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

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.

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

Substituting and dependences from equations 3 and 4 and doing algebraic simplifications yields

where . Substituting equations A-3 and A-4 into A-2 produces equation 15 in the main text.

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

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 , the effective reflection
angle , the effective velocity , half-offset , and the
reflection traveltime is given by the equation

Using equations B-1, B-2, and B-3, one can explicitly solve for the effective parameters , , and expressing them in terms of the data coordinates and and event slopes and . The solution takes the form

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 ().

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)

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)

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

which turns into equation 17. In the case of a horizontal reflector (), , , 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 |

2013-07-26