The procedures used for Longitudinal Tract Data Base (LTDB) 2000-2010 estimates are described in detail by Logan, Xu, and Stults (2014). They involve a combination of area and population interpolation, using a land/water dichotomy as ancillary data. The interpolation procedures was based on Topological Faces layer of the TIGER/Line shapefiles produced by the Census Bureau (2011), which presents the intersection between blocks and tracts (and many other geographic layers) as defined in the 2000 and 2010 censuses. This file is available to be downloaded from the website TIGER/Line Shapefiles and TIGER/Line Files. census geography comprises multiple nested scales, of which the most commonly used are the state, county, census tract, block group, and block. The face polygons created by the intersection of these multiple geographic boundaries are in effect the smallest possible sub-block unit in census geography and refer to it as a 'fragment.' Each 'fragment' is uniquely identified by a topological face ID (TFID), and it includes several useful attributes: total area, an indicator of whether the face polygon is water or land, and all geocodes (from block ID to state FIPS code) in both the 2000 and 2010 census. These fragments from the Faces file can be dissolved to the tract and block layers for 2000 and 2010.
The first step in the procedure of producing 2000-2010 weight estimates is to allocate reported tract level population counts in 2000 to blocks within the tract. The LTDB bases this allocation on the block's share of the total tract population in 2000. This procedure avoids having to assume that population was uniformly distributed through the tract. It then estimates what share of the 2000 block population lies in each fragment within that block. This step (land-only areal weighting at the block level) is solely based on the fragment's share of the block's land area, disregarding portions of fragments that are covered with water. It is then straightforward to aggregate populated fragments to the 2010 census tracts.
A detailed description of the methodology used for obtaining LTDB interpolation weights can be found in the article published by Professor Logan and his colleague in The Professional Geographer and the following validation study also published by Professor Logan in Annals of the American Association of Geographers.

For exmaple, simple areal weighting can allocate population from the tract as defined in 1990 year to a tract area defined in year 2000 tract area directly in proportion to the share of its area that lies within that 2000 tract. Therefore, areal interpolation requires only that we have an accurate overlay of the tract boundaries in two years. The Longitudinal Tract Data Base (LTDB) interpolation weights for 1970-1990 are based on tract boundaries from the National Historic Geographic Information System (NHGIS) and with these a tract-level equivalent of a Topological Faces relationship table for 1970-2000 were created.
In short, the first step was to overlay the 2000 tract boundary file onto the 1990 boundary file and merge these into a single layer. For each tract that did not change between 1990 and 2000, the result is a single polygon and data record. For tracts that changed, multiple records exist in the new layer. This was followed by merge of 1990 census data with this new layer using 1990 state, county, and tract codes, and apportion of the 1990 counts to each fragment of the split tract using the area proportions as weights was done. The same process was repeated for 1970 and 1980, again using the 2000 tract file as the overlay. Finally, the population and area based interpolation method described previously was used to adjust the data from 2000 tract boundaries to 2010 tract boundaries.
A similar approach was used by Neighborhood Change Data Base (NCDB) 1980, first linking source year tracts to 1990 blocks, and then interpolating from those blocks to 2000 tracts. NCDB used area-weighted interpolation using spatial data from Tiger/Line 1992. A less precise area weighting was used for 1970 that relied on the Census Bureau's tract correspondence file between 1970 and 1980. Every 1970 tract contributing to a 1980 tract was weighted equally. Then 1980 tracts were linked to 1990 blocks, and in a final step to 2000 tracts. Researchers should be aware of the potential for error in interpolation that is based only on area weights because population density is not uniformed which is the major assumption when using this type of interpolation.