Geology 5660/6660 Applied Geophysics 10 Feb 2016 © A.R. Lowry 2016 Last Time: Seismic Reflection Travel-Time Cont’d Dix Equations for multiple layers: is an approximation to the true physics that neglects a t -to-the-first-power dependence… Normal Moveout (NMO) revisited :  For single layer case:  First-order binomial series approximation: (& second-order): (Useful to write travel-time in terms of only V & observed t 0 )

This can also be applied to multiple layers using the Dix equation approximation for V rms : h1h1 V1V1   Recall our dipping layer problem: We can write

Using the same approach used earlier for a horizontal layer, can be expanded and truncated as x2x2 t2t2 mean slope down-dip limb up-dip limb Recall the x 2 – t 2 plot for a dipping layer: Could average to get a line with slope 1/V 1 2, and the difference between the line and the limbs was related to the dip angle:

x T NMO mean down-dip limb up-dip limb We can similarly use the difference between up-dip & down-dip NMO to estimate dip via: Binomial series approximation:

Some practical considerations: For x 2 – t 2 methods & Dix Equations: Dix equation approximation of V rms is most valid for small-offset (near-vertical) rays; validity of approximations decreases with distance For reflection seismic in general: NMO approximations in terms of t 0 require x << 2h 1 (i.e., small-offset, near-vertical rays) Reflections from shallow interfaces will be overwhelmed by other arrivals at small-offsets

Burger refers to optimal window at distances beyond interference from low- V waves, but also must get beyond direct & refracted wave interference to observe confidently “Optimal window” (Industry seismic)

Practical considerations cont’d: Multiples are waves that travel the two-way travel path between two interfaces more than once: Recognizable as having: About twice the travel-time of the primary The same apparent velocity but twice the thickness from x 2 – t 2 analysis Less move-out and lower amplitude than the primary Primary Multiple

V1V1 V2V2 Diffractions : Recall diffractions in the refraction method: Observationally, impossible to distinguish diffracted arrival associated with a step from smooth thickness change over one wavelength (10–20 m for 100 Hz)

V1V1 Diffractions : h1h1 h2h2 x The diffraction travel time is time t s = r s /V 1 from the source to the step edge plus time t d from step to geophone at a distance x g from the step rsrs xgxg

Worth noting: In migration of seismic data, we “migrate” travel-times to what they would be for a zero-offset source by correcting for NMO. After correcting, (layer with no offset) (layer with offset) diffraction twtt

In typical seismic images we plot two-way travel-time increasing downward (to simulate depth) so “smiles” are flipped upside-down to form “frowns” So these can be a diagnostic tool to help recognize faults, salt diapirs and other lateral changes in velocity…