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![Page 1: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/1.jpg)
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.
![Page 2: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/2.jpg)
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.
![Page 3: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/3.jpg)
![Page 4: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/4.jpg)
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.
![Page 5: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/5.jpg)
![Page 6: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/6.jpg)
![Page 7: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/7.jpg)
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?
![Page 8: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/8.jpg)
![Page 9: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/9.jpg)
The outer core is a low velocity layer
![Page 10: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/10.jpg)
![Page 11: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/11.jpg)
Amplitude
In general, the wave amplitude decreases with distance from the source.
Note the reinforcement of the surface waves near the antipodes.
![Page 12: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/12.jpg)
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.
![Page 13: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/13.jpg)
Amplitude
• Geometrical spreading.
• Anelastic attenuation.
• Energy partitioning at the interface.
![Page 14: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/14.jpg)
Geometrical Spreading
For surface waves we get :
For body waves, on the other hand, we get:
Amplitude r 1 .
Amplitude r 1/ 2 .
![Page 15: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/15.jpg)
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:
with being the absorption coefficient.
Amplitude exp r ,
The effect of anelastic attenuation and geometrical spreading combined is:
Amplitude r 1 exp r .
![Page 16: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/16.jpg)
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:
Now consider a sine wave propagating in the x-direction, we have:
where w is the frequency, t is time, and k is the wave-number. The particle velocity is:
and the kinetic energy density is:
. 2
1 2UEk
, )sin(wtAU
, )cos(wtAwU
. )]cos([2
1 2wtAwEk
![Page 17: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/17.jpg)
Since the mean value of cos2 is 1/2, the mean kinetic energy is:
In a perfectly elastic medium, the average kinetic and potential energies are equal, and the total energy is:
Thus, the total energy density flux is simply:
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:
The product of and C is referred to as the material impedance.
E k 1
4[Aw]2 .
. ][2
1 2AwEEE pktotal
˜ E total 1
2C[Aw]2 ,
A1
A2
2C2
1C1
.
![Page 18: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/18.jpg)
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:
The transmission coefficient of a normal incidence is:
Areflected
Aincoming
2C2 1C1
2C2 1C1
.
Atransmitted
Aincoming
21C1
2C2 1C1
.
Energy partitioning at the interface
![Page 19: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/19.jpg)
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%.
![Page 20: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/20.jpg)
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.
![Page 21: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/21.jpg)
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?
![Page 22: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/22.jpg)
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.
![Page 23: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/23.jpg)
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.
![Page 24: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/24.jpg)
Processing: common depth gather
For a horizontal flat layer on top of a half-space, the common mid-point 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.)
![Page 25: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/25.jpg)
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.
![Page 26: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/26.jpg)
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.
![Page 27: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/27.jpg)
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).
![Page 28: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/28.jpg)
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-to-noise ratio. To overcome this problem we perform normal moveout corrections followed by trace stacking. This results in a zerro-offset stack.
![Page 29: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/29.jpg)
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.
![Page 30: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/30.jpg)
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 expt, where is the attenuation constant.
![Page 31: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/31.jpg)
Processing: static (or datum) correction
Time-shift of traces in order to correct for surface topography and weathered layer.
Corrections:
where:Es is the source elevationEr is the receiver elevationEd is the datum elevationV is the velocity above the datum
t E s E r 2Ed
V ,
![Page 32: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/32.jpg)
Processing: static (or datum) correction
An example of seismic profile before (top) and after (bottom) the static correction.
![Page 33: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/33.jpg)
Processing: deconvolution of the source
Seismograms are the result of a convolution between the source and the subsurface reflectivity series (and also the receiver).
Mathematically, this is written as:
where the operator denotes convolution.In order to remove the source effect, one needs to apply deconvolution:
where the operator denotes deconvolution.
source wavelet reflectivity series output series
seismogram = source reflectivity ,
reflectivity = seismogram source ,
![Page 34: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/34.jpg)
Processing: deconvolution of the source
Seismic profiles before (top) and after (bottom) the deconvolution.
Note that the deconvolved signal is spike-like.
![Page 35: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/35.jpg)
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.
![Page 36: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/36.jpg)
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.
![Page 37: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/37.jpg)
Processing: curved interface
A syncline with a center of curvature that is located below the surface results in three normal incidence reflections.
![Page 38: The ray parameter and the travel-time curves P flat and P radial are the slopes of the travel time curves T-versus-X and T-versus- , respectively. While.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649e8a5503460f94b8f699/html5/thumbnails/38.jpg)
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.