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```direct
z
zS
z1 x
(a) (b)
Fig. 2.7. Wavefronts at a plane interface with a velocity increase. As Figure 2.6,
with the extra head wave indicated by a dashed-dotted line. The partial and total
reflection are shown dashed. Wavefronts at four times are shown: before the direct
wave reaches the interface; and before, at and after the critical angle is reached.
(Figure 2.6). Because the velocity increases, the transmitted wavefront spreads out
more rapidly and is ahead of the corresponding sphere. Initially when the direct
wavefront is incident on the interface, the intersection point moves along the inter-
face with infinite velocity. As the incident angle increases, the velocity of this inter-
section point drops until at grazing angle, i.e. at infinite range, it has reduced to c1.
Therefore, at some point it will match the velocity in the second medium c2 > c1.
At this point, the critical angle, the transmitted wavefront in the second medium
will be perpendicular to the interface. The wavefront in the second medium will
now continue to propagate sideways with a velocity c2, while the incident wave-
front will propagate slower. Up to the critical angle, the three wavefronts intersect
at the interface satisfying Snell’s law as in Figure 2.2. After the critical angle, the
transmitted wavefront breaks away as it is propagating faster. Wavefronts don’t
just stop discontinuously, so the transmitted wavefront continues to be connected
with the reflected wavefront, by the so-called head wave (the dashed-dotted line),
illustrated in Figure 2.7. In the plane illustrated, the head wave is straight. As the
diagram is a cross-section of an axially symmetric wavefront, the complete head-
wave wavefront is part of a cone (it is sometimes called the conical wave). The
head wave joins the end of the transmitted wavefront with the critical point on the
reflected wavefront. Later we will investigate in detail the generation and proper-
ties of the head wave (Chapter 9). The critical point divides the reflected wavefront
(the dashed line) into two parts: near normal reflection, the wave is a partial reflec-
tion and at wide angles, it is a total reflection.
16 Basic wave propagation
2.3 Travel-time function in layered media
In the previous section we have seen that rays can be traced in the direction of
the slowness vector p, orthogonal to the wavefronts defined by t = T (x). We use
this concept to calculate the travel-time function, T (x), in a layered medium. The
purpose of this section is to outline how this function is calculated, and to describe
the morphology of rays, the travel-time and related functions.
In a layered medium, the slowness parallel to the interfaces is conserved. Let us
write the slowness vector as
p =

 p0
± q

 , (2.3.1)
to avoid subscripts on the components. We have defined the axes so z is perpendic-
ular to the layers and the slowness is in the x–z plane (so py = 0). The slowness
component p is conserved for the ray. In Figure 2.8, we have illustrated the ray
(slowness vector) propagating through a layer. We shall refer to the x direction as
horizontal and the z direction as vertical, positive upwards, the directions we will
always set up axes in a flat, layered Earth, e.g. Figure 0.1.
Let us first consider a model of homogeneous, plane layers. If the velocity in a
layer with thickness �zi is ci , in this layer we have
p = 1
ci
sin θi (2.3.2)
qi =
(
c−2i − p2
)1/2 = 1
ci
cos θi , (2.3.3)
where θi is the angle between the ray and the z axis (Figure 2.8). From the ge-
ometry of the ray segment in the layer, we can easily calculate how far it goes
zi
zi+1
�zi
�xi
θi
p
Fig. 2.8. A ray (slowness vector) propagating through the i-th layer.
2.3 Travel-time function in layered media 17
horizontally (the range) and the travel time
X =
∑
i
�xi =
∑
i
tan θi �zi =
∑
i
p �zi
qi
(2.3.4)
T =
∑
i
�Ti =
∑
i
�zi
ci cos θi
=
∑
i
�zi
c2i qi
, (2.3.5)
where the summation is over all layers along the ray. In general the ray may be
reflected or transmitted at any interface. All layers traversed are included (with
�zi positive for either propagation direction) and for rays that reflect, a layer may
be included multiple times. Later, we will need the derivative of the range function
∂ X
∂p
=
∑
i
∂ (�xi )
∂p
=
∑
i
ci�zi
cos3 θi
=
∑
i
�zi
c2i q
3
i
=
∑
i
�Xi
p
(
1 + p
qi
�xi
�zi
)
.
(2.3.6)
It is straightforward to extend these results to a continuous stratified velocity
function, i.e. c(z). Letting �z → dz in equations (2.3.4) and (2.3.5), the range and
travel time are
X (p) =
∮
p dz
q
(2.3.7)
T (p) =
∮ dz
c2q
, (2.3.8)
where the slowness components are
p = sin θ(z)
c(z)
(2.3.9)
q = cos θ(z)
c(z)
= q(p, z) =
(
c−2(z) − p2
)1/2
. (2.3.10)
The notation
∮
is used as a shorthand to indicate integration over all segments of
the ray, arranged so as to give positive contributions. Thus for the ray illustrated in
Figure 2.9, the complete result is
X (p) =
∮
p dz
q
=
∫ z1
zR
+
∫ z1
z2
+
∫ zS
z2
p dz
q
. (2.3.11)
Typically we write the receiver coordinate as zR and write the range and
travel time as X (p, zR) and T (p, zR). Notice that we have not obtained T (x).
This would require the elimination of the parameter p. Only in simple circum-
stances is this possible. Normally we have to be satisfied with the parameterized
18 Basic wave propagation
z
zS
z1
z2
zR
Fig. 2.9. A ray with a reverberation in a layer.
functions – the conserved horizontal slowness (normally the layers are horizontal),
p, is commonly called the ray parameter. The functions X (p, zR) and T (p, zR)
are commonly called the ray integrals, and T (X), the travel-time curve.
In order to describe the possible morphologies of the travel-time and related
functions, it is useful to know the derivatives of the ray integrals. Provided the end-
points of the integrals are fixed (we discuss below in Section 2.3.1 the case when
the end-point is a function of the ray parameter p), we can easily differentiate the
integrands to obtain
dX
dp
=
∮ dz
c2q3
(2.3.12)
dT
dp
=
∮
p dz
c2q3
= p dX
dp
. (2.3.13)
From this final result, we obtain
p = dT
dX
, (2.3.14)
a result that is more generally true (cf. p = ∇T (2.2.9)). It can also be proved
geometrically. Consider two neighbouring rays with parameters p and p + dp
2.3 Travel-time function in layered media 19
z
xdX
p
p + dp
θ
θ
c dT
Fig. 2.10. Two rays with parameters p and p + dp, extra range dX and extra ray
length c dT .
(Figure 2.10). The extra length of ray is c dT and the extra range dX . From the
geometry of the ray and wavefront, we have c dT/dX = sin θ which, with expres-
sion (2.3.9), gives result (2.3.14).
A useful function is
τ(p, z) = T (p, z) − p X (p, z) =
∮
q dz. (2.3.15)
Clearly as p is the gradient of the travel-time T (X) function (2.3.14), τ is the in-
tercept of the tangent to the travel-time curve with the time axis (Figure 2.11). The
function (2.3.15) is known as the tau-p curve, or the intercept time (or sometimes
the delay time although this is open to confusion). Differentiating either expression
in the definition (2.3.15), it is straightforward to prove that
dτ
dp
= −X. (2.3.16)
Rearranging the definition so
T (p, z) = τ(p, z) + pX (p, z), (2.3.17)
it is clear that the tangent to the tau-p curve intercepts the τ axis at T (Figure 2.12).
The relationship between two functions such as the travel time, T (X), and in-
tercept time, τ(p), illustrated in Figures 2.11 and 2.12, is known as a Legendre,
20 Basic wave propagation
T
τ
T
X
X
p
Fig. 2.11. A simple travel-time```