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Error functions are used to measure the difference between a ground motion record (or suite) and the target design response spectrum. Typically, an error "Erri" is computed at each of several selected sampling periods (Ti), discretized over a pre-defined range [Tmin,Tmax]. The individual errors at the different periods are then combined using one of several error functions. QuakeManager provides four different options for the error function:
1.Sum of the Squares = ∑(Erri2)
2.Square Integral = ∫ ∑(Erri2) dT = ∑(Erri2.ΔTi)
3.Sum of Absolute Error = ∑(|Erri|)
4.Abs Error Integral = ∫ ∑(|Erri|) dT = ∑(|Erri|.ΔTi)
In the above, ΔTi = Ti+1 - Ti, is the difference between consecutive sampling periods.
Option 1 above (Sum of the Squares) is most commonly used. It tends to weigh the error measure more towards the short period range when the sampling periods are log-spaced, and more towards the longer periods when linear-spacing is used. Option 2 provides a compromise between those two extremes, and results in an error measure that is independent of the sampling period spacing method (i.e. log or linear).
Options 3 and 4 are similar to Options 1 and 2, but use the absolute value instead of the square of the error. Options 1 and 2 place a rapidly increasing (power of 2) penalty on large individual errors, while with Options 3 and 4, the overall error function increases linearly with individual errors.
The Error function is typically selected in the "GM Fitting Criteria Dialog".
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