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

In general, the wave amplitude decreases with distance from the source.

Note the reinforcement of the surface waves near the antipodes.

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 :

For body waves, on the other hand, we get:

Amplitude r 1 .

Amplitude r 1/ 2 .

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 .

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

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

.

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

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 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.)

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-to-noise ratio. To overcome this problem we perform normal moveout corrections followed by trace stacking. This results in a zerro-offset 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 expt, 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:

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 ,

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).

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 ,

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.