Assessing Raster Representation Accuracy Using a Scale Factor Model

Home , Mollweide projection, Robinson projection

Jeong Chang Seong and E. Lynn Usery

Abstract using six equal-area projections: Interrupted Goode Homolos- Raster datasets of global and continental extent are subject to ine, Interrupted Mollweide, Wagner IV, Wagner W, Lambert error resulting from projection transformation. This paper Azimuthal Equal Area, and Oblated Equal-Area projections. examines the error problem from a theoretical perspective and They quantified and graphically depicted shape and scale dis- develops a model to calculate the extent of the errors. The tortions caused by reprojection. However, because their theoretical examination indicates that error results in two research used sample grids that were drawn on already pro- forms, areal size change of pixels and categorical error re- jected maps, the application is limited since no theoretical sulting from loss or duplication of pixels. A scale factor model, background or models to estimate and simulate the pixel value based on the horizontal and vertical scale factors of the changes in various projection change situations were pre- projection, is developed to provide a computation of the sented. It is the purpose of this research to investigate the effect resulting error from specific projections. The model is of projection distortion on raster representation at a global scale experimentally tested with the cylindrical equal area, and develop a scale factor model to simulate the effect. The sinusoidal, and Mollweide projections. Results indicate that next three sections of this paper provide a theoretical approach the model predicts error within one percent of actual values to the assessment of raster representation accuracy based on a and that the sinusoidal projection is subject to smaller errors scale factor model. The fifth section develops an experimental in projecting raster data than the other projections tested. design and tests this approach using three specific projections. After the extension of the model to other types of projections is discussed, some conclusions based on this work are provided in Introduction the last section. With society's increasingly global perspective and global modeling needs, raster databases of geographic phenomena and Errors in Raster Representation and Transformation processes at continental and global scales have been con- The transformation of features at regional and global extents structed using a number of different map projections. These using the raster data structure brings two distinct problems raster databases exist at a variety of resolutions and have been which may be labeled as areal size and category. The areal-size generated from many different data sources. For example, problem results from a transformation which causes the area Steinwand (1994) coded algorithms of the Interrupted Goode represented in the raster data format to be unequal to the size of Homolosine projection for coarse-resolution global data sets. the same area represented in the vector data format unless the Tobler et al. (1995) used unprojected spherical quadrilateral pixel resolution is infinitely small. Practically, rather than grids to map global population data. Also, Lowman et al. infinity, vector and raster resoll~tionsconverge to the same rep- (1999)used the Robinson projection to map tectonic and volca- resentation at a pixel size equivalent to the smallest number nic activity around the world. The U.S. Geological Survey also which can be represented in the computer, which also corres- distributes digital raster datasets to the public to assemble ponds to the smallest possible pixel size and the smallest vector global databases of vegetation, climate, land cover, and eleva- distance which can be represented between two points. tion at resolutions of one degree, one-half degree, one kilome- The category problem is that some categories may be lost or ter, and 30-arc seconds, respectively. Dobson et al. (2000) duplicated when vector features are represented in raster for- developed the LandScan population database for the world on mat. There is no loss or gain of categories among equal-area pro- a 30-arc-sec grid. These publicly available data may be in geo- jections in the vector data structure because original features graphic coordinates or a projected coordinate system. In either will remain after projection. However, the situation is quite dif- case, users who wish to combine datasets through geographic ferent in the raster data structure. Suppose there is araster data information system (GIS) software are faced with a problem of layer with a 1-km pixel resolution in the Interrupted Goode map projection transformation and little guidance concerning Homolosine projection with 255 categories and the data are the accuracy of the projection transformation process (Cliffe, reprojected to the Albers conic equal-area projection with the 2000). same 1-km pixel resolution. Unlike the flexibility of vectorrep- Usery and Seong (2001) have demonstrated that continen- resentation, the rigidity of raster representation and the require- tal areas are subject to significant distortion if raster data are ment of nearest-neighbor interpolation for categorical data projected. Steinwand et al. (1995) identified the effects of map cause the frequency of each category to be changed, sometimes projection properties on data quality in global change studies significantly. As an example, a subset of the Eurasia dataset of

J.C. Seong is with the Department of Geography, Northern Photogrammetric Engineering & Remote Sensing Michigan University, 1401 Presque Isle Ave., Marquette, MI 49855 ([email protected]). Vol. 67, No. 10, October 2001, pp. 1185-1191. E.L. Usery is with the Department of Geography, University 0099-1112/01/6710-1185$3.00/0 of Georgia, 204 GGS Building, Athens, GA 30602 (usery@ 0 2001 American Society for Photogrammetry arches.uga.edu; [email protected]). and Remote Sensing

PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING October 2001 1185 the global land-cover characterization (GLCC)data (Eidenshink representing the original feature's size, which yields an 80 per- and Faundeen, 1994;Lauer and Eidenshink, 1998;Brown, cent representation accuracy. 1999) was reprojected from the Interrupted Goode Homolosine The error occurs when the original shape is distorted. lf a projection to the Albers equal-area conic projection and the projection changes an original shape severely, we may expect geographic latitude and longitude system. The subset covered an increased error. The number of categories and spatial auto- the latitude range of 47.1" N through N 53.7"and the longitude correlation also affect the error. If there is only one category, the range of 15.3"E through E 32.0"with 37 global ecosystem land- highest possible spatial autocorrelation, the frequency will be cover categories. The reprojection result showed 2.7 percent the same as the total number of pixels. On the contrary, if there and 35.5 percent errors when the data were reprojected to the are many categories in low spatial autocorrelation, projection Albers equal-area conic projection and geographic latitude1 may bring significant frequency changes. Error varies directly longitude system, respectively. with the spatial autocorrelation of the image; lower spatial The difference among categorical value frequencies autocorrelation yields lower accuracy and higher spatial auto- occurring at the projection stage is not easily visible because correlation yields higher accuracy. At small spatial extents, as the total area is retained in the projected image if an output pro- with local raster databases, the error will not be as significant jection is an equal-area projection. However, the error is obvi- as with global databases because of relatively small shape dis- ous when the frequency of each category is compared between tortions. In global-scale databases, significant error is expected the two images (Usery and Seong, 2001). Categorical values due to the severe shape distortion required to maintain areal may be gained or lost depending on the resulting shape of the equivalency. reprojected features and the arrangement of raster pixels. Modeling the Accuracy of Raster Representation of Equal-Area Characteristics of Categorical Errors in Raster Representation Projectsons In an equal-area projection, a feature on the globe projected to a Equivalency can be maintained by distorting shape. Specifi- flat surface should be represented with its original size. If the cally, when the multiplication of the vertical and horizontal feature's size is not represented correctly, the difference scale factors is 1.0,equivalency is maintained (Bugayevskiy et between the original size and the projected size is the extent of al., 1995; Yang et al., 2000). In most map projections, the hori- the error. zontal and vertical scale factors are represented as mathemati- In Figure 1, assume the symbol 'x' is the center of each cell. cal equations. The change of horizontal and vertical scales If a rasterization algorithm uses the feature in the vector repre- results in shape change of features. If a vector data structure is sentation occurring at the center of a pixel for representing the used, the change of scales does not affect the representation of pixel in the raster representation, features may be ignored. In categories because all features will be represented in the pro- the case of projection A, the XI,x2, ~4,and x5 features are lost. jected vector database. However, if a raster database is used, Projection B shows a feature loss of only ~5.However, features severely reduced features may be ignored because they are not also may be over-represented as in projection B, where ~2is large enough to be represented by a pixel that is much larger duplicated. Therefore, the total error extent is composed of the than the reduced feature size. Figure 2 shows the relationship pixels that are ignored and those that are over-represented. In between scale change and raster representation. Original fea- other words, the accuracy of raster representation is the size of tures are shown in distorted shape in thinner lines. New raster- correct categorical representation. Projections A and B show ized pixels are shown in thicker lines. As shown in the figure, the same number of pixels after projection, which means that the reduction or expansion along one axis is compensated by the total area is still maintained. In the case of projection A, the other axis with exactly the same reciprocal extent of scale- only one pixel is correct, which provides an accuracy of 20 per- factor change, which maintains the equivalency. For example, cent. In the case of projection B, there are four pixels correctly if a horizontal scale factor is 4.0,the corresponding vertical scale factor must be set to 0.25 in order to keep the original size of an area. Because raster representation is affected by the distortion of the original shape, the error will be dependent on the change of scale factors. Also, because the scale factors are

Representation of original pixels = 50 pixels (50%) Loss and gain = 100%(5W loss + 50% gain)

b) X scale factor = 5.0. Y scale factor = 0.2 Repmentation of original pixels = 20 pixels (20%) Loss and gain = 160%(80% loss + 80% gain) Figure 2. Scale factors and raster representation Figure 1. Effects of distortion on raster representation. accuracy. Thin lines represent original features and In Projection A, the pixel value of x3 is duplicated, but thick lines represent projected pixels in raster for- other pixel values are ignored. In Projection B, the pixel mat. One can see that pixel values are duplicated value of x2 is duplicated and that of x5 is ignored. as a function of X or Y scale factors.

1 1286 October 2001 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING reciprocal, the error can be represented as a function of the maximum scale factor at any point on the globe: i.e., where E is the error, is latitude, ,is longitude, S,, is the horizontal scale factor, and Syis the vertical scale factor. From Fig- ure 2, one can see that the error is proportional to the maximum scale factor. Assuming 100 original pixels, if the Xscale factor is 10.0 $s2 and the Y scale factor is 0.10, the number of the 100 original pixels represented correctly will be ten pixels (10 percent). This means that 90 original pixels will be ignored and 90 pixels a) Skew angle - 22.5 degrees b) Skew angle - 45 degrees will be duplicated. Therefore, the total error considering the loss and gain will be 180 percent (90 percent loss and 90 percent gain). The above results show that accuracy of raster representation is

Accuracy of raster representation = l/Sm, X 100. (2)

Also, the total error percentage that is involved in the transformation is twice the percentage of error pixels: i.e., Figure 3. The effect of skewing on raster representation. Percent of total error = 2 X (1 - l/Sm,) X 100. (3) Even though three illustrations have the same horizontal and vertical scale factors, raster representation accuracy varies When the maximum scale factor becomes S,,, the pixel as skew angle changes. One can see a perfect representa- may be represented by [l/Sm, x 1001 percent, which can be tion at the skew angle of 22.5 degrees, even though its easily identified in Figure 2. Because [l/S,,] of original fea- vertical scale factor is 2.0. tures are represented, the other features are ignored, and the percent ignored is [(I- l/Sm,) x 1001. Also, pixels are repeated S,, times, which makes [(S,, - l)/Sm, x 100 percent] duplicated pixels. Therefore, total error becomes [2 x (1 - 11 with a scale factor of 10.0. In this context, the repeat period becomes smaller than the maximum scale factor, which means S,,) X 1001 percent. Thus, one can see that the accuracy of ras- that the maximum possible error is the maximum scale factor. ter representation is equal to the percent of unduplicated origi- In this research, the concept of the maximum possible error nal pixels in total pixels. Equations 1and 2 are based on the was used. Under the maximum possible error condition, only orthogonal transformation without any rotation or skewing. the horizontal and vertical scale factors were considered for Equations 1and 2 may be tested with cylindrical projections in which no rotation is involved. Many global projec- modeling the raster representation accuracy. tions, however, frequently use straight lines along only Experimental Design and Model Testing parallels. The projections with only straight parallels make To test the scale factor model in Equations 1and 2, a raster data non-rectangular shapes after transformation. The resulting set was prepared with approximately 16- by 16-km spatial res- shapes are parabolic forms. Examples include the Mollweide, olution to cover the northeastern hemisphere of the spherical Sinusoidal, and Eckert VI projections which involve skewing of original shapes. Figure 3 shows the effect of skewing. As shown in the figure, the percent of correct pixels varies depending on skew angles even though three illustrations in the figure have the same horizontal and vertical scale factors of 0.5 and 2.0, respectively. The maximum error would occur when the angles are 0,45, and 90 degrees. Figure 3c shows error equal to the effect of no skewing. However, Figure 3a shows no pixel duplication. In this context, we can see that pixels will be duplicated to a maximum of the scale factor. The behavior of duplication can be modeled through mea- suring the repetitiveness of the center of pixels. As shown in Figure 4 (assuming the intersections represent the center of pixels), the repeat period is different depending on the angle. From the lower left corner, the vertical/horizontal distance of the repeat period is one in the case of 0,45, and 90 degrees. When all points are considered, the relationship between repeat period and angle is shown in Figure 5. The repeat period is shorter at higher skew angles. Figure 5 Figure 4. Repeat period examination. The intersections of also shows the minimum repeat period. The repeat is a binary the grid represent the center of each raster pixel. Slant lines occurrence. If the repeat period is 1.0 and maximum scale fac- indicate skew angles. At 0, 45, and 90 degrees of skewing, tor is 2.0, there will be repeat of pixels to a maximum of two. every pixel is represented, which brings the maximum repeat On the contrary, if repeat period is 2.0 and maximum scale fac- period of 1.0. In these angles, if a scale factor is 3.0, pixels tor is 1.5, there will be no repeat of pixels. Also, if the scale fac- may be duplicated three times. tor is 2.0, a less frequent repeat period is expected than the case

PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING October ZOO/ 1187 based on the location of the standard parallel. The Larnbert, 12 - Behrmann, Gall, and Edwards projections are used frequently .- + + + + ++++++ where the standard parallels are 0°, 30°, 45", and about 51°, respectively (Bugayevsky and Snyder, 1995). lo-- + + + ++ In the case of the cylindrical equal-area projection, only the .- + + + ++ ++ scale factors affect the error. The amount of correct representa- 8 -- + + ++ tion, therefore, can be calculated as the function of the largest scale factor. In the case of Lambert's cylindrical equal-area pro- B'd + + + + ++++ jection with a standard parallel at the Equator, the scale factors 6-- + ++ + are 3 + + ++ + m = COS(~) local linear scale factor along meridian (4) g 4.- + + ++ d 2. + ++ * n = sec(q5) local linear scale factor along parallel (5) 2 -- + + ++ + + +++* In this case, the vertical scale factor is the same as the local linear scale factor along a meridian. Also, the horizontal scale fac- 0 :I:;::::::::::::: tor is the same as the local linear scale factor along a parallel. 0 10 20 30 40 50 60 70 80 90 Because cos(q5) is always smaller than sec(c,b),using Equation 2, the percent of correct representation will be Skew Angles (degrees) Correct representation O/O = lIsec(q5) X 100 Figure 5. Relationship between repeat periods and = cos(q5) x 100. skew angles. Figure 6 shows the result of the experiment with 16-km hyperboloids, that the accuracy is a function of latitude, and it decreases toward the North Pole. The comparison between the globe. To create the data, horizontal strips were prepared first experimental results and the model estimations showed only along the parallels, resulting in a meridian arc length of 16krn. 0.3 percent differences on average. This indicates that the Then, the strip was divided by meridians, resulting in a paral- model, which uses horizontal and vertical scale factors, lel arc length of 16 km at the center. Therefore, each polygon is explains most errors very accurately. not exactly a square cell, but rather a square-shaped hyperboloid. There are 497,807 hyperboloids in total for a quarter of the Case 2: Slnwoidal Projection globe with an average area of 255.9997 km2.The minimum size On this projection, often referred to as the Sanson-Flemsteed, is 255.2981 km2 (98.1 percent of 256-km2grid), and the maxi- the poles are points but often truncated to appear as horizontal mum size is 256.3096 km2 (100.1 percent of 256-km2grid]. lines. The Sanson-Flemsteed sinusoidal projection has been Also, the standard deviation is 0.01. Each hyperboloid was used widely in atlases for small-scale general world maps and assigned a unique number from 1to 497,807. maps of South American and Africa (Bugayevsky and Snyder, The hyperboloids were projected to three projections in 1995). In the projection, the horizontal scale factor is always ArcView1 (ESRI, 2000): cylindrical equal-area, sinusoidal, and 1.0. It also skews original shapes severely in high latitude areas. Mollweide. The projected theme was rasterized using a raster- Considering the infinitesimal of a square on a globe, the sinus- izing function in the same software. The rasterized grid was oidal projection will distort the shape with horizontal and ver- converted into a text file in x, y, and z format . Finally, a C+ + tical scale factors of 1.0. The resulting shape will be skewed program was developed to calculate 5- by 5-degree zonal aver- and will show shapes similar to parallelograms. ages of raster representation accuracy. The zonal averages were Therefore, the maximum scale factor S,, is 1.0 in the prepared in x, y, and z format and imported into ArcView. To Sinusoidal projection. This fact means that a pixel value that model a continuous change over latitude and longitude, an has a scale factor of 1.0 cannot be repeated unless a skew angle interpolation using a triangulated irregular network (TIN) is 90 degrees. The 90-degree skew angle, however, does not method was used. Using the TIN and contouring module, iso- occur in the Sinusoidal projection. Because there will be no line maps were created. gain or loss of pixels in the Sinusoidal projection, the accuracy Based on the model in Equation 1,C+ + programs were of representation of original features will be 100 percent. developed to calculate the representation accuracy at the cen- Figure 7 shows the result of the experiment. It clearly ter of each 5- by 5-degree zone. For example, latitudes and lon- shows that the accuracy of representation is 100 percent in all gitudes of 2.5 degrees were used to calculate the model error. places. Also, the model estimation using horizontal and verti- C+ + programs were developed to calculate the model result. cal scale factors exactly coincides with the experimental result. The model results and the experimental results were compared to calculate the differences between accuracies. Finally, Case 3: Mollweide ProJection an average value was calculated using the differences. The Mollweide projection has been used frequently for world maps, especially for mapping oceans, because it represents Case i: Cylindrical Equal-Area Projection oval areas at mid-latitude regions accurately. Also, along with The cylindrical equal-area projection is the simplest form to the Sinusoidal projection for lower latitudes, the Mollweide compare the scale factor model output with actual experimen- projection composes the high latitude areas of the Interrupted tal results. There are several cylindrical equal-area projections Goode Homolosine projection. In the Mollweide projection, all of the meridians are ellipses, except the central meridian, which is a straight line, and the 90-degree meridians, which are 'Use of specific vendor software does not constitute endorsement by circles. The main characteristic of the Mollweide projection is the authors, Northern Michigan University, the University of Georgia, that the parallels are carefully spaced to maintain equivalency or the U.S. Geological Survey. so that the areal scale factor equals 1.0.

1188 October 2001 PHOTOGRAMMETRIC ENGINEERING 81 REMOTE SENSING Accuracy (I) 90 0-5 5-10 80 10-15 15-20 70 20-25 25-30 60 30 - 35 a, 35-40 40-45 e3 50 45-50 w 50-55 Q 40 55-60 J W -65 30 65 - 70 70-75 20 75-80 80-85 85-90 10 SO-$5 95-100 0 0 10 20 30 40 50 60 70 80 90 100 110 720 i3o 140 1W I60 170 180 Longitude

Figure 6. Experimental results with Lambert's cylindrical equal-area projection.

Because the Mollweide projection uses straight parallels, latitude ranges between ?40° 44', but horizontally beyond the the scale factor along the parallel can be used for examining range, which also implies that about 67 percent of the pixels raster representation accuracy. The horizontal scale factor in may be duplicated in high latitude areas. Considering the maxi- the Mollweide projection is calculated as follows (Bugayevsky mum possible errors, the minimum accuracy of raster repre- and Snyder, 1995): sentation in the Mollweide projection can be calculated using the scale factor n and its reciprocal [llnl. n = (2 -\/2/n-]cos(a] set(+] (7) Figure 8 shows experimental results of raster representation in the Mollweide projection. It shows that the accuracy is where n is the scale factor along the parallel, 2a + sin(2aI = very high around the 40-degree latitudes. Also, the accuracy is n-sin(+], and +is the latitude. Because the 2a + sin(2al = primarily a function of latitude as expected from the model. n- sin(+] is a transcendental equation, +was calculated using The average difference between the model and experimental Newton-Raphson iteration with the following equation: accuracies is 0.8 percent. This means that the model, which uses horizontal and vertical scale factors, explains more than an+,= an + [?rsin(+] - 2an - sin(2an)]/[2+ 2 cos(2an)]. 99.2 percent of errors. (81 Discussion The iteration converges rapidly if the initial guess for anis given The scale factor model used in this research was tested with the value of q5 (Pearson, 1990). three projections that have straight parallels, which are cylin- The horizontal local scale factor, n, becomes 1.0 at ?40° drical and pseudo-cylindrical projections. However, many pro- 44'. It is smaller than 1.0at the latitudes between +40° 44' and jections do not have straight parallels or straight meridians, larger than 1.0 beyond the range reaching around 3.0 at the when flat or conic projection surfaces are used, such as the poles. This means that pixels will be duplicated vertically at cases of the azimuthal equal area, Bonne, and Albers projec-

90 Accuracy (%)

80 100

a, 60 3 50 .-u C, Q 40 J 30

0 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 Longitude Figure 7. Experimental results with the Sinusoidal projection.

PHOTOGRAMMETR\C ENGINEERING & REMOTE SENSING (3 L r / 1189 Accuracy (Om) 90 0-5 5- 10 80 10 - 15 15-20 70 20-25 25-30 60 ua, 35 - 40 s50u 45-50 RJ RJ 40 1;:: 55 - 60 60-65 30 65-70 70 - 75 20 75 - 80 80-85 10 85-90 90 - 95 95 100 0 - 0 10 20 30 40 50 60 70 80 90 100 110 120130 140 150 160 170 180 I Longitude Figure 8. Experimental results with the Mollweide projection.

tions. In pseudo-cylindrical projections, it is relatively easy to difference of longitude (AA) is replaced by the deviation of the calculate the horizontal and vertical scale factors with the graticule intersection from a right angle on the map. assumption of parallelograms on hyperboloids. However, if In the case where the areal scale factor is not 1.0, which parallels and meridians are curved, the raster representation occurs with conformal, equidistant, and arbitrary projections, accuracy of the scale factor model will be decreased. the error will be increased by the extent of the areal expansion. In the cases of other cylindrical projections, such as those For those non-equal area projections, it is necessary to calcu- by Gall, Behrmann, and Edwards, mentioned above, the scale late the vertical and horizontal scale factors independently, factor model may be applied with the following scale factors because one is not the reciprocal of the other. Because equiva- (Bugayevsky and Snyder, 1995): lent area is preserved when the product of the horizontal and vertical scale factors equals one, subtracting 1.0 from the mul- Vertical scale factor (m) = COS(+)/COS[+~) tiplication of horizontal and vertical scale factors yields the extent of the areal expansion. riorizontal scale factor (n) = cos(+,l/cos(+) (10) The number of categories may also affect raster representation accuracy. This research tried to find the maximum possi- where is the latitude of the standard parallel. The horizontal ble error, or the minimum accuracy. In that context, the scale factor n, in this case, is not always larger than the vertical experimental data were labeled to unique category numbers. scale factor m. Pseudo-cylindrical equal-area projections, such This seldom happens in the real world. If there is only one cate- as the Mollweide, Sinusoidal, Eckert VI, Wagner I, Wagner IV, gory, the problem will be just the areal-size problem, so that the and Urmayev projections, represent parallels with straight categorical error becomes zero. If there is more than one cate- lines. Therefore, the horizontal scale factor can be calculated gory, the raster representation accuracy will be affected by the using the local linear scale factor along the parallel (n). Also, spatial pattern of the categories. the vertical scale factor becomes the reciprocal of n, which is Also, this research suggests that duplicated error pixels [lln].By using the local linear scale factors along the parallel, exist in current global raster databases that are built on some equal-area pseudo-cylindrical projection representation errors projections. According to the findings so far, supported with can be modeled. empirical work by Usery and Seong (2001),it is highly possible For conic equal-area projections in their normal aspects, that significant numbers of features were ignored and repeated the local linear scale factor along a meridian, m, may be used in the database, which will decrease the accuracy significantly. for approximating the vertical and horizontal scale factors Conclusions using the following equation: This research investigated the effect of projection distortion on raster representation at a global scale, and suggests a scale fac- Vertical scale factor = m X cos(AA) tor model to simulate the effect of using horizontal and vertical scale factors. Results show that the raster representation accu- Horizontal scale factor = ll[m x cos(AA)] racy is a function of those two local scale factors. When three global equal-area projections were tested-the cylindrical where m is the scale factor along the meridian, AA = [A, - A[, equal-area, Sinusoidal, and Mollweide-the differences and AA is the difference of longitude between the central between the experimental results and model results are less meridian (A,), which is parallel to the Y-axis of the raster grid, than 1.0 percent. Interestingly, the Sinusoidal projection shows and the local longitude (A). no error; all the original categories were preserved in the raster The Albers conic equal-area projection may be modeled dataset in projected form. The cylindrical equal-area projection with this approach. Also, the polar azimuthal projections, such shows decreasing accuracy toward the poles. The Mollweide as the Lambert Polar Azimuthal Equal-area projection, may be projection shows very high accuracies around the latitude of 40 modeled using similar equations. If both the parallels and degrees, and shows decreasing accuracy toward the poles and meridians are projected to cumed lines, the above equations the Equator. Considering that projections other than the Sinus- may be used with lower accuracy; however, in this case, the oidal are frequently used for coarse-resolution global data-

1190 October ZOO1 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING I bases, such as the Interrupted Goode Homolosine, Robinson, ESRI, 2000. Arcview, Version 3.2, Environmental Systems Research and even non-projected geographic coordinates,this research Institute, 380 New York Street, Redlands, California, (CD-ROM]. implies that the use of those projections for raster databases Lauer, D.T., and J.C. Eidenshink, 1998. Mapping the global land surface should be reconsidered. using 1km AVHRR data, Space Technology, 18(1-2):71-76. Lowman, P., J. O'Leary, D. Salisbury, J. Yates, P. Masuoka, and B. Mont- gomery, 1999. A digital tectonic activity map of the Earth, Journal Acknowledgments of Geoscience Education, 47(5):430-437. The authors thank anonymous reviewers for their comments Pearson, F. 11, 1990. Map Projections: Theory and Applications, CRC that have helped to improve the manuscript. This research was Press, Inc., Boca Raton, Florida, 372 p. supported by the Geographic Research Award from the U.S. Steinwand, D.R., 1994. Mapping raster imagery to the Interrupted Geological Survey, Award Number 01CRAG0004. Goode Homolosine Projection, International Journal of Remote Sensing, 15(17):3463-3471. Steinwand, D.R., J.A. Hutchinson, and J.P. Snyder, 1995. Map projec- References tions for global and continental data sets and an analysis of pixel Brown, J.F., T.R. Loveland, D.O. Ohlen, and Z. Zhu, 1999. The Global distortion caused by reprojection, Photogrammetric Engineering & Land-Cover Characteristics Database: The users' perspective, Pho- Remote Sensing, 61(12):1487-1497. togrammetric Engineering & Remote Sensing, 65(9):1069-1074. Tobler, W., U. Deichmann, J. Gottsegen, and K. Maloy, 1995. The Global Bugayevskiy, L.M., and J.P. Snyder, 1995. Map Projections: A Reference Demography Project, Technical Report TR-95-6, National Center Manual, Taylor & Francis, London, 328 p. for Geographic Information and Analysis, University of California, Cliffe, J.C., 2000. Datums and Projections for Remote Sensing, GIs, Santa Barbara, California, 69 p. (URL: ftp://ncgia.ucsb.edu/pub/ and Surveying, CRC Press, Boca Raton, Florida, 150 p. Publications/tech reportsl95195-6). Dobson, J.E., E.A. Bright, P.A. Coleman, R.C. Durfee, and B.A. Worley, Usery, E.L., and J.C. Seong, 2001. All equal area map projections are created equal, but some are more equal than others, Cartography 2000. Landscan: A global population database for estimating pop- and Geographic Information Systems, in press. ulations at risk, Photogrammetric Engineering & Remote Sens- ing, 66(7):849-857. Yang, Q., J.P. Snyder, and W.R. Tobler, 2000. Map Projection nansfor- motion, Taylor and Francis, London, 367 p. Eidenshink, J.C., and J.L. Faundeen, 1994. The 1-km AVHRR Global Land Data Set: First stages in implementation, International Jour- (Received 14 September 2000; accepted 07 February 2001; revised 25 nal of Remote Sensing, 15(20):3443-3462. April 2001)

CALL FOR PAPERS Special Issue: Characterlzlng and Modellng Landscape Dynamics

The September, 2002 issue of Photogrammetric Engineering & Remote Sensing (PEUSI will focus on Characterizing and Modeling Landscape Dynamics. The co-editors of the issue are Professor Ling Bian, Department of Geography, University of New York at Buffalo and Professor Stephen J. Walsh, Department of Geography, University of North Carolina at Chapel Hill. This issue will focus on the characterization, analysis, and modeling of landscape dynamics within the framework of geographic information science, with particular emphasis on remote sensing, geographic information systems, and spatial analysis. Landscape dynamics involve sale, pattern, and process that extend across social, biophysical, and geographical domains through their spatial and temporal interactions. Often these interactions are represented through land use and land cover dynamics. These changes, continuous or discrete, are neither random nor indepen- dent. The effective characterization and modeling of spatio-temporal dynamics of landscape can help improve our understanding of the principal drivers of landscape dynamics and the possible feedback and thresholds involving patterns and processes of scale-dependent natural and social systems. New concepts, data, and methods, emergent in geographic information science in recent years, have presented scientists with new opportunities to gain fresh insights into the study of landscape dynamics. This Call for Papers particularly encourages submission of manuscripts that give special attention to seminal approaches in using remote sensing, GIs, and spatial analysis for the characterization, analysis, and modeling of landscape dynamics. The deadline for manuscript submission is December 15, 2001. All manuscripts should be prepared according to the "Instructions to Authors" published in each issue of PE&RS and at the ASPRS web site http://www.asprs.org/ publications.html. Papers will be peer-reviewed in accordance with established ASPRS policy.

Please send manuscripts to: Professor Ling Bian Stephen J. Walsh Department of Geography, University of North Carolina, Chapel Hill, NC 27599-3220 [email protected] [email protected]

PHOTOGRAMMETRIC ENGINEERING 81 REMOTE SENSING October 2001 1191