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Topics covered in this unit

• What is a field?
• What types of fields are there?
• What is the difference between fields and discrete entities?
• How are fields represented?

Intended learning outcomes

• define a field and cite examples
• determine whether a conceptualization satisfies the requirements of a field, a collection of discrete entities, or neither
• argue for and against field conceptualizations
• describe the common methods for representing fields in GIS
• explain the relationships between terminologies in GIS and other areas

Full Table of Contents

Instructors' Notes

Metadata and Revision History

1. What is a field?

• a conceptual model of geographic variation
  • one of several such models
  • the differences between this and other models are conceptual, that is, they exist in the human mind
• a model of variation within a spatio-temporal frame
  • at every point in the frame there exists a single value of a variable
    • e.g. a field of temperature
    • e.g. a field of land surface elevation
    • e.g. a field of land ownership
  • the variable may be measured on any scale
    • temperature - degrees Celsius
    • elevation - meters above sea level
    • land ownership - name of the owner, parcel ID
  • for geographic information, the frame may be defined by:
    • two spatial dimensions (x,y)
    • three spatial dimensions (x,y,z)
    • spatial dimensions and time (e.g. x,y,t)
  • the field variable can be thought of as a function of these dimensions
    • e.g. z(x,y) might denote elevation as a function of two spatial dimensions
    • generally, z(x) is the value at the location defined by the vector x, which will have as many components as there are dimensions of the spatio-temporal frame
  • in GIS applications, x and y are usually interchangeable
    • they have similar characteristics
    • rotation is possible to new axes in the plane
  • however, z and t are not similar
    • it would make little sense to rotate to new axes that are oblique to t
    • z is often sampled at different density to x and y
    • t is expressed in different units of measurement
• a vector field has a vector (direction and magnitude, or possibly just direction) rather than a single value, at every point
  • a field having a single value is also known as a scalar field
  • examples of vector fields with geographic relevance include
    • gradient of the land surface (aspect and slope)
    • wind (direction and magnitude)
    • flow direction, perhaps also speed
  • a vector field could be represented by storing two linked fields
    • either one for direction, the other for magnitude
    • or one for the x component of the vector, the other for the y component
  • vector fields can be displayed using arrays of arrows
• it is possible to think of geographic variation entirely in terms of fields
  • geography consists of a number of variables with single values everywhere on the Earth's surface
    • these variables keep recurring in GIS applications:
    • land elevation
    • population density
    • vegetation cover type
    • soil type
    • surficial geology
    • depth to water table
    • soil pH
    • mean annual precipitation
    • January mean rainfall
    • etc etc.
  • the plates in the front section of any atlas
• in general, it is impossible to achieve a perfect representation of a field
  • fields can be infinitely complex
    • it could take an infinite amount of information to represent a field perfectly
  • representations of fields must be approximate
    • the space available in a digital computer is always limited
  • often, representations capture only the coarser aspects of variation
    • the details or high-resolution elements are not captured
    • they constitute part of the uncertainty of the representation

2. What types of fields are there?

• scalar and vector fields already covered
   
• types of fields based on scales of measurement
  • nominal
    • a measurement is nominal if the measured value has no meaning other than identification
    • a social security number serves only as identification
    • it has no significance as a number, does not establish quantity
    • adding, multiplying, dividing make no sense
    • a larger number indicates nothing, so social security numbers do not establish order
    • examples: name of owner, class of vegetation cover, class of land use, name of street
  • ordinal
    • a measurement is ordinal if its value establishes order
    • e.g. 2 might be between 1 and 3
    • e.g. yellow might be between red and green
    • ordinal information is common in studies of preferences, market research
  • interval
    • a measurement is interval if differences on the scale make sense
    • e.g. it makes sense to say that 80 degrees is 10 degrees hotter than 70 degrees
    • the difference between 80 and 70 is equal to the difference between 70 and 60
  • ratio
    • a measurement is ratio if division makes sense
    • a 200 kg person is twice as heavy as a 100 kg person
    • 1 km is ten times as far as 100m
    • but 80 degrees Celsius is not twice as hot as 40 degrees Celsius
    • to be ratio, a scale must have an absolute zero point
    • negative values are often not meaningful for ratio data
    • Kelvin temperature is ratio, but Celsius and Fahrenheit are only interval
• implications for GIS
  • it makes no sense to perform certain operations on certain kinds of variables
    • arithmetic on nominal data is meaningless
    • a nominal variable doesn't have a mean
    • instead, use the commonest or modal value
    • for ordinal data, use the median, not the mean
    • it makes no sense to divide a ratio variable by an ordinal one
  • GIS software won't protect the user from meaningless operations like these
    • software doesn't keep track of the type of a variable
• the strange case of cyclic data
  • e.g. aspect, measured as a direction from 0 to 360
    • consider averaging aspects of 5, 10, 355, 350
    • all are close to North
    • total and divide by 4, the result is 180
  • cyclic data is a special type, encountered in GIS:
    • aspect
    • flow direction
    • wind direction
  • best to use a vector field, even though there is no variation in magnitude
    • but many GIS do not support vector fields
  • be careful with cyclic variables in GIS
• the term continuous
  • a field is spatially continuous by definition
    • values exist everywhere
  • the term continuous' can also be applied to the variable
    • implying that it is measured on a continuous scale
    • all values of the variable are possible
    • between limits if any exist
    • in this sense, continuous' implies ordinal, interval, or ratio
    • a nominal variable can't have a continuous scale
    • a nominal variable must be measured on a discrete scale where only certain values are possible
• types of fields based on continuity properties
  • defined only for fields of continuous variables (ordinal, interval, ratio)
  • a field can be smooth or rugged
    • locally smooth or locally rugged
  • cliffs may or may not be allowed
    • cliffs are zero-order' discontinuities
    • a field with no cliffs is zero-order continuous
    • mathematically, z(x+dx) - z(x) tends everywhere to zero as dx tends to zero if there are no cliffs
  • sharp ridges and valleys and breaks of slope may or may not be allowed
    • breaks of slope are second-order discontinuities

3. Fields and discrete entities

• the main competitor to the field conceptualization
  • geography consists of an otherwise empty space littered with discrete entities
  • as with fields, this is a question of conceptualization, not digital representation
  • a point can lie in any number of entities, including zero
  • entities can be points, lines, areas, or volumes in three or more dimensions
  • entities can have any number of characteristics (attributes) associated with them
    • the attributes apply to the entire entity
  • the discrete entity conceptualization is the subject of the unit on Representing Discrete Objects

3.1. Examples to clarify the dichotomy

3.1.1. Weather forecasting

• an example of the use of fields
• the processes operating in the atmosphere can be described by physical laws
• the include the Gas Law, the Navier Stokes equation governing the behavior of a viscous fluid, and others
• the laws are valid at the levels of resolution appropriate to the atmosphere, but break down at very detailed molecular scales
• these laws are written in terms of fields
  • many of them are partial differential equations governing the rates of change of field variables in time and space
  • the variables include pressure, temperature, wind
  • such variables can be defined at any point in the spatio-temporal frame
  • that is, they are fields
• these days, computer models are used to predict the behavior of the atmosphere
  • they must work with digital representations
  • important decisions must be made about the level of detail of the representation
  • is it sufficient to represent the atmosphere by sampling every 100km horizontally, every 100m vertically, every 1 hour in time, or is greater detail needed for accurate forecasting?
  • how will errors accumulate and limit the time horizon of the forecast?
  • today, forecasting is limited to about 5 days for reliable estimates
• the inputs to these models are representations of fields
  • as are the outputs
• however, weather forecasts for consumption by the general public translate these fields into more understandable terms
  • e.g. for pressure fields, highs and lows and fronts
  • forecasts may be described as apparent behaviors of these discrete entities
  • e.g., this front will stall
  • e.g., this high will weaken
  • scientific models may work with fields, but people may find discrete entities more acceptable, more easily understood
  • natural language provides much better ways of talking about discrete entities
  • it is comparatively difficult to describe a field
  • as Helen Couclelis writes, "People manipulate objects, but cultivate fields" (Couclelis, 1992)

3.1.2. Lakes in Minnesota

• the Minnesota license plate refers to 10,000 lakes
• who counted them? a student hired for a summer?
• what scale of map was used?
  • the result will depend on the scale
  • with more detailed maps, the total would surely be higher
• who defined a lake?
  • how to tell when one lake with a narrow part is really two?
  • how to tell when a swamp is a lake?
• the result will vary from one person to another
  • it is impossible to define lake' with sufficient accuracy to have one right answer
  • the task boils down to trying to count discrete geographic entities conceived as littering an otherwise empty space
• what is the equivalent field conceptualization?
• define a variable lakeness' with a single value everywhere in the state
  • L=0 for well-drained sandy soils that are never waterlogged
  • L=2 for areas that are swampy in Spring
  • L=5 for permanent swamp
  • L=8 for areas inundated except in very dry summers
  • L=10 for deep, permanent water
• this is a field, rigorously defined at every point
  • two people could agree on its value
  • maps of the field would be useful to others
• its general properties could be computed
  • its mean value over the state
  • the license plate might refer to mean lakeness
  • welcome to Minnesota, mean lakeness 2.8
  • 13% of Minnesota has lakeness 9 or higher

3.1.3. Benefits and disbenefits of fields

• what can we learn from these examples about the benefits, disbenefits of fields?
• fields can be well-defined
  • if the variable is well-defined
• they are the basis of much physical modeling
  • models of social systems using fields are rare
  • population density is a notable exception
  • a good reference is Angel and Hyman (1976)
  • social systems are mostly modeled as collections of discrete entities
• fields are hard to describe in natural language
  • as humans we prefer to reason, remember, describe our surroundings in terms of discrete entities, not continuously varying fields
  • a tourist headed for Minnesota is more likely to be attracted by the idea that Minnesota contains a large number of discrete lakes than to know what percentage of the state is covered by them, even though scientific rigor and rationality appear to favor the other side

4. How are fields represented?

• there are many ways of representing fields
  • not all are implemented in GIS
  • different terminologies exist in different disciplines
  • this discussion begins with what is normal in GIS, discusses other disciplines at the end
• six major representations, with example uses in each case
  • this discussion deals mostly with two-dimensional frames
    • see later discussion for higher dimensionality
  • some of these six representations give values for the field at all points (they are complete)
    • some define the field only at certain points, require additional methods to make estimates elsewhere (they are incomplete)

4.1. Rectangular cells

• see Figure 1(a)
• value in each cell is an average, total, or some other aggregate property of the field within the cell
  • the representation defines a value everywhere, so is complete
  • however, all within-cell variation is lost
  • if necessary, it must be reconstructed by some method of intelligent guesswork
• e.g. remote sensing data and other kinds of digital imagery
• see the unit on Rasters for more on grids and cells

4.2. Rectangular grid of points

• see Figure 1(b)
• e.g. measurements of land surface elevation in a digital elevation model (DEM)
• spacing of measurements is critical to accuracy of representation
  • all variation between sample points is lost
  • elevations at other points must be estimated by some method of intelligent guesswork (the representation is incomplete)

4.3. Irregularly spaced points

• see Figure 1(c)
• the field's value is defined at a set of sample points scattered in the frame
  • values of the field at other points must be interpolated
  • representation is incomplete
• e.g. weather data, available at scattered weather stations
• accuracy depends on the density of points
  • it is not clear what measure best defines accuracy - density per unit area, minimum distance between sample points, maximum distance

4.4. Digitized contours

• see Figure 1(d)
• the field is represented as a set of isolines, each connecting points of constant value
  • representation is incomplete
• the scale of measurement of the variable must be at least ordinal
  • isolines cannot be defined for nominal data
• each isoline is represented as a polyline
• e.g. data obtained from topographic maps
• accuracy depends on:
  • the number of contoured values, or the contour interval
  • the density of polyline points

4.5. Polygons

• see Figure 1(e)
• the frame is partitioned into irregular areas (volumes for 3 or more dimensions)
• value in each area is an average, total, or some other aggregate property of the field within the area
  • the representation is complete
  • all variation within areas is lost
• e.g. data obtained from maps of vegetation cover class, soil type
• the boundaries of areas are continuously curved lines
  • represented digitally as polylines - an ordered sequence of points connected by straight lines
  • the denser the points, the more accurate the polyline as a representation of a continuous curve
  • accuracy depends both on the size of polygons and on the density of polyline points
  • it is not clear what measure of polygon size - average, minimum - best defines accuracy
• every point in the frame lies in exactly one polygon
  • except for points on the boundaries
  • the polygons cannot overlap, must exhaust the frame
  • they are said to tesselate the space, they form an irregular tesselation
• see the unit on Polygon Coverages

4.6. Triangulated irregular networks (TINs)

• see Figure 1(f)
• the frame is covered with a mesh of irregular triangles
  • every point lies in exactly one triangle, or on a triangle edge
• the value of the field is known at every triangle vertex
  • within triangles and along edges it is assumed to vary linearly
  • the representation is complete
  • contours drawn across triangles will therefore always be straight and parallel
  • across triangle edges there will be breaks of slope, but not cliffs
  • contours will kink at edges
• the scale of measurement of the variable must be at least interval
  • variation within triangles cannot be defined for nominal or ordinal variables
• accuracy depends on:
  • how carefully the vertices were located on the surface
  • how well the planes defined within each triangle fit the actual surface
  • the sizes of triangles
  • but it is not clear what property of triangle size best defines accuracy - average, smallest, largest

5. Other representations

• in modeling outside the context of GIS
  • grids and cells are often known as finite difference methods
  • finite element methods use irregular primitive elements
    • TINs are a form of finite element method
  • both are loosely described as grids
  • adaptive grids redefine the grid during execution of the model
    • e.g. after every step or iteration of a dynamic model
    • in GIS the representation normally remains constant
• TINs are known as triangular meshes in many areas of computer graphics
  • used for visualizing solid objects
  • the sharp breaks of slope in a TIN make the triangles visible to the eye, which may be undesirable
    • instead, the field within each triangle is sometimes described by a higher-order mathematical function that achieves first-order (no break of slope) continuity across triangle edges
• many of the concepts discussed above are valid for fields in frames with three or more dimensions
  • note, however, the earlier comments about the lack of equivalence of z and t with x and y
  • grids and cells extend naturally to three and four dimensions, with appropriately defined sampling intervals
  • with a third spatial dimension, polygons become polyhedra with polygonal faces
    • e.g. in subsurface geology
    • see the unit on Representing time and storing temporal data for discussion of adding the time dimension
  • with a third spatial dimension, isolines become isosurfaces and are widely used in 3D GIS
    • it's more difficult to imagine the equivalent of isolines when time is added
  • irregular point samples can be taken in three or four dimensions
  • TINs in three spatial dimensions are widely used in computer graphics for visualizing solid objects as objects covered with triangular meshes
    • TINs don't extend as easily to the temporal dimension

6. References

• Angel, S. and G.M. Hyman (1976) Urban Fields: A Geometry of Movement for Regional Science. London: Pion.
• Couclelis, H. (1992) People manipulate objects (but cultivate fields): beyond the raster-vector debate in GIS. In A.U. Frank, I. Campari, and U. Formentini (editors) Theories and Methods of Spatio-Temporal Reasoning in Geographic Space. Lecture Notes in Computer Science 639. Berlin: Springer-Verlag, pp. 65-77.
   

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7. Exam and discussion questions

1. Give other examples to illustrate the use of fields in scientific research, and discrete entities in human cognition and reasoning.
    
2. After studying this unit and unit 065, make and illustrate a list of the most viable methods for representing fields in two spatial dimensions and time.
    
3. "There appear to be no viable uses of digitized isolines" - discuss.
    
4. If you were asked to design a GIS to handle representations of vector fields, what functions would you want it to perform, and what applications could you find for it?

8. Exercise

Exercise in Digital Elevation/Terrain Models: From Point to Mathematics

by Ahmad S. Massasati, United Arab Emirates University

This paper presents a practical way to teach about elevation models. These currently include solutions geographers refer to as digital elevation/terrain models such as point data, contour lines, triangular irregular networks, and mathematical models. The apparent complexity of data transfer in these methods, however, seems difficult to students and other first time users. In the author’s classroom experience, the pyramids of Egypt have proved to be an excellent and efficient example for teaching digital elevation/terrain models.

Evaluation

Citation

Michael F. Goodchild. (1997) Representing Fields, NCGIA Core Curriculum in GIScience, http://www.ncgia.ucsb.edu/giscc/units/u054/u054.html, accessed [today's date].