## Abstract

Conventional Discrete Global Grid Systems are well suited for storing and indexing data on the Earth’s surface, but not for data above and below the surface. To properly support volumetric data, a 3D version of this data structure is needed. One promising approach for this is the Spheroid Degenerate-Octree Grid (SDOG), first proposed by Yu and Wu in 2009. Compared to other methods, this grid does a good job of ensuring cells have close to equal volume, which is important for ensuring a consistent spatial resolution for the entire Earth. In this paper, we introduce modifications that can be made to the original SDOG subdivision method in order to further improve its volume preserving properties. We perform a brief analysis of the number of cells in an SDOG grid and use this analysis to develop both a stationary and non-stationary modified subdivision scheme. To index the resulting grids, we derive a closed form mapping between conventional SDOG and the grids resulting from our modified subdivision rules. We evaluate the effectiveness of our modifications using two different measures of volume preservation and measure the affect these modifications have on the compactness of cells. A weighting factor allows us to balance the trade off between volume preservation and cell compactness to best meet the needs of different applications. Our method can produce a grid where all cells, except those at the poles, have exactly equal volume.

This is a preview of subscription content, log in to check access.

## References

- 1.
Alderson T, Mahdavi-Amiri A, Samavati F (2016) Multiresolution on spherical curves. Graph Model 86:13–24

- 2.
Alderson T, Mahdavi-Amiri A, Samavati F (2018) Offsetting spherical curves in vector and raster form. Vis Comput 34(6-8):973–984

- 3.
Alderson T, Samavati F (2019) Multiscale nurbs curves on the sphere and ellipsoid. Computers & Graphics

- 4.
Amiri AM, Samavati F, Peterson P (2015) Categorization and conversions for indexing methods of discrete global grid systems. ISPRS International Journal of Geo-Information 4(1):320–336

- 5.
Bahrdt D, Seybold MP (2017) Rational points on the unit sphere: Approximation complexity and practical constructions. In: Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation, ACM, pp 29–36

- 6.
Bassin C (2000) The current limits of resolution for surface wave tomography in North America. EOS Trans. AGU. 81: Fall Meet. Suppl., Abstrac 81:F897

- 7.
Bellman RE, Cooke KL (1963) Differential-difference equations

- 8.
Braun J, Thieulot C, Fullsack P, DeKool M, Beaumont C, Huismans R (2008) Douar: a new three-dimensional creeping flow numerical model for the solution of geological problems. Phys Earth Planet Inter 171(1-4):76–91

- 9.
Calhoun DA, Helzel C, LeVeque RJ (2008) Logically rectangular grids and finite volume methods for pdes in circular and spherical domains. SIAM Rev 50(4):723–752

- 10.
Gang W, Xuefeng C, Feng L, Ke L (2013) Sphere shell space 3d grid. International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences - ISPRS Archives 40:77–82. https://doi.org/10.5194/isprsarchives-XL-4-W2-77-2013

- 11.
Goodchild MF (2018) Reimagining the history of gis. Ann GIS 24(1):1–8

- 12.
Jupp P, Mardia K (1989) A unified view of the theory of directional statistics, 1975-1988. In: International Statistical Review/Revue Internationale de Statistique, pp 261–294

- 13.
Kageyama A, Sato T (2004) “Yin-Yang grid”: An overset grid in spherical geometry. Geochemistry, Geophysics, Geosystems 5(9). https://doi.org/10.1029/2004GC000734

- 14.
Kageyama A, Yoshida M (2005) Geodynamo and mantle convection simulations on the earth simulator using the yin-yang grid. In: Journal of physics: Conference series, vol 16, IOP Publishing, p 325

- 15.
Mahdavi-Amiri A, Alderson T, Samavati F (2015) A survey of digital earth. Computers & Graphics 53:95–117

- 16.
Morton GM (1966) A computer oriented geodetic data base and a new technique in file sequencing

- 17.
OGC (2017a) Discrete global grid systems abstract specification. http://docs.opengeospatial.org/as/15-104r5/15-104r5.html. Accessed on 2018-08-02

- 18.
Raskin RG (1994) Spatial analysis on the sphere: A review (94-7)

- 19.
Rohlf JH, Hancher MD System and method for storing and retrieving geospatial data (2014). US Patent 8,650,220

- 20.
Sahr K, White D, Kimerling AJ (2003) Geodesic discrete global grid systems. Cartogr Geogr Inf Sci 30(2):121–134

- 21.
Snyder JP (1987) Map projections–A working manual, vol 1395. US Government Printing Office

- 22.
Tackley PJ (2008) Modelling compressible mantle convection with large viscosity contrasts in a three-dimensional spherical shell using the yin-yang grid. Phys Earth Planet Inter 171(1):7–18

- 23.
Wadell H (1935) Volume, shape, and roundness of quartz particles. J Geol 43 (3):250–280

- 24.
Yoshida M, Kageyama A (2004) Application of the yin-yang grid to a thermal convection of a boussinesq fluid with infinite prandtl number in a three-dimensional spherical shell. Geophysical Research Letters 31(12). https://doi.org/10.1029/2004GL019970

- 25.
Yu J, Wu L (2009) On coding and decoding for sphere degenerated-octree grid. Geogr Geo-Inf Sci 25:5–9

- 26.
Yu J, Wu L, Li Z, Li X (2012) An sdog-based intrinsic method for three-dimensional modelling of large-scale spatial objects. Ann GIS 18(4):267–278

- 27.
Yu J, Wu L, Zi G, Guo Z (2012) Sdog-based multi-scale 3d modeling and visualization on global lithosphere. Science China. Earth Sci 55(6):1012

- 28.
Yu J, Wu L (2009) Spatial subdivision and coding of a global three-dimensional grid: Spheoid degenerated-octree grid. In: Geoscience and Remote Sensing Symposium, 2009 IEEE International, IGARSS 2009, vol 2. IEEE, pp II–361

- 29.
Zhao D (2004) Global tomographic images of mantle plumes and subducting slabs: insight into deep earth dynamics. Phys Earth Planet Inter 146(1-2):3–34

## Acknowledgments

This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC). We wish to thank Troy Alderson and Lakin Wecker for editorial comments, and John Hall for insightful technical discussion.

## Author information

## Additional information

### Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Appendices

### Appendix A: Approximate Form for Ideal Latitudinal Splitting Surfaces

Considering first the case of LG cells north of the equator where *ϕ*_{2} is always \(\frac {\pi }{2}\). By Eq. (15), the ideal location of latitudinal splitting surfaces for these LG cells is given by

We wish to analyze the behavior as *ϕ*_{1} approaches \(\frac {\pi }{2}\), so we re-parameterize in terms of \({\Delta }\phi = \frac {\pi }{2} - \phi _{1}\) to get

Truncating the Taylor expansion of *ϕ*_{s}, approaching zero from the right, to a second-order approximation gives

and substituting back in for *ϕ*_{1} we get

which is equivalent to a convex combination with a splitting factor of one half. The derivation follows the same for LG cells where \(\phi _{2} = -\frac {\pi }{2}\), except we do the Taylor series expansion approaching from the left.

### Appendix B: Derivation of Mappings for Non-Stationary Subdivision

Recall our goal of defining a function \(M \colon (\phi , r) \rightarrow (\phi , r)\) that converts a latitude and radius in a conventional SDOG grid to the corresponding latitude and radius in the modified grid. Comparing the convex combinations used in conventional SDOG (Eq. (9)) to the volume preserving functions used for our non-stationary modification (Eqs. 10 and 11), we notice that these equations all parameterize the range of an SDOG cell in the given spherical coordinate from zero to one. Therefore, a point (*λ*,*ϕ*,*r*) inside an SDOG cell can be parametrized in terms of the maximum and minimum of that cell in each spherical coordinate, which we find by solving these equations for *α* and *p*. For conventional SDOG we get

and for the modified grid we get

for radius and latitude respectively. Furthermore, if a point is inside a cell where all its children use the same formulations for splitting surfaces as the cell itself (i.e. NG cells), then this parametrization will be consistent with the ones given by the children, and therefore be consistent at all levels of subdivision.

To create this parameterization then, we need to find the boundaries of the coarsest NG cell that contains the given point. Referring back to Figs. 6 and 7, this is equivalent to finding the boundaries of the spherical shells and zones that contain the point. We use *u* and *ℓ* to refer to the maximum and minimum of these boundaries respectively, expressed in the parameter domain. For the spherical shell these values are the same in both the conventional and modified grids, since they both use the same radial splitting surface for SG cells. For the spherical zone, however, these values will differ between the two grids due to the different latitudinal splitting surfaces used for SG and LG cells. Thus, we distinguish between these values in the conventional and modified grids by using subscripts *c* and *v* respectively.

We can now begin to derive the mappings. Since NG cells use \(\alpha = \frac {1}{2}\) in conventional SDOG and \(p = \frac {1}{2}\) in the modified grid, we can directly equate these two parameterizations. We use *d* as the common parameter and get

and

All that is left is to find values of *u* and *ℓ*; we start with the radius case. We first normalize the radius to the range of the grid (\(\frac {r}{R_{m}}\)). Since SG cells use a radial splitting factor of one half, we know *u* and *ℓ* will be an integer power of one half. To find these exponents, we first find the exponent *x* that gives exactly \(\frac {r}{R_{m}}\):

By taking the floor and ceiling of *x*, we can find the closest powers of one half to our target \(\frac {r}{R_{m}}\). The smaller exponent (floor) will give the larger final value and vice versa. Therefore, we conclude with

The latitude case for the forward mapping follows similarly, the main difference being that SG and LG latitudinal splitting surfaces are located at one minus integer powers of one half, instead of the powers themselves. Again, we normalize the latitude (\(\frac {2\phi }{\pi }\)) and solve for x:

The smaller exponent (floor) now results in the smaller value and vice versa. Therefore, we conclude with

We now need to find the equivalent values *u*_{c} and *ℓ*_{c} in the modified grid. We already know that for the conventional grid, these surfaces are located at one minus integer powers of one half. For the modified grid, this is instead one minus integer powers of one quarter (refer to Eq. (11)). Thus,

Rearranging and solving we get

We can now say

and the forward mapping is complete.

The inverse mapping follows similarly to the forward one. For radius we now have

with *u* and *l* the same as the forward. For latitude we have

We still have *u*_{v} and *ℓ*_{v} defined the same as the forward, however *u*_{c} and *ℓ*_{c} are now calculated differently. We use Eq. (20) to map *ϕ* to the appropriate parameter space, where \(p(x) = {\sin \limits } \phi \). Solving again for x:

The values of *u*_{c} and *ℓ*_{c} then follow directly, giving

## Rights and permissions

## About this article

### Cite this article

Ulmer, B., Samavati, F. Toward volume preserving spheroid degenerated-octree grid.
*Geoinformatica* (2020). https://doi.org/10.1007/s10707-019-00391-w

Received:

Revised:

Accepted:

Published:

### Keywords

- Discrete global grid systems
- Volumetric digital earth
- Space partitioning
- Data structures