Earthquakes and Earth Structure (EPSC-330)
Midterm Test (Take home)
Prof. Olivia Jensen Assigned: Feb. 19, 2016
Due: 16h30 Feb. 23, 2016
Instructions
Answer both questions. Each question is worth 15 marks towards your final grade. This midterm and other useful and necessary materials can be found here:
Take-home_Midterm
Question 1. Hooke’s law for isotropic elastic continua describes a linear homogeneous relationship between stress and strain. Gravitational and thermal forces within the Earth provide the stresses which cause tectonic deformations or strains and provide the energy or work source for earthquakes. While rock in the Earth’s crust is not homogeneous and isotropic, Hooke’s law does hold reasonably well for the relatively small stresses and strains in the crust.
We have only explored, in situ, to about 12.6 km of depth into the Earth’s crust. The Kola Superdeep Hole, drilled in northern Russia jammed the drilling bit and stem at 12.6 km.
Consider a unit cube of rock emplaced in the hole at that depth, with each face 1 x 1 dm. Under these lithostatic conditions, in situ, unloaded of the 12.6 km of overlying rock, it cannot deform laterally. That is, it is dilates such that the lateral strains exx = eyy = 0. The rock, though, is partially relieved of strain vertically by the removal of the overlying rock column and the infiltration of the hole with lower density ( = 1000 kg/m3) water. Recall that hydrostatic pressure at depth h in the water column is PH = gh.
Diagram:
pzz
pzz And recall = exx + eyy + ezz.
i. Assuming an average crustal density of = 2500 kg/m3, an average gravitational acceleration g = 9.8 m/s2 and at a depth h = 12.6 km, determine the change of stress condition on the now unloaded cube and calculate its vertical ezz strain as compared to its originally load-stressed state. Assume that our rock emplaced at depth is a granite of density = 2600 kg/m3, with a bulk modulus k = 88 x 109 Pa and a shear modulus of = 22 x 109 Pa. What is the difference in the lateral constraining stresses, exx = eyy , on the rock at depth as compared to the original lithostatic stresses which have been partially relieved by unloading?
ii. Determine the volumetric dilatation of this rock if removed to a stress-free condition as compared to its original lithostatically load-stressed state. What would its density difference be if it were now again constrained under a properly lithostatic load; i.e. with pxx = pyy = pzz = gh?
iii. Determine the seismic velocities for P- and S-waves in this rock and its value for Poisson’s ratio, .
iv. We have been lucky enough to withdraw a core of the rock which we can now submit to stresses with presses in our lab. We apply a uniaxial compressional stress to this rock but do not constrain it laterally; that is we apply a pzz stress but with pxx = pyy =0. The applied compressional stress, pzz = gh, is exactly equal to that in situ at 12.6 km of depth, as above. What strains, exx,eyy, and ezz are obtained under this compression.
v. Exhumation of rocks in the Canadian Shield may well be revealing rock masses that were emplaced at a depth even greater than that of the Kola hole. In view of the change in lateral stress condition resulting from unloading, what structural features might you expect to see in surface rocks of the Canadian Shield?
Note: Properly when drilling, we try to maintain hydrostatic pressure equilibrium in the hole with the lithostatic pressure at depth through use of dense drilling muds for otherwise, the hole would slowly deform to close itself.
Question 2: A little seismogram play
Mw 7.7 - Haida Gwai 2012-10-28 (52.742°N, 132.131°W)
I have abstracted the three-component, long-period seismograms obtained from the BDS Network’s CMB (Columbia College Seismic Pier: coordinates (38.0345°N 120.3865°W) station. Note that LHZ designates the vertical component (graphical up = positive/up motions); LHN designates the north-south oriented seismometer (up = north motions); LHE designates the east-west oriented seismometer (up = east motions). The necessary seismograms are available in the Take-home_Midterm directory (linked here) on the course website.
- Before reaching into part ii, following, estimate the distance from the CMB site to this event using the P-to-S interval time that you "pick" from the seismograms. A travel-time plot (ttgraph.pdf) useful for identifying seismic phasesis available in the directory noted above. As well the IASPEI standard terminology used in designatingseismic phases is available online. Show your first "picks" for the P- and S-wave arrival phases. Note that the if the P-wave (actually Pn by conventional designation at this close distance) first motion is "up", the P-wave radiated as a compression and towards the CMB seismograph. What is the first-motion displacement direction for the P-wave and S-wave arrivals? You might use a little vector diagram to show the azimuthal direction (relative to geographical coordinates) of this initial S-wave displacement. The S-wave path from the earthquake should be normal to this displacement.
- Indicate on the seismograms, as many seismic phases as you can find. The travel-time chart shows arrival times for P, S and Love phases. For this particular event, the earthquake to seismic station distance is so short that mantle-travelling body waves are not especially distinct. Still, you might be able to recognize core-bounced phases... PcP, PcS and ScP, ScS buried in the surface wave train. Qualitatively, describe the directions of motions of the first pulse of each phase you identify. You might do this on the seismogram records themselves.
- What is the direction of the vector displacement of the initial pulses of the surface-travelling LQ (Love) wave? The Love wave might be distinguished as the first large amplitude motions in the seconds following the S-wave (actually Sn) arrival. Remembering that the Love waves are really just assemblies of surface-guided S-wave phases with horizontal wave displacements normal to the direction of wave travel, is this initial displacement vector direction consistent with the azimuthal direction to this earthquake from CMB? Use great-circle distance/azimuth calculator available on the GPSVisualizer site to compare your expectation.
- Given the fault-plane solution "moment tensor beach ball", show that your first motions are consistent with the P- and S-wave radiation from the earthquake. You will need the great-circle azimuthal direction from the CMB site to the Haida Gwai event.
- Ideally, Love waves travel across a flat layered surface. Topography can much affect the particle surface motions of the wave through lateral scattering causing surface motions that are not explicitly horizontal. The extreme topography along the path from Haida Gwai to CMB as the wavefield crosses the Cascades is shown in the diagram (CMB-to-Haida_Gwai-surface-elevation.png) found in the Take-home_Midterm directory. Comments?