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2
(a) Transform the isotropic ray tracing system
dp
dσ
= S(x) ∇S (1)
dx
dσ = p (2)
dT
dσ = S2 (x) (3)
into an equivalent system that uses ξ instead of σ, where ξ is constrained
by equation (4):
dp
dξ
dx
dξ
dT
dξ
What are the physical units of ξ?
= ∇S
S 2 (x)
(4)
= (5)
= (6)
(b) Suppose you are given T (x) – the traveltime from the source to all points
x in the domain of interest. Your task is to find ξ(x) for all x. Derive a
first-order partial differential equation that connects ∇ξ and ∇T .
2. The so-called “parabolic” or 15 ◦ eikonal equation (Tappert, 1977; Claerbout,
1985; Bamberger et al., 1988) has the form
∂T
+ 1
∂x 1 2 S(x)
( ∂T
∂x 2
) 2
= S(x) (7)
where x = {x 1 , x 2 } is a point in space, T (x) is the traveltime, and S(x) is
slowness.
(a) Derive the ray tracing system for equation (7)
dx 2
dx 1
= (8)
dp 1
dx 1
= (9)
dp 2
dx 1
= (10)
dT
dx 1
= (11)
where p 1 represents ∂T/∂x 1 and p 2 represents ∂T/∂x 2 .
(b) Assuming constant slowness S(x) ≡ S 0 , solve the ray tracing system for a
point source at the origin {x 1 , x 2 } = {0, 0}.