fundamental sesmic
TRANSCRIPT
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Seismic waves and Snells law
A wave frontis a surface connecting all
points of equal travel time from the
source.
Raysare the normals to the
wavefronts, and they point in the
direction of the wave propagation.
While the mathematical description of
the wavefronts is rather complex, that
of the rays is simple. For manyapplications is it convenient to consider
rays rather than wavefronts.
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But before proceeding it is important
to understand that the two approachesare not exactly equivalent.
Consider a planar wavefront passing
through a slow anomaly. Can this
anomaly be detected by a seismic
network located on the opposite side?
wavefront
seismic networknegative anomaly
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negative anomaly
negative anomaly
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With increasing distance from the
anomaly, the wavefronts undergohealing (show animation). This effect is
often referred to as theWavefront
Healing.
On the other hand, according to the ray
theory the travel time from point A to
B is given by:
TBA =
BA
dS
C(s),
where dSis the distance measured
along the ray, andCis the seismicvelocity.
Thus, a ray traveling through a slow
anomaly will arrive after a ray travelingthrough the rest of the medium.
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Just like in optics
The angle of reflection equals the angleof incidence, and the angle of refraction
is related through the velocity ratio:
sin i
incoming
air
Vair=sin i
reflected
air
Vair= sin i
refracted
glass
Vglass
Seismic rays too obey Snells law. But
conversions fromP to Sand vice versa
can also occur.
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Phase conversions
Consider a down-going P-wave arriving
to an interface, part of its energy is
reflected, part of it is transmitted to
the other side, and part of the reflected
and transmitted energies are converted
into Sv-wave.
The incidence angle of the reflected and
transmitted waves are controlled by an
extended form of the Snells law:
sin i
1=
sin
1=
sin i
2=
sin
2 P
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The ray parameter and the horizontal
slowness
The ray parameter,P, is constant along
the ray, and is the same for all rays
(reflected, refracted and converted)
originated from the same incoming ray.
Consider a plane wave that propagates
in the k direction. The apparent
velocity c1, measured at the surface is
larger than the actual velocity, c.
c1= c
sin i> c
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sin i= dsdx1
= cdtdx1
= cc1
P sin ic
= 1
c1
Thus, the ray parameter may be
thought as the horizontal slowness.
Snells law for radial earth
The radial earth ray parameter is given
by:
P R sin iV
Next we present a geometrical proof
showing that P is constant along the
ray.
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A geometrical construction showing
that R sin i/V is constant along the ray.
i1
1ii1
i2
V1V2
B
R2R1
From the two triangles:
B =R2 sin i1=R1 sin i1
From Snells law across a planeboundary:
sin i1
sin i2=
V1V2
R sin iV
= constant = ray parameter
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How canPbe measured?
P
Q
N
Deta/2
dDelta/2
N
Q
P
ii
R
dDelta/2
sin i=QN
QP =
V dT /2
Rd/2
dTd
= R sin iV
=P
So P is the slope of the travel time
curve (T-versus-). While the units of
the flat earth ray parameter is s/m,
that of the radial earth is s/rad.
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The bottoming point
With this definition for the ray
parameter in a spherical earth we can
get a simple expression that relates P
to the minimum radius along the raypath. This point is known as the
turning pointor thebottoming point.
Rmin sin 90V(Rmin)
= RminV(Rmin)
=P
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Travel time curves
The ray parameter of a seismic wavearriving at a certain distance can thus
be determined from the slope of the
travel time curve.
The straight line tangent to the travel
time curve at can be written as a
function of the intercept time and theslopeP.
P = dT
d T() =+ dT
d =+P.
This equation forms the basis of what is
known as the - P method.
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P, the local slope of the travel time
curve, contains information about the
horizontal slowness, and the intercepttime , contains information about the
layer thickness.
Additional important information
comes from the amplitude of the
reflected and refracted waves. This andadditional aspects of travel time curves
will be discussed next week.
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Reflection and refraction from a
horizontal velocity contrast
Consider a seismic wave generated near
the surface and recorded by a seismic
station at some distance.
In the simple case of a 2 layer medium,the following arrivals are expected:
the arrival of thedirect wave
the arrival of thereflected wave
the arrival of therefracted wave
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Next we develop the equations
describing the travel time of each ray.
The travel time of the direct wave issimply the horizontal distance divided
by the seismic velocity of the top layer.
t= XV0
This is a surface wave!!!
The travel time of the reflected waveis given by:
t= 2V0
h20+
X2
2
t2 = 2h0V0
2
+ X
V02
So the travel time curve of this ray is a
hyperbola!!!
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The refracted wave traveling alongthe interface between the upper and the
lower layer is a special case of Snells
law, for which the refraction angle
equals 90 deg. We can write:
sin icV0
=sin90V1
sin ic= V0V1
, (1)
where ic is the critical angle. The
refracted ray that is returned to the
surface is ahead wave.
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The travel time of the refracted wave is:
t= 2h0V0 cos ic
+X 2h0 tan ic
V1=
2h0
V21 V2
0
V0V1+
X
V1So this is an equation of a straight line
with a slope of 1/V1, and the interceptis a function of the layer thickness and
the velocities above and below the
interface.
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Refracted waves start arriving after a
critical distance Xcrit, but they overtakethe direct waves at a crossover distance
Xco.
The critical distance is:
Xcrit= 2h0 tan ic
The crossover distance is:
XcoV0
=Xco
V1+
2h0
V21 V2
0
V1V0
Xco= 2h0
V1+ V0
V1 V0Note that at distances greater than Xco
the refracted waves arrive before the
direct waves even though they travel agreater distance. Why?
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Reflection in a multilayerd medium
For a single layer we found:
T2 =T20
+
X
V0
2,
where: T0= 2h0/V0.
Similarly, for a multilayerd medium:
T2n =T2
0,n+ X
Vrms,n2
,
where:
T0,n=n
2hnVn
,
and:
V2rms,n=
n V
2
n2hnVn
n
2hnVn
On aT2-versus-X2 plot, the reflectors
appear as straight lines with slopes that
are inversely proportional to V2rms,n.
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So how do we do it?
Data acquisition:
Next, the traces from several geophonesare gathered:
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And here is a piece of a real record: