Geophys. J. Int. (2021) 225, 1653–1671 doi: 10.1093/gji/ggab055 Advance Access publication 2021 February 10 GJI Seismology

Hypothesis tests on Rayleigh wave radiation pattern shapes: a theoretical assessment of idealized source screening

Joshua D. Carmichael Joshua D Carmichael EES-17, Los Alamos National Laboratory, P.O. Box 1663,MSD446, Los Alamos, NM 87545, USA. E-mail: [email protected]

Accepted 2021 February 8. Received 2021 January 31; in original form 2020 September 16 Downloaded from https://academic.oup.com/gji/article/225/3/1653/6132273 by guest on 25 September 2021

SUMMARY Shallow seismic sources excite Rayleigh wave ground motion with azimuthally dependent radiation patterns. We place binary hypothesis tests on theoretical models of such radiation patterns to screen cylindrically symmetric sources (like explosions) from non-symmetric sources (like non-vertical dip-slip or non-VDS faults). These models for data include sources with several unknown parameters, contaminated by Gaussian noise and embedded in a layered half-space. The generalized maximum likelihood ratio tests that we derive from these data models produce screening statistics and decision rules that depend on measured, noisy ground motion at discrete sensor locations. We explicitly quantify how the screening power of these statistics increase with the size of any dip-slip and strike-slip components of the source, relative to noise (faulting signal strength) and how they vary with network geometry. As applications of our theory, we apply these tests to (1) find optimal sensor locations that maximize the probability of screening non-circular radiation patterns and (2) invert for the largest non- VDS faulting signal that could be mistakenly attributed to an explosion with damage, at a particular attribution probability. Finally, we quantify how certain errors that are sourced by opening cracks increase screening rate errors. While such theoretical solutions are ideal and require future validation, they remain important in underground explosion monitoring scenarios because they provide fundamental physical limits on the discrimination power of tests that screen explosive from non-VDS faulting sources. Key words: Theoretical seismology; Surface waves and free oscillations; Earthquake moni- toring and test-ban treaty verification; Probability distributions.

‘faulting signal’ on the radiation pattern shape is significantly non- 1 INTRODUCTION circular or has a low signal-to-noise ratio (SNR). Smaller yield ex- Seismic sources produce Rayleigh wave radiation patterns that indi- plosions (∼101−103kg) that do not trigger tectonic release may still cate focal mechanism asymmetry (Fig. 1). Teleseismic and regional accompany fracturing events or mechanical sliding on rock joints records of large underground nuclear explosions (∼101−103kT) that output shear energy as bulk damage (Steedman et al. 2016). emplaced in pre-stressed geologies, for example, have historically These bulk effects do not appear to significantly distort surface revealed that tectonic release of strain can superimpose with the wave radiation patterns when they are symmetric about a vertical isotropic explosion source to produce Love and Rayleigh waves axis. In particular, local distance records from a series of conven- with non-circular radiation patterns (Toksoz¨ & Kehrer 1972;Ek- tional explosives conducted through the source physics experiment strom¨ & Richards 1994). These observations contrast with simple (SPE, Snelson et al. 2013) demonstrate that low yield, shallowly models of cylindrically symmetric, shallow sources hosted in verti- buried sources that cause damage are vertical-axis symmetric, and cally stratified geologies that predict azimuthally constant Rayleigh thereby trigger Rayleigh wave radiation patterns that are circular wave amplitudes and absent Love waves. Instead, such observa- within a factor of 1.25 (Larmat et al. 2017). These explosions, tions are partially consistent with models of faulting motion pre- which were emplaced in the granites of Climax stock on the Nevada sented by tectonic release, which produces asymmetric deformation National Security Site (NNSS), also demonstrate that body waves about a vertical axis that excites a Rayleigh-wave field with a non- radiated by the same source in a similar frequency band show ap- circular radiation pattern (Richards & Ekstrom¨ 1995). Observers parent azimuthally asymmetry that is likely sourced by refraction risk misidentifying large, shallow explosions that accompany such effects (Darrh et al. 2019). Experiments with low-yield explosives tectonic release as shallow earthquakes when the imprint of such a emplaced in boreholes within New Hampshire granite demonstrate

C The Author(s) 2021. Published by Oxford University Press on behalf of The Royal Astronomical Society. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. 1653 1654 J.D. Carmichael Downloaded from https://academic.oup.com/gji/article/225/3/1653/6132273 by guest on 25 September 2021

Figure 1. Surface sensors (triangles) record Rayleigh waves from a shallow explosion that is buried to a depth h in a horizontally-stratified half-space that has moment MI, which creates damage with moment MCLVD that each superimpose with faulting of moment M0. The damage occurs when constructive interference between the incident and reflected waves create a conically shaped zone of failure. (a) An explosion source capable of producing damage that is symmetric about a vertical axis, but that includes no faulting (no tectonic release) radiates a circular radiation pattern REX described at top. (b) A source that includes non-VDS faulting near the explosion point radiates a non-circular radiation pattern REQ described at top. Each radiation pattern shape associates with a deterministic scalar (bottom) that quantifies the power of the hypothesis test between data that record circular versus non-circular radiation patterns (at top). Section 4.1 defines each symbol. more complex effects. Namely, these latter experiments reveal that is therefore more earthquake-like than explosion-like. A discrimi- observationally confirmed radial fracturing in hard rock accom- nant that tests radiation pattern distortion from local to teleseismic panies asymmetric, short period (∼6 Hz), guided Rayleigh waves ranges can therefore benefit both the source physics community (Rg). Body waves sourced by these same New Hampshire explo- and support nuclear test-ban treaty surveillance, if justified on both sions can exhibit even greater azimuthal dependence (Stroujkova theoretical and experimental grounds. 2018). This work determines if a discriminant for shallow source types Both the SPE and New Hampshire experiments cumulatively in- that tests Rayleigh wave radiation pattern shapes is justified on dicate two substantive effects of the seismic source on radiation theoretical grounds. To confront this challenge, we apply binary hy- pattern shape can be observed from local distances. First, verti- pothesis tests to competing data models of noisy radiation patterns cal axis-symmetric damage does not significantly distort Rayleigh and assume several source parameters are unknown. Our models wave radiation pattern shapes from circularity, whereas a struc- also use planar geologies and therefore isotropic propagation of ture at depth can distort body wave patterns. Secondly, signifi- Rayleigh waves; we do not consider Love waves. Under this limited cant radial fracturing and shearing motion at the shallow explo- scope, we derive screening curves from the hypothesis tests that sion source measurably distorts both Rayleigh wave and body bound an observers’s capability to discriminate non-vertical dip- wave radiation patterns. These observations indicate that az- slip (non-VDS) shallow faulting sources from symmetric sources imuthal variability of Rayleigh waves sourced by small yield ex- (like shallow explosions that create damage that is symmetric about plosions indicate faulting or shearing motion at the source, whereas a vertical axis). The screening power of these tests is completely body wave asymmetry appears non-uniquely attributed to source defined by the product of a faulting-signal term, and a term that and path. depends on deployment parameters. Using this test, we illustrate A forthcoming test within the SPE series will provide a unique op- several applications of our theory that include finding deployment portunity to study asymmetry in radiation patterns that are sourced locations for sensors to supplement an existing network and that by explosions and superimposed with historic, tectonic records of maximize our source-type screening probability. We conclude that fault slip. This effort will detonate a conventional explosive in a Rayleigh wave radiation pattern discriminant shows a high proba- tectonic regions and monitor its seismic signatures for compari- bility PrD of success (PrD > 0.9) when a sufficient number of sensors son against earlier earthquake signatures that output radiation that (>12) with a limited azimuthal gap (<90◦) records radiation from shares wavefront paths (Walter et al. 2012). If Rayleigh waves out- sources with moderate strike-slip faulting signals (SNR >20). This put by this explosion remains circular and mismatches historical probability diminishes with more ambiguous focal mechanisms that records, theory and the aforementioned observations imply that resemble certain historical events recorded near the Lop Nor nuclear the resultant radiation pattern shape provides a hopeful discrimi- test site in China. Our results suggest that a screening statistic that nant. It remains unclear, however, ‘how much’ non-circularity of tests Rayleigh wave radiation pattern shapes for faulting signals the Rayleigh wave radiation pattern is sufficient to indicate that requires very dense sensor coverage to achieve good efficacy, and a seismic source is dominated by shearing or fault motion, and is probably impractical in most passive monitoring scenarios. We Hypothesis tests on radiation patterns 1655 lastly consider how radiation pattern distortion from opening cracks displacement discontinuities as sets of force couples. Such a mo- and wavefield homogenization can inflate screening errors. ment tensor source excites the frequency-domain, Rayleigh wave T We emphasize that our treatment is purely theoretical and ideal- displacement u = [ux , 0, uz ] , that solves Newton’s second law of ized. Our models require that future efforts provide any evidence motion in this stratified half-space, subject to traction-free surface that this approach has a practical utility as a screening technique in boundary conditions. The scalar product of M with the gradient non-ideal settings. In particular, focusing effects (Selby & Wood- of the (Hermitian) Rayleigh wave Green’s tensor GRAY (Aki & house 2000;Linet al. 2012), tectonic pre-stress (Toksoz¨ et al. Richards 2002, eq. 3.23) compactly represents such a general dis- 1971;Ekstrom¨ & Richards 1994), surface roughness (Steg & Kle- placement: mens 1970), path-dependent attenuation (MacBeth & Burton 1987), = ·∇ RAY. transmission losses through vertical interfaces (Napoli & Russell u M G (1) 2018) and scattering from topography that is comparable in relief When the source is emplaced in half-space with non-trivial strati- to Rayleigh wavelengths (Snieder 1986; Ichinose et al. 2017)often fication (not pure half-space), the displacement solution of eq. (1) dominate the azimuthal variability of observed surface wave radi- is a linear combination of eigenvectors r that sum over distinct ation. Stratified half-space models that support idealized Rayleigh wavenumber modes (Aki & Richards 2002, eq. 7.150-7.151). The wave propagation are likely limited to very local seismic measure- dispersion relationship k = k(ω) for this Rayleigh wave field defines Downloaded from https://academic.oup.com/gji/article/225/3/1653/6132273 by guest on 25 September 2021 ments of SPE shots, cyroseismic studies interior to crevasse-free these modes and couples the radiation pattern R of u to the range- ice sheets and glaciers (Carmichael et al. 2015; Lough et al. 2015; and depth- dependent part of the displacement solution g.Ifthe Lindner et al. 2019; Hudson et al. 2020), and to engineered struc- moment tensor source is shallow enough that the free-surface trac- tures (Lu et al. 2007). We assert that our study remains useful in tion free boundary conditions σ · eˆz = 0 also apply at the source’s geophysics, however, because it helps define the physical limits on depth, then the radiation pattern effectively decouples from g (Aki an observer’s capability to screen Rayleigh wave sources, using & Richards 2002, p. 328); here, σ is the elastic stress tensor, eˆz is a only their radiation pattern shapes. Such limits support previous ef- unit vector pointing vertically upward and 0 is a three element vec- forts that exploit Rayleigh wave radiation pattern geometry to better tor of zeros. More formally, this condition requires that the source is characterize underground explosions (Toksoz¨ et al. 1971; Yacoub buried at a depth that is small compared to the dominant wavelength 1981), and supports present computational and observational work of the seismic radiation triggered by the source–time function. These to understand earthquake radiation (Rosler¨ & van der Lee 2020). cumulative conditions mean the Rayleigh wave field has the simple form and is a product of R and g:

2 REFERENCE SOURCES AND u = R (φ) · g(x,ω) (2) HISTORICAL EVENTS = R¯ + R1 cos (2φ) + R2 sin (2φ) · g(x,ω). Our assessment will compare cylindrically symmetric sources that Vector g in eq. (2) depends on source depth and observation range, include explosions with damage along the vertical axis against a set but not φ;termR¯ is the mean radiation pattern; Rm (m = 1, 2) of four distinct non-VDS, tectonic earthquake sources. One such are non-circular radiation pattern coefficients that all have units of faulting source is the strike-slip earthquake. This particular double- moment and depend on moment tensor M;andφ is the azimuthal couple source is maximally dissimilar from any isotropic source, as angle (see Aki & Richards 2002, fig. 4.20). The general displace- plotted on a Hudson diagram (Hudson et al. 1989) or lune (Tape ment in eq. (1) and the traction free boundary conditions relate the &Tape2019). We also consider historical, tectonically sourced mean radiation pattern and azimuthally dependent coefficients to events that are shallow and locate near former underground testing linear functions of the moment tensor elements: sites. These events include the Rock Valley fault located within β2 the Nevada National Security Site complex in the United States, a 1 2 R¯ = Mxx + Myy − 1 − Mzz regionally recorded seismic event beneath the Kara Sea near No- 2 α2 vaya Zemlya, and a shallow source located near the Chinese Lop 1 (3) R = M − M Nor nuclear test site. Fig. 2 pairs the beach ball representation of 1 2 xx yy the focal mechanisms for these sources with noisy representations R2 = Mxy, for the non-circular part of their radiation patterns. The confidence region for the hypocentral depth of each observed source includes where Mij is the force couple aligned with the Cartesian direction depths shallow enough to be reached by current drilling capabilities i separated by direction j. We note that Rayleigh wave radiation (within 7 km of Earth’s surface). We shall often refer to such shal- patterns do not resolve terms Mxz and Myz that indicate vertical low, non-VDS tectonic earthquakes as simply non-VDS faulting shearing motions directed parallel to Earth’s surface. sources. Similarly, we shall refer to explosions that may or may not accompany damage that is symmetric about a vertical axis as just an explosive source. Any such damage preserves the circularity of 3.1 Colocated faulting, explosion and damage sources Rayleigh wave radiation patterns. The most general symmetric moment tensor for a single point source is a superposition of an explosion with isotropic moment MI that effectively colocates with a non-VDS faulting source of moment 3 SHALLOW SOURCE RAYLEIGH M and a damage source, which is equivalent to a compensated WAVES 0 linear vector dipole (CLVD) with moment MCLVD. Moment MCLVD We consider Rayleigh wave motion present in a cylindrically sym- is positive if explosions create dilatation along the vertical axis of metric, horizontally layered half-space triggered by a buried seismic damage, and negative if explosions drive contraction along that axis. source (Fig. 1). This source may be described by a point force or We assume that the fault is parametrized by strike (φs), rake (λ)and a symmetric moment 3 × 3 moment tensor M that summarizes dip (δ) angles. The radiation pattern coefficients from eq. (2)are 1656 J.D. Carmichael

Figure 2. Three focal mechanism solutions for shallow earthquakes that locate near underground test sites and the noisy, non-circular component of the radiation pattern for each source. (a): The MW = 3.7, 1993-05-30 event from the Rock Valley Sequence, termed ‘Event 1’ (depth ≈ 2 km) (Smith et al. 1993). (b): The ML = 3.3, 1997-08-16 event from the Kara Sea (0 km < depth < 23 km) that located tens of km from Novaya Zemlya (Hartse 1998;Bowers2002; Schweitzer & Kennett 2007). (c): The mb = 4.8, 2003-03-13 tectonic event located near the Lop Nor test site (5 km < depth < 7 km, Selby et al. 2005). In each pattern, Gaussian noise superimposes with a deterministic radiation pattern solution to mimic a finely sampled observation with an overall SNR of 20. We note that the noise on the non-circular component of the radiation pattern increases from left to right, as the sources produce Rayleigh waves less efficiently. Downloaded from https://academic.oup.com/gji/article/225/3/1653/6132273 by guest on 25 September 2021 We further assume that the isotropic component of the source is zero in each case, and within the bound of any uncertainties in the focal mechanism estimate. Here, MW is the moment magnitude, ML is local or Richter magnitude, and mb is body wave magnitude. then (algebra omitted): to a threshold to measure evidence that the data record a non- circular radiation pattern. Specifically, our null hypothesis is that 2β2 3α2 − 4β2 3α2 − 4β2 R¯ = M − M − DS an azimuthally distributed set of sensors samples a circular radia- α2 I α2 CLVD α2 2 tion pattern with zero non-VDS faulting moment (M = 0). Our R = φ − φ (4) 0 1 DScos ( s ) SSsin ( s ) alternative hypothesis states that these same sensors sample a non-

R2 = SScos (φs ) + DSsin (φs ) , circular radiation pattern. The competing hypotheses are (without noise): i where we have used a moment decomposition (Patton & Taylor H R = R¯ 2011, eq. 15). In eq. (4), scalar SS quantifies the strength of the 0 : M = 0 M = 0 0 0 (8) strike-slip faulting component, whereas scalar DS quantifies the H R = R¯ + ψ + ψ , 1 : > = DScos(2 ) SSsin(2 ) strength of the dip-slip component. These terms are (Ekstrom¨ & M0 0 M0 0

Richards 1994, eqs 21 and 22): where δ, λ and M0 parametrize DS and SS, and are unknown. We 1 now add a random variability models to eq. (8) in two respects: DS = M sin (2δ) sin (λ) first, we superimpose Gaussian measurement noise to R,andsec- 2 0 (5) ond, we model R itself to be a sum of the deterministic component = δ λ , SS M0 sin ( ) cos ( ) and a zero mean random process. To include measurement noise, where the equivalent expression for SS elsewhere (Patton & Taylor we note that observations often show that moment estimates are 2011, eq. 16) is missing a factor of two in the dip angle. We express log-normally distributed if such moment is measured from station the full Rayleigh wave displacement by combining eqs (2), (4) magnitudes (Godano & Pingue 2000; Kagan 2002). Moment es- and (5). The superposition of the explosive, damage and non-VDS timates that fit a horizontal line to the low-frequency component faulting sources thereby creates a Rayleigh wave displacement field of waveform’s displacement spectrum reveal Gaussian residuals, that is: because normally distributed errors dominate seismic waveform records (Sˇ´ıleny et al. 1996; Walter et al. 2009). We therefore as- = R¯ + ψ + ψ · , ` u DScos(2 ) SSsin(2 ) g (6) sume waveform noise is the dominant source of random error in where ψ = φ − φs is the difference between the azimuthal and strike scalar moment estimates and model a single observation at fixed angles; because we use ψ hereon, we refer to it as the ‘azimuthal’ azimuth as a Gaussian random variable. To modify the signal por- angle, despite abuse in terminology. Regardless, the function that tion of the radiation pattern, we add a zero-mean Gaussian process 2 scales g is the radiation pattern R, and includes the same functional with covariance σR to the deterministic, non-zero mean of R.We form as the scalar factor that appears in eq. (2): indicate that R is a random variable with mean μ and variance σ 2 with notation R ∼ N μ, σ 2 ,where∼ indicates ‘is distributed as’. R = R¯ + ψ + ψ . DScos(2 ) SSsin(2 ) (7) Eq. (8) then tests between two competing probability distributions Eq. (4) with eq. (7) demonstrates that R¯ is azimuthally constant, for R (with noise): δ = and that azimuthal variability arises from non-VDS faults where H R ∼ N R ,σ2 0 : = M ±π/2, λ =±π/2. Eq. (7) is also the starting point for estimating the M0 0 (9) optimal receiver sampling of the Rayleigh wave radiation pattern of H R ∼ N R ,σ2 + σ 2 . 1 : > M R shallow sources (Section 4.3). M0 0 Eq. (9) applies to one observation ψ. For many azimuthally dis- tributed observations, our hypothesis test includes a vectorized set 4 STATISTICAL HYPOTHESIS TESTS of radiation pattern samples R where the kth sample is Rk : To determine if observed Rayleigh wave radiation indicates an R [R , R , ··· , R , ··· , R ]T . (10) isotropic or mixed non-VDS source, we evaluate a hypothesis test 1 2 k N that compares two distinct models for deterministic signals R.This The competing hypotheses (eq. 9) form a general Gaussian detection test forms a generalized likelihood ratio statistic from N obser- problem, applied to a multisample radiation pattern (Kay 1998, vations of R at different azimuthal locations and compares them section 5.6). To explicitly define each radiation pattern term, we Hypothesis tests on radiation patterns 1657 concatenate the unknown statistical parameters into a single vector. observer must estimate the probability that this event is explosive We then express the mean μ of R under H1 as a linear system that or tectonically sourced by a non-VDS fault (hypothesis H0 or H1). equates with eq. (4) so that component m (row m)ofμ is: To determine which hypothesis Rayleigh wave records likely ⎡ ⎤ R¯ describe, we use a generalized log-likelihood ratio test (log-GLRT). This test compares the logarithmic ratio of the alternative hypothesis μ = 1 (cos(2ψ ) − c) sin(2ψ ) · ⎣ DS⎦, (11) m m m PDF to the null hypothesis PDF, where each density is evaluated at SS H(m,:) maximum likelihood estimates (MLE) of its unknown parameters θ to form a test statistic. The log-GLRT is equivalent to a conditional (I ) R where decision rule on this scalar test statistic Lθ ( ):

β2 α2 − β2 H1 2 3 4 (I ) R ≷ η R¯ R¯ + cDS = MI − MCLVD Lθ ( ) where: α2 2α2 H (12) 0 ⎡ ⎤ α2 − β2 3 4 max{ fR (R; H1) } (17) c θ,σ2 α2 (I ) ⎢ M ⎥ Lθ (R) = ln ⎣ ⎦ . { R H } Downloaded from https://academic.oup.com/gji/article/225/3/1653/6132273 by guest on 25 September 2021 , max fR ( ; 0) and where H(m :) in eq. (11)definesrowm of matrix H.Thefirst θ,σ2 parameter R¯ weights the circular portion of the radiation field M whereas the other parameters weight the non-circular portion. The We compute the MLEs of the parameters in Appendix B using covariance between the observations Ri and R j at two azimuths projector matrices and reduce the scalar statistic into a ratio of is zero (eq. A5). This independence in observations implies that two scaled, statistically independent terms. This provides a test R the probability density function (PDF) for vector R under H1 has statistic Lθ ( ) equivalent to eq. (17) for screening circular from μ = θ σ 2 + σ 2 H non-circular radiation patterns: mean H and covariance ( R M )I. The PDF under 1 is therefore:   σ 2  R2 H   M P X 1 ||R − θ||2 Lθ (R) = (N − 3) ·   ≷ η. (18) 1 H σ 2 + σ 2  ⊥ 2 H fR (R; H ) = exp − , M R P R 0 1 N σ 2 + σ 2 (13) H π σ 2 + σ 2 2 2 R M 2 ( R M ) The rank-1 projector matrix P X in the numerator of eq. (18)is: H = = σ 2 = where I is the identity matrix. Under 0, DS SS R 0. T −1 T P X = X X X X , where: The mean μ of R under H0 is also linear system, but θ is now − (19) constrained to make R circular: X = H HT H 1 AT. − H0 : Aθ = 0, where: A = 011. (14) T 1 T The rank-3 projector matrix P H = H H H H projects vec- R σ 2 H { } The PDF for vector with covariance M I under 0 is then: tors onto the subspace spanned by H,spanH . Projector matrix   P ⊥ = I - P in the denominator of eq. (18) then projects vec- 2 H H 1 ||R − Hθ|| { } fR (R; H ) = exp − . tors onto the space orthogonal to span H . We thereby interpret the 0 N σ 2 (15) πσ2 2 2 M Aθ=0 screening statistic Lθ (R) in eq. (18) as a scaled ratio of the radiation 2 M pattern energy due to non-VDS faulting (note structure of A), di- These competing PDFs provide the basis for a binary hypothesis vided by the residual radiation pattern energy not attributable to the test between circular versus non-circular radiation patterns. The Rayleigh wave model (the system H). Probability theory predicts hypothesis test in eq. (8) that uses such vectorized data is: that this quotient has a non-central-F distribution with one, and N H R ∼ N θ,σ2 , θ = − 3 degrees of freedom. The density for Lθ (R) is then shaped by 0 : H M I A 0 (16) a scalar non-centrality parameter if R includes Gaussian noise H : R ∼ N Hθ, σ 2 + σ 2 I . k 1 M R (1 ≤ k ≤ N) and the radiation pattern is non-circular. We write its To exploit these tests, we consider two cases. This first case (Case I) distributional dependence as Lθ (R) ∼ F1,N−3 (L,>0) and the considers the variability σR in R from random processes as known, cumulative, non-central F distribution for Lθ (R) as: but the stochastic variability σ M as unknown (note we use the term { θ (R)} = , − ( ,) . variability loosely to indicate standard deviation) . The second case CDF L F1 N 3 L (20) σ σ (Case II) considers M as approximately known, but R as unknown, These distributions have a well-defined variance when their sec- though in much less detail than Case I. Parameter θ is unknown in ond degree of freedom exceeds four, so that the screening statistic σ both cases. Case I and Case II both include the scenario that R is Lθ (R) is only well-defined when N ≥ 8. This sampling implies that zero as a special example. at least two sensors sample each quadrant of the Rayleigh radiation pattern (given full azimuthal coverage by sensors), which mitigates spatial aliasing of four-lobed radiation patterns. This statistic is iden- 4.1 Case I: σR known, θ and σ unknown M tical in form to the subspace detector (Scharf & Friedlander 1994) We first assume a network of N ≥ 8 sensors records a Rayleigh that is often applied in seismic monitoring, and has completely anal- 2 wave radiation pattern with known process variance (σR in eq. 16), ogous interpretations for waveform detection. Consequently, many unknown faulting parameters (θ in eq. 11), and unknown stochastic standard detection theory texts document the properties and analyti- σ 2 variance ( M in eq. 16). This case idealizes explosion monitoring cal form for F1, N − 3(L, ), and computational packages like MATLAB scenarios in which an underground testing complex colocates with include routines to calculate random variables and parameters of the a shallow fault of unknown geometry. This case also includes the non-central-F distribution. 2 scenario that all variability is stochastic in origin (σR = 0) as an Crucially, this theory shows that the scalar in eq. (20) com- example. Under this general model, a sensor network then records a pletely quantifies the screening capability of Lθ (R) to test between low magnitude seismic event that originates near a test site, and an H0 and H1 at fixed N;eq.(B5) shows its general analytical form. 1658 J.D. Carmichael

Here, we use those results to write as the product of one factor that depends on deployment azimuth ψ and a second factor that depends on source and noise parameters:

1 (DS + SS)2 = · . T −1 T σ 2 + σ 2 A H H A M R (21) Deployment faulting SNIR This factorization shows that is the product of a scalar term − −1 A HT H 1 AT that stores the N sensor deployment locations, + 2 / σ 2 + σ 2 and a term (DS SS) ( M R) that quantifies the non-VDS faulting signal energy, relative to noise and random process vari- ance. It is analogous to the signal-to-noise plus interference ratio (SNIR) of waveform processing. Explosive or VDS sources that Downloaded from https://academic.oup.com/gji/article/225/3/1653/6132273 by guest on 25 September 2021 2 trigger no Rayleigh waves include parameters σR = DS = SS = 0, Figure 3. Probability density functions that describe the source screening and = 0 (as expected). Strike-slip faults maximize among statistic Lθ (R) (horizontal axis) in eq. (18). The statistic describes N = 10 other faulting sources, given equal sensor placement, because receivers randomly distributed over a circle centred at the hypocentre of a 2 2 2 2 2 2 shallow seismic source (orange star) that sample a source radiation pattern (DS + SS) / σ + σR ≤ M / σ + σR (see eq. 5). Impor- M 0 M R tantly, eqs (18)and(21) demonstrate that discriminating between . The leftmost thick, grey curve indicates a circular radiation pattern ( = > a circular and non-circular Rayleigh wave radiation pattern under 0) and rightmost curves indicate non-circular radiation patterns ( 0). The darkest shaded area under the null PDF curve ( = 0) indicates a PrFA the Case I assumptions is entirely equivalent to a signal detection = × −3 2 5 10 false screening probability. The lighter shaded area below the problem. We further note the random process variance σR acts to alternative PDF ( = 10) measures the screening probability PrD = 0.81. simply inflate the effective noise variance and reduce any observed, The threshold is η ≈ 5(eq.23). non-VDS faulting signal SNR under H1. This asymmetry in vari- ance between the null and alternative hypotheses means that a high We use eq. (22) to write this average as:  SNR faulting signal with high random signal variance is indistin- −1 Pr¯ =|S () | 1 − F , − (η, ) . guishable from a lower SNR faulting signal with zero random signal D 1 N 3 (24) |S | 2 ( ) variance. We will assume σR = 0 hereon, so that and Lθ (R) are each simplified from their more general forms. This simplification in which |S () | is the size of the set (e.g. the number of realiza- preserves our physical insight into the radiation pattern testing prob- tions in a Monte Carlo simulation). The sum in eq. (24) equates lem without adding cumbersome notation. The faulting SNIR then to an integration when the subset of permissible values of is a reduces to just SNR. continuum (e.g. |ψ|≤3π/4) and gives a marginal distribution for ¯ ¯ = / π +3π/4 ψ − η, Under these simplifying (but still rather general) assumptions, PrD (e.g. PrD 4 (3 ) −3π/4 d (1 F1,N−3 ( ))). the probability PrD of detecting any general faulting signal, and Figs 4, 5 and 6 each show qualitatively similar, mean screening then deciding that R records a non-circular radiation pattern is curves (eq. 24) that each index distinct deployment and source pa- the probability PrD of measuring a value of Lθ (R) that exceeds a rameters. Fig. 4 shows the probability that eq. (24) will screen strike- threshold η (eq. 18): slip earthquakes from explosions with N = 12 sensors deployed over three distinct azimuthal ranges against faulting signal (pic- PrD = 1 − F1,N−3 (η, > 0) . (22) tured at right). Each of these three curves show the dependence of + 2 /σ 2 screening probability on the faulting signal SNR (DS SS) M , We select a threshold η for deciding a radiation pattern is non- averaged over 100 individual screening curves. Scalar uniformly circular using the Neyman–Pearson criteria. This criteria uses a samples receiver deployments confined to these azimuthal ranges π constant false-attribution probability PrFA to invert for η: δ = λ = + 2 = where source parameters are 2 , 0, (DS SS) M0 and = 2/σ 2 faulting SNR M0 M . The three curves demonstrate that screen- PrFA = 1 − F1,N−3 (η, = 0) , (or) (23) ing probability markedly decreases as azimuthal gap increases with η = −1 − ,= , F1,N−3 (1 PrFA 0) a fixed number of stations, even as station density over deployment azimuth (sensor number per arc length) increases. Shading about −3 −1 • wherewesetPrFA to an acceptable value (e.g. 10 )andF1,N−3 ( ) is the top blue curve associated with full azimuthal sampling marks the inverse of the central F distribution with one and N − 3degrees the one-standard deviation (±σ ) variability. of freedom. The screening probability PrD in eq. (22) monotoni- Fig. 5 illustrates that earthquake signals triggered by other focal cally increases with at fixed N and η. The locus of points (, mechanisms are more difficult to screen from explosions relative PrD) thereby defines performance curves that measure the screen- to strike-slip events (δ =±π/2, λ = 0, π). Specifically, eq. (18) ing capability of the decision rule in eq. (18) versus SNR of the illustrates predicted curves that quantify the capability of statistic faulting signal at fixed PrFA (Fig. 3). Deployment constraints and Lθ (R) to screen distinct earthquakes from explosions using the tectonic settings can each limit the permissible parameter space of same deployment strategy with N = 12 sensors. In this case, each to a subset S (). These limits include the azimuthal deployment curve is an average of 100 screening curves that randomly sample a range of sensor numbers and locations that quantify the leftmost circular instrument deployment around three different sources (focal factor in eq. (21), and variability in the source focal mechanism, mechanisms at right). These curves demonstrate that the radiation which quantifies the rightmost factor in eq. (21). In such cases, we pattern test screens non-strike-slip earthquakes from explosions less estimate the expected screening capability Pr¯ D of eq. (18)asthe effectively than strike-slip earthquakes, even with identical station average of the detection rate PrD over such parametric constraints. coverage. Seismic events with a Rayleigh wave radiation pattern Hypothesis tests on radiation patterns 1659 Downloaded from https://academic.oup.com/gji/article/225/3/1653/6132273 by guest on 25 September 2021

Figure 4. Averaged screening curves (eq. 24), plotted against faulting signal SNR (horizontal axis; eq. 21). Each curve presents an estimate of 100%× the probability PrD (vertical axis) that the decision rule in eq. (18) will correctly screen a strike-slip earthquake (beach ball, top left) from an explosion. The test statistic Lθ (R)[eq. (18), left-hand side] for each curve associates with N = 12 sensor azimuths that sample the Rayleigh wavefield R under distinct deployment −3 constraints (right); all thresholds η correspond to PrFA = 10 (eq. 23). The topmost curve (blue) presents an average of |S () |=100 individual decision rule curves that each measure screening performance when N = 12 sensors uniformly sample random azimuths along the right, topmost dashed circle that surrounds a source (orange star). The shaded region indicates ± one standard deviation estimate (±σ ) from the mean curve. The second curve (red) presents a similar average of 100 curves that associate with sensors deployed with uniformly random azimuth along the red dashed arc at right surrounding the same source; in this case, the azimuthal range is restricted to 2π/3 radians. The third curve (green) shows a similar average, but for sensor azimuths restricted to π/2 radians (green dashed arc). The filled circles superimposed on the dashed deployment arcs at right show particular, random sensor locations in each case. Some circular sensor markers overlap.

Figure 5. Similar to Fig. 4, but curves now present 100%× the probability PrD (vertical axis) that eq. (18) will screen earthquakes with three distinct focal mechanisms from an explosion, using N = 12 sensors, all plotted against faulting signal SNR (horizontal axis; eq. (21)). Each curve is an average of 100 screening curves (eq. (24)) that associate with sensor deployments that uniformly sample all azimuths along the dashed circle centred at the hypocentre of a shallow source (orange star); the filled circles mark a particular, random deployment. The topmost curve (blue) presents such an average for faulting source like that of ‘Event 1’ from the Rock Valley earthquake; the second curve (red) presents a similar average for faulting source like that of the Kara Sea event; and the bottom curve presents a similar average for a faulting source like that of the 2003 Lop Nor event. The shaded region indicates ± one standard deviation −3 from the top curve mean. All curves correspond to PrFA = 10 . like that of the 2003 Lop Nor tectonic event that is sampled by an = 8, 16, 24). Each deployment randomly samples a circle centred unrestricted deployment of N = 12 sensors is particularly difficult about a shallow source like that of the 2003 Lop Nor tectonic event. to screen from an explosion. Shading about the curve that associates In this case, a screening statistic that uses N = 24 sensors can screen with full azimuthal sampling marks the ±σ variability in screening non-VDS faulting from explosion sources at roughly the same mean sources like Event 1 from the Rock Valley earthquake. rate as a statistic that uses 12 receivers to screen sources like Event An increase in sensor deployments (N > 12) correspondingly in- 1 from the Rock Valley earthquake sequence (Fig. 5). While a 24 creases screening probability between non-VDS faulting and explo- sensor deployment screens shallow sources like that show in Fig. 6 sion sources. Fig. 6 shows three distinct averages of 100 screening with relatively low variance, the test statistic obviously requires curves that each associate with three distinct sensor deployments (N twice the number of observations. 1660 J.D. Carmichael Downloaded from https://academic.oup.com/gji/article/225/3/1653/6132273 by guest on 25 September 2021

Figure 6. Similar to Fig. 4, but curves now estimate 100%× the probability that eq. (18) (vertical axis) will screen earthquakes whose focal mechanisms match that of the 2003 Lop Nor event (beachball, upper left) from an explosion, using three distinct sensor deployments without azimuthal constraint. As in Figs 4 and 5, each curve presents an average of 100 curves (eq. 24) that associate with sensors deployed with uniformly random azimuth along the dashed circles −3 (circular markers, right) surrounding the source (orange star). All curves correspond to a PrFA = 10 false attribution probability and plot against faulting signal SNR (horizontal axis; eq. 21). Filled sensor circles mark a particular, random deployment.

4.2 A note on screening curve interpretation PrD where the total differential dPrD peaks:

Our interpretation of screening curves that compare probabilities ψˆ +1 = argmax {dPrD} ψ|H1 like Pr¯ D against faulting signal SNIR require comment (see Figs 4,   ∂Pr ∂Pr (25) 5 and 6). The binary decision rules necessarily decide between = argmax D η + D . ψ|H a continuum of source types that are not identically explosions or 1 ∂η ∂ faults. For example, the decision rule in eq. (18) will screen data that The decision threshold decreases by an increment η: records Rayleigh motion sourced by an explosion that superimposes η = −1 − ,= with non-zero contributions from a non-VDS faulting signal with a F1,N−2 (1 PrFA 0) probability that much greater than the false-attribution probability − −1 ( − ,= ) , −3 F1,N−3 1 PrFA 0 (26) PrFA = 10 . A proper false attribution probability constraint that considers an acceptable amount of faulting signal with an explosion whereas the faulting signal changes by increment : would more appropriately marginalize out some faulting effects (DS + SS)2 with a prior on from this continuum (Berger & Delampady 1987). = σ 2 We address such source-type errors and ambiguities in Section 5.4  M  of our Discussion. The binary hypothesis testing theory that we 1 1 × − . (27) use here is standard, however, and we proceed with the caveat that T −1 T T −1 T A H+1 H+1 A A H H A neither the H0 nor H1 in our models fully capture the continuum of source type hypotheses. The argmax argument on the right-hand-side eq. (25) is a directional We next explore if more optimal sensor placement can improve derivative. It is maximal at parameter increments [ η, ] in! the ∂ ∂ ∂ screening rates of target sources that are particularly challenging to direction of gradient ∇Pr = PrD , PrD , where sign PrD = " # D ∂η ∂ ∂η screen from cylindrically symmetric sources. ∂ − PrD =+ 1 and sign ∂ 1, and only increment depends on ψ +1. Moreover, the quadratic forms in (27) are both strictly positive, − since HT H 1 is positive definite. This collective system structure 4.3 Case I application: optimal deployment of auxiliary implies that the bracketed difference between the geometric factors receivers is always non-negative (e.g. adding more sensors cannot reduce the source screening probability), so that: We suppose an existing N ≥ 8 receiver network with limited az- imuthal coverage requires an auxiliary receiver to maximize its ψˆ +1 = argmax {dPrD} ψ|H capability to screen shallow non-VDS sources from explosions, 1 = { } through application of Lθ (R) as a discriminant. To develop an ob- argmax (28) ψ|H1 jective function for this discriminant, we add a row [1, cos (2ψ) − ! = T −1 T . c,sin(2ψ)] to H and make a new matrix H+1. We then maximize argmin A H+1 H+1 A ψ|H1 PrD over additional sensor deployment azimuths ψ +1 that we es- timate as ψˆ +1. Such an optimal auxiliary sensor location estimate The last term of eq. (28) defines the objective function for Rayleigh supplements the existing network to form an updated N + 1net- wave sampling. It omits explicit dependence on data R. Therefore, work that maximizes the probability of detecting a non-VDS signal the optimal strike-relative deployment azimuth estimate ψˆ +1 for a σ 2 in the radiation pattern data R. Specifically, this new observation receiver when M is unknown depends only on current receiver α β point updates the discriminant in eq. (18), while eq. (23) updates the placement and the relative seismic wave structure ( and ). Other η T −1 T < threshold to maintain the same false attribution probability PrFA . values of sensor placement where A H+1 H+1 A 1 increase The optimal solution ψˆ +1 depends on both η and , and maximizes from its prior value (eq. 21). Auxiliary sensor azimuths where Hypothesis tests on radiation patterns 1661 T −1 T A H H+ A > 1 decrease from its prior value. To quantify non-central-F CDF for : +1 1 $ % the change in screening power, we reapply eq. (23) to estimate 1 (DS + SS)2 the updated threshold η and reapply eq. (22) to compute source 1 − PrD = F1,N−3 η; = . (29) T −1 T σ 2 screening rates with N + 1 sensors; we then update the decision A H H A M rule (eq. 18). The term 1 − Pr is equivalent to a ‘miss’ in waveform detection Fig. 7(a) shows π-periodic optimal and suboptimal solutions D theory. The faulting signal SNR that solves eq. (29) relates to non- for extra receiver locations that supplement 16 existing receivers centrality parameter estimate ˆ through: uniformly deployed along a 75◦ azimuthal arc (grey circles, " # Fig. 7b). The objective function shows local and global extrema ˆ = argmin |1 − PrD − F1,N−3 (η; ) | . (30) T −1 T of A H+1 H+1 A . The maxima occur at deployment azimuths that minimize and therefore the probability of screening binary Our estimate for the source term of ˆ , as measured by a given source types; these sensor locations provide the poorest improve- sensor deployment locations ψ m (m = 1, 2, ···, N) that are stored in ment in source-type screening. Local minima (triangles) provide H,is: a locally optimal sensor placements near these worst-case sensor Downloaded from https://academic.oup.com/gji/article/225/3/1653/6132273 by guest on 25 September 2021 locations. These local minima solutions may apply to scenarios in  2 (DS + SS) − = A HT H 1 AT .ˆ (31) which deployment outside restricted azimuthal ranges is impossi- σ 2 ble (e.g. inaccessible areas). Our global solutions (circles) confirm M expectation that optimal receiver deployment should include points Fig. 8 illustrates three solution curves to eq. (31) for a strike-slip that include no current receiver coverage. These optimal deploy- faulting signal when 1 − PrD = 0.1 (for a 0.9 correct attribution ment locations are not orthogonal to the mid-point of the azimuthal probability), which depicts averages over 100 random, azimuthally arc, but depend on the relative values of the body wave speeds (pa- constrained sensor deployments. These solutions provide a lower rameter c). Fig. 7(b) shows screening curves for the decision rule bound on threshold source size, since is largest for strike-slip (18) when a receiver network contains the backbone of existing + 2 /σ 2 = 2/σ 2 events ((DS SS) M M0 M ). Vertical differences between sensors (inset grey circles), supplemented with poorly placed sen- distinct curves suggest that false attribution rates of a non-VDS sors (squared one and two), versus optimally placed extra sensors faulting signal to an explosion source (via eq. 18) increases with (circled 1 and 2). The optimally placed sensor that supplements azimuthal gap. this 16 sensor backbone network forms a 17 sensor network the Fig. 9(a) better quantifies this gap effect on screening explosive improves screening rates of the faulting source by >6 × that of from strike-slip sources. Each family of curves show the maximum, the backbone network, for sources with the greatest faulting signal mean and minimum solutions to eq. (31) per azimuthal gap value. shown (circled 1). An additional, optimally placed sensor (circled The vertical range of SNR values between each curve measures 2) provides a greater gain in screening performance, relative to an the variability that results from the 100 randomly placed sensors. 18 sensor network that includes two poorly placed auxiliary sensors In particular, these solution curves indicate that strike-slip source (squared 2). The screening performance our test achieves with this = 2/σ 2 size (SNR M0 M ) increases exponentially as deployment gap azimuthally limited, 18 sensor arrangement outperforms average increases beyond 90◦ (equivalent to a 270◦ azimuthal coverage). test that uses the 24 sensor deployment in Fig. 6. Moreover, the variability in such threshold estimates is asymmetric: Finally, we emphasize that the update to η maintains a consistent the log-scale maximum threshold estimates (top curve) depart from false-attribution probability for the discriminant before and after the mean (middle curve) significantly more than do log-scale min- adding any additional receivers. Failure to update this threshold imum threshold estimates (bottom curve). Noisy radiation patterns when < 1 could lead an observer to erroneously conclude that that associate with these solutions in Fig. 9(b) show that the decision ψ adding receivers at particular values of +1 (shaded domain of defined by eq. (18) can mistake (with probability 1 − PrD = 0.1) the Fig. 7a) can reduce the screening power of the same test. subtle four-lobed shape for a noisy circular radiation pattern, even with full azimuthal sensor coverage.

4.4 Case I application: threshold faulting moments and 4.5 Case II: σ R and θ unknown; σ M known missed detections We progress from Case I and now assume that a sensor network We now quantify the largest faulting source signal (as defined by records a Rayleigh wave radiation pattern with unknown random + 2 /σ 2 σ 2 θ (DS SS) M ) that could be mistakenly attributed to an explo- process variability [ R in eq. (16)], unknown faulting parameters ( σ 2 sion, at a fixed false attribution probability PrFA . This application in eq. 11), and a stochastic variance M that is either known, or has a 2 is particular important to underground explosion monitoring ap- significantly lower estimator variance than than that ofσ ˆR. This case plications because it provides fundamental physical limits on the applies to idealized monitoring scenarios in which an underground the discrimination power of eq. (16) tests to screen explosive from testing complex colocates with a shallow fault of unknown geome- non-VDS faulting sources. try and radiation variability, but with well-characterized stochastic σ 2 To compute this signal, we first note that the probability of mistak- variability M that is perhaps quantified from historical records of ing a non-VDS faulting source for an explosive source is equivalent proximal explosions. Sensor networks that record a low magnitude to the event that Lθ (R) <ηwhen H1 is true (eq. 18). Here, thresh- seismic event that originates from this test site faces the same chal- old η relates to the CDF under hypothesis H0 through eq. (23). lenge as that in Case I, but the screening statistic has a distinct The probability of miss-attributing a non-VDS faulting source to form. We treat Case II in less detail than Case I because it appears an explosion then equates to 1 − PrD and relates to the faulting less practically applicable than Case I while remaining more com- signal through a non-linear equation that requires inversion of the plicated. Appendix C derives several results that we state here. For 1662 J.D. Carmichael Downloaded from https://academic.oup.com/gji/article/225/3/1653/6132273 by guest on 25 September 2021 Figure 7. Optimal and suboptimal azimuthal solutions for extra sensor locations to supplement an existing sensor deployment. (a) Solution curves to eq. (28) that show the dependence of optimization parameter on the geometric deployment factor (left factor in eq. (21); vertical axis) on azimuth ψ (horizontal axis). Points below the dashed, ‘stationary’ line marks azimuthal regions where both and η increase. The global minima (circles) mark solutions that maximize source screening parameter . The local minima (triangles) mark solutions for the best location of an extra receiver that is near the worst deployment location (squares). (b) Screening probability curves PrD (vertical axis) for the decision rule in eq. (28) when deployments include one and two extra sensors to supplement an existing N = 16 station network (grey markers, upper left), compared against the scaled faulting signal (horizontal axis). The radiation pattern source has the same focal mechanism as the 2003 Lop Nor tectonic event (beach ball at top right). The thick grey curve labelled with a grey, circled 0 marks the performance of the screening statistic that associates with the original instrument deployment (grey circles). The other numbered curves show the screening performance of the optimally deployed N = 17 and N = 18 networks. These curves associate with the sensor locations that we depict with deployment schemes (upper left) and minima in (a). Squares mark curves that show worse-case deployment azimuths that correspond to maxima on the curve in (a).

where eqs (14)and(15)define fR (R; H0) and eq. (13)defines 2 fR (R; H1). The MLE for σR is (algebra omitted): ||R − Hθˆ ||2 σ 2 = 1 − σ 2 ˆ R M N (33) ||P ⊥ R||2 = H − σ 2 N M in which the text that follows eq. (19) defines the projector matrix ⊥ θˆ θ P H .Oureq.(B2) defines the MLEs k for parameter vectors k (k = 0, 1). We input these parameters to eq. (33) and reuse the symbol Lθ (R) to write the resultant, equivalent GLRT statistic (algebra omitted):   ||R − θˆ ||2 || ⊥ R||2 H 0 P H Lθ (R) = − N ln . (34) Figure 8. Three solutions curves to eq. (31) that quantify the largest strike- σ 2 σ 2 M M slip faulting source signal that the decision rule in eq. (18) can mis-identify − = to originate from an explosion with probability 1 PrD 0.1. In each case, We use the MLEs for θˆ to express Lθ (R) as a sum of two statisti- = k N 12 sensors confined to three distinct azimuthal deployment ranges (at cally independent random variables (X and Y): right) sample the strike-slip Rayleigh wave radiation pattern (beach ball         inset). Each curve compares faulting signal SNR (vertical axis) against false  ⊥ R2  ⊥ R2  R2 P H P H P X Lθ (R) = − N ln + . (35) attribution probability (horizontal axis), and error bars measure standard σ 2 σ 2 σ 2 error from 100 random sensor deployments. Filled circles (right) mark a M M M particular, random deployment (we use standard error here because standard X Y deviation markers reduce readability). ⊥ R 2//σ 2 We interpret the first quadratic form ( P H M )ineq.(35)as the residual radiation pattern energy not attributable to the Rayleigh wave model, relative to random noise variance. The second term R 2/σ 2 ( P X M ) is energy due to non-VDS faulting, relative to the same noise variance. The decision rule that tests the function of these two terms in eq. (35)is:         Case II, the log-GLRT equivalent to eq. (17)is:  ⊥ 2  ⊥ 2  2 H P R P R P X R 1 H − N ln H + ≷ η.ˆ (36) σ 2 σ 2 σ 2 H H M M M 0 (II) R ≷1 η Lθ ( ) where: η H The threshold ˆ in eq. (36) is strictly an estimate that maintains 0 ⎡ ⎤ a false-attribution probability PrFA , consistent with Case I. We de- max{ fR (R; H1) } (32) ⎢ θ,σ2 ⎥ scribe our estimation strategy for this threshold in Appendix C (II) R = R , Lθ ( ) ln ⎣ ⎦ (see eq. C5). To compute the screening performance for Case II max{ fR (R; H0) } θ (SNIR versus PrD), we note that the terms X and Y are independent, Hypothesis tests on radiation patterns 1663 Downloaded from https://academic.oup.com/gji/article/225/3/1653/6132273 by guest on 25 September 2021

Figure 9. Solutions to eq. (31) for strike-slip sources (Case I). (a) Moving window statistics of threshold faulting SNR values (vertical axis) that associate with 1 − PrD = 0.1 and N = 12 sensors and that are confined by a grid of azimuthal gaps (horizontal axis). The noise free strike-slip beach ball appears at right. Markers on the horizontal axis associate with Rayleigh wave radiation patterns in (b). (b) Three noisy radiation patterns of a strike-slip source (eq. 16) with three corresponding azimuthal gaps in sensor sampling. The decision rule in eq. (18) samples these radiation patterns at N = 12 locations to falsely attribute the source of each pattern to an explosion, with probability 1 − PrD = 0.1. A four-lobe Rayleigh wave radiation pattern is evident in the case that sensors sample the radiation field without azimuthal constraint (blue pattern). We emphasize that the decision rule in eq. (18) mistakes this pattern as circular (with a 0.1 probability). piecewise invertible functions. In particular, the term X is piecewise over 100 deployment realizations. Each such realization effectively ⊥ − = R 2/σ 2 = −1 1 invertible in argument s P H M since function x(s) s recomputes the deployment factor A HT H AT of and − Nln (s) monotonically decreases over s < N and monotonically reconvolves the PDFs f (•)andf (•) in the frequency domain as a increases when s > N. The ratio s ∼ χ 2 (0), where χ 2 (0) is Y X N−3 N−3 spectral product. We document the additional numerical details that chi-squared distribution function with N − 3 degrees of freedom we implemented to construct the screening curves shown by Fig. 10 and a zero non-centrality parameter; the distributional form of s in text following eq. (C5). is identical for both circular and non-circular radiation patterns. Importantly, these curves demonstrate that the Case II decision Unlike the Case I test statistic, these PDFs are well defined if N rule provides a superior source-screening capability, at least for the ≥ 4 sensors rather than eight sensors. The ratio Y ∼ χ 2()isa 1 same faulting signal SNIR and thresholds required to maintain a chi-squared distribution function with 1 degrees of freedom and a false attribution probability of Pr = 10−3.WeexpecttheCaseII non-zero non-centrality (>0) under H . We use these results to FA 1 test statistic to provide such an improved, average performance be- compute the PDF of the sum Z = X + Y through a variable transfor- cause the Case I statistic additionally requires an MLE of the noise mation on the original, statistically independent random variables. variance (σ ˆ 2 ) under H ; in general, test statistics with unknown pa- Given that X and Y, respectively, have PDFs f (x)andf (y), the PDF M 0 X Y rameters show poorer screening performance than competing test f (z; H ) for Z is their convolution f (x)∗f (y) under hypothesis H Z i X Y i statistics with fewer unknown parameters (Kay 1998, p. 195). The (i = 0, 1). Omitting the explicit algebra, we symbolize the PDF variability in screening performance with sensor placement over f (z; H ) as: Z i permissible azimuths (note the 90◦ gap), however, is comparable & ∞ ds between both Case I and Case II (note shading). Both tests are also f (z; H ) = fχ2 (s(x)) · fχ2 (z − x) dx. (37) Z i − similar in the relative order of screening rates with focal mechanism. −∞ N 3 dx 1 In particular, the Case II statistic shows a qualitatively similar order- fY (z−x) fX (x) ing in screening probability when compared to the Case I statistic. We compute eq. (37) in the frequency domain with the characteristic The Case II decision rule (eq. 36) that tests radiation pattern data function method (eq. C4). The PDF fχ2 (•) includes a finite non- that resembles the Rock Valley earthquake (for example) shows a 1 centrality parameter under the alternative hypothesis (i = 1). The much higher probability of correctly screening such events from theory we detail in Appendix C demonstrates that this parameter isotropic sources than, say, data that records an event that resembles equates to the same scalar in eq. (21) that we derived from Case I. the 2003-03-13 Lop Nor earthquake. This means that completely quantifies the screening capability of Lθ (R) to test between H and H .ThePDFf (x) is identical under 0 1 X 5 DISCUSSION H0 and H1. We interpret its effect on f Z (z; Hi ) as a smoothing window in the convolution operation in eq. (37) that is independent Our binary hypothesis test (eq. 16) quantifies an observer’s idealized of the Rayleigh wave radiation pattern model. ability to screen shallowly buried explosion sources from non-VDS Fig. 10 shows the screening power of the Case II test statistic faulting sources. These tests effectively measure deviations from compared against that of the Case I test statistic, for three source circularity in Rayleigh wave radiation patterns that any faulting types and a fixed deployment constraint (an azimuthal gap of 90◦) terms induce at the source location (Fig. 1). While the binary test for N = 12 sensors. The graph format is identical to that shown by provides only a commensurate, binary decision on the significance Figs 4, 5 and 6, in which each curve represents an average (eq. 24) of any non-circular contributions to a measured radiation pattern, it 1664 J.D. Carmichael Downloaded from https://academic.oup.com/gji/article/225/3/1653/6132273 by guest on 25 September 2021

Figure 10. Screening curves for Case II (thick, solid curves) compared with Case I test statistics (thin, dashed curves). The plot format is similar to Fig. 5: Case I and Case II curves present 100%× the probability PrD (vertical axis) that eq. (18) and will screen earthquakes with three distinct focal mechanisms from an explosion, using N = 12 sensors, all plotted against faulting signal SNIR (horizontal axis; eq. 21). Each curve is an average of 100 screening curves (eq. 24) that associate with sensor deployments that uniformly sample azimuths with a 90◦ gap along the dashed circle centred at the hypocentre of the shallow source (orange star); the filled circles mark a particular, random deployment. The topmost curve (blue) presents such an average Case II screening curve for an earthquake with a focal mechanism that matches that of ‘Event 1’ from the Rock Valley earthquake; the second curve (red) presents a similar average for an earthquake with a focal mechanism like that of the Kara Sea event; and the bottom curve presents a similar average for an earthquake with a focal mechanism like that of the 2003 Lop Nor event. The shaded region indicates ± one standard deviation (±σ ) from the top curve mean. All curves correspond to PrFA = 10−3. affords significant insight into an observer’s capability to distinguish (v) Radiation pattern screening probabilities depend on three pa- source types. rameters: azimuthal gap, source radiation pattern, and the number of deployed sensors. Figs 4, 5 and 6 show that certain combinations of these three parameter sets produce qualitatively similar radiation pattern screening curves. 5.1 Case I (vi) An observer can use the screening test to estimate the az- imuthal placement of additional sensors that supplement an ex- We first consider our results under the general assumptions of Case isting network, and which maximize the probability of screening I, when σ 2 = 0: R circular from non-circular radiation patterns. Moreover, deploying (i) An observer can use azimuthally distributed receivers that additional sensors at any azimuth can never reduce screening rates. sample a noisy Rayleigh wave radiation pattern to form a test statis- (vii) Lastly, observers have a non-negligible probability of mis- tic and decision rule (eq. 18). This rule screens explosive from attributing non-circular radiation patterns of faulting sources to non-VDS faulting sources at a given threshold η that maintains a explosions when they measure these patterns with sparse sensor fixed false attribution probability Pr . deployments. The SNR of such a faulting signal increases exponen- FA ∼ ◦ (ii) We interpret this screening statistic as a ratio between radia- tially as the azimuthally gap in sensor deployment exceeds 90 . This SNR can exceed an order of magnitude as gaps increase from tion pattern energy due to non-VDS faulting and residual radiation ◦ ◦ pattern energy that is not captured by our Rayleigh wave model. 0 to 135 (Fig. 9). This test statistic has a non-central F-distribution that quantifies We next consider our (more limited) results under the general this observer’s screening capability, and is well-defined only when assumptions of Case II. an observer deploys eight or more receivers. (iii) Our test between circular versus non-circular Rayleigh radi- ation patterns is equivalent to a general Gaussian signal detection problem (eq. 18). The product of a faulting SNR term and a scalar 5.2 Case II deployment term (eq. 21) defines the test screening power. The first term measures the squared sum of the dip- and strike-slip faulting The assumptions under Case II explicitly hypothesize that an un- + 2/σ 2 known, random process superimposes with a deterministic radiation components, relative to moment noise: (DS SS) M . The de- − −1 pattern signal and imprints on the total pattern. It also assumes that ployment term A HT H 1 AT depends on azimuthal sensor stochastic variability within the radiation pattern (noise) is effec- distribution and the body wave speeds of the host medium (param- tively known under each hypothesis. This latter assumption remains eter c). less practical because the seismic noise field is often insufficiently (iv) Strike-slip sources provide the highest faulting signal stationary to be well-characterized, at least at multiple azimuths. 2/σ 2 strength M0 M for a given faulting moment M0.Eq.(18)there- We can, however, report several outcomes: fore screens strike-slip from explosive and vertical dip-slip faulting = − η, 2/σ 2 sources with the highest probability PrD 1 F1,N−3 M0 M , (i) The Case II statistic increases with Rayleigh wave en- among other sources with the same faulting moment M0 and con- ergy sourced by non-VDS faulting, relative to noise power η = −1 − , R 2/σ 2 stant false attribution threshold F1,N−3 (1 PrFA 0). ( P X M ). It is non-monotonic in residual energy that is not Hypothesis tests on radiation patterns 1665 captured by our Rayleigh wave model and normalized by noise Fourth and lastly, our noisy Rayleigh wave models impart ‘ragged ⊥ R 2/σ 2 power ( P H M ). edges’ to the graphical radiation pattern representations (Fig. 9). (ii) The screening power of the Case II decision rule (eq. (36)) de- These distortions from their deterministic patterns that are sourced pends on a scalar function of faulting signal and sensor deployment. by noise may be relatively small when compared to distortions This scalar term is identical to that under Case I (eq. 21). sourced by deterministic focusing or other scattering effects not in- (iii) A spectral product of two PDFs quantifies the screening cluded in this model (in a real Earth). Such deterministic distortion performance of the Case II test statistic. The resulting PDF does not would shape the radiation pattern from isotropic source (a pure ex- appear to have a known standard form. It’s CDF predicts that the plosion) so that this pattern would appear to have a partial ‘lobe’. Case II decision rule (eq. 36) is more effective at screening circular Our decision rules (eq. 18 or eq. 36) would then show increased from non-circular radiation patterns than the Case I decision rule Type 1 errors, that is, a higher probability of falsely deciding that (eq. 18), because it requires fewer MLEs. the data include a non-VDS faulting signal (choosing H1 when H0 (iv) The Case II test statistic is (as opposed to Case I) well defined is true). Alternatively, scattering of Rayleigh wave energy from sur- with N ≥ 4 sensors (rather than eight sensors). face topography that is comparable to Rayleigh wavelengths might homogenize the radiation pattern from a source with a considerable

Both the Case I and Case II results have implications to screen Downloaded from https://academic.oup.com/gji/article/225/3/1653/6132273 by guest on 25 September 2021 faulting signal. We expect this homogenization far from the source SPE-triggered explosion-plus-damage signals from historical fault- through energy conservation and divergence arguments: more en- ing signals that are each sourced in the NNSS testing complex. We ergy is available to be directed from antinodal azimuths to nodal suggest that an observer can use N = 12 sensors deployed around azimuths from any non-zero scattering effects. This example would the source with an azimuthal gap of 90◦ or less to screen faulting elevate Type 2 error rates, that is, the probability that our decision signals that match such historically recorded mechanisms, provided rules incorrectly decide that a tectonic source is explosive in origin these data have sufficient SNR (Fig. 10). A confirmation of screen- (choosing H when H is true); this latter example would imply that ing power from such a validation exercise remains necessary if this 0 1 eq. (18) will mistake sources with larger faulting signals than shown screening statistic holds efficacy in non-idealized settings. in Fig. 9 as explosions more often than we predict. In the language of signal detection, these collective physical processes present in a real Earth will inflate false alarm and missed detection rates over 5.3 Caveats our predicted rates. Our results remain subject to at least four more specific caveats. First, our work applied a planar model applicable to local distances that concern records from sources like the SPE or New Hampshire 5.4 A physical source of a screening error shot series. More established observational and theoretical work that quantifies the imprint of faulting motion on Rayleigh wave We explicitly treat an analytical example of an erroneous screening radiation patterns (e.g. tectonic release) considers far-regional to test that we apply to a radiation pattern that is induced by a vertical teleseismic data. Such Rayleigh waveforms are dominated by long crack (see comments in Section 4.2) opening within a stratified period spectral content and propagation distances of many hundreds half-space. This example produces a non-circular radiation pattern to thousands of km. We assert hypothesis tests with our model re- without the faulting signal that is modeled by hypothesis H1.To main applicable to such global problems. Specifically, we argue that construct this pattern, we first consider the moment density tensor we can rotate the data at each sensor location through multiplica- for a shallow displacement discontinuity (Pujol 2003, eq. 10.6.6.). tion by a unitary matrix to a locally planar coordinate system; this We then write the (full) moment tensor for such a discontinuity that idempotent matrix multiplication does not change the distributional is source by a vertically oriented surface crack of area A, located at ξ  ξ  form of such quadratic forms, nor the expected performance of each the source point 0, that opens in tension a distance u( 0) in the test statistic. Cartesian y direction (algebra omitted): Secondly, we did not explore the relative contributions of the ex- ⎛ ⎞ α2 plosion versus the CLVD source to the circular part of the radiation − 20 0 ⎜ β2 ⎟ pattern in our model. This CLVD term can exceed the explosion = ρ  ξ β2 α2 . M A u( 0) ⎝ 0 β2 0 ⎠ (38) term in size, which differs in sign so that Rayleigh waveforms re- α2 00− 2 verse polarity. While the Case I and Case II test statistics exclude β2 any such circular terms ( excludes R¯ ), we concede that a zero- Eq. (38) omits the frequency-domain representation for the source mean damage signal may appear in the data as a random process. time function that would describes the history of the crack-face Physically, this means dilatation that is symmetric about the vertical displacement and appear as a factor of M. We note that this moment axis may manifest as a late time CLVD-source. Our linear model tensor is a sum of a tensor from an explosion source, and a tensor does not accommodate such moment-dependent changes in the to- from a force couple that is aligned with the opening crack. We use tal data variance. The possible presence of such a CLVD signal the M in eq. (38) and the radiation pattern that appears in both motivates our future consideration of the Case II test statistic. eqs (2)and(3) to write the Rayleigh surface displacement u as Third, we assumed that noise recorded at distinct sensor azimuths (again, omitting the source–time function): is uncorrelated, largely for mathematical convenience. Correlated noise reduces the effective degree of freedom that parametrize the 3α2 − 4β2 u(ξ,ω) = ρ Au(ξ )β2 distribution of each test statistic. This means that our theoretical 0 α2 screening rates will exceed our observed detection rate if our method units: energy does not accommodate for noise correlation. We can make such $ % α2 cos 2ϕ accommodations by computing empirically-validated estimates for × 1 − g(ξ,ω). (39) 3α2 − 4β2 the effective degrees of freedom in observed data (Carr et al. 2020, eqs A1 and A3). This argument applies to both Case I and Case II. crack radiation pattern 1666 J.D. Carmichael Downloaded from https://academic.oup.com/gji/article/225/3/1653/6132273 by guest on 25 September 2021

Figure 11. A range of predicted screening curves (p-box) superimposed with the mean observed screening curve for the Case I decision rule (eq. 18), which we apply to a synthetic radiation pattern collected from azimuthal records of Rayleigh waves triggered by an opening surface crack (beachball at centre, right; see Section 5.4). Curves present 100%× the probability PrD (vertical axis) that eq. (18) will use N = 24 sensors to decide that a fracture source produces a non-circular radiation pattern and includes a faulting signal, plotted against radiation pattern SNR (horizontal axis). The shaded region shows a measure of the = R 2/σ 2 − σ range of theoretical screening curves. The curve that marks the lower limit of this range is parametrized by non-centrality parameter P X d M , σ σ 2 where is the standard deviation marking expected variability of estimates for , which originate from M . The curve that marks the upper limit of this range = R 2/σ 2 + σ × 4 is similarly parametrized by non-centrality parameter P X d M . The dark curve presents an average of 2 10 synthetic decision rule counts, −3 that is, the number of times that eq. (18) chooses hypothesis H1. All curves correspond to PrFA = 10 . The synthetic ‘ragged edged’ radiation patterns with σ 2 variance M at bottom associate with the labelled SNR values. White circles mark azimuthal sensor locations.

The angle ϕ in eq. (39) measures azimuth from the linear crack synthetic data (now R = Rd + n,wheren is noise) with our Case face. We pair each term with the deterministic radiation pattern I test statistic Lθ (R) and decision rule (eq. 18) over a grid of crack coefficients that associate with trigonometric basis functions in the release energies and SNR values. In particular, we considered ratios R ≤||R ||2/σ 2 ≤ = noisy description of in eq. (9): of 20 d M 400 and N 24 sensors. Fig. 11 shows that the Case I test erroneously screens opening cracks as explosions for 3α2 − 4β2 R¯ = ρ  ξ β2 low SNR, or as non-VDS faulting sources at higher SNR values. A u( 0) α2 Either decision may be interpreted as an error. R = ρ  ξ β2 · ϕ (40) 1 A u( 0) cos (2 )

R2 = 0 · sin (2ϕ)

· ρ  ξ β2 6 CONCLUSIONS AND FUTURE WORK which have units of moment (F m). The term A u( 0) repre- sents the elastic energy released by volumetric crack growth. Such The goal of this work was to determine if a discriminant for shallow sources are common in seismogenic hydrofracture events within source types that tests Rayleigh wave radiation pattern shapes is glaciers (Carr et al. 2020; Taylor-Offord et al. 2019) and ice sheets justified on theoretical grounds. We conclude that this approach (Carmichael et al. 2015). is likely not practical in most passive monitoring scenarios that We now suppose N sensors distributed at distinct azimuths along require sampling the Rayleigh wave field over a non-horizontally a circle that is centred at the source crack samples the radiation stratified medium with heterogeneities and substantial topographic pattern in eq. (40), which we store in vector Rd (subscript d in- relief (a real Earth) for two reasons. First, an observer maintains dicates deterministic). These sensor azimuths populate the system a good discrimination capability between circular and non-circular matrix H in eq. (19), and the non-centrality parameter equivalent radiation patterns only with a large number of sensors with good R 2/σ 2 to eq. (B5) becomes P X d M ; here, eq. (19)definesP X and azimuthal coverage and high SNR in a cylindrically symmetric σ 2 M is Gaussian noise variance. We then add noise with this vari- medium. Secondly, any distortion to the Rayleigh radiation pattern ance to the fracture-generated radiation pattern and process these sourced by deterministic processes (not just noise) will increase Hypothesis tests on radiation patterns 1667

Type 1and Type 2 error rates in the decision rules that provide a our tests with locally recorded data prove successful, we will begin source-type discrimination capability, over that of the ideal cases. testing it on regional, low frequency data. If our method instead In particular, our hypothesis tests on radiation pattern shape fails, then we will quantify the geophysical source of this disagree- show our tests achieve good, but idealized screening power over ment to gain physical insight into how Rayleigh radiation pattern only certain regions of parameter space that include deployment az- shape can or cannot be tested as a discriminant. imuth, focal mechanism, and sensor number. Wedemonstrated that a Rayleigh wave radiation pattern test that exploits a sufficient number of sensors (>12) with limited azimuthal gap (<90◦) to record radi- 7 DATA AND RESOURCES ation from sources with moderate strike-slip faulting signals (SNR Our theoretical study used no data. Fig. 2 provides references that >20), like the historical Rock Valley Event 1, show a high proba- document estimates for hypocentral solutions and magnitude esti- > bility PrD of success (PrD 0.9). The probability of such success mates for reference sources. MATLAB version 2019b output all com- diminishes with more ambiguous focal mechanisms that resemble putational results and plots. The corresponding author may provide certain historical, shallow earthquakes that locate near the Lop Nor scripts upon request, under the auspices of the United States De- nuclear test site in China. Our further technical developments re- partment of Energy (U.S. DOE). quire validating the screening capability of this method against SPE Downloaded from https://academic.oup.com/gji/article/225/3/1653/6132273 by guest on 25 September 2021 shots that source near historical, tectonic sources of non-circular Rayleigh wave radiation. To overcome challenges of limited explo- ACKNOWLEDGEMENTS sion data (single shots), we will perform semi-empirical tests on This research was funded by the National Nuclear Security Admin- existing explosion records. These tests will infuse amplitude-scaled istration, (NNSA). The author acknowledges important interdis- Rayleigh waveforms triggered by a known source into long noise ciplinary collaboration with scientists and engineers from LANL, records, thousands of times, and will recompute screening curves LLNL, MSTS, PNNL and SNL. We thank Dale Anderson, Pat from the resultant, semi-synthetic data. This process thereby will Brug, Leslie Casey and Brian Paeth for their support. We thank provide a set of ‘observed’ screening curves that we can directly Carene Larmat, Howard Patton, Mike Begnaud and Neill Symons compare to our predicted screening curves that test radiation pattern for content suggestions. Don Montoya constructed Fig. 1 at Los circularity. Alamos National Laboratory. We thank one anonymous reviewer In achieving the goal of this work, our hypothesis testing approach for pointing out an energy divergence argument that homogenizes did provide two unexpected, ancillary benefits to experimental plan- the radiation pattern through scattering in real Earth. We are in- ning of seismic deployments. First, we can quantify how a particular debted to Dr. Mark Fisk for a particularly thorough review. This geographical placement of additional sensors to supplement an ex- manuscript has been authored with number LA-UR-20-27299 by isting dense network can drastically increase our capability to screen Triad National Security under Contract with the U.S. Department the non-VDS faults from the explosive sources. We conclude from of Energy, Office of Defense Nuclear Nonproliferation Research this case that optimal sensor deployment does not depend on data and Development. The United States Government retains and the records, only prior sensor locations and the relative compressional publisher, by accepting the article for publication, acknowledges versus shear wave speed. Secondly, our screening statistic quantifies that the United States Government retains a non-exclusive, paid-up, the largest faulting signal that an observer could expect to record irrevocable, world-wide license to publish or reproduce the pub- and mis-attribute to an explosion source (for some probability). The lished form of this manuscript, or allow others to do so, for United size of this source increases rapidly with azimuthal gap in receiver States Government purposes. deployments that exceed 90◦. Our estimates are applicable to test- ban treaty verification scenarios that require attributing discriminant evidence to source identity, but (as stated) we must first assess if REFERENCES such a test even has marginal screening power with real data. Aki, K. & Richards, P.G., 2002. 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Transmission and reflection of fundamental-mode Rg signals from atmospheric and underground ex- plosions transmission and reflection of fundamental-mode Rg signals, APPENDIX A: DETERMINISTIC Bull. seism. Soc. Am., 108(6), 3590–3597. Patton, H.J. & Taylor, S.R., 2011. The apparent explosion moment: in- RADIATION PATTERN MOMENTS ferences of volumetric moment due to source medium damage by We compute deterministic moments of a Rayleigh wave radiation underground nuclear explosions, J. geophys. Res., 116(B3), B03310, pattern R that is produced from buried, shallow sources; we em- doi:10.1029/2011JD016518. phasize that ‘moments’ in this appendix often refer to probabilistic Pujol, J., 2003. Elastic Wave Propagation and Generation in Seismology, Cambridge Univ. Press. One Liberty St, New York, NY 10006 United moments, so we explicitly label seismic moments. The probabilistic R¯ States. moments include the (1) deterministic mean and (2) the deter- σ 2  R Richards, P.G. & Ekstrom,¨ G., 1995. Earthquake activity associated with ministic variance R of . We also bound their behavior with 2 2 underground nuclear explosions, in Earthquakes Induced by Underground their RMS values R ,andσR to measure the expected devi- 2 Nuclear Explosions, pp. 21–34, Springer. ation of the radiation pattern from R¯ +σR. We assume sensors Hypothesis tests on radiation patterns 1669 Downloaded from https://academic.oup.com/gji/article/225/3/1653/6132273 by guest on 25 September 2021

Figure A1. Radiation patterns and first two pattern moments for four different faulting geometries of tectonic release and four distinct ratios of isotropic = 1 to tectonic release seismic moments (MI/M0). Seismic moment ratio panels are separated by thick, blue lines. Each panel (e.g. top left, M0 10 MI )shows Rayleigh wave radiation patterns (black curves) with their RMS values R¯ (purple curves), superimposed with R¯ ±σR (red, dashed lines). The total seismic moment (M0 + MI) scales all radiation pattern plots. We emphasize that figure labels σR in this figure exclude  (due to lack of visibility), but differs from the variance present in an unknown random process, as considered under our Case II analyses. are deployed with equal probability around the source so that ψ is Comparing to eq. (A1), we conclude: uniformly distributed over [0, 2π]. To compute both moments, we 1 assume the source is embedded within a vertically stratified half- σ 2 = DS2 + SS2 . (A3) R 2 space and surrounded by an imaginary, unit radii cylinder to a depth equal to the dominant wavelength λ of the radiated elastic energy; σ = R , R The covariance Ri R j cov i j similarly quantifies how the current context provides little risk to confusing wavelength with samples of the deterministic radiation pattern correlate at distinct screening parameter and we therefore reuse this conventional azimuths ψ i and ψ j: symbol. Regardless of symbolism, we then integrate the squared & & 2π 2π radiation pattern R2 over our cylinder, and normalize the integrand σR R ∝ dψi dψ j Ri − R¯ i R j − R¯ j over the cylinder surface area. This process is analogous to using i j &0 &0 Gaussian surfaces in electrostatics to compute electric fields (Grif- 2π 2π = ψ ψ ψ + ψ fiths 2017, chapter 2). In our case, this integrated radiation pattern d i d j (DScos(2 i ) SSsin(2 i )) 0 0 is (from eq. 7): × DScos(2ψ ) + SSsin(2ψ ) . (A4) j j 2 ¯ 2 2 R = R +σR Products like cos (2ψ )sin (2ψ ) also integrate to zero. Therefore, i i  for i = j: λ 2π ψ R¯ + ψ + ψ 2 (A1) 0 d DScos(2 ) SSsin(2 ) = √ . σR R =0. (A5) 2πλ i j The radiation coefficient weights [1, cos (2ψ), sin (2ψ)] in eq. (A1) Radiation pattern observations collected from distinct azimuths are are mutually orthogonal over the azimuthal interval [0, 2π]. Hence, therefore statistically independent, when the prior distributions on ψ ψ cross terms such as DScos (2ψ) · SSsin (2ψ) integrate to zero. The i and j are uniform. Because this model is a mathematical ide- remaining terms reduce eq. (A1)to: alization, we concede that observations collected from sufficiently proximal azimuths are likely to be correlated. We can conclude, 1 however, that the total covariance in an observed radiation field R2= R¯ 2 + (DS2 + SS2). (A2) is representable as σ 2 I when sensor deployments are sufficiently 2 M 1670 J.D. Carmichael sparse. Fig. A1 illustrates relationships between deterministic radi- statistics that describe cylindrically symmetric ( = 0) and faulting ation pattern moments for several fault types. sources (>0). To factor into its source-dependent and deploy- − ment depend parts, we distribute the scalar A HT H 1 AT from (DS + SS)2. This factorization results in eq. (21). APPENDIX B: DERIVATION OF THE CASE I SCREENING STATISTIC We apply the hypothesis test of eq. (16) and combine eq. (13) APPENDIX C: THE DENSITY (2) FUNCTION FOR CASE II through eq. (17) to compute an equivalent GLRT Lθ (R): This appendix details the development and performance of the Case ||R − θˆ ||2 (2) H 0 R Lθ (R) (N − 3) · . (B1) II test statistic Lθ ( ) (we redundantly label the Case I statistic with ||R − θˆ ||2 H 1 the same symbol). We first note that the multivalued inverse of x(s) The MLEs for the unknown parameters are (Kay 1993, p. 252, eq. that we write as s(x) in the PDF arguments of eq. (37) is not a 8.52): proper function. Rather, x(s) = s − Nln (s) is minimized at s = N, < >

monotonically decreases for all s N, and increases for s N. Downloaded from https://academic.oup.com/gji/article/225/3/1653/6132273 by guest on 25 September 2021 θˆ = T −1 TR 1 [H H] H The Lambert function W(•) compactly represents this multivalued  − −1 inverse s(x): ˆ ˆ T −1 T T 1 T ˆ θ 0 = θ 1 − [H H] A A H H A Aθ 1 $ % (B2) − x e N 2 ||R − Hθˆ i || s(x) =−NW − , where: x > N − N ln(N)(C1) σˆ 2 = max − σ 2 , 0 N Mi N R in which the properties of W(•) constrain s(x) to be non-negative. = in which i 0, 1 indicates distinct noise variance estimates under To implement the change-of-variables in eq. (37), we write the PDF 2 (2) hypothesis i,andσR = 0wheni = 0. To reduce Lθ (R) into f X (x; Hi ) as a two-term sum over each single valued branch of the a more interpretable test statistic, we rewrite the numerator and Lambert function (i = 0, 1): denominator of eq. (B1) using projector matrices. These matrices are documented in eq. (19) and the text that immediately follows ds−1 ds0 f X (x; Hi ) = fχ2 (s−1(x)) + fχ2 (s0(x)) (C2) eq. (19). With these definitions, the Case I test statistic is a ratio of N−3 dx N−3 dx two statistically independent quadratic forms:   in which s−1(x) is the upper branch of the function in eq. (C1)  ⊥ 2 P R + P R (Fig. C1(a), solid curve) and s0(x) is the lower branch of the function (2) R = − · H X , Lθ ( ) (N 3)   (B3) in eq. (C1)(Fig.C1(a), dashed curve). We write the two derivatives P ⊥ R2 H as (algebra omitted): ⊥ $ % so that P projects onto rank-1 subspace span{X}.MatricesP − x X H e N = and P X project onto orthogonal spaces since P H X X.The Wk  ⊥ R + R2 = dsk N Pythagorean identity therefore implies that P H P X = $ %     − x (C3)  ⊥ R2+ R2 dx e N P H P X . This factorization simplifies the screening Wk + 1 statistic into a reduced form Lθ (R) that is ratio of two statistically N independent terms. We then scale Lθ (R) by the ratio of the alterna- σ 2 + σ 2 /σ 2 in which k =−1, 0. The functional form of the PDF for chi-square tive hypothesis to null hypothesis signal variance ( M R) M : statistic fχ2 (•) includes a zero non-centrality parameter under   N−3 σ 2  R2 M P X both hypotheses on radiation pattern shape. We explain this by Lθ (R) = (N − 3) ·   (B4) ⊥ 2 2 2 ⊥ 2 R /σ 2 σ + σR  R expanding the numerator of ratio P H M . First, we note that M P H the radiation pattern R is a sum of a linear model Hθ and noise. The test statistic Lθ (R) has non-central-F distribution parametrized Three of the four terms in this expansion include products of the ⊥ by scalar when Rk includes additive Gaussian noise (1 ≤ k ≤ N). projector matrix P H and H. Each of these terms are zero. This ⊥ We write its distributional dependence as Lθ (R) ∼ F1,N−3 (>0) T leaves n P H n,inwhichn is a vector of zero-mean noise with the and the cumulative, non-central F distribution for Lθ (R) as R χ 2 same dimensions as . Such quadratic forms have central N−3(0) eq. (20). Scalar has the general quadratic form when Aθ = b /σ 2 distributions when scaled by 1 M .  H 2 under 0: The ratio with quadratic form P R /σ 2 determines the PDF  X M −1 • H θ − T T −1 T θ − for fY ( ; i ). In this case, it is identical to the numerator of eq. (B4) (A 1 b) A H H A (A 1 b) χ 2 / σ 2 + σ 2 = (B5) and is distributed as 1 ( ) when scaled by 1 M R . The scalar σ 2 + σ 2 H H M R is zero under 0, and equates to eq. (B6) under 1. Given each PDF (eq. (C2)and fχ2 (•)), we compute the convolution of eq. (37) in which θ is the parameter vector under H .Appliedtothecurrent 1 1 1 numerically in the Fourier domain with the characteristic function problem in which b = 0andeq.(11)definesθ = [ R¯ , DS, SS]T, 1 method (Carmichael et al. 2020, eq. 12): this scalar becomes:  − −1 − 1 f Z (z; Hi ) = F {F { f X (x; Hi )} F { fY (y; Hi )}} . (C4) (DS + SS) A HT H 1 AT (DS + SS) (B6) − = in which F is the fast Fourier transform (FFT) and F 1 is its inverse. σ 2 + σ 2 M R The computational grid and resolution of fast Fourier transforms The scalar completely quantifies the screening capability of (fft.m and ifft.m in MATLAB) control the precision of eq. (C4). Lθ (R) to test between H0 and H1. Specifically, increasing val- Consequently, our estimateη ˆ for a threshold η that maintains a con- ues of decrease the overlap between the density functions for test stant false attribution probability is necessarily inexact. To compute Hypothesis tests on radiation patterns 1671

Figure C1. Mathematical and statistical properties of the arguments and outputs of the Case II screening statistic. (a): Both branches of the Lambert-W − x function. The upper branch s−1(•) and lower branch s0(•) of the function shown in eq. (C1), with argument −e N /N. This example uses N = 12 sensors. (b): Downloaded from https://academic.oup.com/gji/article/225/3/1653/6132273 by guest on 25 September 2021 The density function fZ (z; Hi ) for the Case II test statistic under both H0 and H1. The non-centrality parameter matches that for Case I.

ηˆ, we treat numerically unique values of the null PDF f Z (z; H0) out- when is not too large (Carmichael et al. 2020, section S2). Sources put by eq. (C4) as an independent variable and their corresponding that trigger radiation patterns with high SNR faulting signals will be grid values of z as a dependent variable. We then linearly interpolate obvious to screen, and we therefore do not consider this restriction z at the grid value for PrFA to produce the estimateη ˆ and symbolize on of practical significance. To compute screening probability these numerical operations for reference: values for each of the focal mechanisms shown in Fig. 10,weinte- grate the PDFs in eq. (C5) over a 512 point grid to construct a CDF ηˆ = interp ( f (z; H ) , z , Pr ) , (C5) Z 0 l l FA for each source focal mechanism. We then repeated this integration in which index l indicates a numerically unique grid value. over a 500 point grid of faulting SNIR values (horizontal axis). We Fig. C1(b) illustrates eq. (C4) estimates for PDFs under the compet- then estimate each screening probability by interpolating the CDF ing hypotheses. We selected = 10 under H1 and a false attribution at a screening threshold that is consistent with a false alarm value −3 = −3 probability of PrFA = 5 × 10 . The FFT inversion appears stable of PrFA 10 .