The University of California, San Diego, operates the Pinon Flat Observatory ˜
(PFO)in the mountains northeast of San Diego (near Anza). Instruments include
high-quality strain meters for measuring crustal deformation.
(a) Assume, at 5 km depth beneath PFO, the seismic velocities are α = 6 km/s
and β = 3.5 km/s and the density is ρ = 2.7 Mg/m3. Compute values for
the Lame parameters, ´ λ and µ, from these numbers. Express your answer in
units of pascals.
(b) Following the 1992 Landers earthquake (MS = 7.3), located in southern
California 80 km north of PFO (Fig. 2.7), the PFO strain meters measured a
large static change in strain compared to values before the event. Horizontal
components of the strain tensor changed by the following amounts: e11 =
−0.26 × 10−6, e22 = 0.92 × 10−6, e12 = −0.69 × 10−6. In this notation 1
is east, 2 is north, and extension is positive. You may assume that this strain
change occurred instantaneously at the time of the event. Assuming these
strain values are also accurate at depth, use the result you obtained in part (a)
to determine the change in stress due to the Landers earthquake at 5 km, that
is, compute the change in τ11, τ22, and τ12. Treat this as a two-dimensional
problem by assuming there is no strain in the vertical direction and no depth
dependence of the strain.
(c) Compute the orientations of the principal strain axes (horizontal) for the
response at PFO to the Landers event. Express your answers as azimuths
(degrees east of north).
(d) A steady long-term change in strain at PFO has been observed to occur in
which the changes in one year are: e11 = 0.101×10−6, e22 = −0.02×10−6,
e12 = 0.005 × 10−6. Note that the long-term strain change is close to simple
E–W extension. Assuming that this strain rate has occurred steadily for the
last 1000 years, from an initial state of zero stress, compute the components
of the stress tensor at 5 km depth. (Note: This is probably not a very realistic
assumption!) Don’t include the large hydrostatic component of stress at 5 km
depth.
(e) Farmer Bob owns a 1 km2 plot of land near PFO that he has fenced and
surveyed with great precision. How much land does Farmer Bob gain or
lose each year? How much did he gain or lose as a result of the Landers
earthquake? Express your answers in m2.
(f) (COMPUTER) Write a computer program that computes the stress across
vertical faults at azimuths between 0 and 170 degrees (east from north, at 10
degree increments). For the stress tensors that you calculated in (b) and (d),
make a table that lists the fault azimuth and the corresponding shear stress
and normal stress across the fault (for Landers these are the stress changes,
not absolute stresses). At what azimuths are the maximum shear stresses for
each case?
(g) (COMPUTER) Several studies (e.g., Stein et al., 1992, 1994; Harris and
Simpson, 1992; Harris et al., 1995; Stein, 1999; Harris, 2002) have modeled
the spatial distribution of events following large earthquakes by assuming that
the likelihood of earthquake rupture along a fault is related to the Coulomb
failure function (CFF). Ignoring the effect of pore fluid pressure, the change
in CFF may be expressed as:
where τs is the shear stress (traction), τn is the normal stress, and µs is the
coefficient of static friction (don’t confuse this with the shear modulus!). Note
that CFF increases as the shear stress increases, and as the compressional
stress on the fault is reduced (recall in our sign convention that extensional
stresses are positive and compressional stresses are negative). Assume that
µs = 0.2 and modify your computer program to compute CFF for each fault
orientation. Make a table of the yearly change in CFF due to the long-term
strain change at each fault azimuth.
(h) (COMPUTER) Now assume that the faults will fail when their long-term
CFF reaches some critical threshold value. The change in time to the next
earthquake may be expressed as
where CFFa is the annual change in CFF, CFF1000 is the thousand year change
in CFF, and CFF1000+L is the thousand year + Landers change in CFF (note
that CFF1000+L # CFF1000 + CFFL). Compute the effect of the Landers
earthquake in terms of advancing or retarding the time until the next earth-
quake for each fault orientation. Express your answer in years, using the
sign convention of positive time for advancement of the next earthquake and
negative time for retardation. (Warning: This is tricky.) Check your answer
against the values of shear stress on the fault. Generally (but not always)
the earthquake time should advance when the long-term and Landers shear
changes agree in sign (either both positive or both negative), and the time
should be delayed when the shear stress changes disagree in sign.
(i) No increase in seismicity (small earthquake activity) has been observed near
PFO following the Landers event. Does this say anything about the validity
of the threshold CFF model?
Hint: Getting the signs correct in parts (f)–(h) can be complicated, particularly
for part (h). Stresses can be either positive or negative. To help get it right, define
two unit vectors for each fault azimuth, one parallel to the fault (ˆf) and one perpendicular to the fault (pˆ). Compute the traction vector by multiplying the stress
tensor by pˆ. Then resolve the traction vector into shear stress and normal stress by
computing the dot product with ˆf and pˆ, respectively. Naturally, ˆf and pˆ must be
of unit length for this to work.