Yamartino method

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The Yamartino method is an algorithm for calculating an approximation to the standard deviation σ_θ of wind direction θ during a single pass through the incoming data.^[1] 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.^[2]

[edit] Algorithm

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

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

and average cos θ

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

The average wind direction is then given as

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

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

$\sigma_\theta = \sin^{-1} (\varepsilon) \left[1+\left(\tfrac{2}{\sqrt 3} -1\right)\varepsilon^3\right],$

where

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

The use of $\varepsilon$ alone produces a result close to that produced with a double-pass when the dispersion of angles (in radians) is small, but by construction it is always between 0 and 1. Taking the arcsine then produces the double-pass answer when there are just two equally common angles: in the extreme case of an oscillating wind blowing backwards and forwards, it produces a result of $\tfrac{\pi}{2}$ radians, i.e. a right angle. The final factor adjusts this figure upwards so that it produces the double-pass result of $\tfrac{\pi}{\sqrt{3}}$ radians for an almost uniform distribution of angles across all directions, while making minimal change to results for small dispersions.

The theoretical maximum error against the correct double-pass σ_θ is therefore about 15% with an oscillating wind. Comparisons against Monte Carlo generated cases indicate that Yamartino's algorithm is within 2% for more realistic distributions.