Geology 5640/6640Introduction to Seismology

10 Apr 2015

© A.R. Lowry 2015

Last time: Reflection Data Processing

• Source deconvolution (like filtering methods) is done in the frequency domain (where it is a simple division)

• Migration redistributes reflection energy from dipping structures to its “true” location in two-way travel-time, using redundant sampling by multiple sources… Depth migration also removes velocity effects

Read for Mon 13 Apr: S&W 157-176 (§3.4–3.5)

€

s t( ) = w t( )⊗ r t( ) = w t − τ( )r τ( )dτ−∞

∞

∫ ⇔ S ω( ) = W ω( )R ω( )

Ray Theory in a Spherical EarthMost of us (with a few exceptionsfound mostly in legislative bodies)would agree that the Earth is betterapproximated by a sphere than aflat surface.

At distances >20°, we have to takethis into account somehow…

Two common approaches are

• Build a spherical geometry into the equations used, or

• Assume a Cartesian geometry but then change the layer velocities and thicknesses to give travel-times equivalent to those of a sphere (called an “Earth flattening transformation”)

Let’s consider Snell’s Law in a spherical geometry:

Here the ray path is in red;velocity is constant withinspherical shells and increasesat each layer boundary.

At point A, locally the boundaryis ~flat & we can use regularSnell’s Law at the interface:

But '1 does not equal !Consider the right triangles formed by d, r1 and r2:

2

1

1

1 sinsin

υ

υ ′

=

Since sine = opposite/hypotenuse, sin'1 = d/r1 andsin2 = d/r2, or d = r1sin'1 = r2sin2.

2

1

1

1 sinsin

υ

υ ′

=

€

r1 sinθ1

υ1

=r2 sinθ2

υ 2

Snell’s Law in a spherical geometry:

Now, multiply both sides ofSnell’s Law:

by r1:

Since:

we can write Snell’s Lawin spherical coordinatesas:

€

r1 sinθ1

υ1

=r1 sin ′ θ 1

υ 2

€

d = r1 sinθ1' = r2 sinθ2

Snell’s Law in a spherical geometry:

We can alternatively write

in terms of slowness as:

which for an arbitrary ith layeris simply:

Thus the r multiplier correctsfor the change in orientation ofthe interface normal as the rayapproaches the center of the Earth.

€

r1 sinθ1

υ1

=r2 sinθ2

υ 2

€

u1r1 sinθ1 = u2r2 sinθ2 = p

€

p = uiri sinθ i

Δ=

d

dTp

Snell’s Law in a spherical geometry:

But haven’t we changed theunits of the ray parameter?

Short answer: yes! p has unitsof [time/distance]*radius, but recall that arc length on a circleis equal to radius * angle inradians. Thus p actually can bethought of as having units of[time]/[ in radians].

In other words (and thisshouldn’t come as a surprise):

Δ

r

rΔ

More rigorously, let’s consider two rays with slightly different ray parameters:Ray 1

p1 = pT1 = TΔ1 = Δ

Ray 2p2 = p + dpT2 = T+dTΔ2=Δ+dΔ

In the limit as dΔ goes to zero,the “triangle” at right satisfies:

and thus:

Similar to the Cartesian casewhere the ray parameter is theinverse of the apparent velocity cx along the surface, we have:

€

sin i =v0dT

r0dΔ

€

dT

dΔ=

r0 sin i

v0

= p

€

1

cx

=1

dΔdT

=dT

dΔ= p

With some easy math, which we’ll skip (but you can see p159eqns 9-15), we can show that travel-time is:

where: r0 = surface radiusrp = deepest point on ray path (“turning radius”) = r/v;

Angular distance is:

And in this format,

€

T p( ) = uds∫ =ds

v∫ = 2

ζ 2dr

r ζ 2 − p2rp

r0

∫

€

Δ p( ) = 2pdr

r ζ 2 − p2rp

r0

∫

€

τ p( ) = T p( )− pΔ p( ) = 2ζ 2 − p2

rdr

rp

r0

∫

€

dp

dΔ=

d 2T

dΔ2< 0

For a slowly increasingvelocity with depth,slope of the travel-timecurve (= p) decreaseswith Δ and the intercept( τ) increases with Δ(so decreases with p).

With a sufficiently steepgradient in velocity overa small distance, canget triplication (as in theCartesian case).

(Recall what the Cartesian case looked like:)