Spherekit: Demos |
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- 1. "Smart" Interpolation
- Elevation data are used to assist a multiquadric interpolation of temperature.
- 2. Global Interpolation
- Global land temperatures are interpolated using the GHCN data set.
- 3. Error Analysis
- A comparison of two interpolation algorithms (thin-plate splines and Cressman's method) using cross-validation.
- 4. Spatial Variability
- Spatial variability of a dataset is explored using the semivariogram.

The first diagram below shows the standard, interpolated temperature field using the multiquadric method. The fit fails to take into account the large variations in topography that produce very low temperatures at high altitudes.

- Reduce temperatures to sea-level using the environmental lapse rate.

(SeaLevTemp= Temp + Env LapseRate * Elevation) - Interpolate the "sea-level" temperature field
to a one-degree grid using the multiquadric method.
- Reintroduce elevation effect on the interpolated field.

(Temp= SeaLevelTemp - EnvLapseRate * Elevation)This final step was carried out automatically by Spherekit, by inverting the operations in Step 1.

Standard Multiquadric Interpolation (Degrees C) |
Topographically Informed Interpolation (Degrees C) |
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This "smart" interpolation captures the climatological influences
of topography. The low temperatures associated with the mountains of western
China are now visible, despite the lack of high altitude temperature
stations

This example focuses on interpolation over the entire globe. A large data base containing mean temperatures from 2,456 GHCN weather stations in January 1990 is used. The observation network is not uniformly distributed about the globe, as unpopulated regions tend to be underrepresented. The temperature field is interpolated to a one degree global grid. Note that the use of spherical geometry ensures realistic isotherms throughout the polar region.

The interpolation method used is inverse distance weighting. The user specifies which of several inverse distance functions are to be used and specifies the radius of influence. This radius can be either fixed in radius, fixed in number of points, or as an average radius based upon a specified number of points. The user can select two additional options. An extrapolation correction is available to prevent peaks from falling at the data points, based upon the gradient of the interpolated field. Secondly, a spatial bias correction adjusts the distance weights based upon the angular distribution of the points in the neighborhod. The influence of both the extrapolation and spatial bias corrections may be independently scaled.

The orthographic projection shown below reimpliments Shepard's (1968) interpolator, with an average of seven nearest neighbors, a limited extrapolation capability, and an angular correction.

Global Interpolation of GHCN Data (Degrees C)

MAE 0.593 MBE -0.13 RMSE 0.873 MIN -4.568 MAX 3.173 COUNT.....1,248 |
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MAE 1.068 MBE -0.335 RMSE 1.379 MIN -4.407 MAX 3.437 COUNT.....1,248 |
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Further analyses may be carried out using Spherekit's matrix math capabilities. The image below displays the difference field (Cressman-Spline) of the above error plots. It is interesting to note that the standard deviation of the difference field is greater than both mean average errors.

MIN -4.999 MAX 4.176 AVG -0.223 STD 1.410 COUNT.....1,248 |
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The first figure shows an isotropic semivariogram of the datset. There is a plateau in the semivariogram in the 2000-4000 km range and a sharp rise thereafter. This calculation is repeated using anisotropic semivariograms in the east-west and north-south directions. The second figure (the east-west semivariogram) displays the plateau more prominently. This characteristic corresponds to the common notion that zonal variations are relatively small. The north-south variations vary at shorter distances, as would be expected. Interestingly, the semivariogram falls after reaching a peak; presumably this is due to a return to the same latitude zone at these distances.

Isotropic semivariogram of GHCN Data |
East-West anisotropic semivariogram |
North-South anisotropic semivariogram |
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Deutsch, C. V. and A. G. Journel (1992) GSLIB: Geostatistical Software Library and User's Guide, New York, Oxford University Press.

Shepard, D. (1968) A two-dimensional interpolation function for irregularly-spaced data, Proc. 23rd National Conference ACM, ACM, 517-524.