U.S. Department of the Interior

U.S. Geological Survey

Open-File Report 2015–1241

A Multidimensional Representational Model of Geographic 

Features

A Multidimensional Representational 

Model of Geographic Features

By E. Lynn Usery, George Timson, and Mark Coletti

Open-File Report 2015–1241

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

SALLY JEWELL, Secretary

U.S. Geological Survey

Suzette M. Kimball, Director

U.S. Geological Survey, Reston, Virginia: 2016

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Suggested citation:

Usery, E.L., Timson, George, and Coletti, Mark, 2015, A multidimensional representation model of geographic features: U.S. Geological Survey Open-File Report 2015–1241, 10 p., http://dx.doi.org/10.3133/ofr20151241.

ISSN 2331-1258 (online)
iii

Contents

Abstract...........................................................................................................................................................1

Introduction.....................................................................................................................................................1

Theory of Geographic Feature Representation ........................................................................................2

System Design and Implementation...........................................................................................................3

System Design Overview.....................................................................................................................3

Representation of Changes in Spatial Configuration with Time ...................................................4

Representation of Thematic Change with Time...............................................................................4

Representing Relationships ................................................................................................................7

Representing Features in Raster Data ..............................................................................................8

Populating the Feature Library ....................................................................................................................8

Conclusions.....................................................................................................................................................8

References Cited............................................................................................................................................9

Figures

1. Diagram showing representation of geographic entities as computer objects

with attributes and relationships of space, theme, and time................................................2

2. Diagram showing the implementation of the conceptual model as a Feature

Library provides support for multiple representations and multiple applications.............3

3. Diagram showing the top level diagram for the Feature Library design.............................5

4. Diagram showing coastline features for St. Simons and Cumberland Islands in

Georgia from two different time periods represented in the Feature Library ....................6

5. Diagram showing representation of time attributes in the Feature Library .......................7

6. In this example of relationship implementation in the Feature Library, four

relationship instances are shown by the octagons and five feature instances are

shown by the double ellipses .....................................................................................................8

A Multidimensional Representational Model of 

Geographic Features

By E. Lynn Usery,1 George Timson,1 and Mark Coletti2

1

U.S. Geological Survey.

2

Oak Ridge National Labs.

Abstract

A multidimensional model of geographic features has 

been developed and implemented with data from The National 

Map of the U.S. Geological Survey. The model, programmed 

in C++ and implemented as a feature library, was tested with 

data from the National Hydrography Dataset demonstrating 

the capability to handle changes in feature attributes, such as 

increases in chlorine concentration in a stream, and feature 

geometry, such as the changing shoreline of barrier islands 

over time. Data can be entered directly, from a comma separated file, or features with attributes and relationships can be 

automatically populated in the model from data in the Spatial 

Data Transfer Standard format.

Introduction

Geographical phenomena, termed features, exist in the 

real world as individual, unique entities and can be modeled as 

objects in a computer representation. To support human abilities to conceptualize features at a variety of resolutions, scales, 

and with various attributes and relationships, there are many 

representations (objects) for any one geographical entity. This 

conceptualization of geographic phenomena as single entities 

with multiple representations can be effectively represented 

within a framework of feature objects that have spatial, temporal, and thematic attribute types that, in turn, are interconnected by way of explicit semantic relationship objects (Berry, 

1964; Sinton, 1978; Usery, 1996a; Yuan, 1996).

Traditionally, geographic representation has been based 

on a space-dominant approach in which entities in the real 

world are modeled geometrically, as vector coordinates or raster 

matrices, with thematic attributes attached to basic geometric 

elements of points, lines, areas, or pixels. Berry (1964) established this approach with the concept of a geographical matrix. 

Current commercial geographic information system (GIS) 

software implements this concept. The structure of temporal attributes and of thematic and temporal relationships has 

received little attention until recently, but even these recent 

efforts use the spatial (geometric) characteristics of entities as 

the primary basis of features (Molenaar, 1991). Many spatiotemporal data models have been developed by researchers since 

the late 1980s, with each model representing a single type of 

geographical phenomenon. For example, Armstrong (1988) proposed snapshots for polygonal layers with invariant borders and 

structures were developed for cadastre by Al-Taha (1992) and 

Chen and Le (1996), for wildfire and thunderstorms by Yuan 

(1997, 2001), for the atmosphere by Peuquet (1994, 2002), for 

public boundaries by Wachowicz (1999), and for transportation by Koncz and Adams (2002). In many of these models, an 

object representation is used for the geographic phenomena, 

but usually the object is defined by geometry using a spacedominant view as defined by Wachowicz (1999). Extension to 

data with fields of objects (Cova and Goodchild, 2002) and to 

objects with field-like properties (Yuan, 1996, 1999) have also 

been added to the basic geometric and object models. Again, in 

these cases, specific applications and data types are the primary 

purpose of the representational model. The purpose of this 

report is to present a generic model that can be used to represent 

objects, fields, or features with both types of characteristics and 

that is capable of serving a variety of applications.

The objectives of this research are to present a theory 

based on the dimensions of space, theme, and time in which 

geographic features exist in the real world, and to provide an 

appropriate representation of this theory that can be used to 

implement geographic entities and processes. The theory is 

grounded firmly in geographic and cartographic abstraction 

and modeling principles (Peuquet, 1984; Peuquet, 1988; Guptill and others, 1990) and in cognitive category theory (Mark, 

1993; Usery, 1993, 1996; Frank, 1998). The theory builds 

from basic research into the three aspects of geographical 

entities defined by Berry (1964) and used by many researchers since that time (Sinton, 1978; Usery, 1993,1996; Peuquet, 

1994; Peuquet and Duan, 1995; Yuan, 1996, 1999). This 

research is a unique contribution because the developed theory 

supports a variety of data types including objects, fields, and 

objects with field-like properties and multiple applications 

from a single theoretical structure. This structure has been 

implemented in a completely object-oriented environment.

A design and an implementation of the theory developed 

to support extraction of geographic features from multimodal 

sources for the U.S. Geological Survey’s (USGS) The National 

Map are included in this report. In the report, the section titled 
2 A Multidimensional Representational Model of Geographic Features

“The Theory” presents the basic theory and further places this 

work in the context of other similar research. The basis of the 

system design and implementation is discussed in the “System 

Design and Implementation” section. Methods of populating 

the feature instances in the system with data are then presented 

in the “Populating the Feature Library” section, detailing the 

application of building a feature database for The National 

Map. The “Conclusions” section draws some conclusions from 

this work and addresses future research needs.

Theory of Geographic Feature 

Representation

To support multiple representations including different 

spatial geometries and resolutions, multiple thematic attributes 

and relationships, and multiple temporal attributes and relationships for single instances of geographical entities, a separation of the entity in the real world from the representation is 

required. This separation follows the definition of a feature as 

specified in the Spatial Data Transfer Standard (SDTS); that 

is, a feature consists of two parts, the entity that exists in the 

real world and the object that is the computer representation of 

that entity (U.S. Geological Survey, 2006). A framework for 

the space, theme, and time dimensions of geographic features 

is presented in Usery (1993, 1996). A similar model with 

three domains was proposed and implemented by Yuan (1996, 

1999). Peuquet (2002) attempted to extend the framework by 

positing existence as a fourth element in a pyramidal structure. 

The resulting model assumes existence is equivalent to the 

other three dimensions of space, theme, and time. In reality, 

existence is determined by the presence of attributes and relationships of space, theme, and time, which is the representation used in this report (fig. 1). Thus, this theory of geographic 

feature representation is that unique entities exist in the real 

world and are represented as single objects with multiple 

associations of attributes and relationships along three basic 

dimensions of space, theme, and time.

Attributes are characteristics that are associated with 

single objects and thus apply to individual geographical entities (for example, a road has two lanes). Relationships are 

linkages or interactions between objects and thus involve two 

or more geographical entities (for example, one road, Independence Road, intersects another road, Tenth Street). Egenhofer 

and Franzosa (1991) developed the nine-intersection model of 

spatial relationships to account for the possible combinations 

of spatial objects. This seminal work by Egenhofer and Franzosa (1991) does not include other spatial relationships, such 

as distance and direction, but accounts well for topological 

combinations of spatial objects. A different model is needed 

to include other spatial relationships and relationships along 

thematic and temporal dimensions.

Historically, representations of geographical phenomena, implemented in GIS, were designed to model the spatial 

characteristics and relationships (topology) of entities with 

thematic attributes attached to the spatial representation. This 

space-dominant representation is a logical progression from 

Entity

Object

Attributes

 Space

Theme

 Time

Relationships

Real-world

geographic 

phenomenon

Computer

representation

Figure 1. Representation of geographic entities as computer objects with attributes and 

relationships of space, theme, and time.
System Design and Implementation 3

converting analog map representations, also a space-dominant 

approach, into digital form. Sinton (1978) concluded that in 

an analog map, of the three dimensions, time is held constant, 

theme is controlled, and space is measured. Galton (2001) 

agreed that spatial entities have constant time and a controlled 

theme. This space dominant approach was copied directly 

in the implementation of digital systems for cartography and 

GIS. Research in geographic representation during the past 

15 years has struggled to remove the constraints of the spacedominant model and incorporate theme and time as equivalent 

dimensions (Peuquet, 2002).

With the space-dominant model of conventional GIS, 

thematic relationships and temporal attributes and relationships are not handled. For example, the layer approach in 

GIS includes a rudimentary capability to handle the “is_a” 

thematic relationship. The school ‘is_a” building relationship 

is included in the layer approach by including all buildings in 

a single layer, which is an indirect method to handle a relationship resulting from categorizing geographical phenomena 

into arbitrary layers. The layer approach does not handle 

other thematic relationships, such as “contains” or “part_of.” 

For example, a lake may be a “part_of” a river system, and 

although the two entities may exist on the same hydrography 

layer, the “part_of” relationship must be determined from 

geometry or topology; it is not explicitly modeled as a thematic relationship.

Temporal aspects of geographical entities received 

early examination by Langran (1992), and since that time 

many alternatives have been proposed for discrete phenomena (Al-Taha and others, 1994; Chen and Le, 1996; Chen 

and Jiang, 1998) and continuous geographical processes 

(Peuquet, 1994; Yuan, 1997; Mark and others, 2003). Each 

of these approaches handles one aspect, such as event-based 

temporal modeling (Peuquet and Duan, 1995). The representation developed in this research includes the temporality of 

geographical entities.

System Design and Implementation

System Design Overview

Because the theory captures human cognitive abilities for 

multiple representation, a conceptual model has been developed to support attributes and relationships of geographical 

phenomena, including geometric representations such as vector coordinate lists and topology (Tang and others, 1996) and

raster matrices (Usery, 1994a, 1994b; Yuan, 1996), spectral 

response curves, and mathematical formulas, which can be 

used to depict change in geometric and thematic attributes 

with time. The conceptual model structures a three-dimensional framework of space, theme, and time with associated 

attributes of individual features and relationships among 

features as shown in figure 1.

The objective of the implementation is to capture the 

conceptual model and support multiple representations of individual features including attributes and semantic and temporal 

relationships. For example, a road may be represented as a 

vector string of coordinates with attributes of width, date of 

construction, and others; topological relationships including 

“connected-to” other roads; and semantic relationships such 

as “composed-of” other roads 

and “is_a” interstate highway. 

Alternatively, the same road 

can be represented simply as a 

series of transportation pixels 

in a raster representation of 

land cover or as a series of 

spectral reflectances from a 

digital image. These multiple 

representations allow support 

for a variety of applications 

as shown in figure 2. The 

implementation of the Feature 

Library from figure 2 with a 

sufficiently encompassing data 

model allows support of the 

multiple representations and 

multiple applications.

Figure 2. The implementation 

of the conceptual model as a 

Feature Library provides support 

for multiple representations and 

multiple applications.
4 A Multidimensional Representational Model of Geographic Features

This conceptual model is comprehensive and flexible 

enough to handle all types of geographic entities, their attributes, and relationships. Single or multiple feature, attribute, 

and relationship objects are instantiated, as needed, to represent real-world entities. Attribute objects are connected to 

entity objects or to other attributes to adequately describe their 

respective entities. Relationship objects explicitly interconnect entities and they accommodate one-to-one, one-to-many, 

many-to-one, and many-to-many semantics. The model can 

be extended to add new operations and attribute types, such 

as those for geographic operations. Currently, this “feature 

space” model has been implemented in C++; its design is 

shown in figure 3.

The unique feature identifier (“id” in fig. 3) is the item 

in the library that identifies a particular occurrence of a 

geographical entity and separates one feature instance from 

other instances in the feature space. Although the identifier is 

arbitrarily assigned, it provides a convenient handle, currently 

implemented with pointers, for accessing characteristics of a 

feature instance. Each feature instance has attributes associated with it that can exist in the variety of forms specified 

in the conceptual model (for instance, numbers, characters, 

time stamps and ranges, equations, and spatial and geospatial graphic elements; Usery, 2000). These types of attributes 

potentially allow structuring of all characteristics of space, 

theme, and time of a geographical entity with the feature 

identifier. It is important to note that in this design, the spatial 

dimension is of no more relevance than the thematic or temporal dimensions; that is, the entity exists in geographic reality. 

The spatial configuration of the entity is simply an attribute 

of the entity at a particular time and in this representation, 

becomes an attribute of the feature. The representation of the 

spatial configuration of an entity can be as vector objects of 

points, lines, or polygons (Tang and others,1996); or raster 

GIS and field or field-like objects (Usery, 1994a, 1994b; Usery 

and Pape, 1995; Yuan, 1996,1999).

The top-level diagram (fig. 3) gives an overview of the 

Feature Library design and provides a glimpse of the potential 

power of such a structure. Feature is the core class, represents 

a single geographical entity, and has an arbitrary but unique 

identifier. The feature is shown to have zero or more attributes which may, in turn, have other attributes; thus, it is possible for a feature object to have a complex tree of attribute

objects that describe the corresponding entity. Each attribute

is a wrapper for a value that could be a number, an event, a 

time range, a string, a function, or a geospatial value, among 

other things; the design gives a subset of possible value types 

and can be readily expanded to include new ones. Each 

feature can also be in a relationship with one more other features. The relationship object represents semantics between 

features; each feature is also aware of each relationship to 

which it belongs. Features, attributes, and relationships are 

typed objects; that is, they have a corresponding type. Types

have a name and possibly an associated, terse description. 

For example, a type object could be for a “ROAD,” “AIRPORT,” or “SAMPLING STATION.” Types themselves can 

be part of a type hierarchy. Each type can have an optional 

parent type as well as a possible set of “children,” or subordinate, types. These type parent/child relationships reflect the 

semantic relationships present in source data that would otherwise be lost. For example, one dataset might specify that 

“road” types have “number of lanes” and a “name,” whereas 

another dataset might also specify that “roads” have “surface 

composition.”

In the following sections, we examine the details of 

specific parts of the Feature Library design, including structure of attributes and relationships and the ability to handle 

multiple instances of attributes and relationships for a single 

instance of the entity or feature. These multiple instances 

of attributes and relationships may be from different time 

periods, with different spatial configurations and different 

thematic attributes. In all cases, a single feature instance is 

used; only the attributes and relationships (spatial, thematic, 

and temporal) change. Real-world data from applications in 

GIS analysis and modeling are used in the examples. Shorelines are used as the geographical entities and their spatial 

configurations and other characteristics at two different points 

in time are compared.

Representation of Changes in Spatial 

Configuration with Time

An example of two shoreline features and their associated attributes are provided in figure 4. Note that each 

shoreline has a different geometric configuration in 1990 

from that in 1958. Each shoreline feature is represented by 

a double ellipses. Saint Simons Island shoreline is feature 

ID 18 and Cumberland Island shoreline is feature id 6. Both 

shorelines have two time attributes, one valued 1958 and one 

valued 1990. For each time attribute value, there are further 

attributes of Position, Accuracy, and F_code. In this example, 

the Position attribute values have changed from 1958 to 

1990 for both shoreline features. The Accuracy and F_code 

values have not changed during this time. From an analysis 

and modeling perspective, a tool now exists that provides 

multiple instantiations of the attributes and relationships of 

the same geographic entity in various times and with various 

characteristics.

Representation of Thematic Change with Time

Time is implemented in the system as attributes of time 

stamps, which provides a basis for multiple representations 

of the same feature at different times (fig. 5). Change with 

time then can be modeled as difference in attributes with 

time stamp dates. In this example from a simulated water 

quality modeling study, feature id P1 is of type “Sampling 

Station” and has two attributes of type “Sampling Period.” 

From November 11, 1996, to November 17, 1996, the average 

chlorine (Cl) reading was 13.00 milligrams per liter (mg/L), 

and the average phosphate (PO4

-P) reading was 0.473 mg/L. 
System Design and Implementation 5

Figure 3. The top level diagram for the Feature Library design.
6 A Multidimensional Representational Model of Geographic Features

Saint Simons Island

1990

1958

Cumberland

1990

1958

Figure 4. Coastline features for St. Simons and Cumberland Islands in Georgia from two different time periods 

represented in the Feature Library.
System Design and Implementation 7

Figure 5. Representation of time attributes in the Feature Library. The feature, P1 in the double ellipse, has 

attributes for two different time periods with differing concentrations of chlorine and phosphate.

From November 18, 1996, to November 24, 1996, the average 

Cl reading was 12.87 mg/L and the average PO4-P reading was 

0.461 mg/L. Note that the position and geometry of the feature 

have not changed, but the thematic attributes of chemical concentration have and are modeled in the Feature Library.

Dynamic implementation of time as an equation also is 

supported in the model. In this implementation method, the 

spatial and thematic attributes (characteristics) change as a 

function of a mathematical equation. For example, we model 

the change in PO4-P concentration as a function of time, 

assuming that the changes between two readings at different 

times occur at a constant dilution rate, 0.002 mg/L per day. 

With a starting reading on November 18, 1996, of 0.473 mg/L, 

we model the reading on November 24 as follows:

(PO4-P)Nov24 = (PO4-P)Nov18 – (24-18)*0.002 (1)

where

Nov18 and Nov24 indicate the phosphate values for the dates 

of November 18 and November 24, 

respectively.

Evaluation of the equation yields 0.461 mg/L, corresponding to the readings in figure 5.

Once the equation is implemented, the attribute for 

phosphate concentration is changed to the computed value. 

Because the rate is daily, the equation can be implemented 

for any or all days and provide a dynamic computation of the 

change in phosphate concentration on a daily basis. The series 

of values provide a direct method to implement visualization 

in the form of animation. This simple example indicates the 

potential available from the ability to include an equation as an 

attribute of the feature directly in the Feature Library.

Representing Relationships

Relationships represent interfeature semantics. Typically, 

relationships represent connections such as connected_to or 

flows_into, and containment, such as is_a or part_of. The Feature Library design has a single relationship class that allows 

for any type of relationship. Relationship objects can be topological, thematic, or temporal in nature and may be deleted if 

the relationship changes. Relationships may be of any cardinality with regard to their respective features: one-to-one, 

one-to-many, many-to-one, and many-to-many. An example 

of relationships in the system design is shown in figure 6; the 

feature instances were populated from the National Hydrography Dataset (NHD) of the USGS automatically from SDTS 

format files. The diagram represents how a REACH feature in 

the NHD consists of other features using relationship objects 

(U.S. Geological Survey, 2015). In this case, the REACH 

feature consists of two STREAM/RIVER features and two 

ARTIFICIAL PATH features.

There are four relationship instances represented by octagons and five feature instances represented by double ellipses 

in figure 6. The relationships are all of the composed_of type. 

The diagram can be read as REACH feature id 2069058 is 

composed_of ARTIFICIAL PATH feature id 2064800, ARTIFICIAL PATH feature id 2064802, STREAM/RIVER feature 

id 2063420, and STREAM/RIVER feature id 2064454. The 

dataset used to create this example, NHD, models feature 

relationships using one-to-one cardinality; therefore, four 

relationship objects are used to model the REACH feature 

composition. Because the Feature Library design allows for 

multiple cardinality relationships, one composed_of relationship could have modeled this situation if desired. With time, if 
8 A Multidimensional Representational Model of Geographic Features

Figure 6. In this example of relationship implementation in the Feature Library, four relationship instances are 

shown by the octagons and five feature instances are shown by the double ellipses.

additional features become part_of the REACH, these can be 

depicted as new relationship objects. Likewise, if a feature is 

no longer part_of the REACH, the relationship can be deleted, 

but the deletion is recorded so the entire history of the feature 

is always available.

Representing Features in Raster Data

A feature in a raster dataset, whether continuous (such as 

elevation and temperature) or categorical (such as land cover 

and soils), can be a single pixel, a line of pixels, a contiguous 

set of pixels (polygon), or a set of non-contiguous individual 

or aggregations of pixels (Usery, 1994a). For example, a water 

well might be represented as a single pixel, a road by a line of 

pixels, a lake by a contiguous set of pixels, and a tank farm as 

a set of non-contiguous pixels. Storage of feature geometry 

may be as a listing of pixels with association of geographic 

coordinates, either implicitly or explicitly; a raster image mask 

defining the feature pixel set; or through a formula that, when 

applied to a stored raster layer, yields the appropriate set of 

pixels representing the feature. Because many geographic 

features have fuzzy boundaries, some pixels need to be represented with partial membership when specifying the geometry 

of a feature.

Methods to extract features from raster datasets have 

been developed for geographic entities with crisp (Usery, 

1994b; Usery and Pape, 1995) and fuzzy boundaries (Usery, 

1996b). Once extracted, these features can be stored in the 

Feature Library as individual pixel locations, lists of pixels, a 

binary raster mask, or as a vector outline after conversion of 

the individual pixels or pixel boundary along with the associated attributes and relationships along the three dimensions: 

space, theme, and time.

Populating the Feature Library

Populating the Feature Library requires a list of features 

with associated attributes and relationships; that is, a comprehensive ontology of geographic features. There are a variety 

of formats of geographic data in use, and many include feature 

lists and associated attributes. Some formats, such as SDTS 

and NHD, incorporate relationships of specific types. Because 

one target application of the Feature Library is The National 

Map, and no comprehensive ontology for it exists at this point, 

our approach is to use existing formats and datasets, such as 

Digital Line graph–Enhanced (DLG–E), Digital Line Graph–

Feature (DLG–F), and NHD, that provide needed elements 

of the ontology and can be loaded with file import capability 

(U.S. Geological Survey, 2000).

Creating actual instances of features with all the associated attributes and relationships is a tedious, time-consuming, 

and expensive process. To facilitate this process, we have 

included capabilities in the implementation to import spatial data from standard formats such as SDTS and GeoTIFF; 

import tabular information from comma-separated-value 

(CSV) files; directly enter instance data, including all attributes 

and relationships through an interactive session; and interface 

to existing commercial software systems, such as ERDAS 

Imagine and Quantum GIS. For cases such as NHD in SDTS 

format, the import function automatically populates appropriate 

attribute and relationship classes as shown in figure 6.

Conclusions

A multidimensional representational model for geographic features was developed and implemented as a Feature Library. The feature model has significant capability for 

storing attributes and relationships of geographic features, 

allowing multiple representations of geometry and thematic 

characteristics at differing times or scales. The implementation of the model uses object constructs and was programmed 

in C++. Testing of the model for multiple representation and 

multiple applications was performed in several ways. First, the 

automatic import and storage of attributes and relationships 

from available data in SDTS and NHD formats was achieved. 

The capability for handling temporal change was tested in two 

ways: (1) spatially with a coastline change example including 

maintenance of appropriate relationships and (2) as thematic 

change over time with an example from water quality analysis 

and change in concentration of chlorine and phosphate over 
References Cited 9

time. In all cases, the model and implementation successfully 

met the requirements of multiple representations and multiple 

applications. Limitations of the implemented methodology 

include persistence in a database sense, good links to traditional geographic processing systems such as commercial GIS 

software, and query capabilities.

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Usery and others—A Multidimensional Representational Model of Geographic Features—Open-File Report 2015–1241

ISSN 2331-1258 (online)

http://dx.doi.org/10.3133/ofr20151241