GLG490/598--Tectonic Geomorphology
Simple dislocation models for earthquake displacements and the earthquake cycle
In this set of exercises, you will apply some simple dislocation models to study the earthquake cycle as manifest in different tectonic environments. Turn in all materials as specified below by 5 pm Friday March 21.
You should recall the notes and lecture elements in these presentations from the last couple of weeks:
Introduction to the San Andreas Fault System and Simple model for earthquake cycle: elementary dislocation theory
Dip-slip dislocations--lots of U here!
Application to Wells Nevada earthquake
Loma Prieta addition
Little bit on subduction zones
In addition to anything specified below, provide short to the point, can-be-bulleted, answers to the questions, selected annotated (can be by hand) graphics, and the electronic version of your calculation tools (i.e., spreadsheet or matlab or C codes). Make sure they are well commented so I can figure out what is going on. Try to use variables in as many places as possible. Do not "hardwire" values if you can help it. Keep the tools flexible.
Part 1: Depth of faulting in and after the 1999 Izmit earthquake in Turkey
Images above from http://quake.wr.usgs.gov/research/geology/turkey/images.html
The above photos show damage and the setting of the 1999 event along the North Anatolian fault which accommodates right-lateral motion between Anatolia and the Black Sea region to the north.
Modified from Reilinger, et al, 2000
The above maps show the GPS-measured displacement vectors associated with the earthquake directly (coseismic, left) and for the 75 days after the earthquake (post seismic, right).
From the boxes shown in red, I have measured the distance normal to the fault and the displacement parallel to the fault. We can assume that here in the middle, a 2D approximation to the motion will be ok. Positive is north for location (in km) and to the east for displacements (in m):
Coseismic:
Dist. (km) Uparallel (m)
35.71 0.37
4.29 1.48
-2.86 -1.85
-5.71 -1.04
-28.57 -0.37
-31.43 -0.30
-34.29 -0.22
Post seismic:
Dist. (km) Uparallel (m) (values *0.1 on 3/6/08)
35.71 0.030
34.29 0.024
15.71 0.042
10.00 0.051
4.29 0.033
-4.29 -0.030
-7.14 -0.036
-31.43 -0.018
Tasks:
- Organize the equations. Rewrite the equation from Thatcher for displacement rate normal to a strike-slip fault (see slide 12 at this link if necessary). Rewrite it for just displacement, taking out the time. Note that D is measured from the Earth's surface down to the top of the fault rupture from where the rupture continues to minus infinity.
- Build the computational tools. Using your computational tool of choice (calculator, Excel, Matlab, C, etc.) to build a "calculator" to compute the displacements perpendicular to a strike slip fault at the surface for a given depth of rupture and displacement discontinuity (slip). You will need to include the second dislocation to simulate the effect of a displacement discontinuity along a discrete depth of fault (such as 0 to 10 km). Note that the upper one will have a shallower (or approximately 0) D and the sign of slip will be in the direction desired. The lower one will have a D that is at the bottom of the desired patch of slip with the same magnitude but opposite sign of slip as the upper one. Add the displacements together at each location perpendicular to the fault (each y in Thatcher's notation) to get the resulting displacement profile. Remember that the upper one will actually be at D = 0.001 or something small but not 0 (note where D is in the fractions).
- Coseismic displacements. Using the displacements above and the knowledge that a peak of 6 meters of slip was measured at the fault at the surface in the Izmit earthquake in the area from where I selected the displacements, build a model for the event and determine the best fitting depth of coseismic rupture. Compute the values at each of the measured distances of displacements, but also show the continuous curve out to plus/minus 50 km. Show the best fit and one each worse fitting shallower and deeper displacement curves. What does changing the depth do to the displacement distribution? Be sure to justify the best fit quantitatively (see RMS below). Make sure you use a two panel approach: one graph showing Ux versus fault normal distance y and the other showing the fault patch(es) versus depth with the same y range as the first plot. How does the depth you get compare with what you know from the San Andreas Fault, for example? Think of 1906 example and look through some of the chapters in U.S. Geological Survey, Professional Paper 1515 titled The San Andreas Fault System, California.
- Postseismic displacements. In a similar procedure, determine the depth of rupture and approximate slip in the postseismic period based on the vectors above. Compute the values at each of the measured distances of displacements, but also show the continuous curve out to plus/minus 50 km. Show the best fit and one each worse fitting shallower and deeper displacement curves. Be sure to justify the best fit quantitatively (see RMS below).
Hint: The postseismic one is obviously trickier because you really need to figure out three things: 1) the slip, 2) the depth to the top of the postseismically sliping patch, and 3) the depth to the bottom of the postseismically slipping patch. This is a tweak on our simple 2 stage earthquake cycle. Things are more complicated in real life. Have a look at U.S. Geological Survey, Professional Paper 1515 titled The San Andreas Fault System, California--the Thatcher chapter, and in particular ponder the last figure. My suggestion is to do this by trial and error to begin with and see what is the best set of combinations. You can do a fine job to present and discuss the tradeoffs with some simple one or two dimensional RMS versus parameter plots. If you really want to be fancy (for extra credit) compute the isosurface of 25 cm RMS for the 3 unknowns.
Small aside: Root Mean Square Error as a measure of goodness-of-fit for a model
This is a topic for another class, but a simple way to have a single number as a measure of how good your model fits the observations is to compute the Root Mean Square Error:
Where n is the number of observations and you compute the difference between an observation and a model calculation, square it, sum the squares, compute their mean, and then take the square of the mean.
Part 2: Strike-slip earthquake cycle "heuristics"
Generalize the above tools to show how the depth of the deformation source (basically D) for the coseismic, the interseismic, and the intermediate depth postseismic cases roughly (or exactly?) controls the spatial extent (width) and amplitude of the horizontal displacements. Is this easier to see with the horizontal strains? Write a one paragraph statement that you could post on your bulletin board to remind you of these rules of thumb and produce a couple of annotated plots (one for each case--should probably be the dual panel case).
Part 3: 1964 Great Alaska earthquake coseismic and interseismic displacements
The March 26, 1964 Great Alaska earthquake was one of the largest earthquakes ever recorded with Mw9.2 and resulted from slip along an 850 km long portion of the Alaskan-Aleutian subduction zone. It released stress associated with the accumulation of 6 cm per year of convergence between the Pacific and North American plates. In a famous effort, the vertical displacements from the earthquake were mapped by George Plafker of the USGS and colleagues by studying uplifted and drowned features and tidal organisms that were moved from their standard depth range with respect to tide. His map of displacements is shown in the above map overlain on the satellite imagery and bathymetry of the region on which you can clearly see the trench. Data were extracted from that map along the red line and are shown below (Distance is relative to arbitrary origin at the northwest end and uplift is positive):
Dist. (km) Uplift (m)
88.46 0.00
169.23 -0.61
203.85 -1.22
238.46 -1.83
261.54 -1.83
273.08 -1.22
280.77 -0.61
284.62 0.00
292.31 0.61
300.00 0.61
307.69 1.83
315.38 2.44
334.62 4.57
353.85 9.14
365.38 4.57
423.08 3.05
442.31 2.44
461.54 1.83
500.00 0.00
The above sketch shows a very simple model for the subduction zone faults approximately along the red line on the map above.
Tasks:
- Organize the equations. Write out the equations for the surface displacements due to a single edge dislocation. Write the equations out for the two edge dislocations with opposite slip required for dip slip along a patch of finite length (e.g,. from Du, et al., 1994 and Arrowsmith, 1995). Include a diagram. See slide 13 at this link for a reminder.
- Build the computational tools. Write a computational toolkit that computes the surface displacements (vertical and horizontal) for the pair of edge dislocations given a slip, dip, position of the top of the fault, faulted length, and position along the surface profile. (Hint: use trial and error to figure out the sign conventions and remember that all of the length units will have the same scale; also test it on the examples from this and related lectures).
- Coseismic displacements--downdip length and slip unknown. Using the geometric model defined above in the sketch (which has the key locations in the same coordinates as the displacement data given above), use those functions to determine the best fitting combination of rupture downdip length and slip in the 1964 earthquake by comparing with the displacement data given. Assume that the upper portion of the fault has a constant dip of 15 degrees, that the trench is at 500 km distance and a depth of 5 km, and that the rupture tip is downdip at a position of 354 km (below the peak vertical displacement). How long is the red fault and what was the magnitude of slip that best fits the Plafker data? Make a plot of the best fitting model and the Plafker observations. Include a couple of other model combinations that don't fit as well. What part of the displacement field is sensitive to the slip? How about the length of rupture? With respect to the horizontal displacements, which part of the profile shows the highest amount of shortening? For extra credit compute the 2D error surface around the best fitting slip and downdip length (but only after trial and error so you know roughly what the answer is).
- Interseismic displacements Now for the interseismic vertical displacements. Knowing what you know about the coseismic fault surface upper and lower ends, use your dislocation models to compute the steady vertical displacements due to steady 6 cm/yr interseismic motion along the dashed lines: the updip portion at 15 degree dip to the trench, and the downdip continuation at 45 degrees to a depth of 100 km. Given that combination of steady motion, what is the rate of subsidence or uplift at Anchorage during the interseismic time (that is since 1964 at a distance 173 km in our model)? Make a plot of the interseismic displacement profile and locate Anchorage. Note that this will take two patches or 4 dislocations.
Part 4: Dip-slip earthquake cycle "heuristics"
As you did in part 2, generalize the above tools to show how the depth, slip, and dip of the deformation source for the coseismic and the interseismic (keep it simple, only include the downdip interseismic slip, not shallow creep as we modeled for the subduction example above) cases roughly (or exactly?) controls the spatial extent, position of maxima and minima, and amplitude of the displacements. Try to figure out how to plot the vectors which require you to put the vertical and horizontal components together. Write a one paragraph statement that you could post on your bulletin board to remind you of these rules of thumb and produce a couple of annotated plots (one for each case--should probably be the dual panel case).
Last modified: March 6, 2008