Fundamental Sesmic

8/10/2019 Fundamental Sesmic

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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:

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Fundamental Sesmic