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Toward volume preserving spheroid degenerated-octree grid

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.

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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.

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Correspondence to Benjamin Ulmer.

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Appendices

Appendix A: Approximate Form for Ideal Latitudinal Splitting Surfaces

Considering first the case of LG cells north of the equator where ϕ2 is always \(\frac {\pi }{2}\). By Eq. (15), the ideal location of latitudinal splitting surfaces for these LG cells is given by

$$ \phi_{s} = \sin^{-1} \left( \frac{3}{4} + \frac{1}{4} \sin\phi_{1} \right). $$

We wish to analyze the behavior as ϕ1 approaches \(\frac {\pi }{2}\), so we re-parameterize in terms of \({\Delta }\phi = \frac {\pi }{2} - \phi _{1}\) to get

$$ \begin{array}{@{}rcl@{}} \phi_{s} & =& \sin^{-1} \left( \frac{3}{4} + \frac{1}{4} \sin \left( \frac{\pi}{2} - {\Delta}\phi \right) \right) \\ & =& \sin^{-1} \left( \frac{3}{4} + \frac{1}{4} \cos {\Delta}\phi \right). \end{array} $$

Truncating the Taylor expansion of ϕs, approaching zero from the right, to a second-order approximation gives

$$ \frac{\pi}{2} -\frac{1}{2}{\Delta}\phi + 0 \cdot {\Delta}\phi^{2}, $$

and substituting back in for ϕ1 we get

$$ \frac{\pi}{2} -\frac{1}{2} \left( \frac{\pi}{2} - \phi_{1} \right) = \frac{\pi}{4} + \frac{1}{2} \phi_{1}, $$

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

$$ \alpha = \frac{c - c_{1}}{c_{2} - c_{1}}, $$

and for the modified grid we get

$$ p_{r} = \frac{r^{3} - {r_{1}^{3}}}{{r_{2}^{3}} - {r_{1}^{3}}} \quad \text{and} $$
$$ p_{\phi} = \frac{\sin \phi - \sin \phi_{1}}{\sin \phi_{2} - \sin \phi_{1}} $$

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 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

$$ M_{r}(r) = R_{m} \sqrt[3]{ d u^{3} + \left( 1 - d \right) \ell^{3} }, $$
$$ d = \frac{\frac{r}{R_{m}} - \ell}{u - \ell}, $$

and

$$ M_{\phi}(\phi) = \sin^{-1} \left( d u_{v} + \left( 1 - d \right) \ell_{v} \right), $$
$$ d = \frac{\frac{2\phi}{\pi} - \ell_{c}}{u_{c} - \ell_{c}}. $$

All that is left is to find values of u and ; 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 will be an integer power of one half. To find these exponents, we first find the exponent x that gives exactly \(\frac {r}{R_{m}}\):

$$ \left( \frac{1}{2} \right)^{x} = \frac{r}{R_{m}}, $$
$$ x = \log_{0.5} \left( \frac{r}{R_{m}} \right). $$

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

$$ u = \left( \frac{1}{2} \right)^{\left\lfloor \log_{0.5} \left( \frac{r}{R_{m}} \right) \right\rfloor} \quad \text{and} $$
$$ \ell = \left( \frac{1}{2} \right)^{\left\lceil \log_{0.5} \left( \frac{r}{R_{m}} \right) \right\rceil}. $$

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:

$$ 1 - \left( \frac{1}{2} \right)^{x} = \frac{2\phi}{\pi}, $$
$$ x = \log_{0.5} \left( 1 - \frac{2\phi}{\pi} \right). $$

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

$$ u_{c} = 1 - \left( \frac{1}{2} \right)^{\left\lceil \log_{0.5} \left( 1 - \frac{2\phi}{\pi} \right) \right\rceil}, \quad \text{and} $$
$$ \ell_{c} = 1 - \left( \frac{1}{2} \right)^{\left\lfloor \log_{0.5} \left( 1 - \frac{2\phi}{\pi} \right) \right\rfloor}. $$

We now need to find the equivalent values uc and 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,

$$ \alpha(x) = 1 - \left( \frac{1}{2} \right)^{x} \quad \text{and} \quad p(x) = 1 - \left( \frac{1}{4} \right)^{x}. $$

Rearranging and solving we get

$$ x = \log_{0.5} \left( 1 - a \left( x \right) \right), $$
$$ \begin{array}{@{}rcl@{}} p(x) & =& 1 - \left( \frac{1}{4} \right)^{\log_{0.5} \left( 1 - a \left( x \right) \right)} \\ & = &1 - \left( 1 - a(x) \right)^{\log_{0.5} \left( \frac{1}{4} \right)} \\ & =& 1 - \left( 1 - a(x) \right)^{2} \\ & =& 2 a(x) - a(x)^{2}, \quad \text{and} \end{array} $$
(19)
$$ a(x) = 1 \pm \sqrt{1 - \ p(x)}. $$
(20)

We can now say

$$ u_{v} = 2u_{c} - {u_{c}^{2}} \quad \text{and} \quad \ell_{v} = 2 \ell_{c} - {\ell_{c}^{2}}, $$

and the forward mapping is complete.

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

$$ M^{-1}_{r}(r) = d u + \left( 1 - d \right) \ell, \quad \text{where} $$
$$ d = \frac{ \left( \frac{r}{R_{m}} \right)^{3} - \ell^{3}}{u^{3} - \ell^{3}} $$

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

$$ M^{-1}_{\phi}(\phi) = \frac{\pi}{2} \left( d u_{c} + \left( 1 - d \right) \ell_{c} \right), \quad \text{where} $$
$$ d = \frac{\sin \phi - \ell_{v}}{u_{v} - \ell_{v}}. $$

We still have uv and v defined the same as the forward, however uc and 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:

$$ 1 - \left( \frac{1}{2} \right)^{x} = 1 - \sqrt{1 - \sin \phi}, $$
$$ x = \log_{0.5} \left( \sqrt{1 - \sin \phi} \right). $$

The values of uc and c then follow directly, giving

$$ u_{c} = 1 - \left( \frac{1}{2} \right)^{\left\lceil \log_{0.5} \left( \sqrt{1 - \sin \phi} \right) \right\rceil} \quad \text{and} $$
$$ \ell_{c} = 1 - \left( \frac{1}{2} \right)^{\left\lfloor \log_{0.5} \left( \sqrt{1 - \sin \phi} \right) \right\rfloor}. $$

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

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Keywords

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