What is epsilon?
by Jack Baker, 2005
This page is somewhat obsolete and will be updated at some point. In the meantime, see this page for recent related publications.
This page is provided as a quick reference for researchers who are interested in the ground motion parameter epsilon. You are welcome to use any of the information and Matlab scripts provided here, as long as you acknowledge their use appropriately. The following paper explains much of the information presented here:
Introduction
Epsilon (ε) is a parameter associated with an earthquake ground motion that has recently been identified as having an ability to predict structural response. It is defined as the number of standard deviations by which an observed logarithmic spectral acceleration differs from the mean logarithmic spectral acceleration of a ground-motion prediction (attenuation) equation. The equation corresponding to this definition is:
where lnSa(T) is the natural logarithm of the spectral acceleration value of your record at a specified period T, and 
and 
are the estimated mean and standard deviation of logarithmic spectral acceleration, as predicted by an attenuation equation. (Attenuation equations sometimes provide the median Sa value, but taking the natural log of this value gives you the mean of the logarithm under the assumption that spectral acceleration is lognormally distributed. The standard deviation is typically provided directly in the form needed.)
The quickest explanation as to why epsilon is important is that it is an indicator of "peaks" and "valleys" in the response spectrum. Examine the picture below:
On the left, two response spectra are plotted, along with their associated attenuation predictions (mean and mean +/- one standard deviation). Epsilon is defined as the distance from the mean prediction to the actual spectrum, in terms of number of standard deviations. So the record on the top left is roughly a "2 epsilon" record at 0.8 seconds, while the record on the bottom left is roughly a "-1 epsilon" record. Now look at what happens when we scale these two records to the same spectral acceleration at 0.8 seconds, as seen on the right. The negative epsilon record has much larger spectral acceleration values than the positive epsilon record at most other periods. Thus, we would expect the negative epsilon record to produce larger responses in a structure (we expect it to cause larger higher mode responses, as well as larger nonlinear response once the structure "softens" and its effective period increases).
If you would like to learn more, I encourage you to read the documents listed in the Papers section of this web site.
Calculating Epsilon
Calculating the epsilon value of a ground motion you are using for analysis is quite simple. All that you need is the spectral acceleration value of the record at your period of interest, and the predictions of a ground motion attenuation equation at the same period. Then simply plug these values into the equation above to find epsilon.
The spectral acceleration value of the record can be easily calculated by measuring the peak displacement of a single-degree-of-freedom oscillator with the specified period, and multiplying by the square of the natural frequency (remember that while we often say "spectral acceleration" we really mean "pseudo-spectral acceleration"). These values are also provided in tables for all records in the PEER ground motion library (http://peer.berkeley.edu/ngat/).
The remaining required parameters are the mean and standard deviation of logarithmic spectral acceleration, as provided by an attenuation relationship. For your convenience, I have provided Matlab scripts for several popular attenuation laws below. To use them, simply provide the relevant parameters (period, magnitude, distance, soil type, etc.) for your ground motion to the script of your choice. Comments are provided at the beginning of the scripts to explain their use.
The following models for spectral acceleration (Sa) were published in Volume 68, Issue 1 of Seismological Research Letters, in 1997.
- Abrahamson and Silva, horizontal Sa
- Boore et al., horizontal Sa
- Campbell, horizontal Sa
- Sadigh et al, horizontal Sa
- Abrahamson and Silva, vertical Sa
- Campbell, vertical Sa
Example calculation
- A simple Matlab script is provided here to illustrate use of the attenuations for calculation of epsilon. By calling the attenuation script from inside a loop, it is possible to very quickly compute epsilon values for a suite of records and/or a range of periods.
Frequently asked questions
Does it matter which attenuation model I use?
For a given ground motion record, ε is a function of the ground motion attenuation model used, because the mean and standard deviation of lnSa(T) vary somewhat among models. My recent research indicates, however, that the choice of attenuation function will not affect results significantly. Although there are slight variations among attenuation predictions, for most records the predictions are very similar. There is essentially zero probability that one attenuation model will give you an epsilon value of -1, while another model will give you +1. A more likely situation would be for one model to give you an epsilon value of 1.0, while another gives you 1.1. This small difference is unlikely to affect any resulting analysis.
To be safe, if you would like to use ε in a vector intensity measure to compute a drift hazard (this paper), the attenuation model used to compute ε should be the same as the model used to perform the ground motion hazard assessment. However, the USGS averages several attenuation models in its ground motion hazard computations, so it may be more practical to just use the epsilon value from a single attenuation model, given that the results will not vary significantly.
Why do you include "arbitrary component" and "average component" standard deviations in your attenuation scripts?
Many ground motion prediction equations provide mean and standard deviation values for the average lnSa(T) of the two horizontal components of a ground motion. However, in many engineering analysis projects, we model a two-dimensional structure and thus use only one (arbitrarily chosen) component of a given ground motion. Thus, we use lnSa(T) of an arbitrary horizontal component of the ground motion as our intensity measure. To reflect this, the standard deviation from the attenuation model should be inflated relative to the published value, and this inflated number used for the epsilon computation. A more complete explanation of the issue is given in this paper. The Boore et al attenuation provides an inflated standard deviation, but for the other models I have developed my own inflation factor. The derivation of this inflation factor will be printed as part of a paper currently in preparation.
I calculated epsilon values for my records. What should I do with them?
If you have already analyzed your multiple-degree-of-freedom nonlinear structure using your records, then an interesting test is to see whether epsilon predicts structural response. This is most easily seen if you have scaled your records such that they all have the same lnSa(T) value. Plot your response parameter versus the epsilon value of each record (noting that epsilon is defined with respect to the lnSa(T) value of the unscaled record). You should see something like this:
where the Y axis can display any response parameter you are interested in, and the X axis displays the epsilon values of the records. Each point on this plot is associated with a nonlinear analysis for a single record. We see here that larger epsilon values tend to be associated with smaller levels of structural response.
The effect of epsilon can also be seen in unscaled records, but the analysis is more difficult because lnSa(T) will also vary for each record, and it is more complicated to separate the effects of lnSa(T) from the effects of epsilon.
If you are considering epsilon values before you have selected records for analysis, you might try to choose records with epsilon values that match the disaggregation from your hazard analysis, in the same way that today we select records to match the magnitudes and distances from disaggregation (see, e.g., this paper).