# Wyrobek’s refraction interpretation method

Series Geophysical References Series Problems in Exploration Seismology and their Solutions Lloyd P. Geldart and Robert E. Sheriff 11 415 - 468 http://dx.doi.org/10.1190/1.9781560801733 ISBN 9781560801153 SEG Online Store

## Problem 11.11

Sources ${\displaystyle C}$, ${\displaystyle D}$, ${\displaystyle E}$, ${\displaystyle F}$, and ${\displaystyle G}$ in Figure 11.11a are 5 km a part. The data in Table 11.11a are for three profiles ${\displaystyle CE}$, ${\displaystyle DF}$, and ${\displaystyle EG}$ with sources at ${\displaystyle C}$, ${\displaystyle D}$, and ${\displaystyle E}$, no data being recorded for offsets less than 3 km. For profiles from ${\displaystyle F}$ and ${\displaystyle G}$ the intercepts were 1.52 and 1.60 s, respectively. Use Wyrobek’s method (Wyrobek, 1956) to interpret the data.

### Background

Wyrobek’s method is based on a series of unreversed profiles such as those shown in Figure 11.11a. The steps in the interpretation are as follows:

Figure 11.11a.  Unreversed refraction profiles.
1. The traveltimes are measured, corrected, and plotted, and apparent velocities and intercepts are measured. If ${\displaystyle V_{1}}$ cannot be measured, ${\displaystyle \theta _{c}}$ is calculated from an assumed value.
2. The total delay times ${\displaystyle \delta }$ are calculated [see equation (11.8b)] for each geophone location for each profile. The curves for the different profiles are displaced up or down to obtaina composite curve covering the entire range.
3. The half-intercept times are plotted at the source locations and a curve drawn through them. This curve is compared with the composite curve in (d); if the curves are not sufficiently parallel, ${\displaystyle V_{2}}$ is adjusted to achieve parallelism. The composite delay-time curve is also used to interpolate or extrapolate the half-intercept curve to cover the complete range. Delay times are now converted into depths using equation (11.9a), i.e., by multiplying half-intercept times by ${\displaystyle V_{1}/\left(\cos \theta _{c}\right)}$.
Table 11.11a. Time-offset data for three refraction profiles.
${\displaystyle x}$ (km) ${\displaystyle t_{CE}}$ (s) ${\displaystyle t_{DF}}$ (s) ${\displaystyle t_{EG}}$ (s) ${\displaystyle x}$ (km) ${\displaystyle t_{CE}}$ (s) ${\displaystyle t_{DF}}$ (s) ${\displaystyle t_{EG}}$ (s)
3.00 1.18 1.20 1.19 6.60 1.90 2.12 2.49
3.20 1.22 1.29 1.28 6.80 1.94 2.16 2.54
3.40 1.24 1.38 1.35 7.00 1.97 2.20 2.57
3.60 1.28 1.45 1.43 7.20 2.01 2.25 2.60
3.80 1.35 1.54 1.50 7.40 2.06 2.30 2.65
4.00 1.38 1.60 1.58 7.60 2.10 2.33 2.68
4.20 1.41 1.70 1.68 7.80 2.14 2.37 2.71
4.40 1.47 1.74 1.76 8.00 2.17 2.41 2.74
4.60 1.51 1.77 1.82 8.20 2.20 2.45 2.77
4.80 1.53 1.80 1.89 8.40 2.24 2.47 2.82
5.0 1.58 1.82 2.00 8.60 2.30 2.52 2.85
5.20 1.63 1.85 2.06 8.80 2.32 2.55 2.89
5.40 1.65 1.91 2.15 9.00 2.35 2.61 2.93
5.60 1.69 1.95 2.21 9.20 2.38 2.64 2.97
5.80 1.74 1.97 2.29 9.40 2.44 2.68 3.00
6.00 1.78 1.99 2.38 9.60 2.47 2.73 3.04
6.20 1.82 2.03 2.43 9.80 2.50 2.78 3.07
6.40 1.87 2.08 2.46 10.00 2.54 2.82 3.10

### Solution

The traveltimes in Table 11.11a are plotted in the upper part of Figure 11.11b. The values of ${\displaystyle V_{1}}$ and ${\displaystyle V_{2}}$ have different accuracies since different numbers of points are used for each value, so we obtain weighted averages using as weights the horizontal extent of the data for each value. Thus,

{\displaystyle {\begin{aligned}V_{1}&=\left(2.50\times 1+2.52\times 3\right)/4=2.52\ {\rm {km/s}},\\V_{2}&=\left(5.13\times 7+5.08\times 6+5.59\times 4\right)/17=5.22\ {\rm {km/s}},\\{\hbox{so}}\quad \quad \theta _{c}=\sin ^{-1}\left(2.52/5.22\right)=28.9^{\circ }.\end{aligned}}}

Figure 11.11b.  Time-distance plot (top half ) and plot of delay-times and half-intercept times (bottom).
Table 11.11b. Delay times for profiles ${\displaystyle CE}$, ${\displaystyle DF}$, and ${\displaystyle EG}$.
1 2 3 4 5 6 7 8 9
5.22 6.25 7.7 5.6
${\displaystyle x}$(km) ${\displaystyle \delta _{CE}}$(s) ${\displaystyle \delta _{DF}}$(s) ${\displaystyle \delta _{EG}}$(s) ${\displaystyle \delta _{CE}}$(s) ${\displaystyle \delta _{DF}}$(s) ${\displaystyle \delta _{EG}}$(s) ${\displaystyle \delta _{CE}}$(s) ${\displaystyle \delta _{DF}}$(s)
3.0 0.61 0.70
3.2 0.61 0.71
3.4 0.59 0.70
3.6 0.59 0.70
3.8 0.62 0.74
4.0 0.61 0.74
4.2 0.61 0.90 0.74 1.03
4.4 0.63 0.90 0.77 1.04
4.6 0.63 0.89 0.77 1.03
4.8 0.61 0.88 0.76 1.03
5.0 0.62 0.86 0.78 1.02 0.93
5.2 0.63 0.85 0.80 1.02 0.92
5.4 0.62 0.88 0.79 1.05 0.95
5.6 0.62 0.88 0.79 1.05 0.95
5.8 0.63 0.86 0.81 1.04 0.93
6.0 0.63 0.84 0.82 1.03 0.92
6.2 0.63 0.84 1.24 0.83 1.04 1.44 0.92
6.4 0.64 0.85 1.23 0.85 1.06 1.44 0.94
6.6 0.64 0.86 1.23 0.84 1.06 1.43 1.04 0.94
6.8 0.64 0.86 1.24 0.85 1.07 1.45 1.06 0.95
7.0 0.63 0.86 1.23 0.85 1.08 1.45 1.06 0.95
7.2 0.63 0.87 1.22 0.86 1.10 1.45 1.07 0.96
7.4 0.64 0.88 1.23 0.88 1.12 1.47 1.10 0.98
7.6 0.64 0.87 1.22 0.88 1.11 1.46 1.11 0.97
7.8 0.65 0.88 1.22 0.89 1.12 1.46 1.13 0.98
8.0 0.64 0.88 1.21 0.89 1.13 1.46 1.13 0.98
8.2 0.63 0.88 1.20 0.89 1.14 1.46 1.14 0.99
8.4 0.63 0.86 1.21 0.90 1.13 1.48 1.15 0.97
8.6 0.65 0.87 1.20 0.92 1.14 1.47 1.18 0.98
8.8 0.63 0.86 1.20 0.91 1.14 1.48 1.18 0.98
9.0 0.63 0.89 1.21 0.91 1.17 1.49 1.18 1.00
9.2 0.62 0.88 1.21 0.91 1.17 1.50 1.19 1.00
9.4 0.64 0.88 1.20 0.94 1.18 1.50 1.22 1.00
9.6 0.63 0.89 1.20 0.93 1.19 1.50 1.22 1.02
9.8 0.62 0.90 1.19 0.93 1.21 1.50 1.23 1.03
10.0 0.62 0.90 1.18 0.94 1.22 1.50 1.24 1.03

The intercept times from the data in Table 11.11a are ${\displaystyle t_{C}=0.60}$ s, ${\displaystyle t_{D}=0.82}$ s, ${\displaystyle t_{E}=1.31}$ s, and we are also given ${\displaystyle t_{F}=1.52}$ s, ${\displaystyle t_{G}=1.60}$ s. Obviously the refractor is dipping down from ${\displaystyle C}$ towards ${\displaystyle G}$ and ${\displaystyle V_{2}}$ above is in fact ${\displaystyle V_{d}}$. However, initially we shall ignore dip and use ${\displaystyle V_{2}=5.22}$ km/s.

The calculated delay times are listed in Table 11.11b; ${\displaystyle x}$ is the offset distance from the sources for profiles ${\displaystyle CE}$, ${\displaystyle DF}$, and ${\displaystyle EG}$, while ${\displaystyle \delta _{CE}}$, ${\displaystyle \delta _{DF}}$ and ${\displaystyle \delta _{EG}}$ are total delay times. These were obtained in the same way as ${\displaystyle \delta _{AR}}$ and ${\displaystyle \delta _{BR}}$ in Table 11.9b using the value ${\displaystyle V_{2}=5.22}$ km/s to get columns 2, 3, and 4 in Table 11.11b.

The delay times can also be obtained by drawing straight lines through sources ${\displaystyle C}$, ${\displaystyle D}$, and ${\displaystyle E}$ with slopes ${\displaystyle 1/V_{2}}$ (the lines ${\displaystyle HJ}$, ${\displaystyle KL}$, and ${\displaystyle MN}$ in Figure 11.11b) and then measuring the time differences between these lines and the observed times.

The delay times in columns 2, 3, and 4 are plotted in the lower part of Figure 11.11b using small circles (o). The half-intercept times for sources ${\displaystyle C}$, ${\displaystyle D}$, and ${\displaystyle G}$ are also plotted (solid line at top of the lower figure) but using a different scale from that used for delay times.

The next step is to shift the delay-time values to form a continuous composite curve; we achieve this by moving the ${\displaystyle CE}$ curve up and the ${\displaystyle EG}$ curve down. Since this is merely a preliminary step we do not move individual values but displace the average straight lines through the points, giving the composite curve ${\displaystyle PQ}$.

The delay-time curve is not parallel to the half-intercept line and, to achieve parallelism, we must change ${\displaystyle V_{2}}$ to increase the delay times at large values of ${\displaystyle x}$ relative to those at small values. For profile ${\displaystyle CE}$ we need to change ${\displaystyle V_{2}}$ so that ${\displaystyle J}$ moves downward about 0.2 s more than ${\displaystyle H}$; this gives the curve ${\displaystyle H'J'}$ with slope equal to ${\displaystyle 1/V_{2}=1/6.25}$ km/s, the other two curves becoming ${\displaystyle K'L'}$ and ${\displaystyle M'N'}$. We recalculate the delay times using ${\displaystyle V_{2}=6.25}$ km/s; the new values are given in columns 5, 6, and 7 of Table 11.11b and plotted as ${\displaystyle x's}$ in Figure 11.11b. The new curves do roughly parallel the half-intercept curve, and we obtain a new composite delay-time curve by moving ${\displaystyle \delta _{DF}}$ and ${\displaystyle \delta _{CE}}$ upward by 0.2 s and 0.3 s, respectively, to join the ${\displaystyle \delta _{EG}}$ values to form a continuous curve. The values agree exactly except for the first and last overlapping values, which differ by 2 ms; we used the average values at these two points.

Comparison of the composite delay-time curve with the half-intercept time curve shows reasonably good agreement at the two ends but significant divergence in the central part. We might assume that the intercept time at source ${\displaystyle E}$ is in error but the value 1.31 s would have to decrease to about 1.15 s (for a half-intercept time of about 0.58 s) to agree with the delay-time curve. Although the ${\displaystyle EG}$-curve is short, it is regular so that it is difficult to fit a line having an intercept of 1.15 s. A more likely source of error is variations of velocity; these could be of two kinds: (i) the actual value of ${\displaystyle V_{2}}$ could be 6.25 at the two ends but higher than 6.25 in the range ${\displaystyle 7 km and lower than 6.25 in the range ${\displaystyle 10 km, (ii) velocity changes due to dip (the intercepts show an overall dip down from ${\displaystyle C}$ to ${\displaystyle G}$, so ${\displaystyle V_{2}}$ is the apparent velocity ${\displaystyle V_{d}}$. While velocity variations due to changes in dip are the more likely explanation, we can proceed with the interpretation without deciding which velocity effect is the cause.

To reduce the gap between the two curves, we change ${\displaystyle V_{2}}$ so that the difference between the values of ${\displaystyle \delta _{CE}}$ at ${\displaystyle X_{C}=10.0}$ and ${\displaystyle x_{C}=6.6}$ km increases by 0.1 s. Letting ${\displaystyle V}$ be the required velocity and using equation (11.8b), we get

{\displaystyle {\begin{aligned}\left(2.54-10.0/V\right)-\left(1.90-6.6/V\right)=\left(0.94-0.84\right)+0.10,\\V=3.4/\left(0.64-0.20\right)=7.7\ {\rm {km/s}}.\end{aligned}}}

We also need a new velocity that will increase ${\displaystyle \delta _{DF}}$ about 0.1 s more at ${\displaystyle x_{C}=5.0}$ than at ${\displaystyle x_{C}=10.0}$. Thus

{\displaystyle {\begin{aligned}\left(1.82-5.0/V\right)-\left(2.82-10.0/V\right)=\left(1.02-1.22\right)+0.10,\\V=5.0/0.90=5.6\ {\rm {km/s}}.\end{aligned}}}

These two velocities were used to calculate revised delay times in columns 8 and 9 of Table 11.11b, and the revised values are plotted in Figure 11.11b (using small squares).

The final interpreted curve is represented by inverted triangles (${\displaystyle \nabla }$) from ${\displaystyle x_{c}=3.0}$ to ${\displaystyle x_{c}=15.0}$ and by crosses ${\displaystyle \left(\times \right)}$ from ${\displaystyle x_{c}=16.2}$ to 20.0. The values can be changed to depths by multiplying the half-intercept times by ${\displaystyle V_{1}/\left(\cos \theta _{c}\right)}$ [see equation (11.9b)].

We now get approximate dip ${\displaystyle \xi }$ by finding depths at ${\displaystyle C}$ and ${\displaystyle G}$ using equation (11.9b); then we use ${\displaystyle \xi }$ and ${\displaystyle V_{d}}$ to calculate ${\displaystyle V_{u}}$, ${\displaystyle V_{2}}$, and ${\displaystyle \theta _{c}}$ which give a more accurate depth factor ${\displaystyle V_{1}/\cos \theta _{c}}$. Thus, we have

{\displaystyle {\begin{aligned}V_{1}=2.52\ {\rm {km/s}},\quad V_{2}=5.22\ {\rm {km/s}},\quad \theta _{c}=28.9^{\circ },\quad \delta _{c}=0.60/2,\quad \delta _{G}=1.60/2.\end{aligned}}}

Using these values, the depths become

{\displaystyle {\begin{aligned}h_{C}&=0.30\times 2.52/\cos 28.9^{\circ }=0.86\ {\rm {km}},\\h_{G}&=0.80\times 2.52/\cos 28.9^{\circ }=2.30\ {\rm {km}},\\\xi &=\tan ^{-1}\left[\left(2.30-0.86\right)/20\right]=4.1^{\circ }.\end{aligned}}}

Using ${\displaystyle V_{d}=6.25}$ km/s, we solve equation (4.24d) for ${\displaystyle V_{u}}$, giving

{\displaystyle {\begin{aligned}\xi =\left(1/2\right)[\sin ^{-1}\left(2.52/6.25\right)-\sin ^{-1}(2.52/V_{u})],\\{\hbox{so}}\quad \quad 4.1^{\circ }&=\left(1/2\right)[23.8^{\circ }-\sin ^{-1}(2.52/V_{u})]\\\sin ^{-1}\left(2.52/V_{u}\right)&=23.8^{\circ }-8.2^{\circ }=15.6^{\circ },\\\left(2.52/V_{u}\right)&=\sin 15.6^{\circ }=0.269,V_{u}=9.37\ {\rm {km/s}},\\V_{2}\approx {\frac {1}{2}}\left(V_{d}+V_{u}\right)&={\frac {1}{2}}\left(6.25+9.37\right)=7.81\ {\rm {km/s}},\\\theta _{c}&\approx \sin ^{-1}\left(2.52/7.81\right)\approx 18.8^{\circ },\;\cos \theta _{c}\approx 0.947,\\{\hbox{depth factor}}&\approx V_{1}/\cos \theta _{c}\approx 2.52/0.947\approx 2.66.\end{aligned}}}

Thus, the refractor is nearly flat over the region where we used ${\displaystyle V_{2}=6.25}$ km/s, so local dip is mainly in the places where we carried out the second revision using velocities of 7.7 and 5.6 km/s.

We shall not refine our interpretation further because of the limited acccuracy of the data.