Transcript Document

The ray parameter and the travel-time curves
Pflat and Pradial are the slopes of the travel time curves T-versus-X
and T-versus-, respectively.
While the units of the flat ray parameter is S/m, that of the radial
earth is S/rad.
The T-X curves and the velocity structures
Steady increase in wave speed:
The rays sample progressively deeper regions in the Earth, and
arrive at progressively greater distances.
High velocity layer:
The rays are reflected at the layer, causing different paths to
cross. For some distance range there are three arrivals: the direct
phase, the refracted phase and the reflected phase. This
phenomena is referred to as the triplication point.
Low velocity layer:
The decrease in ray speed causes the ray to deflect towards the
vertical, resulting in a shadow zone.
Question: Were are the low velocity layers in the Earth?
The outer core is a low velocity layer
Amplitude
Note the reinforcement
of the surface waves
near the antipodes.
In general, the wave
amplitude decreases with
distance from the source.
Also, a major aftershock (magnitude 7.1) can be seen at the
closest stations starting just after the 200 minutes mark. Note the
relative size of this aftershock, which would be considered as a
major earthquake under ordinary circumstances, compared to the
mainshock.
Amplitude
• Geometrical spreading.
• Anelastic attenuation.
• Energy partitioning at the interface.
Geometrical Spreading
For surface waves we get :
Amplitude r1/ 2 .
For body waves, on the other hand, we
get:
Amplitude r1 .


Anelastic attenuation
Rocks are not perfectly elastic; thus, some energy is lost to heat
due to frictional dissipation. This effect results in an amplitude
reduction with distance, r, according to:
Amplitudeexpr ,
with  being the absorption coefficient.

The effect of anelastic attenuation and
geometrical spreading combined is:
Amplitude r1 expr .
Energy partitioning at an interface
Energy:
The energy density, E, may be written as a sum of kinetic energy
density, Ek, and potential energy density, Ep.
The kinetic energy density is:
1 2
E k  U .
2
Now consider a sine wave propagating in the x-direction, we have:
U  A sin(wt ) ,
where w is the frequency, t is time, and k is the wave-number. The
particle velocity is:
U  Awcos(wt ) ,
and the kinetic energy density is:
1
Ek   [ Aw cos( wt )] 2 .
2
Since the mean value of cos2 is 1/2, the mean kinetic energy is:
1
E k  [Aw]2 .
4
In a perfectly elastic medium, the average kinetic and potential
energies are equal, and the total energy is:
1
Etotal  Ek  E p   [ Aw ]2 .

2
Thus, the total energy density flux is simply:
1
˜
E total  C[Aw]2 ,
2
were C is the wave speed.
If the density and the wave speed are position dependent, so is
the amplitude. In the absence of geometrical spreading and

attenuation, we get:
A1
2C2

.
A2
1C1
The product of  and C is referred to as the material impedance.
Energy partitioning at the interface
In conclusion, the amplitude is inversely proportional to the square
root of the impedance.
Reflection and transmission coefficients:
The reflection coefficient of a normal incidence is:
Areflected 2C2  1C1

.
Aincoming 2C2  1C1
The transmission coefficient of a normal incidence is:

Atransmitted
2 1C1

.
Aincoming 2C2  1C1
Energy partitioning at the interface
The amplitudes as a function of incidence angle may be computed
numerically (see equations 4.81-84 in Fowler’s book).
Figure from Fowler
• Note the two critical angles at 300 and 600.
• Phases reflected from the critical angles onwards are of larger
amplitude.
• For normal incidence, the reflected energy is <1%.
Energy partitioning at the interface
• Pre-critical angle, i<ic: Reflection and transmission.
• Critical incidence, i=ic: The critically refracted phase travels along
the interface, emitting head waves to the upper medium.
• Post-critical incidence, i>ic: No transmission, only reflection. The
amplitude of the reflected phase is therefore close to the
amplitude of the incoming wave.
Processing: zero-offset gathers
The simplest data collection imaginable is one in which data is
recorded by a receiver, whose location is the same as that of the
source. This form of data collection is referred to as zero-offset
gathers.
• Advantage: Easy to interpret.
• Disadvantage: Impractical. Why?
Processing: common shot gathers
Data collection in the form of zero-offset gathers is impractical,
since very little energy is reflected by normal incidence. Thus, the
signal-to-noise ratio is small.
Seismic data is always collected in common shot gathers, i.e.
multiple receivers are recording the signal originating from a
single shot.
Processing: common midpoint gathers
Common midpoint gathers: Regrouping the data from multiple
sources such that the mid-points between the sources and the
receivers are the same.
Processing: common depth gather
For a horizontal flat layer on top of a half-space, the common midpoint gather is actually a common depth gather.
In that case, the half offset between the shot and the receiver is
located right above the reflector. (Next you will see that this is a
very logical way of organizing the data.)
Processing: normal moveout correction
Step 1: The data is organized into common mid-point gathers at
each mid-point location.
Step 2: Coherent arrivals are identified, and a search for best
fitting depth and velocity is carried out.
Processing: normal moveout correction
Step 3: The arrivals are aligned in a process called normal
moveout correction (NMO), and the aligned records are stacked.
If the NMO is done correctly, i.e. the velocity and depth are
chosen correctly, the stacking operation results in a large increase
of the coherent signal-to-noise ratio.
Processing: plotting the seismic profile
The next step is to plot all the common mid-point stacked traces at
the mid-point position. This results in a zero-offset stacked seismic
section.
At this stage, the vertical axis of the profile is in units of time (and
not depth).
Processing
The above section may be viewed as an ensemble of experiments
performed using a moving zero-offset source-receiver pair at each
position along the section.
In summary, in reflection seismology, the incidence angle is close
to vertical. This results in a weak reflectivity and small signal-tonoise ratio. To overcome this problem we perform normal moveout
corrections followed by trace stacking. This results in a zerrooffset stack.
Processing: additional steps
Additional steps are involved in the processing of reflection data.
The main steps are:
• Editing and muting
• Gain recovery
• Static correction
• Deconvolution of source
The order in which these steps are applied is variable.
Processing
Editing and muting:
• Remove dead traces.
• Remove noisy traces.
• Cut out pre-arrival nose and ground roll.
Gain recovery: “turn up the volume” to account for seismic
attenuation.
• Accounting for geometric spreading by multiplying the amplitude
with the reciprocal of the geometric spreading factor.
• Accounting for anelatic attenuation by multiplying the traces by
expt, where  is the attenuation constant.
Processing: static (or datum) correction
Time-shift of traces in order to correct for surface topography and
weathered layer.
Corrections:
t 
E s  E r  2E d
,
V
where:
Es is the source elevation
Er is the
receiver elevation
Ed is the datum elevation
V is the velocity above the datum
Processing: static (or datum) correction
An example of seismic profile before (top) and after (bottom) the
static correction.
Processing: deconvolution of the source
Seismograms are the result of a convolution between the source
and the subsurface reflectivity series (and also the receiver).
source wavelet
reflectivity series
output series
Mathematically, this is written as:
seismogram = source  reflectivity ,
where the operator  denotes convolution.
In order to remove the source effect, one needs to apply
deconvolution:

reflectivity = seismogram  source ,

where the 
operator  denotes deconvolution.
Processing: deconvolution of the source
Seismic profiles before (top) and after (bottom) the deconvolution.
Note that the deconvolved signal is spike-like.
Processing: 3D reflection
The 3D reflection experiments came about with the advent of the
fast computers in the mid-1980’s.
In these experiments, geophones and sources are distributed over
a 2D ground patch.
For example, a 3D
reflectivity cube of data
sliced horizontally to reveal
a meandering river channel
at a depth of more than
16,000 feet.
Processing: inclined interface
The reflection point is right below
the receiver if the layer is
horizontal. For an inclined layer,
on the other hand, the reflection
bounced from a point up-dip.
Thus the travel-time curve will
show a reduced dip.
Processing: curved interface
A syncline with a center of
curvature that is located
below the surface results in
three normal incidence
reflections.
Processing: migration
Reflection seismic record must be corrected for non-horizontal
reflectors, such as dipping layers, synclines, and more. Migration
is the name given to the process which attempts do deal with this
problem, and to move the reflectors to their correct position. The
process of migration is complex, and requires prior knowledge of
the seismic velocity distribution.