Ray theory and scattering theory

Ray concept is simple: energy travels between sources and receivers only along a “pencil-thin” path (perpendicular to wavefronts)

Simple prediction of traveltime is

(Equation (1) in our notes)

Ray theory and scattering theory

This picture is a fiction: Seismic energy is scattered in all directions, in a complicated way

Ray picture is a very simplified (and very useful) model The model derives directly from the wave equation …

Derivation of ray equation The ray integral can be obtained directly from the

acoustic wave equation:

or, in the frequency domain

assume the shape of the pulse does not change an impulse (delta) changes in amplitude and time

Derivation of ray equation

constant time surfaces (wavefronts) are rays are normal to these surfaces

transforming into the frequency domain this implies no dispersion (no frequency dependence)

Derivation of ray equation We now substitute

into the wave equation

This yields (after division by ω2)

No approximations made so far …

We now make a key approximation in:

We assume that ω approaches infinity (this is the fundamental reason for the limitation of ray methods)

This assumption allows us to retain only zeroth order terms, leaving:

(the “Eikonal” equation)

Derivation of ray equation

Derivation of ray equationThe “Eikonal” equation

implies that is a unit vector.

This vector must be normal to wavefronts (constant time surfaces)

Thus by definition is parallel to rays

Derivation of ray equation The unit vector can be written as (where

dr is a a tangent along the ray, with length ds).

Thus

Take dot product with :

The LHS is

and the RHS is simply

Derivation of ray equation The unit vector can be written as (where

dr is a a tangent along the ray, with length ds).

Thus

or

Since τ defines the constant traveltime surfaces (the wavefronts), integration of this equation along the ray gives us

(Equation (1) in our notes, again)

Projection slice theorem

f ^ is the “projection data”, f is the “model”, and the straight line L is defined by

has the same form as

this is a “Radon transform”

Projection slice theorem find the inverse Radon transform

using Fourier transforms (of the model, and of the data)

1D Fourier transform of the data:

kp

Projection slice theorem 2D Fourier transform of the model:

Projection slice theorem This can be written:

we now take the 1D Fourier transform of both sides (i.e., of the data) …

Projection slice theorem

the 1D Fourier transform of both sides (i.e., of the data):

the “Projection Slice Theorem”

Projection slice theorem

this is an inverse Radon transform (in the Fourier domain) to find the model, fill-in the model Fourier space along a series of lines

with enough projections, we eventually know the model

finally, carry out an inverse Fourier transform

concept is fundamental in tomography

there are problems with this approach in practice, so other methods are more common

Projection slice theorem

Recall our two major assumptions

1. high frequency, “asymptotic ray” assumption – Geometrical optics

2. assumption of straight rays (homogeneous background medium)

curved ray traveltime tomography is the standard approach we now investigate how we may relax the asymptotic ray

assumption

Scattering of seismic waves Return to the acoustic wave equation:

add a point source term

(the response to the point source is the “Green’s function”)

Scattering of seismic waves the Green’s function for a single frequency is the response to a point source

vibrating at that frequency

Scattering of seismic waves For problems with an extended source:

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Scattering of seismic waves

What is the change if we perturb the velocity? “split” the model into two parts:

o(x) is the “object” function, or “scattering potential” introduce Go as the solution to:

Scattering of seismic waves

We now “split” the wavefield into two parts:

substituting these into the wave equation yields:

Scattering of seismic waves the new wave equation describes the scattered field – the

scattering potential “creates” the field as if there were a distribution of sources

there is also an integral form for this equation:

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Scattering of seismic waves

the integral form is non-linear (total field appears on the RHS) standard approximation is to replace the total field on the RHS, with

the background (“incident”) field G0

this is the first Born approximation, valid for weak scattering no restriction on background velocity, provided we can calculate

the background field

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Waveform tomography

s2(r) ω

150 m

300 m

Frequency 300 Hz

Wavelength = 10 m

Waveform tomography150 m

300 m

Frequency 300 Hz

Wavelength = 10 m

s2(r) ω

Waveform tomography

“Seismic wavepath”

(Woodward and Rocca, 1989, 1992)

Fresnel zone width

L 38 m (300 Hz)

150 m

300 m

Frequency 300 Hz

Wavelength = 10 m

s2(r) ω

Generalized projection slice theorem

as before, we want to invert the relationship and find the target (in this case, the scattering potential) with the Radon transform we assumed straight rays here we assume homogeneous background far from the sources/receivers, G0 are plane waves

s and r are unit vectors from the source and receiver locations

ko=ω/co

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Generalized projection slice theorem

i.e., we multiply the target by a sinusoid, and integrate this is Fourier transformation (extraction of a particular

wavenumber of the target)

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with yields

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z

k r

k (r - s)

- k s

kz

kx

Born scattering in (kx,kz)

(e.g., Wu and Toksoz, 1987)

k s

Born scattering in the frequency domain

x

Generalized projection slice theorem

the integral is a Fourier transformation, so

O(kx,kz) is the 2D Fourier transform of the object thus the single frequency, single plane wave response yields one Fourier component of the target by using additional plane wave experiments we eventually can know the Fourier transform of the model finally, carry out an inverse 2D Fourier transformation

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Generalized projection slice theorem

This result is the “Generalized Projection Slice” theorem the waves are scattered in a range of directions, sweeping out

“Ewald” circles

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Projection slice theorems

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