Introduction to Seismology Geology 5640/6640 Introduction to Seismology 3 Apr 2015 Last time: Reflection Seismic Equations for a reflection two-way travel-time from a layer-over-halfspace impedance contrast can be written: • For deeper layers, we can approximate the two-way travel-time as: where: ; • From these we get velocity & thickness from a layer with bracketing reflections via Dix’ Equations: Read for Mon 6 Apr: S&W 157-176 (§3.4–3.5) © A.R. Lowry 2015

Reflections Cont’d: Dipping Layer and Tau-P A reflection from a dipping layer will have two-way travel- time: so updip/downdip limbs are symmetric about a line in x2-t2 • Using ray theory we can show that travel-time relates to slowness as: • By writing the multi-layer reflection equations as a dot-product of a slowness vector with a path vector, we get the tau-P relation: where t(x) = px + (p)

But p is a constant so We can define a variable so that t(x) = px + (p) . This is the tau-P relation used in the intercept-slowness method, and one will note that it looks like the equation for a line (except that  is changing with p)… So this is a description of a travel-time in terms of its slope and intercept at any given distance x!

Let’s convert a layer-over-halfspace reflection to τ(p) form: Squaring this gives: This is the equation of an ellipse with axes along the  and p coordinate axes: The ellipse intersects the -axis at and the p-axis at:

We can look at the relationship with x(p) by recognizing: This means that at p = 1/v0 (where τ = 0), This makes sense if we consider that the asymptote to a reflection hyperbola (i.e. the slope it approaches as x  ∞) is x = 1/v0.

Now consider the head wave: Our travel-time equation is: and the intercept for the wave arrival is Thus, the ray parameter is and plugging p into the equation for (p), which corresponds to the travel-time at the critical distance xc!

So the travel-times in tau-P look like ellipses for reflections; points for refractions (& direct & surface waves!)

What if there are multiple layers? For multiple layers, the reflections are all ellipses, and head waves occur at the intersections of ellipses.

Multi-Channel Geometry: Much of the success of reflection imaging techniques stems from experiment design: Reflections have subtle expression in data from a single source, so signal- to-noise ratios in the data are increased by stacking. Data are collected with many, many sources and receivers, and then data can be categorized by gathers for a common shot, common receiver, common midpoint, common offset…

Common Offset Gathers: Common offset gathers are particularly handy for small-scale projects in which the objective is to find the top of a near- surface layer (especially if it has variable depth) by using the expectation that travel-time follows: Vw V1 V2

Perhaps most important for seismic reflection imaging however are Common Midpoint (CMP) Gathers, because these can be used to suppress energy arriving in head waves, direct waves, surface waves, and multiple reflections (e.g. below) to emphasize desired primary reflections.

The change in reflection travel-time with offset is given by the Normal MoveOut (NMO) equation: For a reflection from a given layer, NMO correction stacks CMP gathers using different assumed velocities and looks for the stacking velocity that maximizes the stacked energy