Interpretation by the plus-minus method

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	11
Pages	415 - 468
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	SEG Online Store

Problem 11.13

Interpret the data in Table 11.13a using the plus-minus method.

Background

Fermat’s principle (problem 4.13) states that the raypath between two points $A$ and $B$ is such that the traveltime is either a minimum (e.g., direct waves, reflections and head waves) or a maximum. Therefore, the raypath between $A$ and $B$ is unique so that $t_{AB}=t_{BA}=t_{r}=reciprocaltime$. As a result, when we have reversed profiles, we can locate the refractor by drawing wavefronts from the two sources $A$ and $B$; when the sum of the traveltimes for two intersecting wavefronts equals $t_{r}$, the point of intersection must lie on the refractor (see problem 11.14c). This is the basic concept of the plus-minus method (Hagedoorn, 1959).

Construction of wavefronts is discussed in problem 11.14c.

Based on the recorded data, we draw and label wavefronts at intervals $\Delta$ as in Figure 11.13a. If the dip is zero, they are at the angles $\pm \theta _{c}$ to the refractor and the intersections give diamond-shaped parallelograms. The horizontal diagonal of a parallelogram is $V_{2}\Delta$ and the vertical diagonal is $V_{1}\Delta /\cos \theta _{c}$. Lines of constant sum of the traveltimes minus $t_{r}$ (plus values) gives the refractor configuration and differences (minus values) give a check on the value of $V_{2}$. The refractor lies at plus value = 0.

Table 11.13a. Time-distance data for plus-minus interpretation.

$x$ (km)	$t_{A}$(s)	$t_{B}$(s)	$x$ (km)	$t_{A}$(s)	$t_{B}$(s)
0.0	0.00	2.30	6.0	1.30	1.32
0.4	0.15	2.23	6.4	1.33	1.28
0.8	0.28	2.15	6.8	1.40	1.24
1.2	0.44	2.09	7.2	1.51	1.18
1.6	0.52	2.04	7.6	1.57	1.10
2.0	0.63	1.98	8.0	1.60	1.04
2.4	0.70	1.92	8.4	1.72	0.96
2.8	0.76	1.85	8.8	1.78	0.90
3.2	0.84	1.80	9.2	1.80	0.83
3.6	0.91	1.72	9.6	1.91	0.76
4.0	0.95	1.64	10.0	1.93	0.66
4.4	1.04	1.60	10.4	2.04	0.52
4.8	1.12	1.55	10.8	2.07	0.39
5.2	1.16	1.47	11.2	2.17	0.25
5.6	1.25	1.40	11.6	2.20	0.12
—	—	—	12.0	2.30	0.00

Figure 11.13a.  Illustrating the plus-minus method.

Figure 11.13b.  Solution by the plus-minus method.

Solution

The traveltime curves are shown in Figure 11.13b. From the figure we obtained the following values: $V_{1}=2.90$ km/s, $V_{2}=6.25$ km/s, $t_{r}=2.28$ s, $\theta _{c}=\sin ^{-1}\left(2.90/6.25\right)=27.6^{\circ }$.

We next draw straight-line wavefronts at $\pm 27.6^{\circ }$ spaced at intervals $\Delta =0.20$ s. Because $t_{r}=2.30$ s and the refraction from source $B$ starts around 0.8 s, we draw wavefronts for source $B$ for $t_{B}=0.80$, 1.00, 1.20, 1.40, and 1.60 s. For source $A$ we draw wavefronts for $t_{A}=1.48$, 1.28, 1.08, 0.88, and 0.68 s. We interpolate to find the starting points of these wavefronts.

The horizontal and vertical diagonals of the parallelograms have lengths of 1.24 and 0.66 km, so

${\begin{aligned}V_{2}\Delta =1.24,\quad V_{2}=1.24/0.20=6.20\ {\rm {km/s}},\\V_{1}\Delta /\cos \theta _{c}=0.66,\quad V_{1}=0.66\cos 27.6^{\circ }/0.20=2.92\ {\rm {km/s}}.\end{aligned}}$

These values agree with the values in Figure 11.13b within the limits of error.

Table 11.14a. Refraction time-distance data.

$x$(km)	$t_{A}$(s)	$t_{B}$(s)	$t_{A}^{*}$(s)	$x$(km)	$t_{A}$(s)	$t_{B}$(s)	$t_{B}^{*}$(s)
0.00	0.000	3.310		6.40	2.330	2.003
0.40	0.182	3.182		6.80	2.422	1.862
0.80	0.320	3.140		7.20	2.504	1.743
1.20	0.504	3.063		7.60	2.602	1.622
1.60	0.680	2.917		8.00	2.658	1.610
2.00	0.862	2.839		8.40	2.720	1.482	1.561
2.40	0.997	2.714		8.80	2.744	1.329	1.440
2.80	1.170	2.681	1.682	9.20	2.760	1.140	1.288
3.20	1.342	2.570	1.760	9.60	2.855	1.018	1.202
3.60	1.495	2.505	1.858	10.00	2.920	0.863	1.177
4.00	1.677	2.442	1.881	10.40	2.980	0.660	1.082
4.40	1.821	2.380	1.962	10.80	3.065	0.503
4.80	1.942	2.318	2.053	11.20	3.168	0.340
5.20	2.103	2.220		11.60	3.230	0.198
5.60	2.150	2.125		12.00	3.310	0.000
6.00	2.208	2.030

The refractor is indicated in Figure 11.13b by the dashed line. The variation in the spacing of the vertical minus lines is very slight so that we can assume that $V_{2}$ is constant.

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Also in this chapter

• Salt lead time as a function of depth
• Effect of assumptions on refraction interpretation
• Effect of a hidden layer
• Proof of the ABC refraction equation
• Adachi’s method
• Refraction interpretation by stripping
• Proof of a generalized reciprocal method relation
• Delay time
• Barry’s delay-time refraction interpretation method
• Parallelism of half-intercept and delay-time curves
• Wyrobek’s refraction interpretation method
• Properties of a coincident-time curve
• Interpretation by the plus-minus method
• Comparison of refraction interpretation methods
• Feasibility of mapping a horizon using head waves
• Refraction blind spot
• Interpreting marine refraction data

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