12.2.2.2 Barnes analysis  12.2.2 Empirical linear interpolation  12.2.2 Empirical linear interpolation

12.2.2.1 CRESSMAN ANALYSIS

George P. Cressman (1959) introduced an interpolation method which corrects the background gridpoint value (obtained from a forecast model) by a linear combination of residuals (corrections) between predicted and observed values. The residuals are weighted depending only upon the distance between the gridpoint and the observation. The scheme begins with a background field from a numerical forecast. The background value at each grid point is successively adjusted on the basis of nearby observations in a series of scans (usually four to six) through the data. The radius of influence (the size of the circle containing the observations which influence the correction) is reduced on successive scans in order to build smaller scale information into the analysis where data density supports it.

The version of the Cressman scheme given here is the one used at the National Meteorological Center in the early 1970s. Three types of observations are distinguished, as in Fig. 12.1:

Fig. 12.1:  Three kinds of observation handled by the Cressman successive corrections scheme: height only, wind only, and height and wind together. R is the scan radius and d is the distance from the gridpoint at the center of the circle to the observation.: height alone (i index), height and wind (j index), and wind alone (kindex). The radius of influence is R; its value decreases on successive scans.

Corrections are applied at the grid point (x) as follows:

 

Observation:	Correction (residual):
Height only	ci = (zo - zc)i
Height and wind	$c_j = (z_o + {\Delta z \over \Delta r}d)_j - z_b$
Wind only	$c_k = (z_c + {\Delta z \over \Delta r}d)_k - z_b$

 where z_o is the observed height, z_c is the height interpolated to the observation location from the nearest four gridpoints, z_b is the background value at the gridpoint where a height estimate is desired, $\Delta z \over \Delta r$is the gradient implied by the observed wind projected onto the line connecting the observation and gridpoint (positive or negative), and d is the distance from observation to gridpoint.

The total correction to the background value on a given scan is a linear combination of all the residuals, given by

$$c_t={\alpha_1 \sum_{i=1}^l w_i c_i + \alpha_2 \sum_{j=1}^m w......l w_i + \alpha_2 \sum_{j=1}^m w_j + \alpha_3\sum_{k=1}^n w_k}$$ (12.11)

 

where

$$w_i = {R^2 - d_i^2 \over R^2 + d_i^2} \qquad d_i^2\leq R^2$$ (12.12)

 

$$w_i = 0 \qquad d_i^2 > R^2$$ (12.13)

 

and $\alpha_1 ,\alpha_2$ and $\alpha_3$ are adjustable constants.

The advantages of the Cressman scheme made it very popular in the 1960s and 1970s:

• The method is simple and computationally fast. (The speed depends upon the number of scans.)
• The method incorporates forecast information in the background field. (The forecast is the source of the first guess.)
• The results are generally pleasing.

The disadvantages are:

• The Cressman scheme is not well-suited for diverse observations because observational error is not accounted for.
• It does not account for the distribution of observations relative to each other.
• The scale (detail) of the result varies with observation density.
• There is no obvious way to analyze the wind field based upon height observations.
• Optimum scan radii have to be determined by trial and error.

   
 12.2.2.2 Barnes analysis  12.2.2 Empirical linear interpolation  12.2.2 Empirical linear interpolation 

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