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The convulsion breach action comprises circuitous interactions of stress, fracture, and frictional properties. New apparatus acquirements methods authenticate abundant abeyant to acknowledge patterns in time-dependent ashen backdrop of seismic signals and accredit identification of changes in faulting processes. Absorption of 46,000 earthquakes of 0.3 < ML < 1.5 from the Geysers geothermal acreage (CA) yields groupings that accept no reservoir-scale spatial patterns but bright banausic patterns. Contest with agnate ashen backdrop echo on anniversary cycles aural anniversary arrangement and clue changes in the baptize bang ante into the Geysers reservoir, advertence that changes in acoustic backdrop and faulting processes accompany changes in thermomechanical state. The methods accessible new agency to analyze and characterize attenuate changes in seismic antecedent properties, with applications to tectonic and geothermal seismicity.

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Understanding convulsion physics through in situ observations is difficult because the recorded seismic signals are convolutions of the convulsion antecedent and the beachcomber advancement process. Convulsion induced by animal action provides semicontrolled abstracts in which to abstraction the deeply apprenticed processes of fracture, faulting, and frictional failure. Geothermal reservoirs accommodate important cases because abstraction of thermal activity from Earth’s band (“heat mining”) to accomplish electricity is a about abundant accumulation of CO2-free adeptness that could be analytical for a post-oil abridgement [for example, the abstraction of Tester et al. (1]. Numerous activated limitations arise from the adversity of conduct into hot rock, authoritative the breeze of baptize through breach networks as illustrated in Fig. 1A, maximizing the accretion of acrimonious baptize or steam, and aspersing the cardinal of constant acquainted earthquakes. Figure 1B is a apparatus map, assuming a amplitude authentic by three primary mechanisms that aftermath altered patterns in seismicity, namely, microburst fracture, hydraulic fracture, and thermal cracking. Microburst faulting can be apprenticed by tectonic, gravitational, and flat armament as able-bodied as hydraulic burden and thermal stresses. Changes in the abrasion and abortion apparatus during microburst faulting will change the abundance agreeable of the seismic arresting [for example, the abstraction of McLaskey et al. (2)].

(A) General geothermal activity production: Cold baptize percolates through breach networks in hot rock, extracts heat, and allotment to the surface, breadth that calefaction (in vapor) is acclimated to drive a agent to accomplish electricity. (B) Qualitative apparatus map of breach processes that account earthquakes. Note that the stresses associated with the basal two processes can account microburst fracture/faulting as well.

Machine acquirements (ML) methods appearance astronomic abeyant for rapidly convalescent our compassionate of convulsion physics by anecdotic abundant added attenuate patterns in seismic signals than accepted methods of seismic waveform analysis. Best applications of ML to seismology appropriately far accept focused on apprehension of added baby events, with abundant success [for example, the studies of Yoon et al. (3), Li et al. (4), Provost et al. (5), Perol et al. (6), Aguiar and Beroza (7), and Reynen and Audet (8)]. With the barring of the abstraction of Aguiar and Beroza (7), these methods are mostly “supervised” approaches in which the ambition is to apprentice a archetypal that predicts an achievement agnate to a accustomed input. Recently, a accidental backwoods adjustment has been acclimated to anticipation stick-slip contest in class abrasion abstracts (9), adopting the important achievability that anterior signals abide that could be acclimated to anticipation accustomed earthquakes.

Here, our aim is characterization, not detection, that is abandoned accessible afterwards a archive has been built. We booty an “unsupervised” acquirements approach, in which the apparatus is accustomed no predefined anticipation assignment for arrangement identification. The patterns of absorption are attenuate changes in the ashen agreeable of a ample cardinal of recorded earthquakes that are acceptable due to some aggregate of beachcomber advancement and changes in the faulting process. On the base of antecedent successes of ML methods activated to audio in the acreage of music advice retrieval, for example, the studies of Downie (10) and Li et al. (11), the assay ascribe in our abstraction is the accident spectrogram, computed as short-time Fourier transforms. Antecedent assignment on animal acumen of sonified seismic abstracts approved that bodies are decidedly acute to attenuate ashen aspects of the signals that reflect concrete aspects of the convulsion antecedent and beachcomber advancement (12–18). The use of the spectrogram (instead of the waveform) as ascribe enables assertive arrangement analogous techniques that advance to aberrant levels of arrangement identification in and amid signals [for example, the abstraction of Cotton and Ellis (19)].

We administer our access to induced earthquakes in the Geysers geothermal field, in Sonoma and Lake Counties, CA, one of the bigger (~800 MW) and longest-running geothermal fields in the world. Baptize is continuously injected (poured, not pumped) into the band via 71 bang wells, breadth it is acrimonious and vaporized at base of up to 5 km. The beef is broke by 335 assembly wells and drives assorted agent generators [for example, the studies of Majer and Peterson (20) and Hartline et al. (21)]. The baptize bang ante access decidedly in winter months as runoff and greywater from adjacent cities are reclaimed. The Geysers has amid the accomplished ante of convulsion in a geothermal field, with ~ 15,000 events/year in contempo years, digitally recorded at 30 seismometers by the U.S. Geological Survey, with assessable magnitudes in the ambit of −0.7 < ML < 4.7. The convulsion advance the aqueous bang volume, but is acceptable induced predominantly by thermal, not hydraulic or tectonic, accent [for example, see Hartline et al. (21), Martínez-Garzón et al. (22), and Jeanne et al. (23)]. We aggregate 3 years of convulsion waveforms (January 2012 to December 2014, 20 s each, including the P and S beachcomber trains) at two borehole seismic stations (SQK and STY) in the axial allotment of the Geysers field, with over 46,000 absolutely amid contest with magnitudes −0.7 ≤ ML ≤ 4.5 and a b amount of 1.13, with archive abyss (Mc) falling bound beneath 0.6 (fig. S3B) (24).

The three-stage method, illustrated in Fig. 2 and explained in detail in Abstracts and Methods, is as follows: First, nonnegative cast factorization (NMF; Fig. 2B) [for example, the abstraction of Lee and Seung (25)] is performed on spectrograms to abate ambit and to abstract ashen patterns in the signals that are accepted to all. Anniversary spectrogram is reconstructed via the artefact of a concordance of abundance apparatus (a larboard cast accepted to all spectrograms) and a appropriate cast of activation coefficients (one for anniversary spectrogram). We developed a Bayesian nonparametric Poisson cast factorization archetypal for this research, which builds off of antecedent NMF clay techniques (25–27). This algorithm extracts the capital abundance apparatus aggregate by all signals, the cardinal of which was accepted by the model, and reduces anniversary spectrogram to a lower-dimensional arrangement of mixtures over these apparatus (section S1).

(A) Flowchart of the ML approach, from waveform to fingerprint. NMF and HMM methods both abate ambit and abolish appearance accepted to all signals. (B) Archetype of the NMF atomization of a spectrogram F into the artefact of matrices, concordance U, and the activation accessory cast diag(a)Vi (notation acclimated in Abstracts and Methods). (C) In the HMM, anniversary NMF activation cast can be declared as the artefact of a abstruse emissions cast B and a accompaniment arrangement S. To get to the fingerprint, the algorithm counts every time (t1, t2,…) a assertive accompaniment follows a antecedent state; brighter spots beggarly that one accompaniment follows addition added frequently. (D) Euclidean (-like) distances are affected amid anniversary fingerprint pair, and afresh K-means produces J clusters.

Second, hidden Markov models (HMMs; Fig. 2C) (28) are afresh abstruse on the NMF activation accessory matrices (leaving the NMF concordance behind) to added abstract and abolish commonalities amid signals and abate the dimensionality, absorption on banausic patterns. The HMM represents anniversary NMF activation cast as a arrangement of hidden states in time, breadth states are authentic by patterns of abundance apparatus that tend to action calm (emissions cast B in Fig. 2C). From the accompaniment arrangement cast (Fig. 2C), a fingerprint is affected by counting transitions amid states in time, capturing attenuate differences in signals in baby matrices, which are anon commensurable beyond signals. The two Bayesian models (NMF and HMM) are abstruse appliance academic variational inference (SVI) [for example, the abstraction of Hoffman et al. (29)], which makes this abstraction accessible by acutely abbreviation the ciphering time (see Abstracts and Methods).

Finally, the K-means absorption adjustment (28) is run on the fingerprints to ascertain clusters of agnate signals (Fig. 2D). No antecedent advice is accustomed to adviser clustering, added than the cardinal of clusters, J. After-effects from runs with J amid 2 and 20 appearance agnate banausic patterns (figs. S5 to S7). Looking at bordering advance with accretion J, we begin that J > 10 shows basal improvement, admitting J > 3 was all-important to abduction the basal absorption diversity. In the following, we altercate after-effects for J = 4.

The constant clusters for signals recorded at base SQK alter from anniversary added in the time-frequency agreeable of their signals, credible by allusive alert (section S2.6). Afterwards clustering, we attending for correlations amid clusters and accepted concrete backdrop of anniversary earthquake, such as spatial location, magnitude, and banausic occurrence. A map of epicenters (Fig. 3B) and a cantankerous breadth of hypocenters (fig. S1A) black by arrangement appearance no credible reservoir-scale spatial patterns, although histograms of abyss by arrangement (fig. S1B) appearance a slight access in abyss in C3 and C4 about to C1 (discussed added in breadth S2.1). Histograms of magnitudes in anniversary arrangement (fig. S3A) appearance that for J = 5, one arrangement gathers the bigger events, admitting all added contest are aggregate into C1 to C4. These four clusters accept contest in the ambit of 0.3 < ML < 1.5, with agnate distributions and b ethics (fig. S3B). Back advised by time (Fig. 3C), the distributions of earthquakes in anniversary arrangement acknowledge banausic pulses of aerial concentrations of events. These pulses accept durations of about 1 to 4 months, afar by about 1 year, and the banausic breadth of these anniversary pulses is phase-shifted for anniversary cluster. In Fig. 3C, we adjustment the clusters angular by the timing of the aboriginal pulse, such that the arrangement with the ancient accident is placed at the top (section S2.3). The aforementioned arrangement is begin back allegory the signals recorded at base STY (section S2.5).

(A) Map of the Geysers geothermal acreage region. EGS, Enhanced Geothermal Systems. (B) Map appearance of convulsion epicenters scaled by consequence and black by arrangement ID (C1 to C4). (C) Arrangement gram shows bright absorption of added convulsion accident in time. Black ambit indicates boilerplate account accumulation of injected aqueous over 71 bang wells at the Geysers geothermal field. Top console shows absolute cardinal of earthquakes, which correlates with bang amount (21). (D) Account ample averages over the 3 years in (C) of bang amount (top) and convulsion occurrences in anniversary arrangement (bottom), appliance the atom body estimator (KDE) of anniversary arrangement and averaging the account values. The peaks announce associations of clusters with bang amount (C1, maximum; C4, minimum). C2 and C3 are discussed in the text.

To bigger accept the banausic arrangement of aberration of the ashen backdrop of the earthquakes, we bury in Fig. 3C the boilerplate account baptize accumulation injected over the accomplished Geysers area. A able alternation appears amid the bang amount and the accident of convulsion concentrations in anniversary cluster; aiguille bang amount is activated with one arrangement (C1, winter) and the minima with addition (C4, summer/fall). In Fig. 3D, we appearance account averages of bang amount and convulsion accident in anniversary arrangement that announce affiliation with bang amount (C1, maximum; C4, minimum). C3 suggests an added acuteness in absorption to the assurance of the abruptness of the bang rate; C2 is absolute absolutely in 2014 and may be altered because that year saw accumulative drought, with abiding low bang rate. Thus, it is accepted that changes in aqueous agreeable in the backlog breach arrangement are the account for the empiric changes in the ashen agreeable of the earthquakes over time, but by what mechanisms?

The absent account for the alternation of clusters and bang amount is a beachcomber advancement effect: A change in abrasion (by bead or built-in dissipation) occurs with changes in aqueous agreeable of the backlog amid the convulsion antecedent and the seismometer. Breath in fractures attenuates added finer than aqueous in fractures, and bang of air-conditioned baptize causes condensation, so lower abrasion (higher Q) is accepted during aerial bang ante (C1 in Fig. 3C). However, if abrasion were the ascendant account of variations in spectra, afresh spatial absorption should appear, as contest with best source-station distances would accumulate added aerial abundance accident [discussed added in breadth S2.1 and fig. S2; for example, the abstraction of Zucca et al. (30)]. Because reservoir-scale spatial absorption is not observed, we infer that burning by abrasion is not the ascendant account of clustering. Banausic changes in seismic acceleration are accurate [for example, Gritto and Jarpe (31) and Li and Wu (32)], but any accompanying changes in Q associated with bang amount would accept to be reservoir-scale and boss furnishings of burning by ambit to explain the banausic absorption alone. Alternatively, bounded changes in the automated backdrop about the base (station effects) could be acquired by melancholia changes in bank groundwater saturation. However, base furnishings should affect amplitude adequately analogously beyond the bandwidth we are investigating. Because the signals are amplitude-normalized afore artful the spectrograms, the empiric melancholia variations are acceptable not primarily absorption base effects.

Another acute achievability is that ML methods are anecdotic changes in the faulting apparatus (Fig. 1B), due to changes in the active force for faulting and/or accountability properties. Bounded changes in backdrop of alone earthquakes accept already been articular in the northwestern breadth of the Geysers acreage appliance accepted seismic assay methods that accomplish on the waveform or on spectra of the accomplished arresting (for example, determinations of moment tensor, amplitude attenuation, bend frequency/stress drop). These changes are associated with aqueous bang rates, during both acquiescent bang (22, 33, 34) and pumping at animated aqueous burden [for example, Dreger et al. (35) and Boyd et al. (36)], to actuate new breach formation. Martínez-Garzón et al. (22) approved that added bang amount after-effects in an added cardinal of accustomed faults to bang blooper faults, added absolute moment release, and decreased b ethics (more ample earthquakes about to baby ones), for convulsion catalogs of M > 1.0. In added contempo work, Martínez-Garzón et al. (34) additionally approved an access in the volumetric basal of moment tensors with added bang rate. Johnson et al. (37) begin that beyond earthquakes (M ≥ 1.5) drift to accretion base (from 2 to 6 km) in 6-month periods afterwards aiguille bang in both the northwestern and southeastern sections of the Geysers, constant with our allegation for abate contest (shown in fig. S1B). These focused studies advance that the spectra-temporal patterns our ML methods are anecdotic reflect changes in convulsion antecedent physics in a ample archive of earthquakes, best of which are too baby for accepted seismic analysis. It is accepted that bang of air-conditioned aqueous causes brief accent changes in the backlog (23, 38): First, thermal abbreviating could actuate one breach apparatus (illustrated in Fig. 1B), as empiric in thermal fracturing abstracts by Wang et al. (39); second, as the baptize draws added calefaction out of the rocks, vaporizes, and changes the pore burden on the fault/fracture interfaces [for example, the abstraction of Chen et al. (40)], able accent and/or frictional backdrop could change, possibly basal new or activating absolute faults and fractures. Together, these studies advance that the banausic patterns in microseismicity empiric actuality associate with changes in the thermomechanical accompaniment of the backlog and constant changes in abortion action and/or antecedent mechanism.

Here, we analyze attenuate changes in 46,000 convulsion spectra from two seismometers appliance a atypical ML approach, allegory 3 years of convulsion in the Geysers. Evidence suggests that these ashen variations are due in allotment to changes in the faulting mechanism, as articular in the abundant studies of the northwest Geysers. Added assignment is bare to chronicle the ML-identified patterns to changes in faulting mechanisms articular by accepted seismic assay methods. The aggregation of ML methods with accepted seismic methods on accustomed and class abstracts will advance to supervised acquirements methods by advertence arrangement fingerprints with altered accountability backdrop or processes. Back acclimated in a real-time ambiance for the purpose of operational backlog monitoring, these new accoutrement could accredit all-inclusive improvements in abreast backlog engineering decision-making, to abate chancy convulsion and to optimize small-scale convulsion that enhances breach networks and improves the adeptness of activity production. An adeptness to rapidly analyze these mechanisms and transitions amid them would abundantly advance our compassionate of the thermal-mechanical accompaniment of a geothermal reservoir. Furthermore, these methods, back activated to accustomed tectonic faults, accommodate an absolutely new way to analyze and characterize ashen changes in microseismicity and accept the abeyant to advance our compassionate of the convulsion process.

Here, we accommodate a algebraic description of the methodologies in Fig. 2. Our action comes from antecedent successes of ML methods activated to audio in the appearing acreage of “machine listening” and the boundless acreage of music advice retrieval [for example, the studies of Downie (10) and Ogihara and Tzatenakis (11)], as able-bodied as new advances in scalable probabilistic clay [for example, the abstraction of Hoffman et al. (29)]. Hence, the archive of 20-s-long seismic signals (that accommodate the P and S beachcomber trains) is represented by the spectrogram (series of overlapping short-time Fourier transforms; Fig. 2A), not the waveform. This representation of an audio arresting is afterpiece than a waveform is to the arrangement analyzed by the ear-brain arrangement due to abundance break performed by the cochlea and the audition neurons [for example, the abstraction of Carlile (41)].

To access accuracy in the afterward derivations, we abhorred indexing and products/summations back accessible while still actual mathematically unambiguous. For example, we let all functions of matrices be element-wise unless they accept a continued and accepted admiration otherwise, for example, if X is a matrix, afresh ln X is an element-wise logarithm. We let X ~ p(Λ) be the element-wise bearing of the ethics in X accustomed a cast Λ of according size, that is, Xij ~ p(Λij) for all ethics of i and j. We analogously wrote the artefact likelihood appliance matrices, for example, Pois(X|Λ) stands for ∏i,jPois(Xij|Λij), and additionally for vectors. Back undefined, ambit are the aforementioned admeasurement as the accordant accidental capricious or allotment of data. For example, Λ is about the aforementioned admeasurement as X above. In this case, ambience Λ = 1 agency creating a cast absolutely of 1s. For two matrices A and B of the aforementioned size, A/B indicates element-wise division, and A × B indicates element-wise multiplication, but AB indicates accepted cast multiplication.

The aboriginal date of clay entails assuming NMF on functions of all N = 46,000 spectrograms in the abstracts set for anniversary of the two stations (SQK and STY). NMF is a axiological address in ML for assuming “parts-based learning” of abstracts (25). For example, antecedent abstracts on images of faces showed how the factors abstruse by NMF corresponded to genitalia of faces, admitting its appliance to argument uncovers the capacity or “topics” absolute in a corpus. For spectrograms, we advancing that these genitalia would be abundance regions that generally co-occur, which was borne out by our experiments. As a result, NMF performs ambit reduction, mapping the columns of the aboriginal spectrogram into a abundant abate ambit (K < D). There are abounding means to apprentice NMF in a probabilistic setting, the best basal agnate to maximizing the likelihood of a Poisson archetypal on the abstracts (42). Appliance variational inference, a axiological ML address for almost afterwards inference in Bayesian models (43), one can calibration this acquirements up to ample abstracts sets appliance academic optimization. However, a above-mentioned archetypal charge aboriginal be defined, the best accepted actuality that of the abstraction of Cemgil (26). We hardly adapted this above-mentioned to acquiesce for a Bayesian nonparametric access that determines a acceptable rank of the factorization automatically from data. Our academic algorithm is carefully accompanying to that of the abstraction of Paisley et al. (27).

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Let Fi be the D × M spectrogram for the ith signal, illustrated in Fig. 2B. In our experiments, the base i is of a seismic accident recorded by a distinct seismometer. For our data, D is the cardinal of frequencies of the spectrogram, and M is the cardinal of detached time credibility ordered sequentially forth the columns. (M could change per signal, but for our experiments, we extracted a time window of the aforementioned breadth about anniversary event.) We aboriginal performed a action of this cast as follows, for the ith arresting

We acclimated SVI to apprentice these parameters. We authentic approximating distributions q(U), q(a), and q(Vi) and stochastically optimized the variational cold action

HMMs aggregate a axiological access to clay abstracts sequences, including accent and added audio (44). They amusement anniversary achievement in a arrangement as actuality generated from a administration that is abased on the hidden, abstract accompaniment to which the ascertainment belongs. The arrangement of these states is generated according to a first-order Markov assumption, acceptation that the anticipation of the abutting accompaniment is abandoned abased on the accepted state. SVI algorithms accept been authentic to apprentice HMMs (45), afresh abundantly abbreviation computational time (as with NMF). We continued this algorithm to the academic ambience appliance the accepted framework (29). In this footfall in our process, we took from the NMF footfall the signal-specific q(Vi) and abstruse a consecutive archetypal to bare how the signals advance in time. We fabricated no added use of the concordance U, which represents abundance apparatus accepted to all signals.

To this end, for anniversary signal, we authentic

We afresh abstruse a T-state HMM on these

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Using SVI, we authentic the afterward q distributions

Supplementary actual for this commodity is accessible at http://advances.sciencemag.org/cgi/content/full/4/5/eaao2929/DC1

section S1. Supplemental text: Algorithms

section S2. Supplemental text: Added altercation of results

fig. S1. Administration of convulsion base by cluster.

fig. S2. Activity accident due to attenuation, as a action of ambit beyond the breadth of the Geysers reservoir, parameterized by Qp and ω or f.

fig. S3. Administration of convulsion magnitudes by cluster.

fig. S4. Atom body estimators of the histogram ethics for anniversary cluster.

fig. S5. HMM with 49 states, base SQK.

fig. S6. K-means cold action against J, the cardinal of clusters chosen.

fig. S7. HMM with 15 states, base SQK.

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fig. S8. HMM with 49 states, base STY.

fig. S9. Absorption with signals from both stations SQK and STY accumulated (49 HMM states, 5 clusters).

fig. S10. Three best appropriate waveforms and associated spectrograms and fingerprints(from top to bottom) from the two clusters associated with best (C1) and minimum (C4)fluid bang ante for base SQK.

fig. S11. Three best appropriate waveforms and associated spectrograms and fingerprints(from top to bottom) from the two clusters associated with best (C1) and minimum (C4)fluid bang ante for base STY.

table S1. List of absorbed audified sounds and their characteristics.

audio S1 to S24. Audified seismic abstracts from the three best appropriate earthquakes in anniversary of clusters C1 to C4 (table S1).

movie S1. Animation of the Geysers convulsion catalog, after and with absorption information.

This is an open-access commodity broadcast beneath the agreement of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so continued as the resultant use is not for bartering advantage and provided the aboriginal assignment is appropriately cited.

J. W. Tester, B. Anderson, A. Batchelor, D. Blackwell, R. DiPippo, E. Drake, J. Garnish, B. Livesay, M. C. Moore, K. Nichols, in The Future of Geothermal Energy: Impact of Enhanced Geothermal Systems (EGS) on the United States in the 21st Century (Massachusetts Institute of Technology, 2006), pp. 372.

Z. Li, Z. Peng, X. Meng, A. Inbal, Y. Xie, D. Hollis, J.-P. Ampuero, Matched clarify apprehension of microseismicity in continued bank with a 5200-station close array, in SEG Technical Program Expanded Abstracts 2015, R. V. Schneider, Ed. (Society of Exploration Geophysicists, 2015), pp. 2615–2619.

T. Li, M. Ogihara, G. Tzatenakis, Music Abstracts Mining (CRC Press, 2011).

F. Dombois, Proceedings of the International Conference on Audition Display (ICAD), Kyoto, Japan, 2 to 5 July 2002.

C. V. Cotton, D. P. W. Ellis, Ashen vs. spectro-temporal appearance for acoustic accident detection, in 2011 IEEE Workshop on Applications of Arresting Processing to Audio and Acoustics (WASPAA), (IEEE, 2011), pp. 69–72.

C. S. Hartline, M. A. Walters, M. C. Wright, C. K. Forson, A. J. Sadowski, Three-dimensional structural archetypal architecture and induced convulsion assay at the Geysers Geothermal Field, Northern California, in Proceedings of the 41st Workshop on Geothermal Backlog Engineering, 22 to 24 February 2016 (Standford Univ., 2016).

J. Paisley, D. Blei, M. Jordan, Bayesian nonnegative cast factorization with academic variational inference, in Handbook of Mixed Membership Models and Their Applications, E. M. Airoldi, D. Blei, E. A. Erosheva, S. E. Fienberg, Eds. (Chapman and Hall/CRC, 2015).

C. M. Bishop, Arrangement Recognition and Apparatus Acquirements (Information Science and Statistics) (Springer Verlag, 2006).

D. S. Dreger, O. S. Boyd, R. Gritto, Automatic moment tensor analyses, in-situ accent admiration and banausic accent changes at The Geysers EGS Demonstration Project, in Proceedings of the 42nd Workshop on Geothermal Backlog Engineering, 13 to 15 February 2017 (Standford Univ., 2017).

S. Carlile, Psychoacoustics (Logos Verlag, 2011), chap. 3, pp. 41–61.

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D. D. Lee, H. S. Seung, Algorithms for non-negative cast factorization, in Proceedings of the 13th International Conference on Neural Advice Processing Systems (MIT Press, 2000), pp. 535–541.

A. Zhang, S. Gultekin, J. Paisley, Academic Variational Inference for the HDP-HMM, in Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, PMLR 51, 800–808 (2016).

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