Accuracy of normal-moveout calculations

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Problem 4.1a

Show that, for constant velocity $ V $, the traveltime $ t $ for the reflection path SCR in Figure 4.1a is


$ {\begin{aligned}t=\left(1/V\right)(x^{2}+4h^{2})^{1/2}.\end{aligned}} $ (4.1a)

Background

Where the velocity is constant, raypaths are straight lines. In Figure 4.1a, a wave travels from the source $ S $ to the receiver $ R $ after being reflected at $ C $, the angle of incidence $ \alpha $ being equal to the angle of reflection. The image point I (or virtual source) is the point on the perpendicular from $ S $ to the reflector as far below the reflector as $ S $ is above. The line IR is equivalent to the actual path SCR.

The difference between the traveltime $ t_{0} $ for a receiver at the source 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/":): S and the 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 for a receiver at an 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 (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 being the source-to-geophone separation) is called the normal moveout, 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_{\rm NMO} . We can get an approximate value 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/":): \Delta t_{\rm NMO} by expanding equation (4.1a) in an infinite series 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/":): t , then subtracting 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} .

An expression 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/":): (1+\phi)^{n} , $ \left|\phi \right|<1 $, can be expanded as a binomial series [see, e.g., Sheriff and Geldart, 1995, equation (15.40)]:


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} (1+\phi)^{n} &=1+n\phi +\frac{n\left(n-1\right)}{2!} \phi^{2} +\frac{n\left(n-1\right)\left(n-2\right)}{3!} \phi^{3} + \cdots \\ &\quad +\frac{n\left(n-1\right)\ldots \left(n-r+1\right)}{r!} \phi^{r} +\cdots. \end{align} (4.1b)

This series converges 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/":): \left|\phi\right|<1 and is infinite except 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/":): n is a positive integer.

While the velocity is assumed to be constant in this problem, equations such as (4.1 a,c) are also used when the velocity varies (usually increasing with 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/":): V being replaced by a suitable velocity such as the average 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/":): \bar{V} (see problem 4.13), the root-mean-square 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_{\rm rms} (problem 4.13), or the stacking 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_{s} (problem 5.12).

Figure 4.1a.  Geometry of NMO.

Solution

The virtual path 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/":): IR 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/":): SCR and thus equals $ Vt $. 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/":): SI is normal to 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 -axis, 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} (Vt)^{2} = x^{2} + (2h)^{2}, \end{align}

i.e., 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/":): (t, x) curve is an hyperbola. Taking the square root, we get equation (4.1a).

Problem 4.1b

Show that 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/":): 2h>x the normal moveout is approximately


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 t_{\rm NMO} \approx x^{2} /2V^{2} t_{0} \approx x^{2} /4Vh. \end{align} (4.1c)

Solution

We write equation (4.1a) as

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=\left(2h/V\right)[1+(x/2h)^{2} ]^{\frac{1}{2}}. \end{align}

If 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/":): 2h>x , we use equation (4.1b) to expand this expression to get the series


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 = \frac{2h}{V} \left\{1+\frac{1}{2} \left(\frac{x}{2h}\right)^{2} +\left[\frac{1}{2} \left(-\frac{1}{2} \right)\frac{1}{2!} \left(\frac{x}{2h}\right)^{4} \right]+\cdots\right\}. \end{align} (4.1d)

Neglecting terms higher than $ (x/2h)^{2} $ (i.e., taking the first approximation) and noting 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_{0} =2h/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\approx t_{0} \left[1+\frac{1}{2} (x/2h)^{2} \right]. \end{align} (4.1e)

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/":): \Delta t_{\rm NMO} = t-t_{0} \approx t_{0} x^{2}/8h^{2} = x^{2} /2V^{2} t_{0} =x^{2} /4Vh .

Problem 4.1c

Calculate the normal moveout 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_{\rm NMO} for geophones 600, 1200, and 3600 m from the source for a reflection at 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} =2.358 s, given that the 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/":): {}=2.90 km/s. What is 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 ?

Solution

From equation (4.1c) we have 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=600 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/":): \begin{align} \Delta t_{\rm NMO} = x^{2}/2V^{2} t_{0} = 0.600^{2} /\left(2\times 2.90^{2} \times 2.358\right)=9.10\ {\rm ms} = 9\ {\rm ms}. \end{align}

Because the NMO varies as $ x^{2} $, the value 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=1200 m will be 4 times that 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=600 m, that is, 36 ms, and 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=3600 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/":): (x \approx h) , 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_{\rm NMO} =328 ms. 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/":): h=(2.358/2)2.90 = 3420 m.

Problem 4.1d

Typical uncertainties in measurements 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/":): x , 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 , 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_{0} be 0.6 m, 0.2 km/s, and 5 ms. Calculate the corresponding uncertainty in $ \Delta t_{\rm {NMO}} $. What do you conclude about the accuracy 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/":): \Delta t_{\rm NMO} calculations?

Solution

The uncertainty in 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 is about 0.1%, that in 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 is about 7%, and that in 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} is about 0.2%. The uncertainties in 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^{2} 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^{2} are 0.2% and 14%. Since the three factors are multiplied or divided 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/":): \Delta t_{\rm NMO} , the uncertainties add so that the uncertainty in 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_{\rm NMO} is about $ 14.4\%\approx 14\% $. This uncertainty is due mainly to the error in 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 .

Problem 4.1e

Show that a more accurate normal moveout 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} \Delta t_{\rm NMO}^{*} \approx \Delta t_{\rm NMO} \left(1-\Delta t_{\rm NMO} /2t_{0} \right). \end{align} (4.1f)

How much difference is there between 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_{\rm NMO}^{*} 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_{\rm NMO} 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=1200 and 3600 m? Taking into account the uncertainties in 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 , $ V $, 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} , when is this equation useful?

Solution

If we go to the second approximation in the expansion of equation (4.1d), equation (4.1e) 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} t=t_{0} \left[1+\frac{1}{2} \left(\frac{x}{2h}\right)^{2} -\frac{1}{8} \left(\frac{x}{2h}\right)^{4} \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{Then,} \qquad \Delta t_{\rm NMO}^{*} &= t-t_{0} = \frac{t_{0}}{2} \left(\frac{x}{2h}\right)^{2} -\frac{t_{0} }{8} \left(\frac{x}{2h}\right)^{4} \\ &=\frac{t_{0}}{2}\left(\frac{x}{2h}\right)^{2} \left[1-\frac{1}{4} \left(\frac{x}{2h}\right)^{2} \right]=\Delta t_{\rm NMO} \left[1-\frac{1}{4} \left(\frac{x}{2h}\right)^{2} \right] \\ &=\Delta t_{\rm NMO} \left[1-\left(\frac{x^{2}}{4Vh}\right)\left(\frac{V}{4h}\right)\right]=\Delta t_{\rm NMO} \left(1-\Delta t_{\rm NMO} /2t_{0} \right). \end{align}

Equation (4.1f) reduces the uncertainty 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/":): \Delta t_{\rm NMO} /2t_{0} =(x/4h)^{2} . To affect the result significantly, this increase in accuracy should be at least 1%, i.e., 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/4h)^{2} \ge 0.01 or 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\ge 0.4 h. Accordingly, equation (4.1f) is useful when 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 is greater than about one-half the reflector depth. 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/":): x=1200 m, the difference in 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_{\rm NMO}^{*} is less than 1 ms but for $ x=3600 $ m, the difference is 22 ms. However, the effect of velocity errors in real situations is usually more important than the approximation error.

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