The difference between the observed and predicted times is:
Loss functions:
where
We seek to find the
How do we define the "best" location?
The average least square residual:
is called the variance of the residuals, where
A common term is variance reduction (VR), which is defined as:
Based on least squares and L2 norm, we define:
where
The
The
The probability density function of the
The 90% confidence interval of the
Table for
ndf | |||
---|---|---|---|
5 | 0.412 | 4.35 | 11.1 |
10 | 3.94 | 9.34 | 18.3 |
20 | 10.9 | 19.3 | 31.4 |
50 | 34.8 | 49.3 | 71.4 |
100 | 77.9 | 99.3 | 129.6 |
Note that the
The estimated data uncertainty
where
Then we can use the estimated
Case: Earthquakes located along a fault will often be mislocated if the seismic velocity changes across the fault.
Case: Earthquake locations for events outside of a network are often not well constrained.
Mitigations:
In the common situation where the location error is dominated by the biasing effects of unmodeled 3-D velocity structure, the relative location among events within a localized region can be determined with much greater accuracy than the absolute location of any of the events.
where
Review: clusering

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- [Bayesian inference](https://en.wikipedia.org/wiki/Bayesian_inference) - [Monte Carlo simulation](https://en.wikipedia.org/wiki/Monte_Carlo_method)