Full text of DOI:10.1111/0033-0124.00327

The Sinusoidal Projection:A New Importance in Relation
to Global Image Data*
Jeong Chang Seong
Northern Michigan University
Karen A. Mulcahy
East Carolina University
E. Lynn Usery
University of Georgia
The reprojection of image data causes the loss or duplication of original pixel values. This research investigated the
feasibility of using the sinusoidal projection for global image database construction. Specifically, reprojection accura-
cies were tested with geographic latitude and longitude coordinates, and the Hammer-Aitoff, Eckert IV, Mollweide,
and sinusoidal projections. Reprojections between these five global projections and the Universal Transverse Merca-
tor (UTM) projection and referencing system were performed using fifty-four sample datasets. A statistical analysis
of categorical accuracy, a measure describing the omission of pixel values during reprojection, was conducted. Geo-
graphic coordinates and the sinusoidal projection both showed very high accuracy rates (100.0 percent and 99.5 per-
cent respectively) when sample datasets were reprojected from UTM. The geographic coordinates, however, showed
very low accuracy (65.3 percent) when sample datasets were reprojected to the UTM projection, while the sinusoidal
projection showed the highest accuracy (98.4 percent). The results strongly suggest that the sinusoidal projection is
very accurate and efficient for building global image databases. Key Words: GIS, global image data, scale factor
model, sinusoidal projection.
new digital environment (Yang, Snyder, and
Tobler 2000; Seong and Usery 2001).
Introduction
s more and more image data are available at
One challenge is the raster representation of
A
global scales, the best representation of features distorted due to the introduction of
the globe in raster format, specifically for ar- map projections (Mulcahy 1999, 2000; Seong
chiving satellite imagery and developing global and Usery 2001; Usery and Seong 2001). Tradi-
image maps, has become an important issue for tionally, map projections have been presented
researchers from various scientific fields (Stein- in terms of the map properties of equivalency,
wand, Hutchinson, and Snyder 1995; Mulcahy equidistance, equidirection, and conformality.
1999, 2000; Seong 1999; Seong and Usery Recent research (Seong and Usery 2001) sug-
2001; Usery and Seong 2001). The representa- gests that categorical accuracy must also be
tion of the ellipsoidal globe on a flat surface has considered along with the other projection is-
been an enduring and significant topic in car- sues when using a raster representation for
tography, geography, surveying, and geodesy. A geospatial data. Categorical accuracy indicates
long history of endeavor for best representing that categorical pixel values in an image dataset
the globe has produced numerous map projec- can be significantly duplicated or omitted due
tions tailored to suit various map compilation to map projection transformations.
purposes as well as the size, shape, and location
Categorical accuracy can be defined as the
of area of interest. The rapidly changing carto- proportion of the number of pixel values repre-
graphic environment, however, challenges re- senting original features correctly among the
searchers in cartography, geographic informa- entire set of pixels.1 If the pixel resolution in a
tion systems (GIS), and remote sensing to target image was to be dictated by the change
rediscover the role of map projections in the of scale factors due to map projection transfor-
* The authors thank Joel Sabin, Carey Crankshaw, Dr. Stephen Degoosh, and anonymous reviewers for their help. The first and last authors were
supported by a grant from the U.S. Geological Survey (Geographic Research Award–01GRA0004). All opinions and any errors remain the
responsibility of the authors.
The Professional Geographer, 54(2) 2002, pages 218–225 © Copyright 2002 by Association of American Geographers.
Initial submission, August 2001; revised submission, November 2001; final acceptance, December 2001.
Published by Blackwell Publishing, 350 Main Street, Malden, MA 02148, and 108 Cowley Road, Oxford, OX4 1JF, UK.

Sinusoidal Projection: A New Importance in Relation to Global Image Data
219
mation in order to represent the most reduced a continuous dataset, but in the resolution (1-
feature accurately, the categorical accuracy km pixels) and size (10 by 10 pixels) of the 723
problem would not occur. However, changing study areas examined, no change occurred.
the output pixel resolution dynamically to rep-
Does all this mean that the sinusoidal projec-
resent the most-reduced feature is hardly prac- tion is superior to other projections for build-
tical, especially when a study area covers the ing global image databases? To answer the
entire globe or at least a continent. Recent re- question, it is necessary to test various re-
search by Mulcahy (1999, 2000) and Seong and projection situations. This research, specifi-
Usery (2001) examined categorical accuracy in cally, will test the categorical accuracy of re-
many projections.
projections from the sinusoidal projection, and
Seong and Usery (2001) found that the sinu- of those to the sinusoidal projection. Also, four
soidal projection represents the original cate- other global map projections will be tested for
gorical values almost perfectly. Specifically, comparison with the sinusoidal projection.
they found that the minimum categorical accu- After examining the sinusoidal projection in
racy is a function of horizontal and vertical the next section, testing methods, results and
scale factors [categorical accuracy ꢀ 1/MAX discussion, and conclusions will follow.
(horizontal scale factor, vertical scale factor)].
They also found that spatial autocorrelation,
The Sinusoidal Projection
fewer categories, and skewing effects would in-
crease the categorical accuracy.
Projections in which parallels are straight have
The very high categorical accuracy of the si- been used since classical Greek times (Snyder
nusoidal projection occurs because the vertical 1987). Jean Cossin of Dieppe made the first re-
and horizontal scale factors are always 1.0 ported use of the sinusoidal projection for a
(Seong and Usery 2001). The fact that the hor- world map in 1570. The origin of the projec-
izontal scale factor along parallels is always 1.0 tion has been assigned to a variety of inventors
in the sinusoidal projection (which means no with various names. Later uses of the projec-
scale distortion) provides a very interesting tion led to the acquisition of yet more names.
characteristic. In equal-area projections, the Due to its use in editions of Mercator’s atlases
multiplication of horizontal and vertical scale from 1606 to 1609, the projection is sometimes
factors should be 1.0 to meet the equivalency referred to as the Mercator equal-area (Deetz
condition. Because the sinusoidal projection is and Adams 1938). The appellation of Sanson-
an equal-area projection with horizontal paral- Flamsteed developed from separate uses by
lels, the vertical scale factor must also be 1.0 in Frenchman Nicolas Sanson d’Abbeville (1600–
order to meet the equivalency condition.2 Even 67) for maps of continents and by Englishman
though original features are skewed severely at John Flamsteed (1646–1719) for astronomical
high latitude areas, the horizontal and vertical maps (Snyder 1993).
scale factors always become 1.0. This suggests
Recently, the sinusoidal projection has been
that the raster representation of a distorted fea- used for archiving global satellite imagery such
ture in the sinusoidal projection will be 100 as Moderate Resolution Imaging Spectroradi-
percent accurate because there is no change of ometer (MODIS) products (USGS EROS
vertical and horizontal scale factors, even Data Center 2000), Advanced Very High Reso-
though the scale factors along meridians are lution Radiometer (AVHRR) Ocean Path-
not 1.0 and change dramatically.
finder products, and the International Satellite
Mulcahy (1999, 2000) describes the distor- Cloud Climatology Products (Campbell, Blais-
tion introduced by projection of discrete raster dell, and Darzi 1995). Although this is referred
datasets in terms of pixel loss and pixel duplica- to as the integerized sinusoidal projection
tion as separate measures. She examined sev- (ISIN) mapping grid, mathematical solutions
eral equal-area world map projections and de- in it are the same as the traditional sinusoidal
termined that the sinusoidal projection was projection. The primary distinction between
unique among those studied. It was the only the two is that the ISIN is a data-storage (or
projection that did not exhibit any pixel loss or binning) scheme designed for a discrete raster
duplication for the study areas. This research data structure, and not a specific map projec-
cannot conclude that no change would occur in tion. Making horizontal strips along parallels

Volume 54, Number 2, May 2002
220
with a specific vertical pixel size creates the beyond ꢁ40ꢂ 44ꢃ and the sinusoidal projection
ISIN mapping grid. The pixel size is approxi- within ꢁ40ꢂ 44ꢃ. The two projections match
mately 9.28 km in the case of the sea-viewing perfectly at the latitudes ꢁ40ꢂ 44ꢃ because their
wide-field-of-view sensor (SeaWiFS) (Camp- horizontal scale factors are the same at that lat-
bell, Blaisdell, and Darzi), and one kilometer in itude. The sinusoidal projection is not used
the case of the MODIS products (USGS near the poles due to extensive angular distor-
EROS Data Center 2000). The number of pix- tion of features.
els in each strip varies by latitude, with the
The construction of the sinusoidal projec-
greatest number in the rows immediately above tion combines important qualities with an ex-
and below the equator. The number of pixels in tremely simple method (Figure 1). The projec-
each strip is determined by dividing the length tion is equal-area and the “scale along the
of parallel by the output of the division of the central meridian and along every parallel of lat-
length of parallel by the vertical bin size. The itude is correct. Parallels of latitude are shown
number of pixels per row decreases away from straight, parallel, and equidistant” (Snyder
the equator by approximately a cosine func- 1993, 50). To manually construct the map, the
tion. In effect, the ISIN takes on the form of parallels of latitude are laid down, the central
the sinusoidal projection, without rigorously meridian is drawn perpendicular to it, and me-
maintaining the property of equal pixel sizes. ridians are marked off at their true distances
To maintain an integer number of pixels per along each parallel. These points are then con-
row, the x-length of each pixel is adjusted to be nected by sine curves (Snyder 1993). The
either larger or smaller depending on strips. mathematic construction of the projection is
The poles are represented by triplets of pie- nearly as simple where the coordinates (x, y) of
shaped wedges (Campbell, Blaisdell, and Darzi). a point are obtained by
The sinusoidal projection is also used for ar-
x ꢀ R(ꢄ ꢅ ꢄo) cos ꢆ
chiving AVHRR datasets for the world, be-
y ꢀ R ꢆ
(1)
cause the sinusoidal projection is a component
part of the interrupted Goode Homolosine
projection used for referencing those data
(Steinwand, Hutchinson, and Snyder 1995).
The interrupted Goode Homolosine projec-
tion uses the Mollweide projection at latitudes
where R equals the radius of the sphere, ꢄ
equals longitude east of Greenwich, ꢄo equals
longitude east of Greenwich of the central me-
ridian of the map, and ꢆ equals north geo-
graphic latitude (Snyder 1987).
Figure 1 The sinusoidal projection, with the equator and the prime meridian as the standard parallel and
central meridian respectively.

Sinusoidal Projection: A New Importance in Relation to Global Image Data
221
Fifty-four sample areas were randomly se-
lected from the northeast quarter of the globe
(Figure 2). At each location, a raster dataset was
prepared that had 100-by-100 pixels with
unique numbers in 30-meter pixel resolution.
After projecting the datasets between global
projections and the UTM, the ratio of unique
pixel numbers to the original pixel numbers
(10,000) was calculated as the categorical accu-
racy. The reprojection was performed using
ERDAS ImagineTM (ERDAS 2001). The near-
est neighbor resampling method was used with
a rigorous transformation method that does
not use polynomial approximations, but, in-
stead, uses true transformation equations.
Methodology
Two situations were tested in this research: the
reprojection of sample datasets from an accu-
rate local projection to global projections in-
cluding the sinusoidal, and the reprojection of
sample datasets from global projections to an
accurate local projection. The Universal Trans-
verse Mercator (UTM) projection system3 was
selected as an accurate local projection because
scale factor changes less than 0.0004 within
each zone of the system. The small-scale factor
change produces the maximum categorical er-
ror of up to 0.08 percent (ꢀ 1.0000 ꢅ 1.00042).
The categorical accuracy during reprojection
between global projections and the UTM,
therefore, will be primarily a result of the dis-
tortion inherent in the global projections, and
not the UTM system.
Results and Discussion
Four more global projections were tested to When it is necessary to reproject a very accu-
make comparison with the sinusoidal projec- rate local image dataset to a global projection,
tion: Mollweide, Hammer-Aitoff, Eckert IV, the geographic coordinate system can be a via-
and geographic latitude and longitude. All ex- ble option. In this research, the geographic co-
cept the geographic are equal-area projections. ordinate system showed a categorical accuracy
The geographic latitude and longitude coordi- of 100.0 percent (Table 1). This is possible be-
nate system was tested because a similar con- cause there is no reduction of scale factors
cept has been used for archiving global data when a local dataset is reprojected to the geo-
such as the global 30 arc-second elevation graphic coordinate system. In the geographic
dataset (GTOPO30) from the U.S. Geological coordinate system, vertical scale factors are al-
Survey and digital terrain elevation data ways 1.0, and horizontal scale factors along
(DTED) from the National Imagery and Map- parallels increase toward the poles. These
ping Agency (NIMA).
scale-factor changes will bring only pixel dupli-
Figure 2 The location of fifty-four sample datasets.

Volume 54, Number 2, May 2002
222
Table 1 Categorical Accuracy during Reprojection between UTM and Global Projections
Original
Projection
Target
Projection
Minimum
Accuracy (%)
Maximum
Accuracy (%)
Mean
Accuracy (%)
Standard
Deviation
UTM
UTM
UTM
UTM
sinusoidal
Mollweide
Eckert IV
Hammer-Aitoff
geographic
UTM
UTM
UTM
UTM
UTM
98.5
48.2
19.7
53.2
100.0
90.5
68.0
29.9
59.9
9.2
99.9
99.6
99.8
100.0
100.0
100.0
99.8
99.3
100.0
99.9
99.5
85.8
76.9
87.0
100.0
98.4
89.5
85.2
87.2
65.3
0.4
13.5
22.0
10.9
0.0
2.2
7.4
11.9
10.3
29.3
UTM
sinusoidal
Mollweide
Eckert IV
Hammer-Aitoff
geographic
cation along parallels without any pixel loss, re- olations during the resampling process. Table 1
sulting in 100.0 percent categorical accuracy shows what happens when datasets in the geo-
(see Table 1). However, the duplication of pixel graphic coordinate system are reprojected to
values is a disadvantage of using the projection. the UTM with a pixel size of 30 m, which is the
For example, at the latitudes of 41 and 80 de- same as the size of an edge of a grid cell around
grees, 32 percent and 600 percent of pixel du- the equator.
plications were observed, respectively, result-
Figure 3 shows the latitudinal pattern of cat-
ing in significant inefficiency for storing global egorical accuracy. In the case of the reprojec-
image data. The sinusoidal projection avoids tions from the UTM to Eckert IV, Mollweide,
the pixel duplication problem and showed a and Hammer-Aitoff projections, categorical
very high categorical accuracy of 98.5 percent accuracies decrease as latitude increases. How-
on average. The Mollweide, Eckert IV, and ever, categorical accuracies are very high and
Hammer-Aitoff projections showed significant stable when UTM data are reprojected to the
loss of original pixel values. The Eckert IV pro- geographic coordinates and the sinusoidal pro-
jection showed the worst accuracy, with a mean jection. Also shown is the latitudinal decrease
of 76.9 percent and a minimum of 19.7 percent. of categorical accuracy when image samples
When converting datasets from global pro- were reprojected from global projections to the
jections to the UTM, the sinusoidal projection UTM. Dramatic decrease of accuracy is shown
showed the highest categorical accuracy of 98.4 at the geographic coordinate system.
percent. The geographic coordinate system
As shown in Table 1 and Figure 3, the sinu-
showed the worst mean accuracy of 65.3 per- soidal projection is significantly better than the
cent, with a minimum of 9.2 percent. In the other projections. The results strongly suggest
case of the geographic coordinate system, the that the sinusoidal projection should be used
meaning of accuracy should be interpreted for archiving global data. A better accuracy
carefully depending on what pixel resolution is than the Mollweide projection also indicates
used for the target image. For instance, sup- that the sinusoidal projection is better than the
pose there is a dataset in a three-arc-second interrupted Goode Homolosine projection,
grid like DTED-1 (Dobson 2001). A grid in which uses both the Mollweide and the sinu-
the dataset around the equator will cover an soidal projections as component parts. The se-
area of approximately 92.7 m by 92.7 m in the lection of the ISIN mapping grid for NASA’s
spherical globe model, with a radius of recent MODIS products is very reasonable.
6,371,000 m. However, the area of a grid will However, the ISIN has varying pixel sizes, which
decrease as latitude increases. In this case, the makes the calculation of secondary information
dataset can be reprojected with a pixel size of from the projection very difficult. Because the
92.7 m or less. Which resolution is to be se- shape of the end of each horizontal strip in the
lected? If the target image does not cover the integerized sinusoidal projection is not vertical
entire globe, a smaller pixel resolution may be but skewed, the advantage of integerization
selected. However, if the target image covers may not be significantly larger than using the
the entire globe, a 92.7 m pixel resolution regular resampling algorithm with the same
would be the most appropriate to avoid extrap- pixel sizes.

Sinusoidal Projection: A New Importance in Relation to Global Image Data
223
ellite imagery such as Landsat Thematic Map-
per data. This approach is very convenient to
data providers because they do not need to
consider the loss of pixel values. The scale fac-
tors of datasets of small geographic extent with
appropriate projection referencing are usually
very close to 1.0. However, if an image patch
covers a large area, local map projections are
not an efficient scheme for archiving data. For
example, the UTM projection is not appropri-
ate for the AVHRR data. Instead, a projection
should be selected that provides horizontal and
vertical scale factors that are always greater
than or equal to 1.0 in order to avoid the loss of
pixel values. If scale factors are less than 1.0,
pixel values will be lost. Because projections
such as geographic, sinusoidal, Mercator, and
equidistant or Plate Carrée have horizontal and
vertical scale factors that are close to 1.0 or
greater than 1.0, image database constructions
with the projections will bring little or no
pixel loss.
When a scale factor is greater than 1.0, pixel
values are duplicated (Seong and Usery 2001).
If a global dataset is allowed to have the dupli-
cated pixel values, the geographic latitude and
longitude coordinate system will be one of the
best choices because there will be no loss of
pixel values. In this case, the horizontal and
vertical scale factors are always greater than or
equal to 1.0. However, the efficiency of storing
data decreases toward the poles, because pixels
are duplicated severely. The efficiency of stor-
ing data cannot be overlooked, because global
datasets are voluminous. Furthermore, the der-
ivation of secondary information such as area is
very difficult with duplicated and exaggerated
datasets. Also, the reprojection of duplicated
datasets should be performed very carefully. If
Figure 3 Latitudinal change of categorical ac-
curacy (A) from UTM to global projections, and (B)
from global projections to UTM.
These results do not necessarily mean that all pixel values need to be preserved in the tar-
all global datasets should be archived using the get image, the minimum actual pixel size
sinusoidal projection. The selection of a map should be used as the resolution of the target
projection for global data must consider many image, which may lead to very large, impracti-
criteria, such as the need for merging smaller cal datasets.
image patches into larger regions, the permissi-
bility of pixel duplication, and the objective of cal accuracy, the user objectives of the global
creating global images.
image data should be considered during the se-
In addition to the consideration of categori-
If there is no need to composite smaller im- lection of a map projection. Depending on the
age patches to make a regional or global dataset, user requirements, map projection properties
it would be best to reference the smaller-area such as equidistance, equal angle, equal area, or
images with local map projections such as equal shape will guide the choice of a projection.
UTM that have very little distortion. This ap- The content of the global image datasets—such
proach is already used for high-resolution sat- as hydrologic, seismic, administrative, histori-

Volume 54, Number 2, May 2002
224
along the meridian. In the case of the sinusoidal
projection, n is always 1.0, and m equals sec(ε)
(Bugayevskiy and Snyder 1995, 67). The vertical
scale factor of the sinusoidal projection, therefore,
becomes [sec(ε)ꢇcos(ε)], which is 1.0. This research
focuses on the horizontal and vertical scale factors.
3 The U.S. Army developed the UTM system in 1947
to provide a global system of rectangular coordi-
nates for large-scale topographic military mapping.
The UTM consists of a series of sixty zones, each six
degrees wide in longitude between latitudes 84ꢂN
and 80ꢂS, and two more polar zones. The transverse
Mercator projection is applied to each zone and a
false easting and northing is applied to each zone.
Proceeding east from the 180th meridian, the zones
are numbered from one to sixty (Snyder 1987).
cal, demographic, economic, and navigational
features—should be considered as well. As a
global projection, the sinusoidal projection has
been used for equal-area maps and for regional
maps of Africa and South America.
Conclusions
The sinusoidal projection shows neither pixel
duplication nor pixel loss, so its use is highly
recommended for archiving global image data,
particularly when equivalency is a requirement.
In this research, two reprojection situations
were tested with fifty-four sample datasets: the
first from the UTM to the global projections,
and the second from the global projections to
the UTM. Results indicate that the sinusoidal
projection is the most efficient for building
global datasets, with categorical accuracies of
99.5 percent and 98.4 percent for the UTM-
to-sinusoidal and sinusoidal-to-UTM re-
projections, respectively. The reprojection
from the UTM to the geographic coordinate
system showed 100.0 percent accuracy. How-
ever, the duplication of pixel values was signifi-
cant, and the accuracy of the reprojection back
to the UTM was low. Results suggest that the
sinusoidal projection not only is appropriate
for building global databases with a require-
ment for area equivalency but also is suitable
for retaining original pixel values. If a global-
image dataset were to be published covering
the entire world, using the sinusoidal projec-
tion would bring together very high categorical
accuracy and efficiency. ꢀ
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Notes
1 Categorical accuracy can also be defined as [1 ꢅ
omission ratio]. For the selection of the best projec-
tion for global image datasets, the duplication ratio
must be considered along with the categorical accu-
racy. The Results and Discussion section of this ar-
ticle discusses the duplication issue.
2 In the case of a sphere, the condition for equal area
transformation is [mꢇnꢇcos(ε) ꢀ 1.0], where m is the
linear scale factor along the meridian, n is the linear
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the map (Bugayevskiy and Snyder 1995, 22). Be-
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[mꢇcos(ε)] produces the vertical scale factor. Here,
the vertical scale factor means the scale factor along
the right angle on the map, not the scale factor

Sinusoidal Projection: A New Importance in Relation to Global Image Data
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clude global GIS database construction, map projec-
tions, GIS and society, spatial analysis, and various
aspects of GIS and remote sensing theories.
KAREN A. MULCAHY is an Assistant Professor in
the Department of Geography at East Carolina Uni-
versity, Greenville, NC 27858. E-mail: mulcahyk@
mail.ecu.edu. Her research interests include com-
puter and analytical cartography, GIScience, map pro-
jections, spatial data quality, and Web cartography.
U.S. Geological Survey (USGS) EROS Data Center.
2000. Data projection. http://edcdaac.usgs.gov/
modis/faq_projection.html (last accessed 24 Janu-
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projection transformation. London: Taylor and
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E. LYNN USERY is an Associate Professor in the
Department of Geography at the University of
Georgia, Athens, GA 30602, and a research geogra-
pher at the Mid-Continent Mapping Center at the
United States Geological Survey, Rolla, MO 65401.
E-mail: usery@uga.edu or usery@usgs.gov. His re-
search interests include feature-based approaches to
GIS, three-dimensional visualization, geographic
feature representation, watershed management and
water quality, and the applications of GIS, remote
sensing, and GPS to precision agriculture.
JEONG CHANG SEONG is an Assistant Professor
in the Department of Geography at Northern Mich-
igan University, Marquette, MI 49855. E-mail:
jseong@nmu.edu. His current research interests in-