Earthquake Location and Relocation

Notebooks: codes/earthquake_location.ipynb

How to locate an earthquake?

How to locate an earthquake?

Optimization (Inverse) problem

• Minimize the difference between observed and predicted values

How to solve an optimization/inversion problem?

• Forward function
• Objective/Loss function
• Gradient
• Optimizer

Locating earthquake using absolute arrival times

notebook

Earthquake location problem:

• Given:
  • Observed arrival times at multiple stations
  • Velocity model
• Goal:
  • Locate the hypocenter and origin time of the earthquake

Forward function:

where is the predicted arrival time at station , is the forward non-linear function (e.g., ray tracing or eikonal equation), and is the model parameter (e.g., source location, origin time, and velocity).

For a uniform velocity:

where is the location of the -th station, is the location of the source, is the origin time, and is the uniform velocity.

Objective/Loss function:

The difference between the observed and predicted times is:

Loss functions:

• Mean squared error (MSE):
• Absolute error:
• Huber loss:

Iterative location methods

where is the initial model, is the perturbation of .

Iterative location methods

We seek to find the that

can be obtained using standard least squares. Next, we set to and repeat the process until the locatin converges.

How to evaluate the results of earthquake location?

How do we define the "best" location?

The average least square residual:

is called the variance of the residuals, where is the number of degrees of freedom.

A common term is variance reduction (VR), which is defined as:

How to define the uncertainty in the location?

Based on least squares and L2 norm, we define:

where is the uncertainty of the -th residual.

The distribution approximate the degree of freedom of the residuals .

The distribution

The distribution is a probability distribution that describes the sum of the squares of independent standard normal random variables.

The probability density function of the distribution is:

90% confidence interval of

The 90% confidence interval of the distribution is bounded by:

Table for :

How to apply to real data?

Note that the are critical in the analysis, which is based on the assumption that the data misfit are random, uncorrelated, and have a Gaussian distribution.

The estimated data uncertainty is often estimated from the residual of the best location:

where is the best-fitting location.

Then we can use the estimated to calculate the value; then obtain an estimate of the 95% confidence ellipse for the solution.

Challenges: unmodeled velocity heterogeneity

Case: Earthquakes located along a fault will often be mislocated if the seismic velocity changes across the fault.

Challenges: trade-off between event dpeth and origin time

Case: Earthquake locations for events outside of a network are often not well constrained.

Mitigations:

• time can be used to estimate the source-receiver range at each station
• Adding depth phase (using the differential time ) can help constrain the depth

Locating earthquake using relative arrival times

notebook

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.

HypoDD: Double-difference earthquake location

where and are the observed and predicted arrival times at the -th station for the -th earthquake, respectively.

HypoDD: Double-difference earthquake location

GrowClust: A Hierarchical Clustering Algorithm for Relative Earthquake Relocation

Review: clusering

More on: Uncertainty

• Aleatoric uncertainty
  • The irreducible part of the uncertainty
  • Uncertainty due to inherent randomness, e.g., the outcome of flipping a coin
• Epistemic uncertainty
  • The reducible part of the uncertainty
  • Uncertainty due to lack of knowledge, e.g., lack of data

Uncertainty Quantification

• Standard deviation of slope and intercept of linear regression
• Bootstrapping
• Markov Chain Monte Carlo (MCMC)
• Hamiltonian Monte Carlo (HMC)
• Stein Variational Gradient Descent (SVGD)
• Dropout as a Bayesian Approximation

HypoSVI: Hypocentre inversion with Stein variational inference