Refraction interpretation by stripping

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Problem 11.6a

Solve problem 11.5 by stripping off the shallow layer.

Background

Stripping is a method of interpreting refraction data by removing the effect of upper layers, the removal being accomplished by reducing the traveltimes and distances so that in effect the source and geophones are located on the interface at the base of the “stripped” layer. Stripping can be accomplished by calculation or graphically, or by a combination.

Solution

We wish to compare our results with those of problem 11.5, so we use the same measurements, namely $ V_{1}=2.02 $ km/s and

$ {\begin{aligned}V_{d2}=3.73\ {\rm {km/s}},\quad V_{u2}=4.51\ {\rm {km/s}},\quad t_{u1}=0.92\ {\rm {s}},\quad t_{d1}=0.46\ {\rm {s}};\\V_{d3}=4.29\ {\rm {km/s}},\quad V_{u3}=5.81\ {\rm {km/s}},\quad t_{u2}=1.28\ {\rm {s}},\quad t_{d2}=0.66\ {\rm {s}}.\end{aligned}} $

(To avoid triple subscripts, we denote intercept times at downdip and updip source locations by $ d $ and $ u $.)

We start by using equations (4.24f) to get $ V_{2} $:

$ {\begin{aligned}1/V_{2}=\left(1/V_{d2}+1/V_{u2}\right)/2,\quad V_{2}=4.08\ {\rm {km/s}}.\end{aligned}} $

Equations (4.24b,d) can be written

$ {\begin{aligned}\sin \left(\theta _{c1}+\xi _{2}\right)=V_{1}/V_{d2},\qquad \sin(\theta _{c1}-\xi _{2})=V_{1}/V_{u2};\\{\hbox{so}}\quad \quad \sin(\theta _{c1}+\xi _{2})=2.02/3.73,\qquad (\theta _{c1}+\xi _{2})=32.8^{\circ };\\\sin(\theta _{c1}-\xi _{2})=2.02/4.51,\qquad (\theta _{c1}-\xi _{2})=26.6^{\circ };\end{aligned}} $

hence $ \theta _{c1}=29.7^{\circ } $, $ \xi _{2}=3.1^{\circ } $. These are the same as those in problem 11.5.

Next we calculate the distances perpendicular to the first refractor at $ A $ and $ B $ (Figure 11.6a). We use equation (4.24b) to get $ h_{d} $ and $ h_{u} $:

$ {\begin{aligned}h_{d}=\left(V_{1}t_{d1}\right)/2\cos \theta _{c1}=2.02\times 0.46/2\times \cos 29.7^{\circ }=0.53\ {\rm {km}};\\h_{u}=\left(V_{1}t_{u1}\right)/2\cos \theta _{c1}=2.02\times 0.92/2\times \cos 29.7^{\circ }=1.07\ {\rm {km}}.\end{aligned}} $

These results are identical with those in problem 11.5. We verify the dip using these depths:

$ {\begin{aligned}\xi _{2}=\tan ^{-1}\left[\left(1.07-0.53\right)/10.0\right]=3.1^{\circ }.\end{aligned}} $

The first step in stripping is to plot the shallow refractor; we do this by swinging arcs with centers $ A $ and $ B $ and radii 1.07 and 0.53 km, the refractor being tangent to the two arcs. To get the “stripped” time values, we subtract the times down to and up from the first refractor, i.e., traveltimes along 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/":): AA' 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/":): BB' for sources 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 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 . Although maximum accuracy would be achieved by stripping times for all geophones, the curves for the shallow refraction are so nearly linear that we calculate the stripped times only for each source and one intermediate point on each profile (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/":): Q 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/":): N ). We obtain the required distances by measuring the paths in Figure 11.6a. Calculation of the stripped times is given below. Path lengths: 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/":): AA'\approx 1.30 , 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/":): MN\approx 0.83 , 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/":): BB'\approx 0.65 , 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/":): PQ\approx 1.08 km.

Figure 11.6a.  Stripping for refraction interpretation. The numbers below the zero-time line are distances from 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' .

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{Time along path:}\quad \quad \ AA'\approx 0.64, \ MN\approx 0.41, \ BB'\approx 0.32, \ PQ\approx 0.53\ {\rm s} \\ \hbox{Time along path:}\quad \quad \ AA'MN =2.50\ {\rm s} \\ \hbox{Stripped time for}\quad \quad \ AA'MN \approx 2.50-\left(0.64+0.41\right)\approx 1.45\ {\rm s}\ ({\rm point}\ E) \\ \hbox{Time along path}\quad \quad \ AA'B'B =3.00\ {\rm s} \\ \hbox{Stripped time for}\quad \quad \ AA'B'B \approx 3.00-\left(0.64+0.32\right)\approx 2.04\ {\rm s}\ ({\rm points}\ F, H) \\ \hbox{Time along path}\quad \quad \ BB'P Q=2.30\ {\rm s} \\ \hbox{Stripped time for}\quad \quad \ BB'PQ \approx 2.30-\left(0.32+0.53\right)\approx 1.45\ {\rm s}\ ({\rm point}\ G) \end{align}

Stripping off the first refractor in effect moves sources 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 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 down 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/":): A' 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/":): B' and geophones 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/":): Q 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/":): N down 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/":): P 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/":): M , so the stripped times are plotted above these shifted points, the new traveltimes curves being 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/":): EF 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/":): GH . Measurements on these stripped curves give the following:

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_{d3} =4.65\ {\rm km/s},\quad V_{u3} =5.13\ {\rm km/s},\quad t_{d3} =0.19\ {\rm s},\quad t_{u3} =0.30\ {\rm s}. \end{align}

Table 11.6a. Comparison of results of Adachi’s and stripping method.
Item Problem 11.5 Problem 11.6 Difference
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_{3} ({\hbox {km/s}}) 4.92 4.88 0.8%
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/":): \theta _{c2} 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/":): 56.0^{\circ} 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/":): 56.7^{\circ} 1%
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 _{3} 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/":): 5.8^{\circ} 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/":): 5.7^{\circ} 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/":): h_{2}^{*} ({\hbox {km}}) 1.32 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/":): 1.20^{*} 10%
*Vertical depth measured at source A.

We now 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} V_{3} &=[\left(1/4.65+1/5.13\right)/2]^{-1} =4.88\ {\rm km/s},\\ \theta _{c2} &=\sin ^{-1} \left(4.08/4.88\right)=56.7^{\circ}, \\ h_{u2} &=V_{2} t_{u3} /2\cos \theta _{c2} =4.08\times 0.30/2 \cos 56.7^{\circ} =1.11\ {\rm km}, \\ h_{d2} &=4.08\times 0.19/2 \cos 56.7^{\circ} =0.71\ {\rm km}, \\ \xi _{3} &=\tan^{-1} \left[\left(h_{u2} -h_{d2} \right)/A'B'\right]=\tan^{-1} \left[\left(1.11-0.71\right)/8.90\right]=2.6^{\circ}. \end{align}

This dip is relative 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/":): A'B' , so the total dip 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/":): \xi _{2} +\xi _{3} =\left(3.1^{\circ} +2.6^{\circ} \right)=5.7^{\circ}

Problem 11.6b

Compare the solutions by stripping with those using Adachi’s method (problem 11.5).

Solution

To compare depths, we measured vertical depths below A. Results for the first layer are the same for both methods, those for the next layer are given in Table 11.6a.

Problem 11.6c

What are some of the advantages and disadvantages of stripping?

Solution

Advantages of stripping:

  • Easy to understand
  • Straightforward in application
  • Can be used with beds of different dips if the strike is the same
  • As rapid as other methods when done graphically
  • Can be used to interpret irregular or curved surfaces

Disadvantages:

  • Very sensitive to velocity errors
  • Like most methods, assumes the same strike for all refractors
  • Difficult to apply when dips are steep
Figure 11.7a.  Refractors with the same strike but different dips.

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