Towards an Algebra of Grid Functions

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Towards an Algebra of Grid Functions

• Bill Howe

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Claims

• All our data can be modeled using
• grids (discretizations of a domain) and
• functions over the discrete elements
• All our data products can be specified as
  manipulations of these grid functions
• bind
• subgrid
• regrid

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Vocabulary Grids

• A k-dimensional cell, or k-cell can be thought of
  as just a polygon
• 0-cell vertex c
• 1-cell edge (c1, c2)
• d-cell element (or just cell) (c1,
  c2,...,cm)
• A grid G is a sequence of sets of k-cells
  C0,C1,...
• k-cell(G) denotes the (k1)th set in the
  sequence, Ck.
• dim(G) is the highest k for which k-cell(G) ? ?.
• G1 ? G2 is the cross product of G1 and G2

2-cell
1-cell
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Vocabulary Grid Functions

• GFk a ( G, f ), where
• G is a grid,
• f k-cell(G) ? a
• where a is some type. (scalar float, vector of
  int, etc.)
• grid(GFk a) G
• dom(GFk a) kcell(G)

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Examples of f

• geometry 0-cell(G) ? (x,y,z)
• salinity 2-cell(G) ? float
• neighbors(i, j) i-cell(G) ? j-cell(G)
• measure(i) i-cell(GF0 (x,y,z)) ? float
• //length, area, etc.
• column k-cell(G1) ? k-cell(G1 ? G2)

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Bind
b
(b,x)
Bind(f)
a
c
(c,y)
(a,w)
d
(d,z)
gfZ bind(vgrid, depths) gfXY bind(hgrid,
geometry) gfBath bind(gfXY, bath)

• bind( f )GFk a GFk (a,b)
• where f k-cell(G) ? b

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Subgrid

• subgrid(p)GFk a GFk a
• where p a ? Bool

gfXY bind(hgrid, nodes) gfZ bind(vgrid,
depths) gfBath bind(gfXY, bath) crossGrid
gfXYb ? gfZ corieGrid subgrid(d gt b?, crossGrid)
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Regrid

• Three components
• target grid, G
• cell mapping, g
• where g k-cell(G) ? j-cell( grid( GFk b ) )
• aggregate function, f
• where f a ? b
• Regrid( G, g, f )GFj a GFk b

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Example max salinity

• target grid hgrid
• cell mapping g e ? e ? vgrid
• where e ? 2-cell(GF)
• aggregate max

gfSalt bind(salt)hgrid ? vgrid gfMaxSalt
regrid(hgrid, g, max)gfSalt
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Example salinity gradient
salinity

1. target grid same
2. cell mapping halo
3. aggregate function gradient

gfAreas bind(measure(2))hgrid ? vgrid gfGeo
bind(geometry)gfAreas gfSalt
bind(salinity)gfGeo gfSaltGrad regrid(G,
halo, gradient)gfSalt
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Example plume volume
float

• target grid unit_grid
• cell mapping g
• where g e ? 3-cell(G)
• aggregate sum

gfElev bind(elev)corieGrid gfOcean
subgrid(ocean?)gfElev gfPlume
subgrid(saltlt26)gfOcean gfVol
bind(measure(3))gfPlume gfPlumVol
regrid(unit, g, sum)gfVol
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Grid Functions

• Grid functions can model
• Spatial data sets
• Image data
• Plots
• Timeseries
• Independent of representations
• Arrays
• Objects
• Relations
• File Formats (CORIE, NetCDF, etc.)