Estimating Seismic Moment Tensors based on Bayesian Machine Learning, Steinberg et al., @EGU2021

Estimating Seismic Moment Tensors based on Bayesian Machine Learning

Andreas Steinberg¹*, Hannes Vasyura-Bathke²*, Peter Gaebler¹, Matthias Ohrnberger² and Lars Ceranna¹

1 Federal Institute for Geosciences and Natural Resources (BGR), B4.3 Federal Seismological Survey, Nuclear-Test Ban

2 University of Potsdam, Institute for Earth and Environmental Sciences

* equal contribution

Fast estimation of seismic moment tensors necessary for:

Earthquake early warning systems, e.g. mechanisms based hakemaps
Large catalogs
Monitoring of geothermal stimulations (SEIGER project)

Machine Learning Terminology

Training input: images with data to be learned
Labels : associated (source) parameters
Network: layered architecture of neurons, which can activate connected neurons
Output: model able to predict labels given unseen data

Machine Learning - what has been done

detection and location: e.g. Kriegerowski et al., 2019
first motion polarity: e.g. Ross et al. 2019
full waveform DC mechanism determination: e.g. Kuang et al. 2021

Machine Learning - what has not been done

full MT
data driven consideration of uncertainties
recent advances allow to learn distributions of weights (Bayesian Neuronal Networks, BNN)

Training Input

training on synthetic waveforms from a volume of potential source locations
advantage: control over source and targets
using pre-calculated Green's function stores (Pyrocko and QSEIS)

most of ML optimised for images
Amplitudes vary for different sources-distances

Pre-processing of waveforms

filter waveforms (bandpass 1-5.4 Hz)
cut 1s before first phase onset
cut 3.5s after first phase onset
sort waveforms by station and channel name

Normalized input image for a single earthquake source

Labels

Using the Lune paramterziation of the MT

only 5 unique paramters (κ, σ, h, w, v)
cartesian coordinate system
v=0 and w=0 -> isotropic source
no scaling with ρ because of normalized input

Network architecture and variational inference

$$ \text{Approximate Kullback-Leibler divergence } \text{by drawing Monte Carlo samples } $$

$$ \mathbf{w}^{(i)} \text{ from variational distribution } q(\mathbf{w} \lvert \boldsymbol{\theta}) $$

$$ \text{with prior distribution } p(\boldsymbol{w}) \text{ and data likeliehood } p(\mathcal{D}|\boldsymbol{w}) $$

$$ \mathcal{F}(\mathcal{D},\boldsymbol{\theta}) \approx {1 \over N} \sum_{i=1}^N \left[ \log q(\mathbf{w}^{(i)} \lvert \boldsymbol{\theta}) - \log p(\mathbf{w}^{(i)}) - \log p(\mathcal{D} \lvert \mathbf{w}^{(i)})\right] $$

$$ \boldsymbol{\theta} = (\boldsymbol{\mu}, \boldsymbol{\sigma}) $$

Bayes by Backprop

Disclaimer: this is a sketch!

Bayes by Backprop

Disclaimer: this is a sketch!

Single Bayesian Neuronal Network design

First two layers 1xN kernel over time
Third layer 3x1 kernel over station components
Pooling layer resampling 3x4 (stations and time)
fully connected layer
non-activated output layer

Design inspired by Kriegerowski et al., 2019

Single Bayesian Neuronal Network design

loss function: neg. log-likeliehood
optimizer: Adam, RELu activations
Flipout layers (Wen et al., 2018), Gaussian priors from backpropgation
0.2 dropout to simulate missing data
Learning rate and batch size important

Variational inference from multiple BNN's

learn individual BNN's for a volume of possible source locations (grid)
learn individual BNN's for different Earth structure models
cut out input waveforms with respect to timing and location uncertainties

➜ errors in the Earth structure, timing and location can to be taken into account

Test and validation with unseen data of the 2019 Ridgecrest sequence

Variational inference (Bayesian) Machine Learning approach

Training one BNN for each grid point

Stations Grid points

Some statistics

41 stations up to 150km away
grid horizontally 10.5 by 10.5 km; step size: 1.5 km
grid vertical from 2 to 10 km; step size: 2 km
discretization of 0.1 π κ ; 0.2 for σ, h and w; 0.02 for v
171.600 waveforms per source location or individual BNN
588 BNNs (196 grid points times three Earth structures) are trained
including "Mojave" Earth structures from SCEDC

Comparison of predictions with SCEDC catalog moment tensors

8 reference mechanisms available (red)

Grid points

Omega angle measure of MT similarity

\[\begin{aligned} d = \frac{1}{2}\Bigg[1-\frac{U_{1}\cdot U_{2}}{||U_{1}||||U_{2}||}\Bigg] = \frac{1}{2}\Bigg[1-\frac{\sum U_{1_{ij}}\cdot U_{2_{ij}}}{(\sum U^2_{1_{ij}})^{\frac{1}{2}} (U^2_{2_{ij}})^{\frac{1}{2}}}\Bigg] \end{aligned} \]

after Tape and Tape, 2012 and Cesca et al., 2014

Ensemble of 6000 MT predictions for four earthquakes

Examplary waveform fits for Mw 4.1 on 2019/07/11 23:45:19

Forward calculated for 20% of the predicted ensemble, observed waveforms (black), MAP in red

Ensemble of 6000 source parameter predictions for Mw 3.8 on 2019/07/06 12:00:05

increase in v with Earth structure consideration (Vasyura-Bathke et al., 2021)

Density plot comparing predictions with 196 SCEDC catalog DC's

with v= and w=0

magnitude dependent performance
choice of filters/time windows/SNR

Location of Mw 3.9 at 2019-07-06 17:59:15

location not learned
but: showing same waveforms to all BNN's results in

➡ correlation of the LLK values with distance to the centroid location (black star)

Conclusions

Cons

gridded locations
non-transferable to other areas or magnitude ranges
computional up-front costs (cpu 2-3 months)
no moment magnitude

Conclusions

Pros

fast: single evaluation takes ms
representative ensembles in up to a few s
flexible variational inference consideration of model and data errors

Conclusions

ML can be applied to determine full MTs
sucessfully reproduced indepently determined MTs for subset of 2019 Ridgecrest earthquakes
future application to Landau/Insheim area and potentially to Morsleben

Paper and software

Pre-print available at: https://www.essoar.org/doi/10.1002/essoar.10506663.1

We make the software available as a jupyter notebook at: https://github.com/braunfuss/BNN-MT

Thank you!

Application to recent Landau event


																								Time: 2020-11-09 21:47:07.800
																								Latitude [deg]: 49.149
																								Longitude [deg]: 8.131
																								Depth [km]: 4
																								Magnitude: 1.5