On the Dropout of a Bed in a Three-Layer Section in Seismic Exploration

V. A. Andreev

(Presented by the Department of Geophysical Exploration Methods)

The dropout of a bed is understood as the case in which the refracted wave corresponding to some refracting boundary is never recorded in the region of first arrivals.

In a three-layer section, for given velocity ratios \(\frac{v_1}{v_2}\) and \(\frac{v_2}{v_3}\), the appearance of the refracted wave from the first boundary \(R_1\) (Fig. 1) in the first arrivals (hodograph \(t_2\)) or in the second arrivals (hodograph \(t_2'\)) depends on the magnitude of the ratio \(\frac{h_2}{h_1}\). As the value of \(\frac{h_2}{h_1}\) decreases, the hodograph of refracted waves corresponding to the first interface passes from the region of first arrivals into the region of second arrivals.

When working by the CMPV method, bed dropout presents no danger, since refracted waves by this method can be traced not only in the first arrivals and, consequently, interpretation of the observations obtained will make it possible to draw the correct conclusion about the structure of the medium. When working by the first-arrival method, dropout of a bed leads to significant errors in determining depths down to the second interface, since instead of a three-layer section the section is represented as a two-layer one, consisting of beds with velocities \(v_1\) and \(v_3\).

In the literature [1, 2] values are given for the minimum magnitudes of \(\frac{h_2}{h_1}\) at which bed dropout begins to occur (hodograph \(t_2'\), Fig. 1) for certain special cases of the structure of a three-layer medium. However, the minimum values of \(\frac{h_2}{h_1}\) are of interest for various magnitudes of the velocities \(v_1\), \(v_2\), \(v_3\), especially when working by the first-arrival method, since it is not always necessary to associate the disappearance of a refracted wave from the region of first arrivals with the wedging-out of a bed.

In Fig. 2 a graph is given for determining the minimum values of \(\frac{h_2}{h_1}\) for values of \(\frac{v_1}{v_2}\) and \(\frac{v_2}{v_3}\) from 0.2 to 0.8 in the case of horizontal boundaries in a three-layer section. For calculating the graph, the formula was used

\[ \frac{h_2}{h_1} = \frac{\cos i_{12}(1-\sin i_{13})}{\cos i_{23}\sin i_{12}(1-\sin i_{12})} - \frac{\cos i_{13}}{\cos i_{23}\sin i_{12}}, \tag{1} \]

which can be obtained for \(\varphi=\psi=0\) from the equation

\[ \frac{h_2}{h_1} = \frac{\cos i_{12}\left[1-\sin(\alpha_1+\varphi)\right]} {\cos i_{23}\sin i_{12}\left[1-\sin(i_{12}+\varphi)\right]} - \frac{\cos\beta_1+\cos\alpha_1}{2\cos i_{23}\sin i_{12}} -\operatorname{tg}\psi\,\sin\varphi . \tag{2} \]

Fig. 1. Possible relation of hodographs of direct and refracted waves in the case of a three-layer medium.
\(t_1\)—hodograph of the direct wave; \(t_2, t'_2, t''_2\)—hodographs of refracted waves obtained from boundary \(R_1\) for different ratios \(h_2/h_1\); \(t_3\)—hodograph of refracted waves arising at boundary \(R_2\).

Fig. 2. Graph for determining the minimum value of \(h_2/h_1\), at which dropout of the bed begins to occur in the case of a horizontal three-layer medium \(\varphi=\psi=0\). Dashed lines give curves for the particular case: \(z=5^\circ\), \(\psi=0\), \(v_0/v_1=0.6\). The upper curve is for down-dip; the lower, for up-dip.

where \(i_{12}\) and \(i_{23}\) are critical angles,

\[ i_{12}=\arcsin\frac{v_1}{v_2},\qquad i_{23}=\arcsin\frac{v_2}{v_3}, \]

\(\varphi\)—angle of dip of the first boundary,
\(\psi\)—angle between the bounding interfaces,
\(\alpha_1\) and \(\beta_1\)—angles of refraction at the first boundary,

\[ \alpha_1=\arcsin\left[\frac{v_1}{v_2}\sin(i_{23}-\psi)\right], \]

\[ \beta_1=\arcsin\left[\frac{v_1}{v_2}\sin(i_{23}+\psi)\right]. \]

For conformable bedding of the boundaries \((\psi=0)\),

\[ \alpha_1=\beta_1=\arcsin\frac{v_1}{v_3}=i_{13}. \]

Formula (2) can be obtained from the equations of the hodographs \(t_1\), \(t_2\), and \(t_3\) under the equality \(x_{12}=x_{13}=x_{23}\).

From the graph presented it is clear that, for certain velocity ratios, dropout of a bed may occur at fairly large values of \(\dfrac{h_2}{h_1}\) (up to 1.2 and more). This quantity increases when the velocities in the first and second beds differ only slightly and when there is a greater difference between the velocities in the second and third beds.

As was already indicated, dropout of a bed in the first-arrivals method leads to large errors in determining the depth to the surface of a layer characterized by the wave-propagation velocity \(v_3\).

In determining the depth to the boundary \(R_2\) (Fig. 1), not knowing of the existence of the second bed in the absence of the hodograph \(t_2\) in the region of first arrivals, we would proceed from the following formula

\[ h^*=\frac{v_1 t_{02}}{2\cos i_{13}}. \]

The depth \(h^*\) will be the apparent depth to the boundary \(R_2\); it is less than the true depth \(h_1+h_2\). Figure 3 gives the dependence of the quantity

\[ \frac{h_1+h_2}{h^*} \]

for various values of \(\dfrac{v_1}{v_2}\) and \(\dfrac{v_2}{v_3}\) in the case of horizontal boundaries. The graph was constructed from the formula

\[ \frac{h_1+h_2}{h^*} = \left(\frac{h_2}{h_1}+1\right)\cos i_{13} \left(1-\frac{v_1}{v_2}\right), \tag{3} \]

where \(\dfrac{h_2}{h_1}\) are the minimum values of the thickness ratios at which bed dropout begins.

Fig. 3. Graph for determining the error in determining the depth to the second boundary when a bed drops out in the case of a horizontal three-layer medium.

Formula (3) can be obtained from (1) and the time ratio \(\dfrac{t_{02}}{t_{01}}\) (Fig. 1) for \(x_{12}=x_{13}=x_{23}\):

\[ \frac{t_{02}}{t_{01}}=\frac{h^{*}\cos i_{13}}{h_{1}\cos i_{12}} = \frac{\dfrac{1}{v_{1}}-\dfrac{1}{v_{3}}}{\dfrac{1}{v_{1}}-\dfrac{1}{v_{2}}}, \]

It follows from Fig. 3 that errors in determining the depth to the second boundary increase as the values of the velocity ratios \(\dfrac{v_{1}}{v_{2}}\) and \(\dfrac{v_{2}}{v_{3}}\) decrease. Even under the most favorable velocity relationships (the velocities \(v_{1}, v_{2}, v_{3}\) being close in value), the error in determining the depth is about \(20\%\).

It is clear from equation (2) that the dropout of a bed depends not only on its thickness and on the velocity ratio, but also on the angles \(\varphi\) and \(\psi\). In the case of dipping, conformable beds \((\varphi \ne 0, \psi = 0)\), the minimum values of \(\dfrac{h_{2}}{h_{1}}\) for boundary updip will be smaller than the analogous values for boundary downdip. In Fig. 3, dashed curves are given for determining the minimum values of \(\dfrac{h_{2}}{h_{1}}\) for the particular case \(\dfrac{v_{2}}{v_{3}} = 0.6\); \(\varphi = 5^\circ\), \(\psi = 0^\circ\) (the upper curve is for downdip of the boundaries, the lower for updip).

Owing to the influence of the dip angles of the boundaries on bed dropout, cases are possible in which, along one of the directions, no dropout of the bed will occur, while along the other the bed will drop out; therefore opposing hodographs may have different numbers of branches.

REFERENCES

1. G. A. Gamburtsev, Seismic Methods of Exploration, Part II, ONTI, 1938.

2. G. A. Gamburtsev, B. V. Deryagin, et al., Applied Geophysics, issue 2, ONTI, 1934.

3. G. A. Gamburtsev, Yu. V. Riznichenko, I. S. Berzon, et al., Correlation Method of Refracted Waves, Publishing House of the Academy of Sciences of the USSR, 1952.