modeling post-depositional mixing of archaeological...

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Journal of Anthropological Archaeology 26 (2007) 517–540

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Modeling post-depositional mixing of archaeological deposits

P. Jeffrey Brantingham a,*, Todd A. Surovell b, Nicole M. Waguespack b

a Department of Anthropology, University of California, Los Angeles, 341 Haines Hall, Los Angeles, CA 90095, USAb Department of Anthropology, 1000 E. University Avenue, University of Wyoming, Laramie, WY 82071, USA

Received 12 April 2007; revision received 6 August 2007

Abstract

We develop a series of simple mathematical models that describe vertical mixing of archaeological deposits. The modelsare based on assigning probabilities that single artifact specimens are moved between discrete stratigraphic layers. A recur-sion relation is then introduced to describe the time evolution of mixing. Simulations are used to show that there may beimportant regularities that characterize the mixing of archaeological deposits including stages of dissipation, accumulationand equilibration. We discuss the impact of post-depositional mixing on the apparent occupation intensities at modeledstratified archaeological sites. The models may also help clarify some of the problems inherent in making general inferencesabout the nature culture change based on mixed archaeological deposits. We demonstrate the modeling approach by devel-oping a post-depositional mixing model for the Barger Gulch Folsom site.� 2007 Elsevier Inc. All rights reserved.

Keywords: Post-depositional mixing; Mathematical models; Simulation; Archaeological site formation processes; Folsom

Archaeologists have long-recognized that post-depositional mixing of archaeological deposits mayintroduce substantial biases into the character ofarchaeological assemblages (Schiffer 1987; Woodand Johnson 1978). As such, post-depositional mix-ing may make archaeological inferences problem-atic, particularly those concerning the nature andpace of culture change (Brantingham, 2007; Lyman2003; Morin, 2006). In the simplest case, for exam-ple, the downwards mixing of archaeological speci-mens into older deposits could be the source of anerroneous inference that a cultural attribute or

0278-4165/$ - see front matter � 2007 Elsevier Inc. All rights reserveddoi:10.1016/j.jaa.2007.08.003

* Corresponding author. Fax: +1 310 206 7833.E-mail addresses: [email protected] (P.J. Brantingham),

[email protected] (T.A. Surovell), [email protected](N.M. Waguespack).

behavior appeared earlier than previously thought.Here it is clear that mixing leads to a breakdownof the Law of Superposition. In more realistic situ-ations, where archaeological specimens may bemoving en masse over long periods of time, theinferential impact may be not only more severe,but also much more complex. For example, the mix-ing of artifact specimens between discrete strati-graphic units might lead us to question the resultsof frequency seriation or, indeed, any specific infer-ences about cultural change than hinge on the rela-tive frequencies of different artifact types.

A battery of methods are available for diagnos-ing the presence and even severity of post-deposi-tional mixing. These methods fall generally intothree categories: (1) comparisons of the degree ofoverlap in assemblage characteristics based on an

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518 P.J. Brantingham et al. / Journal of Anthropological Archaeology 26 (2007) 517–540

a priori assumption that the pre-taphonomic assem-blages were completely non-overlapping in theirattributes (Albert et al. 2003; Karavanic and Smith,1998; Morin et al., 2005; Rowlett and Robbins,1982); (2) examination of the distribution of refitstone, bone or ceramic specimens between discretestratigraphic units (Audouze and Enloe, 1997; Bol-long, 1994; Delagnes and Roche, 2005; Hofman,1986; Kroll, 1994; Morin et al., 2005; Surovellet al., 2005; Villa, 1982); and (3) experimental orobservational characterization of the behavior ofindividual taphonomic agents and the developmentof criteria to aid in recognizing them in the field(Araujo and Marcelino, 2003; Balek, 2002; Bocek,1986; Erlandson, 1984; Johnson, 1989; Lavilleet al., 1980; Morin, 2006; Van Nest, 2002). Usingthe first set of methods, assemblage overlap—forexample, in faunal species representation, ceramicor lithic types—is taken as evidence for the presenceof post-depositional mixing, but may also serve as aproxy for the magnitude of the disturbance if thedegree of overlap can be established. The reliabilityof the approach is dependent, however, on the valid-ity of the a priori models describing the undisturbedassemblage characteristics. The presence of mixingalso may be estimated using the second set of meth-ods if there is evidence for refits between discretestratigraphic units. Here we are often safe in assum-ing that the specimens comprising a refit set werediscarded together and that their separation withina stratigraphic profile is the result of some form ofpost-depositional mixing. The magnitude of the sep-aration between members of a refit set may providea measure of the magnitude of mixing (Morin et al.,2005; Villa, 1982), though questions may stillremain as to whether a small sample of refits is rep-resentative of the disturbance experienced by amuch larger assemblage. Using the last collectionof methods, mixing is suspected if there is sedimen-tological or stratigraphic evidence that a specifictaphonomic agent such as bio- or cryoturbationhas been active at a site. Such suspicions are con-firmed with evidence of the preferential orientationof artifacts or specific forms of artifact damage(Esdale et al., 2001; Karavanic and Smith, 1998;Lenoble and Bertran, 2004; McPherron, 2005;Surovell et al., 2005; Villa, 1982). Despite the quan-titative nature of these data, however, assessing themagnitude of disturbance using these measure maystill be problematic (Lenoble and Bertran 2004).

While these methods are clearly indispensable, itour intention to take a step back from the empirical

record to examine the general dynamics of post-depositional mixing. The goal is to provide a simplequantitative basis for describing mixing and lay afoundation for more complex archaeological mod-els. The focus in this paper is on a series of relativelysimple mathematical models which describe themixing of archaeological deposits by a genericpost-depositional taphonomic agent. The first sec-tion of the paper considers an hypothetical archae-ological scenario involving the vertical distributionof artifacts in a sequence of stratigraphic units.The structure of the scenario is meant to be consis-tent with typical forms of evidence collected in theexcavation of archaeological sites, namely threedimensional distribution of archaeological speci-mens in a stratigraphic section. Drawing on modelsdeveloped to study sea floor bioturbation and sedi-mentary diagenesis (e.g., Boudreau, 1997; Meysmanet al., 2003; see also Rowlett and Robbins, 1982;Shull, 2001), the components of a discrete mathe-matical model describing the probabilistic transportof archaeological specimens within a sedimentaryprofile are presented. Of the formalisms available(see Meysman et al., 2003), discrete transport mod-els are both relatively easy to understand andappropriate for treating distributions of archeologi-cal materials in buried contexts. Ordinary and par-tial differential equation models are preferred formodeling the effects of bioturbation on radioactivetracers (Boudreau, 1997; see also Pendall et al.,1994; Trumbore, 2000), but they also tend to befar more difficult to analyze.

The second section of the paper shows how thebasic model components fit together to describegeneric post-depositional mixing of archaeologicalmaterials between discrete stratigraphic units. Themodel includes terms to describe the transport ofarchaeological specimens from one or more strati-graphic units into a so-called focal stratum andtransport out of the focal stratum to one or moreother stratigraphic units. Key distinctions are madehere between local and non-local mixing, on the onehand, and symmetrical and asymmetrical mixing,on the other.

The third section then turns to simulating theimpact of different mixing processes on severalhypothetical archaeological scenarios. Here we seein a very general, but practical way how post-depo-sitional mixing may impact inferences about tempo-ral patterns of occupation intensity. Local and non-local mixing models are applied to sites that haveboth single and multiple discrete occupation hori-

P.J. Brantingham et al. / Journal of Anthropological Archaeology 26 (2007) 517–540 519

zons. A more complex scenario is also examinedwhere mixing is applied to a pre-taphonomic expo-nential increase in artifact densities meant to mimica case of increasing occupation intensity at a site. Itis shown that there are substantial regularities to theevolution of artifact density profiles in each of thesecases, suggesting that the signatures of post-deposi-tional mixing may be both easily recognized andpredictable. We consider in the discussion howsmall sample sizes might impact our ability to recog-nize these regularities.

The fourth section examines a specific case ofmixing at the Barger Gulch Folsom site (Mayeret al., 2005; Surovell et al., 2005; Waguespack,2005). The simple models developed here provideinsights into the nature of post-depositional mixingat Barger Gulch and provide possibly a quantitativebasis for estimating the magnitude of disturbance.

The final section of the paper discusses some lim-itations to and possible extensions of the models. Inparticular, more complete formal models of archae-ological site formation and diagenesis shouldinclude a consideration of how site burial, differ-ences in sediment characteristics and variable mate-rial characteristics impact mixing. We brieflyintroduce one such extension where mixing involvesa material that degrades through time (e.g., bone).

A model stratigraphic section

Consider a hypothetical stratigraphic sectiondivided into a series of discrete stratigraphic units(Fig. 1a). The stratigraphic units are numbered frombottom to top beginning with Layer 1, which is bydefinition the base of the sequence. For convenience

Fig. 1. Hypothetical stra

it is assumed that each stratum is the exact samethickness Dd and each is also identical in terms ofsedimentary characteristics. These are simplifica-tions adopted for the purposes of modeling. Archae-ological materials may be found buried within anyof the strata and a large archaeological site mayconsist of a number of such stratigraphic sections.In the present case we are not concerned with theinitial deposition and burial of archaeological mate-rials in the section, though a more complete modelwould certainly include these processes. Rather,we are interested only in modeling the process bywhich materials buried initially within differentstratigraphic units may be subsequently transportedbetween units (see Johnson, 1989).

If we start with the assumption that artifacts canonly move vertically between stratigraphic units—that is, we do not consider horizontal transportbetween individual vertical sections—then it is pos-sible to represent the stratigraphic column in Fig. 1aas a one-dimensional line with nodal points for eachstratigraphic unit. We label each node with a num-ber corresponding to its stratigraphic layer from1,2,. . ., m, where m is the total number of strati-graphic units in a section. In analyzing post-deposi-tional mixing processes between stratigraphic unitsit will be convenient to talk about movement ofspecimens into and out of some focal stratum j(Fig. 1b). Other strata in the section may be referredto either in terms of their position relative to thefocal stratum (e.g., j + 1), or using a label k =1,2,. . ., m, where we are sometimes careful to specifyk 5 j. The number of specimens of a single artifactclass present in a focal stratum j at any point in timeis given as nj and in any other stratum as nk.

tigraphic sequence.


Given this simple mathematical structure, it ispossible to model the movement of archaeologicalspecimens between stratigraphic units using so-called hopping probabilities. The probability thatany one specimen in a class of artifacts is mixedfrom the focal stratum, represented by node j, to adifferent stratum, represented by node k, is givenas ajk. The probability that a specimen is mixedfrom stratum k to the focal stratum j is then akj.The exact mechanism of transport is intentionallyleft unspecified so that many different mixing agentsmight be modeled. For example, ajk might describethe active movement of archaeological specimensbetween layers by burrowing animals (Erlandson,1984; Johnson, 1989) or frost heave (Hilton 2003),or passive movement as might occur if archaeologi-cal specimens migrate through open cracks or bur-rows (Johnson, 1989).

For an entire stratigraphic section consisting ofm total stratigraphic units, the probabilities thatspecimens are mixed between any two strata isdescribed by a m · m transition matrix (Boudreau,1997; Shull, 2001). For example, for the hypotheti-cal stratigraphic section shown in Fig. 1a, the tran-sition matrix is given as:

A ¼

a11 a12 a13 a14 a15

a21 a22 a23 a24 a25

a31 a32 a33 a33 a34

a41 a42 a43 a44 a45

a51 a52 a53 a54 a55

0BBBBBB@

1CCCCCCA

ð1Þ

The entries in A give the probabilities of movementbetween any two stratigraphic layers. For example,the row entry a21 is the probability that a specimenof a particular artifact class present in stratum j = 2is moved by some taphonomic processes to stratumk = 1. Because these are probabilities, the rows (butnot necessarily the columns) must sum to one. Notethat when j = k the resulting term ajk represents theprobability that a specimen remains within the focalstratum. For example, a11 is the probability that aspecimen of a particular class present in Stratum 1remains in Stratum 1. In some studies this probabilityis given as sj (Shull, 2001) and is calculated as

sj ¼ 1�Xm

k¼1k 6¼j

ajk ð2Þ

The transition matrix given in Eq. (1) may be usedto represents both local and non-local mixing ofarchaeological deposits. Local mixing is said to

occur only between adjacent strata. The sedimentchurning activities of so-called mixmaster species,for example, may lead to transfers of materials with-in and between adjacent strata (Johnson et al.,2005). In Fig. 1c, movement of any specimen fromthe focal stratum j to either of the adjacent strataj ± 1 is considered local (see Boudreau, 1997). Onlythe central diagonal and two adjacent diagonals inA have non-zero entries in this case.

By contrast, non-local mixing may occur if anyspecimen in a stratigraphic layer j can be trans-ported directly to a non-adjacent layer k > j ± 1.Burrowing organisms, for example, may movematerials directly between non-adjacent strata pro-ducing so-called conveyor-belt mixing (Boudreau,1997; Johnson et al., 2005). In Fig. 1d, we see a casewhere specimens from the focal stratum j can betransported directly to a non-adjacent stratum atposition j � 2, or from stratum j + 2 to the focalstratum. The transition matrix in this case wouldhave additional entries beyond the main diagonaland those immediately above and below. Note,however, that non-local mixing is not the onlymechanism by which archaeological specimens canmake their way from a focal stratum to non-adja-cent layers. Local mixing may lead to the movementof specimens between non-adjacent strata over time.For example, if a specimen was initially deposited ina stratum j and at some point was mixed to the nextlower stratum in the section j � 1, then this samespecimen might at some later time be mixed fromstratum j � 1 to the next lowest stratum, j � 2. Ifwe were to follow this specimen over the course ofmany time steps the process of stratigraphic mixingwould be described as Markovian. In other words,the location of the specimen at time t is dependentonly upon its location at time t � 1 and the associ-ated probabilities of mixing between stratigraphicunits. In the case of strict local mixing, over timea specimen may migrate from one stratum intonon-adjacent strata through one or more intermedi-ate transport events. The time dynamic of mixing isa simple random walk between layers.

Modeling mass movement

To model the movement of multiple archaeolog-ical specimens out of a focal stratigraphic unit j wedefine ajk nj as the total number of specimens leavingj and being deposited in stratum k in a single mixingevent (Boudreau, 1997; Meysman et al., 2003). Theexpected number of specimens being moved to all


other strata k = 1,2,. . ., m from stratum j in a singleevent is then

E½njk� ¼ nj

Xm

k¼1k6¼j

ajk ð3Þ

Similarly, to model the movement of multiplespecimens into the focal stratigraphic unit we firstdefine akj nk as the total number of specimens mov-ing from any other stratum k to stratum j. Theexpected number of specimens moving into j in asingle event is then given by

E½nkj� ¼Xm

k¼1k 6¼j

akjnk ð4Þ

It is important in both Eqs. (3) and (4) to specifythat the sums are taken over all stratigraphic unitsexcluding the focal unit (i.e., j 5 k). This ensuresthat the equations refer exclusively to taphonomicmixing between layers, rather than mixing withinlayers. This is a distinction that is sometimes madein identifying post-depositional mixing using refitstone or bone specimens (Morin et al. 2005).

Given these components it is possible to write arecursion relation to study the time evolution ofthe number of specimens of a given class of artifactpresent in a focal stratigraphic layer

n0j ¼ nj þ Dnj ð5Þ

Eq. (5) states that the number of specimens in thefocal stratum in the next time interval n 0j (the primeindicating a that a change has occurred) is simplythe number of specimens in the previous time stepnj plus any change Dnj in the number of specimensthrough taphonomic mixing. Using Eqs. (3) and(4) we can write

ð6Þ

giving the master equation

n0j ¼ nj þXm

k¼1k6¼j

akjnk � nj

Xm

k¼1k6¼j

ajk ð7Þ

The second and third terms on the right hand side ofEq. (7) give some clues to the dynamics of tapho-nomic mixing. The second term is always greaterthan or equal to zero and therefore will tend to

increase the number of specimens of a given classof artifact within the focal stratum j. By contrast,the third term is always less than or equal to zeroand therefore will tend to decrease the number ofspecimens represented in the focal stratum j.

Simulated post-depositional mixing

It is reasonably straightforward to simulate theeffects of post-depositional mixing on archaeologi-cal deposits by iterating Eq. (7) and using a transi-tion matrix like Eq. (1) defined a priori. Here weconsider several baseline cases including: (1) sym-metrical local and non-local mixing, where theprobabilities of mixing upward through a strati-graphic section are the same as mixing downward;(2) asymmetrical local mixing, where the probabilityof mixing downward through the section is greaterthan upward; (3) local and non-local mixing oftwo discrete occupations; and (4) local mixing ofan exponentially increasing occupation. The base-line cases are not meant to be exhaustive, nor arethey tied to any specific archaeological case. Ratherthey serve to illustrate some of the general dynamicsand patterns that may result from post-depositionalmixing.

All of the simulation models have several proper-ties in common. First, the models examine an hypo-thetical stratigraphic profile with 51 discretestratigraphic units or layers (Fig. 2). Mixing dynam-ics are therefore described by a 51 · 51 transitionmatrix. All stratigraphic units are of the same thick-ness and composition, a simplifying assumption ofthe abstract specification above. The models arenot dependent upon adhering to these properties,however (see Boudreau, 1997; Meysman et al.,2003; Shull, 2001). Stratum 1 is at the base of theprofile and is assumed to be above sediment or bed-rock that is impermeable to post-depositional mix-ing. Stratum 51 is the unit directly below thesediment–surface interface. It is assumed that thereis no mixing of archaeological materials betweenthe surface and subsurface.

In all of the following simulations the number ofspecimens within each stratigraphic unit is measuredin discrete vertical sections at points distributedalong the length of the profile. These vertical sec-tions might be regularly spaced, as is frequentlythe case when excavating a site using a samplinggird. In the present case there are 100 randomly dis-tributed vertical sections along the simulated strati-graphic profile. Given such a large sample of

Fig. 2. A stratigraphic profile showing the basic elements of the simulations.


vertical artifact densities it is generally possible toinfer the profile-wide distribution of archaeologicalmaterials. In Fig. 2, the distribution of artifactsobserved in 20 of the 100 randomly spaced verticalsections leads the inference that there is a single,compact occupation horizon centered around Stra-tum 25.

Since the initial distribution of archaeologicalmaterials in the simulated stratigraphic section isknown a priori it is possible to look at a numberof different measures of how post-depositional mix-ing impacts the character of archaeological deposits.First, in some cases we are interested in the changein the mean stratigraphic position (depth) of anarchaeological occupation through multiple stagesof post-depositional mixing. This measure is partic-ularly relevant for tracking the impact of mixing onarchaeological deposits that were initially unimodal;i.e., deposits that display only one density peak inthe section. Second, we are also interested in howspecimens are spread throughout a stratigraphicsection over the course of mixing. Here we considerchanges in the standard deviation of the distributionof specimens across stratigraphic units and the max-imum distances over which individual specimens aremixed. Finally, we ask whether there are distinctstages that a deposit goes through during mixing.

Simple mixing scenarios

Symmetrical local mixing

The simplest possible mixing scenario to consideris symmetrical local mixing of a discrete occupation.

Here we begin with artifacts buried within Stratum25 and observed in each of 100 vertical sectionsalong the profile. For simplicity there are initially20 specimens in each of the 100 sections giving atotal of 2000 specimens for the entire profile. Mixingis both symmetrical and local, meaning that theprobabilities of mixing upward and downward inthe stratigraphic column are equal and that individ-ual specimens can move only to immediately adja-cent stratigraphic units. More specifically, theprobability that a single specimen moves up ordown one stratigraphic unit in a single time step isajk = 0.01, when k = j ± 1. Mixing to non-adjacentstrata is prohibited so that ajk = 0, when k > j ± 1.The probability that a specimen remains in placeduring a single time step is sj = 1 �

Pajk, k 5 j =

0.98. These terms comprise a transition matrix witha main diagonal set of entries and diagonalsabove and below. All other entries in the matrixare zero.

Fig. 3 shows the distribution of archaeologicalspecimens across the 51 hypothetical stratigraphicunits after post-depositional mixing lasting differentlengths of time. Prior to mixing all of the specimensare contained within Stratum 25, as shown by thesingle frequency spike at time step zero. With thestart of mixing, specimens begin to migrate to posi-tions above and below the original occupation hori-zon and within 50 time steps more than half of thespecimens are found outside of their original strati-graphic unit. The maximum displacement after 50time steps is �4 stratigraphic units above andbelow, demonstrating that repeated local mixingcan lead to migration of artifacts between non-adja-

Fig. 3. Symmetrical local mixing of a single occupation horizon over a short period of time. The panels on the left show the verticalposition of observed specimens along the face of the profile before taphonomic mixing (bottom) and after 25, 50, 100 and 250 simulationtime steps. The panels on the right show the frequency distribution of all specimens across all strata for the same times.


cent strata (Table 1). The original occupation hori-zon continues to lose specimens and the distributionof artifacts across the strata approaches a symmetri-cal Gaussian form, which is particularly evident by250 time steps. At this point, the maximum displace-ment of specimens from their original depositionalcontext is �9 units above and below and 35% ofthe stratigraphic units (18 of 51) contain specimensmixed from the original layer. Not surprisingly, thestandard deviation of the distribution of specimensacross strata increases with mixing (Table 1). Themean stratigraphic position, however, remainsapproximately stationary.

As Fig. 4 makes clear, however, a symmetrical,Gaussian distribution of specimens is only character-istic of symmetrical local mixing over relatively shorttime periods. The actual equilibrium state of the sys-tem is a uniform distribution of specimens across allstratigraphic units. The pronounced mode apparentin the distribution of specimens after limited mixingdissipates and is difficult to identify after 10,000 sim-ulation time steps. At 25,000 time steps we approachmaximum variance in the distribution, but the meanremains at the center of the profile (Table 1). Themaximum displacement of any one specimen fromthe original depositional context is 25 layers.

Asymmetrical local mixing

If there is a bias in the movement of archaeolog-ical specimens towards positions farther down (or

farther up) a stratigraphic profile, then we are deal-ing with asymmetrical mixing. If this mixing is alsolocal, meaning that movement of specimens is onlybetween adjacent stratigraphic units, then it is rela-tively straightforward to modify the above model todeal with the asymmetrical case. Fig. 5 illustrates amixing system where the probability that a singlearchaeological specimen is mixed from a focal stra-tum j to the next lower stratum k = j � 1 isajk = 0.04. The probability of mixing from j to thenext higher stratum k = j + 1 is ajk = 0.01. Thereis thus a weak bias for downward mixing. Overthe short term, specimens spread out from the origi-nal occupation horizon. The observed distributionafter 250 simulation time steps is skewed towardshigher stratigraphic levels, though it is still visuallysimilar to a Gaussian normal distribution. Greaterbiases in the probability of downward mixing tendto enhance the skew (e.g., Morin 2006). What is alsoapparent, in contrast with symmetrical local mixing,is that the distribution migrates down the profile.The mean stratigraphic position of all specimensdeclines from Stratum 24, after 25 simulation timesteps, to Stratum 7, after 250 simulation time steps(Table 1). The variance and maximum deviationsfrom the original occupation horizon also increase.After 250 simulation time steps, 23 stratigraphicunits separate the uppermost and lowermost speci-mens and the maximum deviation from the originaloccupation horizon is �25 layers. Eventually thispattern is reversed and the variance and maximum

Table 1Summary statistics for mixing models involving a single initial occupation horizon

Simulationtime step

Mean stratigraphicposition

Standarddeviation

Minimum stratigraphicposition

Maximum stratigraphicposition

Symmetrical local mixing of a single occupation

0 25.00 0.00 25 2525 24.99 0.70 22 2850 24.97 1.01 21 29100 25.01 1.44 19 30250 24.97 2.27 17 34500 24.92 3.17 14 371000 24.91 4.54 9 395000 24.62 10.04 1 5125,000 23.99 12.57 1 5120,000 23.92 14.40 1 51

Asymmetrical local mixing of a single occupation horizon

0 25.00 0.00 25 2525 24.19 1.16 19 2850 23.44 1.60 17 29100 21.84 2.28 12 29250 17.04 3.57 5 28500 9.29 4.79 1 271000 1.78 1.75 1 215000 1.31 0.64 1 6

Symmetrical non-local mixing of a single occupation horizon

0 25.00 0.00 25 2525 24.79 0.00 6 4450 24.79 6.10 3 50100 24.47 8.62 1 51500 24.37 13.75 1 51


deviations decrease as more and more specimensaccumulate at the base of the section (Fig. 6 andTable 1). There is a trailing ‘‘tail’’ of specimensmaintained in a few strata above the base of the sec-tion by the small probability of upward movementof specimens. Mixing continues, but the distributionis stationary.

More complex mixing scenarios

Symmetrical non-local mixing

Non-local mixing by definition involves themovement of specimens directly between non-adja-cent strata. To model such systems it is necessaryto define transition probabilities ajk (and akj) fork > j ± 1. There are many possible ways in whichwe could define these probabilities. Illustrative inthe present case is to make the transition probabili-ties between a focal stratum j and another non-localstratum k a function of the distance between thestrata

ajk ¼ ajld�l; l ¼ j� 1 and k ¼ j� d ð8Þ

Here stratum k is located d strata above or belowthe focal stratum j, ajl is the baseline local mixingprobability and l is an constant that falls in therange 1 < l 6 3 (see also Reible and Mohanty,2002). Note that because the strata are all of equalthickness d is a ratio scale metric. Eq. (8) states thatthe probability that a single specimen mixes betweenany two strata is a negative power of the distancebetween them (Fig. 7). In general, most mixingevents are between adjacent strata but occasionallythere are mixing events over longer distances be-tween non-adjacent strata. Note that when d = 1,Eq. (8) returns the local mixing probability (i.e., aj-

k = ajl), which from the symmetrical case above is0.01. When d > 1 the probability of mixing betweenlayers declines rapidly, but over many mixing eventseven low probability will tend to occur. For exam-ple, with l = 1.2 and the local mixing probabilityajl = 0.01, the movement of a specimen fromStratum 25 to Stratum 15 (i.e., d = 10) is expectedto occur with a probability ajk = 0.006, or approxi-mately once every 1667 mixing events.

Fig. 8 shows the results of non-local mixing inour hypothetical stratigraphic profile when

Fig. 4. Symmetrical local mixing of a single occupation horizon over a long period of time. The panels on the left show the verticalposition of observed specimens along the face of the profile after 500, 1000, 5000, 10,000 and 25,000 simulation time steps. The panels onthe right show the frequency distribution of all specimens across all strata for the same times.

Fig. 5. Asymmetrical local mixing of a single occupation horizon over a short period of time. The panels on the left show the verticalposition of observed specimens along the face of the profile before taphonomic mixing (bottom) and after 25, 50, 100 and 250 simulationtime steps. The panels on the right show the frequency distribution of all specimens across all strata for the same times.


ajl = 0.01 and l = 1.2. The broad dynamics of thesystem are the same as for local mixing, namely thatspecimens from the original occupation horizon inStratum 25 spread out to form a unimodal distribu-tion and then ultimately converge on a uniform dis-tribution across all strata. However, there areseveral important quantitative differences withnon-local mixing. First, compared with the Gauss-ian distribution seen with local mixing, the non-

local mixing distribution is initially strongly concavewith long tails. In fact, very early in the process ofmixing, at 25 time steps, the distribution of speci-mens above and below the original occupation hori-zon follows approximately the probabilitydistribution shown in Fig. 7. The maximum dis-placement of specimens from the original occupa-tion horizon after 25 time steps is �19stratigraphic layers and the maximum separation

Fig. 6. Asymmetrical local mixing of a single occupation horizon over a long period of time. Shown on the left is the vertical position ofobserved specimens along the face of the profile after 5000 time steps of simulated mixing. The panel on the right shows the frequencydistribution of all specimens across all strata for the same mixing period.

Fig. 7. A Levy distribution describing the probability of mixingbetween adjacent and non-adjacent stratigraphic units.


of any two specimens is 38 stratigraphic layers(Table 1). Second, by 100 time steps, the distribu-tion appears more Gaussian-like in shape, thoughthe standard deviation of the distribution (non-local: r = 8.62) is approximately six times largerthan with local mixing at the same time step (local:r = 1.44). Third, the system approaches a uniformdistribution at much faster rate, 500 simulation time

steps compared with the �25,000 time steps it tookwith local mixing.

Local and non-local mixing of two discrete

occupations

Arguably, post-depositional mixing of a singlediscrete occupation only tends to impact our per-ception of the duration and intensity of occupation.While the associated dynamics are important tounderstand, they do not necessarily provide obviousexpectations for what happens in mixing multiplediscrete occupations. Consider a case of symmetri-cal local mixing identical to that described above,but here starting with two discrete occupationslocated in strata 17 and 34, respectively (Fig. 9). Atotal of 18 stratigraphic layers separate the twooccupations prior to mixing. Ten specimens areobserved in each horizon in each of the 100 ran-domly spaced vertical sections (see Fig. 2). Thus,the sampled horizons each contain 1000 specimensand combined have 2000 specimens. With the onsetof mixing, specimens begin to spread out from eachof the discrete occupations in a manner identical tothe single occupation case discussed above. Both areapproximately Gaussian and the means of eachremain centered on the original occupation strata.However, the tails of the distribution start toapproach one another and, within 250 simulationtime steps, specimens that have migrated down-

Fig. 8. Symmetrical non-local mixing of a single occupation horizon over a short period of time. The panels on the left show the verticalpositions of observed specimens along the face of the profile before taphonomic mixing (bottom) and after 25, 50, 100 and 500 simulationtime steps. The panels on the right show the frequency distribution of all specimens across all strata for the same times. The probability ofmixing between layers is calculated from Eq. (8) with ajl = 0.01 and l = 1.2.

Fig. 9. Symmetrical local mixing of two discrete occupations located initially in strata 17 and 34. The panels on the left show the verticalpositions of observed specimens along the face of the profile before taphonomic mixing (bottom) and after 25, 50, 100 and 250 simulationtime steps. Each occupation horizon originally contained ten specimens at each of the measured vertical sections. The panels on the rightshow the frequency distribution of all specimens across all strata for the same times.


wards from the upper occupation are found in thesame strata as specimens that have migratedupwards from the lower occupation (Fig. 9). At thispoint, the distributions are no longer discrete, ornon-overlapping. In a strict sense, the stratigraphicsequence appears to contain a continuous occupa-tion with a bimodal distribution of occupationintensities.

Fig. 10 illustrates that, as mixing continues, thetwo distinct modes gradually disappear and, by timestep 5000, merge into a unimodal distribution with amean centered midway between the two originaloccupation horizons. By this point specimens havealso migrated to the top and bottom of the strati-graphic section. After time step 5000, the singlemode begins to dissipate and the frequency of spec-

Fig. 10. Symmetrical local mixing of two discrete occupationhorizons at 500, 1000, 5000, 10,000 and 25,000 simulation timesteps. Shown are the frequency distributions of all specimensacross all strata for each time step.

Fig. 11. Symmetrical non-local mixing of two discrete occupa-tions located initially in strata 17 and 34. Each occupationhorizon contains ten specimens at each of the measured verticalsections. Shown are the frequency distribution of all specimensacross all strata before taphonomic mixing (bottom) and after 25,50, 100 and 500 simulation time steps.


imens across all stratigraphic levels converges on auniform distribution. The endpoint of this processinvolving two discrete occupation horizons is indis-tinguishable from the case involving one discreteoccupation horizon (compare Fig. 4 and 10).

The same general conclusion may be drawn aboutnon-local mixing of two discrete occupation horizons

(Fig. 11). Using the parameters for non-local mixingemployed above, it is clear that the two discrete occu-pations follow the trajectory of a single occupationduring the early stages of mixing (compare withFig. 8), namely the development of strongly‘‘peaked’’ and concave distributions centered on theoriginal occupations. The two modes eventuallymerge to form a single unimodal distribution, occur-ring here at around 100 simulation time steps. Even-tually the single mode dissipates to form a uniformdistribution of archaeological specimens across allstrata. The time to arrive at a uniform distribution(�500 simulation time steps) is again much fastercompared with local mixing (�25,000 simulationtime steps) (compare Fig. 10 and 11).

Local mixing of an exponentially increasing

occupation

A final hypothetical case to examine involves apre-taphonomic occupation that spans strata 10through 40. The occupation is characterized by anexponentially increasing numbers of specimens fromtwo specimens observed in Stratum 10 up to 22specimens in Stratum 40 in each of the vertical sec-tions (Fig. 12). Above Stratum 40 the number ofspecimens drops to zero. This is a vertical distribu-tion of artifacts that one might expect if occupationintensity was increasing through time followed bysite abandonment at the peak of occupation (butsee Surovell and Brantingham, 2007).

Symmetrical local mixing using the parameter val-ues introduced above eventually eradicates any traceof the initial exponential occupation pattern andrapid site abandonment (Fig. 12). By time step5000, the occupation appears to increase and thendecrease smoothly through time. The mode of the dis-tribution is also displaced downwards in the sectionsuch that the apparent peak of occupation intensityat time step 5000 is in Stratum 34, as opposed to Stra-tum 40 before mixing. As was the case in all of theother examples of symmetrical mixing presentedabove, the representation of specimens across allstrata ultimately converges on a uniform distribu-tion, seen clearly in simulation time step 25,000.Occupation intensity appears to remain constantthrough time.

Vertical mixing at the Barger Gulch Folsom site

We demonstrate how to develop and apply a gen-eral mixing model like those presented above using

Fig. 12. Symmetrical local mixing of an occupation that is originally distributed across strata 10 to 40 and exponentially increasingthrough time. Shown are the frequency distributions of archaeological specimens across all strata before post-depositional mixing (bottomleft), after 25, 50, 100, 250 (left panels), and after 500, 1000, 5000, 10,000, and 25,000 simulation time steps (right panels).


data from the Barger Gulch site, a series of latePleistocene and early Holocene archaeologicallocalities associated with Barger Gulch, a tributaryof the Colorado River in Middle Park, Colorado(USA). Since 1997, the University of Wyominghas completed eight seasons of excavation at Local-ity B, a large single component Folsom campsite(Kornfeld and Frison 2000; Kornfeld et al. 2001;Mayer et al. 2005; Surovell et al., 2005; Waguespacket al., 2006). Through the 2006 field season, morethan 60,000 chipped stone artifacts have been recov-ered and more than 12,000 of these have beenmapped with mm precision in three dimensions.

Spatial data from the site provide an excellent casefor evaluating the potential of the above models toreplicate a real-world vertical artifact distributions.

Chipped stone artifacts are found in relativelydiscrete clusters associated with hearth features(Surovell and Waguespack, in press; Waguespacket al., 2006). Stratigraphically, the Barger GulchFolsom occupation is shallowly (0–60 cm) buriedin primary/and or secondary aeolian sedimentsheavily modified by pedogenesis (Surovell et al.,2005). It is extremely difficult to identify lithostrati-graphic units at the site since all sediments are textur-ally similar (silt loam), and therefore stratigraphic


designations are based in large part on pedostratig-raphy (Surovell et al., 2005). Detailed descriptionsof stratigraphy and post-depositional artifactdispersal can be found in Surovell et al. (2005)and Mayer et al. (2005), and only a brief summaryof the geomorphic history of the site is providedhere.

Aeolian silts began slowly aggrading on a lag sur-face of Miocene bedrock residuum in the late Pleis-tocene. The ground surface was slowly aggrading orrelatively stable at the time of human occupation(ca. 10,450 14C yr BP) because the occupation sur-face is associated with the upper contact of a welldeveloped Btkb soil horizon. Following the occupa-tion, the site experienced mild erosion, stripping theA horizon of the Pleistocene soil. Depositionresumed by 9400 14C yr BP. During the early Holo-cene as much as 40–50 cm of silt accumulated bury-ing the Folsom occupation. During the early tomiddle Holocene, the surface stabilized and a welldeveloped soil formed. In the late Holocene, this soilwas partially truncated by erosion and then buriedby up to 30 cm of silt loam. Although stratigraphicsections across the site are fairly consistent, themagnitude of erosion and soil formation varieslaterally.

In Fig. 13, we present five randomly chosen ver-tical artifact distributions from the site. Each graphshows lithic artifact counts by 5 cm excavation levelfor individual 1 m2 excavation unit. Artifacts aregenerally found throughout the late Quaternarydeposits. In typical excavation units (e.g.,

Fig. 13. Vertical distribution of lithic specimens in five randomly

Fig. 13b–e), artifact counts gradually increase fromthe surface downward reaching their maximum atwhat we interpret to be the Folsom occupation sur-face. Below the occupation horizon, artifact countsrapidly drop to zero, usually within 10–15 cm. Inroughly 10% of excavation units, vertical artifactdistributions show considerably more noise and/ormultimodality (e.g., Fig. 13a). When viewed incross-section, the zone of high artifact density mark-ing the Folsom occupation surface is evident withconsiderable movement of artifacts upward andminimal downward movement (Fig. 14).

Producing a general vertical artifact density dis-tribution for the entire site requires standardizingabsolute artifact elevations to a common strati-graphic unit. The most readily identifiable deposi-tional event at the site is the Folsom occupationitself. Thus, artifact elevations by excavation unitwere standardized to the Folsom occupation sur-face, identified as a weighted (by artifact count)average of elevation for the three contiguous 5 cmlevels with the greatest artifact count. We are confi-dent that this method accurately identifies the occu-pation surface. This surface is consistentlyassociated with the largest artifacts (>100 g), whichshould be least likely to experience post-deposi-tional movement, and it is bracketed by radiocar-bon ages which place it in the correct time framefor a Folsom occupation (Surovell et al., 2005).Finally, pit and hearth features originating from thissurface have been identified (Surovell et al., 2005;Waguespack et al., 2006). For the purposes of this

chosen excavation units at the Barger Gulch Folsom site.

Fig. 14. Composite profile of the Barger Gulch site showing the vertical and horizontal distribution of lithic specimens for a 1 m thicktransect through on portion of the site from N1474 to N1475.


paper, it is of no consequence that the original Fol-som occupation surface may have been somewhatdeflated by erosion, since the resulting lag wouldhave been a single surface with a vertically con-strained distribution at the time of burial.

To minimize the effects of size-dependent mixing,we limit this to analysis to piece-plotted lithic arti-facts with maximum length of 10–25 mm (Surovellet al., 2005). This includes a total of 7018 specimensfrom the 75 m2 excavated area. The composite ver-tical distribution of artifacts at the site is shown inFig. 15. The distribution in unimodal with the upperand lower tails both strongly concave. The uppertail of the distribution is much longer than the lowertail towards the base of the section. Artifacts havedispersed upwards as much as 55 cm and down-wards as much 35 cm.

Many of the empirical features of the BargerGulch artifact distribution may be recognized inthe simple simulations presented above. In particu-

Fig. 15. Vertical frequency distribution of lithic specimens from a75 m2 excavation at the Barger Gulch site. The elevation ofspecimens (in meters) is given relative to the inferred Folsomoccupation level (±0 m). Shown is the distribution for 7018 lithicspecimens 10–25 mm in maximum dimension.

lar, the concave distributions above and below theFolsom occupation may be diagnostic of non-localmixing (compare Fig. 8 with 13e and 15). In otherwords, one or more mixing agents at Barger Gulchmay have been active in transporting artifactsdirectly between non-adjacent strata. It is also rea-sonable to propose that Barger Gulch has experi-enced only early stage mixing since, under manyconditions, the concave distributions generated bynon-local mixing are replaced by Gaussian-likeand ultimately a stable (e.g., uniform) distributionsat intermediate and later stages of mixing, respec-tively. However, unlike in the simple simulationsabove, there is a global asymmetry in the verticaldistribution of specimens centered on the occupa-tion horizon. Not only have more specimens movedupwards through the section, but they also havemoved over a greater vertical distance. Differencesin sediment characteristics are the obvious causeand consequently modifications to the above modelsare necessary to account for the asymmetry.

We simulated mixing of the Barger Gulch assem-blage lithic assemblage by calibrating the modelagainst observed artifact counts. We assume, asrequired by the simple models, that each specimenin the size class 10–25 mm has the same probabilityajk of mixing between a focal stratum j and anotherstratum k. Individual mixing probabilities are prob-ably very different for specimens of different sizeclasses (Baker 1978; Bocek 1986; Villa 1982). Surov-ell et al. (2005) have demonstrated that the largestartifact specimens at Barger Gulch cluster verticallyat the inferred Folsom occupation horizon com-pared with smaller artifacts. We assume that allspecimens in any given 1 m2 excavation unit wereinitially deposited at the Folsom occupation sur-face. Therefore, all specimens start with a relativevertical elevation of ±0 centimeters. In accordancewith the excavation strategy used at the site, mixingis modeled between discrete stratigraphic units each5 cm in thickness. Thus, mixing occurs in incre-


ments of 5 cm above and below the occupationsurface.

For simplicity, we also assume that there hasbeen no horizontal movement of artifacts betweenexcavation units subsequent to their initial burial(but see Balek, 2002; Surovell et al., 2005). Thereis spatial variability in the number of specimenspresent in any simulated vertical section (excavationunit), but the number of specimens in each verticalsection is conserved over the course of mixing. Forexample, a vertical section starting with 23 speci-mens at the Folsom occupation surface will stillhave 23 specimens, possibly distributed over multi-ple stratigraphic units, after mixing. Using the Bar-ger Gulch data, the mean number of specimens persimulated vertical section is 93.6 (median = 48) andthe standard deviation is 113.3. The minimum num-ber of specimens in any of the simulated sections is 2(Units N1467, E2433 & N1476, E2436) and themaximum is 556 (Unit N1475, E2448).

Examination of the Barger Gulch vertical artifactdistribution suggests that the most appropriate non-local mixing models has two essential features.First, we propose that the probability of mixingbetween non-adjacent strata decreases at a constantrate with increasing distance from the Folsom occu-pation surface. The corresponding probability ajk isgiven by a negative exponential distribution (seeFaure and Mensing, 2005; Surovell and Branting-ham, 2007)

ajk ¼ ajle�kðk�1Þ; k ¼ j� d 6 r ð9Þ

where, as above, ajl is the baseline local mixingprobability between focal stratum j and the nextadjacent stratum above or below (i.e., l = j ± 1).The parameter k is a decay constant which drivesa decline in the probability of mixing between j

and k as the distance between j and k increases.The parameter d is the distance in stratigraphic unitsbetween the focal stratum j and the recipient stra-tum k, which at Barger Gulch corresponds to 5 cmintervals. For example, d = 1 applies to the proba-bility of a single specimen moving from one strati-graphic unit to either of the adjacent units ±5 cmaway. By contrast, d = 5 applies to the probabilityof mixing between one stratigraphic unit and units±25 cm away. Note that Eq. (9) returns the baselinelocal mixing probability ajl when d = 1 and that thevalue of ajk decreases from this maximum towardszero as d increases. Finally, the parameter r is therange over which non-local mixing is allowed to oc-cur. For example, if r = 4, then the maximum non-

local mixing distance is between any focal stratum j

and those four stratigraphic units above or below(i.e., k = j ± 4). If r = 10, then mixing could occurbetween a focal stratum and strata 10 stratigraphicunits above or below (i.e., k = j ± 10). Mixing isnot allowed between strata outside of the range ofmixing. For example, if r = 4 then the probabilityof mixing between a focal stratum and one fivestratigraphic units away is zero.

Second, we recognize that there is a major differ-ence the nature of mixing above and below theinferred Folsom occupation horizon. More exten-sive mixing has occurred above the occupation thanbelow. The lower permeability to mixing below theoccupation surface is likely related to the well-devel-oped Btkb soil horizon found at 0 to �5 cm (Surov-ell et al., 2005). We therefore allow separate modelparameterizations for the stratigraphic units above(+5 to +60 cm) and below (�5 to �60 cm) the occu-pation surface (Table 2). In particular, we allow fora smaller decay constant (k1 = 1) in the strata abovethe Folsom occupation surface compared with thestrata below (k2 = 2). The range of mixing is thesame (r1 = r2 = 6). In practice, this means that indi-vidual specimens have a higher probability of non-local mixing in the zone above the Folsom occupa-tion. For example, using the model parametersgiven in Table 2, the probability of mixing betweenthe Folsom occupation surface and a position threestratigraphic units above is ajk = 0.00135. Mixingbetween the Folsom occupation surface and a posi-tion three stratigraphic units below isajk = 0.000025, about two orders of magnitude lesslikely. These differences are driven by the differentvalues of k.

Figs. 16 and 17 show the results of the simula-tion. The exponential non-local mixing model forBarger Gulch produces mixing dynamics that paral-lel the example non-local mixing model in someways (see Fig. 8). The initial stages of mixing (e.g.,after 50 steps) generate a vertical distribution ofspecimens that is concave both above and belowthe occupation surface (Fig. 16). Over longer peri-ods of time, the distribution appears more Gaussian(e.g., after 100 steps) and eventually approaches astable form that encompasses all of the stratigraphicunits (e.g., after 5000 time steps). Importantly, dur-ing the early stages of simulated mixing, the mode ofthe distribution remains centered on the initial occu-pation surface. During later stages of simulatedmixing, the density of specimens follows a step func-tion centered on the original occupation horizon.

Table 2Barger Gulch simulation model parameters

Variable Value Units Description

ajk <ajl Probability Probability of single specimen mixing from j to k

ajl 0.01 Probability Baseline local mixing probability between j and the adjacent stratum above or below(l = j ± 1)

k1 1 Stratigraphicunits�1

Decay constant for vertical positions above the Folsom occupation surface

k2 3 Stratigraphicunits�1

Decay constant for vertical positions below the Folsom occupation surface

r1 6 Stratigraphicunits

The maximum non-local mixing span (in stratigraphic units) for vertical positions abovethe Folsom occupation surface

r2 6 Stratigraphicunits

The maximum non-local mixing span (in stratigraphic units) for vertical positions belowthe Folsom occupation surface

e 2.718282 Scale free Base of the natural logarithm

Fig. 16. Simulated mixing at the Barger Gulch site using a non-local exponential model with mixing faster in the zone above theoccupation horizon. Model parameters given in Table 2.


The step distribution is clearly a product of thegreater permeability of the deposits to mixing abovethe occupation horizon and lower permeabilitybelow. Above the occupation horizon, the late stageuniform distribution is perhaps consistent withmodels of an well-sorted, isotropic biomantle (John-son et al., 2005).

Repeated simulations find a best fit between theexponential non-local mixing model and the BargerGulch data after 41 simulated mixing events or timesteps (Fig. 17a and b). The mean number of speci-mens remaining in the occupation stratum overten separate simulations is very similar to theobserved number at Barger Gulch (unmixed:nsim = 2826.4, rsim = 41.6, nobs = 2830). The mean

number of specimens moving upward from the Fol-som occupation in the ten simulations is slightlygreater than observed (mixing up: nsim = 2710.2,rsim = 32.8, nobs = 2419), whereas the mean movingdownward is slightly lower than observed (mixingdown: nsim = 1481.4, rsim = 37.32, nobs = 1769).These deviations are driven primarily by differencesbetween the model and observed distributions in thetwo strata immediately above the occupation sur-face and the one immediately below (Fig. 17b).The maximum upward movement in ten simulationsis between the occupation stratum and the stratumat +55 cm, identical to that observed at BargerGulch. The maximum downward movement in thesimulations is between the occupation stratum andthe stratum at �40 cm, one stratigraphic unit morethan observed. Despite these differences, the twodistributions are statistically very similar (Kolmogo-rov–Smirnov Z = 0.566, p = 0.906) (Fig. 17b). Thesimulation results may thus provide some supportfor the observations offered above, namely: (1) thatthe vertical movement of artifacts seen at BargerGulch is representative of relatively early stages ofnon-local mixing; and (2) that the nature of the geo-logical deposits at the site (and possibly the tapho-nomic agents involved) produced a significant biasagainst downward mixing of specimens.

There are a number of caveats that must accom-pany these conclusions. First, the fact that we find agood statistical fit between our model and the Bar-ger Gulch data does not prove causation. Alterna-tive models could be proposed to explain theobserved vertical artifact distribution at BargerGulch. For example, it is theoretically possible thatthe observed vertical distribution represents a caseof asymmetrical mixing with the Folsom occupationsurface serving as a sink for accumulation of arti-

Fig. 17. Simulated mixing at the Barger Gulch site using a non-local exponential model with mixing both faster and vertically moreextensive in the zone above the occupation surface. (a) Simulated vertical frequency distribution of the Barger Gulch lithic specimens(n = 7018) relative to the occupation surface at ±0 (elevation in centimeters). Shown are the mean counts from 10 separate simulations ineach 5 cm stratigraphic unit. (b) Quantitative comparison of simulated (+) and observed distributional data (s). See text for discussion.Model parameters given in Table 2.


facts moving with a downwards mixing bias. In thiscase, the Folsom occupation surface might repre-sent a subsurface stone line in the sense of Johnson(1989). However, the pedostratigraphy, distributionof features, stone refits and, particularly, radiocar-bon dates argue strongly against this interpretation(see Surovell et al., 2005). These data suggest ratherthat lithic specimens are mixing upwards into muchyounger Holocene sediments.

Second, since the model deals with mixing at avery abstract level, it is not immediately clear whichspecific mixing agents could be responsible for theobserved and simulated patterns. Numerous distur-bance agents were likely active at Barger Gulchbecause the site is shallowly buried and always hasbeen (Surovell et al., 2005). Bioturbation is evidentin the form of krotovina of fossorial mammalsand invertebrates. Also, modern roots and rootletsof small shrubs and grasses are regularly encoun-tered and, though no trees grow on the site today,dispersed pine charcoal and burned roots indicatethe early Holocene presence of conifers, which canphysically displace artifacts by root growth and dur-ing tree throw events. Cryoturbation may haveaffected the vertical artifact distribution as site sed-iments experience regular freeze–thaw cycles in thehigh mountain valley of Middle Park, though recentexperiments suggest it probably did not play a

major role (O’Brien 2006). Though clay content isfairly low (�10%), argilliturbation may have beenan important disturbance process as modern desic-cation cracks occur on the surface, and identical fea-tures are occasionally encountered subsurface aswell. As in many archaeological contexts, multipledisturbance agents have contributed to the observedvertical (and horizontal) spatial patterning at BargerGulch. Since the aggregate impact of the manyimplicated mixing agents is uncertain, we argue thatthe abstract, general model presented above isappropriate for investigating mixing at BargerGulch.

Third, while mixing above and below the Folsomoccupation horizon is simulated as occurring simul-taneously, it is at least possible that mixing mayhave occurred at different times in these two con-texts; i.e., mixing below the occupation horizonmay have occurred primarily in the time periodbetween initial deposition and the inferred erosionevent during the early Holocene, while mixingabove the occupation horizon may have occurredprimarily during late Holocene (see Surovell et al.,2005). It is also difficult therefore to be more precisethan simply suggesting Barger Gulch has experi-enced only early stages of mixing. One would needto calibrate the time scale of mixing either basedon experimental observation of different tapho-


nomic agents, or by comparing multiple sites of dif-ferent ages from similar sedimentary contexts to saymore than this. Neither of these alternatives arepresently available for Barger Gulch.

Finally, it is clear that differences in sedimentarycharacteristics may have played an important role inthe broad mixing dynamics at Barger Gulch. Thesedifferences were accommodated within our modelby proposing that baseline mixing probabilities ajl

were modified by exponential functions with param-eterizations favoring more extensive mixing abovethe occupation horizon than below. However, amore realistic model would need to include a morecomplete characterization of sedimentary character-istics and perhaps explicit rendering of these charac-teristics in the transition probability matrix (Eq.(1)).

Discussion

The simple models presented here provide a gen-eral, but rigorous approach to studying two of theprimary processes involved in post-depositionalmixing. Mixing of specimens out of individualstratigraphic units is described by Eq. (3), while

Fig. 18. Major stages in post-depositional mixing of discrete occupationAsymmetrical local mixing of a single occupation horizon. (c) Symmet

mixing into the same units is described by Eq. (4).These are combined in a master equation thatdescribes the simultaneous operation of both pro-cesses [Eq. (7)]. Using these equations in combina-tion with general rules for assigning mixingprobabilities between strata [Eq. (1)], it is possibleto identify at a general theoretical level the commonstages and patterns that may accompany post-depo-sitional mixing (Fig. 18a).

When archaeological materials are confined tosome subset of the total strata in the sequence—inother words, they form discrete occupations—thenmixing may involve two stages: (1) dissipation;and (2) establishment of a mixing equilibrium. Thisis very apparent in the case of symmetrical localmixing of a single occupation horizon (Fig. 18a).Dissipation is signaled first by the development ofa Gaussian normal distribution of specimens acrossstratigraphic units. This distributional pattern maybe diagnostic of relatively minor amounts of sym-metrical local mixing, though future empirical workis needed to verify this claim. Dissipation is pre-dicted where the cumulative probability of mixingout from focal units with large number of specimens(e.g., the stratigraphic mode) is greater than the

s. (a) Symmetrical local mixing of a single occupation horizon. (b)rical local mixing of two separate occupations.


probability of mixing in from units with small num-bers of specimens (see Eq. (6)). As dissipation con-tinues, the amplitude of the mode declines and,over long periods of time, disappears completelyleaving a uniform distribution (Fig. 18a). This is aso-called mixing equilibrium since mixing may beongoing, but it leads to no net change (beyond smallstochastic fluctuations) in the number of specimensin any of the stratigraphic layers.

In cases where there is either asymmetrical mix-ing, or there are multiple occupation horizons in aprofile, an accumulation stage may appear betweendissipation and the establishment of a mixing equi-librium. In the case of asymmetrical mixing with adownward bias, for example, an initial dissipationphase is followed by gradual accumulation of spec-imens in the lowest unit that allows in mixing(Fig. 18b). Trailing specimens may take some timeto complete their transit to base of the profile.But, when they finally do, the system arrives at atype of secular equilibrium (see Faure and Mensing,2005); the rate of any further downward mixing iscontrolled by the rate at which specimens are mixedup from the bottom of the profile. A similar accu-mulation dynamic may also occur above the baseof a section if there are major changes in sedimenttexture; for example, a subsurface carbonate hori-zon (caliche) may serve as a barrier to mixing ofspecimens below a certain depth (see Johnson,1989; Shull, 2001).

Accumulation may also characterize post-deposi-tional mixing of multiple discrete occupations(Fig. 18c). However, the cause of the dynamic in thiscase is quite different. In general, dissipation duringthe early stages of mixing may cause specimens fromdiscrete occupations to merge, producing a unimo-dal distribution from a bimodal (or multimodal)distribution. A Gaussian-like normal distributionmay result if the two merging distributions arethemselves Gaussian. The system may shift backto dissipation, which is reflected in the progressivefall in amplitude of the single mode, and then con-verge on a mixing equilibrium (Fig. 18c).

Non-local mixing proceeds through all of thesame general stages, though typically much fasterand with different characteristic distributionalshapes apparent in each stage.

The emergence of approximately Gaussian nor-mal distributions in cases of symmetrical local mix-ing is not surprising since the mixing process asmodeled is a simple random walk equivalent ofone dimensional diffusion (Boudreau, 1997; Denny

and Gaines, 2002). A Gaussian distribution also ispredicted in the case of symmetrical non-local mix-ing if there is some limit to how far specimens canmove (i.e., a maximum mixing distance). In bothcases, the Central Limit Theorem applies (Dennyand Gaines, 2002). Specifically, since the total dis-tance traveled by a given specimen within a strati-graphic section is the sum of the distances traveledin each of many independent random mixing events,the aggregate distribution of distances traveled forall specimens will be approximately normally dis-tributed. The situation becomes more complicatedwhen we are dealing with asymmetrical local mix-ing. Here the appropriate comparison is with biasedrandom walk models (Turchin 1998), which lead tonormal diffusion with drift in the direction of theasymmetry.

For the simplest case of symmetrical local mixingit is also possible to specify quantitatively how thevertical spread of specimens evolves with time.Defining a as the sum of the probabilities that a sin-gle specimen is mixed from a focal stratum j to thestrata immediately above or below (i.e,a = ajj+1 + ajj�1), then the standard deviation ofthe distribution of specimens after t mixing eventsis (Denny and Gaines 2002)

r ¼ffiffiffiffiatp

ð10Þ

The standard deviation relative to an initial occupa-tion horizon is proportional to the square root ofthe amount of time exposed to mixing. Estimatesof the spread of specimens in cases of non-local mix-ing may also be obtained (e.g., Zanette and Alema-ny 1995), but their form is strongly dependent onthe specific nature of the transport process.

The appearance of a uniform mixing equilibriumis also expected in cases of symmetrical mixing. Thisis easiest to see if we write a simplified version of Eq.(6) dealing with a focal stratum and only the twoadjacent strata above and below

Dnj ¼ njþ1ajþ1;j|fflfflfflfflffl{zfflfflfflfflffl}n from stratum above

þ nj�1aj�1;j|fflfflfflfflffl{zfflfflfflfflffl}n from stratum below

� njaj;jþ1|fflfflffl{zfflfflffl}n to stratum above

� njaj;j�1|fflfflffl{zfflfflffl}n to stratum below

ð11Þ

Because the mixing probabilities are symmetricaland equal in all cases we may replace the variousterms aj+1,j, . . ., aj,j�1 with a single term ajk. Simpli-fying the resulting equation gives

Dnj ¼ ajkðnjþ1 þ nj�1 � 2njÞ ð12Þ

Fig. 19. Simulated symmetrical local mixing involving small numbers of archaeological specimens. Shown are the vertical distributions ofspecimens in ten separate sections after 500 mixing events. For large numbers of specimens the distributions should be Gaussian but arehighly variable because of small sample sizes.


The system can be in equilibrium (i.e., Dnj = 0) ifajk = 0, meaning that there is no mixing betweenstratigraphic layers, or if nj+1 = nj�1 = nj, meaningthat there is an equal number of specimens presentin each layer. In other words, the mixing equilib-rium is a uniform distribution of specimens acrossall strata.1 The same conclusion obtains in the caseof non-local mixing provided that the mixing prob-abilities at distances d above and below a focal stra-tum are also symmetrical.

We must temper the above observations with therecognition that the small sample sizes typical ofreal archaeological contexts might impede our abil-ity to recognize both different mixing systems anddifferent mixing stages. Fig. 19 shows nine simulatedvertical artifact distributions for small samples ofspecimens (n = 20) after 500 time steps of symmetri-cal local mixing with ajk = 0.01, k = ±1. Theexpected result under these conditions is a Gaussian

1 This is strictly true only for cases where there is no flux ofspecimens across the boundaries, which should hold for moststratigraphic sections and material types. Flux boundary condi-tions might be more appropriate for studying isotopes or organicsthat may be lost through the base of the section. Flux boundaryconditions allow for linear decreasing or increasing equilibriumsolutions.

normal distribution of specimens centered on theinitial occupation horizon (see Fig. 4). At smallsample sizes, however, there is substantial variabil-ity in empirically observed patterns and individualstratigraphic sections might be easily misinter-preted. For example, in none of the nine simulatedsections is there an obvious single peak in occupa-tion intensity. In some instances (e.g., Section 1)specimens remain largely clustered vertically, sug-gesting continuous occupation, whereas in othersthere appear to be clusters with intervening occupa-tion hiatuses (e.g., Section 9). There is also greatvariability in the distances over which specimenshave moved (e.g., Section 1 vs. 5). In all of the sec-tions one would be hard pressed to infer that mixingwas local and symmetrical and it is not obviouswhether the distributions represent a mixing equilib-rium or some intermediate mixing stage. Only at lar-ger sample sizes do these patterns become apparent.

A number of criticisms might be voiced about theresearch presented here and we address some ofthese now. First, it is important to recognize thatthe simulations and empirical case presented aboveare not meant to be exhaustive of all of possibletypes of post-depositional mixing that might be seenin the archaeological record. Nor is it necessarily thecase that the generic mixing models considered here


are equally likely to occur in empirical settings. Forexample, it may be the case that asymmetrical mix-ing is far more common than symmetrical mixing inthe vast majority of archaeological settings. Whilearchaeological site formation and diagenesis is acomplex, multivariate process, we also believe thatthere are many advantages to occasionally steppingback from a complex empirical reality to considersimple models. These are often easier to analyzeand may provide a clear picture of fundamentaldynamics.

Second, of the many ways in which these modelsmight find empirical application, we opted formodel-fitting as a reasonable first approach.Another approach would be to quantitatively char-acterize the short-term effects of different individualmixing agents and produce a calibrated transitionmatrix. Simulation could then be used to examinethe theoretical impact of mixing by each agent typeover ecological and geological time scales. Compar-isons between simulations and empirical examplesmight reveal which of the many possible tapho-nomic agents were likely responsible for observedpatterns. The advantage of simulation over tradi-tional experimental approaches in this context isthat ecological and geological time scales can beinvestigated (Wilke and Adami 2002). Unfortu-nately, this approach is not easily accomplished atBarger Gulch because many different mixing agentsmay have been active and may have had overlap-ping effects.

Third, it is clear that there are a number ofimportant variables that have been excluded fromthe current models. These include the impact of dif-ferences in sediment characteristics on mixing(Johnson et al., 2005; Shull, 2001), variability inthickness of stratigraphic units (Meysman et al.,2003; Shull, 2001), active sedimentation and burial(i.e., advection) (Balek, 2002; Boudreau, 1997), sed-iment compaction (Andrews, 2006), horizontal mix-ing, which might occur at the surface before burialor within a sedimentary column after burial(Audouze and Enloe 1997; Enloe 2006; Surovellet al. 2005), and decay or destruction of materialsas a function of time or sediment characteristics(Pendall et al. 1994; Surovell and Brantingham2007; Trumbore 2000; Wang et al. 1996). The factthat we needed to use different model parameteriza-tions above and below the Folsom occupation sur-face at Barger Gulch to arrive a reasonabledescription of mixing dynamics there illustrates thatmost empirical applications may need to move

beyond the simple examples presented here. How-ever, there are clear avenues for modeling each ofthese added complexities. For example, in contrastto the simulations presented above where the num-ber of specimens is conserved over time, other typesof materials (e.g., organics) might be graduallydestroyed in addition to being exposed to mixing.One might incorporate time (and depth) dependentdestruction of specimens by modifying Eq. (6) toread

Dnj ¼Xm

k¼1k6¼j

akjnk � nj

Xm

k¼1k 6¼j

ajk � cjnj ð13Þ

where cj is a decay constant associated with focalstratum j. The last term in Eq. (13) thus introducestime-dependent exponential decay in the number ofspecimens present in stratum j. Similar modifica-tions to Eq. (6) may allow one to consider the im-pact of active burial into or variable sedimentarycharacteristics on mixing dynamics (Boudreau1997; Meysman et al. 2003; Shull 2001). The ab-sence of these and other complexities from the cur-rent models, however, should not detract from thegeneral conclusions that there may be substantialregularities that emerge from post-depositional mix-ing of archaeological deposits.

Conclusions

Post-depositional mixing of archaeologicaldeposits may be modeled as a stochastic processthat assigns probabilities to the event that a singlespecimen of a given artifact class is mixed betweenany two discrete stratigraphic layers. The changein the number of artifact specimens in a single strati-graphic layers is described by a simple recursionrelation that includes terms for the movement ofspecimens both out of and into a so-called focalstratum. Mass movement of artifacts within a strati-graphic section is then the cumulative number ofsuch mixing events between all strata.

Simulations were used to characterize symmetri-cal local and non-local mixing dynamics. Both pro-cesses appear to have characteristic dynamics. Theearly and intermediate stages of symmetrical localmixing may be recognized by the development ofGaussian-like vertical distributions of artifactswithin a stratigraphic column. Symmetrical non-local mixing, by contrast, may be characterized bythe development of vertical distributions of artifactsthat follow closely the underlying single-specimen


probabilities of transport between strata. As in thesymmetrical case, intermediate stages of symmetri-cal non-local mixing may be characterized by thedevelopment of a Gaussian-like vertical distribu-tions of specimens. The vertical distributions of arti-facts generated by the late stages of symmetricallocal and non-local mixing are typically uniform.

Simulations were also used to characterize asym-metrical local mixing, where there is directional biasin the movement of single specimens downwards (orupwards) in the section. Here we see the develop-ment of skewed vertical distributions of specimensin section and the regular migration of the densitypeak (distribution mode) in the direction of the bias.The equilibrium distribution of specimens in thiscase shows most artifacts accumulated in the lowest(or highest, if the bias is upwards) stratigraphic unitthat allows mixing. It was shown that many of thesegeneral dynamical features of post-depositionalmixing appear even under more complex initial dis-tributions of archaeological materials in section. Inparticular, we considered mixing of more than onediscrete initial occupation and a case of an exponen-tially increasing distribution of artifact specimens.

Examination of the Barger Gulch Folsom sitesuggested that non-local mixing agents may haveplayed a role in generating the vertical distributionof artifacts seen there. The preservation of stronglyconcave vertical distributions of artifacts bothabove and below the initial Folsom occupation isindicative of early stage mixing. It was proposedalso that there is a global asymmetry characterizingmixing at the site; more extensive mixing occurredin the strata above the original occupation surfacecompared with below. A calibrated simulation,using a negative exponential model for the mixingprobabilities, was able to statistically approximatethe vertical distribution of artifacts at Barger Gulch.We view this result as preliminary support for thegeneral validity of the models developed herein forstudying post-depositional mixing of archaeologicaldeposits.

Acknowledgments

We thank David Madsen, Charles Perreault, Da-vid Rhode, and several anonymous reviewers fortheir insightful comments on earlier versions of thispaper. Work at Barger Gulch is supported by theNational Science Foundation (BCS-0450759)awarded to Waguespack and Surovell. Any opin-ions, findings, and conclusions or recommendations

expressed are those of the authors and do not neces-sarily reflect the views of the National ScienceFoundation.

References

Albert, R.M., Bar-Yosef, O., Meignen, L., Weiner, S., 2003.Quantitative phytolith study of hearths from the Natufianand middle palaeolithic levels of Hayonim Cave (Galilee,Israel). Journal of Archaeological Science 30, 461–480.

Andrews, B.N., 2006. Sediment consolidation and archaeologicalsite formation. Geoarchaeology—An International Journal21, 461–478.

Araujo, A.G.M., Marcelino, J.C., 2003. The role of armadillos inthe movement of archaeological materials: an experimentalapproach. Geoarchaeology 18, 433–460.

Audouze, F., Enloe, J.G., 1997. High resolution archaeology atVerberie: limits and interpretations. World Archaeology 29,195–207.

Baker, C.M., 1978. Size effect-explanation of variability in surfaceartifact assemblage content. American Antiquity 43, 288–293.

Balek, C.L., 2002. Buried artifacts in stable upland sites and therole of bioturbation: a review. Geoarchaeology 17, 41–51.

Bocek, B., 1986. Rodent ecology and burrowing behavior-predicted effects on archaeological site formation. AmericanAntiquity 51, 589–603.

Bollong, C.A., 1994. Analysis of site stratigraphy and formationprocesses using patterns of pottery sherd dispersion. Journalof Field Archaeology 21, 15–28.

Boudreau, B.P., 1997. Diagenetic Models and Their Implemen-tation: Modeling Transport and Reactions in Aquatic Sedi-ments. Springer, Berlin.

Brantingham, P.J., 2007. A unified evolutionary model ofarchaeological style and function based on the price equation.American Antiquity 72, 395–416.

Delagnes, A., Roche, H., 2005. Late Pliocene hominid knappingskills: the case of Lokalalei 2C, West Turkana, Kenya.Journal of Human Evolution 48, 435–472.

Denny, M., Gaines, S., 2002. Chance in Biology: Using Proba-bility to Explore Nature. Princeton University Press,Princeton.

Enloe, J.G., 2006. Geological processes and site structure:assessing integrity at a late Paleolithic open-air site innorthern France. Geoarchaeology 21, 523–540.

Erlandson, J.M., 1984. A case-study in faunalturbation-delineat-ing the effects of the burrowing-pocket-gopher on the distri-bution of archaeological materials. American Antiquity 49,785–790.

Esdale, J.A., Le Blanc, R.J., Cinq-Mars, J., 2001. Periglacialgeoarchaeology at the Dog Creek site, northern Yukon.Geoarchaeology—An International Journal 16, 151–176.

Faure, G., Mensing, T.M., 2005. Isotopes: Principles andApplications. John Wiley and Sons, Hoboken.

Hilton, M.R., 2003. Quantifying postdepositional redistributionof the archaeological record produced by freeze–thaw andother mechanisms: an experimental approach. Journal ofArchaeological Method and Theory 10, 165–202.

Hofman, J.L., 1986. Vertical movement of artifacts in alluvialand stratified deposits. Current Anthropology 27, 163–171.

Johnson, D.L., 1989. Subsurface stone lines, stone zones, artifact-manuport layers, and biomantles produced by bioturbation


via pocket Gophers (Thomomys Bottae). American Antiquity54, 370.

Johnson, D.L., Domier, J.E.J., Johnson, D.N., 2005. Reflectionson the nature of soil and its biomantle. Annals of theAssociation of American Geographers 95, 11–31.

Karavanic, I., Smith, F.H., 1998. The middle/upper Paleolithicinterface and the relationship of Neanderthals and earlymodern humans in the Hrvatsko Zagorje, Croatia. Journal ofHuman Evolution 34, 223–248.

Kornfeld, M., Frison, G.C., 2000. Paleoindian occupation of thehigh country: the case of Middle Park, Colorado. PlainsAnthropologist 45, 129–153.

Kornfeld, M., Frison, G.C., White, P., 2001. Paleoindianoccupation of Barger Gulch and the use of Troublesomeformation chert. Current Research in the Pleistocene 18, 32–34.

Kroll, E.M., 1994. Behavioral-implications of pliopleistocenearchaeological site structure. Journal of Human Evolution 27,107–138.

Laville, H., Rigaud, J.-P., Sackett, J., 1980. Rock Shelters of thePerigord: Geological Stratigraphy and Archaeological Suc-cession. Academic Press, New York.

Lenoble, A., Bertran, P., 2004. Fabric of Palaeolithic levels:methods and implications for site formation processes.Journal of Archaeological Science 31, 457–469.

Lyman, R.L., 2003. The influence of time averaging and spaceaveraging on the application of foraging theory in zooar-chaeology. Journal of Archaeological Science 30, 595–610.

Mayer, J.H., Surovell, T.A., Waguespack, N.M., Kornfeld, M.,Reider, R.G., Frison, G.C., 2005. Paleoindian environmentalchange and landscape response in Barger Gulch, Middle Park,Colorado. Geoarchaeology 20, 599–625.

McPherron, S.J.P., 2005. Artifact orientations and site formationprocesses from total station proveniences. Journal of Archae-ological Science 32, 1003–1014.

Meysman, F.J.R., Boudreau, B.P., Middelburg, J.J., 2003.Relations between local, nonlocal, discrete and continuousmodels of bioturbation. Journal of Marine Research 61, 391–410.

Morin, E., 2006. Beyond stratigraphic noise: unraveling theevolution of stratified assemblages in faunalturbated sites.Geoarchaeology 21, 541–565.

Morin, E., Tsanova, T., Sirakov, N., Rendu, W., Mallye, J.B.,Leveque, F., 2005. Bone refits in stratified deposits: testing thechronological grain at Saint-Cesaire. Journal of Archaeolog-ical Science 32, 1083–1098.

O’Brien, M., 2006. Cryoturbation effects of upheaving on verticaldeposition of lithic artifacts. Unpublished M.A. Thesis,Department of Anthropology, University of Wyoming, Lar-amie, WY.

Pendall, E.G., Harden, J.W., Trumbore, S.E., Chadwick, O.A.,1994. Isotopic approach to soil carbonate dynamics andimplications for Paleoclimatic interpretations. QuaternaryResearch 42, 60–71.

Reible, D., Mohanty, S., 2002. A levy flight-random walk modelfor bioturbation. Environmental Toxicology and Chemistry21, 875–881.

Rowlett, R.M., Robbins, M.C., 1982. Estimating originalassemblage content to adjust for post-depositional verticalartifact movement. World Archaeology 14, 73–83.

Schiffer, M.B., 1987. Formation Processes of the ArchaeologicalRecord. University of New Mexico Press, Albuquerque.

Shull, D.H., 2001. Transition-matrix model of bioturbation andradionuclide diagenesis. Limnology and Oceanography 46,905–916.

Surovell, T.A., Brantingham, P.J., 2007. A note on the use oftemporal frequency distributions in studies of prehistoricdemography. Journal of Archaeological Science 34, 1868–1877.

Surovell, T.A., Waguespack, N.M., in press. Folsom hearth-centered use of space at Barger Gulch, locality B. In:Brunswig, R.S., Pitblado, B. (Eds.), Emerging Frontiers inColorado Paleoindian Archaeology. University of ColoradoPress, Boulder.

Surovell, T.A., Waguespack, N.M., Mayer, J.H., Kornfeld, M.,Frison, G.C., 2005. Shallow site archaeology: artifact dis-persal, stratigraphy, and radiocarbon dating at the BargerGulch locality B Folsom Site, Middle Park, Colorado.Geoarchaeology—An International Journal 20, 627–649.

Trumbore, S., 2000. Age of soil organic matter and soilrespiration: radiocarbon constraints on belowground Cdynamics. Ecological Applications 10, 399–411.

Turchin, P., 1998. Quantitative Analysis of Movement: Measur-ing and Modeling Population Redistribution in Animals andPlants. Sinauer Associates, Sunderland.

Van Nest, J., 2002. The good earthworm: how natural processespreserve upland Archaic archaeological sites of westernIllinois, USA. Geoarchaeology—An International Journal17, 53–90.

Villa, P., 1982. Conjoinable pieces and site formation processes.American Antiquity 47, 276–290.

Waguespack, N.M., 2005. The organization of male and femalelabor in foraging societies: implications for Early Paleoindianarchaeology. American Anthropologist 107, 666–676.

Waguespack, N.M., Surovell, T.A., Laughlin, J.P., 2006. The2004 and 2005 Field Seasons at Barger Gulch, Locality B(5GA195). Technical Report No. 41, George C. FrisonInstitute of Anthropology and Archaeology, Department ofAnthropology, University of Wyoming, Laramie, WY.

Wang, Y., Amundson, R., Trumbore, S., 1996. Radiocarbondating of soil organic matter. Quaternary Research 45, 282–288.

Wilke, C.O., Adami, C., 2002. The biology of digital organisms.Trends in Ecology and Evolution 17, 528–532.

Wood, W.R., Johnson, D.L., 1978. A survey of disturbanceprocesses in archaeological site formation. Advances inArchaeological Method and Theory, 1.

Zanette, D.H., Alemany, P.A., 1995. Thermodynamics of anom-alous diffusion. Physical Review Letters 75, 366–369.

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