Juan A. Barceló, Laura Mameli, Alfredo Maximiano, Oriol Vicente, Departmento de Prehistoria, Facultat de Lletres,

Universidad Autónoma de Barcelona, Catalonia, Spain 8193 Bellaterra, Spain

ARCTIC ANTHROPOLOGY, Vol. 46, Nos. 1–2, pp. 203–214, 2009 ISSN TK© 2008 by the Board of Regents of the University of Wisconsin System

New Computational and Mathematical Methods for Archaeological Fieldwork at the Extreme South of the Populated World

Juan A. Barceló, Laura Mameli, Alfredo Maximiano, and Oriol Vicente

Abstract. We present the methodology of computer modeling of an archaeological excavation. The method allows the study of archaeological formation, modifi cation and transformation pro-cesses. Our main goal has been to build a computer model of midden-site formation processes. The main purpose of these models is converting excavation data to visual elements (lines, sur-faces, and solids) which can be used as a representation of that data. Furthermore, geostatistics and other quantitative spatial analysis methods are also discussed to characterize the processes having generated the spatial distribution of archaeological data.

Fieldwork along the Beagle Channel

Along the northern shore of Beagle Channel and neighboring areas in South America’s extreme South (54° degrees South), the main archaeolog-ical evidence consists of massive accumulations of shell and other materials generated by hunter- gatherer populations. Defi ned as shell middens, these sites were constituted by the accumula-tion of a large quantity of mussel and other mol-lusk shells, discarded tools, charcoal, stone fl akes, the bones of fi sh, birds, sea and terrestrial mam-mals, etc. Within the middens, remains of dif-ferent social activities such as subsistence prac-tices and resource processing, fi res for cooking and/or heat, lithic reduction, bone tool making, etc., have been documented. Cooking and warm-ing fi res were tended inside huts and the refuse was tossed outside the built space, creating or contributing to rapidly accumulating refuse mid-dens. In the landscape, these structures may ap-

pear as domes, ring-shaped structures, thin lenses, or a combination of these. The hut features contain an inordinate amount of charcoal and humus in comparison with the adjacent midden consisting mainly of shells. Overall, the accumulations pres-ent hummocky topography (Estévez and Vila 1995, 2006, 2007; Estévez et al. 2001; Orquera and Piana 1989–90, 1992, 1999, 2000).

The distance between two neighboring sites is usually less than a few hundred meters, ex-cept in cases of non-habitable, very steep or wall-like coasts. There seems to be no difference among the sites in density, size, or aspect. A preliminary analysis of archaeological site locations (Barceló, Piana, and Matinioni 2002) concluded:

• The majority of shell middens are near or very near the actual shoreline (81% of all sites are closer than 100 meters). Most ar-chaeological sites (63%) overlay an ancient pebble/cobble beach, which suggests that the shoreline was also a main locational factor in ancient times.

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• These sites generally seem related to modern beaches with smooth slope (67%), which lie near a fresh water source (76% of all sites closer than 100 meters).

• Most of the archaeological sites have been discovered near moraine deposits (69% of all sites are closer than 100 meters).

• In many cases, the moraine deposits behind sites seem to have provided protection from the strongest winds (56%).

According to these preliminary results, the main spatial attractors for human settlement activities seem to have been:

• The location and slope of the shoreline. This is related to easier access to basic resources (seafood) and the proper use of the most rel-evant work instruments (canoes);

• Fresh water sources; and• Moraine deposits, and other landscape

features used as protection from weather conditions.

In general, multivariate statistical analysis sug-gests that the most infl uential factors in explaining the patterns of spatial variability are:

• Geomorphological features of the site location,

• Distance to the shoreline,• Relation to woods, and• Wind protection.

The Nature of the Archaeological ProblemA shell midden can be described in terms of a se-quence of accumulations of diverse refuse mate-rial. Therefore, we should assume their shape and extent will vary according to deposition and the microtopography of the ground surface over which the refuse material was accumulated. This means that garbage accumulates over previous accumu-lations and that new deposition episodes bury the occasional formation of natural soils at tempo-rarily abandoned settlements. As a result, natural and anthropogenic events are integrated into the accumulation/deformation history of the site.

Our main goal has been to build a computer model of a midden site formation process (Barceló 2005, Barceló et al. 2003, Barceló, Maximiano, and Vicente 2005; Vicente 2005). The main goal of this model is converting excavation data to visual el-ements (lines, surfaces, and solids) which can be used as a representation of those data. Such a vi-sual model will compress many individual data into one single picture so it can reveal correlations between different archaeological features in space and time.

Michael Leyton (1992) argues that a trajectory of changes (a history) can be described as a dis-continuous sequence composed of a minimal set

of distinguishable actions. Consequently, what ap-pears to be different in the present speaks about some action in the past that generated such a dif-ference. Translating this general principle into ar-chaeological research, we can defi ne an archae-ological site as a sequence of fi nite states of a temporal trajectory, where an entity (ground sur-face) has been modifi ed successively. Natural and human processes modify physical space and as a result we are able to distinguish components, which can be used as analytical units. A compo-nent is a region in space delimited by perceived discontinuities in color, shape, texture, or topol-ogy, where the probability of a specifi c forma-tion process is the highest. Therefore, if we can distinguish variation in archaeological space we can follow the successive transformation of the ground surface where social action was originally performed.

We assume that the past will be “seen” within the present, if, and only if, some different states within the present can be distinguished, and these states are ordered according to some kind of directionality. This last assumption is very impor-tant, because not just any observed difference in the present speaks about the past. The true nature of social activity is expressed in such directional-ity of the sequence of human induced changes be-tween spatio-temporally successive events. Direc-tionality is the result of interpreting a persistent state or the discovery of an order in a sequence or trajectory of multiple different states. We can ex-plain a trajectory of changes by imposing a tempo-ral slicing on archaeologically perceived disconti-nuities (Barceló and Vicente 2004). The intention of such a slicing will be to visually represent the transitions between events. In this way, we would simulate the actual occurrences of the events in an historical sequence. Such a trajectory of events would be “explanatory” because the same occur-rence of an event within the trajectory, and its spatio-temporal relationship with the preceding and successive event would serve as the explicans of what happened in the past.

Decomposing Archaeological Space: A Case Study

We have used data from the excavation of two ar-chaeological sites from Beagle Channel (Túnel VII, Shamakush VIII). Those sites have been described as shell middens, and they have been explained as the material consequence of the social activities of human groups having intensively exploited the littoral and maritime resources, including a large amount of mollusks (primarily mussels). Túnel VII is a multicomponent site, which has been dated to the nineteenth century, contemporaneous with the fi rst European settlement in the area (Estévez and

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Barceló et al.: New Computational and Mathematical Methods 205

Vila 2006, 2007; Vila et al. 2005). Shamakush VIII is a single component site dated 1400 ± 90 B.P. (AC 1678) and 1380 ± 115 B.P. (AC 1681) (Piana and Vázquez 2005).

The excavation covered the full extension of an occupation unit in order to investigate spatial horizontal synchronies. The differences between subunits were described according to a formalized and standardized questionnaire of the sediment structure and composition that was confi rmed by the analysis and quantifi cation of these variables in the sediment composition. The limits of every subsequent subunit were set and registered, tak-ing a general contextual picture of the subunit and some detailed pictures of each square meter be-fore and after the extraction. The subunits were extracted in the inverse order of their deposition. The depth of the surfaces was also measured fol-lowing a fi fty square centimeter grid. The formal-ized and standardized recording of the sediment structure was completed with the notation of the 3D location and relative situation (geographic ori-entation, gradient, and archaeological/anatomical position) of anthropogenic residues which means, for instance, all identifi able bone fragments lon-ger than 3 cm. Sediments were screened in the ex-

cavation at a millimeter scale. There was a signif-icant variation in the composition of the different components of the subunits (fi ne sediments, gran-ules and small pebbles, mollusks, charcoal, bones, and lithic fl akes) (Fig. 1).

We assume that the original ground surface was modifi ed successively by accumulating refuse material on it, by deforming a previous accumula-tion, or by direct physical deformation (building, excavation) (Barceló 2002, 2005). Thus, the aim of the visualization analysis is to represent those characteristic properties of physical space that were modifi ed by human labor through time.

We decompose the physical space where hu-man work took place—the archaeological site—in the geometrical sense of the word, that is to say, using two kinds of perceived discontinuities: physical modifi cations (structures) and differen-tial accumulations (deposits). In the specifi c case of the Shamakush VIII excavation, most of the ar-chaeologically perceived discontinuities consisted of strata, layers, and material concentration areas. They can be described and analyzed using two different kinds of variables: form and frequency. Form is a synonym for shape. We measure shape, when we refer to the interfacial boundaries of a

Figure 1. Excavating at the Shamakush VIII site. The pictures depict the process of single accumulations identifi ca-tion, delimitation, and documentation of spatial frequencies using a 50 cm. grid. Photographs by J. A. Barceló.

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206 Arctic Anthropology 46:1–2

perceived modifi cation of ground surface. We con-sider the frequency aspect of archaeological fea-tures when we describe the qualitative and quanti-tative composition of what has been accumulated on the ground surface; of course, those proper-ties are just a way of analyzing archaeological re-ality. That means that any archaeological feature can be studied in terms of its interfacial boundary, or in terms of some quantitative or intensity mea-sure. Form decomposition and frequency decom-position should be considered as separated analyt-ical strategies.

A Visual Model of Spatial Shape DeformationFrom a formal viewpoint, we have considered the archaeological site as a semi-infi nite continuum made up of discrete, irregular, discontinuous vol-umes defi ned by characteristics, which in turn in-fl uence the spatial variation of an archaeological or geological feature. These volumes, called ar-chaeological phases, are the building blocks for the visual model, and they should be understood as the events of the site history. They are expres-sions of the fact that differentiated regions of phys-ical space correspond to areas where a social, geological, or biological event generated some de-formation of the ground surface.

A sequence of different temporal occupations within the midden should be detected and sepa-rated like the pages of a book. They are the conse-quence of a change in the formation process act-ing on a specifi c location. To distinguish contacts between the “pages,” we were guided by how dif-ferent spatial areas were separated from each other by clearly defi ned interfacial boundaries. Observ-able criteria such as the state of the shells (whole, fragmented, orientation), the consistency, texture, disposition, and color of the soil matrix, and par-ticularly, the identifi cation of distinct contact sur-faces between different accumulations, allow the defi nition of interfacial boundaries between forma-tion units. When possible we distinguished strat-ifi cation plans for their mechanical properties (cohesion, density, continuity). Interfacial bound-aries between formation units are translated geo-metrically into polygons or polylines, and loaded into a GIS software system for subsequent pro-cessing (Fig. 2). However, interfacial boundar-ies between different spatio-temporal phases are multi- dimensional in nature. This aspect cannot be properly simulated in a standard GIS approach, given the bi-dimensional nature of their vector or raster building blocks. We need additional data to transform a drawn polygon into a virtual unit which correctly represents the interfacial bound-ary. Additional micro-topographic x, y, z values

should be measured within the region delimited by observed interfacial boundaries. The result is a three- dimensional matrix representing the three dimen sional characteristics of each contact surface (Fig. 3). A geometric surface was interpolated us-ing measured z values, and the 3D view for each surface contact was computed.

Because an archaeological site is a sequence of events, to represent this process, we must corre-late interfacial boundaries between observed dis-continuities in the site formation process (Fig. 4). This implies the representation of spatio- temporal changes in terms of a number of particular geo-metrical models, each for a time slice. A series of correlated surfaces of this type is referred to as a volumetric dataset (Fig. 5). It can be represented by a series of volumes, each containing a similar n dimensional data array. Collectively, these

Figure 2. Identifying a single formation unit (sub- midden), and computer representation of its geometri-cal features using a GIS software system (using the Manifold tool, http://www.manifold.net).

Figure 3. A graphical display of a three-dimensional matrix, with a photograph of the archaeological contact surface overlaid.

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fi les are interpreted as a single array of n+1 dimensions.

Processing a volumetric dataset begins by stacking the slices of a given dataset in com-puter memory according to the interelement (dis-tances between slices as between different layers), both within a single image and between different layers—so that the data are converted into a mul-tidimensional geometric representation which ac-curately refl ects the real world dimensions of the originally sampled volume. The next step is to cre-ate additional slices to be inserted between the dataset’s actual slices so that the entire volume, as it exists in computer memory, is represented as one solid block of data. The number of slices needed

to fi ll in the blanks is based on the dataset’s inter-pixel and interslice spacing and the slices needed are created through interpolation (Barceló 2005; Barceló et al. 2003; Barceló and Vicente 2004.

The purpose of this visual model is to repre-sent transitions between depositional events. Dis-continuities in the model are related to measured interfacial boundaries in the fi eld, which were dy-namically constructed, and hence conformable through space and time. The visual model allows an interpretation of the spatio-temporal variabil-ity of the material consequences of social action in terms of discrete, contiguous, irregular surfaces, with uniform value throughout each volume.

A Visual Model of Spatial FrequenciesWe consider the spatial frequency aspect of any ar-chaeological feature when we describe it as an ac-cumulation of some material items on the ground surface where the action took place, or as the in-tensity of the action (Barceló 2002, 2005; Barceló and Maximiano 2007; Barceló, Maximiano, and Vicente 2005; Maximiano 2005, 2007). Formally, spatial frequencies may be thought of as consisting of a set of locations (s1, s2, etc.) in a defi ned “study region,” R, at which the material consequences of some social action performed in the past (archaeo-logical event) have been recorded. We can use sca-lar fi elds to represent this data structure (Fig. 6).

A scalar fi eld is a name we give to a func-tion which takes in points in a two or three di-mensional space (R2 or R3) and outputs real num-bers. It is a collection of scalar values together with a mathematical function by which those val-ues are shown to be related to one another by de-fi ning a distance measure. The scalar fi eld is a concept spawned from the natural and physical

Figure 4. A graphical representation of correlated in-terfacial boundaries between successive archaeological phases. Shamakush VIII data set (Barceló et al. 2003).

Figure 5. Graphical representation of a volumetric data set. Shamakush VIII data set (Barceló, Maximiano, and Vicente 2005).

Figure 6. An example of a scalar fi eld. Data: Bird bone remains from the Túnel VII archaeological site (Barceló and Maximiano 2007; Mameli 2004).

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sciences since they often deal with a region of an abstract space with a function attached to it. For example, the function that gives the tempera-ture of any point in the room in which you are sit-ting is a scalar fi eld. Archaeologically, the function that gives the quantity of archaeological materi-als (bones, lithics, shells, or any other kind of ar-chaeological evidence) at any location of the site is a scalar fi eld. In most shell midden sites along the Beagle channel site (Túnel VII, Lanashuaia, Shamakush VIII, etc.) a location was defi ned as a spatio- temporal sampling area of 50 × 50 cm and a volume delimited by its upper contact surface with the temporally successive layer.

Our approach is based on the idea that site components should be characterized by the inten-sity of the social activity performed at each place and moment, associated with its probability. We assume the probability that a social action oc-curred at a specifi c location is related to the fre-quency of its material effects (the archaeological record) at nearby locations. When the frequency of the archaeological feature at some locations in-creases, the probability that the social action was performed in its neighborhood will converge to-wards the relative frequency at adjacent loca-tions. Then, assuming that a measure of spatial fre-quency is a function of the probability an action was performed at that point, we will say that the area where spatial density values are more con-tinuous (stationary) is the most likely place where a social action was performed. This can be eas-ily computed by estimating the spatial probabil-ity density function associated with each loca-tion. Obviously, the method gives no explanation of the causal nature of the deposition. It is just a blind visualization of an observed accumulation, and not an interpretation of primary or secondary refuse processes, which should be included in the model as a priori probabilities in a Bayesian model (Buck, Cavenaugh, and Litton). In the same way, the degree of emptiness can be represented follow-ing the same approach, provided the spatial scale of observed discontinuities at the spatial level is coherent. More details about the limitations of the approach are given by Barceló and Maximiano (2007).

A three dimensional histogram is the sim-plest way of visualizing such a density function, emphasizing how the frequency of material conse-quences of the social action may vary across space (Fig. 7). A more complex representation of spatial probabilities is possible if we identify the mathe-matical model which produced the observed spa-tial distribution of scalar values (Fig. 8). In the fi g-ure, the interpolated surface of a random spatial distribution is characteristically fl at and irregu-lar (Fig. 8a). When the spatial distribution is aggre-gated and concentrated, the interpolated surface

appears regular and centered around the spatial mean (Fig. 8b). In this way, a spatial process can be defi ned in terms of the function that predicts the intensity or frequency (Z) at each location (co-ordinates x, y) (Bailey and Gattrell 1995; Fother-ingham, Brunsdon, and Charlton 2000; Haining 2003; Lloyd and Atkinson 2004; Orton 2005). It is simply a mathematical representation of a polyno-mial surface showing the global spatial pattern.

The purpose of such visual models of spa-tial frequencies is to predict the probability of a social activity at any imaginable spatial coordi-nate. The result is a probabilistic map for the spa-tiality of social actions. In such a map near things appear to be more related than distant things (To-bler’s Law), and this is so because the synchronic-ity of social actions states that, all else being equal, activities that occur at the same time will tend to increase the joint frequency of their effects. In the same sense, all else being equal, elements that are located within the interfacial boundary defi ned by some previous spatio-temporal gradient will be spatially related, confi guring a common region.

These interpolation models of spatial pro-cesses can be understood as composed by a deter-

Figure 7. a) Simple descriptive visual model of random distributed data. b) Simple descriptive visual model of spatially normalized data.

Figure 8. a) Interpolated visual mode of randomly dis-tributed data. b) Interpolated visual model of spatially normalized data (4th order polynomial interpolation) (Maximiano 2007).

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ministic component (spatial trend) and a stochas-tic variation. In the easiest case—often not the case with real data—we have:

Z = ax2 + by2 + dxy + fx + ry + f

Spatial trend Stochastic variation

The spatial trend represents what we know about the process—bio-geological or anthropo-genic, depositional or post-depositional—that formed the observed frequencies. The presence of stochastic variation is a direct consequence of un-certainty associated with the spatial process re-sponsible for the properties of that particular dis-tribution. Many surface interpolation algorithms are available, and most of them allow us to dis-tinguish the stochastic component from the spa-tial trend as a residual of the interpolated surface (Schabenberger and Gotway 2005).

Note that we need to know the polynomial parameters of the trend (a, b, d, f, r) to describe it. If we do not have previous information about how z (i.e., the frequency of the variable of inter-est at a given coordinate) might have been formed, then estimates of these parameters must be gener-ated from our actual observations of archaeologi-cal frequencies. Nevertheless, most of the time it is impossible to provide a single equation to charac-terize the spatial process. We should take into ac-count that the spatial trend contains both the pro-cess that generated the original frequencies prima facie, and all post-depositional processes that al-tered the original values. That means that we can never hope to fully characterize the process, but we can investigate some properties that represent important aspects of what generated the observed frequencies of archaeological features. In those cases, many aspects of spatial trends may be char-acterized in terms of the so-called fi rst-order and second-order properties of the spatial distribution. Very informally, the fi rst-order properties describe the way in which the expected frequency of mate-rial consequences of the social action varies across space, while second-order properties describe the covariance (or correlation) between frequencies at different regions in space (Barceló and Maximiano 2007; Maximiano 2007).

In seeking to understand “pattern” in ob-served spatial data, it is important to appreci-ate that this might arise either from area-wide “trends” (fi rst-order variation) or from correlation structures (second-order variation), or from a mix-ture of both. In the fi rst-order case, frequencies of archaeological features vary from location to loca-tion due to changes in the underlying properties of the local environment. For example, frequencies of accumulated refuse material may be infl uenced by variations in terrain. In the second-order case, fre-quencies of archaeological data vary from location

to location due to local interaction effects between observations. An additional example: material consequences of social action tend to happen in areas where the social action has been performed. We should assume a second order pattern in the data is due to some process that varies spatially. That means that patterns arise due to variations in social actions performed at discrete locations.

First and second order spatial analysis al-lows us to discover the spatial modality of social activities and bio-geological processes which oc-curred at the site. The question that now arises is whether the observed frequencies display any sys-tematic spatial pattern or departure from random-ness either in the direction of clustering or regular-ity. More interesting questions include:

• Is observed clustering due mainly to natural background variation in the population from which intensities arise?

• Over what spatial scale does any clustering occur?

• Are clusters merely a result of some obvious a priori heterogeneity in the region studied?

• Are clusters associated with proximity to other specifi c features of interest, such as the location of some other social action or pos-sible point sources of important resources?

Discriminating between random, clustered, and regular patterns of observed frequencies of ar-chaeological features is a fundamental concern, be-cause it will help us to understand the nature of the causal process (social actions) involved. Ran-domness at the spatial level can be the result of post-depositional alteration, and should be de-tected before social action at the spatial level can be explained. We need tools and methods to differ-entiate the different spatial ways that an action can be performed at different places. The actual evi-dence of the presence of a social action should be statistically different from the random location of its material evidences in or through different spa-tial and temporal locations. The idea is to inves-tigate the possibilities of relevant discontinuities in the general distance pattern. If such discontinu-ities exist and can be discovered, then we would conclude that actions were placed relative to the placement of other actions. This point has been discussed many times in the archaeological liter-ature (Blankholm 1991; Hodder and Orton 1977; Kintigh and Amermann 1982; Orton 2005), but it is still forgotten in many practical applications that before being able to answer any kind of “Where?” question, we should answer the “How?” one, be-cause the actual placement of any action depends of the way it is performed. The spatial features of the archaeological record provide us with indica-tions about the spatial modality of the social ac-tion, but not about the nature of the action itself.

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210 Arctic Anthropology 46:1–2

Put another way, an accumulated assemblage of archaeological evidence will show some hints of having been generated by an intentional social ac-tivity if the observed spatial frequencies at a dis-tinct location are associated with the value of the same kind of material evidence at neighboring points. If a single action generated the spatial pat-tern of frequency values of archaeological materi-als we are observing, then at lesser distances the differences should be statistically dependent on each other in frequency, and at greater distances the differences should be statistically indepen-dent. In the latter case, the probability that the so-cial action took place at one location would be equal to its probability value at any other location. Simply put, intentionality at the spatial level is the most probable cause of the tendency for a random error to be similar to its neighbors, and exhibit what has been called second-order stationarity.

Integrating Form and Frequency Variation in Space and TimeThe spatial intentionality of social actions can be explained in terms of the spatio-temporal “infl u-ence” an action performed at a location has over all locations in the proximity. An action can gen-erate the reproduction of similar actions around it, or it can prevent any other similar action in the same vicinity. Some of the actions performed in the vicinity of the location increase the chances of one type of action and decrease the chances of oth-ers. What we are looking for is whether what hap-pens (and happened) in one location is the cause of what happens (or happened) in neighboring locations (Barceló 2002, 2008). The analysis then attempts to examine if the characteristics in one spatio-temporal location have anything to do with characteristics in a neighboring location, through the defi nition of a general model of spatio- temporal dependencies. Once we know whether social ac-tions at neighboring locations are similar or not, we should explain why the location of social ac-tions is homogeneous or heterogeneous in the area defi ned by the performance of those actions.

A simple and descriptive visual depiction of this relationship consists in overlaying the inter-polated model of deformed ground surface and the interpolated model of spatial frequencies. In that sense, each temporal step model will display the 4-dimensional variation of shape and frequency. In Figure 9, grey level is used to represent differ-ent spatial frequency variations at different x, y, z, spatial coordinates.

A temporal stack is a display of multiple tem-porally differentiated 4D scalar maps in a single window. Stacks can be viewed from different per-spectives, treating the layers of the stack as an-other spatial dimension. If we have sampled three

temporal periods, we can integrate all data into a single relational model (Fig. 10).

This model allows the visualization and un-derstanding of:

• how the spatial distribution of an action has an infl uence over the spatial distribution of (an)other action(s),

• how the temporal displacement of an action has an infl uence over the spatial distribution of (an)other action(s),

Figure 9. A 4D representation of a single archaeologi-cal time step. Shamakush VIII data set. Grey levels are used to represent different spatial frequencies of ani-mal bones at different spatial coordinates. The higher the frequency, the darker the level (Barceló and Vicente 2004).

Figure 10. A spatio-temporal stack showing three dif-ferent time steps. Shamakush VIII data set. Grey lev-els are used to represent different spatial frequencies of animal bones at different spatial coordinates. The higher the frequency, the darker the level (Barceló and Vicente 2004).

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• how the temporal displacement of an action has an infl uence over the temporal displace-ment of (an)other action(s), and

• how the spatial distribution of an action has an infl uence over the temporal displacement of another action (actions).

The key idea is that effective discontinuities in the spatial probabilities of a social action often coincide with important limits in causal process having modifi ed physical space. The material con-sequences of social action performed at a house-hold scale will then appear as discontinuities in the spatial variation of frequency values of an ar-chaeological feature. Clearly there is a relationship between the places in an abstract scalar fi eld (mea-surement space) where the frequency of social ef-fects show a change in value, and the places of physical space where the material effects of social activities differ. Domestic space, as any other kind of socially modifi ed physical space, will be made real when interfacial discontinuities between suc-cessive events can be detected. Its study is then a matter of reporting at what spatial locations a change in the observable frequency of some ar-chaeological feature leads to a change in the prob-ability of its causal action or process.

Discontinuity detection is essentially the operation of detecting signifi cant local changes among spatially sampled values of some physi-cal properties. What can be done just qualitatively in form and shape analysis should be computed quantitatively in the present spatial frequency case. Formally, such a discontinuity in the spatial probabilities of the social action is defi ned as an observable edge in the fi rst derivative of the math-ematical function that describes the archaeolog-ical frequencies over space. This task can be ap-proached by calculating the spatial gradient in the data array—that is, the direction of maximum rate of change of the perceived size of the depen-dent values, and a scalar measurement of this rate (Barceló 2008). This spatial gradient describes the modifi cation of the density and the size of ar-chaeologically measured values such that regu-larity patterns in spatial variation can be deter-mined. The gradient is calculated by fi nding the position of maximum slope in its intensity func-tion (a graph of the value of the dependent vari-able as a function of space). Thus, the intensity profi le of spatial frequencies can be graphed as a curve in which the x axis is the spatial dimension and the y axis corresponds to the dependent vari-able (for instance, the quantity of some archaeolog-ical material at each sampled location). Likewise, the directivity of such a probability gradient (or “aspect” of the scalar fi eld) is simply the polar an-gle described by the two orthogonal partial deriv-atives. This procedure is depicted in Figure 11. In

the graph, arrows show the directivity of bird bone accumulations in the Túnel VII site and hence, the direction of human work at the site (probably, cleaning activities). Peaks are the most probable locations of single accumulations. Directivity in-ferences allow us to understand the relationship between neighbor accumulations and how some parts of the space were “cleaned,” suggesting the most probable location of residence activities.

ConclusionsVisual geometrical models of spatial form and fre-quency support the idea of attraction as a very ap-propriate analogy for studying how social agents, the products of their work, and the physical en-vironment in which activities took place are re-lated. What we are really studying are archaeo-logical events as places of attraction in space and time. Each identifi ed spatial gradient constitutes a localized event in space and time, be it an individ-ual, a collective action, or a series of actions, and develops together with its environment as a com-plex network of dialectical relationships at mul-tiple levels, conditioning the performance of the action and successive actions performed in the neighborhood. On the one hand, the gradient vi-sualization materializes a complex fi eld of attrac-tion, radiation, repulsion, and cooperation around this activity, producing the necessary energy for the functioning and even the existence of the so-cial system. It is easy to see that in those circum-stances, a deterministic model is an oversimplifi -cation of the spatial process.

Figure 11. A representation of the spatial process which relates bird bone remains at the Túnel VII site. This probability map is a superposition of a gradient and a directional model, with arrows at each grid node pointing uphill according to steepness of gradient; that is, the spatial points where attraction has its highest intensity. (Barceló and Maximiano 2007; Mameli 2004).

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212 Arctic Anthropology 46:1–2

We have used spatio-temporal gradients of the frequency of social effects to defi ne the limits and intensity of an attraction fi eld, which repre-sents how the action has modifi ed physical space around it. As we have seen previously, the spatial process (activities in the household) can be rep-resented in terms of probability surfaces interpo-lated at the locations of where the effects of the spatial process have been quantifi ed.

We have created different visual models that allow understanding of how archaeological sites can be construed as the places where social action was originally performed and as a consequence the ground surface was modifi ed, both qualitatively and quantitatively, by accumulating refuse mate-rial, by deforming a previous accumulation, or by direct physical deformation (building, excavation). We have argued that shell middens along the Bea-gle channel can be explained in terms of discontin-uous sequences of minimal sets of distinguishable components or phases. Such components have been modeled as regions in a multidimensional space delimited by perceived discontinuities in form and frequency, where the probability of a spe-cifi c formation process is the highest.

We have stressed that the past will be “seen” within the present, if and only if some different states within the present can be distinguished, and these states are ordered according some kind of di-rectionality. This last assumption is very impor-tant, because not every observed difference in the present speaks about the past. The true nature of social activity is expressed in such directivity of the sequence chain of human-induced changes be-tween spatio-temporal successive events. Direc-tivity is the result of interpreting a persistent state or the discovery of an order in a sequence or tra-jectory of multiple different states. We have ex-plained the archaeological site’s history by im-posing a temporal slicing on archaeologically perceived discontinuities. The intention of such a slicing will be to visually represent the transitions between events. In this way, we have simulated the actual occurrences of the events in an histori-cal sequence.

Acknowledgments. These computer techniques have been applied in the southernmost part of America (Beagle Channel in Tierra del Fuego) in collaboration with Ernesto Piana and Argentinean archaeologists from Centro Austral de Investiga-ciones Científi cas, and Spanish archaeologists from the Institució Milà i Fontanals (CSIC). The investigation presented here constitutes a joint col-laboration between Universitat Autònoma de Bar-celona and the Institució Milá i Fontanals (Spanish Research Council). It is a part of an ongoing joint project on the Archaeology of Coastal and Marine Environments between both institutions and is

funded by the Spanish Ministry for Education and Research, and the Catalan Government Com-mission for Research (Research Grant GRS 00829 awarded to the AGREL research group). Fieldwork at the Samakush VIII was effectively co-organized with the Centro Austral de Investigaciones Cientí-fi cas (CONICET, Argentina), and funded by the Spanish Ministry of Research and Education, and the Ministry of Culture. Alfredo Maximiano also acknowledges his grant from the Program of Formation of Investigators F.I. 2007, managed by AGAUR (Generalitat de Catalunya). Ernesto Pi-ana contributed decisively in most aspects of the fi eldwork and his archaeological experience in the region helped us to understand the complexity of the shell midden formation processes. We also ac-knowledge Martín Vázquez and Eduardo Moreno for work during excavation, and Jordi Estévez, Estela Mansur, Raquel Piqué, and Asumpció Vila for discussions on the aims and methods. Aspects of this research have been funded by the Spanish Ministry for Education and Research (Research Grant HUM2006-01129/HIST). We extend a special thank you to Hans Peter Blankholm, Bryan Hood and Susan Kaplan for suggestions that improved the fi nal version of this paper.

Figures 9 and 10 are grey-scale reproductions of original color images. The difference between blue (low density), violet (extremely low density), and red (high density) disappear in this black, white, and grey image. This is an important alteration but the proper color is not what is important in this case. In the text the authors use these fi gures only to show how an interpolated density map (color isoareas) does not necessary fi t the digital micro-topography of occupation fl oors. The grey-level image can be used as an example of multivari-ate data representation. To see the original color images go to http://prehistoria.uab.cat/Barcelo/TyTEspacial.html.

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