Velocity-independent time-domain seismic imaging using local event slopes

Acknowledgments

I would like to thank Jon Claerbout and Antoine Guitton for inspiring discussions. Huub Douma, Gilles Lambaré, Isabelle Lecomte, and one anonymous reviewer provided thorough and helpful reviews.

This publication is authorized by the Director, Bureau of Economic Geology, The University of Texas at Austin.

Appendix A

Mathematical derivation of oriented Dix inversion

The famous Dix inversion formula (Dix, 1955) can be written in the form

$$v_i^2 = \frac{d}{d\,t_0}\,\left[t_0\,v^2(t_0)\right]\;,$$ (21)

where $v_i$ is the interval velocity corresponding to the zero-offset traveltime $t_0$ and $v(t_0)$ is the vertically-variable root-mean-square velocity. By a straightforward application of the chain rule, I rewrite the Dix equation in the form

$$v_i^2 = \frac{d\,\left[t_0(t)\,v^2(t)\right]/d\,t} {d\,t_0/d\,t}.$$ (22)

Substituting $t_0(t)$ and $v(t)$ dependences from equations 3 and 4 and doing algebraic simplifications yields

$$\frac{d\,\left[t_0(t)\,v^2(t)\right]}{d\,t} = \frac{l}{p^2\,t}\, \frac{p\,l (p + t\,p_t) - 2 p_t\,t^2}{2\,t_0}\;,$$ (23)

$$\frac{d\,t_0}{d\,t} = \frac{2\,t - l\,(p + t\,p_t)}{2\,t_0}\;,$$ (24)

where $p_t = \partial p/\partial t$. Substituting equations A-3 and A-4 into A-2 produces equation 15 in the main text.

Appendix B

Mathematical derivation of oriented time-domain imaging operators

rays
Figure B-1. Reflection ray geometry in an effectively homogeneous medium (a scheme).

The mathematical derivation of oriented time-domain imaging operators follows geometrical principles. Consider the reflection ray geometry in Figure B-1. Making a hyperbolic approximation of diffraction traveltimes used in seismic time migration is equivalent to assuming an effective constant-velocity medium and straight-ray geometry. The geometrical connection between the effective dip angle $\alpha$, the effective reflection angle $\theta$, the effective velocity $v$, half-offset $h$, and the reflection traveltime $t$ is given by the equation

$$t = \frac{2\,h}{v}\,\frac{\cos{\alpha}}{\sin{\theta}}\;,$$ (25)

which follows directly from the trigonometry of the reflection triangle (Fomel, 2003b; Clayton, 1978). Additionally, the two angles are connected with the traveltime derivatives $p_h=\partial t/\partial h$ and $p_y=\partial t/\partial y$ according to equations

$$p_h = \frac{2\,\cos{\alpha}\,\sin{\theta}}{v}\;,$$ (26)

$$p_y = \frac{2\,\sin{\alpha}\,\cos{\theta}}{v}\;.$$ (27)

Using equations B-1, B-2, and B-3, one can explicitly solve for the effective parameters $\alpha$, $\theta$, and $v$ expressing them in terms of the data coordinates $t$ and $h$ and event slopes $p_h$ and $p_y$. The solution takes the form

$$\tan^2{\alpha} = \frac{h\,p_y^2}{p_h\,(t-h\,p_h)}\;,$$ (28)

$$\sin^2{\theta} = \frac{h\,p_h}{t}\;,$$ (29)

$$v^2 = \frac{4\,h\,(t-h\,p_h)}{t\,\left[t\,p_h + h\,(p_y^2-p_h^2)\right]}\;.$$ (30)

Note that equation B-6 is equivalent to equation 20 in the main text. It reduces to equation 4 in the case of a horizontal reflector ($p_y=0$).

With the help of equations B-4, B-5, and B-6, one can transform all other geometrical quantities associated with time-domain imaging into data attributes. The vertical two-way time is (Sava and Fomel, 2003)

$$\tau = t\,\frac{\cos^2{\alpha} - \sin^2{\theta}}{\cos{\alpha}\,\cos{\theta}}\;,$$ (31)

which turns, after substituting equations B-4 and B-5, into equation 18 in the main text. The separation between the midpoint and the vertical is (Sava and Fomel, 2003)

$$y - x = h\,\frac{\sin{\alpha}\,\cos{\alpha}}{\sin{\theta}\,\cos{\theta}}\;,$$ (32)

which turns, after substituting equations B-4 and B-5, into equation 19 in the main text. Additionally, the zero-offset traveltime is (using equation B-7)

$$t_0 = \frac{\tau}{\cos{\alpha}} = t\,\frac{\cos^2{\alpha} - \sin^2{\theta}}{\cos^2{\alpha}\,\cos{\theta}}\;,$$ (33)

which turns into equation 16. Finally, the separation between the midpoint and the zero-offset point is (using equations B-1 and B-8)

$$y - y_0 = y - x - \frac{v\,\tau}{2}\,\tan{\alpha} = h\,\tan{\alpha}\,\tan{\theta}\;,$$ (34)

which turns into equation 17. In the case of a horizontal reflector ($\alpha=0$), $y=y_0=x$, $t_0 = \tau$, and the zero-offset traveltime reduces to the NMO-corrected traveltime in equation 3.

Non-hyperbolic and three-dimensional generalizations of this theory are possible.

Velocity-independent time-domain seismic imaging using local event slopes