Identification of seismic phases

2008, May 12, M7.9, Eastern Sichuan, China

A stack of (long period) data from a global network

InterpretationData

Travel times Ray paths

• Reflected phases include: PcP and PcS.• Refracted phases include: P, S and PKP.

Snell’s law and the ray parameter: reminder

Flat Earth:

We have seen that:

Thus, the ray parameter may be thought as the horizontal slowness.

Radial Earth:

Similarly, we have seen that:

Thus, the radial ray parameter too is a slowness parameter, and may help to infer Earth velocity structure!

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Pradial ≡Rsin i

V .

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Pflat =dT

dXhorizontal

=1

Vhorizontal

.

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Pflat ≡sin i

V .

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Pradial =dT

dΔ=

Rmin

Vmin

.

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.

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

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.

Factors affecting seismograms

• The challenge of source (i.e., earthquake) seismologists is to infer the source time function. Isolation of the source effect is obtained via removal of the propagation, site and instrument effects.

• Global seismologists are interested in imaging earth structure, and their challenge is to remove the source, site and instrument effects.

• The objective of exploration seismologists is to image the subsurface structure on a scale that is relevant for the industry. They use controlled sources, such as dynamite gun shots, weight drop and hammers.

Source, global and exploration seismologists

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

• Energy partitioning at the interface.

• Anelastic attenuation.

• Geometrical spreading.

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:

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Ek =1

2ρ ˙ U 2 .

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U = Asin(wt − kx) ,

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˙ U = Aw cos(wt − kx) ,

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Ek =1

2ρ[Aw cos(wt − kx)]2 .

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 mean energy is:

Thus, the average 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.

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E k =1

4ρ[Aw]2 .

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E total = E k + E p =1

2ρ[Aw]2 .

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˜ E total =1

2ρC[Aw]2 ,

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A1

A2

=ρ 2C2

ρ1C1

.

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:

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Areflected

Aincoming

=ρ 2C2 − ρ1C1

ρ 2C2 + ρ1C1

.

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Atransmitted

Aincoming

=2ρ1C1

ρ 2C2 + ρ1C1

.

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.

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.

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Amplitude∝ exp−αr ,

Geometrical Spreading

For surface waves we get :

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

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Amplitude∝ r−1 .€

Amplitude∝ r−1/ 2 .

Finally, the effect of anelastic attenuation and geometrical spreading combined is:

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Amplitude∝ r−1 exp−αr .