MAPPING RASTER IMAGERY TO THE INTERRUPTED GOODE HOMOLOSINE PROJECTION

D.R. Steinwand, Hughes STX Corporation*, Sensor Systems, EROS Data Center, Sioux Falls, South Dakota 57198, U.S.A.

Abstract. Because of the increasing emphasis on global monitoring, processing remotely sensed raster image data onto global map projections has become an important issue. One class of map projections, interrupted equal-area projections, is especially useful for this purpose. The use of the Interrupted Goode Homolosine map projection for the Global Land Advanced Very High Resolution Radiometer (AVHRR) l-km project and the AVHRR Pathfinder project has produced a map that is both attractive to the viewer and useful for data analysis. This interrupted, composite, equal-area map projection uses the Sinusoidal projection for low latitudes and the Mollweide projection for high latitudes and is broken into 12 regions, each with its own central meridian. This combination of projections helps to preserve the shape of the land masses and results in a map that has less distortion than an uninterrupted global map. The use of the interrupted projection also simplifies the processing and management of larger data sets, because the data can be processed either separately in the component projections that make up the interrupted projection or directly onto the interrupted projection.

* Work performed under U.S. Geological Survey contract 1434-92-C-40004

1. Introduction

Because of the increasing emphasis on global monitoring, processing remotely sensed raster data to global projections has become an important issue. One class of projections, interrupted equal-area projections, is especially useful for this purpose, although special processing of raster imagery is required because of the discontinuities in the interrupted areas of the map. Several equal-area map projections have been suggested for global and continental studies (Steinwand and others), including the Interrupted Goode Homolosine, the Interrupted Mollweide, the Wagner IV, and the Wagner VII for maps of the world and the Lambert Azimuthal Equal-Area for maps of hemispheres and continents. The Interrupted Goode Homolosine, however, is fairly common, has visual appeal, and has favorable distortion characteristics for global land studies.

The Interrupted Goode Homolosine map projection was chosen for the Global Land Advanced Very High Resolution Radiometer (AVHRR) 1-km Data Set project (Townshend 1992) and for the AVHRR Pathfinder project (see article in this issue). The characteristics of the Goode projection and two methods for projecting raster image data to the Goode projection, interrupted in the standard places to show land masses, are presented. The first method maps the imagery in the component parts that make up the interrupted projection, and the second method maps the imagery directly onto the interrupted projection.

2. The Interrupted Goode Homolosine Projection

The Interrupted Goode Homolosine projection, developed by J.P. Goode in 1923, is an equal-area pseudocylindrical composite map projection which is interrupted to reduce distortion in the major land masses. This projection merges the Mollweide projection for higher latitudes (also called the Homolographic projection) and the Sinusoidal projection for lower latitudes (Goode 1925). The two projections join at 40 44'11.8" North and South; this is where the linear scale of the two projections match. The projection used in each region, the central meridians, and the geographic extents are given in table 1. All the major continents, with the exception of Antarctica, are intact. The projection is often presented with repeated sections so that Greenland and the eastern tip of Siberia are not interrupted.

In both the uninterrupted Sinusoidal projection and the uninterrupted Mollweide projection, the areas of maximum distortion are symmetrically positioned in the outer portions of the projections' four quadrants. In the Interrupted Goode Homolosine, these regions of higher distortion are avoided by using the central portions of these projections. In addition, combining the Sinusoidal projection, which presents shapes more accurately in the equatorial regions, with the Mollweide projection, which presents shapes more accurately in the higher latitudes, further minimizes distortion (Robinson and others 1988). Figure 1 (8,803 bytes) shows an ellipse of distortion plot for the Interrupted Goode Homolosine projection. The ellipse of distortion is used graphically to depict distortion on map projections by showing what happens to an infinitesimally small circle when it is projected from the spherical Earth to a flat map.

In relation to data processing, the advantage of using this projection is the manner in which the geometric processing can be broken up, based on the characteristics of the projection. The projection can be treated as a single projection or split into the 12 regions given in table 1. Processing regions as separate maps on different computer systems or at different times lessens the amount of disk space required. Another data processing consideration is that of presentation data versus analysis data. The minimized distortion and equal-area characteristics of the Goode projection make it a good choice for analysis purposes, and also lend to its visual appeal. This is important because the sheer size of the data sets in the Global Land AVHRR 1-km project and the large number of data sets in the AVHRR Pathfinder project make reprojection for display purposes a complex and resource-consuming task.

The two map projections upon which the Interrupted Goode Homolosine is based are fairly common. The Sinusoidal projection is supported in the General Cartographic Transformation Package (GCTP), and both the Sinusoidal and the Mollweide are supported by the ARC/INFO geographic information system. In addition, the U.S. Geological Survey's EROS Data Center has developed a C language version of the GCTP that includes the Sinusoidal, Mollweide, and Interrupted Goode Homolosine projections. A software utility that converts between image coordinates and geographic coordinates in either the Global Land 1-km AVHRR data set or the AVHRR Pathfinder data set has been developed by this author. This software is available to users of either data set to aid in the use of these data.

3. Method 1: Mapping in Parts--the Global Land 1-km AVHRR Data Set Project

At 1-km resolution, a raster image of global extent in the Interrupted Goode Homolosine map projection consists of 17,347 lines of imagery with each line containing 40,031 pixels. In addition, sections are repeated to prevent Greenland and Siberia from being interrupted on the map. With 5 bands of two-byte integer data and 5 bands of derived data in the one-byte data type, each 1-km data set is just under 10 gigabytes.

To process efficiently this large amount of data, it is necessary (as is common in other image geometric rectification systems) to interpolate geometric locations between points that were determined with a satellite geometry model. Such a transformation is referred to as grid-based mapping, and a one-to-one mapping is assumed. The mapping between satellite space and an interrupted projection with repeated regions is not, however, one-to-one everywhere. Also, the size of the image makes computation difficult, requiring vast amounts of disk space and computer resources.

The interrupted nature of the projection can be used to overcome these limitations. The resulting image, projected to the Interrupted Goode Homolosine map projection, can be processed region by region, which results in a one-to-one mapping, even with repeated areas. The regional pieces can be computed with standard AVHRR geometric processing systems in the Sinusoidal and Mollweide projections. A rectangular image results for each of the 12 regions (Figure 2) (44,262 bytes). The number of lines in each region is about 5,000, with the number of pixels per line varying depending upon region. Table 1 lists image sizes and projection parameters necessary for processing. Another benefit of this method is that the regions need not be processed on the same computer or at the same time. This allows for flexible scheduling of computers and personnel.




Table 1.  Interrupted Goode Homolosine Projection Parameters







   Radius of Sphere:       6370997 meters



   Total Number of Lines:    17347



   Total Number of Samples:  40031







   Top of Image in Goode's meters:  8673000 meters  (Maximum Y)



   Bottom:                         -8673000 meters  (Minimum Y)



   Left Side:                     -20015000 meters  (Minimum X)



   Right Side:                     20015000 meters  (Maximum X)











         LATITUDE        LONGITUDE                   CENTRAL    GOODEs WINDOW        LOCAL PROJECTION (Y,X)    GOODE's PROJECTION (Y,X)



REGION    RANGE            RANGE        PROJECTION  MERIDIAN    (SL,SS,NL,NS)           Upper Left                Upper Left



====== ==============   ============    ==========  ========  ==================     =======================   ========================







  1    90N to 40 44'N   180W to 40W     Mollweide    100W    1,2156,4144,12572       9009410.83  -6740512.57   8673000.00  -17860000.00







  2    90N to 40 44'N   40W to 180E     Mollweide     30E    1,17454,4144,18815      9009410.83  -5897846.23   8673000.00   -2562000.00







  3    40 44'N to 0     180W to  40W    Sinusoidal   100W    4145,1,4530,15568       4529000.00  -8895512.57   4529000.00  -20015000.00







  4    40 44'N to 0     40W to 180E     Sinusoidal    30E    4145,15569,4530,24463   4529000.00  -7782846.23   4529000.00   -4447000.00







  5    0 to 40 44'S     180W to 100W    Sinusoidal   160W    8675,1,4529,8897          -1000.00  -2223820.11     -1000.00  -20015000.00







  6    0 to 40 44'S     100W to  20W    Sinusoidal    60W    8675,8898,4529,8895       -1000.00  -4446307.54     -1000.00  -11118000.00







  7    0 to 40 44'S     20W to  80E     Sinusoidal    20E    8675,17793,4529,11120     -1000.00  -4446897.48     -1000.00   -2223000.00







  8    0 to 40 44'S     80E to 180E     Sinusoidal   140E    8675,28913,4529,11119     -1000.00  -6670282.40     -1000.00    8897000.00







  9   40 44'S to 90S    180W to 100W    Mollweide    160W    13204,540,4144,6741    -4866410.83  -1684820.11  -4530000.00  -19476000.00







 10   40 44'S to 90S    100W to  20W    Mollweide     60W    13204,9974,4144,6741   -4866410.83  -3370307.54  -4530000.00  -10042000.00







 11   40 44'S to 90S    20W to  80E     Mollweide     20E    13204,18870,4144,8426  -4866410.83  -3369897.48  -4530000.00   -1146000.00







 12   40 44'S to 90S    80E to 180E     Mollweide    140E    13204,30528,4144,8426  -4866410.83  -5055282.40  -4530000.00   10512000.00



After all processing (geometric, atmospheric, and NDVI compositing) is complete, the individual regions are masked to obtain the proper interruptions--the "Goode" shape (Figure 3) (38,609 bytes). After the masking process, the regions are abutted using either line and sample offsets or projection x,y offsets, as given in table 1. The result is the completed image in the Interrupted Goode Homolosine projection (Figure 4) (36,965 bytes). Map projection transformation equations, both forward and inverse, for the Sinusoidal and Mollweide projections are given in Snyder (1987).

4. Method 2: Mapping as One Projection--the AVHRR Pathfinder Project

At 8-km resolution, an image of global extent in the Interrupted Goode Homolosine map projection consists of 2,168 lines with each line containing 5,004 pixels. Because this is a significantly smaller data set, the data management of separate regions is not desirable and the masking process is an added complication. Furthermore, with fewer pixel locations to compute, the satellite model (giving raw line, sample to latitude, longitude) and map projection transformations were computed for each pixel. As a result, the masking operation is built into the projection transformation. In addition, the designers of this data set decided not to extend Greenland and Siberia, and so repeated areas are not present on the map.

The algorithm to process as a single map projection was derived by this author from Goode (1925), Snyder (1987), and Snyder and Voxland (1989). It is derived to be accurate for the 1-km data set and was used to determine the 1-km mapping offsets given above. These offsets also apply to the 8-km dataset; the two datasets are therefore geometrically compatible. The algorithm, which combines the Mollweide and Sinusoidal projections along with the necessary offsets and scaling factors, is presented in the appendix.

5. Conclusions

The use of the Interrupted Goode Homolosine map projection for both the Global Land 1-km AVHRR data set project and the AVHRR Pathfinder 8-km project has led to a visually pleasing product that is still useful for data analysis. The interruption and the combination of the Sinusoidal projection for low latitudes and the Mollweide projection for high latitudes helps to preserve the shape of the land masses and provides lower distortions than a noninterrupted global map. The use of the interrupted projection also simplifies the processing and data management of the larger, 1-km project.

Due to the large number of pixels that are fill-data (ocean pixels and those in the interrupted areas), this author and others are investigating lossless compression techniques and multi-resolution decompression techniques which should significantly reduce the data volume for distribution and analysis purposes.

Acknowledgments

The author would like to thank John Hutchinson of the EROS Data Center for preparing the distortion diagram figure, Kent Lethcoe of the EROS Data Center for checking mapping offsets, and John Snyder, U.S. Geological Survey, for his guidance during this project.

Appendix

The algorithm to process the Interrupted Goode Homolosine map projection as a single projection was derived from Goode (1925), Snyder (1987), and Snyder and Voxland (1989). It is derived with enough precision to be accurate for both the 1-km and 8-km data sets and was used to determine the 1-km mapping offsets given in the table 1. This algorithm pertains to method 2 in the text. Users of method 1 may consult Snyder (1987) for the equations for the individual Sinusoidal and Mollweide projections. The following variables are used:




            Variables used in algorithm



A. Initialization Routine

The initialization routine stores the central meridians for each region and calculates, in terms of false eastings, offsets into the resulting Goode's projection space. These x-axis offsets for each region are merely the longitude of each central meridian (in radians) multiplied by the radius of the reference sphere, R.

B. Forward Mapping

The forward mapping equations map latitude and longitude to projection (x,y) coordinates. The algorithm first determines in which region the latitude, longitude coordinate resides.

For those points lying in the Sinusoidal regions, that is between 40 44' 11.8" north and south, the following equations apply:




                Equation for Sinusoidal regions



For those points lying in the Mollweide regions, the following equations apply. (Note that this is an iterative solution.)




                Equation for points in lying in the Mollweide Regions



Iterations Iterate using the Newton-Raphson method, 30 iterations maximum




              Equation using the Newton-Raphson method






Newton-Raphson method






                Equation continued



C. Inverse Mapping

The inverse mapping equations map projection (x,y) coordinates to latitude and longitude. As with the forward solution, the algorithm first determines which region the point is in.

Next, the X coordinate is adjusted back into the appropriate region:




                 x coordinate



For those points lying in the Sinusoidal regions, that is between 40 44' 11.8" north and south, the following equations apply:




                points lying in Sinusoidal regions



Equation if close to pi/2




                Equation if close to pi/2



Otherwise




                Equation if not close to pi/2



For those points lying in the Mollweide regions, the following equations apply.




                Points in Mollweide regions



Alpha greater than 1Longitude less than negative piAlpha greater than 1




                Alpha greater than 1






Alpha greater than 1Longitude less than negative piAlpha greater than 1






                Points in Mollweide regions



More pointsAlpha greater than 1Longitude less than negative pi




                Alpha greater than 1



As a last step to the inverse equations, a final check is done to see if the coordinate is in an interrupted area. If so, the status code "IN_BREAK" is returned to the caller.

References




GOODE, J.P., 1925, The Homolosine projection: a new device for



     portraying the Earth's surface entire.  Association of



     American Geographers, Annals, 15, p. 119-125







ROBINSON, A.H. and others, 1988, Choosing a World Map--Attributes,



     Distortions, Classes, Aspects, (American Congress on Surveying



     and Mapping)







SNYDER, J.P., 1987, Map Projections--A Working Manual, U.S. Geological



     Survey Professional Paper 1395 (United States Government Printing



     Office, Washington D.C.)







SNYDER, J.P.,  and VOXLAND, P.M., 1989, An Album of Map Projections,



     U.S. Geological Survey Professional Paper 1453 (United States



     Government Printing Office, Washington D.C.)







STEINWAND, D.R., HUTCHINSON, J.A., and SNYDER, J.P., Map Projections



     for Global and Continental Data Sets and an Analysis of Pixel



     Distortion Caused by Reprojection, in press in Photogrammetric



     Engineering and Remote Sensing.







TOWNSHEND, J.R.G., editor, 1992, IGBP Global Change Report No. 20 -



     Improved Global Data for Land Applications.  A Proposal for a



     New High Resolution Data Set (The International Geosphere-



     Biosphere Programme, Stockholm).



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