So far, we have discussed how to collect geographic data, how to manage and manipulate it in a database, and how to represent thematic data in map form. This chapter will explore the various geographic approaches to representing Earth’s surfaces. We will begin the chapter describing topographic maps, from their historical use to their current applications. Next, we will consider different approaches to storing, creating, and representing Earth’s elevation data. Finally, we end the chapter by considering surfaces that are not land-based: bathymetry, the measurements of oceanic depths, or the varying sea floor elevations.

Objectives

Students who successfully complete Chapter 8 should be able to:

1. identify different techniques and approaches to representing the Earth’s surfaces;
2. describe how topographic data are compiled from aerial imagery;
3. calculate an interpolated spot elevation based on neighboring elevations;
4. understand how continuous surfaces are created from a set of measured values at discrete locations through interpolation;
5. given a regular or irregular array of spot elevations, construct a triangulated irregular network, interpolate contour intervals and draw contour lines;
6. compare vector and raster representations of terrain elevation;
7. acquire and view digital elevation data from the National Elevation Dataset.

Table of Contents

• Topographics Maps
• Elevation 
• Beyond Terrain Surfaces: Bathymetry
• Summary
• Glossary
• Biblography

Chapter lead author: Jennifer Smith.
Portions of this chapter were drawn directly from the following text: 

Joshua Stevens, Jennifer M. Smith, and Raechel A. Bianchetti (2012), Mapping Our Changing World, Editors: Alan M. MacEachren and Donna J. Peuquet, University Park, PA: Department of Geography, The Pennsylvania State University.

Since the eighteenth century, the preparation of a detailed basic reference map has been recognized by the governments of most countries as fundamental for the delimitation of their territory, for underpinning their national defense, and for management of their resources (Parry, 1987).

Specialists in geographic information recognize two broad functional classes of maps: reference maps and thematic maps. As you recall from Chapter 3, a thematic map is usually made with one particular purpose in mind. Often, the intent is to make a point about the spatial pattern of a single phenomenon. Reference maps, on the other hand, are designed to serve many different purposes. Like a reference book -- such as a dictionary, encyclopedia, or gazetteer -- reference maps help people look up facts. Common uses of reference maps include locating place names and features, estimating distances, directions, and areas, and determining preferred routes from starting points to a destination. Reference maps are also used as base maps upon which additional geographic data can be compiled. Because reference maps serve various uses, they typically include a greater number and variety of symbols and names than thematic maps. The portion of the United States Geological Survey (USGS) topographic map shown below is a good example.

Figure 8.1. A typical reference map. A portion of a USGS topographic quadrangle map showing Bellefonte, PA.

Credit: USGS, 1971.

The term topography derives from the Greek topographein, "to describe a place." Topographic maps show, and name, many of the visible characteristics of the landscape, as well as political and administrative boundaries. Topographic map series provide base maps of uniform scale, content, and accuracy (more or less) for entire territories. Many national governments include agencies responsible for developing and maintaining topographic map series for a variety of uses, from natural resource management to national defense. Affluent countries, countries with especially valuable natural resources, and countries with large or unusually active militaries, tend to be mapped more completely than others.

8.1.1 Legacy Data: USGS Topographic Maps

The systematic mapping of the entire U.S. began in 1879, when the U.S. Geological Survey (USGS) was established. Over the next century, USGS and its partners created topographic map series at several scales, including 1:250,000, 1:100,000, 1:63,360, and 1:24,000. The diagram below illustrates the relative extents of the different map series. Since much of today’s digital map data was digitized from these topographic maps, one of the challenges of creating continuous digital coverage of the entire U.S. has been to seam together all of these separate map sheets. The current process for topographic mapping in the U.S. is organized as The National Map (NationalMap.gov [1]). But, since the process still relies on some data collected in traditional ways using the sheet-based organizational structure, we begin with a description of past topographic mapping practice.

Figure 8.2. Relative extents of several USGS quadrangle map series.

Credit: Thompson, 1988.

Map sheets in the legacy 1:24,000-scale series are known as quadrangles or simply quads. A quadrangle is a four-sided polygon. Although each 1:24,000 quad covers 7.5 minutes longitude by 7.5 minutes latitude, their shapes and area coverage vary. The area covered by the 7.5-minute maps varies from 49 to 71 square miles (126 to 183 square kilometers), because the length of a degree of longitude varies with latitude.

Figure 8.3. Topographer compiling topographic map using a plane table and alidade.

Credit: NOAA, 2007.

Through the 1940s, topographers in the field compiled by hand the data depicted on topographic maps. Anson (2002) recalls being outfitted with a 14 inch x 14 inch tracing table and tripod, plus an alidade [a 12 inch telescope mounted on a brass ruler], a 13 foot folding stadia rod, a machete, and a canteen (p. 1). Teams of topographers sketched streams, shorelines, and other water features; roads, structures, and other features of the built environment; elevation contours, and many other features. To ensure geometric accuracy, their sketches were based upon the geodetic control network (of about 240,000 locations of known position as described here: Horizontal Control PDF [2]), as well as positions and spot elevations they surveyed themselves using alidades and rods. Depending on the terrain, a single 7.5-minute quad sheet might have taken weeks or months to compile. In the 1950s, however, photogrammetric methods (discussed in Chapter 7) permitted topographers to make accurate stereoscopic measurements directly from overlapping pairs of aerial photographs providing a viable and more efficient alternative to field mapping.

8.1.2. Scanned Topographic Maps

Many digital data products have been derived from the USGS topographic map series. The simplest of such products are Digital Raster Graphics (DRGs). DRGs are scanned raster images of USGS 1:24,000 topographic maps. DRGs are useful as backdrops over which other digital data may be superimposed. For example, the accuracy of a vector file containing lines that represent lakes, rivers, and streams could be checked for completeness and accuracy by plotting it over a DRG (subject to the age of the data on the DRG).

Figure 8.4 Portion of a Digital Raster Graphic (DRG) for Bushkill, PA.

DRGs are created by scanning paper maps at 250 pixels per inch resolution. Since at 1:24,000 1 inch on the map represents 2,000 feet on the ground, each DRG pixel corresponds to an area about 8 feet (2.4 meters) on a side. Each pixel is coded from 0 to 12; the numbers stand for the 13 standard DRG colors. Like the paper maps from which they are scanned, DRGs comply with National Map Accuracy Standards (Standards and Specifications [3]).

Figure 8.5. Magnified portion of a Digital Raster Graphic (DRG) for Bushkill, PA.

To investigate DRGs in greater depth, visit the USGS DRG site [4] or search the Internet on “USGS Digital Raster Graphics”.

By 1992, the series of over 53,000 separate quadrangle maps covering the lower 48 states, Hawaii, and U.S. territories at 1:24,000 scale was completed, at an estimated total cost of $2 billion. However, by the end of the century, the average age of 7.5-minute quadrangles was over 20 years, and federal budget appropriations limited revisions to only 1,500 quads a year (Moore, 2000). As landscape change has exceeded revisions in many areas of the U.S., the USGS topographic map series has become legacy data outdated in terms of format as well as content. The paper quad-based topographic map series has been replaced by the National Map program. The National Map is designed to produce a multi-scale digital map for the country; this is discussed in Section 1.3 below. First, we discuss map accuracy, which is a topic that applies to both the legacy paper map products and the new digital products of the National Map.

8.1.3 Accuracy Standards

Errors and uncertainty are inherent in geographic data. Despite the best efforts of the USGS Mapping Division and its contractors, topographic maps include features that are out of place, features that are named or symbolized incorrectly, and features that are out of date.

The locational accuracy of spatial features encoded in USGS topographic maps and data are guaranteed to conform to National Map Accuracy Standards. The standard for topographic maps states that horizontal positions of 90 percent of the well-defined points tested will occur within 0.02 inches (map distance) of their actual positions (thus, 10% of points can vary by more than this). Similarly, the vertical positions of 90 percent of well-defined points tested are to be true to within one-half of the contour interval. Both standards are scale-dependent. For example, at 1:24,000, 0.02 inches equals 40 feet (thus 90% of points tested at this scale must be within 40 feet of their true location; in contrast, at 1:250,000, the tolerance is 416.7 feet).

Objective standards do not exist for the accuracy of attributes associated with geographic features. Attribute errors certainly do occur, however. A chronicler of the national mapping program (Thompson, 1988, p. 106) recalls a worried user who complained to USGS that "My faith in map accuracy received a jolt when I noted that on the map the borough water reservoir is shown as a sewage treatment plant."

The passage of time is perhaps the most troublesome source of errors on topographic maps. As mentioned in the previous page, the average age of the original USGS topographic map series was over 20 years at the turn of the century when the decision was made to stop updating maps on a quad-by-quad basis. Geographic data quickly lose value (except for historical analyses) unless they are continually revised. The sequence of map fragments below shows how frequently revisions were required between 1949 and 1973 for the quad that covers Key Largo, Florida. Revisions are based primarily on geographic data produced by aerial photography.

8.1.4 USGS National Map

Executive Order 12906 decreed that a designee of the Secretary of the Department of Interior would chair the Federal Geographic Data Committee. The USGS, an agency of the Department of Interior, has lead responsibility for three of the seven National Spatial Data Infrastructure (NSDI) framework themes--orthoimagery, elevation, and hydrography, and secondary responsibility for several others. In 2001, USGS announced its vision of a National Map that "aligns with the goals of, and is one of several USGS activities that contribute to, the National Spatial Data Infrastructure" (USGS, 2001, p. 31). A 2002 report of the National Research Council identified the National Map as the most important initiative of USGS’ Geography Discipline at the USGS (NRC, 2002). Recognizing its unifying role across science disciplines, USGS moved management responsibility for the National Map from Geography to the USGS Geospatial Information Office in 2004. (One reason that the term "geospatial" is used at USGS and elsewhere is to avoid association of GIS with a particular discipline, i.e., Geography.) In 2001, USGS envisioned the National Map as the nation’s topographic map for the 21st Century (USGS, 2001, p.1). According to Characteristics of the National Map (USGS, 2001, p. 11-13), improvements over the original topographic map series were to include:

• Currentness
  Content will be updated on the basis of changes in the landscape instead of the cyclical inspection and revisions cycles now in use [for printed topographic map series]. The ultimate goal is that new content be incorporated within seven days of a change in the landscape.
• Seamlessness
  Features will be represented in their entirety and not interrupted by arbitrary edges, such as 7.5-minute map boundaries.
• Consistent classification
  Types of features, such as "road" and "lake/pond," will be identified in the same way throughout the nation.
• Variable resolution
  Data resolution, or pixel size, may vary among imagery of urban, rural, and wilderness areas. The resolution of elevation data may be finer for flood plain, coastal, and other areas of low relief than for areas of high relief.
• Completeness
  Data content will include all mappable features (as defined by the applicable content standards for each data theme and source).
• Consistency and integration
  Content will be delineated geographically (that is, in its true ground position within the applicable accuracy limit) to ensure logical consistency between related features. For example, ... streams and rivers [should] consistently flow downhill...
• Variable positional accuracy
  The minimum positional accuracy will be that of the current primary topographic map series for an area. Actual positional accuracy will be reported in conformance with the Federal Geographic Data Committee’s Geospatial Positioning Accuracy Standard.
• Spatial reference systems
  Tools will be provided to integrate data that are mapped using different datums and referenced to different coordinates systems, and to reproject data to meet user requirements.
• Standardized content
  ...will conform to appropriate Federal Geographic Data Committee, other national, and/or international standards.
• Metadata
  At a minimum, metadata will meet Federal Geographic Data Committee standards to document ... [data] lineage, positional and attribute accuracy, completeness, and consistency.

As of 2012, USGS’ ambitious vision has not yet been fully realized. Insofar as it depends upon cooperation by many federal, state, and local government agencies, the vision may never be fully achieved. Still, elements of a National Map do exist, including national data themes, data access and dissemination technologies such as the Geospatial One Stop portal (GeoPortal [7]) and the National Map Viewer [8], and the U.S. National Atlas [9]. A new Center of Excellence for Geospatial Information Science (CEGIS) was established in 2006 under the USGS Geospatial Information Office to undertake the basic GIScience research needed to devise and implement advanced tools that will make the National Map more valuable to end users. The data themes included in the National Map are shown in table 8.1.

The NSDI Framework Introduction and Guide (FGDC, 1997, p. 19) points out that "elevation data are used in many different applications." Civilian applications include flood plain delineation, road planning and construction, drainage, runoff, and soil loss calculations, and cell tower placement, among many others. Elevation data are also used to depict the terrain surface by a variety of means, from contours to relief shading and three-dimensional perspective views.

The NSDI Framework calls for an "elevation matrix" for land surfaces. That is, the terrain is to be represented as a grid of elevation values. The spacing (or resolution) of the elevation grid may vary between areas of high and low relief (i.e., hilly and flat). Specifically, the Framework Introduction states that:

Elevation values will be collected at a post-spacing of 2 arc-seconds (approximately 47.4 meters at 40° latitude) or finer. In areas of low relief, a spacing of 1/2 arc-second (approximately 11.8 meters at 40° latitude) or finer will be sought (FGDC, 1997, p. 18).

The elevation theme also includes bathymetry--depths below water surfaces--for coastal zones and inland water bodies. Specifically,

For depths, the framework consists of soundings and a gridded bottom model. Water depth is determined relative to a specific vertical reference surface, usually derived from tidal observations. In the future, this vertical reference may be based on a global model of the geoid or the ellipsoid, which is the reference for expressing height measurements in the Global Positioning System (FGDC, 1997, p. 18).

USGS has lead responsibility for the elevation theme of the NSDI. Elevation is also a key component of USGS' National Map. The next sections consider how heights and depths are created, how they are represented in digital geographic data, and how they may be depicted cartographically.

8.2.1 Vector and Raster Approaches

The terms raster and vector were introduced back in Chapter 4 to denote two fundamentally different strategies for representing geographic phenomena. Both strategies involve simplifying the infinite complexity of the Earth's surface. As it relates to elevation data, the raster approach involves measuring elevation at a sample of locations that are evenly spaced. The vector approach, on the other hand, involves measuring the locations of a sample of elevations and depicting the surface with elevation contours.

Figure 8.8. Vector and raster representations of the same terrain surface.

The illustration above compares how elevation data are represented in vector and raster data. On the left are elevation contours, a vector representation that is familiar to anyone who has used a USGS topographic map. Contours are a kind of isarithm, from the Greek words for "same" and "number." A contour line, then, is a line along with an elevation value (number) that remains equal (the same). There are many kinds of isarithm, with the variants reflecting the kind of thing depicted (an isobaths is a line of equal bathymetry, or depth under water; an isotherm is a line of equal temperature). Contours are one of the few isarithmic line types with a name that does not include “iso” as a prefix.

As you will see later in this chapter, when you explore Digital Line Graphs, elevations in vector data are encoded as attributes of line features. The distribution of locations with precisely specified elevations across the quadrangle is therefore irregular. Raster elevation data, by contrast, consist of grids at which elevation is encoded at regular intervals at each intersection. Raster elevation data are what is called for by the NSDI Framework and the USGS National Map. Contours can now be rendered easily from digital raster data. However, much of the raster elevation data used in the National Map was produced from digital vector contours and hydrography (streams and shorelines). For this reason, we will consider the vector approach to terrain representation first.

8.2.2 Contours

Figure 8.9. Contour lines trace the elevation of the terrain surface at regularly-space intervals.

Credit: Raisz, 1948, McGraw-Hill, Inc. Used by permission.

Drawing contour lines is a way to represent a terrain surface with a sample of elevations. Instead of measuring and depicting elevation at every point, you measure only along lines at which a series of imaginary horizontal planes slice through the terrain surface. The more imaginary planes, the more contours, and the more detail is captured with a smaller the contour interval (the magnitude of difference from one contour to the next).

Figure 8.10. Contour lines representing the same terrain as in Figure 8.9, but in plan view.

Credit: Raisz, 1948. McGraw-Hill, Inc. Used by permission.

Until photogrammetric methods came of age in the 1950s, topographers in the field sketched contours on the USGS 15-minute topographic quadrangle series. Since then, contours shown on most of the 7.5-minute quads were compiled from stereoscopic images of the terrain, as described in Chapter 7. Today computer programs draw contours automatically from the spot elevations that photogrammetrists compile stereoscopically.

Although it is uncommon to draw terrain elevation contours by hand these days, it is still worthwhile to know how to develop an understanding of how automated methods work and of the kinds of error they can produce. In the next few pages, you'll have a chance to practice the technique, which is analogous to the way computers do it.

8.2.3 Contouring by Hand

This page will walk you through a methodical approach to rendering contour lines from an array of spot elevations (Rabenhorst and McDermott, 1989). To get the most from this exercise, we suggest that you print the illustration in the attached image file [12]. Find a pencil (preferably one with an eraser!) and a straightedge, and duplicate the steps illustrated below. A "Try This!" activity will follow this step-by-step introduction, providing you a chance to go solo.

Figure 8.11. Beginning a triangulated irregular network.

Starting at the highest elevation, draw straight lines to the nearest neighboring spot elevations. Once you have connected to all of the points that neighbor the highest point, begin again at the second highest elevation. (You will have to make some subjective decisions as to which points are "neighbors" and which are not.) Taking care not to draw triangles across the stream, continue until the surface is completely “triangulated,” where triangles connect any given three neighbors.

Figure 8.12. Complete TIN. Note that the triangle sides must not cross hydrologic features (i.e., the stream) on a terrain surface.

The result is a triangulated irregular network (TIN). A TIN is a vector representation of a continuous surface that consists entirely of triangular facets. The vertices of the triangles are spot elevations that may have been measured in the field by leveling, or in a photogrammetrist's workshop with a stereoplotter, or by other means. (Spot elevations produced photogrammetrically are called mass points.) A useful characteristic of TINs is that each triangular facet has a single slope degree and direction. With a little imagination and practice, you can visualize the underlying surface from the TIN even without drawing contours.

Wonder why we suggest that you not let triangle sides that make up the TIN cross the stream? Well, if you did, the stream would appear to run along the side of a hill, instead of down a valley as it should. In practice, spot elevations would always be measured at several points along the stream, and along ridges as well. Photogrammetrists refer to spot elevations collected along linear features as breaklines (Maune, 2007). We omitted breaklines from this example just to make a point.

You may notice that there is more than one correct way to draw the TIN. As you will see, deciding which spot elevations are "near neighbors" and which are not is subjective in some cases. Related to this element of subjectivity is the fact that the fidelity of a contour map depends in large part on the distribution of spot elevations on which it is based. In general, the density of spot elevations should be greater where terrain elevations vary greatly, and sparser where the terrain varies subtly. Similarly, the smaller the contour interval you intend to use, the more spot elevations you need. In the example below, we use a contour interval of 100.

There are algorithms for triangulating from irregular arrays of point elevations that produce unique solutions. One approach is called Delaunay Triangulation which, in one of its constrained forms, is useful for representing terrain surfaces. The distinguishing geometric characteristic of a Delaunay triangulation is that a circle surrounding each triangle side does not contain any other vertex.

Figure 8.13. Tick marks drawn where elevation contours cross the edges of each TIN facet.

Now, draw ticks to mark the points at which elevation contours intersect each triangle side. As noted above, we will use a contour interval of 100 feet in this example, with each contour line representing some increment of 100. For instance, see the triangle side that connects the spot elevations 2360 and 2480 in the lower left corner of the illustration above? One tick mark is drawn on the triangle where a contour representing elevation 2400 intersects. Now find the two spot elevations, 2480 and 2750, in the same lower left corner. Note that three tick marks are placed where contours representing elevations 2500, 2600, and 2700 intersect.

This step should remind you of the equal interval classification scheme you read about in Chapter 3. The right choice of contour interval depends on the goal of the mapping project. In general, contour intervals increase in proportion to the variability of the terrain surface. It should be noted that the assumption that elevations increase or decrease at a constant rate is not always correct, of course. We will consider that issue in more detail later.

Figure 8.14. Threading elevation contours through a TIN.

Finally, draw your contour lines. Working downslope from the highest elevation, thread contours through ticks of equal value. Move to the next highest elevation when the surface seems ambiguous.

Keep in mind the following characteristics of contour lines (Rabenhorst and McDermott, 1989):

• Contours should always point upstream in valleys.
• Contours should always point downridge along ridges.
• Adjacent contours should always be sequential or equivalent.
• Contours should never split into two.
• Contours should never cross or loop.
• Contours should never spiral.
• Contours should never stop in the middle of a map.

How does your finished map compare with the one we drew below?

Figure 8.15. Final map with contour intervals.

8.2.4 Digital Line Graph (DLG)

Identification

Digital Line Graphs (DLGs) are vector representations of most of the features and attributes shown on USGS topographic maps. Individual feature sets (outlined in the table below) are encoded in separate digital files. DLGs exist at three scales: small (1:2,000,000), intermediate (1:100,000) and large (1:24,000). Large-scale DLGs are produced in tiles that correspond to the 7.5-minute topographic quadrangles from which they were derived (Digital Line Graphs [4]).

Credit: USGS, 2006.

Figure 8.16. Portion of three Digital Line Graph (DLG) layers for USGS Bushkill, PA quadrangle; imaged with Global Mapper (dlgv32 Pro) software. Transportation features are arbitrarily colored red, hydrography blue, and hypsography brown. The square symbols are nodes and the triangles represent polygon centroids.

Data quality

Like other USGS data products, DLGs conform to National Map Accuracy Standards. In addition, however, DLGs are tested for the logical consistency of the topological relationships among data elements. Similar to the Census Bureau's TIGER/Line, line segments in DLGs must begin and end at point features (nodes), and line segments must be bounded on both sides by area features (polygons).

Spatial Reference Information

DLGs are heterogenous in terms of the projection they are based upon. Some use UTM coordinates, others State Plane Coordinates. Some are based on NAD 27, others on NAD 83. Elevations are referenced either to NGVD 29 or NAVD 88 (USGS, 2006a).

Entities and attributes

The basic elements of DLG files are nodes (positions), line segments that connect two nodes, and areas formed by three or more line segments. Each node, line segment, and area is associated with two-part integer attribute codes. For example, a line segment associated with the attribute code "050 0412" represents a hydrographic feature (050), specifically, a stream (0412).

Distribution

Not all DLG layers are available for all areas at all three scales. Coverage is complete at 1:2,000,000. At the intermediate scale, 1:100,000 (30 minutes by 60 minutes), all hydrography and transportation files are available for the entire United States. At 1:24,000 (7.5 minutes by 7.5 minutes), coverage remains spotty. The files are in the public domain, and can be used for any purpose without restriction.

Large- and Intermediate- scale DLGs are available for download through EarthExplorer system (EarthExplorer [20]). You can plot 1:2,000,000 DLGs on-line at the USGS' National Atlas of the United States (National Atlas [9]).

Digital Line Graph Hypsography

In one sense, DLGs are as much "legacy" data as the out-of-date topographic maps from which they were produced. Still, DLG data serve as primary or secondary sources for several themes in the USGS National Map, including hydrography, boundaries, and transportation. DLG hypsography data are not included in the National Map, however. It is assumed that GIS users can generate elevation contours as needed from DEMs.

Figure 8.17. Portion of the hypsography and hydrography layers of a large-scale Digital Line Graph (DLG). USGS Bushkill, PA quadrangle; imaged with Global Mapper (dlgv32 Pro) software.

Hypsography refers to the measurement and depiction of the terrain surface, specifically with contour lines. Several different methods have been used to produce DLG hypsography layers, including:

• scanning contour lines on photographic film or paper maps, converting the scanned raster data to vectors, then editing and attributing the vector features;
• manually digitizing and attributing contour lines on photographic film or paper maps;
• producing contours by photogrammetric processes; and
• deriving contours from LiDAR (discussed later in section 8.2.8).

Figure 8.18. The highlighted contour line has been selected and its attributes reported in a Global Mapper window. Notice that the line feature is attributed with a unique Element ID code (LE01, 639) and an elevation (1000 feet).

8.2.5 Digital Elevation Model (DEM)

The term "Digital Elevation Model" has both generic and specific meanings. Generically, a DEM is any raster representation of a terrain surface. Specifically, in relation to the NSDI, a DEM is a data product of the U.S. Geological Survey. Here we consider the characteristics of DEMs produced by the USGS. Later in this chapter, we'll consider sources of global terrain data.

Identification

USGS DEMs are raster grids of elevation values that are arrayed in series of south-north profiles. Like other USGS data, DEMs were produced originally in tiles that correspond to topographic quadrangles. Large scale (7.5-minute and 15-minute), intermediate scale (30 minute), and small scale (1 degree) series were produced for the entire United States. The resolution of a DEM is a function of the east-west spacing of the profiles and the south-north spacing of elevation points within each profile.

DEMs corresponding to 7.5-minute quadrangles are available at 10-meter resolution for much, but not all, of the United States. Coverage is complete at 30-meter resolution. In these large scale DEMs, elevation profiles are aligned parallel to the central meridian of the local UTM zone, as shown in Figure 8.19, below. See how the DEM tile in the illustration below appears to be tilted? This is because the corner points are defined in unprojected geographic coordinates that correspond to the corner points of a USGS quadrangle. The farther the quadrangle is from the central meridian of the UTM zone, the more it is tilted.

Figure 8.19. Arrangement of elevation profiles in a large scale USGS Digital Elevation Model.

Click for a text description of Figure 8.19

A graphic representation to illustrate the fact that elevation profiles are aligned parallel to the central meridian of the local UTM zone, with an overlay showing a DEM polygon (7.5-minute quadrangle) that appears to be tilted because the corners are defined in unprojected geographic coordinates that correspond to the corner points of a USGS quadrangle.

Credit: USGS, 1987.

As shown below, the arrangement of the elevation profiles is different in intermediate- and small-scale DEMs. Like meridians in the northern hemisphere, the profiles in 30-minute and 1-degree DEMs converge toward the North Pole. For this reason, the resolution of intermediate- and small-scale DEMs (that is to say, the spacing of the elevation values) is expressed differently than for large-scale DEMs. The resolution of 30-minute DEMs is said to be 2 arc seconds and 1-degree DEMs are 3 arc seconds. Since an arc second is 1/3600 of a degree, elevation values in a 3 arc second DEM are spaced 1/1200 degree apart, representing a grid cell about 66 meters "wide" by 93 meters "tall" at 45º latitude (the width expands to over 80 meters in the southern US).

Figure 8.20. Arrangement of elevation profiles in a small scale USGS Digital Elevation Model.

Credit: USGS, 1987.

DEMs are produced from a wide range of sources, using the highest quality date available for each location. The sources in order of descending priority are:

1. High-resolution data, typically derived from lidar or digital photogrammetry, and often with edited water bodies. If collected at a ground sample distance no coarser than 5 meters, such data may also be offered within the NED at a resolution of 1/9th arc-second.
2. Moderate-resolution data, other than that compiled from cartographic contours. These data may also be derived from lidar or digital photogrammetry, or less often by Interferometric Synthetic Aperture Radar IFSAR. A typical ground sample distance is 10 meters, though it is commonly called “1/3 arc-second data”.
3. 10-meter DEMs derived from cartographic contours and mapped hydrography. Most often, such data are produced by or for the USGS as a standard elevation product, and they currently account for the bulk of the NED.
4. 30-meter cartographically derived DEMs. Similar in most respects to their 10-meter counterparts, though usually of lower overall quality.
5. 30-meter photogrammetrically derived DEMs. These are the oldest DEMs in the 7.5-minute series. These data were derived directly from stereo photography, either by a human operator or by an early form of electronic image correlation. They are badly marred by production artifacts that are addressed to the greatest practical extent by digital filtering within the NED production process.
6. 2-arc-second DEMs are a standard USGS product. They are derived from cartographic contours at a scale of 1:63,360 over the state of Alaska, and a scale of 1:100,000 elsewhere.
7. 1-arc-second Shuttle Radar Topography Mission (SRTM) data, to date, are only used in preference to 3 arc-second data in the Aleutian Islands.
8. 3-arc-second DEMs are another standard USGS product and are generally only used within the NED as a source of fill values over large water bodies.

The list above comes from the NED site, which is no longer in service. 

Some older DEMs were produced from elevation contours digitized from paper maps or during photogrammetric processing, then smoothed to filter out errors. Others were produced photogrammetrically from aerial photographs.

Data quality

The vertical accuracy of DEMs is expressed as the root mean square error (RMSE) of a sample of at least 28 elevation points. The target accuracy for large-scale DEMs is seven meters; 15 meters is the maximum error allowed.

Spatial Reference Information

Like DLGs, USGS DEMs are heterogeneous in terms of their relationship to position on the Earth. They are cast on the Universal Transverse Mercator projection used in the local UTM zone. Some DEMs are based upon the North American Datum of 1983, others on NAD 27. Elevations in some DEMs are referenced to either NGVD 29 or NAVD 88.

Entities and attributes

Each record in a DEM is a profile of elevation points. Records include the UTM coordinates of the starting point, the number of elevation points that follow in the profile, and the elevation values that make up the profile. Other than the starting point, the positions of the other elevation points need not be encoded, since their spacing is defined. (Later in this lesson, you'll download a sample USGS DEM file. Try opening it in a text editor to see what we are talking about).

Distribution

DEM tiles (subregions divided into areas for easier download) are available for free download through many state and regional clearinghouses. You can find these sources by searching GeoData.Gov [24].

As part of its National Map initiative, the USGS has developed a "seamless" National Elevation Dataset [25] that is derived from DEMs, among other sources. NED data are available at three resolutions: 1 arc second (approximately 30 meters), 1/3 arc second (approximately 10 meters), and 1/9 arc second (approximately 3 meters). Coverage ranges from complete at 1 arc second to extremely sparse at 1/9 arc second. An extensive FAQ on NED data is published at: NED FAQ [26]. The second of the two following activities involves downloading NED data and viewing it in Global Mapper.

8.2.6 Interpolation

When DEMs are derived from contours and when other surface representations (e.g., see bathemitry mapping below) are derived from sample data at points, the process used is interpolation. In general, interpolation is the process of estimating an unknown value from neighboring known values. It is a process used to create gridded surfaces for many kinds of data, not just elevation (an example will be shown below).

Figure 8.21. A USGS 7.5-minute DEM and the DLG hypsography and hydrography layers from which it was produced.

The elevation points in DLG hypsography files are not regularly spaced. DEMs need to be regularly spaced to support the slope, gradient, and volume calculations they are often used for. Grid point elevations must be interpolated from neighboring elevation points. In Figure 8.22, below, for example, the gridded elevations shown in purple were interpolated from the irregularly spaced spot elevations shown in red.

Figure 8.22. Elevation values in DEMs are interpolated from irregular arrays of elevations measured through photogrammetric methods, or derived from existing DLG hypsography and hydrography data.

Elevation data are often not measured at evenly-spaced locations. Photogrammetrists typically take more measurements where the terrain varies the most. They refer to the dense clusters of measurements they take as "mass points." Topographic maps (and their derivatives, DLGs) are another rich source of elevation data. Elevations can be measured from contour lines, but obviously contours do not form evenly-spaced grids. Both photogrammetry and topographic maps give rise to the need for interpolation.

Figure 8.23. Interpolating an intermediate value on a number line.

The illustration above shows three number lines, each of which ranges in value from 0 to 10. If you were asked to interpolate the value of the tick mark labeled "?" on the top number line, what would you guess? An estimate of "5" is reasonable, provided that the values between 0 and 10 increase at a constant rate. If the values increase at a geometric rate, the actual value of "?" could be quite different, as illustrated in the bottom number line. The validity of an interpolated value depends, therefore, on the validity of our assumptions about the nature of the underlying surface.

As was mentioned in Chapter 1, the surface of the Earth is characterized by a property called spatial dependence. Nearby locations are more likely to have similar elevations than are distant locations. Spatial dependence allows us to assume that it is valid to estimate elevation values by interpolation.

Many interpolation algorithms have been developed. One of the simplest and most widely used (although often not the best) is the inverse distance weighted algorithm. Thanks to the property of spatial dependence, we can assume that estimated elevations are more similar to nearby elevations than to distant elevations. The inverse distance weighted algorithm estimates the value z of a point P as a function of the z-values of the nearest n points. The more distant a point, the less it influences the estimate.

Figure 8.24. The inverse distance weighted interpolation procedure.

As indicated above, interpolation is used for many kinds of data beyond elevation. One example is to generate a temperature estimate from sample values at weather stations. The map below shows how 1995 average surface air temperature differed from the average temperature over a 30-year baseline period (1951-1980). The temperature anomalies are depicted for grid cells that cover 3° longitude by 2.5° latitude.

Figure 8.25. 1995 Surface Temperature Anomalies.

Credit: National Climatic Data Center, 2005.

The gridded data shown above were estimated via interpolation from the temperature records associated with the very irregular array of 3,467 locations pinpointed in the map below.

Figure 8.26. The Global Historical Climate Network.

Credit: Eischeid et al., 1995.

8.2.7 Slope

Slope is a measure of change in elevation. If you have ever ridden a bike up a hill, you have an understanding of slope. Slope is also a crucial parameter in several well-known predictive models used for environmental management, including the Universal Soil Loss Equation (that deal with soil erosion driven by water, which moves faster with steeper slope) as well as for agricultural non-point source pollution models (that deal with agricultural run-off, which is also obviously influenced by slope).

One way to express slope is as a percentage. To calculate percent slope, divide the difference between the elevations of two points by the distance between them, then multiply the quotient by 100. The difference in elevation between points is called the rise. The distance between the points is called the run. Thus, percent slope equals (rise / run) x 100.

Figure 8.27. Calculating percent slope. A rise of 100 feet over a run of 100 feet yields a 100 percent slope. A 50-foot rise over a 100-foot run yields a 50 percent slope.

Click for a text description of Figure 8.27

Percent slope for Rise=100, Run =100, 100%

= (rise/run) x 100

= (100÷100) x 100

= 100

Percent slope for Rise=50, Run =100, 50%

= (rise/run) x 100

= (50÷100) x 100

= 50

Another way to express slope is as a slope angle, or degree of slope. As shown below, if you visualize rise and run as sides of a right triangle, then the degree of slope is the angle opposite the rise. Since degree of slope is equal to the tangent of the fraction rise/run, it can be calculated as the arctangent of rise/run.

Figure 8.28. A rise of 100 feet over a run of 100 feet yields a 45° slope angle. A rise of 50 feet over a run of 100 feet yields a 26.6° slope angle.

Click for a text description of Figure 8.28

Slope degree for Rise=100, Run=100, 45°

= arctan (rise/run)

= arctan (100÷100)

= 45

Slope degree for Rise=50, Run=100, 26.6°

= arctan (rise/run)

= arctan (50÷100)

= 26.6

You can calculate slope on a contour map by analyzing the spacing of the contours (relatively, on any contour map, the slope is steepest in locations where contour lines are most closely spaced). If you have many slope values to calculate, however, you will want to automate the process. It turns out that slope calculations are much easier for gridded elevation data than for vector data, since elevations are more or less equally spaced in raster grids.

Several algorithms have been developed to calculate percent slope and degree of slope. The simplest and most common is called the neighborhood method. The neighborhood method calculates the slope at one grid point by comparing the elevations of the eight grid points that surround it.

Figure 8.29. The neighborhood algorithm estimates percent slope in cell 5 by comparing the elevations of neighboring grid cells.

Click for a text description of Figure 8.29.

Z1	Z2	Z3
Z4	Z5	Z6
Z7	Z8	Z9

The neighborhood algorithm estimates percent slope at grid cell 5 (Z₅) as the sum of the absolute values of east-west slope and north-south slope, and multiplying the sum by 100. The diagram below illustrates how east-west slope and north-south slope are calculated. Essentially, east-west slope is estimated as the difference between the sums of the elevations in the first and third columns of the 3 x 3 matrix. Similarly, north-south slope is the difference between the sums of elevations in the first and third rows (note that in each case the middle value is weighted by a factor of two).

Figure 8.30. The neighborhood algorithm for calculating percent slope.

The neighborhood algorithm calculates slope for every cell in an elevation grid by analyzing each 3 x 3 neighborhood. Percent slope can be converted to slope degree later. The result is a grid of slope values suitable for use in various soil loss and hydrologic models.

8.2.8 Relief Shading

You can see individual pixels in the zoomed image of a 7.5-minute DEM below. We used dlgv32 Pro's "Gradient Shader" to produce the image. Each pixel represents one elevation point. The pixels are shaded through 256 levels of gray. Dark pixels represent low elevations, light pixels represent high ones.

Figure 8.31. A digital elevation model in which light pixels represent high elevations and dark pixels represent low elevations.

It's also possible to assign gray values to pixels in ways that make it appear that the DEM is illuminated from above. The image below, which shows the same portion of the Bushkill DEM as the image above, illustrates the effect, which is called shaded relief (also referred to as terrain shading or hill shading).

Figure 8.32. Shaded terrain image produced from the same DEM as shown in the above figures, using dlgv32 Pro's Daylight Shader option, with the Surface Color set to gray.

The appearance of a shaded terrain image depends on several parameters, including vertical exaggeration. Click the buttons under the image below to compare the four terrain images of North America shown below, in which elevations are exaggerated 5 times, 10 times, 20 times, and 40 times respectively.

Another influential parameter for hill shading is the angle of illumination. Click the buttons to compare terrain images that have been illuminated from the northeast, southeast, southwest, and northwest. Does the terrain appear to be inverted in one or more of the images? To minimize the possibility of terrain inversion (where hills look like valleys and the reverse), it is conventional to illuminate terrain from the northwest.

8.2.9 LIDAR

For many applications, 30-meter DEMs whose vertical accuracy is measured in meters are simply not detailed enough. Greater accuracy and higher horizontal resolution can be produced by photogrammetric methods, but precise photogrammetry is often too time-consuming and expensive for extensive areas. Lidar is a digital remote sensing technique that provides an attractive alternative.

Lidar stands for LIght Detection And Ranging. Like radar (RAdio Detecting And Ranging), lidar instruments transmit and receive energy pulses, and enable distance measurement by keeping track of the time elapsed between transmission and reception. Instead of radio waves, however, lidar instruments emit laser light (laser stands for Light Amplifications by Stimulated Emission of Radiation).

Lidar instruments are typically mounted in low altitude aircraft. They emit up to 5,000 laser pulses per second, across a ground swath some 600 meters wide (about 2,000 feet). The ground surface, vegetation canopy, or other obstacles reflect the pulses, and the instrument's receiver detects some of the backscatter. Lidar mapping missions rely upon GPS to record the position of the aircraft, and upon inertial navigation instruments (gyroscopes that detect an aircraft's pitch, yaw, and roll) to keep track of the system's orientation relative to the ground surface.

In ideal conditions, lidar can produce DEMs with 15-centimeter vertical accuracy, and horizontal resolution of a few meters. Lidar has been used successfully to detect subtle changes in the thickness of the Greenland ice sheet that result in a net loss of over 50 cubic kilometers of ice annually.

Figure 8.35. Image of Greenland, viewed from the south, showing changes in ice thickness measured by airborne lidar. Ice sheet thickness decreasing at 40-60 cm per year in darker blue areas.

Credit: Goddard Space Flight Center, n.d.

To learn more about the use of lidar in mapping changes in the Greenland ice sheet, visit NASA’s Scientific Visualization Studio Greenland's Receding Ice [27].

8.2.10 Global Elevation Data

This section profiles three data products that include elevation (and, in one case, bathymetry) data for all or most of the Earth's surface.

ETOPO1

Figure 8.36. Shaded and colored terrain image produced from ETOPO1 data.

Credit: National Geophysical Data Center, 2009.

ETOPO1 is a digital elevation model that includes both topography and bathymetry for the entire world. It consists of more than 233 million elevation values which are regularly spaced at 1 minute of latitude and longitude. At the equator, the horizontal resolution of ETOPO1 is approximately 1.85 kilometers. Vertical positions are specified in meters, and there are two versions of the dataset: one with elevations at the “Ice Surface" of the Greenland and Antarctic ice sheets, and one with elevations at “Bedrock" beneath those ice sheets. Horizontal positions are specified in geographic coordinates (decimal degrees). Source data, and thus data quality, vary from region to region. You can download ETOPO1 data from the National Geophysical Data Center at the NOAA ETOPO1 site [28].

GTOPO30

Figure 8.37. Shaded and colored terrain image produced from GTOPO30 data. Data are distributed as 33 tiles.

Credit: USGS, 2006b.

GTOPO30 is a digital elevation model that extends over the world's land surfaces (but not under the oceans). GTOPO30 consists of more than 2.5 million elevation values, which are regularly spaced at 30 seconds of latitude and longitude. At the equator, the resolution of GTOPO30 is approximately 0.925 kilometers -- two times greater than ETOPO1. Vertical positions are specified to the nearest meter, and horizontal positions are specified in geographic coordinates. GTOPO30 data are distributed as tiles, most of which are 50° in latitude by 40° in longitude.

GTOPO30 tiles are available for download from USGS' EROS Data Center at the EROS GTOPO30 site [29]. GTOPO60, a resampled and untiled version of GTOPO30, is available through the USGS products and data site [30].

Shuttle Radar Topography Mission (SRTM)

From February 11 to February 22, 2000, the space shuttle Endeavor bounced radar waves off the Earth's surface, and recorded the reflected signals with two receivers spaced 60 meters apart. The mission measured the elevation of land surfaces between 60° N and 57° S latitude. The highest resolution data products created from the SRTM mission are 30 meters. Access to 30-meter SRTM data for areas outside the U.S. are restricted by the National Geospatial-Intelligence Agency, which sponsored the project along with the National Aeronautics and Space Administration (NASA). A 90-meter SRTM data product is available for free download without restriction (Maune, 2007).

Figure 8.38. Anaglyph stereo image derived from Shuttle Radar Topography Mission data.

Credit: NASA Jet Propulsion Laboratory, 2006.

The image above shows Viti Levu, the largest of the some 332 islands that comprise the Sovereign Democratic Republic of the Fiji Islands. Viti Levu's area is 10,429 square kilometers (about 4000 square miles). Nakauvadra, the rugged mountain range running from north to south, has several peaks rising above 900 meters (about 3000 feet). Mount Tomanivi, in the upper center, is the highest peak at 1324 meters (4341 feet).

Learn more about the Shuttle Radar Topography Mission at websites published by NASA [31] and USGS [32].

There are many other kinds of “surfaces” that methods discussed here are used to represent. They include the ocean depths (bathymetry), atmospheric surfaces in which the concept of a surface is more abstract than that for visible terrain to include any continuous mathematical “field” across which quantities can be measured (e.g., precipitation, atmospheric pressure, wind speed), and even conceptual surfaces such as population density. One example of the latter is this population density surface:

Population density map [33]

Here, we provide one example that is closest to those above, the representation of the surface under water bodies, bathymetry. The term bathymetry refers to the process and products of measuring the depth of water bodies. The U.S. Congress authorized the comprehensive mapping of the nation's coasts in 1807, and directed that the task be carried out by the federal government's first science agency, the Office of Coast Survey (OCS). That agency is now responsible for mapping some 3.4 million nautical square miles encompassed by the 12-mile territorial sea boundary, as well as the 200-mile Exclusive Economic Zone claimed by the U.S., a responsibility that entails regular revision of about 1,000 nautical charts. The coastal bathymetry data that appears on USGS topographic maps, like the one shown below, is typically compiled from OCS charts.

Figure 8.39. "Isobaths" (the technical term for lines of constant depth) shown on a USGS topographic map.

Early hydrographic surveys involved sampling water depths by casting overboard ropes weighted with lead and marked with depth intervals called marks and deeps. Such ropes were called leadlines for the weights that caused them to sink to the bottom. Measurements were called soundings. By the late 19th century, piano wire had replaced rope, making it possible to take soundings of thousands rather than just hundreds of fathoms (a fathom is six feet).

Figure 8.40. Seaman paying out a sounding line during a hydrographic survey of the East Coast of the U.S. in 1916.

Credit: NOAA, 2007.

Echo sounders were introduced for deepwater surveys beginning in the 1920s. Sonar (SOund NAvigation and Ranging) technologies have revolutionized oceanography in the same way that aerial photography revolutionized topographic mapping. The seafloor topography revealed by sonar and related shipborne remote sensing techniques provided evidence that supported theories about seafloor spreading and plate tectonics.

Below is an artist's conception of an oceanographic survey vessel operating two types of sonar instruments: multibeam and side scan sonar. On the left, a multibeam instrument mounted in the ship's hull calculates ocean depths by measuring the time elapsed between the sound bursts it emits and the return of echoes from the seafloor. On the right, side scan sonar instruments are mounted on both sides of a submerged "towfish" tethered to the ship. Unlike multibeam, side scan sonar measures the strength of echoes, not their timing. Instead of depth data, therefore, side scanning produces images that resemble black-and-white photographs of the sea floor.

Figure 3.41. Multibeam and side scan sonar in use for bathymetric mapping.

Credit: NOAA, 2002.

A detailed report of the recent bathymetric survey of Crater Lake, Oregon, USA, is published by the USGS at Crater Lake Bathymetry Survey [34].

This chapter introduced the various techniques for representing Earth’s continuous surfaces, both on land and underwater, often derived from a set of measured values at discrete locations. These representations included various methods for gathering, depicting, and computing elevation from both vector and raster-based strategies such as contouring, digital elevation models, interpolation, relief shading, LiDAR, and bathymetry. We manually drew TINs and contours to understand the vector-based approaches towards representing elevation. The first section discussed the history and current use of topographic maps in the USGS’s National Map. Additionally, the accuracy standards and topographic map improvements were detailed based upon the National Spatial Data Infrastructure (NSDI) Framework under the USGS. Finally, the chapter ended with an overview of other surface representations: bathymetry.

Angle of Illumination: The angle at which the illumination source is directed from. Terrain images are often illuminated from the northeast, southeast, southwest, and northwest. To minimize the possibility of terrain inversion (where hills look like valleys and the reverse), it is conventional to illuminate terrain from the northwest.

Bathymetry: The representation of the surface under water bodies.

Breaklines: Spot elevations collected along linear features.

Contours: A line along with an elevation value (number) that remains equal (the same).

Contour Interval: The interval or difference in magnitude between two sequential contour lines on a map.

Delaunay Triangulation: A circle surrounding each triangle side on a TIN surface does not contain any other vertex.

Digital Elevation Model: Any raster representation of a terrain surface. Specifically, in relation to the NSDI, a DEM is a data product of the U.S. Geological Survey. Here we consider the characteristics of DEMs produced by the USGS.

Digital Raster Graphics (DRGs): Scanned raster images of USGS 1:24,000 topographic maps.

Interpolation: The process of estimating an unknown value from neighboring known values. It is a process used to create gridded surfaces for many kinds of data, not just elevation.

Inverse Distance Weighted: An algorithm that assumes that estimated elevations are more similar to nearby elevations than to distant elevations. The algorithm estimates the value z of a point P as a function of the z-values of the nearest n points. The more distant a point, the less it influences the estimate.

Isarithm: A line that connects locations of the same value.

LIDAR: LIght Detection And Ranging. Like radar (RAdio Detecting And Ranging), lidar instruments transmit and receive energy pulses, and enable distance measurement by keeping track of the time elapsed between transmission and reception. Instead of radio waves, however, lidar instruments emit laser light (laser stands for Light Amplifications by Stimulated Emission of Radiation).

Mass Points: Spot elevations produced photogrammetrically.

Multibeam: A multibeam instrument mounted in the ship's hull calculates ocean depths by measuring the time elapsed between the sound bursts it emits and the return of echoes from the sea floor.

National Spatial Data Infrastructure (NDSI): "The technology, policies, standards and human resources necessary to acquire, process, store, distribute, and improve utilization of geospatial data" (White House, 1994). See Chapter 4.4 for more.

Neighborhood Method: A calculation of the slope at one grid point by comparing the elevations of the eight grid points that surround it (i.e., those in its neighborhood).

Quadrangles: A four-sided polygon where each 1:24,000 quad covers 7.5 minutes longitude by 7.5 minutes latitude, with its shapes and area coverage varying depending on its location on Earth.

Raster: Involves sampling attributes for a set of cells having a fixed size.

Reference Maps: Maps that help people look up facts. They are designed to serve many different purposes such as locating place names and features, estimating distances, directions, and areas, and determining preferred routes from starting points to a destination. They can also be used as base maps.

Shaded Relief: A method of assigning gray values to pixels in ways that make it appear that a DEM is illuminated from above.

Side Scan Sonar: Instruments are mounted on both sides of a submerged "towfish" tethered to the ship. Unlike multibeam, side scan sonar measures the strength of echoes, not their timing. Instead of depth data, therefore, side scanning produces images that resemble black-and-white photographs of the sea floor.

Slope: A measure of change in elevation.

Sonar: (SOund NAvigation and Ranging) Echo sounders for deepwater surveys beginning in the 1920s.

Spatial Dependence: Nearby locations are more likely to have similar elevations than are distant locations.

Tiles: Regions that correspond to the 7.5-minute topographic quadrangles. Users often download data that is bound to the region in a tile for each quadrangle.

Topography: A science that studies the surface of the Earth.

Topographic Maps: Maps that show and name many of the visible characteristics of the landscape, as well as political and administrative boundaries.

Triangulated Irregular Network (TIN): a vector representation of a continuous surface that consists entirely of triangular facets.

Vector: Involves sampling either specific point locations, point intervals along the length of linear entities, or points surrounding the perimeter of areal entities, resulting in point, line, and polygon features.

Vertical Exaggeration: The effect of elevations exaggerated several times to make the terrain more pronounced.