Why Do We Need Discrete Global Grids?

• Atmospheric Aerosol Type and Optical Depth
  
  
• Surface Bi-Directional Reflectance Properties
  
  
• Cloud Properties

    5.1 x 108   x   16     x      36       =     2.9 x 1011

    km2/Earth     1/4 km       channels      basic measurements

• Raw data ~ 40 Gbytes/day;
  
  
• Total Data ~ 300 Gbytes/day
  
  
• 15 - 100 Tbytes/yr for at least 5 years

• Land Surface (Regional Studies)
  
  
• "Cloud" Meteorology (mostly zonal, often regional, usually <~ a week)
  
  
• We didn't have a lot of good "global" data

• Material Transports (Biogeochemical Cycles)
  • Ice Core Data
  • Ozone Hole
  • Pinatubo Sulfate Particles
  
  
  
• Energy and Momentum Transports (fluid phenomena)
  • Teleconnections

• You need to calculate gradients
  
  
• You need to compare time-series globally, with regional spatial resolution

• They could summarize and organize the multiple, non-uniformly spaced measurements over the globe
  
  
• They could allow us to calculate gradients faithfully (e.g., for budget calculations)
  
  
• They could allow us to make comparisons of time-series of globally distributed data (e.g., for detecting climate changes)
  
  
• They could allow us to make statistically meaningful regional comparisons of globally distributed data
  
  
• They could allow us to compare and combine data sets taken at different spatial resolutions, such as data from multiple satellite instruments, and field validation data

An aside on Numerical Models:

• To address the same problems, they must also be global
  
  
• They require discrete grids because most numerical models are based on finite difference equations

• Level 1 Products

  - raw radiances, geometrically and radiometrically calibrated
  
  

• Level 2 Products

  - geophysical parameters at the highest resolution available
  
  
  • Needed where accuracy at high spatial scale more important than uniformity of coverage (e.g., validation, field studies, and other local- and regional-scale problems)
    
    
  • Reported on an orbit-by-orbit basis
  
  

• Level 3 Products

  - globally, temporally uniform data sets
  
  
  • Needed where large-scale, uniform coverage is required (e.g., global-scale budgets, and problems that depend on data sets from multiple sources)
    
    
  • Spatial and temporal scales must be chosen for Level 3 standard products (tradeoffs)
  
  

EOS "Nominal Grid"

• Grid type: Global, equal-angle, based on latitude-longitude lines
  
  
• Projection: Cylindrical
  
  
• Spatial Resolution: 1º (~112 km)
  
  
• Sub-Grids: Equal-angle divisions at 0.5º (~56 km) and 0.25º (~28 km)
  
  
• Grid Box Origin: Upper left corner of box
  
  
• Grid Origin: 90º N, 180º W, progressing from left to right
  
  
• Temporal Grid: Monthly and approximate 10-day resolution (with bins for days 1 to 10; days 11 to 20; and days 21 to end-of-month)
  
  
• Temporal Grid Origin: Restart at the beginning of each month

Intended to satisfy the basic needs of the "global modeling" community, which includes GCM builders, carbon cycle modelers, and some oceanographers, hydrologists, and bio-geo-chemical modelers

Advantages

• The latitude - longitude system is convenient, familiar, and entrenched
  
  
• Zonal characteristics are favored
  
  
• Easy to represent in 2-D arrays

Issues

• Spatial resolution at high latitudes
  
  
• Non-uniform resolution for fine scale, equal-angle subdivisions of any region
  
  
• Representing data at multiple spatial scales
  
  Current Solution: Multiple, discipline-specific grids
  • polar grid
  • local, high-spatial-resolution grids
  
  
  
• Calculating gradients in directions other than zonal
  
  
• "Nearest neighbor" operations are ambiguous

Proposed Grid Scheme from Richard Green & Bruce Weilicki:

• 1.25º "Equal Area" Grid (based on the 2.5º ISCCP grid concept)
  
  
• The Greenwich Meridian and the Equator are region boundaries.
  
  
• Each grid area contains its northern and western boundaries, except at the S Pole.
  
  
• Subgrids are subdivisions by 2, using equal angles (on the sphere)
  
  
• Resulting characteristic lengths: 140, 70, 35,17, 8, 4, 2, 1 km.
  
  
• All nesting is to be done by area-weighted average.
  
  
• Ordering of regions is done by sequential index (1 value) or tiling (4 values):
  
  (N,G,I,J),
  
  N = 140 km ref. grid number (there are 26,410 of these)
  G = sub-grid characteristic length
  I = longitude index from W to E
  J = latitude index from S to N
  

Advantages

• Cells at the basic resolution have nearly equal area over the globe
  
  
• Zonal characteristics are favored
  
  
• Familiar to some workers in the Satellite Cloud and Earth Radiation Budget communities

Issues

• Representing gridded data in 2-D arrays
  
  
• Non-uniform resolution for fine scale subdivisions of cells
  
  
• Representing data at multiple spatial scales
  
  
• Representing data in polar regions
  
  
• Calculating gradients in directions other than zonal
  
  
• Performing any "nearest neighbor" operations
  
  Current Solution: Multiple, discipline-specific grids
  • polar grid
  • local, high-spatial-resolution grids

Usually the arithmetic mean of all points falling into a grid cell

• can be calculated with a running tally

Occasionally the median value of all points falling into a grid cell

• the idea less familiar
  
  
• requires storing and sorting the entire histogram

Area-weighted, bi-linear interpolation used to alter cell size for comparisons among gridded data sets

How Current Binning Techniques Can Degrade Data

	This illustration shows the monthly mean cloud amount for July of 1979 from the HIRS2/MSU data on a 2 degree latitude by 2.5 degree longitude grid. Click on the image to see an enlarged gif image.
	Here the data above is rebinned using area-weighted averaging into a 500 km by 500 km grid that is used for Earth radiation budget studies. Click on the image to see an enlarged gif image.
	Finally, the data in the second picture is resampled back to the 2 x 2.5 degree grid used in the first picture, and this is subtracted from the first picture. Note that the differences are nearly as large as the range of the signal, with both positive and negative values. Click on the image to see an enlarged gif image.

1. Begin with a regular icosahedron inscribed in a sphere.
   
   
2. Each face of the icosahedron is an equilateral triangle; inscribe a hexagon in each triangle, by dividing each edge into thirds
   
   
3. Project these hexagons back onto the sphere using the Inverse Snyder Icosahedral Equal Area Projection.
   
   This yields a coarse-resolution global grid, which we call Grid Resolution 1. It consists of 20 hexagons on the surface of the sphere, and 12 pentagons (each centered on one of the 12 vertices of the icosahedron). All the hexagons have the same area, and each pentagon has an area exactly 5/6 of a hexagon.
   
   
4. To form higher resolution grids, return to the planar representation of the hexagons. Let r_i be the edge length of a hexagon, on the plane, for Grid Resolution i. Tessellate each face of the icosahedron with regular hexagons of r_(i+1) = 2/3 r_i sin(60º).
   
   
   • One hexagon (or pentagon) of the new grid is centered at the same point as a cell from the previous grid
     
     
   • An additional hexagon of the new grid is centered on each vertex of the previous grid cells.
   
   
   The area of each hexagon is sub-divided into thirds by cells at the next higher resolution.

Characteristic Length Scales for the ISEA Nested Global Grid for Earth

Advantages

• No singularities (at the poles or anywhere else)
  
  
  • no need for separate grid scheme for polar studies
  • symmetry about the equator can be preserved
  
  
  
• Infinite nesting of equal-area sub-grids
  
  
  • single grid system for data at all spatial resolutions
  • uniformly represents global density of sampling
  
  Comparisons between high and low latitude, high and low spatial-resolution data are facilitated.
  
  
• Improves the isotropy of finite-difference quantities compared to those calculated for rectangular grid schemes
  
  
  • all nearest neighbors share an edge
  • distances to the centers of nearest-neighbor cells are nearly equal

Issues

• ISEA system is unfamiliar to the user community
  
  
• Representing gridded data in 2-D arrays
  
  
• Some aspects need to be demonstrated
  
  
• Needs to be implemented in a practical way for use with geophysical data

Embed each Level 2 data set in a grid with appropriate spatial resolution

• Select grid resolution for Level 2 data, based on, e.g., spatial distribution and coherence length of data, acceptable degree of interpolation.
  
  
• Bin Level 2 data into this grid, to create Level 3 product
  
  
• Calculate the associated error statistics for the data set at this grid resolution

Make comparisons among data sets with different inherent spatial resolution, automatically carrying along estimates of the formal errors (e.g., field validation studies)

• Aggregate and/or dis-aggregate each data set to a common grid resolution
  
  
• Use the formalism to also derive the associated uncertainties in each data set at the common resolution

Calculate regional and global fluxes of budget quantities, using formalism to evaluate uncertainties

• Need fast translation to and from lat-lon grid
  
  
• Need semi-automatic methods of selecting a grid scale appropriate to a data set and application
  
  
• Need semi-automatic methods of binning Level 2 data to create Level 3 products, including characterization of associated error properties
  
  
• Need efficient codes for aggregation and dis-aggregation among nested grid levels, for both Level 3 data fields and associated error statistics
  
  
• Need efficient labeling, storage, and access schemes for data in nested cells
  
  
• Need to optimize raster display of nested levels of data
  
  
• Need to implement data ingest, selection and display of regions, at different spatial scales, using IDL, some GISs, other common tools
  
  
• Need algorithms for calculating finite-difference quantities and associated errors
  
  
• Need to consider space-time gridding, and the vertical dimension
  
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