Yamartino method

The Yamartino method is an algorithm for calculating an approximation to the standard deviation &sigma;&theta; of wind direction &theta; during a single pass through the incoming data. The standard deviation of wind direction is a measure of lateral turbulence, and is used in a method for estimating the Pasquill stability category.

The typical method for calculating standard deviation requires two passes through the list of values. The first pass determines the average of those values; the second pass determines the sum of the squared differences between the values and the average. This double-pass method requires access to all values, and special consideration must be made for a discontinuous variable such as wind direction.

The single-pass method is used as a rapid way to compute a standard deviation, although this method is not practical for angular data such as wind direction.

The Yamartino method avoids the need to have access to the original n values of wind direction. The United States Environmental Protection Agency (EPA) has chosen it as the preferred way to compute the standard deviation of wind direction.

Algorithm
During the single pass through n values of wind direction measurements (&theta;) two values are computed; the average values of sin &theta; defined as


 * $$s_a=n^{-1}\sum_{i=1}^n \sin \theta_i,$$

and average cos &theta;


 * $$c_a=n^{-1}\sum_{i=1}^n \cos \theta_i.$$

The average wind direction is then given as


 * $$\theta_a=\tan^{-1} \left (\frac{s_a}{c_a} \right ).$$

From twenty different functions for &theta; using variables obtained in a single-pass of the wind direction data Yamartino found the best function to be


 * $$\sigma_\theta = \sin^{-1} \left (\varepsilon) \right [1.0+0.1547 \varepsilon^3], $$

where


 * $$\varepsilon=\sqrt{1-(s^2_a+c^2_a)}.$$

Comparisons against the correct double-pass &sigma;&theta; indicates that Yamartino's algorithm is within 2%.