Periodic seismicity detection without declustering
PPeriodic seismicity detection without declustering
Timothy Park , Franz J. Kiraly , and Stephen J. Bourne Shell Global Solutions International B.V., 1031HW Amsterdam, The Netherlands
January 28, 2021
Abstract
Any periodic variations of earthquake occurrence rates in response to small, known,periodic stress variations provide important opportunities to learn about the earthquakenucleation process. Yet, reliable detection of earthquake periodicity is complicated by thepresence of earthquake clustering due to aftershocks and foreshocks. Existing methodsfor detecting periodicity in an earthquake catalogue typically require the prior removal ofthese clustered events. Declustering is a highly uncertain process, so declustering meth-ods are inherently non-unique. Incorrect declustering may remove some independentevents, or fail to remove some aftershocks or foreshocks, or both. These two types of er-ror could respectively lead to false negative or false positive reporting of periodic seismic-ity. To overcome these limitations, we propose a new method for detecting earthquakeperiodicity that does not require declustering. Our approach is to modify the existingSchuster Spectrum Test (SST) by adapting a test statistic for periodic seismicity to ac-count for the presence of clustered earthquakes within the catalogue without requiringtheir identification and removal.We describe the mathematical basis for this new test statistic, and the practical stepsfor applying this method to an earthquake catalogue. Using simulated earthquake cat-alogues we verify the expected performance of our Modified Schuster Spectrum Test(MSST) and compare it to the existing SST. In the absence of aftershocks, both meth-ods yield similar true positive, false positive and false negative rates, as expected. In thepresence of aftershocks, the existing SST false positive rates increase significantly whilstthe MSST false positive rates remain essentially unaffected. We conclude by applyingboth methods to two observed earthquake catalogues with previously reported evidencefor seasonal seismicity rates. The first is within the New Madrid Seismic Zone where theSST method finds statistically significant periodicities with and without declustering. Our a r X i v : . [ phy s i c s . g e o - ph ] J a n SST method finds no statistically significant evidence for any periodicity. This suggeststhat errors in declustering processes may confound periodicity tests that require after-shock and foreshock removal. Alternative tests that do not require declustering, such asthe MSST, should enable a more reliable and sensitive means of investigating seismic-ity responses to periodic stress variations. The second observed catalogue is taken fromthe Himalayas and has been reported to show evidence of seasonal activity rates afterdeclustering. We applied the SST and MSST to the data before declustering as well as theSST to the data after applying two different declustering techniques. We found that theevidence for seasonality when using the SST depended upon the parameter choices fordeclustering however with the MSST there is no ambiguity and it did not find statisticallysignificant evidence for seasonality. . Introduction Testing for periodicity in an earthquake catalogue is a common and important procedurein state-of-art seismological data analysis Hernandez (1999) - for example, in the studyof tidal / solar periodicities Heaton (1975); Tanaka et al. (2002a); Cochran et al. (2004a),hydrospheric periodicities Ader and Avouac (2013); Johnson et al. (2017a); Craig et al.(2017); Johnson et al. (2020), or blast detection Rydelek and Hass (1994). The earth-quake periodicity testing problem is an instance of the general data scientific problem oftesting for periodicity in an abstract series of events. In this manuscript, we make twomain contributions:• Presenting what is, to our knowledge, the first seasonality test that can cope withearthquake clustering, which is also a formal hypothesis test with provable guaran-tees rather than a complete heuristic,• Validating its practical use by application on a selection of commonly known earth-quake catalogues where seasonality is a question of interest.To our knowledge, all state-of-art testing procedures with formal guarantees are sub-ject to the implicit mathematical assumption of no aftershocks. Aftershocks are a phe-nomenon which is empirically well-validated, considered of practical importance, andscientifically well-studied since more than a century Utsu et al. (1995); in addition, theexistence and severity of the issue was already pointed out, in 1897, in the original paperof Schuster (Schuster, 1897, paragraph 5). As a small thought experiment illustrates, ac-tual presence of aftershocks where none are assumed can lead to both false detection andnon-detection of periodicity: if aftershocks occur on a very short time scale comparedto a yearly testing period, all shocks (incorrectly considered primary shocks) cluster inmonths and may create an illusion of seasonality; whereas if aftershocks occur on a timescale which is long compared to a daily testing period, periodic primary shocks may notbe detected between aftershocks (incorrectly considered primary shocks) spreading moreevenly across the day. More formally, in a periodicity testing procedure, the assumptionof no aftershocks may lead to type I (false positives) and type II (false negatives) errors,caused by null hypothesis mismatch.Current common techniques for detecting earthquake periodicity all rely on core test-ing procedures which assume that there is no earthquake clustering, such as based onthe Schuster test Schuster (1897); Ader and Avouac (2013), after prior heuristic removalof clustered events from the catalogue ensuring applicability of the core testing proce- ures. Here, “heuristic” means that these prior procedures come with no guarantees ofcorrectness, e.g., that no strong bias is introduced. Hence, the state-of-art suffers from se-vere limitations , as declustering is an imprecise procedure that relies on judgement-basedchoices (Molchan and Dmitrieva, 1992; Zaliapin et al., 2008) that typically vary betweenapplications, likely bias the periodic signals that are sought, and weaken any scientificconclusions based on them to the strength of the weakest link in the argumentation chainwhich is typically, declustering choices. The method of Dutilleul et al. (2015) is one exam-ple which can be applied to data without declustering however this is a heuristic methodand is applied to monthly event counts rather than event times.Our proposed testing procedure, MSST, succeeds in removing the strong assumption ofno clustering from the core testing procedure, in favour of a much weaker assumption ofno secondary or higher-order aftershocks. As secondary aftershocks typically account onlyfor a very small fraction of shocks, or can be removed much more easily, this is much less ofa restriction than the far stronger assumption of no clustering, and much less of a problemthan heuristic testing procedures without any proper mathematical guarantees. Thereforeour testing procedure is significantly less susceptible to errors in periodicity detection dueto clustering than existing state-of-the art procedures. Furthermore, we argue that theassumption of no secondary aftershocks is not a restriction in-principle, as it can also betreated in our mathematical framework - though at the cost of substantial mathematicaloverhead in this manuscript which we chose to avoid in favour of readability and clarityof exposition. We leave this for future work. We discuss the two contributions in moredetail below. We discuss our methodological contribution in the context of previous methodologicaladvances. Historically, abstract hypothesis tests for periodicity in event sequences areclosely linked with the seismology application. The most common approach in seismol-ogy applies frequentist hypothesis tests to the event sequence or a derived time series,see Hernandez (1999) for an overview. To our knowledge, this is also the only kind offramework to which formal guarantees have been derived, i.e. , guarantees on type I andtype II errors, or guarantees on quantifiers of evidence arising from the testing procedure.Most variants rely on procedures closely related classical work of Schuster and Fisher Schus-ter (1897); Fisher (1929). Differentiation occurs through the specific approximation ofthe statistic, and whether testing is for a specific, pre-determined period, or multiple eriods at once. The methodological backdrop is somewhat subtle - there are multiplenon-identical procedures which are called “the Schuster test” in literature; and the pro-cedure known under this name today was first introduced by Heaton Heaton (1975) in1975 and is not identical with the original procedure by Schuster and Fisher. Further, toour knowledge, no formal proof for properties of this latter Schuster test exist, thoughthey are not difficult to adapt from the original work of Schuster and Fisher - which wecarry out as a subsidiary contribution in this manuscript.We provide a more technical discussion of the data scientific literature in Section 2,as making precise the differences in methods and discussing the trail of ideas requiresintroduction of some mathematical notation. For our main contribution, it is importantto note that all testing procedures discussed above assume that there is no clustering,i.e., the methodological framework of an inhomogeneous Poisson process. Treatment ofclustering, in literature, is predominantly heuristic, e.g., reliant on windowing or boot-strap simulation procedures without any formal contraposition, null hypothesis, or typeI / II guarantees.Our proposed method, MSST, is to our knowledge the first method to provide suchguarantees in the case of a general process with aftershocks. It is based on theoreticalresults on asymptotic properties of a first-generation inhomogenous Poisson process withsecond-generation aftershocks. The null hypothesis is that the first-generation processis homogeneous, and type I / II guarantees (more precisely: control of type I error givenfixed type II error) is obtained in the common frequentist testing framework of asymptoticapproximation. This contains an additional semi-heuristic step of assuming sufficientregularity in the Schuster spectrum. Technical details and a summary of assumptions arepresented in Section 3.
Recent advances in earthquake data processing using methods such as template matching(Yoon et al., 2015; Li and Zhan, 2018) and convolutional neural networks (Perol et al.,2018; Ross et al., 2019) mean the quality and size of earthquakes catalogues are growingrapidly. This creates new opportunities to detect the amplitude and phase of small peri-odic variations in earthquake nucleation rates driven by cyclic stresses within the earth’scrust. Such observations are important as constraints on frictional fault failure processes(Beeler and Lockner, 2003; Ader et al., 2014), as indicators of mean stress changes Tanakaet al. (2004); Ogata (2005, 1999), and as indicators of proximity to failure in a future arthquake Tanaka (2012).There are multiple sources of cyclic stress that load seismogenic faults: hydrospheric,atmospheric, thermal, tidal, solid Earth tides (e.g. Johnson et al., 2017a,b; Heki, 2003;Tolstoy et al., 2002; Métivier et al., 2009), and anthropogenic activities such as seasonalvariations in water reservoir levels (Simpson et al., 1988; Talwani, 1997) and natural gasextraction, such as the Groningen field (Bourne et al., 2014; Bourne and Oates, 2017;Bourne et al., 2018). For semidiurnal periods, micro-earthquakes regions of seafloorvolcanic and hydro-thermal activity do correlate with ocean tides (Wilcox, 2001; Tolstoyet al., 2002; Stroup et al., 2007), as do shallow tectonic earthquakes (Tanaka et al., 2002b;Wilcock, 2009). However the evidence for seismicity triggered by semidiurnal solid Earthtides is mixed; with reports of weak to no correlations (Shudde and Barr, 1977; Heaton,1982; Vidale et al., 1998; Beeler and Lockner, 2003), strong positive correlations forall focal mechanisms (Métivier et al., 2009) or only thrust-fault mechanisms (Cochranet al., 2004b), or only normal-fault mechanisms (Tsuruoka et al., 1995). Tanaka reportssignificant triggering of earthquakes by semi-diurnal solid Earth tides prior to the 2011 M w reep experiments do show a similar transition under cyclic loads (Chanard et al., 2019),but the scaling of this result to geological faults is uncertain.Methods for detecting earthquake periodicity without declustering would be very use-ful to reduce bias and to improve both temporal and spatial resolution of periodic seis-micity responses to stress changes. In this paper we present what we believe are the firsttheoretical results on the Schuster test statistic in the presence of aftershocks. We presentsome experiments on synthetic data and a real earthquake catalogue, corroborating thatthe impact of omitting the important assumption is not only practically meaningful, butpotentially very serious. We also discuss how to diagnose the severity of the missingaftershock assumption for a given seismological use case, as well as possible pathways(subject to potential alternative assumptions) for adapting the Schuster test to real worldearthquake catalogues with aftershocks. Technical details and assumptions are describedin Section 3. The manuscript is structured as follows: Section 2 gives a brief description of the existingSchuster spectrum algorithm. In Section 3 we present a theoretical description of theeffect of aftershocks on the test statistic for the Schuster spectrum test. We then describea method for correcting the test to account for this effect by making use a a spline basedfit to estimate the unknown aftershock effects. Finally in Section 4 we show the results ofapplying our modified test to a range of simulated earthquake catalogues and to two ob-served example of seasonal seismicity due to hydrospheric loading from the New Madridregion of North America (Craig et al., 2017) and the Himalayas (Ader and Avouac, 2013).
2. Prior Work on the Schuster test
In literature, somewhat inconsistent use of terminology surrounds “the Schuster test”, asin referring to procedures that are closely related mathematically and algorithmically, butnot exactly identical. While this has constituted no severe issue in the relevant applica-tions, it makes a precise literature discussion difficult as multiple terms are used for mul-tiple non-identical procedures, interchangeably. As the latter issue arises from somewhatsubtle, mathematical points, we start by briefly introducing some necessary mathematics. he procedure which we in this manuscript refers to as “the Schuster test” is a test forperiodicity in an event sequence , i.e., a series of observed event times T , . . . , T N (presentedin uniformly random order). Periodicity at a radian frequency ω is tested through aneffect size called “squared Schuster distance”, defined as D ( ω ) : = | D ( ω ) | , where D ( ω ) : = N (cid:88) i = exp ( ı ω T i ) is the “Schuster distance”. Under the null hypothesis of no periodicity, exp ( ı ω T i ) is uni-formly distributed on the complex unit circle, and it can be shown that D ( ω ) is approx-imately Exp ( N ) -distributed, giving rise to a p-value-like test statistic, testing against thealternative of periodicity where D ( ω ) is larger. A detailed discussion with explicit proofsand derivations can be found in Appendix D. There may be small variations in how thefinal p-value is computed, e.g., through a choice of asymptotic approximation, therefore,formally, there are multiple possible variants of this “Schuster test”.A second, closely related method is called the “Schuster / Fisher procedure”, or the“Fisher g-test”. This is a test of periodicity in a time series , i.e., a series of observed realvalues X , . . . , X N , at pre-determined times t , . . . , t N ∈ (cid:82) , which are equally spaced, i.e.,all the differences t i − − t i , i = N − ω , that is. E ( ω ) : = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) i = X i exp ( ı ω t i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) .Under the null hypothesis of no periodicity, it can be shown that E ( ω ) is approximatelyexponential with known mean, giving rise to a suitable test statistic. In addition, if the X i are normally distributed, an exact statistic is available - “Fisher’s g” (which otherwiseonly yields a good approximation).As time series are not the same as event series, the two testing procedures are notidentical - however, they are closely related. If, starting at the symbols in “the Schustertest”, the times T i are all replaced by the closest (“rounded”) t j , and X j is defined asthe number of T i that are replaced by t j , the Schuster distance of the rounded eventseries, and the Fourier spectrum energy of the time series of counts X j observed at t j ,are mathematically identical. Conversely, starting with symbols in “the Schuster / Fisherprocedure”, if all the X j are integers, one can produce an event series where t j appears X j times, and the same identification holds in reverse. Thus, the “event series” setting canbe interpreted as the limit of infinitesimally small bins, arising from a time series of bin ounts. It is also interesting to note that, under these identifications, the mathematicalargumentation, derivations of test statistics and p-values and their properties, largelycoincide.In below discussion, we will refer to the Schuster test or Fisher g-test procedure (noquotes) when referring to the mathematical class of procedures as discussed above. Wewill refer to “the Schuster test”, “Schuster / Fisher procedure”, or “Fisher g-test” whenreferring to the terms, not the procedures, sparingly and only insofar it is necessary forthe discussion.
The Schuster test (as defined in Section 2.1 or in Appendix D), seems to first appear asan original idea in Heaton (1975) as a heuristic algorithmic procedure without proof, ci-tation of earlier sources, or mention of Schuster, or Fisher, or any Schuster / Fisher relatedterminology. The seminal papers of Schuster Schuster (1897) and Fisher Fisher (1929),frequently cited as original sources for “the Schuster test”, introduce in fact not the Schus-ter test, the test for periodicity in event sequences, but the Fisher g-test, that is, the testfor periodicity in time series. Fisher Fisher (1929) greatly extends Schuster’s originalwork Schuster (1897), formulated for time series of counts (arising from earthquakes).Confusingly, despite the Schuster test not appearing in the original work of Schuster andFisher, or any other work of Schuster, most contemporary sources refer to the Schuster testindeed as “the Schuster test” (using Schuster’s name, “Schuster”, as an epithet). Whilethe Fisher g-test has also found applications in many domains outside geoscience, use ofthe Schuster test is mostly localized to geoscience - although, confusingly, both have beenhistorically, and recently, applied to important applications in seismology.As regards formal mathematical proofs, e.g. , on the distribution of the test statisticand its asymptotics, Schuster’s and Fisher’s original work Schuster (1897); Fisher (1929),as well as subsequent theoretical work, seems to focus entirely on equally spaced timeseries, i.e., the Fisher g-test. In contrast, the Schuster test appears in literature mostly asan algorithm and heuristic, with literature frequently (and incorrectly) citing Schuster’sand Fisher’s work (on a different test, the Fisher g-test) as a theoretical justification. Tothe best of our knowledge, our Appendix D is the first complete formal treatment of theSchuster test, its properties, and the first proper justification for its use in its event seriessetting - where, of course, the argumentation is, as expected, largely but not entirelycongruent with the parallel results for the Fisher g-test. ne additional noteworthy area of research and results is the treatment of the multipletesting problem, arising from testing for “some periodicity” (across many radian frequen-cies ω ) as opposed to “on specific periodicity”. While this is now relatively classical for theFisher g-test, see for example Brockwell and Davis (2013), developments for the Schus-ter test are relatively recent, with prevalent miss use in the geoscientific domain, as forexample carefully highlighted by Ader and Avouac Ader and Avouac (2013). In the samearticle Ader and Avouac (2013), a multiple testing correction procedure is suggested forthe special case of sinusoidal variations, but this is not generally valid for all periodicvariations.
3. The Schuster Test Statistic in the Presence ofAftershocks
In this section we describe our proposed aftershock correction for the Schuster test, basedon derivations in the case of a relatively general aftershock model. We start in Section3.1 with the definition of such a process which will be the basis of our theoretical results.In Section 3.3 we present some theoretical results based on this process. These resultsdemonstrate analytically why the standard SST is not suited to the case where there aresignificant aftershocks present in the earthquake catalogue.
For the purpose of this study, we adopt the following mathematical model as an aftershockprocess:• We observe shocks at times T , . . . , T N , all at some time in [ ∞ ) . There is at mostone shock at a given time.• All shocks are either one of M primary shocks, or one of A i aftershocks of the i -thprimary shock. In particular, this implies that N = M + A + . . . A M .• The precise labelling of observed shocks, i.e., whether a shock is a primary shockor an aftershock of the i -th primary shock, is a-priori unknown to the observer.• Aftershocks are always an aftershock of one and only one associated primary shock.Aftershocks themselves have no aftershocks.• There is a fixed observation window [
0, 1 ] for the catalogue (which can be re-scaledor shifted), in which all the primary shocks are observed. Aftershocks, but not rimary shocks, may lie in [ ∞ ) . This is a plausible assumption if the typicalaftershock occurs much closer to their primary shocks than the observation windowis wide. We will discuss and remove this simplifying assumption later on.• The primary shocks are assumed to be sampled from a (possibly heterogeneous)Poisson process, henceforth referred to as the primary shock process , assumed tohave some intensity functional λ (vanishing outside the observation window [
0, 1 ] ).• All aftershocks associated to one given primary shock are assumed to be sampledfrom a (possibly heterogeneous) Poisson process, henceforth referred to as the af-tershock processes .• The statistical dependency between aftershocks is mediated entirely through theirrespective primary shocks; that is, conditional on primary shocks (being known / ob-served), different aftershock processes are statistically independent.• Aftershock processes are assumed causal, i.e., have zero intensity at time beforetheir associated primary shock (that is, the intensity functional λ for the aftershockprocess of a primary shock T i satisfies λ ( t ) = t < T i ).• Aftershock processes are assumed identical in distribution and intensity, on a shiftedtime axis which has the associated primary shock at time zero (that is, the intensityfunctional for the i -th aftershock process, of a primary shock at time T i , is assumedto be t (cid:55)→ λ ( t − T i ) for some functional λ , where the dependency on the primaryshock is through T i and through T i only).A mathematically formal exposition of this setting is found in Appendix E.1. Formally,the process defined through the assumptions above is equivalent to a Hawkes processwhere aftershocks cannot have aftershocks. Hence our setting constitutes a simplification,but it is sufficient to highlight some key phenomena in the presence of aftershocks. The complex unit will be denoted by the symbol ı . Exponential distributions will bedenoted as parametrised by inverse rate or decay parameter, i.e. , Exp ( λ ) denotes theexponential distribution with expectation λ . We continue analysing, as a well-known test statistic, the squared Schuster distance. Thisis defined as D ( ω ) : = | D ( ω ) | , where ω is some radian frequency, and D ( ω ) is the chuster distance, defined as D ( ω ) : = (cid:80) Ni = exp ( ı ω T i ) . Often, D is also seen parametrisedby period length k : = πω instead of frequency ω .Under the common null hypothesis of no seasonality overall, the primary intensityfunctional λ is constant, i.e., λ ( t ) = ν for any t ∈ [
0, 1 ] and some ν ∈ [ ∞ ) . Forseasonality at a specific radian frequency ω , the nulls is weaker - in this case, it is assumedonly that D ( ω ) behaves as if λ were constant, i.e., any angle exp ( ı ω T i ) is uniform onthe complex unit circle. Under this null hypothesis, the following holds: Proposition 3.1.
Assume that the primary aftershock process (as specified above) has noperiodicity at frequency ω , that is, exp ( ı ω T i ) ∼ Unif { x ∈ (cid:67) : | x | = } , for all i.As above, denote by λ the aftershock intensity functional, and by ν is the total rate constantof the aftershock process, i.e., ν = (cid:82) ∞ λ ( t ) d t . Denote by χ the characteristic function to the pdf ν − λ ; equivalently, χ ( t ) : = ν − ( F λ ) (cid:16) − t π (cid:17) , where F λ is the Fourier transform (unitary normalization convention) of the aftershockintensity functional λ . Then:(i.a) it holds for the expected Schuster distance that (cid:69) [ D ( ω )] = (i.b) it holds for the variance of the absolute Schuster distance that Var [ | D ( ω ) | ] = Var [ D ( ω )] = (cid:69) [ D ( ω )] (ii) it holds for the expected squared Schuster distance spectrum that (cid:69) [ D ( ω )] = ν · ν + ν · | + ν · χ ( ω ) | . (1) (iii) if the numbers M of primary shocks and A , . . . , A M of aftershocks are considered fixed,the expected squared Schuster distance (conditional on observed numbers) can be ex-pressed as (cid:69) [ D ( ω ) | M , A , . . . , A M ] = ( N − M ) (cid:0) − | χ ( ω ) | (cid:1) + M (cid:88) m = | + A m · χ ( ω ) | . (2)A proof can be found in Appendix Section E.4, where this Proposition 3.1 appears aspart of Proposition E.2.The term (cid:0) − | χ ( ω ) | (cid:1) is also closely related to the variance of the aftershock process: Lemma 3.2.
Let Z be a random variable distributed according to the pdf ν − λ , as consideredin Proposition 3.1. Then the following hold: i) Var ( exp ( ı ω Z )) = − | χ ( ω ) | (ii) Var ( exp ( ı ω Z )) ≤ ω Var ( Z ) . Proof. (i) is a consequence of Lemma C.4 (i) in the Appendix; (ii) is a consequence ofProposition C.5 (v) in the Appendix.(for convenience, variance of a complex random variable, as used in Lemma 3.2 (i),is introduced and discussed in Appendix B)
Without further methodology, Proposition 3.1 is not practically usable: the objects ν , ν , χ , F λ are unknown to the observer, and are in general not straightforward to estimatewithout knowledge of which shocks are primary or aftershocks. Before proceeding withhow the above is usable for a correction to the Schuster test, we would like to pointout how the quantities are physically interpretable and obtainable from low-assumptionplug-in estimates:(i) ν is the expected number of primary shocks, and νν is the expected number ofaftershocks. As the observed number of shocks, N , is a realisation of the sum ν ( + ν ) , plug-in estimates (cid:98) ν : = γ N and (cid:98) ν : = ( γ − − ) may be obtained for any assumedor estimated fraction of primary shocks γ .(ii) It holds that ν · χ ( ω ) = ( F λ ) (cid:0) k − (cid:1) , the Fourier transform of λ . The latter describesphase and energy of the aftershock process (kernel). This may be directly obtainedfrom an assumed or estimated explicit form of the aftershock process (kernel). It isclosely related to variance of the aftershocks (and aftershock phasors) as discussedin Lemma 3.2, also see Lemma C.4 and Proposition C.5 in the appendix. If it isknown which shocks are aftershocks of the same shock, this term can thus also beobtained from sample variance estimation. We illustrate the impact of aftershocks on the original Schuster procedure, and thus pos-sible corrections to it, by considering a simple example in the context of Proposition 3.1.Within observational seismology, temporal clustering of aftershocks was described by thelong-standing Omori law Omori (1894), λ ( t ) = K ( t + c ) − , where K and c are constantsinferred from earthquake catalogues. More recently this was generalized by the ModifiedOmori law Utsu and Ogata (1995), λ ( t ) = K ( t + c ) − p . The constant c is typically smaller han 1 day, and the constant p varies in the range 0.9–1.9. This inverse-law parame-terization of the aftershock model does not lead to simple algebraic expressions withinthis analysis. For simplicity, we will illustrate our method with exponentially-distributedaftershocks as an initial approximate representation of the Modified Omori law. We an-ticipate these results may be readily extended to include the Modified Omori law as itsFourier transform also has a closed-form expression, albeit somewhat more complex.Consider the aftershock model with an exponential form of λ ( t ) :Let λ ( t ) = A τ · e − t /τ · [ t ≥ ] , with parameters τ , A ∈ (cid:82) + . Then, ( F λ )( ω ) = A ωτ + ı , ν = (cid:90) ∞ λ ( t ) d t = A , (cid:69) [ D ( ω )] = ν · A + ν · (cid:12)(cid:12)(cid:12)(cid:12) + A ωτ + ı (cid:12)(cid:12)(cid:12)(cid:12) , (cid:69) [ D ( ω ) | M , A , . . . , A M ] = N − M + ( ωτ ) − + M (cid:88) i = (cid:12)(cid:12)(cid:12)(cid:12) + A m ωτ + ı (cid:12)(cid:12)(cid:12)(cid:12) (3)For illustration of the impact of aftershocks, we consider some corner cases: No Aftershocks:
In the case where the catalogue does not contain any aftershockevents, we should recover well-known properties of the SST procedure. Mathematically,no aftershocks means A =
0, and A i =
0. In this case E [ D ( ω )] = ν , which is the mainshock rate constant, with the natural plug-in estimate in this case being N = E [ D ( ω ) | M ] ,the total number of earthquakes observed. This is the expectation used in the SST proce-dure, as expected. Long characteristic aftershock duration:
This should intuitively agree with a sit-uation of “maximal decoupling”, i.e., for all practical purposes aftershocks are unrelatedto the primary shocks. Mathematically, if the characteristic duration (also known as theintensity decay constant) of aftershocks, τ , is very long as compared to the period, k ,the intensity function λ will become increasingly flat. In the limit τ → ∞ , the Fouriertransform, F λ will approach a delta function supported at 0, for finite values of k . Inequation (1) the limit of the squared Schuster distance becomeslim τ →∞ E [ D ( ω )] = ν · A + ν · = ν · ( ν + ) (4)which is simply the expected number of shocks, overall. s one may intuitively expect, one recovers the case of no aftershocks when consideringaftershocks as primary shocks (as one should, since they are in fact completely decou-pled). Again, N is the natural plug-in estimate. Short aftershock duration:
This should intuitively agree with “counting the primaryshock with multiplicity”, as aftershocks are infinitesimally instantaneous and virtuallyindistinguishable from the primary shock. Mathematically, if the characteristic durationof aftershocks, τ , is very short as compared to the period, k , the intensity function λ ( t ) approaches A times a delta function. In this case the Fourier transform F λ will approacha constant function with magnitude of A . More formally, the limit of equation (1) can beshown to be equal to lim τ → E [ D ( ω ) | M , A , . . . , A M ] = M (cid:88) i = ( + A m ) , (5)which corresponds to the case without aftershocks, but counting each primary shock withweight ( + A m ) accumulated from counting all its aftershocks as part of the primary shock.Outside such corner cases, that is, for a general choice of λ ( t ) , it is not easy to simplifyequation (1), and therefore the expected value of D ( ω ) will vary with frequency. Thoughthe important point to note is that the transition behaviour between the (long and shortaftershock period) corner cases is continuous, as Equations 3 quantitatively confirm. Thatis, as the aftershock intensity becomes more peaked, the effective multiplicity with whichprimary shocks count grows; whereas, as aftershock intensity becomes flatter, the effectivenumber of primary shocks increases.If the actual primary shocks are correctly identified as such in the theoretical model,approaching the former extreme (“short aftershock duration”) causes type I errors, i.e.,detecting periodicity where there is none. Qualitatively, approaching the latter extreme(“long aftershock duration”) causes type II errors, i.e., not detecting periodicity wherethere is.Interestingly, the “short aftershock duration” corner case was also already discussedin the original manuscript of Schuster (Schuster, 1897, paragraph 5) (for the closelyrelated Schuster / Fisher procedure, see Section 2.2) - as a criticism to, or response to,coefficients derived by Knott KNOTT (1888) about a decade earlier, alleging what inmodern terminology are type I errors. .6. Modification to the Schuster Spectrum Test An aftershock corrected Schuster test may be derived from comparing the (large sample)distribution of the Schuster spectrum in the presence and absence of aftershocks:
Proposition 3.3.
Assume the null hypothesis of no periodicity holds, that is, the primaryshock process is a homogeneous Poisson process (the intensity functional λ is constant).Then, the following approximations hold:(i) if there are no aftershocks, then N D ( ω ) ≈ Exp ( ) .(ii) in the general case, N D ( ω ) ≈ Exp (cid:0) − κ − + κ − | + ν · χ ( ω ) | (cid:1) , where κ = ν + is the expected number of shocks per primary shock.(both in the sense of LHS converging in distribution to RHS as M → ∞ , with λ fixed,conditioning on M, and M increasing.)Proof. (i) follows from the discussion in Appendix D.3.(ii) follows from the discussion in Appendix E.5.As per Proposition 3.3 (ii), we note that the scaling constant of the Schuster spectrum,equal to the parameter of the Exp distribution versus ω , is no longer known a-priori, dueto χ ( ω ) and κ being unknown. In addition, it has a dependence on ω .This means that, in the presence of an aftershock process, it is no longer appropriate tonormalise the Schuster periodogram by the number of earthquakes N .If it is possible to say with certainty which earthquakes are main shocks and which areaftershocks, χ and κ may be directly estimated, as outlined in Section 3.4. Otherwise istherefore not practical in most cases to use equation (1) directly, especially as the labellingof primary vs aftershocks may not even be identifiable.A semi-heuristic correction based on the above theoretical insights may neverthelessbe derived, subject to the following (semi-heuristic) assumptions:(a) the observation window is large, in comparison to the aftershock process (mathe-matically: Var ( Z ) (cid:28) Z as in Lemma 3.2)(b) χ varies smoothly in ω , and deviations from the null appear only in isolated peaks.(c) D ( ω ) , for different close-by values of ω , is close to being independentSubject to the above, we can estimate the rate parameter of the Schuster spectrumby regression on the sample estimate of N D ( ω ) against ω , i.e., subject to smoothnessregularization on ω . Periodicity will then appear as an outlier in this ω . e opt for using an off-shelf class of regression methods with the above behaviour, robustquantile regression, regressing at the ( − exp ( − )) -quantile which is always identical withthe mean of an exponential distribution.The full algorithm for our modified Schuseter spectrum test is as follows: Algorithm 1:
Modified Schuster Spectrum Test
Input : occurrence times, t , . . . , t N periods to consider, k , . . . , k K Output:
Schuster p-values p , . . . , p K for these periods compute D j ← (cid:12)(cid:12)(cid:12)(cid:80) Ni = exp (cid:128) π t i k j (cid:138)(cid:12)(cid:12)(cid:12) , for j in {
1, . . . , K } ; initialize quantile regression model QRM , with quantile 1 − exp ( − ) ≈ fit QRM to sample of covariates k , . . . , k K , targets D , . . . , D K ; use QRM to predict quantiles (cid:98) D , . . . , (cid:98) D K at k , . . . , k K ; compute p j ← exp (cid:128) − D j / (cid:98) D j (cid:138) , for j in {
1, . . . , K } ; return p , . . . , p K The algorithm, as described, returns unadjusted p-values, p i being a p-value for peri-odicity at period length k i , subject to known asymptotics and heuristic assumptions made.This is computed for an earthquake catalogue with observation times t , . . . , t N .In set-up, our algorithm follows closely the SST procedure described in Ader andAvouac (2013) - both compute the p-value at a frequency ω based on the exponentialasymptotic of the Schuster D , normalizing by its expectation / rate parameter D ( ω ) . Themain difference arises through the frequency dependency of D ( ω ) , see Proposition 3.3.We address this by estimating the normalization factor (cid:98) D j , then use this in line 5 in placeof the factor N in the original SST algorithm.As explained in the discussion above, we opt to conduct this estimation by a quantileregression algorithm, on the calculated Schuster spectrum, in lines 2 to 4. This strategyseems to work well with off-shelf quantile regressors, if the periods for the spectrum areequally spaced in frequency (not length) domain.In our own implementation, for the quantile regression method QRM in the algorithm,we use the B-spline quantile regression algorithm from the R cobs package Ng andMaechler (2017), version 1.3-4, which implements the quantile regression method de-scribed in Ng and Maechler (2007). We interface the cobs quantile regressor subjectto the constraint that for very short periods the expected value of D j approaches N , asderived in Section 3.3. Otherwise, we run cobs with unchanged default parameter set- ings. In principle, further constrains can be placed on the spline fit, for example if partialinformation is know about the underlying physical process, however we have chosen notto apply any such constraints to keep this method as general as possible.In all our experiments below, the period length k , . . . , k K are taken to be equallyspaced in the frequency domain, with k = k K = K = (cid:108) k − K − k − T N − T (cid:109) ,following Ader and Avouac (2013). That is, k − i = k − + i · k − K − k − K − for i = K .Significances in the reported results will be subject to multiple testing correction ifmultiple periods lengths are considered. For simplicity, we apply Bonferroni correctionwhich is a procedure developed by Dunn (1961). More precisely, if we apply our test to K different periods, then the corrected p-value for period length k j is K · p k . We choose thismethod due to its simplicity in interpreting the graphically presented results and scru-tinizing the impact of the correction; of course other, more efficient methods to controlfamily-wise error rate (such as Holm / Bonferroni or Hochberg) could be applied here.
A demonstration of the MSST on simulated data can be seen in Figure 1. In this plot wehave simulated earthquake catalogues which include aftershocks. The aftershock processfollows that defined in equation (3) with τ equal to either 1 × − years, 1 /
12 years or10 years. This represents the extremely short and long aftershock durations discussed inSection 3 as well as a medium duration which is more typical of what is seen in real data.Here we have set the parameters ν =
500 and ν =
3, this corresponds to an expectedvalue of N of 2000. The periods we consider range from a minimum of 1 day up to 5years.Figure 1 shows the results of this simulation. We can see from these plots that ingeneral the true value of the Schuster periodogram lies within the 95% uncertainty rageof our estimate. We do note that for very long periods the uncertainty range becomesvery large, this is due to the spline fit being poorly constrained in this region due to alack of observable periods. This could be reduced by either adding some constrains, asdiscussed above, or where possible increasing the observation period. We also note thatfor long periods the mean of the simulated periodograms deviates form the theoreticalexpected value given by equation (1). This is likely due to a finite sample effect causedby the finite observation window. While this is noted we are as yet unable to properlyquantify this effect. D k for 1000simulated earthquake catalogues. The black points show the average of the calculated valuesof D k over the simulations. The green line shows the theoretical mean of D k as calculatedfrom equation (1). The solid red line shows the mean of the fitted values obtained by fitting aspline to each of the 1000 simulations, the dashed red lines show the pointwise 95% quantilesof the fitted values. 19 . Results In this Section we present some results of applying the MSST to both simulated datain Section 4.1 and to real data in Section 4.3. In both cases we present the results ofour method alongside those of the current SST. In the case of simulated data this allowsus to understand the impact of aftershocks, and to investigate how each method is ableto perform in a situation where we know exactly the underlying earthquake generatingmodel and therefore we know what the “optimal” results are. In the case of real data wedemonstrate how the two models perform outside of the idealised setting of simulateddata.
In this section we present the results of applying the MSST to simulated data. We describesimulation procedures for four different data generative scenarios (A,B,C,D), which arelater used to generate independent replicates of experiments, in which one particularscenario is independently re-sampled. One replicate, in each of the four scenarios, isgenerated in the following way: (A)
No Aftershocks and No Seasonality:1. We simulate a catalogue with N earthquakes where N is taken from a Poissondistribution with mean ν = (B) Aftershocks and No Seasonality:1. The number M of primary shocks is sampled from a Poisson distribution withmean ν = A m aftershocks, where A m are sampled i.i.d. from a Poisson distribution with mean ν = τ = (C) No Aftershocks and Seasonality: . The total number of shocks N is sampled from a Poisson distribution with mean ν = + α sin ( π t / t s ) , as a function in t , for t ∈ [
0, 50 ] years, where α = t s = (D) Aftershocks and Seasonality:1. First the number M of primary shock is sampled from a Poisson distributionwith mean ν = + α sin ( π t / t s ) , as afunction in t , for t ∈ [
0, 50 ] years, where α = t s = A m aftershocks, where A m are sampled i.i.d. from a Poisson distribution with mean ν = τ = τ = evel and so would expect a false discovery rate of 5%. The results of this are shown inTable 1. Looking at this table we see that for the case of no aftershocks both methodshave a false discovery rate of just above 7.4%. For the case where there are aftershocksour method gives a very similar false discovery rate, however the current method showsa much higher false discovery rate of close to 97%. Table 1: False positive rates for the current Schuster spectrum test and our new method. Thisis calculated for 1000 simulations both with and without aftershocks.Aftershocks SST MSSTNo 0.063 0.074Yes 0.971 0.07022 .005 0.020 0.100 0.500 2.000 + − − − No Aftershocks − SST
Period (years) P − v a l ue llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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+ − − − No Aftershocks − MSST
Period (years) P − v a l ue 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llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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+ − − Aftershocks − MSST
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llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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− − − Seasonality − MSST
Period (years) P − v a l ue llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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− − − Aftershocks and Seasonality − MSST
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llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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Figure 2:
Each plot shows the results of applying either the standard SST, left hand column, or ourMSST, right hand column, to single simulated catalogues from four different scenarios. In each casethe y axis shows the p-value for each period considered. The black line shows the Bonferroni corrected5% significance threshold for K = .005 0.020 0.100 0.500 2.000 + − − − No Aftershocks − SST
Period (years) P − v a l ue + − − − No Aftershocks − MSST
Period (years) P − v a l ue + − − Aftershocks − SST
Period (years) P − v a l ue + − − Aftershocks − MSST
Period (years) P − v a l ue − − − Seasonality − SST
Period (years) P − v a l ue − − − Seasonality − MSST
Period (years) P − v a l ue − − Aftershocks and Seasonality − SST
Period (years) P − v a l ue − − Aftershocks and Seasonality − MSST
Period (years) P − v a l ue Figure 3:
Each plot shows the results of applying either the standard SST, left hand column, or ourMSST, right hand column, to four different scenarios for 1000 simulations. In each case the y axis showsthe p-value for each period considered. The black line shows the Bonferroni corrected 5% significancethreshold for K = .2. Simulated Data - Recursive Aftershock Process The aftershock process used within Section 4.1 is a simple, one generational, processwhich is not truly representative of the more complex processes which can be observedin nature. We use this simple aftershock process as it allowed us to derive an analyticalexpression for the Schuster distance in Section 3.3. The modified test described in Section3.6 however does not make strong assumptions on the particular aftershock process andso will be robust to different aftershock processes. In this section we demonstrate thisby applying both the SST and our MSST to simulated data which includes both multigenerational aftershocks, i.e. aftershocks can themselves produce further aftershocks, andalso a magnitude depended rate of aftershock generation. These catalogues are simulatedin the following way,1. The number of primary shocks, M , is sampled from a Poisson distribution with mean ν = + α sin ( π t / t s ) , as a function in t , for t ∈ [
0, 50 ] years, where α = t s = µ i from an exponential distributionwith rate b = ν exp (cid:2) a ( µ i − µ min ) (cid:3) where ν = µ min = a = b = τ = ST gives false positive results in the case of Aftershocks but non seasonality whereas theMSST does not. In the scenario where there is both seasonality and aftershocks the SSTgives a positive result for the period of seasonality however it also gives an additionalpositive result for a neighbouring period. The MSST does not show positive result forany frequency. Looking at Figure 5, and comparing with Figure 3, we can see that thisaftershock process has had a larger effect on the SST with a much larger chance of falsepositive results in both scenarios. The MSST however is still able to control the falsepositive rate. For this case we do see that when seasonality is present the power of thetest to discover the true seasonality is reduced, likely this is as a result of the correctingfor the more pronounced aftershock process. + − − Aftershocks − SST
Period (years) P − v a l ue llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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+ − − Aftershocks − MSST
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llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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+ − Aftershocks and Seasonality − SST
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llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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+ − Aftershocks and Seasonality − MSST
Period (years) P − v a l ue llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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Figure 4:
Each plot shows the results of applying either the standard SST, left hand column, or ourMSST, right hand column, to single simulated catalogues from two different scenarios with the morecomplex aftershock process. In each case the y axis shows the p-value for each period considered.The black line shows the Bonferroni corrected 5% significance threshold for K = .005 0.050 0.500 5.000 + − − Aftershocks − SST
Period (years) P − v a l ue + − − Aftershocks − MSST
Period (years) P − v a l ue − − Aftershocks and Seasonality − SST
Period (years) P − v a l ue − − Aftershocks and Seasonality − MSST
Period (years) P − v a l ue Figure 5:
Each plot shows the results of applying either the standard SST, left hand column, or ourMSST, right hand column, to two different scenarios with the more complex aftershock process for1000 simulations. In each case the y axis shows the p-value for each period considered. The black lineshows the Bonferroni corrected 5% significance threshold for K = In this section we apply our method to an observed earthquake catalogue from the NewMadrid Seismic Zone for the period 1 / / / / M c = Earthquake epicentres (black dots) within the New Madrid Seismic Zone (blue polygon)from the CERI catalogue (CERI, 2019) for the period 1 / / / / M ≥ M c = M c = M c > M c values we find the day-night effectfirst disappears for M c = Declustering:
Since the SST is sensitive to aftershocks one approach which can betaken is to first decluster the catalogue and then apply the seasonality test. In theory thisshould remove the aftershocks from the catalogue whilst preserving the seasonal patternin the main shocks. In practice it can be very difficult to apply declustering without aking strong assumptions about the underlying earthquake generating process. For thisexample we apply the following simple declustering to the New Madrid catalogue, usinga standard method based on the space–time linked windows (e.g. Reasenberg, 1985).• Iterate through the catalogue in order of occurrence time.• For the current earthquake any subsequent earthquake is considered an aftershockif:1. It occurs within 5 days of the time of the current earthquake.2. It occurs within 10km of the current earthquake.• Remove all earthquakes fitting the above criteria and continue iterating through thecatalogue, ignoring those which have already been identified as aftershocks.Stochastic declustering methods based on the ETAS model for spatial-temporal clustering(Zhuang et al., 2002) yield a sample of many alternatively declustered catalogues suitablefor probabilistic seismic hazard analysis but not suited to the SST that requires a singlecatalogue.We now apply three tests to the New Madrid data,1. The MST applied to the full catalogue2. The original SST applied to the full catalogue.3. The original SST applied to the declustered catalogue.As with the simulated example we compute the spectrum at equally spaced frequenciesbetween inverse 1 day and 5 years, as described at the end of Section 3.6. The resultsof this can be seen in Figure 7. Looking at Figure 7a, showing results for magnitudesgreater than or equal to 1.4, we see that when we apply both tests to the data withoutdeclustering we find significant seasonality with a period of 1 day. This we attribute to asampling bias in the catalogue which is an indication that the catalogue is not completeand the minimum magnitude if not large enough. It is worth noting that the test appliedto declustered data does not show significant daily periodicity, this could indicate that thedeclustering process has destroyed this daily seasonal signal.Looking at Figure 7b, showing results for magnitudes greater than or equal to 1.9, thefirst thing we note is that in all cases we do not find any significant seasonality. Whilethis may appear to contradict the findings of Craig et al. (2017) we note that this maybe explained by the differences in testing procedures used. Comparing the results of theindividual tests in Figure 7b we note that while there are no significant results for theoriginal Schuster Spectrum the p-values for longer periods are noticeably lower than for ur adapted version. This is likely a result of aftershock effects as they do not coincidewith any expected seasonal pattern. Declustering the data has reduced this effect to someextent however not to the extent of our test which does not require a separate declusteringalgorithm. Reducing the magnitude of completeness to M c = In this section we present the results of applying the SST and our MSST to a data setfrom the Himalayas. This data was also analysed by Ader and Avouac (2013) who alsoapplied the SST and found evidence of seasonal patterns at several different periods. Thedata used is restricted to events in the midcrustal cluster of seismicity with a minimummagnitude of 3 and for the years 1995-2001. This again follows the choices of Ader andAvouac (2013). A map of these events is shown in 8.In their paper Ader and Avouac (2013) first apply declustering to this data using theapproach described in Bollinger et al. (2007). For our comparison we first analyse thisdata using the SST and MSST on the data without declustering. We then apply theSST and MSST to two sets of data which have been declusterd following the methodofBollinger et al. (2007). First with the distance parameter D ≤ D ≤ ? which should allow for a more direct comparison with their results.The results can be seen in Figure 9. We can see that the MSST does not find anysignificant seasonal patter in the data. When applied to data without declustering theSST finds lots of significant seasonal periods, however this is very likely due to falsepositives driven by the aftershock process. For the declustered data the SST shows somesignificant periods for the choice of parameter D ≤ D ≤ a) Minimum magnitude M c = M c = Figure 7:
This figure shows the results of applying both the original SST, left, and our MSST, rightto the New Madrid earthquake catalogue for two choices of minimum magnitude, M c . The periodstested are plotted on the x-axis and the p-values of each test are plotted on the y-axis. In both casesthe dashed line shows the Bonferroni corrected 5% significance threshold for K = Earthquake epicentres (black dots) within the Nepal midcrustal cluster for the period1 / / / / M ≥
3. Colours denote surface elevation in meters and mapcoordinates are in degrees of longitude and latitude.
This figure shows the results of applying both our modified MSST, top left, and the originalSST, top right right, to the Nepal earthquake catalogue. Additionally the SST is also applied to thedeclustered catalogue for two choices of parameter D ≤ D ≤ K =
5. Conclusion
In this paper we have shown that, in the presence of aftershocks, the expected value ofthe Schuster periodogram is given by equation (1). This result demonstrates why theSchuster spectrum test introduced by Ader and Avouac (2013) gives an elevated falsepositive rate when applied to earthquake catalogues which contain aftershocks. Further-more in Section 3 we are able to give some physical meaning to the terms in equation (3)in terms of the expected number of main shocks and aftershocks as well as the character-istic aftershock function. We also describe certain edge cases where this formula can besimplified.In principle these theoretical result could be used to construct an improved hypothesistest which accounts for the effect of aftershocks. However in general it is not possible toestimate the unknown parameters in equation (1) without making strong assumptionsabout the underlying physical process. In Section 3.6 we described an alternative proce-dure which is to first make a non-parametric estimate of the expected value of the Schusterspectrum, D ( k ) using the calculated values of the Schuster spectrum, D k . In Section 4we demonstrated that our improved test is better able to control the false positive rate or catalogues containing aftershocks, while still detecting true periodicity.In conclusion we believe that our Modified Schuster Spectrum test has been shown tooutperform the standard test for earthquake catalogues which contain aftershocks. Thiscomes with the additional advantage that we do not require a separate declustering stepto remove aftershocks. We believe this has a wide ranging applicability because we expectmost observed earthquake catalogues to contain aftershocks. Authors’ contributions
SJB stated the original, high-level research question. The technical content was developedmostly by FJK and TP in discussions. FK carried out the methodological literature review,SJB carried out the domain literature review. FK proved most of the main theoreticalresults, e.g., Propositions 3.1 and 3.3. TP devised the algorithm for the aftershock adaptedtest. SJB planned the experiments on real world data. TP implemented the algorithm andcarried out the experiments. The manuscript was written by all authors, split roughlyacross contribution areas; the entire manuscript was reviewed by all authors, and allauthors are responsible for its content.
Acknowledgements
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A. Order Statistic Property of Poisson Processes
We introduce a well-known result in treatment of events from a Poisson process, andconcomitant notation of which we will make use extensively.
Theorem 1.
The property of being a samples from a Poisson processes, and being the orderstatistics of an i.i.d. sample, are equivalent in the following sense.As the first part of this theorem, the following two statements are true:(i) Consider a Poisson process over a bounded real interval I ⊆ (cid:82) , with Poisson distributedobservation count variable N and τ = ( T , . . . , T N ) a sample. Then, there exists anrandom variable X , absolutely continuous over I, and i.i.d.copies X , X , . . . of X , suchthat ( X , . . . , X N ) is identical in distribution to (cid:0) T S ( ) , . . . , T S ( N ) (cid:1) , where the conditionalS | N is a uniform random permutation of the index set [ N ] . ii) Let X be a random variable, absolutely continuous over a compact real interval I ⊆ (cid:82) , let N be Poisson distributed. Let X , . . . , X N be copies of X , conditionally i.i.d., on N .Let T i : = X ( i ) the i-th order statistic of the sample ( X , . . . , X N ) , ties resolved uniformly.Then, the sample τ : = ( T , . . . , T N ) is a sample from a Poisson process over I.As the second part of the theorem, consider the collections of objects as assumed (in premises)and implied (by consequences) in the statements (i) and (ii).Then, these two collections of objects are canonically isomorphic, i.e., concordant choicesfor premise symbols will yield the same choices for constructed consequence symbols (with“same” meaning “isomorphic as collections of random variables”).Furthermore, the equality λ = ν · p X | N holds between the pdf p X | N of X | N , the rate constant ν of N , and the intensity function λ of τ .Proof. This is directly implied by discussion in Section 7.1, in particular Example 7.1(a),of Daley and Vere-Jones (2003), which phrases the matter in terms of laws of randomvariables, without introducing notation for the random variables themselves. The corre-spondence between our notation and notation ibidem is: our N is n ibidem; our I is A ibidem; our joint conditional law of X , . . . , X N | N (no symbol introduced here) is j n ibi-dem; our p X | N is φ ibidem; our ν is C ibidem.Additional, simplifying assumptions for the situation ibidem are implied by the statementof this theorem: we assume that the domain is univariate, i.e., d = N is Poisson, implying finiteness of the measure λ on A ibidem.The statements in this theorem are then implied by definitions, and the canonicity of theobjects established ibidem.Theorem 1 is sometimes referred to as “Poisson process samples have the order statis-tics property”, though this terminology is slightly inexact in its lack of reference to therandom size of the sample (the random N in the theorem must be Poisson).The original publications around the Schuster spectrum test Ader and Avouac (2013)also make use of this correspondence implicitly by representing the process τ by the pdfof X - which may be somewhat confusing for a reader unfamiliar with Theorem 1, but iseasily recognized by one who is.However, as our situation is more general, with primary and secondary shocks, wherethe secondary shocks are only conditionally Poisson, we need to introduce more stringentnotation as identification and conceptualization is no longer obvious. . Basic results on complex random variables The proofs of our main results heavily make use of complex (number valued) randomvariables which arise in Fourier representations of signals with a suspected periodicity.The results we use are natural generalizations of known results on expectation andvariance of real random variables (univariate and multivariate); for complex randomvariable, many are straightforward generalizations which occasionally appear implicitlyin literature when needed. However, we were unable to find a citable reference for thoseresults we use (the reference Eriksson et al. (2009) has some), therefore we collect andpresent necessary, ancillary, and related statements below.Hence, proofs are provided below.
Definition B.1.
A random variable, taking values in (cid:67) , is any random variable X whichcan be written as X = Y + ı · Z, where Y , Z are random variables taking values in (cid:82) , and ıis the complex unit.In the above case, we write Y : = Re X , and Z : = Im X , calling Re X the real part of X , and Im X the imaginary part of X.
Thus, formally, complex random variables are identified with (cid:82) -valued random vari-ables, via concatenation with the canonical bijection (cid:67) ∼ = (cid:82) , x (cid:55)→ ( Re x , Im x ) whichidentifies each complex number uniquely with its point in the “complex plane”.We identify real variables with complex random variables whose imaginary part is zerowith probability one, and carry through the above identification for conditional randomvariables. Definition B.2.
Let X be a random variable, taking values in (cid:67) .(i) The conjugate of X is defined as X ∗ : = Re X − ı · Im X .(ii) The modulus of X is defined as | X | : = (cid:112) ( Re X ) + ( Im X ) = (cid:112) X · X ∗ .(iii) The expectation of X is defined as (cid:69) [ X ] : = (cid:69) [ Re X ] + ı · (cid:69) [ Im X ] .(iv) The (complex) variance of X is defined as Var [ X ] : = (cid:69) [ X · X ∗ ] − (cid:69) [ X ] · (cid:69) [ X ] ∗ . It is very important to note that the (complex) variance Var [ X ] is not the same asthe variance of the bivariate real random variable ( Re X , Im X ) , which is a ( ) covariancematrix. However, they are closely related: Lemma B.3.
Let X be a random variable, taking values in (cid:67) .The following numbers (possibly infinite) are the same:(i)
Var [ X ] ii) (cid:69) (cid:2) | X | (cid:3) − | (cid:69) [ X ] | (iii) (cid:69) (cid:2) | X − (cid:69) [ X ] | (cid:3) (iv) (cid:69) (cid:2) | X − X (cid:48) | (cid:3) , where X (cid:48) is any independent copy of X(v) Var [ Re X ] + Var [ Im X ] Proof.
This all follows by elementary computation, expanding to real random variablesand then applying rote calculation laws there (e.g., linearity of expectation).Lemma B.3 (v) immediately connects complex variance to properties and results aboutreal variance:
Lemma B.4.
Let X , X , . . . , X N be random variables, taking values in (cid:67) . Then,(i) Var [ X ] ≥ (ii) If X , . . . , X N are mutually independent, then Var (cid:150) N (cid:88) i = X i (cid:153) = N (cid:88) i = Var [ X i ] Proof.
This is a direct consequence of Lemma B.3 (v) and well-known properties of thevariance: for (i), that Var [ Z ] ≥ (cid:148)(cid:80) Ni = Z i (cid:151) = (cid:80) Ni = Var [ Z i ] for mutually independent real random variables Z , . . . , Z N .We further prove some useful results about conditionals expectations and varianes: Lemma B.5.
Let X be a complex random variable, let Y be any random variable such thatthe conditional X | Y is defined. Then:(i) (cid:69) [ X ] = (cid:69) [ (cid:69) [ X | Y ]] (ii) Var [ X ] = (cid:69) [ Var [ X | Y ]] + Var [ (cid:69) [ X | Y ]] .when both sides of the respective equation are finite and well-defined.Proof. This all follows by elementary computation, expanding to real random variablesand then applying rote calculation laws there (e.g., linearity of expectation).Very direct proofs are obtained as follows:For (i), substitute the definition of (cid:69) on both sides, use linearity of (conditional) expec-tation, apply the law of total expectation for real random variables.For (ii), proceed as follows: for all variances of complex numbers, use equivalence be-tween (i) and (v) in Lemma B.3; for all expectations of complex numbers, use the defini-tion. Then, apply the law of total variance (“EVE law”) for real random variables. t should be pointed out that while Lemma B.5 is an easy consequence, it is not animmediate corollary of the real law of total variance: both variances on the RHS arecomplex variances, and the first expectation is real, while the second is complex. C. Phasors, characteristic functions, Fourier transforms,and phase invariance
In our proof, we will also make heavy use of random phases and phasors to encode peri-odicities. The cumulants of these, in turn, are known to be closely related to characteris-tic functions and Fourier transforms of probability densities. We introduce notation andhighlight key correspondences in this section, below.
Definition C.1.
Let a and θ be real numbers or random variables taking values in the reals.We will write a ∠ θ : = a · exp ( ı · θ ) , where ı is the complex unit. The (possibly random)quantity a ∠ θ is called phasor with amplitude a and phase θ (especially in engineering andphysics literature).We also abbreviate ∠ θ by ∠ θ . Advantages of phasor notation are compatibility with modulus and multiplication, i.e., | a ∠ θ | = | a | and ( a ∠ θ ) · ( b ∠ θ (cid:48) ) = ( ab ) ∠ ( θ + θ (cid:48) ) , calculation rules that are used below. Definition C.2.
Let X be a random variable taking values in (cid:82) . The characteristic function (cf) of X is defined as χ X : S → (cid:67) ; t (cid:55)→ (cid:69) [ ∠ ( tX )] , where S is set of t where (cid:69) [ ∠ ( tX )] iswell-defined. S is called the domain of convergence of χ X . As well-known stylized results, it always holds (cid:82) ⊆ S . Furthermore, χ X is distributiondefining for all X with domain of convergence being all of (cid:67) . That is, the law of any such X is uniquely determined by its characteristic function over (cid:82) . This statement becomesfalse when considering χ X with arbitrary domain of convergence. Definition C.3.
Let p : (cid:82) → (cid:82) be an integrable function over (cid:82) , i.e, (cid:82) ∈ L ( (cid:82) ) . The Fouriertransform F p of p is defined as F p : (cid:82) → (cid:82) ; y (cid:55)→ (cid:90) (cid:82) f ( x ) ∠ ( − π x y ) d x . Notationally, the operator F applies before point-wise evaluation, i.e., we will write F p ( y ) instead of ( F p )( y ) . ell-known stylized results from Fourier theory assert that F p ∈ L ( (cid:82) ) as well, and F is unitary (as long as the normalization convention in Definition C.3 is used). It shouldalso be noted that the pdf of (univariate) real random variables are automatically in L ( (cid:82) ) , hence always possesses a Fourier transform. Lemma C.4.
Let X be an absolutely continuous (w.r.t. Lebesgue measure) real randomvariable, with pdf p X and cf χ X . The following equalities hold for any t ∈ (cid:82) :(i) (cid:69) [ ∠ ( tX )] = (cid:69) [( ∠ X ) t ] = F p X (cid:0) − t π (cid:1) = χ X ( t ) (ii) Var [ ∠ ( tX )] = − | χ X ( t ) | = − (cid:12)(cid:12) F p X (cid:0) − t π (cid:1)(cid:12)(cid:12) . Proof.
This is immediate from simple elementary computation, following substitution ofdefinitions.Some basic properties of random phasors’ cumulants, most well-known from the closelyrelated context of distributions on the circle:
Proposition C.5.
Let θ be a random variable, taking values in (cid:82) . Let X : = ∠ θ . Then:(i) Var [ X ] = − | (cid:69) [ X ] | (ii) ≤ | (cid:69) [ X ] | ≤ and ≤ | Var [ X ] | ≤ (iii) | (cid:69) [ X ] | = iff Var [ X ] = iff X is a constant random variable(iv) Var [ X ] + | (cid:69) [ X ] − ∠(cid:69) [ θ ] | = (cid:69) (cid:2) | X − ∠(cid:69) [ θ ] | (cid:3) ≤ Var [ θ ] (v) In particular, Var [ X ] ≤ Var [ θ ] Proof.
For the proofs, write S : = { x ∈ (cid:67) : | x | = } for the unit circle in the complexplane. Note that S is the image of the map (cid:82) → S , α (cid:55)→ ∠ α .(i) is the equivalence of (i) and (ii) in Lemma B.3, observing that | X | = | ∠ θ | = ≤ | Var [ X ] | follows from the equivalence of (i) and (v) in Lemma B.3, and non-negativity of (real) variance. 0 ≤ | (cid:69) [ X ] | is because | . | is a norm. The remaining inequal-ities follow from these inequalities, the equality in (i), and elementary computation.(iii) | (cid:69) [ X ] | = [ X ] = [ X ] = [ Re X ] = Var [ Im X ] =
0. The latter, by basic theory of (real)variance, holds iff both Re X and Im X are constant random variables, which holds iff X is a constant random variable.(iv) The first equality is the complex analogue of the bias-variance decomposition of themean squared error, which one verifies just as in the real case by elementary computation.For the inequality, note that Var [ θ ] = (cid:69) [( θ − (cid:69) [ θ ]) ] . Identifying S , the image of ∠ ( . ) ,with the unit circle in the complex plane (as above in the pre-amble), we observe that X − ∠(cid:69) [ θ ] | = | ∠ θ − ∠(cid:69) [ θ ] | ≤ | θ − (cid:69) [ θ ] | is implied by the schoolbook theorem “chordlength is less or equal arc length” for any circle (and any arc connecting the chord, ir-respective of orientation and multiplicity). Taking squares and expectations yields theclaim.(v) This follows directly from (iv), from comparing leftmost and rightmost side, and ob-serving that | (cid:69) [ X ] − ∠(cid:69) [ θ ] | ≥ | . | is a norm.Angular invariance defines an important class of distributions: Definition C.6.
A complex random variable Z is called circular symmetric if Z and Z · ∠ θ are identically distributed for any θ ∈ (cid:82) . Proposition C.7.
Let Z be a circular symmetric complex random variable. Then,(i) Z and Z · ∠ θ are identically distributed for any real random variable θ .(ii) Z · Y is circular symmetric for any complex random variable Y .(iii) If Y is circular symmetric, so is Y + Z.(iv) If (cid:69) [ Z ] exists, then (cid:69) [ Z ] = .(v) Let X : = ( Re Z , Im Z ) be the bivariate obtained by identification of the domain of Zwith (cid:82) (“the complex plane”). Then, Var [ Z ] = Var [ X ] · I, with I ∈ (cid:82) × being thebivariate identity matrix.Proof. (i) By assumption of circular symmetry, it holds that the conditional distribution Z · ∠ θ | θ is distributed according to Z , and does not depend on the specific value of θ .Therefore, the unconditional distributions Z · ∠ θ and Z must be identically distributed.(ii) By assumption of circular symmetry, ( ∠ θ ) · Z and Z are identically distributed forarbitrary θ ∈ (cid:82) . Hence, (( ∠ θ ) · Z ) · Y and Z · Y are identically distributed. Since θ wasarbitrary, Z · Y is circular symmetric.(iii)By assumption of circular symmetry, ( ∠ θ ) · Z and Z are identically distributed, and ( ∠ θ ) · Y and Y are identically distributed, for arbitrary θ ∈ (cid:82) . Hence, ( ∠ θ ) · ( Z + Y ) and Z + Y are identically distributed. Since θ was arbitrary, Z + Y is circular symmetric.(iv) By the assumption of circular symmetry, (cid:69) [ Z ] = (cid:69) [ Z ] · ∠ x for any x ∈ (cid:82) . There-fore, arg (cid:69) [ X ] = arg (cid:69) [ X ] + x mod 2 π for any x ∈ (cid:82) , if arg (cid:69) [ X ] exists. In particular,0 = x mod 2 π for any x if arg (cid:69) [ X ] exists, therefore arg (cid:69) [ X ] does not exist, and there-fore (cid:69) [ X ] =
0. The claim on (cid:69) [ Z ] follows from definition of Z .(v) From Proposition B.3 (v), Var [ X ] = Tr Var [ Z ] . Circular invariance implies that Z and Z (cid:48) : = ıZ = ∠ π/ · Z have the same distribution, thus X (cid:48) = ( Re Z (cid:48) , Im Z (cid:48) ) = ( Im Z , − Re Z ) nd X must have the same distribution. Thus, Var [ X (cid:48) ] = Var [ X ] , which implies Var ( Im Z ) = Var ( Re Z ) , and Cov ( Re Z , Im Z ) = − Cov ( Re Z , Im Z ) . The latter implies Cov ( Re Z , Im Z ) =
0, the former, together with Var [ X ] = Tr Var [ Z ] , implies Var [ X ] = ( Re Z ) = ( Im Z ) .Altogether, these imply the claim.An important special case is the uniform on the unit circle: Lemma C.8.
Let θ ∼ Unif ([
0, 2 π ]) . Let X : = ∠ θ , let Z : = ( Re X , Im X ) be the bivariateobtained by identification of the domain of X with (cid:82) (“the complex plane”). Then:(i) X ∼ Unif ( { x ∈ (cid:67) : | x | = } ) , and Z ∼ Unif ( { x ∈ (cid:82) : | x | = } ) (ii) X is circular symmetric.(iii) (cid:69) [ X ] = and (cid:69) [ Z ] = (
0, 0 ) (iv) Var [ X ] = and Var [ Z ] = I, with I ∈ (cid:82) × being the bivariate identity matrixProof. (i) this is a direct consequence of the transformation theorem for multi-variaterandom variables (in this case: univariate to bivariate).(ii) This follows from rotation invariance of the uniform distribution on the circle (as apossible defining property).(iii) This is a consequence of Proposition C.7 (iii).The claim on (cid:69) [ Z ] follows from defini-tion of Z .(iv) By (iii) and definition of complex variance, Var [ X ] = (cid:69) [ | X | ] . By (ii), | X | = [ X ] =
1. The statement for Z is a consequence of Proposition C.7 (v). D. The Schuster test
We present the “Schuster test” (also referred to as “Schuster / Fischer procedure”), in aconsensus form, based on implicit use in literature when this term is referred to, in thegeoscience domain.
D.1. Generative setting
The following generative setting is usually assumed for a Schuster test: ( T n ) n ∈ N with N ∼ Poi ( ν ) , assumed a sample from a (possibly heterogeneous) Poisson process, mod-elling the events.By Theorem 1, it is no loss of generality (by identification with the i.i.d. sample presenta-tion) to also assume the following: The sample T , . . . , T N is an i.i.d. sample (conditionalon N ). .2. The Schuster distance The Schuster distance is defined as D ( ω ) : = N (cid:88) i = ∠ ( ω T i ) .Intuitively, this is simply the discrete Fourier transform of the event sequence, i.e., a con-stant signal, but sampled at time points which in general are not from a grid. Testing forperiodicity is modelled as testing for a peak, for some specific frequency ω - as the signalis constant, the peak must be caused by the sampling process.As the T i are i.i.d. conditional on N (as one can assume according to discussion inSection D.1), so are ∠ ( ω T i ) , 1 ≤ i ≤ N , and so are the pairs of real random variables Z i : = ( Re ∠ ( ω T i ) , Im ∠ ( ω T i )) , 1 ≤ i ≤ N , say, Z , . . . , Z N ∼ Z for some generative Z . Thecommon argumentation proceeds by conditioning on N , i.e., considering the sample sizefixed.Writing X : = Re D ( ω ) , Y : = Im D ( ω ) , according to the multivariate central limit the-orem (note that Z i are compactly supported and therefore satisfy assumptions), one has (( X , Y ) − (cid:69) [ Z ]) / (cid:112) N ≈ N (
0, Var ( Z )) , subject to the usual quantitative bounds on the ap-proximation (e.g., Berry-Esseen, and the heuristic that the approximation is “good” if N ≥ T n are drawn is assumedhomogenous, thus ∠ T i ∼ Unif { x ∈ (cid:67) : | x | = } , and Z ∼ Unif { x ∈ (cid:82) : | x | = } .An elementary calculation yields that (cid:69) [ Z ] = (
0, 0 ) , and Var ( Z ) = I , with I ∈ (cid:82) × theidentity matrix. Therefore, under the null, (cid:0) X / (cid:112) N , Y / (cid:112) N (cid:1) ≈ N ( I ) . D.3. The squared Schuster distance and the Schuster p-statistic
The squared Schuster distance is defined as D ( ω ) : = | D ( ω ) | = D ( ω ) D ( ω ) ∗ .Using the above notation, an elementary calculation shows that D ( ω ) / N = ( X + Y ) / N =( X / (cid:112) N ) + ( X / (cid:112) N ) . Continuing under assumption of the null: as (cid:0) X / (cid:112) N , Y / (cid:112) N (cid:1) ≈ N ( I ) , (by virtue of continuous mapping theorems) it also holds that D ( ω ) / N ≈ Exp ( ) (in the sense of convergence in distribution with quantifiable error bounds that aresmall for all practical purposes if N ≥
30) by virtue of the following well-known result:
Lemma D.1.
Let ( U , V ) ∼ N ( a ˙ I ) with a ∈ (cid:82) + , with I ∈ (cid:82) × the identity matrix and the zero of (cid:82) .Then, U + V ∼ Exp ( a ) . roof. This is an elementary calculation, using standard results on multivariate transfor-mation of random variables.On a historical note, that D ( ω ) / N ≈ Exp ( ) (under the null) was probably first de-rived in Rayleigh (1880). The connection to the null of a Schuster-like test is made bySchuster (Schuster, 1897, paragraph 2).To obtain the Schuster test in its contemporary form, as a frequentist hypothesis test,we use another well-known result to derive the p-value: Lemma D.2.
Let U ∼ Exp ( ) . Then exp ( − U ) ∼ Unif [
0, 1 ] .Proof. This is also an elementary calculation, using standard results on univariate trans-formation of random variables.to infer that p : = exp (cid:0) − D ( ω ) / N (cid:1) ≈ Unif [
0, 1 ] .The above chain of arguments shows that p ≤ α iff F P ( | D ( ω ) | ) ≥ − α , for the randomvariable P ∼ Exp ( N ) , where | D ( ω ) | ≈ P is a “good” approximation (if N ≥ p is an approximate p -value for | D ( ω ) | taking uncharacteristically largevalues, with a rejection region of D ( ω ) being in { x ∈ (cid:67) : | x | ≥ (cid:112) N } .It should be noted that the above properties are completely independent of the par-ticular value of ω , except through the rejection region which changes with ω , and thusgives a different test , for each choice of ω . E. The Schuster Test subject to an Aftershock Process
This section contains derivation of the main results of the manuscript, on the Schusterspectrum under an afterschock process.
E.1. Generative setting
We will assume the following generative setting (note that we re-define some symbolsused in Section D.1 in the new scope of this section): τ : = ( τ m ) m ∈ M with M ∼ Poi ( ν ) ,assumed a sample from a Poisson process, modelling the primary shocks. ς m : = ( ς mj } ) j ∈ A m with A m ∼ Poi ( ν ) , for m =
1, . . . , M . Conditional on τ m , it is assumedthat ς m is a sample from a Poisson process, modelling secondary shocks. It is furtherassumed that ϕ mj : = ς mj − τ m are identically distributed, independent of τ , and withnon-negative support (i.e., aftershocks happen after their respective primary shock. y Theorem 1, it is no loss of generality (by identification with the i.i.d. sample presen-tation) to also assume the following: The sample τ , . . . , τ M is an i.i.d. sample, with char-acteristic function χ τ (conditional on M ). The (pooled) sample ϕ mj , j =
1, . . . , A m , m =
1, . . . , M , is an i.i.d. sample, independent of τ (and all A m ), with characteristic function χ ϕ . The pdf g has non-negative support by the assumption on ϕ mj above. We would alsolike to note that, according to Theorem 1, the densities and hence characteristic functions χ τ and χ ς do not depend on the values of M or A m .We further assume that the times of the shocks are known to the observer, but it is notnecessarily known to the observer which shocks are primary shocks, or aftershocks. Forthis, we denote by T , . . . , T N , with N = M + (cid:80) Mi = A i the sample of all time points τ m , ς mj ,in order of their occurrence, i.e., the order statistics of the pooled samples τ , ς , . . . , ς M .Practically and algorithmically usable statements can make reference to this sample only- while theoretical considerations may make reference to the unobserved knowledge ofwhich events are primary shocks, which are secondary shocks, and which secondaryshocks are associated with which primary shock.We note that our setting is equivalent to that of a Hawkes process where aftershockscannot have aftershocks, or of a re-parameterized Hawkes process. E.2. Angle representation of Schuster distance
The Schuster distance is defined as D ( ω ) : = N (cid:88) i = ∠ ( ω T i ) = M (cid:88) m = ∠ ( ωτ i ) + A m (cid:88) j = ∠ ( ως mj ) ,where ω is some (radian) frequency value. Below, we also use the period length param-eterization ω = : π k , and will write ˜ T i : = ω · T i , ˜ τ i : = ω · τ i , and so on, for products with ω .Note that the first expression of D ( ω ) , in terms of T i is algorithmically computable underthe assumptions, while the second in terms of τ i and ς mj is not (as our observer doesnot know which shocks are primary resp. aftershocks). We also make note of two usefulequalities: D ( ω ) = N (cid:88) m = ∠ ˜ T i = M (cid:88) m = ∠ ˜ τ m · (cid:32) + A m (cid:88) j = ∠ ˜ ϕ mj (cid:33) . riting S i : = ∠ ˜ τ i · (cid:128) + (cid:80) A i j = ∠ ˜ ϕ i j (cid:138) , one can write D ( ω ) : = M (cid:88) m = S m = M (cid:88) m = ∠ ˜ τ m · (cid:32) + A i (cid:88) j = ∠ ˜ ϕ mj (cid:33) .The statistical assumptions on ˜ τ i and ˜ ϕ i j imply that S , . . . , S M are an i.i.d. sample condi-tional on M (this is not true when conditioning on the A i ), therefore D ( ω ) | M is a sum ofindependent samples (the S i ).We collect some properties that the S i and D ( ω ) have, under the null of the Schustertest: Lemma E.1.
Under the null assumption of ∠ ˜ τ i ∼ Unif { x ∈ (cid:67) : | x | = } (for all i), it holdsthat:(i) S i is circular symmetric, for any i =
1, . . . ,
M.(ii) D ( ω ) | M = m is circular symmetric, for any m ∈ (cid:78) .(iii) D ( ω ) is circular symmetric.Proof. (i) this is implied by Proposition C.7 (ii) and circularity of the unit circle uniform,also see Lemma C.8 (i) and (ii).(ii) D ( ω ) | M = S + . . . S M , so D ( ω | M ) is circular symmetric by Lemma C.7 (iii).(iii) Two pairs of distributions are identical iff all conditionals are identical, therefore(applying this to copies of D ( ω ) where M remains the same), the fact that m is arbitraryin (ii) implies circular symmetry of D ( ω ) . E.3. Angle representation of squared Schuster distance
The squared Schuster distance is defined as D ( ω ) : = | D ( ω ) | = D ( ω ) D ( ω ) ∗ . By ele-mentary computation (distributive law and linearity of complex conjugate) we observe hat D ( ω ) = N (cid:88) i , j = ∠ (cid:0) ˜ T i − ˜ T j (cid:1) = M (cid:88) m , m (cid:48) = ∠ ( ˜ τ m − ˜ τ m (cid:48) ) + A m (cid:88) j = ∠ (cid:0) ˜ ς mj − ˜ τ m (cid:48) (cid:1) + A m (cid:88) j = A m (cid:48) (cid:88) j (cid:48) = ∠ (cid:0) ˜ ς mj − ˜ ς m (cid:48) j (cid:48) (cid:1) = M (cid:88) m , m (cid:48) = ∠ ( ˜ τ m − ˜ τ m (cid:48) ) + A m (cid:88) j = ∠ (cid:0) ˜ ς mj − ˜ τ m (cid:48) (cid:1) + A m (cid:88) j = A m (cid:48) (cid:88) j (cid:48) = ∠ (cid:0) ˜ ς mj − ˜ ς m (cid:48) j (cid:48) (cid:1) = M (cid:88) m , m (cid:48) = m (cid:54) = m (cid:48) ∠ ˜ τ m · ( ∠ ˜ τ m (cid:48) ) ∗ + A m (cid:88) j = ∠ ˜ ς mj · ∠ ( ˜ τ m (cid:48) ) ∗ + A m (cid:88) j = A m (cid:48) (cid:88) j (cid:48) = ∠ ˜ ϕ mj · (cid:0) ∠ ˜ ϕ m (cid:48) j (cid:48) (cid:1) ∗ · ∠ ˜ τ m · ( ∠ ˜ τ m (cid:48) ) ∗ + N + M (cid:88) m = A m (cid:88) j = ∠ ˜ ϕ mj + A m (cid:88) j , j (cid:48) = j (cid:54) = j (cid:48) ∠ ˜ ϕ mj · (cid:0) ∠ ˜ ϕ mj (cid:48) (cid:1) ∗ ,where for the last equality we have split double summation based on whether indicesare equal or not, and that ∠ = E.4. Expectation of squared Schuster distance
We now proceed with computing the conditional expectation (cid:69) [ D ( ω ) | M , A , . . . , A m ] un-der the null hypothesis assumption of no seasonality at frequency ω , i.e., ∠ ˜ τ i ∼ Unif { x ∈ (cid:67) : | x | = } .Under this null hypothesis, recalling that in every sum, all product factors are statisti-cally independent, all sums with a term ∠ ˜ τ i vanish after taking expectations. For example, (cid:69) [ ∠ ˜ τ m · ( ∠ ˜ τ m (cid:48) ) ∗ ] = (cid:69) [ ∠ ˜ τ m ] · (cid:69) (cid:2) ∠ ˜ τ ∗ m (cid:48) (cid:3) =
0, with first equality from independence of ˜ τ m and ˜ τ m (cid:48) , and second equality from the null assumption. Overall, (cid:69) [ D ( ω ) | M , A , . . . , A m ] = N + M (cid:88) m = A m (cid:88) j = (cid:69) [ ∠ ˜ ϕ mj ] + A m (cid:88) j , j (cid:48) = j (cid:54) = j (cid:48) (cid:69) [ ∠ ˜ ϕ mj ] · (cid:69) (cid:2) ∠ ˜ ϕ mj (cid:48) (cid:3) ∗ = N + ( N − M ) Re χ ϕ ( ω ) + M (cid:88) m = A m ( A m − ) | χ ϕ ( ω ) | = ( N − M ) (cid:128) − (cid:12)(cid:12) χ ϕ ( ω ) (cid:12)(cid:12) (cid:138) + M (cid:88) m = (cid:12)(cid:12) + A m · χ ϕ ( ω ) (cid:12)(cid:12) where the second line follows from Lemma C.4 (i), and the last line from using that − M = (cid:80) Mm = A m .For the unconditional expectation, we use the law of iterated expectation: (cid:69) [ D ( ω )] = (cid:69) (cid:2) (cid:69) [ D ( ω ) | M , A , . . . , A m ] (cid:3) = (cid:69) (cid:150) N + ( N − M ) Re χ ϕ ( ω ) + M (cid:88) m = A m ( A m − ) | χ ϕ ( ω ) | (cid:153) = (cid:69) (cid:150) (cid:69) (cid:150) N + ( N − M ) Re χ ϕ ( ω ) + M (cid:88) m = A m ( A m − ) | χ ϕ ( ω ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M (cid:153)(cid:153) = (cid:69) (cid:150) M + M ν + M ν · Re χ ϕ ( ω ) + M (cid:88) m = ν | χ ϕ ( ω ) | (cid:153) = ν + ν ν + ν ν · Re χ ϕ ( ω ) + ν ν | χ ϕ ( ω ) | = ν ν + ν · (cid:12)(cid:12) + ν · χ ϕ ( ω ) (cid:12)(cid:12) where we have repeatedly used the law of iterated expectation, properties of the Poissonrandom variables M ∼ Poi ( ν ) and A m | M ∼ Poi ( ν ) , which were also be used as identities (cid:69) [ N | M ] = M + M ν , and (cid:69) [ A m ( A m − ) | M ] = ν , to obtain the fourth line from the third.We summarize the above in a proposition Proposition E.2.
Assume that there is no periodicity at frequency ω , that is, ∠ ˜ τ i ∼ Unif { x ∈ (cid:67) : | x | = } for all i. Then:(i.a) it holds for the expected Schuster distance that (cid:69) [ D ( ω )] = (i.b) it holds for the variance of the absolute Schuster distance that Var [ | D ( ω ) | ] = Var [ D ( ω )] = (cid:69) [ D ( ω )] (ii) it holds for the expected squared Schuster distance spectrum that (cid:69) [ D ( ω )] = ν · ν + ν · (cid:12)(cid:12)(cid:12) + ( F λ ) (cid:16) ω π (cid:17)(cid:12)(cid:12)(cid:12) = ν · ν + ν · (cid:12)(cid:12) + ( F λ ) (cid:0) k − (cid:1)(cid:12)(cid:12) , (6) where F λ denotes the Fourier transform (unitary normalization convention) of theaftershock intensity functional λ , and ν is the rate constant of the aftershock process,i.e., ν = (cid:82) ∞ λ ( t ) d t . (iii) if the numbers M of primary shocks and A , . . . , A M of aftershocks are considered fixed,the expected squared Schuster distance (conditional on observed numbers) can be ex- ressed as (cid:69) [ D ( ω ) | M , A , . . . , A M ] = ( N − M ) (cid:128) − (cid:12)(cid:12) ˆ g (cid:0) k − (cid:1)(cid:12)(cid:12) (cid:138) + M (cid:88) m = (cid:12)(cid:12) + A m · ˆ g (cid:0) k − (cid:1)(cid:12)(cid:12) , (7) where ˆ g : = ν − ( F λ ) , i.e., ˆ g is the Fourier transform of the pdf ν − λ (i.e., the functiont (cid:55)→ ν − λ ( t ) , which is the canonical aftershock pdf according to Theorem 1).(iv) if the numbers M of primary shocks is considered fixed, the expected squared Schusterdistance (conditional on observed numbers) can be expressed as (cid:69) [ D ( ω ) | M ] = M ν + M (cid:12)(cid:12) + ( F λ ) (cid:0) k − (cid:1)(cid:12)(cid:12) . (8) Proof. (i.a) This is implied by Lemma E.1 (iii) and Proposition C.7 (iv). (i.b) This isalmost immediate from (i.a) and definition of complex variance; or, more explicitly, from(i.a), and equivalence of (i) and (ii) in Lemma B.3.(ii)-(iv) are proved in the in-text calculation above.
E.5. The aftershock corrected Schuster test
To derive an aftershock corrected Schuster test, we need to derive a test statistic.Writing S i : = ∠ ˜ τ i · (cid:128) + (cid:80) A i j = + ∠ ˜ ϕ i j (cid:138) , as in Section E.2, one can write D ( ω ) = (cid:80) Mm = S m as an sum of the i.i.d. sample S , . . . , S M , conditional on M . The pair of real randomvariables Re D ( ω ) | M and Im D ( ω ) | M , is hence subject to well-known results on sums ofindependent samples, and we can pursue the same strategy as in the original Schustertest, see Section D.2, with M governing the asymptotic instead of N .For this, we define Z i : = ( Re S i , Im S i ) , 1 ≤ i ≤ M . For ease of notation, we alsointroduce generative versions S , Z such that Z , . . . , Z M ∼ Z and S , . . . , S M ∼ S . We needto compute expectation and variance of Z i to use the central limit theorem: Lemma E.3.
For the bivariate real random variable Z, it holds under the null of ∠ ˜ τ i ∼ Unif { x ∈ (cid:67) : | x | = } (for all i) that:(i) (cid:69) [ S ] = , and (cid:69) [ Z ] = (
0, 0 ) (ii) (cid:69) [ D ( ω ) | M ] = (iii) Var [ D ( ω ) | M ] = (cid:69) [ D ( ω ) | M ] (iv) Var [ S ] = (cid:69) [ D ( ω ) | M ] / M = ν + (cid:12)(cid:12) + ( F λ ) (cid:0) k − (cid:1)(cid:12)(cid:12) . (v) Var [ Z ] = (cid:69) [ D ( ω ) | M ] M · I = ν · I + (cid:12)(cid:12) + ( F λ ) (cid:0) k − (cid:1)(cid:12)(cid:12) · I , where I ∈ (cid:82) × is the bivariateidentity matrix. roof. (i) This is implied by Lemma E.1 (iii) and Proposition C.7 (i), observing that S and S are identically distributed.The statement for Z then follows from observing that this implies (cid:69) [ Z ] = (cid:69) [( Re S , Im S )] =(
0, 0 ) , which implies (cid:69) [ Z ] = Z and Z are identically distributed.(ii) This follows from (i) and linearity of expectation.(iii) This is due to (ii) and equivalence of (i) and (ii) in Lemma B.3.(iv) By Lemma B.4 (ii), Var [ S ] · M = Var [ D ( ω ) | M ] . Also, , proving the first equality. Thesecond equality follows from Proposition E.2 (iv).(v) The second equality follows from (ii); the first equality follows from Lemma E.1 (ii)and Proposition C.7 (v).As a direct corollary of Lemma E.3 and the central limit theorem, we obtain: Corollary E.4.
Define Z M : = ( Re D ( ω ) , Im D ( ω )) / M . Under the null hypothesis of ∠ ˜ τ i ∼ Unif { x ∈ (cid:67) : | x | = } (for all i), it holds that:(i) (cid:69) [ Z M | M ] = , and Var [ Z | M ] = Var [ Z ] / M = ν M + M (cid:12)(cid:12) + ( F λ ) (cid:0) k − (cid:1)(cid:12)(cid:12) .(ii) (cid:112) M · Z M | M = m d → N (
0, Var [ Z ]) as m → ∞ In particular, under the null, and writing X : = Re D ( ω ) , Y : = Im D ( ω ) , we can in-fer that ( X / (cid:112) M , Y / (cid:112) M ≈ N ( Var [ Z ]) is a good approximation as long as the actualvalue of M is large (say, ≥ M , not in the number of shocks overall; the covariance matrix in theapproximation is still isotropic, but has value Var [ Z ] = ν · I + (cid:12)(cid:12) + ( F λ ) (cid:0) k − (cid:1)(cid:12)(cid:12) · I , whichin general is different from the aftershock-free value I .Normalizing instead with the easily estimable (by N ) expected number of total shocks, M ( ν + ) , we obtain that ( X , Y ) / (cid:198) M ( ν + ) ≈ N (
0, 12 ( ν + ) Var [ Z ]) = N (cid:129)
0, 12 · I − κ · I + κ (cid:12)(cid:12) + ( F λ ) (cid:0) k − (cid:1)(cid:12)(cid:12) · I (cid:139) ,where we have written κ : = ν +
1, the expected number of shocks per primary shock. Thisexposes two isotropic additive corrections to the aftershock-free variance with oppositesign: a negative term with scale κ − , and a positive term with scale κ − (cid:12)(cid:12) + ( F λ ) (cid:0) k − (cid:1)(cid:12)(cid:12) .In either case, M D ( ω ) | M ≈ Exp (cid:128) ν + (cid:12)(cid:12) + ( F λ ) (cid:0) k − (cid:1)(cid:12)(cid:12) (cid:138) , according to Lemma D.1(and continuous mapping theorems), which exhibits a smooth dependence of the Schusterspectrum on the Fourier spectrum of λ ..