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sahrk / DGGRID

A command-line application that generates and manipulates icosahedral discrete global grids.
GNU Affero General Public License v3.0
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Comparing the ISEA and FULLER projections #15

Open ppKrauss opened 4 years ago

ppKrauss commented 4 years ago

Hi, I am new here, I have not yet installed DGGRID, but maybe I don't need to install it, if there are some more information...

I am looking for a statistical study on the error in the area equivalence. I am interested in comparing the ISEA and FULLER projections. Particularly interested in the quadrilateral grid, in hierarchical levels where the sides of the cell vary from 5m to 1km,

PS: context is the Brazilian "statistical grid" of the Census (population density distribution), which uses Albers projection for the Brazilian territory. We are looking for a geocode system that is based on an equal-area grid (that complies with the DGGS OGC standard), but is no worse than the Albers.

sahrk commented 4 years ago

@ppKrauss I assume by "the error in the area equivalence" you mean the range of cell areas at each resolution? The fundamental factor that drives variation in cell areas is the projection. The ISEA is an equal area projection, so if you use any ISEA grid and work in ISEA projection space you will have exactly equal area cells (to within the precision of your computer). I believe the Albers is also an equal area projection, so the same should hold for it and I am unsure what you mean by "no worse than the Albers".

The ISEA4D is nice because the congruently nested cells would make it easy to work in ISEA projection space. If you work in some other space (like latitude/longitude) then — since the edges of ISEA cells are not great circles — you may need to densify the cell boundaries to get equal area cells to within your desired precision. Or is it that you're asking what the variation in cell areas would be if you create a grid by just drawing great circles between cell vertices without any densification?

A discussion of the variation in cell areas (as well as other metrics) that includes the Snyder (ISEA) and Fuller projections is given in White et. al 1998.

ppKrauss commented 4 years ago

Thanks a lot @sahrk! Your White et. al 1998 citation and explanations were of great help.

@ppKrauss I assume by "the error in the area equivalence" you mean the range of cell areas at each resolution?

Yes, at each level/resolution, the variation in cell areas. For example, White et. al 1998 says something as:

We need more precision in the standard deviation, to compare it with a grid obtained by Albers — we can use as reference region any Icosahedron face that intersects with Brazil. Here a draft result of Brazilian Albers grid at its official FTP distribution:

square side avg_area_m2 dev_area dev_area% samples
1 km 1 km2 0.13900 m2 0.000014% 184039
200 m 40000 m2 0.00893 m2 0.000022% 113550

It was performed by PostGIS. Maybe I can use your examples/gridgenGeoJSON to export a good sample of ISEA4D grid to PostGIS and perform similar estimations.

The fundamental factor that drives variation in cell areas is the projection. The ISEA is an equal area projection, so if you use any ISEA grid and work in ISEA projection space you will have exactly equal area cells

Ok, it is good to confirm that ISEA4D is the best candidate (now I see that FULLER4D is not only a change in the globe orientation).

(to within the precision of your computer).

Hum... I am imagining that, in theory, I can use something as arbitrary-precision arithmetic to reduce the area standard deviation... It is a good news. By other hand, in a benchmark context, I need to compare Albers and ISEA in same computation conditions (same kind of truncation errors)... For example, in the table above I used PostGIS.

I believe the Albers is also an equal area projection,

Yes, the linked SR is a Albers equal-area conic projection with standard parallels at 2°S and 22°S ... It is used by the Brazilian "Official Statistical Grid".

so the same should hold for it and I am unsure what you mean by "no worse than the Albers".

Sorry, the correct meaning is "near same area standard deviation than the Albers".

The ISEA4D is nice because the congruently nested cells would make it easy to work in ISEA projection space. If you work in some other space (like latitude/longitude) then — since the edges of ISEA cells are not great circles — you may need to densify the cell boundaries to get equal area cells to within your desired precision. Or is it that you're asking what the variation in cell areas would be if you create a grid by just drawing great circles between cell vertices without any densification?

When you say "densify" is to use small cells? For example cells of the level 12 of ISEA4D. Well, I was asking "what the variation in cell areas", all of the same level. The ideal statistics would be made not in full globe, but with sample cells in the the Brazilian territory (or an Icosahedron face that intersects there).

A discussion of the variation in cell areas (as well as other metrics) that includes the Snyder (ISEA) and Fuller projections is given in White et. al 1998.

Thank you for your evaluation, patience, good clues and good citation.

NOTES on motivation. It is not only "Official Statiscs Grid" of Brazil, the European countries are also looking for a better projection solution and future DGGS compatibilization. They implemented in the past (with similar problems) the INSPIRE D2.8.I.2_v3.1, and now is time to replace it. Trends and harmonization explained at this 2019 document, "GSGF Europe -Implementation guide for the Global Statistical Geospatial Framework in Europe".

Suggestion for DGGRID project: to publish here, at git, an updated table of basic statistics of each DGGS — a table like this one or using metrics of White et. al. Ideally a table in CSV file and registering its sources at DGGRID/examples. I can help.