Problem Given the number of times in which an unknown event has happened and failed: Required the chance that the probability of its happening in a single trial lies somewhere between any two degrees of probability that can be named. SECTION 1 Definition 1. Several events are inconsistent, when if one of them happens, none of the rest can. 2. Two events are contrary when one, or other of them must; and both together cannot happen. 3. An event is said to fail, when it cannot happen; or, which comes to the same thing, when its contrary has happened. 4. An event is said to be determined when it has either happened or failed. 5. The probability of any event is the ratio between the value at which an expectation depending on the happening of the event ought to be computed, and the value of the thing expected upon it’s 2 happening.

An essay towards solving a problem in the doctrine of chances. [Facsimil]

Determining earthquake hypocenters and focal mechanisms requires precisely measured P wave arrival times and first‐motion polarities. Automated algorithms for estimating these quantities have been less accurate than estimates by human experts, which are problematic for processing large data volumes. Here we train convolutional neural networks to measure both quantities, which learn directly from seismograms without the need for feature extraction. The networks are trained on 18.2 million manually picked seismograms for the Southern California region. Through cross validation on 1.2 million independent seismograms, the differences between the automated and manual picks have a standard deviation of 0.023 s. The polarities determined by the classifier have a precision of 95% when compared with analyst‐determined polarities. We show that the classifier picks more polarities overall than the analysts, without sacrificing quality, resulting in almost double the number of focal mechanisms. The remarkable precision of the trained networks indicates that they can perform as well, or better, than expert seismologists.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2017JB015251

P Wave Arrival Picking and First-Motion Polarity Determination With Deep Learning

Determining earthquake hypocenters and focal mechanisms requires precisely measured P-wave arrival times and first-motion polarities. Automated algorithms for estimating these quantities have been less accurate than estimates by human experts, which is problematic for processing large data volumes. Here, we train convolutional neural networks to measure both quantities, which learn directly from seismograms without the need for feature extraction. The networks are trained on 18.2 million manually picked seismograms for the southern California region. Through cross-validation on 1.2 million independent seismograms, the differences between the automated and manual picks have a standard deviation of 0.023 seconds. The polarities determined by the classifier have a precision of 95% when compared with analyst-determined polarities. We show that the classifier picks more polarities overall than the analysts, without sacrificing quality, resulting in almost double the number of focal mechanisms. The remarkable precision of the trained networks indicates that they can perform as well, or better, than expert seismologists.

P-wave arrival picking and first-motion polarity determination with deep learning

Over a 13?day period magma propagated laterally from the subglacial Barðarbunga volcano in the northern rift zone, Iceland. It created >?30,000 earthquakes at 5–7?km depth along a 48?km path before erupting on 29 August 2014. The seismicity, which tracked the dike propagation, advanced in short bursts at 0.3–4.7?km/h separated by pauses of up to 81?h. During each surge forward, seismicity behind the dike tip dropped. Moment tensor solutions from the leading edge show exclusively left-lateral strike-slip faulting subparallel to the advancing dike tip, releasing accumulated strain deficit in the brittle layer of the rift zone. Behind the leading edge, both left- and right-lateral strike-slip earthquakes are observed. The lack of non-double-couple earthquakes implies that the dike opening was aseismic.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1002/2015GL067423

Strike‐slip faulting during the 2014 Bárðarbunga‐Holuhraun dike intrusion, central Iceland

In this letter, we use deep neural networks for unsupervised clustering of seismic data. We perform the clustering in a feature space that is simultaneously optimized with the clustering assignment, resulting in learned feature representations that are effective for a specific clustering task. To demonstrate the application of this method in seismic signal processing, we design two different neural networks consisting primarily of full convolutional and pooling layers and apply them to: 1) discriminate waveforms recorded at different hypocentral distances and 2) discriminate waveforms with different first-motion polarities. Our method results in precisions that are comparable to those recently achieved by supervised methods, but without the need for labeled data, manual feature engineering, and large training sets. The applications we present here can be used in standard single-site earthquake early warning systems to reduce the false alerts on an individual station level. However, the presented technique is general and suitable for a variety of applications including quality control of the labeling and classification results of other supervised methods.

Unsupervised Clustering of Seismic Signals Using Deep Convolutional Autoencoders

We studied the seismic velocity structure beneath the Krafla central volcano, NE Iceland, by performing 3-D tomographic inversions of 1453 earthquakes recorded by a temporary local seismic network between 2009 and 2012. The seismicity is concentrated primarily around the Leirhnjukur geothermal field near the center of the Krafla caldera. To obtain robust velocity models, we incorporated active seismic data from previous surveys. The Krafla central volcano has a relatively complex velocity structure with higher P wave velocities (V_p) underneath regions of higher topographic relief and two distinct low-V_p anomalies beneath the Leirhnjukur geothermal field. The latter match well with two attenuating bodies inferred from S wave shadows during the Krafla rifting episode of 1974–1985. Within the Leirhnjukur geothermalreservoir, we resolved a shallow (−0.5 to 0.5 km below sea level; bsl) region with low-V_p/V_s values and a deeper (0.5–1.5 km bsl) high-V_p/V_s zone. We interpret the difference in the velocity ratios of the two zones to be caused by higher rock porosities and crack densities in the shallow region and lower porosities and crack densities in the deeper region. A strong low-V_p/V_s anomaly underlies these zones, where a superheated steam zone within felsic rock overlies rhyolitic melt.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1002/2015JB012350

Seismic imaging of the shallow crust beneath the Krafla central volcano, NE Iceland

Earthquake source inversion is highly dependent on location determination and velocity models. Uncertainties in both the model parameters and the observations need to be rigorously incorporated into an inversion approach. Here, we show a probabilistic Bayesian method that allows formal inclusion of the uncertainties in the moment tensor inversion. This method allows the combination of different sets of far-field observations, such as P-wave and S-wave polarities and amplitude ratios, into one inversion. Additional observations can be included by deriving a suitable likelihood function from the uncertainties. This inversion produces samples from the source posterior probability distribution, including a best-fitting solution for the source mechanism and associated probability. The inversion can be constrained to the double-couple space or allowed to explore the gamut of moment tensor solutions, allowing volumetric and other non-double-couple components. The posterior probability of the double-couple and full moment tensor source models can be evaluated from the Bayesian evidence, using samples from the likelihood distributions for the two source models, producing an estimate of whether or not a source is double-couple. Such an approach is ideally suited to microseismic studies where there are many sources of uncertainty and it is often difficult to produce reliability estimates of the source mechanism, although this can be true of many other cases. Using full-waveform synthetic seismograms, we also show the effects of noise, location, network distribution and velocity model uncertainty on the source probability density function. The noise has the largest effect on the results, especially as it can affect other parts of the event processing. This uncertainty can lead to erroneous non-double-couple source probability distributions, even when no other uncertainties exist. Although including amplitude ratios can improve the constraint on the source probability distribution, the measurements are often systematically affected by noise, leading to deviation from their noise-free true values and consequently adversely affecting the source probability distribution, especially for the full moment tensor model. As an example of the application of this method, four events from the Krafla volcano in Iceland are inverted, which show clear differentiation between non-double-couple and double-couple sources, reflected in the posterior probability distributions for the source models.

/pdf/a-bayesian-method-for-microseismic-source-inversion-kq0lainu05.pdf

A Bayesian method for microseismic source inversion

The MTfit code was developed as part of a PhD project (Pugh 2015), funded under a Natural Environment Research Council (NERC) studentship at the University of Cambridge, as a CASE award with Schlumberger. Seismometers were borrowed from the NERC SEIS-UK (loan 842).

/pdf/mtfit-a-bayesian-approach-to-seismic-moment-tensor-inversion-qxwnlj6w6p.pdf

MTfit: A Bayesian Approach to Seismic Moment Tensor Inversion

This work was funded under a Natural Environment Research Council (NERC) studentship as a CASE award with Schlumberger. Seismometers were borrowed from the NERC SEIS-UK (loan 842), who also archive the data.

Automatic Bayesian polarity determination

Seismicity was monitored beneath the Krafla central volcano, NE Iceland, between 2009 and 2012 during a period of volcanic quiescence, when most earthquakes occured within the shallow geothermal field. The highest concentration of earthquakes is located close to the rock-melt transition zone as the IDDP-1 wellbore suggests, and decays quickly at greater depths. We recorded multiple swarms of microearthquakes, which coincide often with periods of changes in geothermal field operations, and found that about one third of the total number of earthquakes are repeating events. The event size distribution, evaluated within the central caldera, indicates average crustal values with b = 0.79 ± 0.04. No significant spatial b-value contrasts are resolved within the geothermal field nor in the vicinity of the drilled melt. Besides the seismicity analysis, focal mechanisms are calculated for 342 events. Most of these short-period events have source radiation patterns consistent with double-couple (DC) mechanisms. A few events are attributed to non-shear faulting mechanisms with geothermal fluids likely playing an important role in their source processes. Diverse faulting styles are inferred from DC events, but normal faulting prevails in the central caldera. The best-fitting compressional and tensional axes of DC mechanisms are interpreted in terms of the principal stress or deformation-rate orientations across the plate boundary rift. Maximum compressive stress directions are near-vertically aligned in different study volumes, as expected in an extensional tectonic setting. Beneath the natural geothermal fields, the least compressive stress axis is found to align with the regional spreading direction. In the main geothermal field both horizontal stresses appear to have similar magnitudes causing a diversity of focal mechanisms.

/pdf/focal-mechanisms-and-size-distribution-of-earthquakes-21lgfxe4dg.pdf

David J. Pugh

Papers

Seismic imaging of the shallow crust beneath the Krafla central volcano, NE Iceland

A Bayesian method for microseismic source inversion

MTfit: A Bayesian Approach to Seismic Moment Tensor Inversion

Automatic Bayesian polarity determination

Focal mechanisms and size distribution of earthquakes beneath the Krafla central volcano, NE Iceland