GG5210/6211
Take-Home Final Exam In
Tectonophysics and Elastic Waves
Autumn
Semester, 2006
Due Friday, December 15, 2006, 5 PM,
in RB Smith’s office.
All
work must be done by yourself, without any help or coaching from others. If you have aquestion ask the instructor or
the TA. The exam problems are intended
to help you understand how one integrates various types of strain, stress, wave
propagation, etc. data to solve a typical geophysical problem as well as examine
your overall understanding the methodology and theory. Make sure to include all
calculations, computer scripts, etc. and anything else in the kitchen sink.
1. 300 points, Strain and stress
determinations across an active normal fault: the Hebgen Lake fault, Montana.
The extensional tectonic regime of the
Basin and Range province is emphasized by the occurrence of the largest
earthquake in Rocky Mountain history, the Ms 7.5 Hebgen Lake, MT,
earthquake (Doser, 1985). This rupture accompanying this earthquake exhibited
more than 5.5 m of vertical normal-fault displacement over a 40 km fault
length. Because of its large size, this
earthquake has been taken as the quintessential normal-faulting earthquake,
i.e., the maximum credible earthquake for the Wasatch Front.
Because of its size and importance, many
studies of this earthquake have been made including earthquake source analyses
and geodetic measurements. Savage et al,
(1993) and Savage, Lisowski, and Prescott (1989) have reported measuring large
strain rates across the Hebgen Lake fault zone, many years after the earthquake
based upon re-observations of horizontal control points by trilateration
measurements using EDM (Electronic Distance Measurements). GPS measurements by the U of U confirm these
rates to as recently as 2001. The strain
rates for this fault are as large as some across parts of the San Andreas
fault.
The following papers are referenced for background material and you are encouraged to read them. You can get copies of the papers in the library.
Barrientos,
S. E., R. S. Stein and S. N. Ward, 1987, Comparison of the 1959 Hebgen Lake,
Montana, and the 1983 Borah Peak, Idaho, earthquakes from geodetic
observations, Bull. Seismol. Soc. Amer., 77, 784-808.
Doser,
D. I., 1985, Source parameters and faulting processes of the 1959 Hebgen Lake,
Montana, earthquake sequence, J. Geophys.
Res. 90, 4537-4555.
Savage,
J. C., M. Lisowski and W. H. Prescott, 1985, Strain accumulation in the Rocky
Mountain states, J. Geophys, Res.,
90, 10310-10309.
*Savage, J. C., M. Lisowski, W. H.
Prescott, W. H. and A.M. Pitt, 1993, Deformation from 1973 to 1987 in the
epicentral area of the 1959 Hebgen lake, Montana, earthquake (Ms = 7.5), J. Geophys. Res., v. 98, p. 2145-2154. (strain rate figure included in exam
material).
The attached map shows the locations of
the benchmarks (horizontal control stations) that have been observed by
EDM (electronic distance measuring
equipment) and GPS measurements (Global Positioning System) to measure lengths between
benchmarks (called baselines) across and near the Hebgen Lake fault. Also included is a list of stations and
coordinates, the observed length lines for several surveys, and a summary of
the changes in the line lengths from 1975 to 1985. Note that because we are using data that are
time dependent, the corresponding displacements, strains, stresses, etc. are
therefore given as rates, i.e. per year or second.
Background
Information: Seismic
studies of the Hebgen Lake area indicate that the average upper-crustal P
velocity of the rocks to 15 km depth is Vp = 5.9 km/sec and r
= 2.6 gm/cc. Assume Young's modulus is 8.5 x 1011 dynes/cm2. Assume that 
is in the e1 direction and 
is in the e2 direction.
STEPS:
The following are the suggested steps to
solve for the stress and recurrence rate on the Hebgen Lake fault. Note that the strain data must be in units of
strain/unit time so equivalently, stress rates are in stress/unit time. We
usually quote rates per year, i.e., stress/yr. Also note that a large earthquake
on the Hebgen Lake fault has not occurred since 1959, therefore time zero
starts at the time of the earthquake.
1)
To determine the long-term strain rates from the deformation rates for
1975-1980 and 1980-1985, plot by eyeball lines that fit the Dl
(L-L0) vs. time curves attached for each baseline and compare
with the USGS results. The original
station data are on the attached tables. The baseline lengths are on the plot
or from the table.
2) Next determine the principal strains by
the strain rate expression derived in class:
a) matching
the expression:
,
b) by a least-squares approach. Be sure to note which data points were used and which were thrown out from the calculations and why.
This calculation gives the principal
(2x2) strain-rate tensor, 
, but is of course only for the horizontal components. Be careful to watch for tensional and
compressional strain components.
3) Next rotate the these tensor components
and 
, into the
direction of north-south for
and east-west for 
using 
4)
From part 3 next calculate the shearing strain rates 
, 
, and the cubical dilatation rate
.
5) As you calculate the strains components, note
if they are logically consistent with the observed geology, i., e., with the
orientation of the Hebgen lake fault zone.
Note
that we assumed the strain field was homogeneous. If not, try to use only those EDM and GPS
stations that you think best obey the homogeneity assumption and go through
steps 2 to 4 again for subsets of the data that best fit the homogeneity assumption.
6)
Next convert the strain rates to stress rates assuming Hooke's law. Note that I have given you the P wave
velocity for the Hebgen Lake area. Now
using the plane-strain assumption for Hooke's law, calculate the stress rates,
and from:
7)
Next, using the velocities given in terms of 
and 
and the expressions
for Young’s modulus and the respective P- and S-wave velocities:
, 
, 
calculate
and the velocities.
These values are needed for later
8) Next calculate, 
and , where 
and 
is the time period of
the observations.
9) Then calculate the components of the lithostatic stress tensor that you recall is given by:.
10)
Calculate the full stress tensor; 
= deviatoric +
lithostatic stress tensors for the 3-D case, recognizing that the fault is
aligned almost EW axis.
11)
Calculate the traction, 
, on the fault for a southerly dip of 60o and at a
depth of 15 km where the normal to the fault surface is 
related by Cauchy’s
formula.
12) Calculate the shear stress on the Hebgen Lake fault at a depth of 15 km. This depth is where most large earthquakes nucleate and the shear stress at this depth is the critical value.. Assume a southward dip of the fault at 60°S that produced the measured strain. This will require an assumption on which components are the principal stresses and an extrapolation of the vertical component to the 15 km depth.
13) Calculate the stress rate on the fault and estimate when it will exceed the strength limit. This gives an estimate of the time of the next earthquake.
14) Calculate the average displacement rate in the direction of maximum horizontal strain. To do this, simply convert the strain to displacement using a simple one-dimensional model.
15)
The Hebgen fault last ruptured in 1959 accompanying the magnitude 7.5
earthquake. Assuming that a shear stress
drop of 150 bars is required to nucleate an equivalent magnitude earthquake,
when would we expect the next M 7.5 earthquake to occur? The
mayor of West Yellowstone is very interested in this answer.
2.
100 points, Seismic Measurements of Rock Properties
Earthquake studies in the areas of active faulting have enabled seismologists to pinpoint dilatant volumes associated with the impending fault rupture, i.e., zones of volumetric increase due to stress increase. In order to determine the rock properties and extent of these possible dilatant regions, drilling and seismic measurements can be done to determine the rock properties.
For this hypothetical problem consider the following seismic field acquisition program. A survey consisted of four seismic recording profiles, 20 km in length, deployed in the area of the dilatant zone as shown in the diagram.
The purpose of an accompanying exploratory drilling program was to obtain cores samples from the dilatant zone in order to determine in situ rock properties from laboratory measurements.
Seismic information obtained from the
seismic data:
Velocity of P-wave through undilated rocks = 6.00 km/sec
Velocity of S-wave through undilated rock = 3.46 km/sec
Travel-time of P-wave through undilated rock for the endpoint 1 and 4 = 3.33 sec
Travel-time of S-wave through undilated rock for the endpoint 1 and 4 = 5.78 sec
Travel-time of P-wave for the endpoint separation for lines 2 and 3 = 3.68 sec.
Core sample analysis information:
Poisson's
ratio, 
, for dilated rock = 0.35
Velocity of P-wave for dilated rock = 5.1 km/sec
From the available information, determine:
a) Poisson's ratio for undilated rock surrounding the dilatant zone which is within the box.
b) The shear wave velocity for the dilated rock.
c) The lateral extent of the dilatant zone.
d) What wavelengths for P-waves will best “detect’ the dilatant zone for the following frequencies (in Hz.) of the source: 0.1, 1, 5, 10, and 100? and explain why.