I would like to download GPW and GRUMP data, but I'd like to know if it is possible to download files by some coordinate extent, rather than by country?

It is possible to download coverage’s based on a given coordinate extent by using WCS getCoverage request. For example:

GRUMPv1 Population Density 2000 with BBOX extent: L:123,T:43.999,B:37,R:130.999

http://sedac.ciesin.columbia.edu/geoserver/wcs?&SERVICE=WCS&VERSION=1.0.0&REQUEST=GetCoverage&Coverage=grump-v1:grump-v1-population-density_2000&CRS=EPSG:4326&BBox=123,37,130.999,43.999&resx=0.0083333&resy=0.0083333&FORMAT=GeoTIFF

 

GPWv3 Population Density 2000 with BBOX extent: L:123,T:43.999,B:37,R:130.999

http://sedac.ciesin.columbia.edu/geoserver/wcs?&SERVICE=WCS&VERSION=1.0.0&REQUEST=GetCoverage&Coverage=gpw-v3:gpw-v3-population-density_2000&CRS=EPSG:4326&BBox=123,37,130.999,43.999&resx=0.041665&resy=0.041665&FORMAT=GeoTIFF

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