1. Chris Kidd UMD/ESSIC & NASA/GSFC
Global Precipitation: a time to rethink the representation of data products?
Chris Kidd
UMD/ESSIC & NASA/GSFC
6th IPWG meeting, October 2012
São José dos Campos, Brazil
Goddard Space
Flight Center

2. Background
The fundamental problem is that no map from the sphere to the plane can accurately represent both angles (and thus shapes) and areas.
In general, area-preserving map projections are preferred for statistical applications, because they behave well with respect to integration, while angle-preserving (conformal) map projections are preferred for navigation.
Ideally we need area-preserving map projections
6th IPWG meeting, October 2012
São José dos Campos, Brazil
Goddard Space
Flight Center

3. Why?
Latitude/longitude (CED) grids are typically the de facto standard for global visualisation
- to date, distortion has not been seen as critical since most products are 60°N-60°S; however, we must address mapping at higher latitudes
- critically, Polar regions are data-rich!
i) Most low-Earth orbiting-observations are equal-area
ii) ground observations (e.g. radar) are essentially equal area
iii) remapping introduces additional errors into estimates
6th IPWG meeting, October 2012
São José dos Campos, Brazil
Goddard Space
Flight Center

4. Lat/lon grid Mapping is easy: lat/lon→x/y conversion simple (!?)
Visualisation easy: locating regions, etc
Interpretation okay: near the Equator reasonably equal area, less so mid-latitudes, but certainly not at high-latitudes
Treatment of higher latitude inconsistent; grid boxes represent smaller areas; improved number of observations but fewer samples/grid box; area smoothing/replication... .
Equal-area pseudo lat/lon grids exist (e.g. Gall-Peters, but these compress the latitudinal range.
6th IPWG meeting, October 2012
São José dos Campos, Brazil
Goddard Space
Flight Center

5. ‘Standard’ Lat/Lon or CED grid
6th IPWG meeting, October 2012
São José dos Campos, Brazil
Goddard Space
Flight Center

6. Lat/lon scale distortion
Equator 1.000
30S
45S
60S
75S
Scale relative to the Equator (=1.00)
6th IPWG meeting, October 2012
São José dos Campos, Brazil
Goddard Space
Flight Center

7. Lat/lon swath distortion
6th IPWG meeting, October 2012
São José dos Campos, Brazil
Goddard Space
Flight Center

8. Lat/lon swath distortion (Gall-Peters)
Equal-area, but greater distortion in the higher latitutes.
6th IPWG meeting, October 2012
São José dos Campos, Brazil
Goddard Space
Flight Center

9. Representing high latitudes
GPM = Global Precipitation Measurement, therefore necessary to provide, represent and present precipitation on a global basis.
Lat/lon grid inappropriate for polar regions, therefore some sort of polar mapping required.
Mapping should become the de facto standard (i.e. one that everyone should use)
6th IPWG meeting, October 2012
São José dos Campos, Brazil
Goddard Space
Flight Center

10. Polar projections Azimuthal Equidistant - preserves distances
Lambert Azimuthal Equal Area - preserves areas
Gnomonic - all great circles become straight lines
Stereographic - conformal, equal angle
Universal Polar Stereographic (UPS) - a specific Stereographic Projection for use with UTM (Universal Transverse Mercator) system
Orthographic - infinite perspective projection
Near-Sided Perspective - finite perspective projection
6th IPWG meeting, October 2012
São José dos Campos, Brazil
Goddard Space
Flight Center

11. Polar mapping examples:
Orthographic
Equal area, ‘distorted’
at low latitudes
Azimuthal
Not equal area, less ‘distorted’
at low latitudes
6th IPWG meeting, October 2012
São José dos Campos, Brazil
Goddard Space
Flight Center

12. Polar mapping examples:
SSMIS F18 10th November 2011
Orthographic
Equal area, distorted
in low latitudes
Azimuthal
Not equal area, less distorted
in low latitudes
6th IPWG meeting, October 2012
São José dos Campos, Brazil
Goddard Space
Flight Center

13. The answer: There isn’t one! There are many…
Ultimately, it depends upon what degree of distortion you (developer & user) are prepared to accept
6th IPWG meeting, October 2012
São José dos Campos, Brazil
Goddard Space
Flight Center

14. Equal distance along latitude centres
*90°
60°
30° 0°
30°
60°
*90°
6th IPWG meeting, October 2012
São José dos Campos, Brazil
Goddard Space
Flight Center

15. Equal degrees of distortion
*90°
76°
60°
41° 0°
41°
60°
76°
*90°
6th IPWG meeting, October 2012
São José dos Campos, Brazil
Goddard Space
Flight Center

16. Polar & Equatorial regionsof
overlap
+ve: Relatively simple mapping
-ve: Large distortions at edges
6th IPWG meeting, October 2012
São José dos Campos, Brazil
Goddard Space
Flight Center

17. Local “true” equal-area mapping
Lighter = more distortion
6th IPWG meeting, October 2012
São José dos Campos, Brazil
Goddard Space
Flight Center

18. Local “true” equal-area mapping
As above, but clipped
6th IPWG meeting, October 2012
São José dos Campos, Brazil
Goddard Space
Flight Center

19. Local “true” equal-area mapping
+ve: controls degree of distortion
overlap region - advection
-ve: less easy to interpret
6th IPWG meeting, October 2012
São José dos Campos, Brazil
Goddard Space
Flight Center

20. Local “true” equal-area mapping
Plotted with SSMIS orbit
6th IPWG meeting, October 2012
São José dos Campos, Brazil
Goddard Space
Flight Center

21. Global-IR remapped (8-10 October 2012)
6th IPWG meeting, October 2012
São José dos Campos, Brazil
Goddard Space
Flight Center

22. Gaussian Grids N48 grid 208 km N128 grid 78 km N640 grid 15 km
+ve: essentially equal area globally & existing usage
-ve: computatively awkward; point vs area
6th IPWG meeting, October 2012
São José dos Campos, Brazil
Goddard Space
Flight Center

23. Summary
No single projection is suitable for mapping global precipitation, so:
Lat/lon grid for whole Earth (i.e. 90°N-90°S)
Polar grid based upon Longitude (bearing) and Latitude (distance), or ‘weighted’ Latitude
Polar grid limited to polewards of ‘x’ latitude.
Local equal-area mapping
Gaussian grids (or other ‘model’ grids)
6th IPWG meeting, October 2012
São José dos Campos, Brazil
Goddard Space
Flight Center