Temperature Spatial Variograms
A simple kriging model can be used to interpolate min and max air temperatures for any location,
based on the distances from measured locations.
This approach uses the covariance in temperature as a function of the distance between measured locations.
For ease of calculation the distance was calculated in raw degrees longitude.
Degrees latitude were corrected (times 1.448) to equal the same distance on the ground,
at least at the center of Montana.
The variograms below show the relationship between temperature covariance and distance between stations.
These are examined separately for minimum and maximum temperatures for the total year and for the 3 seasons.
There are 26,803 pairs of stations, each with up to 3652 daily observations of min and max temperatures.
The spread is often broad, but the covariance at short distances is strongly positive.
This is more evident in the plots of correlation that follow the covariance plots.
The covariance at large distances becomes negative,
presumably reflecting the normal size of weather patterns in Montana.
The kriging model uses the actual daily means of min and max temperature, the demeaned raw data and
the observed variances for each season.
The error variance is assumed to be due only to rounding to the nearest degree (0.0833).
The covariance decay rate (theta) is the best fit (minimum absolute error) of the exponential model.
The values of theta and the observed standard deviations (Std.) are shown in the table below and
the resulting curves are shown on the graphs below.
In order to maintain reasonable computation speed, stations beyond about 2.2 degrees must be ignored.
This limit is also shown on the graphics below.
| | Minimum | Maximum |
|---|
| | Theta | Std. | Theta | Std. |
|---|
| Spring | 1.42 | 7.672 | 1.58 | 7.543 |
| Summer | 0.50 | 5.722 | 0.71 | 5.996 |
| Autumn | 1.08 | 7.255 | 1.21 | 7.048 |
| Annual | 1.11 | 6.814 | 1.20 | 6.809 |
27 AUG 2001, updated on 30 OCT 2001
dlg@rapid.msu.montana.edu