A Bayesian Framework to Rank and Combine Candidate

Recurrence Models for Specific Faults

by Delphine D. Fitzenz, André Jalobeanu, and Matthieu A. Ferry*

Abstract We propose a probabilistic framework in which different types of infor-mation pertaining to the recurrence of large earthquakes on a fault can be combined inorder to constrain the parameter space of candidate recurrence models and provide thebest combination of models knowing the chosen data set and priors.

We use Bayesian inference for parameter and error estimation, graphical models(Bayesian networks) for modeling, and stochastic modeling to link cumulative offsets(CO) to coseismic slip. The cumulative offset-based Bayesian approach (COBBRA)method (Fitzenz et al., 2010) was initially developed to use CO data to further con-strain and discriminate between recurrence models built from historical and archae-ological catalogs of large earthquakes (CLE). We discuss this method and present anextension of it that incorporates trench data (TD). For our case study, the Jordan Valleyfault (JVF), the relative evidence of each model slightly favors the Brownian passagetime (BPT) and lognormal models.

We emphasize that (1) the time variability of fault slip rate is critical to constrainrecurrence models; (2) the shape of the probability density functions (PDF) of paleo-seismic events is very important, in most cases not Gaussian, and should be reported inits complexity; (3) renewal models are in terms of intervals between consecutive earth-quakes, not dates, and the algorithms should account for that fact; and (4) maximum-likelihood methods are inadequate for parameter uncertainty evaluation and modelcombination or ranking. Finally, more work is needed to define proper priors andto model the relationship between cumulative slip and coseismic slip, in particular,when the fault behavior is more complex.

Introduction

In this paper, we propose a probabilistic framework inwhich different types of information pertaining to the recur-rence of large earthquakes on a fault can be combined in or-der to constrain the parameter space of candidate recurrencemodels and provide the best combination of models, know-ing the chosen data set and priors. The aim of such recur-rence models is to (1) set the baseline for the probabilityfor the next event with which probability gains, usingshort-term indicators, can be computed and used for decisionmaking, and (2) be used to build long-term hazard maps use-ful for land-use planning or public education, for example.We discuss the choices we make in terms of target faults (i.e.,faults to which our method can be readily applied) and can-didate recurrence models, and we explain how working with-in a Bayesian framework ensures the reproducibility and theobjectivity of the results once the data and the priors arechosen.

Then, we present our probabilistic approach to themodeling of recurrent large events on a fault, integratingsequentially the catalog of large earthquakes (CLE) from his-torical and archeological records, the dated cumulative off-sets (CO), and finally trench data (TD). We are not comparingstatistics computed from real versus simulated data to testcandidate recurrence models (Scharer et al., 2010) but theplausibility of the candidate models, given the data in theBayesian sense. To this effect, we make use of generativemodels and directed Bayesian networks, a special type ofgraphical models (Jordan, 2004).

Throughout the paper we will present the motivation forthe work and the methodology in as general a manner as pos-sible, so as to allow each reader to adapt it to his/her specificdata sets and requirements. However, we will illustrate eachstep on the case study of the Jordan Valley segment of theDead Sea fault using the wealth of data that was acquired,compiled, or analyzed over the past five years in that area(Ferry et al., 2007; Fitzenz et al., 2010; Ferry et al., 2011;and references therein).

*Now at Geosciences Montpellier, UMR 5243, Université Montpellier 2,France, matthieu.ferry@gm.univ-montp2.fr.

936

Bulletin of the Seismological Society of America, Vol. 102, No. 3, pp. 936–947, June 2012, doi: 10.1785/0120110087

Choosing the Target Type, the Candidate Models,and a Probabilistic Modeling Framework

Target Faults: Structurally Simple

Wework with dated high-resolution cumulative slip datawhenever possible. It is not compatible with imposing a con-straint on the average slip rate (when we have high-resolutiontime-dependent data). Our method is therefore more suited tocase studies of great structural simplicity (i.e., in which largeearthquakes tend to rupture the whole fault segment). This isthe characteristic earthquake case of Schwartz and Copper-smith (1984) without the requirement for constant maximumcoseismic slip from one rupture to the next. We generatesynthetic catalogs of large earthquakes and we use the cumu-lative slip data to build the likelihood function of the recur-rence model parameters. Slip per event in the syntheticcatalog is picked from a wide range of allowed values so thatthe slip is not characteristic, but over some time period thecumulative slip catches up with the measured one (see Fig. 1,which shows the algorithm used in the CO-based Bayesianapproach [COBBRA] method first presented in Fitzenzet al., 2010).

Gathering Candidate Recurrence Models

The recurrence models we choose to use are stochasticin nature because we want them to take into account the

sources of variability that will affect fault behavior (e.g., faultroughness) seismicity in the surroundings. They could bebimodal complex episodic models or quasi-periodic models.Moreover, the data that the models are fitted to are scarce andnoisy, pointing to a need for a probabilistic treatment ofearthquake recurrence. Indeed, proper priors can help definethe parameter space when few data are available.

In probabilistic seismic hazard assessment (PSHA) and inearthquake forecast models, three stochastic models havebeen favored to describe the interevent time distribution forlarge recurrent events on a given fault or fault segment(e.g., Ogata, 1999; Rhoades and Dissen, 2003; WGCEP,2003): Brownian passage time (BPT, or inverse Gaussian,equation 1), lognormal (logN, equation 2), andWeibull (equa-tion 3). Weibull is often used to describe fatigue processes inmaterials and BPT is thought of as a combination of steadyloading and randomwalk-type perturbations (Matthews et al.,2002; Yakovlev et al., 2006; Zöller et al., 2008).

These are the three laws we will consider here, and theyall have two parameters, which we decide to call a and b:

P�xja; b�BPT ��

b2πx3

�1=2

× exp�− b�x − a�2

2a2x

�; (1)

P�xja; b�logN � 1

xb�����������2π�

p × exp�− �log�x� − a�2

2b2

�; (2)

Figure 1. Algorithm to produce synthetic catalogs, adapted from Fitzenz et al. (2010). The shaded rectangles suppose some model. Theinset, after Fitzenz et al.(2010), figure 1, shows the dated cumulated offset data from Ferry et al. (2007) with error bars and simulated COsversus time curves for selected parameters using the BPT, lognormal, and Weibull interevent time distribution. The long-term trend is about4:9 mm=yr, but the rate was observed to change from 3.5 to 11 mm=yr.

A Bayesian Framework to Rank and Combine Candidate Recurrence Models for Specific Faults 937

P�xja; b�Weibull �ba

�xa

�b−1

× exp�−�xa

�b�: (3)

Throughout the study, we will monitor how the two-dimensional parameter space (the ranges of values of a andb to which correspond probabilities larger than zero) of theserecurrence models evolves when we start from prior beliefand then incorporate each data set. The lognormal andBPT models exhibit a similar behavior up to their optimum.However, the BPT model has a much heavier tail (i.e., largerprobabilities at very long return times). The limiting casewhen b � 1 for the Weibull model corresponds to an expo-nential distribution (i.e., the Poisson case). For high values ofb (greater than 15), Weibull tends toward a Dirac distributionand the BPT model tends toward Gaussian.

We want to point out that these models are eitherempirical, or based on simple physical models (i.e., are notthe result of process-based data integrative models of seis-mogenesis). It is therefore (1) not expected that one of thesemodels will be the right one, and (2) necessary to have a rig-orous way of determining the best combination of models inthe Bayesian sense. Such a combination reflects the relativeadequacy of each candidate model and not only the reducedparameter space for each model, leaving the weight to give toeach model to expert decision.

Gathering Usable Earthquake Geology Data

Our aim is to constrain probabilistic recurrence modelsfor given faults. The approach we consider here is to use allavailable evidence of past fault behavior to define the appro-priate parameter space for each candidate model and to rankor combine those models. Qualitative information can beused in the probabilistic framework to help define the properpriors for the model parameters. However, we need quanti-tative evidence of past earthquakes. By quantitative we meandated occurrences of past large earthquakes from historical,archeological, or earthquake geology records (e.g., trenches,coral, turbidites, marshes), with their uncertainties (seeMcCalpin, 1998, for more details). We also mean dated mea-surements of cumulative slip.

Zechar and Frankel (2009) made a review of all possibleways to express the uncertainties on slip rate. They make aseries of recommendations to help ensure that the uncertain-ties are given in journal articles related to the measurement ofoffsets and to their dating, and that they are meaningful.Zechar and Frankel (2009), Biasi et al. (2002), and Goldand Cowgill (2011) also propose methodologies to use theseuncertainties.

Since we are entering an age when high spatial and tem-poral resolution earthquake geology data become available(Ferry et al., 2007; Ludwig et al., 2010; Zielke et al., 2010),such methologies that are able to (1) use these data and theiruncertainties, (2) detect significant variability in apparentslip rate, and (3) use them to understand fault behavior

are becoming increasingly necessary whether they areconcerned with recurrence models for large earthquakes(Fitzenz et al., 2010) or the overall fault slip rate (Gold andCowgill, 2011).

Some pieces of evidence of past fault behavior areindirectly related to the candidate recurrence laws we wishto constrain. In the case of the CO, we have to make assump-tions to relate the slip per event of large earthquakes to theCO recorded in the landscape. For example, (1) it is mostlythe coseismic slip during the full-segment ruptures that getsrecorded in the landscape, meaning that (2) both coseismicplastic deformation and postseismic deformation are negligi-ble compared to the coseismic slip. These hypotheses arequestionable (Wesnousky, 2010) and alternative models needto be built (they can be stochastic or mechanistic in nature) torelate the data to the interval of time between consecutiveevents.

Getting the Most from This Composite DataSet: The Bayesian Approach

We present here the rationale for our Bayesian approach,and the bases of COBBRA, first presented in Fitzenz et al.(2010), that we propose to extend to TD in the present study.

Bayesian methods assess how a prior belief is modifiedonce we take data into account. In other words, if we have aprior belief A about an event, and we get data B, the posteriorprobability of A, knowing B, is expressed using the theoremof Bayes, as shown in equation (4):

P�AjB� � P�A; B�P�B� � P�BjA� × P�A�

P�B� ; (4)

where P�A; B� is the joint probability of A and B, P�BjA� iscalled the likelihood of A and is the conditional probability ofB, given A, P�A� is the a priori probability (prior) of A, P�B�is the prior of B, and P�AjB� is the posterior probability.

We first focus on the reasons for going beyond maxi-mum likelihood methods. When we use the likelihood com-puted with the seismicity data to compute the probabilitydensity function (PDF) for the next event (and go on to thecumulative density function [CDF]), we manipulate compo-site laws. These laws are the weighted averages of the PDFfor the next event over all the possible parameters, theweights being given by the likelihood of the parameters,given the data. We should therefore not expect to get thesame properties as with individual laws taken at the optimumparameter values (Rhoades et al., 1994). For example, aweighted average of, say, lognormal distributions withdifferent parameters is not guaranteed to look anything likea lognormal distribution (see Fitzenz et al., 2010, supple-mental material). The first conclusion we draw is that thewhole parameter space in which the likelihood is greater thanzero matters. BPT, Weibull, and lognormal laws, with theiroptimum parameters, were found to be very similar to eachother until we reach times that are two to three times the peak

938 D. D. Fitzenz, A. Jalobeanu, and M. A. Ferry

time (Matthews et al., 2002). However, Fitzenz et al. (2010)showed that for the composite laws (i.e., the ones whichincorporate all the data) the CDFs are significantly different,as are their uncertainties, even within 300 yr after a peaktime, three times as large. Finding the best model or combi-nation of models is therefore crucial, even at relatively shorttime scales. The second conclusion we draw is that our meth-od needs to be able to perform quantitative model combina-tion or ranking. This is why we chose a Bayesian framework.Finally, Bayesian methods allow the use of priors that willhelp solve the problem when few data are available. The pos-terior is proportional to the likelihood times the prior. So thatwhen the likelihood is not bounded (i.e., when it does nothave a finite integral over the parameter space), the posteriormight become bounded. In that case, it is true that part of thenonzero likelihood space disappears from the posterior, butthis is not done arbitrarily. It results from the choice of a priorand a formal computation, and it is not equivalent to takingonly the couple of parameters for which the likelihood (oreven the posterior) is maximum.

From Bayes’ theorem (equation 4), the posterior prob-ability of each model P�modeljdata� is proportional to theprobability of the data, knowing the model (i.e., the evi-dence), ×the prior on the model. Assuming that we chooseto assign equal priors to each model (we can not tell beforegetting the data which model is more plausible), the differentmodels are ranked by evaluating the evidence (MacKay,2003). It is, by definition, the integral of the product ofthe prior and the likelihood calculated over the entire para-meter space. Note that the plausibility for competing modelsis deemed significantly different only if the evidences aredifferent by at least one order of magnitude. In Bayesianmodel averaging, the relative weights of the different modelsare given directly by the ratio between their evidences, alsocalled Bayes factor.

Finally, in the approach by generative models that weuse throughout the paper, one starts out with hidden variables(for example the true dates of the earthquakes) and builds thePDF for the observations, knowing the hidden variables, as ifthe hidden variables were fixed, and the observations were tobe simulated. It allows the joint probability to be formallyexpressed. Then, using Bayes’ law, one gets the posteriorPDF. We use graphical models (in our case directed Bayesiannetworks) to represent all the random variables (nodes) ofour problems and the presence (edge) or absence (no edge)of conditional dependence assumptions. Hence, graphicalmodels provide a compact representation of the joint probabil-ity distributions (Jordan, 1998). This formalism (combininggenerative models and graphical models) is a well-establishedframework in which people have been solving applied mathe-matics and engineering problems for two decades (Jordan,1998; Jordan and Weiss, 2002; Jordan, 2004). This allowsus to make use of specialized techniques that have been de-veloped in other fields, such as the message passing algorithm(see TD: From Radiocarbon Dates to Recurrence Models andBishop, 2006, chapter 8.4).

For completeness, we summarize here the main featuresof the COBBRA method and Bayesian updating. The readercan find more details in Fitzenz et al. (2010).

(1) Priors. The priors should reflect our state of ignoranceabout the parameters. We choose a uniform distributionfor the mean recurrence time and a flat distribution forthe other parameter b. This translates into a uniform dis-tribution for a, for the BPT and the Weibull distributions,and into exp�a� for the lognormal distribution. Indeed, ifwe call a0 the mean recurrence time for the lognormaldistribution, and a0 follows a flat distribution (e.g.,P�a0� � C), and a � log�a0�, then da0 � da × exp�a�.Because P�a0�da0 � P�a�da, P�a� � C × exp�a�. Formathematical convenience, we choose a lower boundof one year for the mean recurrence interval. The upperbound is set to 200,000 yr in this study because it is highenough to represent infinity.

(2) Parameter Likelihood Knowing the CLE. The CLE con-sists of the list of consecutive (chronologically ordered)instrumental, historical, and archeological events forwhich the date is well known (i.e., uncertainty on thedate is orders of magnitude smaller than the intereventtimes). The CLE likelihood consists of the product overeach interevent time of the product of the functionalform of the model by the distribution of the observedtime interval, integrated with respect to the unknown in-terevent times (equation 5). When we use historical andarcheological data with uncertainties on dates below50 yr, as is the case for the Jordan Valley fault (JVT)data that we use, we can choose Dirac distributionsfor the observed time intervals (i.e., perfectly knowndates with respect to our problem). This terminology re-flects our generative model approach. We suppose a trueinterevent time i that corresponds to the functional formof the candidate recurrence model (function of the para-meters a and b), and from it we build observations ofthis interevent time. From this generative model, weare only interested in the observations and the parameters.Therefore, we start from the joint probability, and we in-tegrate it with respect to the nuisance parameters that arethe true interevent times that we will never know:

LCLE�a; b� �Yi�1;5

ZP�Δtija; b� × P�Δtobsi jΔti�dΔti

�Yi�1;5

P�Δtobsi ja; b�: (5)

We getPCLE�a; b�, the CLE posterior PDF, proportional toLCLE�a; b� × P�a; b�, from Bayes’ rule. We will use thisCLE posterior as an empirical prior in the following dis-cussion. This allows us to impose a more meaningfulprior on the parameter b than the uniform distributionused in the previous step. For large events that have alarge date uncertainty, we refer to the method presentedin TD: From Radiocarbon Dates to Recurrence Models.

A Bayesian Framework to Rank and Combine Candidate Recurrence Models for Specific Faults 939

(3) Parameter Likelihood Knowing CO. Following the ob-servation of Ferry et al. (2007) that fault slip rates couldchange by a factor of 3 over intervals of three to fourinterevent times, Fitzenz et al. (2010) decided to developa method that allowed the integration of earthquakegeology information on a given fault system (in theircase, the JVT). They did so by building synthetic seis-micity catalogs (see flowchart on Fig. 1). It enabledthem to compute the likelihood from the square of dif-ferences between synthetic cumulative slips and obser-vations at the time of the geomorphic markers, averagedover the number of realizations N (number of catalogsproduced for a couple of parameters). We assume themeasurement noise of the ith cumulative slip D followsa Gaussian distribution of standard deviation σD

i , and nis the number of data points:

LCO�a; b� � P�Dobsja; b�≃ 1

N

XNk�1

1������2π

pσDi

× exp�−Xn

i�1

�Dsynthi;k �a; b� −Dobs

i �22�σD

i �2�: (6)

(4) Combined Use of CLEs and CO.We compute the poster-ior probability by using PCLE as a prior and by multi-plying it to the likelihood of the geomorphic markers:

PCLE&CO�a; b� � P�a; bjfDobsi g; fΔtobsi g�

∝ PCLE�a; b� × LCO�a; b�: (7)

This iterative process (the fact of using a posterior as aprior to integrate an additional data set) is called Bayesianupdating.

Hence, we are ready to integrate more data sets as theybecome available. The next section presents such anadditional data set, TD derived from Ferry et al. (2011). Thegraphical model explaining the relative role of all data sets ispresented in Figure 2.

TD: From Radiocarbon Datesto Recurrence Models

In order to perform successful paleoseismological inves-tigations, Ferry et al. (2011) carefully selected trench sites toensure an optimal expression of faulting events, a continuousand detailed sedimentary record, and material suitable forage determinations. They made four trenches.

In the following, we will work with the six well-constrained events they evidenced in trench 4 that are be-tween two erosion surfaces, to avoid preservation issues.Most events younger than the younger erosion surface aredeemed well identified in the historical and archeologicalcatalogs, and the following six consecutive events were in-cluded in the CLE in Fitzenz et al. (2010): three historical

events (AD 1033� 0, AD 749� 1, and 759 BC� 1 Ambra-seys, 2009) and three archeoseismic events (1150 BC� 50,2300 BC� 50, and 2900 BC� 50 [Franken, 1992; Savageet al., 2003]). In the trench, the events that are older thanthe oldest erosion surface have large uncertainties attachedto their age and would not necessarily add a lot to our study.Thus, although it will be necessary in the future to extend themethodology to account for missing data, it is beyond thescope of our present study.

Ferry et al. (2011) computed the calibrated ages of thesamples taken from the stratigraphy in trench 4 using Oxcal(Ramsey, 1995). In the present work, we go through thefollowing steps dictated by the structure of the Bayesian net-work of Figure 3. The goal is to hierarchically integrate outall the unwanted variables, starting from the end of the graph,first with y, then x, and finally t, as shown in the three fol-lowing sections, respectively.

From Calibrated Ages to Radiocarbon Layer Ages

We first present the generative model. Let us call y thetrue radiocarbon date (unknown), yobs the observed radiocar-bon age, and x the calibrated date. The generative model (i.e.,the probabilistic forward model) starts from x, gets y using acalibration curve f, and obtains yobs due to the uncertaintieson the measure of the radiocarbon date and the uncertainty inthe calibration curve. The joint PDF of true and observedradiocarbon dates, given the calibrated age x, is given byP�yjx� × P�yobsjy�. When several samples of organic matterare dated to characterize a same layer and give different re-sults, the PDF of P�yobsjy� may be complex. We show expli-citly here the more simple case of Gaussian P�yobsjy�. Thenboth PDFs P�yjx� and P�yobsjy� are Gaussians, with meansf�x� and y and variances ϵ2 and σ2, respectively. These Gaus-sians are assumed to model the noise in the radiocarbon agemeasurement and the calibration curve (Reimer et al., 2004).

Figure 2. Graphical model summarizing the role of the differentdata sets. M represents any of the candidate recurrence models ofparameters a and b. The CLE is used to compute an empirical prior.Dobs

i is the dated CO, and yobsi− and yobsi� are the radiocarbon ages ofthe layers directly below and directly above an event in the trench(TD), respectively. The color version of this figure is available onlyin the electronic edition.

940 D. D. Fitzenz, A. Jalobeanu, and M. A. Ferry

Therefore, by integrating this product with respect to y, weget the conditional probability P�yobsjx� or the likelihoodof x:

P�yobsjx� �1�������������������������

2π�σ2 � ϵ2�p exp

�− �f�x� − yobs�2

2�σ2 � ϵ2�

�: (8)

Using Bayes’ rule, one could compute the posterior PDFP�xjyobs�, which is proportional to this likelihood if weassume a flat prior P�x�.

Figure 4 shows the calibration step starting from twosample radiocarbon ages and the calibration curve. We seethat a large uncertainty on the calibrated date can come fromseveral sources. The radiocarbon date itself might be uncer-tain (not enough organic matter in the sample, or differentsources of carbon in the same sample). But we can see on theorange set of curves on Figure 4 that the slope of the cali-bration curve also plays a critical role in the width of the PDFof the sample age.

For completeness, we stress that in the case when wehave a prior on y, and we want to translate it into a prioron x, the Jacobian has to be taken into account (the slopeof the calibration curve f).

From Earthquake Ages to Pairs of RadiocarbonLayer Ages

We used the PDFs of the radiocarbon ages of samplestaken in layers bracketing earthquake evidences in the trench(Ferry et al., 2011). We also took into account the expert

opinion of Ferry et al. (2011) when the samples were con-sidered too far up or down the stratigraphy (see their table 3).

We can do the mind experiment (the generative model)that consists in starting from the true date of the earthquakeand assuming that the event can be any time between the truecalibrated ages of the layer deposited before and the layer

Figure 3. Bayesian network exhibiting a Markov chain structure for the hidden nodes ti (earthquake times linked through the recurrencemodel P�ti�1jti; a; b�). Simplified view (left; after the marginalization of the intermediate variables) and expanded view (right) for index i.The color version of this figure is available only in the electronic edition.

Radiocarbon age of layer below UT4

Rad

ioca

rbon

age

, yr

BP

Calibration curve intCal04, interpolated every 5 yrsRadiocarbon age of layer above UT4

Calibrated age of layer below UT4Calibrated age of layer above UT4

Date of event UT4, BP

13000

12000

6000

8000

11000

7000

9000

10000

900013000 12000 11000 100001400015000

Calibrated ages, yr BP

Figure 4. Calibrating layer ages and inferring the date of anearthquake between two sample ages. The PDFs of the carbon datesof samples taken in two layers are shown next to the y axis: the thincurve is the calibration curve f, and the filled PDFs are the calibratedages of the corresponding layers. The outlined PDF shows the in-ferred posterior of the age of the earthquake assuming a uniformprior distribution between each pair of possible layer ages (equa-tion 8; after Biasi et al., 2002). The color version of this figureis available only in the electronic edition.

A Bayesian Framework to Rank and Combine Candidate Recurrence Models for Specific Faults 941

deposited after the event. In this mind experiment, we canconstruct the true carbon dates and the observed carbon datesusing the previous analysis (see the directed graphical model,right part of Fig. 5). We use this construct to express the con-ditional probability of the observed carbon dates of the layersknowing the true event date, by marginalizing with respect tothe calibrated ages of the layers, that we do not need in therest of the study (i.e., nuisance parameters).

The earthquake date t is split into two calibrated datesx− and x�, which are the ages of the layer preceding theevent and that of the layer deposited after the event, accord-ing to the conditional distribution P�x−; x�jt�. This PDFcan be expressed, thanks to Bayes’ rule, as P�x−; x�jt� ∝P�tjx−; x�� and by assuming a flat prior (the earthquakemight have happened at any time between the true dates ofthe layers) such that P�tjx−; x�� � 1=�x� − x−�, with theconstraints x− < t and x� > t.

The joint PDF of the observed radiocarbon dates yobs− andyobs� and the calibrated dates x− and x�, given the date t, isP�yobs− jx−� × P�yobs� jx�� × P�x−; x�jt�, which we integratewith respect to the calibrated dates in order to get the likeli-hood of the two measured layer ages related to an earthquake:

P�yobs− ; yobs� jt�

� CZx�>t

Zx−<t

P�yobs− jx−�P�yobs� jx��1

x� − x−dx�dx−;

(9)

where the constantC does not depend on the recurrence mod-el, so we will ignore it in the following. Equation (3) by Biasiet al. (2002) is similar and corresponds to the posterior PDF oft, given the layer ages and assuming a flat prior P�t�.

Figure 4 shows the date of the earthquake superimposedon the two calibrated layer ages, and Figure 5 synthesizes the

dates found for the six consecutive events (the chronologicalorder is shown by the alphabetical order of the names of theevents) evidenced in trench 4 of Ferry et al. (2011).. We cansee that the PDFs are usually not Gaussian. This is an impor-tant property and we would like to encourage authors to sys-tematically report both the raw data (radiocarbon dates ofeach sample pertaining to the same layer) and these eventdate PDFs in all their complexity, maybe as supplementaryonline material.

From Radiocarbon Layer Ages to the Likelihood ofRecurrence Model Parameters

Let us call t1; t2;…tn the true dates of the consecutiveearthquakes in chronological order and tobsi the observed dateof the event i, from the radiocarbon dates of the layers brack-eting the event, without any other manipulation. Note thatthese are potentially different from a typical Oxcal output.This way, all tiobs are independent from each other and maybe non-Gaussian and may overlap, as is the case for our casestudy. Then P�ti�1jti; a; b� is the recurrence model witharguments �ti�1 − ti�, the time interval between the twoconsecutive events, and the model parameters �a; b�.P�yobs− ; yobs� jt� is the likelihood computed previously from thesample dates. Following the structure of the graphical modelpresented in Figure 3, the marginal probability P�dataja; b�can be computed recursively by passing a series of messagesin the chain from i to i� 1, using a belief propagation meth-od (Bishop, 2006, Chapter 8.4 ). At each step, the hiddennode ti is integrated out.

The first message, from node 1 to node 2, is computed asthe integral with respect to the nuisance parameter t1 of theproduct of the likelihood of t1 times the recurrence model.That is P�t2jt1; a; b�, and

m2�t2; a; b� �Z

P�yobs1− ; yobs1�jt1�P�t2jt1; a; b�dt1: (10)

Then, for all the other nodes, to go from i to i� 1, one needsto compute the integral with respect to the nuisance param-eter ti of the previous message times the likelihood of ti timesthe model, which is the conditional PDF P�ti�1jti; a; b� suchthat

mi�1�ti�1; a; b�

�Z

mi�ti; a; b�P�yobsi− ; yobsi� jti�P�ti�1jti; a; b�dti: (11)

Finally we can see that the marginal PDF, or the likelihoodknowing TD, is given by

LTD�a; b� � P�fyobsi− ; yobsi� gja; b�

�Z

mn�tn; a; b�P�yobsn− ; yobsn�jtn�dtn: (12)

This is a completely different method compared to sam-pling methods used by Parsons (2008) or Scharer et al.

ST4

TT4

UT4VT4

WT4

XT4

0.004

0.003

0.002

0.001

013000 1100015000 9000 7000

Calibrated dates, yr BP

Pro

babi

lity

Figure 5. Calibrated earthquake dates in years before present(BP) for the events evidenced in trench 4 in the JVF (original un-calibrated age layers and constraints from Ferry et al., 2011, table 3).The chronological order is shown by the alphabetical order of thenames or the events. The color version of this figure is availableonly in the electronic edition.

942 D. D. Fitzenz, A. Jalobeanu, and M. A. Ferry

(2010) in which the ordering has to be enforced each time asample interevent time is built, using samples from the ob-served date of consecutive events. Also, whereas our methodjust requires one chain of nested convolution products andintegrations for each candidate recurrence model for eachPDF described by a coefficient of variation and a mean re-currence, Parsons (2008) draws event interval sets at randomand assembles them into earthquake sequences 5 milliontimes. Those that match observed event windows (necessa-rily truncated range of possible event times as constrainedby radio carbon dating assuming a uniform distribution) aretallied.

This probabilistic approach is also different from thestatistical treatments presented by Biasi et al. (2002) andScharer et al. (2010). We do not have to sample the dates,build possible series of interevent times from observed earth-quake dates, build synthetic series of interevent times usingan exponential, nor BPT or lognormal distribution and per-form statistical tests to determine whether they have commoncharacteristics or not.

To sum up, we incorporate the PDFs of the event dates,in the chronological and stratigraphic order, into the model-ing and get the likelihood of the parameters of a given model(lognormal or other), and we can use that likelihood to up-date any prior belief so as to be able to compute the posteriorPDF of the parameters

PALL�a; b� � P�a; bjfΔtobsi g; fDobsj g; fyobsi− ; yobsi� g�

∝ PCLE&CO�a; b� × LTD�a; b�: (13)

In theory, the evidence of the candidate recurrence mod-el (i.e., the joint distribution of all the data sets, given thismodel) should be obtained by integrating the joint PDF ofthe data and model parameters, so the recursive Bayesian up-dating cannot be applied. This joint PDF is the product of thelikelihoods Ldataset�a; b� times the prior P�a; b�. However, inorder to minimize the effect of the subjective prior we havechosen, we prefer to use LCLE�a; b� instead of P�a; b�, whichis more objective because it is data driven. The CLE data setis only used to build an empirical prior and is withdrawnfrom the data used for evidence computation. Thus, we fi-nally consider this conditional evidence,

P�fDobsi g; fyobsi− ; yobsi� gjfΔtobsi g�

�ZZ

PCLE�a; b�LCO�a; b�LTD�a; b�dadb: (14)

It is only on the basis of the evidences that we rank or com-bine the recurrence models.

Conclusion for the JVF

We first recall the conclusions that Fitzenz et al. (2010)had reached using only the CLE and the earthquake geology

data so that we have a basis for comparison with the newresults using TD as well.

After combining a noninformative prior, the catalog ofthe six large earthquakes from historical and archeologicalrecords, and the dated CO (see inset in Fig. 1), Fitzenz et al.(2010) had concluded, using the evidence ratio, that the log-normal, the BPT, and the Weibull model were equally plau-sible. However, they could also use directly the likelihood ofthe parameters for each model to show that some subsets ofthese more general models could be eliminated. Indeed, thesesubsets correspond to regions in the parameter space wherethe likelihood was zero. Neither the Gaussian (special case ofBPT), periodic, nor exponential (special cases of Weibull)was likely. The exponential case (corresponding to a Poissonrecurrence model) was discarded when earthquake geologydata were introduced into the computations. In fact, it is notso much the dated COs that were incompatible with thatmodel, but really the slip rate constraints. We constrainedgroups of four to five consecutive synthetic large events tocorrelate to fault slip rates between 2 and 13 mm=yr. This isa very conservative constraint because earthquake geology ofthe JVF indicates that the mean slip rate over a period of48,000 yr is 4:9 mm=yr and that the known excursions spanthe range from 3.5 to 11 mm=yr (Ferry et al., 2007). We nowwant to combine the information contained in those datasources with that included in the TD and check if that con-clusion still holds.

Discussion of Resulting Likelihoods

Figure 6 displays the likelihoods of the model param-eters, knowing the TD only. We see two interesting features.First, the shapes of the likelihoods for the three modelsstrongly resemble the shapes obtained using the CLE butare shifted toward larger mean values. This reflects the pre-sence of larger intervals in the TD than in the CLE (see Fig. 5).We do not believe this can be an event preservation issue asno erosion surface was identified in the interval that we use.The small intervals that are in the CLE and not in the TD thatwe use are between 300- and 500-yr long. Although there isnever any guarantee, events as much as 300 yr apart shouldbe detectable as separate events if deposition was not inter-rupted. Again, such a case would lead to the formation oferosion surfaces, which are not observed.

The second feature can be seen on the likelihood of theWeibull parameters. The value of the likelihood is large forshape parameters close to one, contrary to the likelihoodobtained with the CLE. This means that when using the TDalone, the Poisson model can not be rejected. Figure 5 showsthat several PDFs of earthquake dates overlap significantly,leading to a high probability of very short interevent times.In addition to the fact that long intervals are also present,this leads to the high likelihood of b � 1. In places wherethe sedimentation rate for the considered period is wellknown, the measured thickness of sediments between two

A Bayesian Framework to Rank and Combine Candidate Recurrence Models for Specific Faults 943

consecutive events could be used to rule out very small in-terevent times.

Apparent Episodicity versus Actual Episodicity

When working with the complete data set for the JVF,one gets a sense that earthquakes seem to happen in shortclusters of two to three events separated by longer intervals.It would therefore be tempting to test a truly episodic modelversus the unimodal models we used so far. There are twoways to look at the apparent episodicity of an earthquakecatalog. The first is to say that it may only reflect the stochas-

tic nature of the (unimodal) recurrence model and the smallsample size. The second is to say that it stems from aninherently episodic recurrence model. Such models can bemultimodal, conditional multimodal, etc. We have checkedthe relationship between an interval and the next one with theidea of identifying systematic trends such as short intervalsfollowed by short intervals or, on the contrary, long intervals.With the CLE and the TD, however, no clear trend was iden-tified. Model comparison is a difficult task because it is notpossible to simply choose the model that fits the data best:more complex models have more free parameters and cantherefore fit the data better. So the maximum likelihoodmodel choice would lead to implausible overparametrizedmodels, which generalize poorly (MacKay, 2003). In theBayesian framework, the model that achieves the greatestevidence is the one that most reduces its parameter spacebetween the prior and the posterior, while achieving a reason-able fit to the data. Because models with a larger number ofparameters are penalized because they have a large priorparameter space, they need to have a greatly reduced poster-ior parameter space to achieve a large evidence. Testing morecomplex models would require having large enough data setsof consecutive events.

Model Comparison or Combination

We use the (normalized) posterior distributions for eachmodel obtained using the COBBRA method as a prior andmultiply them with the corresponding likelihoods computedwith the TD (equation 12). Figure 7 shows that the shapes ofthe posterior distributions with and without TD differ. Theparameter space becomes smaller as we add this data set,showing how important and complementary it is to the datatypes used previously. The relative values of the evidences ofeach model (equation 13) change when the information in-cluded in the TD is taken into account (Table 1). The absolutevalues of the evidence are now 8:79 × 10−25 for the BPTmodel, 18:8 × 10−25 for the lognormal model, and 4:81×10−25 for the Weibull model. The ratio of evidences, alsocalled Bayes’ factor, is deemed significant for values largerthan 10 and shows the unambiguous predominance of onemodel over the others for values above 100 (see the scale inJeffrey, 1961). The evidence ratio between lognormal andWeibull is about four, which means that the Weibull modelis slightly disfavored. But the evidence ratio between lognor-mal and BPT is about two. We can therefore conclude that, inthe light of the complete data set, the BPT and lognormalmodels are slightly more plausible than the Weibull model.The best combination of models is given by 0.27 times theBPT model plus 0.58 times the lognormal plus 0.15 times theWeibull model. A direct implication for the JVF is the factthat the Weibull model becomes less plausible than the log-normal and BPT models, and that it will further decrease theprobability of having the next event in the near future (toabout 50% in the next 300 yr).

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Figure 6. Likelihood of the parameters of the different recur-rence models, knowing the TD (BPT [a], lognormal [b], and Weibull[c]) over the parameter space over which the posterior CLE–CO isdifferent from zero (range useful for the computation of the poster-iors shown on Fig. 7). We see that the shape of the likelihood is veryclose to that obtained using the CLE (Fitzenz et al., 2010, figure 2),but that there is a significant shift to higher a values, in particular forthe Weibull model (see [c] insert: the same Weibull likelihood butfor a large interval for a has the same shape as for the CLE). Thecolor version of this figure is available only in the electronic edition.

944 D. D. Fitzenz, A. Jalobeanu, and M. A. Ferry

Discussion and Perspectives

We presented a methodology to derive the combinationof recurrence models that best represents our knowledgeusing prior beliefs updated by three distinct data sets: theCLE, the earthquake geology data including dated COs, andTD. We were interested in faults that are known to experiencefull-segment events each time they have a very large earth-quake. We do not suppose that the slip is always the same, orthat it propagates always in the same direction. We just saythat when a fault has a relative structural simplicity and isbounded by large pull-apart basins, large events have a ten-dency to continue until they come close to these large naturalbarriers. In the case of the JVF, we allowed slip between 2.5and 4 m, based on expert opinions of earthquake geologists.

We also tried wider or narrower distributions but found thatthe dated COs and the minimum and maximum slip rates(enforced at the scale of four consecutive events) had a muchlarger impact on the overall synthetic fault slip rate than thewidth of the slip per event distribution. Faults with highlyheterogeneous slip per event profile along strike might re-quire a more complex model relating slip per event to COs.For faults which can undergo overlapping large ruptures,other forward models, probably more process-based, will beneeded to produce both the synthetic time lines and the slipper event versus cumulative slip models.

We showed that using a Bayesian framework is verypowerful in that it provides a way to sequentially update ourresults (a.k.a, belief in the probabilistic jargon) as new datasets become available, and it allows us to compute the plau-sibility of a given model, knowing a given data set. In otherwords, we can determine the relative weights of the param-eters of each model and the relative weights of the modelsthemselves, in an objective fashion, once the data sets and thepriors are chosen. Such results are not achievable using max-imum likelihood methods or statistical characteristics com-parisons of distributions (Scharer et al., 2010; Parsons,2008). In this work, we have shown how the parameter spaceshrinks and how the relative weights of each model evolvewhen we first consider the catalog of known large events(from instrumental, historical, and archeological sources)and then update the study using earthquake geology informa-tion and finally TD. In particular, dated COs proved most use-ful, namely in cases where they evidence changes in fault sliprates over short time periods (about four interevent times, inthe case of the JVF). We showed how a Bayesian analysis ofthe TD incorporates the candidate model that is thought toexplain the interval of time between consecutive events. Ithas to be built using an algorithm that goes from each ob-servation (i.e., the uncertain date of the event and its orderin the sequence) to the true” interevent time (that we do notknow) and the next observation, through the recurrencemodel. We emphasize the need to report both the raw data(i.e., the radiocarbon dates of all samples used) and the PDFsfor the observed event dates in their full complexity.Although the algorithm needed to compute the joint prob-ability might look complex, it provides a clear answer in

BPT

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Figure 7. Comparison of the posterior distributions using onlyCLE, the CLE� CO, and the CLE� CO� TD shows the reductionof the relevant parameter space for the different recurrence modelsas we combine data sets. The x axis is in years (i.e., values of a forBPT and Weibull, and of exp�a� for the lognormal). The contourscorrespond to values more than one order of magnitude smaller thanthe maximum. The dash-dotted contour is using the prior and theCLE, the dashed contour is obtained when adding the dated COs,and finally the plain contour is when adding the TD. The color ver-sion of this figure is available only in the electronic edition.

Table 1Evidence of Each Model According to the Data Used

Models CO Evidence CO × TD Evidence

Weibull 7:38 × 10−05 4:81 × 10−25BPT 1:86 × 10−05 8:79 × 10−25Lognormal 4:09 × 10−05 1:88 × 10−24

We compute the CO evidence and CO × TD evidence. Thepriors are the normalized products of the CLE likelihood and auniform prior for Weibull and BPT and log (uniform) for thelognormal distribution. The bounds are 1 and 200,000 for a orexponent 1 and exponent 200,000 for lognormal. The Weibullmodel is slightly disfavored, but no model can be discarded.

A Bayesian Framework to Rank and Combine Candidate Recurrence Models for Specific Faults 945

terms of parameter space (likelihood) and the necessary stepto compute the model evidence. It is therefore very efficientcompared to methods that have to rely on statistical testsperformed on simulated earthquake date sequences usingeither a well-known model (Poisson, lognormal, etc.) orsamples of the data (Scharer et al., 2010). We will provideaccess to the computer code performing this task to all inter-ested colleagues.

Finally, let us stress again that once the data and thepriors were chosen, no expert opinion was involved in thedetermination of the best model or best combination of mod-els, unlike what is usually done in PSHA. This ensures a com-plete reproducibility of the results. Of course, it does notensure that any of the models tested is the right one. This iswhy we encourage the community to continue work both onempirical recurrence models and on physics-based recur-rence models (Fitzenz et al., 2007).

What makes it difficult to use Bayesian methods? Onedifficulty comes from the number of dimensions of the pro-blem at hand. If there are many parameters that are involvedin the joint probability, but we consider them as nuisanceparameters (i.e., we are not interested in them), we need tointegrate them out (marginalize). This can be mathematicallychallenging. An alternative to analytical or numerical inte-gration is Monte Carlo sampling. This brings about other dif-ficulties such as the computer time needed to perform thetask and the test of the convergence of the PDF being builtto the marginal that we are looking for.

Another major difficulty is the choice of a proper prior.Flat or uniform priors are often used. Alternatives includeconjugate priors, used only for mathematical convenience,or priors built using part of a data set or an independent dataset (Nomura et al., 2011). Here we chose flat priors for b anda uniform prior on the mean recurrence time over the intervalbetween 1 and 200,000 yr. Priors do not have much impor-tance when the data sets are large. However, if we were tocompare the evidence of the models using the CLE alone(shown in the first column of values in Table 1) using severalpriors, we would get different values because we have onlyfive data points. It is very important to use a prior, whichreflects our state of ignorance about the parameter.

Finally, working on recurrence models for large earth-quakes is not only a geophysical, mathematical, and compu-ter science challenge but also could have large societalimplications. In the light of the data sets we used for thisstudy, the long-term probability for earthquake occurrencethat should be used for the JVF is the combination of modelsthat we propose and not a Poisson model. This long-termmodel should serve as a basis to compute (1) the long-termhazard maps used for land-use planning and insurance pre-mium computations and (2) the baseline for the computationof the probability gain due to short-term probabilities com-puted using, for example, the seismicity rate or other phe-nomena considered as potential precursors (such short-term probabilities are not in the scope of our work). It is thisprobability gain that is needed in short-term cost–benefit

analyses to make informed decisions, namely about evacua-tion (van Stiphout et al., 2010). We saw that an extensive anddiverse data set can significantly decrease the uncertaintieson the parameters for each model and can help identify fa-vored or unfavored models. However, the models we testedall have an intrinsic variability (width of the PDF), makingprecise forecasting beyond our reach. It is therefore crucialthat, while scientists continue working to find more appro-priate recurrence models, society uses the long-term hazardmaps to increase public awareness and education and opti-mize land-use planning. The idea is not only to increase pre-paredness and resilience but also to diminish vulnerability.

Data and Resources

All data sets used are published and referenced in thisarticle.

Acknowledgments

This research was funded by the Fundação para a Ciência e a Tecno-logia (FCT; PTDC/CTE-GIX/101852/2008 and FCOMP-01-0124-FEDER-009326). D. Fitzenz, A. Jalobeanu, and M. Ferry all benefit from the FCTCiência 2007–08 program.

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Centro de Geosica de ÉvoraUniversidade de ÉvoraRua Romao Ramalho 597002-554 Évora, Portugaldelphine@uevora.ptjalobeanu@uevora.ptmatthieu@uevora.pt

Manuscript received 25 March 2011

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