fz_{m} . Next we use the 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-t plot to find the corresponding 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 . Finally, 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_{v} , the vertical traveltime to 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_{m} , is given by 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_{v} =t/F .

Verify the Blondeau procedure by deriving these relations:

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=z_{m}\sin^{n} i , 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/":): i is the angle of incidence measured with respect to the vertical 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 ;
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=Fz_{m} , 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/":): F = 2n \mathop{\int }\nolimits_{0}^{\pi /2} \sin^{n} i \ \hbox {d}i= function of 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/":): n , hence also of 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/":): B ;
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=\left(G/a\right)(x/F)^{B} , 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/":): G = 2n \mathop{\int }\limits_{0}^{\pi /2} \hbox {sin}^{n-2} i\ \hbox{d}i ; [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/":): G can be obtained from the table 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/":): F by writing 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/":): m=n-2 , finding 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/":): F , and multiplying the value by 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/":): \left(m+2\right)/m=n/(n-2
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/":): \hbox{d}x/\hbox{d}t , the horizontal component of the apparent velocity at any point 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/":): \left(x_{j} ,\; z_{j} \right) of the trajectory, 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/":): V_{m} =x/Bt ;
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_{v} = \mathop{\int }\nolimits_{0}^{t_{m} } \hbox{d}z/V=t/F .

Solution

i) Note that the quantities 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 are the offset and traveltime for the point of emergence. For intermediate points we write 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/":): \left(x_{j} ,\; t_{j} ,\; z_{j} ,\; V_{j} ,\; i_{j} \right) except for the deepest point, where 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/":): x_{m} =x/2, 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_{m} =t/2, 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_{j} =z_{m} , 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_{j} =V_{m} , 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_{m} =\pi /2 .

Solving equation (8.21a) 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_{j} 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_{j} =(V_{j} /a)^{n}. \end{align}

(8.21b)

From Snell’s law we have

{\begin{aligned}\left(\sin t_{j}/V_{j}\right)=1/V_{m}.\end{aligned}}

(8.21c)

Thus, using equations (8.21b,c) 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} \left(z_{j} /z_{m} \right)=(V_{j} /V_{m} )^{n} ={\sin}^{n} i_{j}. \end{align}

(8.21d)

ii) From equations (4.17d), 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_{j}-\int^{z_j}_{0}\tan i\ \mathrm{d}z=\int^{z_i}_{0} (\tan i)[nz_m\sin^{(n-1)} i \cos i]\mathrm{d}i=nz_{m}\int^{i_j}_{0} \sin^{n} i\ \mathrm{d}i, \end{align}

where we have differentiated equation (8.21d) to replace 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 with 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 . If we integrate from 0 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/":): \pi/2 and multiply by 2, the result 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} x=2x_{m} =2nz_{m} \int\limits_{0}^{\pi/2} {\sin}^{n} i\ \mathrm{d}i=Fz_{m}, \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} \mbox {where} \qquad \qquad F=2n\int_{0}^{\pi/2}\sin^{n} i\ \mathrm{d}i. \end{align}

(8.21e)

iii) From Figure 4.17a we have $\Delta t=\Delta z/V\cos i$ , 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/":): \begin{align} t_{j} =\int_{0}^{z_{j} } \mathrm{d}z/V \cos i=\left(nz_{m} \right) \int_{0}^{i_{j}} \frac{\sin^{\left(n-1\right)} i}{V} \mathrm{d}i, \end{align}

where equation (8.21d) was used to replace 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 with 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 . Using equation (8.21c) to eliminate 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 , 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} t_j=(nz_m/V_m)\int^{i_j}_{0}\,\sin^{(n-2)} i\ \mathrm{d}i. \end{align}

Integration from 0 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/":): \pi /2 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/":): t_{m} , 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/":): \begin{align} t=2t_{m} =\left(2nz_{m} /V_{m} \right)\int_{0}^{\pi/2} \sin^{\left(n-2\right)} i\ \mathrm{d}i=\left(z_{m} /V_{m} \right)G. \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/":): V_{m} =az_{m}^{1/n} from equations (8.21a) 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/":): z_{m} =x/F from equation (8.21e), this result 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} t=(G/a)(x/F)^{B}. \end{align}

(8.21f)

iv) Solving equation (8.21f) 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 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} x=F(at/G)^{1/B}; \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} \mbox{then} \qquad\qquad \frac{\mathrm{d}x}{\mathrm{d}t} =\left(F/B\right)(a/G)^{1/B} t^{\left(1/B-1\right)}. \end{align}

(8.2lg)

Equation (8.21f) can be used to eliminate 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/":): \left(a/G\right) , so equation (8.21g) becomes

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} \frac{\mathrm{d}x}{\mathrm{d}t} =\left(F/B\right)t^{-1/B} \left(x/F\right)t^{\left(1/B-1\right)} =x/Bt. \end{align}

(8.21h)

The angle between the wavefront and a horizontal line equals the angle between a ray and the vertical, both angles being the angle of incidence (see Figure 4.2c), so

${\begin{aligned}\sin i_{j}=\left(V_{j}\Delta t_{j}/\Delta x_{j}\right)\end{aligned}}$

[compare with equation (4.2d)]. Therefore,

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} \left(\Delta t_{j} /\Delta x_{j} \right)=\sin i_{j} /V_{j} =1/V_{m} \end{align}

using Snell’s law. Thus the apparent 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/":): \left(\Delta x_{j} /\Delta t_{j} \right)=V_{m} ; since this is a constant, it holds at every point of the trajectory, including the point of emergence. Therefore, from equation (8.21h),

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} V_{m} =\mathrm{d}x/\mathrm{d}t=x/Bt. \end{align}

(8.21i)

v) 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/":): t_{v} = time to travel vertically from the surface to 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_{m} , Thus,

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_{v} =\int_{0}^{z_{m}} dz/V=\int_{0}^{z_{m} } \mathrm{d}z/az^{1/n} =\frac{z_{m}^{\left(1-1/n\right)} }{a\left(1-1/n\right)} =\frac{z_{m} }{aBz_{m}^{1/n} } =\frac{z_{m} }{BV_{m} }, \end{align}

where we have used equation (8.21a) in the first and last steps. From equations (8.21e) and (8.21i) 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/":): z_{m} =x/F 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/":): V_{m} =x/Bt , 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/":): \begin{align} t_{v} =t/F. \end{align}

(8.21j)

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