metrics of scale in remote sensing and gis.pdf

8/11/2019 Metrics of scale in remote sensing and GIS.pdf

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Metrics of scale in remote sensing and GIS

thus having units of length. The symbol S will be usedhere to denote this measure. Unfortunately several of thestandard representations of a field (a variable conceivedas a continuous function of the spatial variables x and y)do not have suitable well-defined metrics of spatial reso-lution. A square raster (Figure IA) provides the simplestinstance, because in this representation of the spatial

variation of a field the spatial resolution is clearly thelength of a cell side, all variation within cells having beenlost. A slight complication occurs because it is impossibleto cover the Earths curved surface with square cells ofuniform dimensions, so specifications of spatial resolu-tion can be at best approximate. In remote sensing spa-tial resolutions are often termed nominal for this reason,and it is common to cite the most detailed spatial reso-lution, often the resolution of the image directly belowthe satellite, as its nominal value, even though this reso-lution is achieved over only a small part of the image. Asquare array of sample points (Figure 1 B), often the basis

for digital elevation models, also leads to a straightfor-ward interpretation of spatial resolution.

From a geostatistical perspective, spatial resolution isrelated to the ability to estimate the variogram or correl-ogram at very short distances. In principle, direct estima-tion of variance or correlation over distances less than Sis impossible since nothing is known about variation oversuch distances, and instead the variogram or correlogrammust be modeled in this region, without benefit ofempirical data. In turn, any variation estimated by krigingor simulation over such distances, for example in the

immediate vicinity of data points, is purely speculativesince it depends on the validity of the modeled spatialcorrelation function.

Much more problematic are the vector representations ofa field illustrated in Figure 1. In Figure IC, spatial varia-tion is represented by capturing the value of the field atirregularly spaced points, This approach is commonly usedin atmospheric science, where the sample points areirregularly spaced weather stations, and in soil survey,where the sample points are soil pits. The shortest dis-tance over which change is recorded is the shortest dis-

tance between neighbors, but this implies a variable spa-tial resolution, ranging from the shortest to the longestdistance between any point and its nearest neighbor.From a geostatistical perspective, estimates of the vari-ogram or correlogram at short distances comparable tothe distances between nearest neighbors may be basedon very small numbers of pairs of points. Similarly inFigure ID all variation between adjacent contours is lost,although contours place upper and lower bounds on vari-ation. In the triangulated irregular network (TIN) model(Figure IE) variation is represented as a mesh of planar,triangular facets, but the irregular size of the triangles

fails to lead to a simple metric of spatial resolution.

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JAG Volume 3 - Issue 2 - 2001

FIGURE 1 Fiveof the representations of a field commonly usedin GIS. The spatial resolution of a square raster (A) is defined bythe length of a cell side, and of a square grid (6) by the pointspacing. Spatial resolution is not obviously defined for irregu-rarly spaced sample points (C}, contours (D), or triangulatedirregular network (TIN) model E).

Measures based on the spatial structure of a representa-tion are also problematic because of the ease with

which modern geographic information systems (GIS) canresample, interpolate, and otherwise change the appar-ent spatial resolution of data. Many methods exist totake the representation shown in Figure IC and to cre-ate a square raster with a user-defined cell size, includ-ing a wide range of geostatistical methods. The appar-ent spatial resolution is now the length of the cell side.But suppose this length is less than the shortest distancebetween any pair of observations. As noted earlier, insuch circumstances the form of the variogram or correl-ogram at short distances is determined entirely by thechoice of the fitted mathematical function, since there

are no actual observations of variance or correlation at

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The scale of a process can be defined by examining theeffects of spectral components on outputs. If somewavelength s exists such that components with wave-lengths shorter than s have negligible influence on out-puts, then the process is said to have a scale of s. It fol-lows that if s is less than the spatial resolution S of theinput data, the process will not be accurately modeled.

While these conclusions have been expressed in terms ofspectra, it is also possible to interpret them in terms ofvariograms and correlograms. A low-pass filter reducesvariance over short distances, relative to variance overlong distances. Thus the short-distance part of the vari-ogram is lowered, and the short-distance part of the cor-relogram is increased. Similarly a high-pass filter increas-es variance over short distances relative to variance overlong distances.

US RATIO

While scaling ratios make sense for analog representa-tions, the representative fraction is clearly problematicfor digital representations. But spatial resolution and spa-tial extent both appear to be meaningful in both analogand digital contexts, despite the problems with spatialresolution for vector data. Both Sand L have dimensionsof length, so their ratio is dimensionless. Dimensionlessratios often play a fundamental role in science (eg, theReynolds number in hydrodynamics), so it is possible thatL/S might play a fundamental role in geographic infor-mation science. In this section I examine some instancesof the L/S ratio, and possible interpretations that provide

support for this speculation.- Todays computing industry seems to have settled on a

screen standard of order 1 megapel, or 1 million pic-ture elements. The first PCs had much coarser resolu-tions (eg, the CGA standard of the early 198Os), butimprovements in display technology led to a series ofmore and more detailed standards. Today, however,there is little evidence of pressure to improve resolu-tion further, and the industry seems to be content withan L/S ratio of order 103. Similar ratios characterizethe current digital camera industry, although profes-sional systems can be found with ratios as high as

4,000.- Remote sensing instruments use a range of spatial res-

olutions, from the 1 m of IKONOS to the 1 km ofAVHRR. Because a complete coverage of the Earthssurface at 1 m requires on the order of 1015 pixels,data are commonly handled in more manageable tiles,or approximately rectangular arrays of cells. For years,Landsat TM imagery has been tiled in arrays of approx-imately 3,000 cells x 3,000 cells, for an L/S ratio of3,000.

- The value of S for a paper map is determined by thetechnology of map-making, and techniques of symbol-

ization, and a value of 0.5 mm is not atypical. A map


sheet 1 m across thus achieves an L/S ratio of 2,000.- Finally, the human eyes S can be defined as the size

of a retinal cell, and the typical eye has order 108 reti-nal cells, implying an L/S ratio of 10,000. Interestingly,then, the screen resolution that users find generallysatisfactory corresponds approximately to the parame-ters of the human visual system; it is somewhat larger,

but the computer screen typically fills only a part of thevisual field.

These examples suggest that L/S ratios of between 103and 104 are found across a wide range of technologiesand settings, including the human eye. Two alternativeexplanations immediately suggest themselves: the nar-row range may be the result of technological and eco-nomic constraints, and thus may expand as technologyadvances and becomes cheaper; or it may be due to cog-nitive constraints, and thus is likely to persist despitetechnological change.

This tension between technological, economic, and cog-nitive constraints is well illustrated by the case of papermaps, which evolved under what from todays perspec-tive were severe technological and economic constraints.For example, there are limits to the stability of paper andto the kinds of markings that can be made by hand-heldpens. The costs of printing drop dramatically with thenumber of copies printed, because of strong economiesof scale in the printing process, so maps must satisfymany users to be economically feasible. Goodchild[ZOO01 has elaborated on these arguments. At the same

time, maps serve cognitive purposes, and must bedesigned to convey information as effectively as possible.Any aspect of map design and production can thus begiven two alternative interpretations: one, that it resultsfrom technological and economic constraints, and theother, that it results from the satisfaction of cognitiveobjectives. If the former is true, then changes in technol-ogy may lead to changes in design and production; but ifthe latter is true, changes in technology may have noimpact.

The persistent narrow range of L/S from paper maps to

digital databases to the human eye suggests an interest-ing speculation: That cognitive not technological or eco-nomic objectives confine L/S to this range. From this per-spective, L/S ratios of more than 104 have no additionalcognitive value, while L/S ratios of less than 103 are per-ceived as too coarse for most purposes. If this specula-tion is true, it leads to some useful and general conclu-sions about the design of geographic information han-dling systems. In the next section I illustrate this by exam-ining the concept of Digital Earth. For simplicity, the dis-cussion centers on the log to base 10 of the L/S ratio,denoted by log L/S and the speculation that its effective

range is between 3 and 4.

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This speculation also suggests a simple explanation forthe fact that scale is used to refer both to L and to S inenvironmental science, without hopelessly confusing thelistener. At first sight it seems counter~ntuitive that thesame term should be used for two independent proper-ties. But if the value of log L/S is effectively fixed, thenspatial resolution and extent are strongly correlated: a

coarse spatial resolution implies a large extent, and adetailed spatial resolution implies a small extent. If so,then the same term is able to satisfy both needs.

THE VISION OF DIGITAL EARTHThe term Digital Earth was coined in 1992 by U.S. VicePresident Al Gore [Gore, 19921, but it was in a speechwritten for delivery in 1998 that Gore fully elaboratedthe concept (www.d~gitalearth.gov~Pl9980131 .html):Imagine, for example, a young child going to a DigitalEarth exhibit at a local museum. After donning a head-

mounted display, she sees Earth as it appears from space.Using a data glove, she zooms in, using higher and high-er levels of resolution, to see continents, then regions,countries, cities, and finally individual houses, trees, andother natural and man-made objects. Having found anarea of the planet she is interested in exploring, shetakes the equivalent of a magic carpet ride through a 3-D visualization of the terrain.

This vision of Digital Earth (DE) is a sophisticated graphicssystem, linked to a comprehensive database containingrepresentations of many classes of phenomena. It implies

specialized hardware in the form of an immersive envi-ronment (a head-mounted display), with software capableof rendering the Earths surface at high speed, and fromany perspective. Its spatial resolution ranges down to 1 mor finer. On the face of it, then, the vision suggests datarequirements and bandwidths that are well beyondtodays capabilities. If each pixel of a 1 m resolution rep-resentation of the Earths surface was allocated an aver-age of 1 byte then a total of 1 Pb of storage would berequired; storage of multiple themes could push this totalmuch higher. In order to zoom smoothly own to 1 m itwould be necessary to store the data in a consistent data

structure that could be accessed at many levels of resolu-tion. Many data types are not obviously renderable (eg,health, demographic, and economic data), suggesting aneed for extensive research on visual representation.

The bandwidth requirements of the vision are perhapsthe most daunting problem. To send 1 Pb of data at 1Mb per second would take roughly a human life time,and over 12,000 years at 56 Kbps. Such requirementsdwarf those of speech and even full-motion video.

But these calculations assume that the DE user would

want to see the entire Earth at Im resolution. The previ-


ous analysis of log L/S suggested that for cognitive (andpossibly technological and economic) reasons userrequirements rarely stray outside the range of 3 to 4,whereas a full Earth at 1 m resolution implies a log L / S ofapproximately 7. A log L/S of 3 suggests that a userinterested in the entire Earth would be satisfied with 10km resolution; a user interested in California might

expect 1 km resolution; and a user interested in SantaBarbara County might expect 100 m resolution.Moreover, these resolutions need apply only to the cen-ter of the current field of view.

On this basis the bandwidth requirements of DE becomemuch more manageable. Assuming an average of 1 byteper pixel, a megapel image requires order 107 bps ifrefreshed once per second. Every one-unit reduction inlog L/S results in two orders of magnitude reduction inbandwidth requirements. Thus a Tl connection seemssufficient to support DE, based on reasonable expecta-

tions about compression, and reasonable refresh rates.On this basis DE appears to be feasible with todays com-munication technology.

DATA VOLUME, PARALLELISM, AND LOG USOne of the technical as distinct from cognitive reasonsfor a limit to log L/S stems from its relationship to datavolume. The number of data elements in an uncom-pressed raster (Figures IA and 1 B) is approximately (L/s)2or 102 log L/s. Little is known about the general propertiesof compression, and for most purposes compression

rates are expressed as simple percentages, unrelated tothe specific characteristics of the data. Again, little isknown about the properties of vector data that deter-mine data volume, and their relationship to spatial reso-lution and extent. If spatial resolution is constant, datavolume is expected to rise with the square of extent,such that a doubling of L results in a quadrupling of datavolume; and similarly, if extent is constant a halving of Sis expected to result in a quadrupling of data volume(although fractal arguments would suggest a more com-plex relationship). Thus it seems reasonable to assumethat the volume of all forms of geographic data is simi-

larly proportional to (L/s)z, with a constant of propor-tionality that depends on the specific representation andother properties of the data.

In recent years there has been much discussion of theso-called firehose problem created by satellite-basedsensors. The series of satellites in NASAs EOS programwill generate more than 1Tb of information per day, allof which may have to be processed and disseminatedthrough a single data system. But if log L/S is effec-tively confined to the range 3-4, an application to theentire Earths surface will be limited to 108 data ele-

ments, and a spatial resolution of approximately 4 km

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(the Earths circumference times 10-d). Of course appli-cations with greater data requirements are possible. Aglobal climate modeler might conceivably wish tomodel global atmospheric fields at resolutions of finerthan 4 km, but this is well beyond the power of currentcomputing systems.

Applications with S less than 4 km can be expected tocover less than the entire surface of the Earth. Systemsthat supply data to such applications could be organizedin parallel. For example, a single global server could dis-seminate data at spatial resolutions coarser than 4 km; anetwork of order 100 servers could disseminate datadown to spatial resolutions of 400 m; and a network oforder 10,000 servers could disseminate data down tospatial resolutions of 40 m. Arrangements such as theseare increasingly common in practice, as local govern-ments establish geographic data servers and clearing-houses for their own jurisdictions.

MAKING SCALE ACCESSIBLEThe use of conventions that are legacies from the era ofanalog paper maps creates problems for users of geo-graphic data who have little understanding of their basis,such as the general public or children. If L and S areimportant characterizations of digital geographic data,then such users should have ways of specifying them thatmake intuitive sense. For example, a user of a majorarchive of geographic data will need to specify bothparameters, in order to search for data relevant to a

given problem. With widespread access to such data pro-vided by the Internet and WWW, it is reasonable toassume that such users will include people with very lit-tle understanding of cartography or geographic informa-tion science.

Currently efforts are being made to provide intuitiveways of defining scale. A request to MapQuest(www.mapquest.com) to provide a map based on astreet address returns a map with an L of approximate-ly 2 km, while a request based on a city name returns amap with an L of approximately 50 km. Since these are

vector maps there is no obvious basis for inferring S,but pixel sizes are approximately 5 m and 120 m,respectively. Although a map is displayed, no attempt ismade to determine its representative fraction, whichwill depend on the users screen size, a parameterunknown to the server. Instead, each map shows a barof length equal to about one tenth of L labeledaccording to its length in both metric and U.S. units.Scale can be changed by clicking on a vertical bar posi-tioned next to the map, marked with 10 discrete levelsthat range in L from approximately 2 km to approxi-mately 12,000 km, with the smallest values of L at the

top (zoom in).


GlobeXplorer (www.globexplorer.com) offers both mapsand images. When a city name or street address is sup-plied, the site displays a MapQuest vector map andalongside it a raster image of the same area. Although Sis well-defined for the latter, the display offers no speci-fication of its value for either part of the display.

One possible solution to the problem of specifying S is toallow the user to identify familiar classes of objects. Forexample, a user might request sufficient detail to seethe buildings, and the system might respond by pro-viding an image with a resolution of 4 m. A request forenough detail to see the cars might result in a 40 cmresolution, whereas the streets might be translatedinto a request for a spatial resolution of 10 m. A requestfor the towns and cities might be satisfied by 1 km res-olution. Linking the scale of S to such familiar objectswould make it possible for users to provide the necessaryspecification, without understanding the more abstract

notion of spatial resolution expressed in dimensions oflength.

CONCLUDING COMMENTSt have argued that scale has many meanings, only someof which are well defined for digital data, and thereforeuseful in the digital world in which we increasingly findourselves. The practice of establishing conventions whichallow the measures of an earlier technology - the papermap - to survive in the digital world is appropriate forspecialists, but is likely to make it impossible for non-spe-

cialists to identify their needs. Instead, I suggest that twomeasures, identified here as the large measure L and thesm ll measure S, be used to characterize the scale prop-erties of geographic data.

The vector-based representations do not suggest simplebases for defining 5, because their spatial resolutions areeither variable or arbitrary. On the other hand spatialvariat;on in S makes good sense in many situations. insocial applications, it appears that the processes thatcharacterize human behavior are capable of operating atdifferent scales, depending on whether people act in the

intensive pedestrian-oriented spaces of the inner city orthe extensive car-oriented spaces of the suburbs. In envi-ronmental applications, variation in apparent spatial res-olution may be a logical sampling response to a phe-nomenon that is known to have more rapid variation insome areas than others; from a geostatistical perspectivethis might suggest a non-stationary variogram or correl-ogram (for examples of non-statjonary geostatisticalanalysis see Atkinson [ZOOI]). This may be one factor inthe spatial distribution of weather observation networks(though others, such as uneven accessibility, and unevenneed for information are also clearly important).

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Metrics of scale in remote sensing and GIS JAG Volume 3 - Issue 2 - 2001

The primary purpose of this paper has been to offer aspeculation on the significance of the dimensionless ratioL/S. The ratio is the major determinant of data volume,and consequently processing speed, in digital systems. Italso has cognitive significance because it can be definedfor the human visual system. I suggest that there are fewreasons in practice why log L/S should fall outside the

range 3 - 4, and that this provides an important basis fordesigning systems for handling geographic data. DigitalEarth was introduced as one such system. A constrainedratio also implies that L and S are strongly correlated inpractice, as suggested by the common use of the sameterm scale to refer to both.

ACKNOWLEDGMENTThe Alexandria Digital Library and its Alexandria DigitalEarth Prototype, the source of much of the inspiration forthis paper, are supported by the U.S. National Science

Foundation.

REFERENCES

Atkinson, P.M., 2001. Geographical information science:Geocomputation and non-stationarity. Progress in PhysicalGeography 25(l): 111-122.

Goodchild, M F 2000 Communicating geographic information in adigital age. Annals of the Association of American Geographers90(2): 344-355.

Goodchild, M.F. J. Proctor, 1997. Scale in a digital geographicworld. Geographical and Environmental Modelling l(1): 5-23.

Gore, A., 1992. Earth in the Balance: Ecology and the Human Spirit.

Houghton Mifflin, Boston, 407~~.Lam, N-S D. Quattrochi, 1992. On the issues of scale, resolution,

and fractal analysis in the mapping sciences. ProfessionalGeographer 44(l): 88-98.

Quattrochi D.A M.F. Goodchild (Eds), 1997. Scale in RemoteSensing and GIS. Lewis Publishers, Boca Raton, 406~~.

RESUMELe terme echelle a plusieurs significations, dont certaines survi-vent la transition de representations de Iinformation de Ianalo-gique vers le numkrique mieux que dautres. En particulier, lepremier systeme metrique dkhelle en cartographic traditionnel-le, b savoir la fraction repkentative na pas un sens bien definipour des donnkes num iques. Une dimension spatiale et une@solution spatiale sont toutes deux pleines de signification pourdes donnees numeriques et leur rapport, represent6 par le sym-bole L/S, est sans dimension. En pratique L/S parait confine dansune endue etroite. Les implications de cette observation sontexplorees dans le contexte de Terre Num ique, une visionpour un systeme dinformation geographique intkgre. II estmontre quen depit des tr grands volumes des donnkes poten-tiellement concern@es, une Terre Numkique est technique-ment rkalisable avec la technologie daujourdhui.

RESUMENEl tkmino escala tiene muchos significados, de 10s cuales algu-nos sobreviven mejor que otros la transickn entre las represen-taciones an6logas de la informacibn y las digitales.Especificamente, la metrica primdria de la escala en cartografiatraditional, la fracci6n representativa, no tiene significado bien

definido en el case de datos digitales. Extensi6n espacial y reso-luci6n espacial tienen sentido para datos digitales, y su relacidnsimbolizada coma US no tiene dimensi6n. La relacidn US pareceser confinada en la pr6ctica a un rango estrecho. Se exploran lasimplicaciones de esta observack en el context0 de TierraDigital, una visibn para un sistema integrado de informackngeogrhfica. Se muestra que, a pesar de 10s amplios vollimenesde datos potencialmente involucrados, Tierra Digital es tecnica-mente factible con la tecnologia actual.

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