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What is Epsilon?

What are epsilon (epsilon) and epsilon0 (epsilon)?

The ground motionY that might be recorded at a specific site from an earthquake with a given magnitude and distance (the "source") is typically modeled as a lognormal variate. That is, the logarithm of Y, which we denote y has a normal or Gaussian distribution, with mean (mu) and standard deviation (sigma). The Greek letter epsilon has been chosen to represent the standardized y at the site from a specific source, epsilon equals y minus mu divided by sigma

In PSHA, we compute the ground motion at the site having a fixed probability of exceedance, p0. We can label this ground motion y0 and the epsilon from a specific source and this specific ground motion epsilon0. Thus, epsilon sub zero equals y sub zero minus mu divided by sigma

To compute the probability that the ground motion from that source occurs in the interval [y0, y1], you would compute the area under the standard normal density curve with integration limits epsilon0 and epsilon1. This computation is performed in PSHA deaggregation, because we bin the contributions for exceedances of y0. The interactive web-site deaggregations use epsilon-bin boundaries at k sigma, k = 0, ±1, ±2

What epsilons are reported in the deaggregations?

Two mean epsilons are reported. The first of these is the mean value of epsilon0 for the sources contributing to a specified p0 at the site. The second is the conditional mean value of epsilon, conditional on yy0. We denote these as "Mean epsilon0" and "Mean epsilon" respectively. A more precise definition of these epsilons is given in a manuscript submitted to BSSA (Harmsen, 2001).

Two modal epsilons are reported. The first of these is the mean value of epsilon0 from the sources in the most likely distance, magnitude bin, denoted by "Modal epsilon0." The second of these is the interval of epsilons corresponding to the most probable distance, magnitude, and epsilon triple in the deaggregation, denoted "Modal epsilon*." Note that the first of these modal epsilons corresponds to a two-dimensional conditional mode, while the second corresponds to a three-dimensional conditional mode. For most sites and most return times, the distance and magnitude in Modal (R*,M*,epsilon*) are almost the same as the distance and magnitude in Modal (R,M, epsilon0), but these (R,M) pairs are sometimes quite distinct.

These means and modes are conditional on yy0. Mean and modal event triples vary with spectral period but there is high correlation among them for a given PE.

Magnitude and distance have practical or physical meaning. Does epsilon?

Yes. Although apparently more abstract than magnitude and distance, epsilon is as fundamental as magnitude and distance for understanding ground motion exceedances.

Suppose an earthquake with magnitude M occurs at distance R from your site. Suppose that your structure was designed to resist horizontal ground acceleration Y0 with logarithm y0. If epsilon0 is zero, then your structure was designed to resist median motion from this source. If epsilon0 is 1, your structure was designed to resist motion 1 sigma greater than the median motion. (1sigma corresponds to a factor of approximately 2)

PSHA considers ground motion from many sources, not one. Therefore, the mean and modal epsilon0 inform you about how your structure's design motion Y0 compares to median motion, in an average sense, from all sources (mean epsilon0) or just the modal sources (modal epsilon0; modal here means sources in the most likely magnitude, distance bin).

Mean epsilon or mean epsilon, has a slightly different meaning. Given that yy0, we may want to know, by how much does y exceed y0 for the PSHA sources? This can only be answered in a probabilistic way (we can't predict the future). For a given source, if epsilon0 is zero then mean epsilon is 0.798. That is, conditional on ground motion having exceeded the median, the quantile of exceedance is 0.798, so that on average, and in round numbers, ground motion would be about median times 20.798, or median times 1.7. For many sources, if mean epsilon is 0.798, ground motions would be distributed with a mean that is probably near 1.7 times the median of a dominating source. Because multimodal distributions are frequently encountered in PSHA deaggregations, you should not attempt to read too much into the relationship between mean epsilon for the set of sources and epsilon0 for any specific source.

Modal epsilon0 from the triple (R,M, epsilon0) also indicates by how much y exceeds y0. For a single source, if epsilon0= -0.5, then the most probable 1 sigma-wide bin is the 0-to-1 sigma bin, given that epsilon bin boundaries occur at k sigma, k=0,±1,±2 What if epsilon0= 0.5? Here, we would be well-advised to consult a table or Web-site of normal probabilities. A useful web-site for doing these sorts of computations is http://www-stat.stanford.edu/~naras/jsm/FindProbability.html .

You will find the interval probability from 0.5 to 1.0 is 0.1499 and the interval probability from 1.0 to 2.0 is 0.1359, so the 0-to-1sigma bin is the most likely for this example as well as for the previous example. In PSHA deaggregations, we accumulate all of these interval probabilities for each source. The graphical and ascii report at our interactive deaggregation web site is the binned (R,M, epsilon) for all sources considered.

Why are there two epsilons: epsilon and epsilon0, not one?

In an important PSHA article, McGuire (1995) suggests that structural engineers want to consider for earthquake-resistant design probabilistic ground motions that equal but do not exceed y0. This led McGuire to the decision to concentrate all of the exceedance probability at the point y = y0 in his deaggregations of magnitude, distance and epsilon. In the USGS deaggregations, we do not concentrate the exceedance probability at y0 but we do compute the statistics, mean and modal epsilon0, as if the probability had been concentrated at y = y0. This implies that modal epsilon0 comes from the two-dimensional (M,R) modal bin.

In a more recent important PSHA article, Bazzurro and Cornell (1999) emphasize that the modal bin of the joint conditional binned distribution of magnitude distance and epsilon, where epsilon has a normal distribution, is the event that most likely exceeds the target ground motion. They strongly recommend using the magnitude and distance from this triplet as the controlling event in various seismic resistant design applications. To solve the problem that (M*,R*, epsilon*) may exceed rather than equal the target motion, they recommend scaling the controlling event record to match the target ground motion. Scaling here means multiplying by the appropriate scalar.

The relative merits of the two approaches to determining the controlling event, one which does restrict e to a limited range (the bin width) (Bazzurro and Cornell, 1999), and the other which does not restrict e, but determines the controlling event from the most likely magnitude, distance pair(McGuire, 1995; Chapman, 1995) may be debated for some time into the future, but in the interest of "robust" strategies for seismic resistant design, the decision to design to resist both of these M,R combinations if they happen to differ seems to have some merit. Other (M,R) combinations may also be worthy of consideration. For example, where knowledge of active Quaternary faults is believed to be satisfactory, the average (M,R) for sources on the fault that contributes more than any other fault to ground motion exceedances may be a good choice for a scenario earthquake. As of 2001, the ASCII output at the interactive deaggregation web site includes hazard deaggregated by fault as well as by M and R.

These examples indicate that where variability and uncertainty in M and R for future events is large, many plausible strategies for defining a scenario event should be considered. The decision on what (M,R) pairs to resist should be arrived at by consultation among the responsible structural engineers, code agencies such as local governments, and informed earth scientists.

References Cited

  • Bazzurro and Cornell, 1999. Disaggregation of Seismic Hazard, Bull. Seism. Soc. Am., 89, pp 501-520. This article's citations include the 1995 references mentioned above.
  • Harmsen, S., 2001. Epsilon in Probabilistic Seismic Hazard Analysis: the Deaggregation of Ground-Motion Uncertainty, submitted to Bull. Seism. Soc. Am.