Constraining the Coefficient of Friction (above)
The two most important factors that contribute to uncertainty in our estimate of the coefficient of friction are the paleo-depth of the sample locality while the fault was active and the size of the single largest earthquake slip event on the San Gabriel fault. The plot above shows how differing assumptions about these values change our estimate of the coefficient of friction. Because we do not observe a frictional heat anomaly, we can only say that the amount of frictional heat generated never exceeded a certain minimum value (otherwise we would record a signal!). The curves show the upper bound on the coefficient of friction that is consistent with the lack of a frictional heat anomaly in fission track ages, for three different assumed depths (see below). The grey area highlights the fact that the coefficient of friction could be anywhere in the region below the appropriate curve.
More Detail
The apparent coefficient of friction is essentially the ratio between shear stress and normal stress during sliding. To constrain the coefficient of friction on the fault, we must therefore estimate both shear and normal stresses at our sample locality. What follows below is a detailed accounting of error sources in our constraints. Some of this information is not included in the published Geology paper about this project because of space limitations.
Shear Stress
The constraint on shear stress comes from the fact that we do not observe any measurable change in fission track ages or track lengths adjacent to the fault. By using forward modeling, we compute the maximum temperature increase that would remain undetected by fission tracks. We feel this estimate is quite robust because the time scale of frictional heat pulses (minutes to hours) is the same time scale used in the laboratory to compute fission track annealing algorithms -- no extrapolation is required.
We use a forward model of heat generation to compute the shear stress that would generate temperature increases below that threshold, for a given magnitude earthquake ("slip event"). Unfortunately, we do not know how many slip events occurred or their magnitude. Due to the nature of fission track annealing, we are most sensitive to the single largest slip event (though thousands of slightly smaller events will make a small contribution). We must assume a reasonable value for the largest magnitude slip event that ever occurred on the San Gabriel fault. As a former plate boundary fault, we feel that 4 m of slip is probably a good estimate, but the graph above shows how different assumptions about the size of the largest event will affect our estimate of the coefficient of friction. The shear stress we compute is therefore the maximum shear stress that would remain undetected by fission tracks for an assumed value of the largest earthquake ever on the San Gabriel fault. If there was never even a single earthquake with more than about 2 m of slip, we cannot provide any reasonable constraints on the coefficient of friction for the fault.
Normal Stress
We assume that the normal stress is equal to the weight of the overlying rocks. By using information about the timing of faulting and the exhumation history of the samples, we can determine the depth that the samples were at while the fault was slipping. Previous geologic studies summarized in great detail by Powell (1993) show that the San Gabriel fault was active from about 13 Ma to 4 Ma. Fission track thermochronology is traditionally used to compute the exhumation history of samples by modeling the distribution of fission track lengths. The result of these models is a complete estimate of the ambient temperature of the samples over time. We find that the best fitting ambient temperature of our samples was between 70-80 degrees C while the fault was active. Different published annealing relationships yield results that differ by less than 10 degrees for the best fitting case. However, acceptable fits to the fission track length distribution for ambient temperatures as much as 20 degrees warmer or cooler. However, significant deviations from the 70-80 degree C best fit are not entirely consistent with the thermal histories determined for nearby samples by Blythe et al. (2000). The precise timing of fault activity is not a large source of uncertainty because the thermal history of the sample is relatively constant from about 20 Ma to about 4 Ma, a period that encompasses all the various estimates of the timing of the San Gabriel fault.
The biggest source of uncertainty in estimating the normal stress probably comes when converting the ambient temperature to a depth. This requires knowing the geothermal gradient at the time the fault was slipping. In the figure above, we plot estimates of the depth given temperatures between 70-80 degrees C at geothermal gradients between 25 - 35 degrees C per kilometer. We feel that this range accurately encompasses the reasonable uncertainty in the paleo-depth of the samples, and therefore of the normal stress.
More About Fission Track Annealing
More About the San Gabriel fault