GG 5210/6211

Problem Set #10

Due December 6, 2006

 

Seismic Wave Propagation In A Layered Media

 

 1. Fermat’s Principle:  In the figure below, point A is 1,732 ft. offshore from a straight coastline, LL', and the point B is 2,000 ft. inland from the coast line.  The point P on LL' nearest A is 4,464 ft. from the point Q on LL' nearest to B.  If one must row at 10,000 ft/hour from A to some point R on the coast line and then walk from that point to B at 20,000 ft/hour, at what point R should one aim (in still water) to get from A to B in the least possible time?  How much time is actually involved in this least time?  Is the answer the same if the direction of travel is from B to A?

 

              

 

2.  Seismic Wave Attenuation--This problem illustrates the attenuation of seismic P wave propagation in the Pierre shale due to intrinsic attenuation (Q).  Compute and graph the losses, in decibels (db), for the effects of; a) absorption including the effects of Q, and b) spherical spreading for 1, 3, 10, 30, and 100 Hz waves at distances of 200 m, 1200 m, 2200 m and 8200 m from the source.  You must do the calculations at 200 m to avoid a singularity at zero distance. I suggest that you use MatLab scripts for calculating and plotting the data. 

 

Recall that I = wave intensity. A = displacement amplitude, A0 is the amplitude at the zero or calibration distance, and I = A2.  The empirical expression for the amplitude variation with distance is:

 

 

Where the phase velocity (c) here is a P-wave velocity of Pierre shale of 4 km/s and the attenuation coefficient, a, is 0.15 db/wavelength. f = frequency and r = distance. 

 

i.       For the problem, compute the amplitude losses for the effects alone, in decibels (db), for:

a) attenuation (or absorption) at 1, 3, 10, 30, and 100 Hz,

b) spherical spreading,

c)  the combined effect of both attenuation and spreading.

 

I suggest making a table of absorption loss (in dB) at the various frequencies given above vs. distance from the shot point for the absorption loss alone and the spreading losses. Then make a single plot of these data with distance on the ordinate axis vs. attenuation (dB) on the abscissa.

 

Start the calculations from a point 200 m from source so that it does not increase to infinity at zero.
xs = the distance to the shotpoint, λ = wavelength, and f is the frequency of the wave in Hz (cycles per sec).  Use the expressions for:


Absorption: Intensity loss in db is = 10 log10(I0/I) = 20 log10(A0/A)

                                                       = 0.3 (x / l) = 0.3 (xs - 200) / l = 0.3 f (xs - 200)/2000

Geometric Spreading: The intensity loss in db is = 10 log10(I0/I) = 20 log10(xs / 200)

ii.  Explain the significance of your findings in guiding seismic investigations.

 

3.  Zoeppritz Equations:  To understand the wave interaction at an interface, Zoeppritz equations are employed to calculate the relative amplitudes of wave potentials, energies, displacements, etc. and interface properties.

 

Use the online Java script for CREWS (University of Calgary, Consortium for Research in Elastic Wave Exploration Seismology) starting at:   http://www.crewes.org/

 

With programs at:  http://www.crewes.ucalgary.ca/Samples/

 

Use the CREWES Explorer and Energy calculators, their plotters, and the Interface calculators, for the versions post 2002.  Produce plots for handing in.

 

Typical interfaces

 

1) a rock-air interface, with the wave incident from the rock.

2) a sea bed overlain by a water later, with the wave incident from the water layer.

3) a shale overlying a coal bed;

4) gas-filled sand over water-filled sand;

5) crust over mantle; and

6) solid rock over magma. 

 

I.  Make up a table of interface properties for the rocks in the table below, such as specific values and ratios of Young’s modulus, bulk modulus, shear modulus, impedance

 

II.  To understand their fundamental employment, use the following values for velocity and density of rock types to examine: the relative amplitudes and energy vs. incident angle and phase.

 

Determine the problem for both P- and Sv-waves in each case using geologically appropriate thicknesses for the top layer

 

Table of Rock Types

 

Vp (m/s)

Vs (m/s)

r gm/cc)

shale

2200

1300

2.6

coal

3000

1800

1.5

gas-filled sand

1500

1000

1.5

water-filled sand

1800

1200

1.9

lower crust

7400

3800

2.8

upper mantle

8200

4500

3.3

solid basalt

5000

2700

2.9

basaltic magma

2600

500

2.7

 

III. Discuss the similarities and differences between these geologically different examples, including critical angles at which the relative amplitudes are maximum, offsets at which refractions will be the first arrival and so on.  How might you use the information gained from this exercise to design a seismic survey in which these interfaces were the targets?