Towards a general theory for representation of geographic ...



Towards a General Theory of Geographic Representation in GIS

May Yuan, Michael F. Goodchild, Thomas J. Cova

Abstract

We outline a general theory of geographic representation in GIS based on points as the fundamental geographic information primitive. Points are combined to build a geographic quartet of geo-objects field objects (f-objects), object fields (o-fields), and geo-fields through aggregation over space, time, or geo-semantics (meaningful information associated with a geographic location). Aggregation is the sole operation used to construct GIS representations of geographic reality and to incorporate interactions among any combination of geo-objects by organizing them into geo-pairs, geo-clusters, geo-hierarchies, and geo-networks. The general theory captures geographic dynamics through interactions among geo-objects, field objects, object fields, and geo-fields in these organizational frameworks of geo-pairs, geo-clusters, geo-hierarchies, and geo-networks. We envision that the general theory of geographic representation will provide a unified framework to GIS data modeling and broaden the dichotomy of object- and field-views of geography. The paper introduces the preliminary framework of the theory and defines the geographic quartet and the four organizations of interactions with examples.

Keywords: geographic representation, space and time, objects and fields, GIS data modeling.

Introduction

While the development of geographic data modeling techniques in geographic information systems (GIS) has thrived in the last five years, the development is largely driven by technology without a solid theoretical foundation. In this paper, we attempt to establish a foundation for a general theory of geographic representation to capture geographic dynamics in GIS data models. Recognizing that extensive GIScience research has addressed conceptual and theoretical issues on geographical space and time (Peuquet 2002), we contend a need for a general theory of geographic representation to underpin the diverse views of space-time integration and to bridge the gap between abstract concepts that address the essence of space and time and practical applications that call for concrete methods to handle spatiotemporal data. The discussions of field ontology that occurred in the 1990s (Couclelis 1992; Goodchild 1992) and in the Project Varenius specialist meeting on the Ontology of Fields (Peuquet et al. 1998) largely ignored the issue of time. Other developments, such as metamaps (Takeyama and Couclelis 1997) and object fields (Cova and Goodchild 2002), also need to be brought within a single unifying framework. A theoretical foundation of rigorous and robust representation is needed to fully support visualization and computation of spatiotemporal information that characterizes both happening and becoming of geographic worlds. In the next section, we inquire into the most fundamental elements of geography and propose a framework as the first step towards a general theory of geographic representation. Following the proposed framework is a discussion on how the proposed framework entails a general theory. Finally, we conclude the paper with a discussion of the novelty that our proposed framework brings to geographic representation and GIS data modeling.

Primitives of Geographic Representation

We begin with three elements: space, time, and geo-semantics. Geo-semantics ([pic]) is used here for “geographic information meaningful to the user” to stress the selective and situational nature of geographies. What is at a geographic location depends upon the purpose, measurement, and interpretation of an observer. For example, for a given area, agricultural scientists are likely to identify soil types and soil properties that are distinctly different from the types and properties of soils that may be relevant to civil engineers, and while botanists see trees, ecologists may see forests. For space, we use a location vector (S) to address spatial dimension and geometry. Location may be referenced by absolute coordinates of x, y, and z, or positions relative to other geographic things of interest in the form, for example, of eastings, northings, and relative elevations. Similarly, we use a time vector (T) to indicate time of interest in multiple dimensions, that may include the time of happening (th), the time of observation (to), the time of recording (tr), and periodic time (tp) in the case of cyclic phenomena.

2.1 Geo-atoms [pic]

All geographic information can be decomposed into point sets or geo-atoms. An individual (i) geo-atom ([pic]) consists of geo-semantics ([pic]) measured, observed, or inferred at a point location[pic]and a given time [pic]. Points are used here to refer to a location within which a piece of geographic information can be associated; hence, cells are a specific type of point. We consider this atomic form (i.e. geo-atoms) of geo-semantics at a location at a given time the most primitive unit of geographic information; all other types of geographic information consist of aggregations of the atomic form, often over infinite point sets. Our premise is that points are the most primitive spatiotemporal element with which information can be associated to space and time. Higher-dimensional properties of lines, areas, and volumes are aggregates of point sets by thresholding geo-semantics or assuming uniform geo-semantics in the aggregation. For example, property lots are aggregates of point sets that have the same ownership, contours are aggregates of point sets based on elevation criteria, and the State of Hawaii is an aggregation of a point set under the administrative authority of the State of Hawaii. Direct measurements of lines (e.g. distance or length) are not considered as geo-semantics at individual locations but are relationships (in the case of distance) between location points or secondary properties (in the case of length) by identifying all linearly aligned location points that satisfy a given geo-semantic threshold.

2.2 Geo-objects [pic]

Geo-objects are what we identify as individuals in geography that cannot be further divided into individuals of the same kind. We consider that a geo-object is a uniquely identified (indicated by ID) location point set (S) and time set (T) in which geo-semantics meet certain requirements;[pic] indicates a qualified set of geo-semantics [pic] over space S and time T of interest. Notation (S) differs from notation (Si) in that S denotes a point set whereas Si marks an individual point. The same convention is applied to other notations throughout the paper. Under the consideration that geo-atoms are the most primitive units of geographic information, a geo-object is a function that aggregates geo-atoms [pic]within space (S) and time (T) of interest. Locations of the point set may be represented by Cartesian coordinates, relative coordinates, or mathematical expressions (e.g., circles, arcs of ellipses, and Bézier curves) and may include disjoint subsets for geo-objects that consist of multiple parts (e.g., multipoints, multipart polylines, multipart polygons as recognized in the OGC Simple Feature Specification, ).

Each geo-object must have a unique identifier (ID) to distinguish itself from the others. In some cases, the spatial location and extent of a geo-object are defined before geo-semantics are measured. Examples are census enumeration zones and lakes. Most current GIS data models take a space-centered approach that recognizes geographic objects by location and ascribes geometry to these objects. In doing so, new object identities are needed when changes occur to location or the geometry of an existing object. Alternatively, the formation of census enumeration zones can be considered as an outcome of a spatial aggregation based on geo-semantics which have been measured previously (e.g., previous census) or are easily distinguished without measurements (e.g., locations of water or not water). From this perspective, a geo-object identity is not determined by location or geometry, but by its intrinsic properties that make it a geo-object of its kind as judged by geo-semantic requirements. “Portage” for example, was used as the county name for two spatially disjoint areas in Wisconsin at different times. When state-county names are used as county identifiers, the identity of Portage county is not tied to a particular geographic location.

In addition to properties assumed uniform over the object (spatially intensive properties), properties at the set level are likely to emerge and may include measures of the point set (e.g., length, area) and integrals of spatially intensive properties. These set measures and integrals are spatially extensive properties which are closely dependent on and cannot be separated from line or area objects. The contrast between spatially intensive and extensive variables is described in (Longley et al. 2001). For example, population counts are a function of the area of the reporting zone. When a county is subdivided into two smaller units, the population density (a spatially intensive property) in its subdivisions may remain the same as the county population density, but population counts (a spatially extensive property) are likely to be different (unless one of the subdivisions has no population). Furthermore, many geo-objects have properties that are transitional in space. Their identities cannot be determined through aggregation of the geo-atoms that result from simply geo-semantic thresholding. Such a geo-object is characterized by its indeterminate boundaries and will be conceptualized here as a fuzzy point set (Burough and Frank 1996).

When time is a consideration, the identities of geo-objects are critical to track through changes in location, geometry, and properties (Hornsby and Egenhofer 2000). Since a geo-object is an aggregate of geo-atoms under a set of geo-semantic criteria, its identity is determined by the defined geo-semantic criteria and spatial and temporal constraints to the aggregation. The defined geo-semantic criteria specify the range or discrete values (e.g. domain) within which geo-semantics can be considered as the properties of a geo-object. Only when a geo-atom has geo-semantic values at [pic] are within the domain, the geo-atom is part of the geo-object. When geo-semantic values meet the geo-semantic criteria at [pic] but not at[pic], the geo-object ceases to exist at[pic]either by moving out of the location, dissipating entirely, or transforming into another geo-object with a different identity.

Once the identity of a geo-object is determined, its spatiotemporal path and behavior can be represented and tracked by lifelines (Mark and Egenhofer 1998); and its spatiotemporal domain of accessibility can be represented by space-time prisms (Miller 1991). For geo-objects with geometry of higher dimensions (e.g. lines or polygons) or with multiple parts, space-time volumes are necessary to represent their spatiotemporal extent as elaborated in the SPAN ontology (Grenon and Smith 2004). While observations of a lifeline or spatiotemporal geo-object are discrete, various interpolation methods (e.g. linear or curvilinear) may be applied to estimate intermediate locations and geometry between temporal observations (Pfoser and Jensen 2001; Stroud et al. 2001; Pfoser et al. 2003). In addition to trajectories, additional parameters are necessary to record geo-objects which may change geometry over time. An example is the helix representation that uses a spline to track the location of a geo-object’s centroid and prongs to record the extension of the geo-object in different directions at each point in time (Agouris and Stefanidis 2003).

Aggregation of geo-atoms to form a geo-object is subject to spatial and temporal constraints by the nature of the geo-object, and such spatiotemporal constraints can be used to select appropriate interpolation methods as discussed above. A geo-object only exists in certain spatial and temporal extents bound by biological, physical, or administrative processes through which geographic entities are formed. At the highest level, no geo-objects on Earth can have a spatial extent greater than the surface of the Earth. Under constraints of physical processes, the largest hurricane recorded (Typhoon Tip) extended out to 1,100 km, and the smallest (Cyclone Tracy) was about 50km in radius. In addition, geo-objects have life expectancy; some may be ephemeral (e.g., rainstorms), but others can be long-lasting (e.g. mountains). Some geo-objects must be conterminous in space and time (e.g., a reservoir or a pollution plume), but others may have spatially or temporally disjoint parts (e.g., a wildfire or a country).

In summary, geo-objects are formed by aggregating geo-atoms under spatial, temporal, and geo-semantic constraints. Identities of geo-objects are recognized by spatial and temporal extents in meeting certain geo-semantic requirements. Changes to a geo-object over time can be tracked based on identities. Some geo-objects may have spatially or temporally disjoint parts. These geo-objects consist of discrete point sets[pic], but these discrete point sets are united by a common geo-object identity. On the one hand, geo-objects are ‘discovered” by spatiotemporal aggregation of locations with qualified geo-semantics; in other words, geo-semantics are measured prior to the identification of geo-objects. On the other hand, geo-objects may be recognized by distinct geo-semantic discontinuity which enforces the perception of boundaries, and consequently the extents over which spatial and temporal aggregation takes place. In such cases, geo-semantics are characterized after the identification of geo-objects.

2.3 Geo-fields: [pic]

A geo-field is a continuous field characterizing the variation of a single spatial variable [pic]of the geo-semantics ([pic]) over space [pic]. Similar to geo-objects, a geo-field can be also be considered as an aggregate function of geo-atoms within geo-semantics: [pic].When time (T) is a constant, the field represents the spatial variation of [pic] at a given time; whereas, when time of interest is a period, we have a space-time field that represents how the state of a spatial property transitions over time. Multivariate fields, such as vector fields, are considered as geo-semantic aggregates of univariate fields. As such, multiple spatial variables (recorded as a vector array) are measured at individual predefined locations.

Geo-fields are distinguished from geo-objects in two fundamental ways. First, the boundary of a geo-field is determined by the interest of an observer over a domain (Peuquet et al. 1998), not by geo-semantic thresholds. Nevertheless, boundaries inside a geo-field may be determined by thresholding geo-semantics, such as the choropleth maps or the area-class map (Mark and Csillag 1989). Hence, spatial aggregation that forms a geo-field takes precedence to the measurement of geo-semantics. In contrast to the identity of a geo-object, locations and their spatial configurations constitute the unit of observation, measurement, or inference (including interpolation). Second, geo-fields over space and time represent changes of geo-semantics at locations in a pre-defined spatial and temporal reference framework. Since all locations are pre-defined and spatial variables are measured as properties at each location, a geo-field contains no empty space and allows no movement. While temporal information of a geo-object may refer to changes to location, geometry, geo-semantics, and identity, temporal information of a geo-field records geo-semantic histories at individual locations.

In GIS practice, locations and spatial configuration of locations can be defined in various ways. Below are six common approaches to constructing a digital representation of a 2D geo-field (Goodchild 1992):

• piecewise constant, such that the variable is constant within each of a set of non-overlapping, space-exhausting polygons (the choropleth map, and the area-class map in the nominal case);

• piecewise linear in a triangular mesh (the triangulated irregular network or TIN);

• piecewise constant in a regular grid (normally rectangular in the two-dimensional case);

• sampled at a set of irregularly spaced points;

• sampled at a set of points in a regular array (normally rectangular in the two-dimensional case);

• sampled along a set of isolines.

The sampling schemes adopted in the last three cases represent only the measurements of the focal spatial variable at the sampled points or lines. Spatial interpolation methods are required to provide estimates of the value of the field away from the sampled points or lines.

The six approaches are clearly not the only possible methods of representing a geo-field. Finite-element meshes (Topping et al. 2003) cover an area with a mixture of triangles and quadrilaterals, modeling the variation of a geo-field within each element as a polynomial function of location, and imposing continuity constraints across element boundaries. The finite-element meshes define a geo-field in two levels: Likewise, the triangular irregular network (TIN) model can be extended to represent geo-fields with two levels of aggregates by adopting similar strategies used in the finite-element meshes to ensure continuity across element boundaries. Nevertheless, the TIN model is conventionally implemented with a spatial variable (such as elevation) as a linear function of location within individual elements of triangles (i.e., with spatially continuous elevations within each triangle); discontinuity of slope occurs across element boundaries in a TIN unless additional constraints are imposed.

2.4 O-fields: [pic] and F-objects: [pic]

As discussed previously, geo-objects are aggregates of geo-atoms over space and time based on geo-semantic thresholds; geo-fields are aggregates of geo-atoms on geo-semantics based on space and time. In other words, for geo-objects, we first identify geo-semantic thresholds and then form geo-objects by spatiotemporal aggregation of geo-atoms meeting the geo-semantic requirements. For geo-fields, we first identify the space and time of interest and then aggregate geo-semantics, one geographic variable at a time. We further adopt the concept of ‘aggregation” to cut across the conventional views of objects and fields in geographic representation such that we aggregate objects over a field to form an object field (o-field), and we aggregate fields within an object to form a field object (f-object).

An o-field maps locations in a field to objects (Cova and Goodchild 2002): [pic] where the field function:[pic] maps a geo-object set [pic]to a set of locations (S) over time (T). Given a point set (S) and time set (T), we can map every location in a geo-field to a viewshed geo-object. Furthermore, we can map viewsheds at locations over time to illustrate how line of sights at locations change over time as the morphological landscape changes, a useful method for archaeological research. What geo-objects are to be associated with each location in an o-field and how these geo-objects may be influenced by the changes in the o-field is determined by two factors: (1) the interest of the observer or the inquiry of a problem domain as different observers or problem domains encounter different geo-objects of interest at locations; (2) the degree to which properties of the field change in the time interval of interest and functional relationships between the properties and the geo-objects. For example, geo-objects of burns by wildfire at a location are insensitive to changes in elevation at locations over time; however, geo-objects of viewsheds or drainage paths are closely related to elevation. Geomorphic erosion and deposition can change viewshed size and drainage patterns at the impacted location. When these geo-objects have uncertain boundaries, an o-field can map locations to measures of set memberships that indicate degrees to which a location in the o-field may be associated with a set of geo-objects.

Likewise, a field object (f-object) emerges when we aggregate geo-fields to form geo-objects. As such, each of the geo-objects inherit properties of geo-fields to represent spatial variation of the geo-object’s properties (Yuan 1999): [pic], where the function f (S,T) maps a set of locations and time points to spatial variables of geo-semantics, and the function is applied to different sets of location points (S) over time (T) to form an f-object [pic]. Like geo-objects, f-objects may also associates with different sets of locations when changes occur to locations, geometry, or parts (e.g. movement, dilation, contraction, split. or merge) F-objects bring an additional dimension of change: changes in internal structure that is unaccounted for in geo-objects, geo-fields, and o-fields. The spatial variation of geo-semantics embedded in an f-object can be indicative of physical dynamics that drive the development of the f-object over space and time. For example, the winds, precipitation, temperature, and pressure fields of a convective storm characterize the dynamics and stability of the storm and how it may evolve under certain atmospheric conditions (Yuan 2001). The set of geo-semantics associated with an f-object is bounded by the spatiotemporal extent of the f-object. As the f-object moves, it carries the embedded geo-fields with it. The changes of geo-fields inside an f-object signify how the f-object interacts with the environment and its dynamics, and how the environment may continue to evolve.

We synthesize the continuum of geo-objects, f-objects, o-fields, and geo-fields in a cube with three dimensions of potential changes (static: no change vs. dynamic: change over time): Geometry, Movement, and Internal Structure (Figure 1). The geographic object-field continuum illustrates the transitional nature of geo-atom aggregation according to the degree of constraints imposed on spatiotemporal extents and geo-semantic thresholding. Judgment of "static" or "dynamic" depends upon the scale and resolution of observations and analysis, and inherently involves uncertainty. The geometry of an object may be considered static at one spatiotemporal scale and resolution, but may be dynamic at another. Similarly, scale and resolution will determine if movement or internal structure should be considered static or dynamic. For example, a forest may be seen as static in its internal structure at a national scale, as a uniform areal feature with attributes (such as the name of the forest, ownership, primary net production, etc.) applicable to the entire area. However, its internal structure becomes important when observed at a local scale, such as in an analysis of the ecological dynamics in the forest, when internal structure is then considered dynamic.

The cube also represents a continuum of objects and fields according to the degree to which geo-atoms are aggregated in space, time, and geo-semantics. The dimension of Internal Structure separates object dominance (static internal structure in which all geo-atoms possess constant values of geo-semantics) from field dominance (evolving internal structure, i.e., with spatial and temporal variations of geo-semantics). On the object-dominance side, we have geo-objects that do not have variation in their internal structure and are distinguished by their ability to change geometry or move over time. Likewise, static geometry denotes geo-objects with rigid shape, while dynamic geometry characterizes elastic geo-objects which may experience shape change over time.

Below are examples of objects in each of the blocks characterized by object dominance (static internal structure):

• Stationary rigid uniform objects: buildings and streets in a city.

• Stationary elastic uniform objects: the seasonal expansion or contraction of a lake when only the area of the lake is considered.

• Moving rigid uniform objects: vehicle tracking and space-time life lines.

• Moving elastic uniform objects: a spreading wildfire when only the burn scar is considered.

Alternatively, a geo-field can represent a property of a pre-defined area (e.g. satellite images) or a geo-object. Conventionally, a geo-field is bounded by a pre-defined quadrilateral area that is determined by instruments, measurements, or area of interest. When geo-fields are embedded in a geo-object, the geometry of the geo-object determines the shape and location of its geo-fields. The geo-object is, therefore, an aggregate of geo-fields in space and time. For example, a temperature field of the United States is bounded by the U.S. territory. As long as the U.S. geometry remains the same, the geometry of the temperature field is constant. However, the geometry of a geo-field will vary as its parent geo-object's shape or location changes. An example is a geo-field of landuse and landcover (LULC) within a metropolitan area. When the metropolitan area expands (e.g. urban sprawl), the LULC field changes its size and shape accordingly. How the embedded geo-fields evolve represents the internal dynamics of the parent geo-objects; yet how the parent geo-object moves (e.g., rotates, jumps) and develops (e.g., splits, merges) reflects the meta-structure of its embedded geo-fields. Furthermore, evolution of the embedded geo-fields and the dynamics of the parent geo-object are inter-dependent: geo-field evolution can drive changes in geo-object dynamics, and vice versa. Below are examples of fields in each of the blocks characterized by field dominance (dynamic internal structure):

• Stationary rigid evolving field: soils in a watershed and digital elevation models.

• Stationary elastic evolving field: temperature of heat-island effects in an urban area, and vegetation cover during desertification.

• Moving rigid evolving field: surface properties of drifting islands.

• Moving elastic evolving field: oil spills and hurricanes.

Overall, the cube offers a conceptual basis to build an integrated geo-ontology by considering different kinds of objects and fields in geography. With the idea that all geo-objects and geo-fields can be interpreted by various forms of aggregation in space, time, and geo-semantics, we obtain a much richer and more simplified GIS framework to represent geography than the conventional approach that stresses a static object-field dichotomy, entity geometry, field variables, or time series of features or raster layers. The proposed geographic representation is richer than the conventional approaches because the degree of aggregation results in a geographic quartet of objects, f-objects, o-fields, and fields that have been overlooked by previous GIS data modeling efforts. In addition, the proposed representation simplifies GIS data modeling by a single concept: aggregation of geo-atoms in space, time, and geo-semantics, to address the entire object-field quartet while providing a means to investigate dynamics involved in the quartet. As such, the proposed framework provides a step towards a general theory for geographic representation for two reasons: (1) It enables a unified view to all geographic in the object-field quartet; and (2) It allows prediction of potential dynamics involved in geography through interweaving elements in space, time, and geo-semantics within geo-objects and geo-fields, and across objects and fields in geography.

Interactions and Aggregation

The ability to represent and predict potential dynamics require understanding both individuals and groups, as well as interactions at both levels. The cube in Figure 1 depicts eight types of “geographic things.” Here we use “geographic things” as a general term for things that can be considered as geo-atoms, geo-objects, f-objects, o-fields, or geo-fields. The idea of aggregation of space, time, or geo-semantics to form geographic things is extendable to the formation of geographic groups. Interactions among geographic things and groups in space, time, or geo-semantics are essential to the understanding of geographic processes, and therefore dynamics because interactions drive certain organizational structures that reflect the dynamics involved at the group level or between individuals and groups. Interactions over space, in the form of flows of people and goods or telephone and Internet traffic, for example, are important in most social and physical processes operating on the geographic landscape. We consider four types of aggregation to capture geographic interactions: geo-pairs, geo-clusters, geo-hierarchies, and geo-networks, depending on the linkages among individuals and groups.

Geo-pairs bond two geographic things because some geo-semantics arise, and can only be meaningfully examined, by looking at pairs. For geo-atoms, a metamap connects pairs of cells and the properties associated with the pairs (Takeyama and Couclelis 1997). Such geo-pairs can be obtained from the Cartesian product of a raster with itself, each pair or element of the product being characterized by a set of attributes. (Goodchild 2000) has generalized this concept to any pair of geo-objects, using the term object-pair. In the Unified Modeling Language (UML) attributes of the relationship between two features would be organized as an association class or relationship class. At the geo-atomic level we might represent this as [pic] to capture that geo-semantics [pic]result from a function of two geo-atoms [pic].

Geo-clusters form un-connected groups based on distance in space, time, geo-semantics, or some combination of the three domains. Various spatial, temporal, or spatiotemporal analysis methods provide the means to acquire clusters based on spatial autocorrelation. Spatialization of geo-semantics (Skupin and Fabrikant 2003; Randviir et al. 2004) computes distances in attribute space to investigate similarity. Geo-clusters often offer a basis to formulate hypotheses to investigate forces which drive geo-clusters or to predict the future development of geo-clusters.

Geo-hierarchies institute a hierarchical structure among geographic things (i.e., geo-atoms, geo-objects, f-objects, o-fields, and geo-fields). Imagery pyramids that build multiple levels of images at different resolutions are examples of geo-field hierarchies. Similarly, geo-hierarchies can form by aggregates of o-fields. A geo-hierarchy may include any combination of geo-atoms, geo-objects, f-objects, o-fields, and geo-fields. For example, a geo-hierarchy may constitute o-fields of elevation and drainage basins, each drainage basin (an f-object) consists of soil moisture (a geo-field), and each location at the geo-field of soil-moisture has a set of sensors (geo-objects) used with a kriging method to estimate the soil moisture at that location. Of most interest in geographic dynamics are hierarchies that result from geographic processes operating at different spatial and temporal scales. Geo-objects, for example, that result from global geographic processes can manifest themselves globally (such as areas of extreme temperatures from greenhouse effects); while those from local processes will be confined at the local scale (such as damage areas from flash floods). Nevertheless, local processes may be triggered by global processes, as when global climate change triggers glacier thinning in Alaska (Motyka et al. 2003). Hierarchies of geographic things form as a result of hierarchies among the processes that produce these geographic things. Hierarchy theory provides a systematic way to analyze and understand hierarchical structures of processes operating at different spatial and temporal scales, as well as geographic things and their behavior in response to hierarchies of processes (Allen and Starr 1982; Ahl and Allen 1996).

Geo-networks connect geographic things in a network structure such that there are no determined parent-child relationships as clearly defined in geo-hierarchies. Transportation and utility applications fit well with such network structures. Additionally, groups of geographic things that exhibit feedback mechanisms likely relate to each other in networks for their cyclic influence. For example, LULC may trigger regional climate change which in turn pushes policy changes to enforce LULC compliances to minimize negative impact on climate. From a system’s perspective, feedbacks are common to most geographic processes, and geographic things, as products or agents of geographic processes, will likely propagate in a network structure. When a geo-network consists of some combination of geo-objects, f-objects, o-fields, and geo-fields, the geo-network can express how driving forces or trigger agents (usually considered as geo-objects or f-objects) act on the environment (usually considered as o-fields or geo-fields) which, meanwhile, promotes or suppresses these forces through modification of agent behavior. Dynamics among geographic things can become apparent in a geo-network. For example, a network of ozone depletion will link geo-fields of climatic parameters (temperature, precipitation, etc.), geo-fields of greenhouse gases, o-fields of upwind sources of ozone precursors at each location, and f-objects of pollution plumes. As to the ozone hole, it can be considered as an f-object with variation of ozone column inside the hole. However, it can also be considered as a geo-object to determine the size of the ozone hole in which all locations have ozone column < 220 Dobson units.

Concluding Remarks

Central to this paper is the emphasis on a unified theory for geographic representation. GIS researchers have long focused on separate object- or field-based technologies for GIS data modeling and analysis. While technological advances contribute greatly to GIS software, a weak theoretical basis hampers real scientific breakthroughs. On the other hand, much conceptual research reveals intrinsic and singular properties of space and time. Our premise posits a need to bridge the gap between technological needs for implementation and conceptual understanding of space and time abstractions. A general theory of geographic representation will serve the purpose by providing a unified, integrated view of geography, an operational framework for data modeling, and a means for computation and prediction.

The proposed framework is a step towards to a general theory of geographic representation. We start with the geographic primitive: geo-atoms, each of which is an observation, a measurement, or an inferred value at a location taken at a time. Geo-atoms are represented by an array of space (e.g. location and geometry), time (different kinds of time), and geo-semantics (geographic observations, measurements, or inferences meaningful to the user). We adopt one single operation, aggregation, by thresholding space, time, geo-semantics, or some combination to form a geographic quartet of geo-objects, f-objects, o-fields, and geo-fields. We understand that o-fields may be better considered as geo-pairs of geo-objects and geo-fields, and consequently aggregates of geo-atom pairs rather than geo-atoms as for geo-objects, f-objects, and geo-fields. On the other hand, o-fields can still be aggregates of geo-atoms if we consider that geo-semantics inlcude geo-objects (that is, geo-objects can be meaningful observations at a location to the user and therefore geo-objects can be properties at locations). In doing so, we are able to unify object- and field-based views of geography by varying degrees of aggregation. We synthesize the geographic quartet in a cube of three parameters: shape, movement, and internal structure and consider if each of the parameters will change over time (i.e. static vs. dynamic). Finally, we outline four types of interaction (i.e., geo-pairs, geo-clusters, geo-hierarchies, and geo-networks) reflecting upon how geographic things can be aggregated into groups. Each type of the interaction aggregates geographic things by some combination of geographic things across the quartet to capture interactions induced by geographic processes and therefore dynamics at the group level.

We consider the framework a basis for theoretical developments for representation of geography because it enables a unified view of geography and allows for structuring and prediction of geographic dynamics. A comprehensive formalization of the proposed framework is necessary to develop such a general theory and integrate it into GIS data modeling. In the meantime, we are building use-cases to prove the concepts presented and to demonstrate their value.

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