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

${\begin{aligned}V=az^{1/n};\end{aligned}}$

(8.21a)

$a$ and $n$ being constants, $n>+1.$

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 $B=\left(1-1/n\right)$ (see Musgrave and Bratton, 1967, 244).

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

Verify the Blondeau procedure by deriving these relations:

1. $z=z_{m}\sin ^{n}i$, where $i$ is the angle of incidence measured with respect to the vertical at depth $z$;
2. $x=Fz_{m}$, where $F=2n\mathop {\int } \nolimits _{0}^{\pi /2}\sin ^{n}i\ {\hbox{d}}i=$ function of $n$, hence also of $B$;
3. $t=\left(G/a\right)(x/F)^{B}$, where $G=2n\mathop {\int } \limits _{0}^{\pi /2}{\hbox{sin}}^{n-2}i\ {\hbox{d}}i$;

   [Note that $G$ can be obtained from the table for $F$ by writing $m=n-2$, finding for $F$, and multiplying the value by $\left(m+2\right)/m=n/(n-2$

4. ${\hbox{d}}x/{\hbox{d}}t$, the horizontal component of the apparent velocity at any point $\left(x_{j},\;z_{j}\right)$ of the trajectory, is $V_{m}=x/Bt$;
5. $t_{v}=\mathop {\int } \nolimits _{0}^{t_{m}}{\hbox{d}}z/V=t/F$.

Solution

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

Solving equation (8.21a) for $z_{j}$ gives

${\begin{aligned}z_{j}=(V_{j}/a)^{n}.\end{aligned}}$

(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

${\begin{aligned}\left(z_{j}/z_{m}\right)=(V_{j}/V_{m})^{n}={\sin }^{n}i_{j}.\end{aligned}}$

(8.21d)

ii) From equations (4.17d), we get

${\begin{aligned}x_{j}-\int _{0}^{z_{j}}\tan i\ \mathrm {d} z=\int _{0}^{z_{i}}(\tan i)[nz_{m}\sin ^{(n-1)}i\cos i]\mathrm {d} i=nz_{m}\int _{0}^{i_{j}}\sin ^{n}i\ \mathrm {d} i,\end{aligned}}$

where we have differentiated equation (8.21d) to replace $z$ with $i$. If we integrate from 0 to $\pi /2$ and multiply by 2, the result is

${\begin{aligned}x=2x_{m}=2nz_{m}\int \limits _{0}^{\pi /2}{\sin }^{n}i\ \mathrm {d} i=Fz_{m},\end{aligned}}$

${\begin{aligned}{\mbox{where}}\qquad \qquad F=2n\int _{0}^{\pi /2}\sin ^{n}i\ \mathrm {d} i.\end{aligned}}$

(8.21e)

iii) From Figure 4.17a we have $\Delta t=\Delta z/V\cos i$, so

${\begin{aligned}t_{j}=\int _{0}^{z_{j}}\mathrm {d} z/V\cos i=\left(nz_{m}\right)\int _{0}^{i_{j}}{\frac {\sin ^{\left(n-1\right)}i}{V}}\mathrm {d} i,\end{aligned}}$

where equation (8.21d) was used to replace $z$ with $i$. Using equation (8.21c) to eliminate $V$, we get

${\begin{aligned}t_{j}=(nz_{m}/V_{m})\int _{0}^{i_{j}}\,\sin ^{(n-2)}i\ \mathrm {d} i.\end{aligned}}$

Integration from 0 to $\pi /2$ gives $t_{m}$, so

${\begin{aligned}t=2t_{m}=\left(2nz_{m}/V_{m}\right)\int _{0}^{\pi /2}\sin ^{\left(n-2\right)}i\ \mathrm {d} i=\left(z_{m}/V_{m}\right)G.\end{aligned}}$

Since $V_{m}=az_{m}^{1/n}$ from equations (8.21a) and $z_{m}=x/F$ from equation (8.21e), this result can be written

${\begin{aligned}t=(G/a)(x/F)^{B}.\end{aligned}}$

(8.21f)

iv) Solving equation (8.21f) for $x$ gives

${\begin{aligned}x=F(at/G)^{1/B};\end{aligned}}$

${\begin{aligned}{\mbox{then}}\qquad \qquad {\frac {\mathrm {d} x}{\mathrm {d} t}}=\left(F/B\right)(a/G)^{1/B}t^{\left(1/B-1\right)}.\end{aligned}}$

(8.2lg)

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

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

(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,

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

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

${\begin{aligned}V_{m}=\mathrm {d} x/\mathrm {d} t=x/Bt.\end{aligned}}$

(8.21i)

v) We have $t_{v}=$ time to travel vertically from the surface to the depth $z_{m}$, Thus,

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

where we have used equation (8.21a) in the first and last steps. From equations (8.21e) and (8.21i) we have $z_{m}=x/F$ and $V_{m}=x/Bt$, so

${\begin{aligned}t_{v}=t/F.\end{aligned}}$

(8.21j)

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Also in this chapter

• Effect of too many groups connected to the cable
• Reflection-point smear for dipping reflectors
• Stacking charts
• Attenuation of air waves
• Maximum array length for given apparent velocity
• Response of a linear array
• Directivities of linear arrays and linear sources
• Tapered arrays
• Directivity of marine arrays
• Response of a triangular array
• Noise tests
• Selecting optimum field methods
• Optimizing field layouts
• Determining vibroseis parameters
• Selecting survey parameters
• Effect of signal/noise ratio on event picking
• Interpreting uphole surveys
• Weathering and elevation (near-surface) corrections
• Determining static corrections from first breaks
• Determining reflector location
• Blondeau weathering corrections

External links

find literature about Blondeau weathering corrections