The ray equation

2.12.1 The ray equation

A reasonable expectation for inhomogeneous media is that individual temporal frequencies can be represented with a mathematical form that is similar to that for a Fourier plane wave. In two or three dimensions, a Fourier plane wave has the form Ae 2πi(f t± k· x) where #k and #x are the wavenumber and position vectors. By analogy, for the variable-velocity scalar wave equation

v (#x) ∂t

a solution will now be assumed in the form

(2.69) where A(#x) and T (#x) are unknown functions describing amplitude and traveltime that are expected

ψ(#x, t) = A(#x)e 2πif (t−T ( x))

to vary with position. In the constant velocity limit, A becomes constant while T still varies rapidly which leads to the expectation that variation in A will often be negligible compared with variation in T . Substitution of equation (2.69) into equation (2.68) will require computation of the Laplacian of the assumed solution. This is done as

∇ 2 ψ(#x, t) = # ∇·# ∇ A(#x)e 2πif (t−T ( x)) =# ∇· e 2piif (t−T ) ∇A − 2πiAe # 2πif (t−T ) ∇T # (2.70) which can be expanded as

∇ ψ(#x, t) = ∇

A − 4πif # ∇A · # ∇T − 4π f A ∇T − 2πifA∇ T e 2πif (t−T ( x)) . (2.71) Then, using this result and ∂ 2 t ψ(#x, t) = −4π 2 f 2 Ae 2πif (t−T ( x)) in equation (2.68) and equating real

and imaginary parts gives the two equations

(2.72) and

∇T −

2 f 2 4π − A v(#x) 2 =0

∇ T−# ∇A · # ∇T = 0.

So far, these results are exact and no simplification has been achieved. However, the second term in equation (2.72) is expected to be negligible when velocity gradients are weak or when frequencies are high regardless of the velocity gradient. When this term is discarded, the result is the nonlinear,

60 CHAPTER 2. VELOCITY

dx P P 

Figure 2.28: A differential raypath element d#x extends from a point P 1 on a wavefront at time T 1 to a point P 2 on a wavefront at time T 2 . The ray segment is perpendicular to both wavefronts and has length |d#x| = ∆s.

partial-differential equation called the eikonal equation

(2.74) Though not exact, for many realistic situations, a solution to the eikonal equation gives accurate

∇T −

v(#x) 2

traveltimes through complex media. Equation (2.73) is called the geometrical spreading equation because its solution can be shown to describe the flow on energy along a raypath.

The differential equation for raypaths is the vector equation that is implied by the eikonal equa- tion (2.74). For isotropic media, raypaths are normal to the wavefronts and the latter are described as surfaces where T (#x) = constant. Therefore, # ∇T must be a vector that points in the direction of the raypath and the eikonal equation shows that # ∇T

−1 . Consider a wavefront at time T 1 (#x 1 ) and another at a slightly later time T 2 (#x 2 ) where T 2 −T 1 is very small. Let P 1 be a point on the

surface T 1 and P 2 be the corresponding point on T 2 along the normal from P 1 (Figure 2.28). Then ∆s = (T 2 (P 2 )−T 1 (P 1 ))/v(P 1 ) is an estimate of the perpendicular distance between these wavefronts, and

v(P 1 ) where ˆ s is a unit vector that is normal to the surface T 1 (P 1 ) or, in other words, ˆ s points along the

raypath. If d#x is the differential vector pointing from P 1 to P 2 then ˆ s may be written

where ds = |d#x| and s is the arclength along the raypath. Combining equations (2.75) and (2.76) gives the raypath equation

∇T (#x) = # s ˆ

1 d#x

v(#x)

v(#x) ds

The traveltime T can be eliminated from equation (2.77) by calculating its derivative with respect to arclength, which is

d# ∇T

d 1 d#x

ds

ds v ds

61 The left-hand-side of this equation can be simplified as follows

2.12. RAYTRACING FOR INHOMOGENEOUS MEDIA

v(#x)

The last step in this equation is justified using equation (2.77) and the identity d/ds = ˆ s·# ∇. Intuitively, this step is valid because # ∇T points along the raypath and therefore dT/ds, that is the scalar derivative with respect to arclength along the raypath, gives the full magnitude of # ∇T . Thus the ray equation is recast as

∇v(#x) = − # d 1 d#x

(#x)

ds v(#x) ds

which is a second order ordinary differential equation for the raypath vector #x. This result can be further recast into a system of first-order equations by modifying the right-hand-side using

v(#x) dt

and defining the slowness vector, # p=# ∇T , that is (from equation (2.77)

v(#x) ds

(#x) dt

These last two equations allow equation (2.80) to be recast as the first-order system

d#x

=v 2 (#x)# p

∇v(#x) #

v(#x)

Verification that these two equations (2.83) and (2.84) are equivalent to equation (2.80) can be done by solving equation (2.83) for # p and substituting it into equation (2.84).

Example 2.12.1. As an illustration of the validityof equations (2.83) and (2.84), it is instructive to show that theyreduce to the v(z) case considered previously. If v(#x) = v(z), then in two dimensions, let #x = [x, z] and # p = [p x ,p z ] so that equation (2.84) becomes

dp x

dp z

1 ∂v(z)

v(z) ∂z

and equation (2.83) is

dx

=v 2 (z)p x and

dz

=v 2 (z)p z .

The first of equations (2.85) immediatelyintegrates to p x = constant. Using this result in the first of equations (2.86) together with v −2 (z)dx/dt = v −1 (z)dx/ds results in

v(z) ds

v(z)

62 CHAPTER 2. VELOCITY

Figure 2.29: A ray is shown traced through v(x, z) = 1800 + .6z + .4x using ode45 as shown in Code Snippet 2.12.1.

where sin θ = dx/ds has been used. Of course, this is Snell’s law for the v(z) medium. Of the remaining two equations in the system (2.85) and (2.86), the first is redundant because p 2 +p x 2 z = v −2 (z) and the second gives

v (z)p z

v(z) cos θ

v(z) 1 − v 2 (z)p 2 x

When integrated, this will result in equation (2.57). In that equation, p is the same as p x here.