Inferring skeletal production from time-averaged ... · Adam Tomašových, Susan M. Kidwell, ... (Olszewski 2004; Tomašových et al. 2014). However,theshapesofAFDsarealmost certainly

Inferring skeletal production from time-averaged assemblages:skeletal loss pulls the timing of production pulses towards themodern period

Adam Tomaovch, Susan M. Kidwell, and Rina Foygel Barber

Abstract.Age-frequency distributions of dead skeletal material on the landscape or seabedinformationon the time that has elapsed since the death of individualsprovide decadal- to millennial-scaleperspectives both on the history of production and on the processes that lead to skeletal disintegration andburial. So far, however, models quantifying the dynamics of skeletal loss have assumed that skeletal pro-duction is constant during time-averaged accumulation. Here, to improve inferences in conservationpaleobiology and historical ecology, we evaluate the joint effects of temporally variable production andskeletal loss on postmortem age-frequency distributions (AFDs) to determine how to detect fluctuations inproduction over the recent past from AFDs. We show that, relative to the true timing of past productionpulses, the modes of AFDs will be shifted to younger age cohorts, causing the true age of past pulses to beunderestimated. This shift in the apparent timing of a past pulse in production will be stronger where lossrates are high and/or the rate of decline in production is slow; also, a single pulse coupled with a decliningloss rate can, under some circumstances, generate a bimodal distribution. We apply these models to deathassemblages of the bivalve Nuculana taphria from the Southern California continental shelf, finding that:(1) an onshore-offshore gradient in time averaging is dominated by a gradient in the timing of production,reflecting the tracking of shallow-water habitats under a sea-level rise, rather than by a gradient in disin-tegration and sequestration rates, which remain constant with water depth; and (2) loss-corrected model-based estimates of the timing of past production are in good agreement with likely past changes in localproduction based on an independent sea-level curve.

Adam Tomaovch. Earth Science Institute, Slovak Academy of Sciences, Dubravska cesta 9, 84005, Bratislava,Slovakia. E-mail: [email protected]

Susan M. Kidwell. University of Chicago, Department of Geophysical Sciences, 5734 S. Ellis Avenue, Chicago,Illinois 60637

Rina Foygel Barber. University of Chicago, Department of Statistics, University of Chicago, 5734 S. UniversityAvenue, Chicago, Illinois 60637

Accepted: 9 June 2015Published online: 30 October 2015Supplemental materials deposited at Dryad: http://dx.doi.org/10.5061/dryad.rv0k6

Introduction

Understanding and quantifying the rates ofproduction and the postmortem fates of skele-tal carbonates is important to many aspects ofgeology, paleobiology, and ecology. The inter-action between carbonate production, loss,reworking, and burial determine the overallcarbonate budget of ecosystems dominated bycarbonate producers (Pandolfi et al. 2011; Perryet al. 2014;Waldbusser and Salisbury 2014), theavailability of skeletal debris in benthic habi-tats and the preservation of reef framework(Powell et al. 2006; Waldbusser et al. 2013), thetemporal resolution and completeness of deathassemblages and accuracy of paleoecologicalindices (Bush and Bambach 2004; Hunt2004; Olszewski 2004; Kidwell et al. 2005;

Tomaovch and Kidwell 2010; Olszewski2012; Berkeley et al. 2014; Tomaovch et al.2014), the reconstruction of baseline statesand onset of anthropogenic habitat changes(Kowalewski et al. 2000; Kidwell 2007), andcarbonate accumulation rates and flux indepositional systems (Scarponi et al. 2013;Kemp and Sadler 2014; Kosnik et al. 2014;Olszewski and Kaufman 2015). Similardynamics apply to the postmortem fates ofany organic remains, such as diatoms, bones,pollen, and wood (Cameron 1995; Kavvadiaset al. 2001; Leorri and Martin 2009; Terry 2010;Terry et al. 2011; Dawson et al. 2014).

Postmortem skeletal loss rates are evaluatedin two ways: (1) by tracking the disintegrationof individual shells through time in laboratory

Paleobiology, 42(1), 2016, pp. 5476DOI: 10.1017/pab.2015.30

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and field experiments, usually over shortperiods of a few days to years (Cumminset al. 1986; Glover and Kidwell 1993; Dittertand Henrich 2000; Powell et al. 2006, 2008;Waldbusser et al. 2011; Ford and Kench 2012);and (2) by measuring frequency distributionsof the radiocarbon-calibrated ages of shells inmixed-layer death assemblages sampled at asingle time (Meldahl et al. 1997; Olszewski1999; Kidwell et al. 2005; Kosnik et al. 2009,2013; Tomaovch et al. 2014). Short-termexperiments are extremely valuable, andcan calibrate preservation indices that trackdown-core changes in preservation (Conanet al. 2002). However, the postmortemage-frequency distributions (AFDs) producedby the latter approach reveal the full temporaldimension of time-averaged accumulation:molluscan, brachiopod, and foraminiferalshells sampled from modern seabeds, forexample, can be hundreds of years to a fewthousands or tens of thousands of years old(for reviews, see Kosnik et al. 2009; Kidwelland Tomasovych 2013). AFDs thus providedecadal- to millennial-scale perspectives onthe net effects of dissolution, bioerosion,diagenetic precipitation and burial in themixed layer. The models used to quantify ratesof skeletal loss using AFDs are representedeither by a simple exponential model where therate of loss is constant in time (Meldahl et al.1997; Kosnik et al. 2009) or by models withtemporally declining loss rates (Olszewski2004; Tomaovch et al. 2014).However, the shapes of AFDs are almost

certainly determined not simply by rates ofdisintegration and burial but also by temporalchanges in biological production (Flessa et al.1993; Krause et al. 2010; Dexter et al. 2014). Bothsimple exponential and time-varying models ofshell loss assume a constant rate of input ofindividuals into a death assemblage over theduration of time-averaged accumulation, and sotheir estimates of loss rates might be inaccurate.Populations can wax and wane on many time-frames, such as in response to annual anddecadal climate cycles, millennial-scale climate-driven changes in range size and location, andanthropogenic stresses such as acidification,nutrification, and the introduction of non-nativespecies. All of these fluctuations can occurwithin

the window of time averaging. Such changes canbe captured by AFDs, thus providing a uniquetool for one of the major goals in conservationpaleobiology and historical ecology, namelyinferring changes in abundance and productivityover the recent past. However, disintegrationand burial likely modulate the signature oftemporal history in production. It is thusimportant to assess how loss and productionjointly affect the shapes of AFDs if we are toaccurately infer past production pulses as well asdisintegration and burial rates from deathassemblages encountered in the surficial mixedlayer of the sedimentary record.

Here, first, we expand three models of shellloss within and below the mixed layer,introduced in Tomaovch et al. (2014), thatassume constant production, using these to assessthe consequences of variable production onestimates of loss rates. These models can estimateloss rates using either: (1) information on thedistributions of ages of dead individuals at thetime of their disintegration, such as derived fromtime-lapse experiments; or (2) information on thedistributions of ages of dead individuals sampledin a mixed layer at a single time (AFDs). Second,we assess how the dynamics of production andloss can be jointly inferred from mixed-layerdeath assemblages sampled from modernseabeds and landscapes. We present twomodels that couple temporal variation in produc-tion with shell loss: (1) a single past pulse ofproduction, following a normal distributiontrajectory, subjected both to constant loss and toa more realistic model where per-capita loss ratesdecline over time; and (2) a scenario of produc-tion that started at a specific time (e.g., initialflooding of a marine habitat) and terminatedat another time (e.g., anthropogenic stress),subjected both to constant loss and to temporallydeclining loss.

With these approaches, we assess: (1) theextent to which estimates of loss rates based onconstant-production models are robust tovariation in production, and (2) the extent towhich temporal variability in production couldbe masked by loss rates, thus providing usefulheuristics for conservation paleobiology todetect true changes in skeletal production. Wefocus on a single species, the aragoniticdeposit-feeding bivalve Nuculana taphria that

INFERRING PRODUCTION FROM AGE-FREQUENCY DISTRIBUTIONS 55



prefers a relatively narrow, 1956m depthzone on the southern California shelf. If thisspecies migrated with post-glacial sea-level rise,tracking its preferred shallow waters, then itsAFDs should vary significantly in shape acrossthe modern shelf if changes in the timing ofproduction control the shape of mixed-layerAFDs.

Material and Methods

Death Assemblages.A total of 253 deadspecimens of Nuculana taphria were acquiredfrom Van Veen grab samples taken at 15 siteson the continental shelf between Santa Barbaraand San Diego during the Southern CaliforniaBight Regional Monitoring Program in 2003(Ranasinghe et al. 2007) and during a sedimentmapping project of the San Diego shelfconducted by the City of San Diego in 2004(Stebbins et al. 2004) (Fig. 1). Each grab had asurface area about 0.1 m2 and penetratedapproximately 10 cm into the seabed. We

randomly selected 25 specimens per grabfrom samples that contained more than 25specimens of N. taphria. We pooled samples tofour assemblages that correspond to four waterdepth intervals along the bathymetric gradient,including: (1) seven sites at 2331m (n= 105shells), (2) three sites at 4048m (n= 66),(3) five sites at 5158m (n= 61), and (4) twosites at 89m (n= 46). A subset of the samplesfrom the 23 to 58 m depth range was analyzedby Tomaovch et al. (2014). Here, we add46 specimens from the two deeper sites (89m),and exclude two shallow-water sites from thePalos Verdes shelf where habitat modificationwas severe in the second half of the 20th century(Supplementary Table 1). Knowledge of thepreferred water depth of N. taphria is basedon our compilation of benthic data from 490stations on the southern California shelfsampled by Van Veen grabs during publiclyfunded surveys in 2003 and 2004 (Ranasingheet al. 2003, Stebbins et al. 2004). The proportionalabundance of living N. taphria peaks between19 and 56 m.

FIGURE 1. Southern California Bight study area and sample sites grouped by water depth interval (coordinates inSupplementary Appendix Table 1). Bathymetric contours in meters.

56 A. TOMAOVCH ET AL.



Radiometric Ages and Calibration of AARRatios.All 253 shells of Nuculana taphria wereanalyzed for the extent of amino acidracemization (AAR) at Northern ArizonaUniversity using the procedures of Kaufmanand Manley (1998). Eleven specimens wereselected for AMS 14C dating at the NOSAMSfacility in Woods Hole, using the screeningcriteria of Kosnik and Kaufman (2008).Radiocarbon ages have been converted tocalendar years using Calib6.0 (Stuiver andReimer 1993) and calibrated with the Marine04data (Hughen et al. 2004), using a regionalmarinereservoir correction (R) of 234 years (SD=96).The calibration curves were constrained to passthrough the origin and the D/L value oflive-collected specimens was used as value thatdetermines zero age. The ages are computed as b*([D/L]e [D/L]alive

e ), where b is a slope and e is apower-law exponent that minimize: (1) absolutedifferences between the measured 14C age andthe age predicted by the linear relationshipbetween D/Le and the calibrated 14C age,divided by the measured 14C age, and (2)absolute differences between age estimatesderived from aspartic and glutamic acid,divided by the measured 14C age. The meancalibrated AAR age (averaged over twocalibrations for aspartic and glutamic D/Lvalues, respectively) is weighted by the inverseof the standard error of the age squared. All agesare calibrated relative to AD 2003, when thesamples were collected from the seafloor(Supplementary Table 2). Details on specimenpreparation, AAR age calibration and the sourcecodes written in R (R Development Core Team2014) are presented in the Supplement and inTomaovch et al. (2014).Model Evaluation.We estimate maximum-

likelihood parameters of three modelsassuming constant production and twotruncated-normal models permitting variationin production. Akaike Information Criterion(AIC) values quantify the level of fit andpenalize for the number of parameters(Burnham and Anderson 2002). However, themixtures of truncated-normal and exponentialdensities in the varying-production modelsrepresent different classes of models.Therefore, we evaluate the goodness of fit ofboth of the two varying-production models

and the three constant-production modelsusing a G-test. The G-statistic measures thedeviation between the observed frequenciesand the frequencies expected under a randomsampling from the distributions fitted by thefive models. Although this test depends on thechoice of age binning, varying bin size between100 and 1000 years leads to similar results, andso we report the values only for bins with 250years. We also compare the observed estimatesof median age and interquartile range with theestimates predicted by models.

ModelsWe consider three scenarios of loss of

skeletal carbonates from the mixed layer (bydisintegration and burial) (Fig. 2), each repre-sented by a probability density function ofshell ages under an assumption of constantproduction. Shells can be lost from the mixedlayer of the seabed, where taphonomic processestend to be most intense (i.e., in the taphonomi-cally active zone, TAZ of Davies et al. 1989), bytwo pathways, which are not mutually exclusive:(1) disintegration (caused by any process thatdamages shells, reducing their taxonomicidentifiability); and/or (2) net burial to a deepersediment zone, eventually to a zone offinal burial(caused by any biological or physical process thatmoves shells downward, away from the TAZ).Loss can also be caused by a third pathway,namely transport to other sites (e.g., Flessa 1998),but this effect is minimized with an increasingspatial extent of sampling: out-of-habitat trans-port becomes less likely over increasingly largeareas. In addition, between-habitat transport ofsignificant numbers of shells is generally limitedto a few high-energy or non-level settings(see reviews by Kidwell and Bosence 1991;Kidwell 2013).

If disintegration rates vary vertically orhorizontally within the mixed layer, then thislayer can be subdivided conceptually into aTAZ with high disintegration rates and asequestration zone (SZ) with low disintegra-tion rates (e.g., Olszewski 2004). Our threemodels differ in the way they account for shellloss via disintegration and/or burial withinand below the mixed layer (left three graphs inFig. 2; adapted from Tomaovch et al. 2014).




The first model is defined by a single para-meter that represents an instantaneous perindividual loss rate that is constant in timewithin the mixed layer (one-phase exponentialmodel), whereas the other twomodels permit agradual (Weibull model) or an abrupt (two-phase exponential model) temporal decline inthe instantaneous per-individual loss rate.Although sedimentological and biologicalevidence implies that the loss-rate parameter isdominated by disintegration rather than byburial below the mixed layer in marine systems(Tomaovch et al. 2014), the relative contribu-tions of disintegration and burial processes tothe loss parameter likely vary as a function oftaxa and environments (Terry and Novak 2015).

All three models assume that production isconstant. To evaluate the joint effects ofvariation in production and shell loss, wecombine our one-phase and two-phase modelsof loss described above with (1) a gradualchange in production following a normaldistribution trajectory; and (2) a stepwisechange in production following a rectangulardistribution trajectory, with productionincreasing sharply at one point and thendeclining sharply at a later point.

Censored age dataWhen using age data (e.g., AFDs) to estimate

the time until the occurrence of some event(here, the timing of loss of a shell from a deathassemblage), those age data are considered tobe censored if the event of interest did notoccur at the actual timewhen the age datawerecollectedthat is, the event of interest eitherpre-dates or post-dates the time of sampling(Kleinbaum and Klein 2005). Right censoringrefers to the scenario where the eventhere,the loss of individuals from the death assem-blagehas not yet occurred: the time to shelllossi.e., the postmortem shell lifespanexceeds the observed age at the time ofsampling. We focus here on right censoringbecause the age of shells sampled on theseafloor pre-dates their loss. Left censoringrefers to a scenario where the loss of indivi-duals occurred before the age data weresampled (i.e., the time to shell loss is smallerthan the observed age). Such censoring wouldapply where rates of fragmentation are esti-mated on the basis of the postmortem ages ofshell fragments (fragmentation could haveoccurred at any time prior to sampling), or intime-lapse experiments where rates of loss are

FIGURE 2. AC, Three changes in loss rate as a function of shell age in the mixed layer, with A, one-phase model ( isconstant), B, Weibull model ( declines gradually with time), and C, two-phase model ( declines abruptly from high 1to a much lower rate 2). Loss of shells from the mixed layer can arise from disintegration and/or from burial to deeperlayers. The depth of burial within the mixed layer may or may not correlate with shell age. DF, Three possiblescenarios of the two-phase model plotted against sediment depth. D, Shells subject to high 1 in the TAZ (white layer)shift by burial to the SZ (gray layer) where they experience much lower 2. Variable corresponds to net burial rate ifshells that are exhumed back to the TAZ but do not retain their 2. Sampling gear penetrates into the SZ. E, The samescenario as in (D), but shells exhumed back to the TAZ retain 2, e.g., if burial in the SZ is coupled with diageneticstabilization. Sampling gear not penetrating into the SZ will still encounter shells that reflect the two-phase model.F, the SZ represents a patchy microenvironment favoring diagenetic stabilization. Diagenetic stabilization still results ina decline of with time but with little or no burial below the sediment-water interface. Final burial zone (FBZ) denotesa position below the sediment-water interface where shells are beyond the reach of further physical and biogenicreworking, and where further disintegration is determined by the realm of late diagenesis.




inferred from the first samplingwhen shells areno longer present rather than from the lastsampling when they are present.Figure 3 visualizes the importance of right

censoring of postmortem lifespans of shellsunder conditions of temporally constant produc-tion (the postmortem lifespans of shells arerepresented by horizontal black lines, eachbounded by the time of input to and loss fromthe death assemblage) and a single time of

sampling (vertical dashed line). The time ofsampling splits postmortem lifespans into right-censored parts (light grey shading) andunsampled parts (dark grey shading). Severalfeatures are evident. First, shell age at the time ofsampling underestimates the full postmortemlifespans of the shells that are sampled in themixed layer. This effect decreases the medianage of sampled shells relative to the median ageof a random selection of shells at the time of theiractual loss. Second, shells with shorter lifespansare less likely to be sampled than are shells withlonger lifespans. This effect increases themedianage of sampled shells relative to the median ageof a random selection of shells at the time of theiractual loss. These two biases cancel each other inmodelswith constant loss (Fig. 3A): the snapshotAFD produced by sampling shells in the mixedlayer at any given moment (gray histogram)provides an accurate estimate of the AFD thatwould be produced by a random sampling ofthe true, full lifespans of shells (black histogramin Fig. 3A), with differences in AFD shapearising purely from the effects of finite sampling.In contrast, if loss rates change over time, thesnapshot AFD provides a biased estimate of theAFD based on a random sampling of fulllifespans (Fig. 3BC; we use a simple Weibullfunction to explore the effects). Older shells aremore likely to be sampled than younger shells inmodels with decreasing loss rates (as in the

FIGURE 3. Visualizing the effects of right censoring on theshape of AFDs sampled at a single time (vertical dashedlines) under constant shell production (with 100,000shells produced) and A, temporally constant loss rates,B, temporally declining loss rates, and C, temporallyincreasing loss rates. Black horizontal lines representindividual shells bounded by the time of their input(death) on the left and by the time of their loss on theright. The lengths of lines thus correspond to fullpostmortem lifespans of shells (for clarity, each plotshows only 2,500 shells). The time of sampling splits thelifespans into: (1) thick light grey bands representing thecensored ages of shells (i.e., their right-censored lifespans),and (2) thick dark gray bands representing the unsampledportions of lifespans. Gray histograms represent snapshotAFDs, i.e., the expected frequency distributions of (right-censored) shells ages at the time of their sampling on theseafloor, and the black histograms represent the expectedfrequency distributions of a random sample of the full,true lifespans of shells. In contrast to a scenario withconstant loss (A), the snapshot AFD provides a biasedestimate of the AFD based on a random sampling of fulllifespans if loss rates change over time (B, C).




Weibull and two-phase models in Fig. 2;Fig. 3B), and older shells are less likely to besampled than younger shells in models withincreasing loss rates (Fig. 3C). Under conditionsof constant production, the right-censoreddistributions will always monotonicallydecrease from young to old postmortem ages.The most frequent shells will thus tend to have azero age, that is will fall in the first age bin,regardless of the shape of the underlyingnon-censored probability density function. Thatfunction can be non-monotonic, for example ifloss rate increases with time (Fig. 3C).

Therefore, AFDs sampled at a single timeare different from the AFD based on a randomsampling of the full postmortem lifespansof shells (i.e., the kind of data more likelyto be generated by time-lapse experiments);only in the special case when the loss rateis constant would a snapshot AFD match theAFD of true lifespans (Fig. 3A). Probabilitydensity functions used to estimate times to losson the basis of censored age data thus differfrom those using non-censored age data.Analyses of survivorship in population biol-ogy (Colchero and Clark 2012) and evolution-ary paleobiology (Gilinsky 1988; Foote andRaup 1996; Ezard et al. 2012) also correct forage censoring. This issue thus has conse-quences for many paleobiological studies thatwork with age data.

Models with Constant ProductionFor all three constant-productionmodels, we

present both: (1) non-censored and (2) rightcensored density functions because both typesof age data can be collected in taphonomicand paleobiologic analyses. Non-censoredfunctions can be used to quantify loss rate bymeasuring shell ages at the time of theirloss. This kind of dataor a very closeapproximation of it, depending upon thespacing of successive samplingsis producedby experiments that track individual shells orcohorts of shells through time (e.g., Cumminset al. 1986; Kotler et al. 1992; Glover andKidwell 1993; Simon et al. 1994; Powell et al.2008, 2011; Waldbusser et al. 2011). Right-censored density functions permit one toestimate loss rates using the frequency

distribution of shell ages that are sampled inthe mixed layer at a single time.

Thus, for each constant-production model,we specify a probability density function f(t)that represents the likelihood of a shell beinglost at time t, and a right-censored probabilitydensity function g(t) that represents the like-lihood of a shell being sampled in the mixedlayer at time t, assuming that time-to-lossfollows the density f(t). In general, to relatef(t) and g(t), we first use f(t) to calculate asurvival function S(t), which is the probabilitythat a shell has not been lost at time t:

St 1s ts 0

f s ds; (1)

where s refers to time intervals between thetime of death (time of input to the deathassemblage), i.e., 0 years, and time of loss t.The ages of the sampled shells then follow adensity that is proportional to S(t). For exam-ple, if half the shells disintegrate by time t1 (thatis, S(t1)= 0.5), then we are twice as likely tosample a shell of age 0 than a shell of age t1(assuming a steady rate of shells entering thesystem). This means that the right-censoreddensity g(t) is

gt 1=C St; (2)where C is a normalization constant ensuringthat g(t) integrates to one,

C 1t 0

St dt: (3)

One-Phase Exponential Model of Constant LossRate.This function is identical to radiometricdecay, with the distribution of time-to-lossf(t) equal to

f t et; (4)where t is shell age and is the maximum-likelihood estimate of the instantaneousper-individual loss rate. The density for aright-censored distribution g(t) has the sameform as f(t) because of a shell is independentof the time that shell has already spent in adeath assemblage (see Fig. 3A). This modelspecifies that is vertically and horizontallyhomogeneous throughout the mixed layer.Burial to deeper layers and transport laterallyto some other site can contribute to the loss of ashell from the mixed layerthe formulation




does not specify that loss must be exclusivelyor largely from disintegration.Weibull Model of Gradual Decline in Loss

Rate.The Weibull density describes theinstantaneous per-individual loss rate usingtwo parameters r and k. The parameter r is abaseline rate in the exponential function thatis raised to the power of k: if k< 1, the lossrate declines over time; if k> 1, the loss rateincreases; and if k= 1, then the Weibull densityis equivalent to the one-phase exponentialdensity and the parameter r is equal to theparameter . The decline in loss rate in thisformulation is thus expected to be gradual, i.e.,progressing through many very smallincremental steps. At small k values (kT

; (7)

and

gt 1C e

1t tTe2tT e1T; t>T

; (8)

where F1 is the disintegration rate for shellsyounger than or equal to T (i.e., shells locatedin the TAZ), F2 is the loss rate for shells olderthan T (encompassing the disintegration rate inthe SZ and processes that bury shells below theSZ), T represents the elapsed postmortem timeuntil shells are subjected to F2, and C repre-sents a normalization constant equal to (1/F1)(1 e F1

T)+ (1/F2)e

F1T.

It is more likely, however, that the time ofsequestration that separates 1 in the TAZ from2 in the SZ varies among individual shells, i.e.each shell follows its own trajectory owing tothe vagaries of burial within the mixed layerand/or diagenetic stabilization. We place anexponential distribution on this random timeto sequestration of shells of a given age and




describe it by a rate . This rate permits varia-tion among shells in the time at which theyabruptly shift to a lower disintegration rate andcan also accommodate scenarios where shellsof different ages are buried at the same time. Incontrast, Terry and Novak (2015) model burialprocesses using a mean burial rate anddiffusive mixing; disintegration rate does notchange with the depth of burial.

To summarize the two-phase exponentialdensity, for each shell that enters the TAZ:

1. We first run two independent exponentialloss processes, both starting at the time thatthe shell enters the TAZ. One process hasrate (representing sequestration to the SZ)and the other has rate 1 (representingdisintegration within the TAZ).

2. If disintegration in the TAZ occurs first (i.e.,before burial or diagenetic stabilization to theSZ), then the shell has disintegrated and doesnot move to the SZ. If sequestration to the SZoccurs first, then the shell has moved to the SZand now we run an exponential loss processwith slower rate 2 to determinewhen the shelldisintegrates and/or is buried to the next layer.

The probability density for this modelin non-censored f(t) and right-censored g(t)forms is

f t 1 + 1 e + 1t + 2e2t; (9)where = /(+ 1 2),and

gt 1 + 1 e + 1t + 2e2t; (10)where = (+ 1)/((+ 1)+ (1 2)2). Largedifferences between 1 and 2 can generate

strongly right-skewed and long-tailed, visiblyL-shaped distributions (Tomaovch et al.2014). Although the two-phase exponentialmodel predicts an abrupt decline in disin-tegration rate for each shell, the mean trajec-tory in disintegration rates nonethelessdeclines gradually because individual shellsmake this transition from TAZ to SZ at differ-ent times. The timing of this decline and thekink in the AFD shape is determined not justby but also by 1 because higher 1 shifts thekink to smaller ages.

Results

Sensitivity of Loss-Rate Parameters to VariableProduction

In reporting the sensitivity of loss rates tovarying production, we focus on: (1) the one-phase model, which provides a useful baselinebecause it is nested within Weibull and two-phase models; and (2) the two-phase model,which has the highest AIC weights and thesmallest G-statistic (see below). The disintegra-tion rates that we test encompass the range ofempirically estimated parameters derived fromdeath assemblages from the Southern CaliforniaBight (adopted from Tomaovch et al. 2014).

In the first scenario, we assess the effects ofregular fluctuations in production on loss-rateestimates, with samples of n=100 individuals.We generate sinusoidal fluctuations in produc-tion with five periods (25, 50, 125, 500 and1000 years) and standardize them so thatproduction varies between 0 and 1 (Fig. 4A).Sampling occurs at the time of minimumproduction, representing the scenario with the

FIGURE 4. Effects of fluctuations in production and of a recent termination in production on estimated rates of shellloss, for three different true rates of loss. A, Hypothetical histories of fluctuation in production (dashed black lines),with each fluctuation ranging from 0 to 1 and recurring at periods from 100 to 1000 y, and of sudden terminations inproduction at varied times in the past (heavy gray lines). To maximize bias, production is at a minimum value at thetime of sampling the death assemblage (0 years). The effects of production period (B) and of the timing in terminationof production (C, D) on loss rate (left column), on 1 under = 0.00025 and = 0.0001 (middle column), and on 2(right column), using three true values of and 1 ranging between 0.05 (solid black lines, half-life= 13.8 years), 0.005(dashed dark gray lines, half-life= 138 years), and 0.0005 (dotted light gray lines, half-life= 1386 years), and using2= 0.0001. If models do not take into account fluctuations in production (dashed lines in A), estimates of and 1 arebiased downward but only if the production period exceeds one disintegration half-life; in those cases, estimates of 1converge towards the very low values of 2. C, in models assuming constant production and not accounting for recenttermination in production, estimates of and 1 are biased downward across almost the entire range of times oftermination in production (gray lines in A) and converge towards values of 2. D, when models assume constantproduction but add a prior Tmin for recent termination in production, for example based on knowledge of recent climateor cultural change, then estimates of loss rates are accurate as long as the duration of zero production does not exceedone disintegration half-life. Thick and thin lines represent means and 95% confidence intervals.







largest potential for bias. We multiply thesefluctuations with survival functions (withpredetermined loss-rate parameters) thatspecify the probability that a shell has not yetdisintegrated at time t in order to generatethe probabilities of sampling individualsthat differ in age. We then simulate AFDs bydrawing shell ages (with replacement) withsuch probabilities, and then estimate fittedmodel parameters.

In the second scenario, we assess the sensitivityof parameters to a sudden termination inproduction (preceded by constant production)at time Tmin=50, 100, and 500 years before thetime of sampling, at n=100 individuals.Although such a decline in production to zerolevels is more likely to be gradual than abrupt,this extreme scenario with an abrupt droprepresents the worst-case scenario for bias inestimating loss rates.

Effects of Variable Production.We find thatall loss-rate estimates are robust tofluctuations inproduction whose periods are at time scalescommensurate with the time scales of half-lives(Fig. 4B). If the fluctuation has a long period, sothat production is at a relatively prolonged lowphase at the time of sampling (i.e., the lastmaximum in production occurred in the distantpast), then few shells are produced recently toregister the high loss rate; the proportion of thedeath assemblage in the second phase of shellloss in two-phase models thus tends to beoverestimated. Once the period of fluctuation inproduction approximately exceeds the durationof the half-life driven by 1, then the initial lossratewill be underestimated, declining towards 2(set to 0.0001 in Fig. 4BD). Similarly, ifproduction stops abruptly after formerlyconstant production, a shortage of shells in theearly high-loss phase (thus having half-lives thatare shorter than the duration of zero production)will lead to marked underestimates of 1(Fig. 4C). This dependence of estimates of or1 on active skeletal production has clearimplications for published estimates of half-lives that have been based, perhapsinadvertently, on populations that havedeclined in production. Death assemblagesproduced under such conditions will haveAFDs that are better fitted by a one-phase thanby a two-phase model, and the calculated half-

life will reflect the loss rate in the SZ (2). In AFDsfrom settingswithout active production of shells,information on the initial disintegrationtrajectory in the TAZ (1) is thus missing.

Effects of Loss on the Detection of VariableProduction

We develop here two models where one-phase and two-phase models of loss are com-bined with temporal changes in production inorder to determine the ability of AFDs to capturean unbiased signal of such changes, exploringtwo different scenarios of production (Fig. 5).First, the history of production is envisioned as anormal distribution that mimics a gradualincrease and/or a decrease in production;shells are then subjected to either a one-phase(Fig. 5A) or a two-phase loss model(Fig. 5B). The fluctuation in production may bebrief (left graph in Fig. 6), e.g. in response to ashort incursion of well-oxygenated water, orprolonged (other graphs in Fig. 6), e.g. reflectingthe tracking of a preferred water depth. Second,we envision a sudden onset of production (e.g.,no production occurs before the time of trans-gression) followed by constant production,which is suddenly terminated to zero levels,generating a rectangular distribution (Fig. 5C).

We find that normal trajectories in produc-tion are transformed into unimodal AFDs withsignificantly different shapes when combinedwith a one-phase model of constant loss (Fig.5A with = 0.001). Single pulses can betransformed into bimodal AFDs when com-bined with a two-phase model of constant loss(Fig. 5B with 1= 0.01 and 2= 0.0005). Incontrast, rectangular trajectories simply trans-late into exponential AFDs, regardless of lossmodel (Fig. 5C). Four empirical AFDs from thesouthern California shelf were fitted to the firsttype of model. The second model requiresindependent estimates of production onset andtermination (see below).

One-Phase Truncated-Normal Model.Thismodel shows how a one phase model of lossfilters the normally distributed trajectorygenerated by a single fluctuation inproduction (Fig. 6). If production over timefollows a normal distribution with mean (i.e., the time of maximum production) and




variance 2 (determined by the rate of increaseand decrease in production), and the time todisintegration follows a one-phase model withloss rate , then the expected distribution ofshells with ages > 0 years is equal to

gt t; 2;

2

; (11)

where refers to the density for a normaldistribution with mean t equal to 2 andwith variance equal to 2, and refers to thecumulative distribution function for a normaldistribution with mean equal to 0 and varianceequal to 1. The frequency distribution of shellages thus follows a truncated-normaldistribution (i.e., only positive shell ages aredrawn from a normal distribution). The higherthe loss rate (thin black lines in Fig. 6, asopposed to thin gray lines) and/or the lowerthe rate of decline in production (decreasesfrom left to right in graphs of Fig. 6), then thelarger the difference between the originalproduction trajectory and the snapshot AFD,and between t and (Fig. 7). As loss ratesincrease and/or the rates of change inproduction decrease, the modes of AFDs will

be shifted to younger age cohorts, causing thetrue timing of past production pulses to beunderestimated (Fig. 6). High loss rates resultin only the most recent part of the productionpulse being captured by the death assemblage.Even when the mode in the AFD is youngerthan the true peak in production, the varianceof its truncated-normal distribution is equal tovariance in the trajectory in production, and socan be used to directly estimate the rate ofincrease or decrease in production.

Two-Phase Truncated-Normal Model.Thismodel shows how a two-phase model of lossaffects the AFD generated by a singlefluctuation in production. If the time todisintegration is a mixture of two phases ofloss, then the expected distribution ofpostmortem ages is

gt t; 22;

22 + 1 t; 1 + 2;

1 + 2

:(12)

The distribution of postmortem ages is thus amixture of two truncated-normal distributions,with means equal to t1= (1+ )

2 and

FIGURE 5. Preservation of production trajectories that follow (A, B) a normal distribution of gradual increase anddecrease, and C, a rectangular distribution with an abrupt onset and termination. Dashed lines show the originalproduction trajectories with production pulses at three different times, and solid lines depict the expected shape of thecorresponding snapshot AFDs. Thin dotted lines are survival curves of shells (i.e., expected trajectories of loss at= 0.01 in a one-phase model and at 1= 0.01 declining to 2= 0.0005 in a two-phase model). In A and B, vertical solidlines represent apparent timing of peak of production, with arrows showing the offset relative to the true timing ofproduction pulses. In B, the decline in loss rates can produce bimodal distributions. In contrast, rectangular trajectories(C) translate into exponential distributions that are not shifted in time relative to the original production trajectories.




FIGURE 6. The joint effects of loss rate () and of rate of change in production (i.e., inverse of standard deviation of theproduction trajectory) on the shape of an AFD, assuming A, a one-phase model and B, a two-phase model of loss.Production is assumed to have peaked 5000 years ago and followed the trajectory of a normal distribution (thick grayline). Each graph shows the effects of varying on the shapes of AFDs under a specific standard deviation of thistrajectory, from a temporally narrowly focused event, e.g. injection of favorable food or oxygen, to more prolongedphases, e.g., tracking a preferred water depth across the site. AFDs do the best job of registering the existence of a pastpeak in production, preserving it as a distinct mode, if: (1) that event was narrowly focused in time and productiondeclined rapidly (left columns), and/or (2) is low (gray dashed and solid lines). As the peak broadens in temporalduration (right columns) and/or loss rate becomes high (black dashed and solid lines), the AFD does not match theoriginal production trajectory. Past pulses of production are better captured by two-phase models with declining lossrate.

FIGURE 7. Effects of shell loss and of rate of change in production on the timing of maximum past production. Assuming aone-phase model of loss, the difference between the true time of maximum production (i.e., mode of the originalproduction trajectory) and the mean of the truncated-normal model estimated on the basis the AFD (i.e., mode of the AFD)increases with A, increasing loss rate (scenarios differing in standard deviation shaded in different colors) and B, increasingstandard deviation of the original production trajectory (i.e., decreasing rate of change in production). The same two effectswill also affect the difference between the true and apparent time of maximum production under a two-phase model ofloss, but this difference will be reduced with increasing (time to attain SZ) and decreasing 2 (loss rate in SZ).




t2= 22. Identical values of t1 and t2imply that the difference between 1+ and 2is negligible, thus supporting the one-phasemodel rather than the two-phase model of loss.As in the one-phase model described above,the relationship between these means and theoriginal time of maximum production depends on loss rates (determined by 1, ,and 2) and on the magnitude of

2: both ofthese parameters shift t, and thus also theobserved AFD mode, towards younger agesrelative to the true timing of the productionpulse (). The timing of maximum production can nonetheless be inferred by using arange of loss rates estimated at sites whereshells are still being produced, i.e., where a keyassumption of the loss-rate model is notviolated.

In contrast to the effects of a one-phasemodel, the two-phase loss model can, undersome circumstances, generate bimodal AFDs.The young mode is composed of shells stillsurviving from recent, post-pulse production,and the older mode is composed of shells sur-viving from the past pulse of production:sequestration in the SZ allows the relicts of pastproduction pulses to persist in death assem-blages even when an AFD is dominated by theyoungest cohorts, which peak at ~0 age. Thisbimodality thus arises under conditions of high1 and very low 2 (thereby minimizing theshift of the production pulse towards moderntime), combined with an intermediate rate ofchange in production. At a very slow decline inproduction, only the recent production ispreserved, whereas at a very rapid decline inproduction, only the past production pulse ispreserved.One-Phase and Two-Phase Models of Abrupt

Increase and/or Termination in Production.Thetime of first colonization of a seafloor orlandscape, for example determined by theflooding of an estuary or human introductionof an alien species, can lead to a rapid increasein production; a natural or anthropogenicdisturbance can also result in an abruptdecline in production to zero levels. If thetiming of first colonization Tmax and/or thetiming of the termination in production Tmincan be estimated from independent data, thenloss can be estimated by the addition of a prior

probability that accounts for times with zeroproduction, so that f(t)= 0 for tTmax,f(t)= 1 for tTmin, and tTmax. The expecteddistribution of postmortem ages for one-phasemodel is

gt 11 e1Tmine1Tmax

1

24

35n

: n

i11 e1t

: (13)

The expected distribution of postmortem agesfor two-phase model is

gt 1e2Tmine2Tmax

2+ 12e

+ 1 Tmine + 1 Tmax + 1

24

35n

Yni1

e2t + 12e + 1t: 14

By adding to the model a prior probability thataccounts for the absence of shells that areyounger than Tmin (Fig. 4D), estimates of lossrate can be estimated with higher accuracythan those obtained from constant-productionmodels (Fig. 4C).

AFDs from the Southern California ShelfOnshore-Offshore Gradient in the Shape of

AFDs.AFDs change markedly in their shapewith increasing water depth. The median shellage increases from 21 y at 2331m, to 141 y at4048m, 4778 y at 5158m, and 12,112 y at89m (Table 1). AFDs of the three shallowestassemblages show a L-shaped, hollow-curvedistribution: they are dominated by shells

Model Fitting.The two-phase model has anAIC weight equal to ~1 and thus outperformsboth the Weibull and the one-phase models inexplaining the shape of the AFDs of the threeshallowest assemblages (Table 2; as also foundwith a larger array of AFDs in this region byTomaovch et al. 2014). The two-phase modelalso does best at explaining these assemblagesusing a G-test and outperforms other models inpredicting the median age and IQR ofthree shallowest assemblages (black points inFig. 9A,B). However, the G-statistic of theassemblage at 5158m is not significantlydifferent from the prediction of the two-phasetruncated-normal model. This model clearlycaptures the bimodal shape of the AFDobserved at this water depth, suggesting thefingerprints of a past production pulse here.

In contrast, based on AIC, the Weibull modeloutperforms a two-phase model in the deepest,89-m assemblage, where it predicts a ratheruniform right-censored distribution withk= 13.6. When k > 1, the Weibull function cangenerate unimodal and even left-skewed non-

censored distributions because shells degrade ata higher rate as they age. However, such adynamic cannot generate a left-skewed right-censored distribution. Based on the G-statistic,the deepest assemblage is best supported byboth of the truncated-normal models that allowproduction to vary, and the expected distribu-tions of these models do not differ significantlyfrom the AFD observed at 89m (Table 2).Truncated-normal models also predict themedian age and IQR of the deepest assemblagefairly accurately (stars in Fig. 9A,B).

Estimates of Disintegration and Sequestration.In the three shallowest assemblages (2358m),1 varies between 0.076 and 0.53 (correspondingto decadal to yearly half-lives, or a annual lossof 7 and 41% of shells in a cohort, Table 3) and istwo orders of magnitude larger than 2, whichvaries between 0.00018 and 0.00031 (half-livesof ~3900 to 2200 years, or annual loss of 1.7 and3%). ranges between 0.00019 and 0.00064,which correspond to the median time tosequestration (stabilization or burial) fallingbetween ~3650 and 1100 years. The variables

TABLE 1. Median age, IQR, total age range, and skewness of age-frequency distributions (AFDs) of Nuculana taphria infour bathymetrically arrayed death assemblages from the southern California continental shelf. Median is in years before2003 AD with bootstrapped 95% confidence intervals in parentheses. N=number of shells dated.

2331m 4048m 5158m 89m

N 80 66 61 46Median age (years) 21 (16582) 141 (191602) 4778 (27935685) 12112 (1070613486)IQR (years) 2731 (10663834) 3136 (17045149) 6908 (40307558) 4880 (32766843)Range (years) 11332 (923011861) 11903 (880612685) 10761 (987011127) 21189 (1532122849)Skewness 1.7 (1.22.4) 1.5 (0.92.2) 0 (0.30.4) 0.3 (0.90.6)

FIGURE 8. AD, Age-frequency distributions (AFDs) of shells of the bivalve Nuculana taphria in death assemblagessampled at four water depths on the Southern California continental shelf, based on amino acid racemization datingand calibrated in years before 2003 AD. With increasing water depth, median shell age increases and the skewness ofAFDs decreases.




and 1 are independent and thus, even when is much smaller than 1, some shells can still besequestered during the first few decades (e.g.,about six out of 1000 shells at = 0.00064), evenwhen they represent a minor proportion of theoriginal production.

The Weibull model also outperforms a one-phase model in shallow assemblages, and itsvery low shape parameter (k= 0.090.20)implies a similar, markedly rapid temporaldecline in loss rates. The baseline rate para-meter r of the Weibull model is extremelyhigh (1061015), but likelihood surfaces ofthe Weibull model (1) yield a very broad

range of rate estimates with very similarlog-likelihood values, and (2) show a negativecorrelation between the estimates of r and k.In contrast, log-likelihood surfaces showthat two-phase model parameter estimatesare uncorrelated and have relatively steep log-likelihood surfaces (Supplementary Figure 1).The parameters of the Weibull models arethus less precise compared to the estimatesof two-phase models. In spite of the increasein median shell age and IQR from 23 to 58m(Table 1), the two-phase model parametersdetects a two order-of-magnitude decline inloss rates in each of the three shallow-water

TABLE 2. Estimated parameters of three models with constant production and two models with varying production(see explanation in text).

2331m 4048m 5158m 89m

1-phase 0.00059 0.00050 0.00023 0.000082Weibull r 1487099.8 9.45 1015 1.9 0Weibull k 0.11 0.07 0.2 13.582-phase 1 0.081 0.532 0.11 0.0572-phase 2 0.000255 0.000314 0.000184 0.0000802-phase 0.00019 0.00052 0.00064 0.003791-phase truncated-normal mean t 2383131 2723876 252 121871-phase truncated-normal st. dev. 63785 73108 5472 43542-phase truncated-normal mean t1 4698 847 6942 121582-phase truncated-normal mean t2 64029 107929 18230 121222-phase truncated-normal st. dev. 2724 2909 3287 4328

FIGURE 9. A, B, Comparison of median ages and interquartile ranges (IQR) observed in death assemblages and thosepredicted by five models of loss. In the three shallowest assemblages (from 23 to 58m), both summary statistics areaccurately predicted by a two-phase model (black circles): these predicted values lie closest to or on top of the diagonalline indicating equivalence with observed values. In contrast, the deepest assemblage (89m; with median= 12,113 yearsand IQR= 4880 years) is best predicted by truncated-normal models (crosses and stars), indicating that production hasnot been constant. Some results plot on top of others so that not all 20 data points are visible. C, Estimates of loss rates1 and 2 (black and gray circles, respectively) and of sequestration rate (white circles) for AFDs along a bathymetricgradient (with bootstrapped 95%confidence intervals). Even though the shapes of AFDs change markedly withincreasing water depth (x-axis), each parameter remains fairly constant along the bathymetric gradient, with 1 and 2differing consistently by two orders of magnitude.




assemblages, and yields water depth-invariantestimates of 1, , and 2 across most of the shelftransect (Fig. 9C).

Estimates of Past Changes in Production.AFDs that are better fit by constantproduction models can nonetheless stilloriginate under variable production becausethe signature of a change in production isessentially removed by high loss rates and/orby slow rates of change in production. Thus,when estimating the timing of a past change inproduction, we focus both on the 89massemblage and on the 5158-m assemblage:the latter is better supported by a two-phasemodel with constant production, but appearsto be bimodal as predicted by the two-phase truncated-normal model, with thesecond mode at ~6000 years. The two-phasetruncated-normal model shows that thestandard deviation of the original normaldistribution is ~3287 years for the 5158massemblage and ~4328 years for the 89massemblage: that is, in both water depths, therise and decline in production occurred on amillennial scale. The estimated means of thetruncated-normal distributions are ~6942 yearsago for the 5158-m assemblage and~12,100 years ago for the 89-m assemblage(slightly older than the observed mode of thatdistribution at 11,591 years). We use (1) the lossrates estimated from the two-phase models

and (2) estimates of and 2 from the two-phase truncated-normal models to computethe timing of maximum production (i.e., themean of the normal distribution) forassemblages at 5158m (gray lines in Fig. 10)and at 89m (black lines in Fig. 10): thetwo-phase dynamic is best supported at theshallow sites with active production. Usingthe range of loss rates estimated from allfour assemblages, the timing of maximumproduction was ~780010,300 years ago for5158m (Fig. 10C) and ~13,65018,050 yearsago for 89m (Fig. 10D).

Discussion

Mechanisms of Disintegration andSequestration.The shallowest deathassemblages with active production ofNuculana taphria shells in southern California,that is those assemblages collected fromseafloors within the preferred 1956 meterdepth range of this species, all possess right-skewed AFDs that are better supported bymodels assuming constant production thanmodels assuming variable production. Inthese assemblages, our analysis gives thestrongest support to models that imply a veryhigh disintegration rate (decadal half-lives),such as commonlymeasured during bench and

TABLE 3. Goodness of fit with AIC weights and the G-statistic for three constant-production models and twotruncated-normal models that allow production to vary. The three shallowest assemblages are best supported by atwo-phase model on the basis of AIC weights and G-test. The assemblage from 89m has stronger support from bothtruncated-normal models.

2331m 4048m 5158m 89m

N 80 66 61 461-phase model AICc 1352 1138 1144 960Weibull model AICc 1157 942 1138 9222-phase model AICc 1070 921 1067 9621-phase model AIC weight 0 0 0 0Weibull model AIC weight 0 0 0 12-phase model AIC weight 1 1 1 01-phase model G-statistic 68.4 55.1 57.9 62.21-phase model p-value 0.01065 0.16908 0.05175 0.95703Weibull model G-statistic 25 26.2 56.4 52.1Weibull model p-value 0.98724 0.98892 0.05503 0.99592-phase model G-statistic 17.8 21 40 61.42-phase model p-value 0.99961 0.99871 0.47047 0.948951-phase truncated m. G-statistic 68.3 55 54.9 39.11-phase truncated m. p-value 0.00832 0.14507 0.07216 0.999982-phase truncated m. G-statistic 26.4 22.8 43.5 39.12-phase truncated m. p-value 0.97096 0.99663 0.32294 0.99998




field experiments, followed by a slow long-term disintegration rate and/or slow net burialrates below the mixed layer (millennial half-lives) that can explain the large window oftime averaging. The rapid decline in loss rateexperienced by shells as they age can beconceptualized as the threshold betweensignificantly different preservation regimes. Forintrinsically fragile shell types (such as very thinand/or organic-rich shells), a fast-track to the SZvia burial may well be the only successful routefor long-term carbonate preservation: this can beachieved if the TAZ is very thin, or if large-scaleburial events are frequent, e.g. via downwardadvection by burrowers or thick storm deposits(e.g., Aller 1982). Refuge in the SZ can also beachieved in sediments where the probability ofdiagenetic stabilization increases abruptly acrossshort physical distances, for example fromincomplete bioturbation that permits pockets ofsaturated or supersaturated porewaters near thesediment-water interface. While in thosepockets, we postulate diagenetic stabilization,for example from intercrystalline cementationand coatings (Perry 1999; Rivers et al. 2008;Jarochowska 2012), annealing (coarsening ofmicrostructure) in the absence of mineralogicalchange (Walter and Morse 1984; Morse andCasey 1988; Hu and Burdige 2007), and/orrecrystallization of aragonite or high-Mg calcite

to thermodynamically more stable minerals(Reid and Macintyre 1998; Brachert and Dullo2000; Hover et al. 2001). Any of these processesmight reduce the likelihood of disintegrationduring reworking. Safety in numbers,whereby a primary concentration of shellsbenefits by self-buffering of porewatersand/or by resisting erosional reworking,might also contribute (e.g., Seilacher 1985;Kidwell 1989), as might bioencrustation andbioimmuration. These diverse mechanisms arenot mutually exclusive, and all have beeninvoked, singly or in combination, to explainthe paradox of preservation of diverseskeletal carbonates, especially aragonite, underconditions of low net seafloor aggradation,although burial is usually favored (e.g., Kidwell1985, 1989; Davies et al. 1989). Preliminary SEManalyses imply that N. taphria shells only a fewthousands years old show coalescence of originalcrystallites, suggesting that the decline in lossrates is related to microstructural changes.

The two-phase model does not explicitlydisentangle burial of shells downward fromexhumation of shells upward (from the SZ intothe TAZ). If sequestration (movement to a SZ)is coupled with burial (greater sediment depthbelow the depositional interface), can corre-spond to a simple burial rate (in the absence ofexhumation) or to a net burial rate when burial

FIGURE 10. Inferring the timing of maximum production based on AFDs from two water depths (5158m and 89m),assuming that the change in production follows a normal distribution trajectory. In A, we use the probability densitiesof the two-phase model estimated from all four assemblages because they are rather depth-invariant (Fig. 8C). B, Thetruncated-normal models fitted to the 5158m (gray line) and 89m (black line) assemblages. C, D, the inferredproduction trajectories at 5158m and 89m, where the means of the distributions are determined from the four two-phase densities in A and from the fits of the truncated-normal models in B. The standard deviations are determinedfrom the fits of the truncated-normal models in B. Past pulses in production (sets of thick lines in C, D) occurred a fewthousand years earlier (arrow) than would be suggested by the modes of the observed truncated-normal distributions(faint dashed line shows fit as determined in B).




events are counteracted by exhumation events.When partitioning of 1 and 2 is generated byvertical separation of TAZ and SZ rather thanby horizontal heterogeneities, two-phasemodels can conceptualize the AFD of a deathassemblage sampled by gear that bites downinto (1) at least some portion of the SZ (Fig. 2D)or (2) into the TAZ only, but where that TAZhas received shells reworked upward aftersome period experiencing low rates of loss in aSZ (Fig. 2E).

In situations where (1) the sampled deathassemblage derives entirely from the uppermostpart of the seabed, presumably just the TAZ, andyet (2) the empirical data support a two-phaseexponential model, then (3) the estimate of 2indicates that shells were in fact reworked upfrom the SZ to the TAZ. At least two naturalscenarios are possible. In the first, shells arediagenetically stabilized irreversibly during theirresidence in the SZ, and thus keep their slow lossrates 2 after theymove back to the TAZ (Fig. 2E).In the second, shells revert to fast disintegrationrates 1 when they are reworked back into theTAZ (Fig. 2D). They are older than other shells inthe TAZ because they had a time-out fromthose fast loss rates but, once back in the TAZ,they resume disintegrating as fast as youngershells there. In this second scenario, 2 in the SZwill be underestimated relative to its true values.Both scenarios produce a set of shells within theTAZ that are much older than they would havebeen had they spent their entire postmortemexistence within the TAZ. They can be thusexpected to produce a strongly right-skewed andlong tailed, characteristically L-shaped AFDwithin the TAZ.

Sampling designs with stratified depth ofburial, presently underway on the southernCalifornia shelf by us (2-cm core increments) andelsewhere by others, will be needed to detectwhether old shells (1) derive from deeper parts ofthe mixed layer corresponding to the SZ, and/or(2) represent shells reworked upward from theSZ. On the basis of bone-age data from a series offully buried and well-stratified small-mammalassemblages, Terry and Novak (2015) suggestthat disintegration rates can be biased upward(i.e., inferred rates will be erroneously high) ifskeletal hardparts are not completely mixedwithin the top sediment layer because hardparts

reworked downward would be more frequentthan those reworked upward. This situationwould arise from a down-core decline inhardpart abundance under constant disintegra-tion; and would increase the slope of an AFD.Their cave-hosted record contrasts with themarine conditions considered here. On thesouthern California shelf, as elsewhere, biologicalprocesses mix sediment at monthly, yearly, anddecadal scales that are rapid relative to centennialor millennial scales of long term burial rates(e.g., Alexander and Lee 2009). Our models thusassume that the TAZ and SZ are vertically orhorizontally separated but internally well mixed.

The Dynamic of Production.With increasingwater depth, southern California AFDs ofN. taphria become less skewed, unimodal, andbetter supported by models with temporallyvariable production. The variance of thetruncated-normal distribution implies amillennial-scale decline in production. Thismillennial-scale offset is almost certainlylinked to N. taphria tracking its preferredshallow-water habitat with post-glacial rise insea level. The onshore-offshore gradient in theshapes of AFDs is thus driven by a gradient inthe timing of active production, with earlieronset and earlier shutdown of production indeeper environments. This conclusion issupported by our finding that estimates ofdisintegration and sequestration do not changemarkedly with water depth (Fig. 9C). Based ondata for live-collected bivalves in the SouthernCalifornia Bight, the proportional abundanceof living N. taphria peaks between 19 and 56m(Fig. 11). Death assemblages from 89m depthon the modern shelf are thus presently out ofthe depth range preferred byN. taphria, and theassemblages at 5158m are on the outer edgeof preferred water depths.

Based on the two-phase truncated-normalmodels of the AFDs at these two sites(Fig. 10CD), maximum production at the5158m sites is estimated at ~780010,300years ago and maximum production at the89m sites is estimated to have occurred13,65018,050 years ago (intervals denoted bygray bands in right-hand graph of Fig. 11).These estimates of maximum productionderived from AFDs are in good agreementwith the independent sea-level curve of Nardin




et al. (1981) for southern California. At thetimes of maximum production estimated fromAFDs, the water depths of these sites wereindeed within the preferred depth range of N.taphria (horizontal dashed lines)the 5158msites were 1956m deep between 0 and 11 ka(thick black line is the water depth trajectory),and the 89m sites were in that depth range~12.5 ka to 17.5 ka (thick gray line). Althoughsea-level change since the last glacial max-imum on the southern California shelf remainspoorly constrained, the estimated times anddepths of maximum production are compar-able to the preferred depth of Nuculana taphria.Similar migrations of species producing localpopulation declines can be predicted for otherspecies on continental shelves, along withanalogous shifts in latitude and, on land, alti-tude with climate change.

Conclusions and Implications

Our new models demonstrate for the firsttime the joint effects of skeletal loss andvariable production on the postmortemage-frequency distributions (AFDs) of deathassemblages sampled in the mixed layer, and

thus show how these key variables can bereconstructed, using molluscan shell assem-blages from the southern California shelf.

First, when interpreting the timing of a pastpeak in production or the timing of a recentdecline, such as for conservation and historicalecology studies, it should be expected that highloss rates and slow declines in production havepulled the modes of AFDs forward toward amore modern time. These factors will also tendto modify unimodal trajectories of production(symmetrical pulses) into right-skewed AFDs,with the result that those AFDs might be betterfitted by constant-production models eventhough production was declining over thecourse of time averaging. Care thus must beexercised, but these biases are predictable indirection magnitude. Specifically, they under-estimate the timing of production pulses.

Second, where populations were extirpatedor their production declined strongly, loss ratesinferred from AFDs will approximate the long-term rates of disintegration within the mixedlayer and/or the rates of burial below themixed layer, and thus the true rates of recyclingin the taphonomically active zone will beunderestimated. In the absence of recently

FIGURE 11. Timing in the production of Nuculana taphria found in death assemblages at 5158m and 89m water depthon the present-day southern California shelf. Left graph: Based on sampling of living macrobenthos on the shelf(20032004), N. taphria has peak abundances at shallow depths, with the 5th and 95th percentiles between 19 and 56m.Right graph: Based on the two-phase truncated-normal models fitted to AFDs from our two depth zones, maximumproduction occurred ~780010,300 years ago at the 5158m sites and 13,65018,050 years ago at the 89m sites (graybands, based on Fig. 10C,D). The brackets on these time-frames of maximum production are based on the minima andmaxima of loss rates of the two-phase model (vertical dashed lines). These estimates are in good agreement with theindependent sea-level curve of Nardin et al. (1981) for southern California, which is reproduced here to show thehistory of deepening at these two sites. Our 5158 m depth zone (thick black line is its water depth trajectory)supported a water column 1956 m deep between 0 and 11 ka, and the 89 m depth zone (thick gray line) supported thepreferred water depths ~12.5 ka to 17.5 ka. At the times of maximum production estimated from the AFDs, theestimated water depths at these two depth zones were thus within the preferred depth range of N. taphria (horizontaldashed lines) at the times of maximum production estimated from the AFDs.




produced shells, the AFD cannot detect theshort-term, initially high loss rates such aswould be detected in short-term experimentaldeployments of shells.

Finally, we find from our field test that,although AFD shapes vary across the southernCalifornia shelf, with increasing median shell ageand age range and loss of skew with waterdepth, estimated rates of disintegration andsequestration do not change. The cross-shelfgradient in time averaging is thus created by agradient in the timing of production (here, thetransgressivemigration of our target species withsea-level rise) rather than by gradients in disin-tegration and burial rates. Regardless of waterdepth, most shells in the mixed layer disintegratequickly, with decadal-scale half-lives, whereas asmall subset of shells persist for millennia byattaining a refuge from high loss rates.This postmortem persistence permits deathassemblages sampled at local scales to carryinformation on composition and diversityaccrued over long temporal scales, permittingthe capture of regional-scale composition(Warme 1969; Kowalewski et al. 1998; Smith2008; Tomaovch and Kidwell 2010; Milleret al. 2014; Hassan 2015) even when most shellsdisintegrate quickly. This consistency indynamics across the shelf, derived from AFDdata, is also encouraging for habitat-scalepaleoecological inference: it suggests that bathy-metric variation in live-dead agreement inspecies composition that is encountered in somemeta-analyses (e.g., Kidwell 2001; Tomaovchand Kidwell 2011) arises from differences inscales of time averaging (and thus environmen-tal condensation) rather than from bias per se(interspecies differences in preservation), adistinction that has not previously been possible.

Acknowledgments

We thank A. Ranasinghe of the SouthernCalifornia Coastal Water Research Project forarranging release of sieve residues from theBight03 survey; R. Velarde, D. Cadien, andT. Parker for helpful insights into the SouthernCalifornia Bight system; D. S. Kaufmanof Northern Arizona University for AARanalyses; the NOSAMS facility in Woods Holefor radiocarbon analyses; and for funding the

University of Southern California Sea GrantProgram (National Oceanic and AtmosphericAdministration), the National Science Founda-tion (EAR-0345897 and EAR-1124189), theSlovak Research and Development Agency(APVV 0644-10), and the Slovak ScientificGrant Agency (VEGA 0136-15). We thank ananonymous reviewer for helpful comments,T. D. Olszewski for a review and help with agraphical presentation of censoring effects, andM. Foote for raising the issue of censoring atthe beginning of our project.

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