Yamartino method
Encyclopedia
The Yamartino method is an algorithm for calculating an approximation to the standard deviation
σθ of wind direction
θ 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 squares of the differences between the values and the average. This double-pass method requires access to all values. A single-pass method can be used for normal data but is unsuitable for angular
data such as wind direction where the 0°/360° (or +180°/-180°) discontinuity forces special consideration. For example, the directions 1°, 0°, and 359° (or -1°) do not average to the direction 120°!
The Yamartino method solves both problems. The United States Environmental Protection Agency
(EPA) has chosen it as the preferred way to compute the standard deviation of wind direction.
A further discussion of the Yamartino method, along with other methods of estimating the standard deviation of wind direction can be found in Farrugia & Micallef.
Then the average wind direction is given via the four-quadrant arctan(x,y) function as
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
where
The key here is to remember that sin2θ + cos2θ = 1 so that for example, with a constant wind direction at any value of θ, the value of
will be zero, leading to a zero value for the standard deviation.
The use of
alone produces a result close to that produced with a double-pass when the dispersion of angles is small (not crossing the discontinuity), 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 
radians, i.e. a right angle
. The final factor adjusts this figure upwards so that it produces the double-pass result of
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 σθ 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.
A variant might be to weight each wind direction observation by the wind speed at that time.
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...
σθ of wind direction
Wind direction
Wind direction is reported by the direction from which it originates. For example, a northerly wind blows from the north to the south. Wind direction is usually reported in cardinal directions or in azimuth degrees...
θ during a single pass through the incoming data. The standard deviation of wind direction is a measure of lateral turbulence
Turbulence
In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic and stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time...
, 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 squares of the differences between the values and the average. This double-pass method requires access to all values. A single-pass method can be used for normal data but is unsuitable for angular
Polar coordinate system
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction....
data such as wind direction where the 0°/360° (or +180°/-180°) discontinuity forces special consideration. For example, the directions 1°, 0°, and 359° (or -1°) do not average to the direction 120°!
The Yamartino method solves both problems. The United States Environmental Protection Agency
United States Environmental Protection Agency
The U.S. Environmental Protection Agency is an agency of the federal government of the United States charged with protecting human health and the environment, by writing and enforcing regulations based on laws passed by Congress...
(EPA) has chosen it as the preferred way to compute the standard deviation of wind direction.
A further discussion of the Yamartino method, along with other methods of estimating the standard deviation of wind direction can be found in Farrugia & Micallef.
Algorithm
Over the time interval to be averaged across, n measurements of wind direction (θ) will be made and two totals are accumulated without storage of the n individual values. At the end of the interval the calculations are as follows: with the average values of sinθ and cosθ defined asThen the average wind direction is given via the four-quadrant arctan(x,y) function as
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
where
The key here is to remember that sin2θ + cos2θ = 1 so that for example, with a constant wind direction at any value of θ, the value of
will be zero, leading to a zero value for the standard deviation.The use of
alone produces a result close to that produced with a double-pass when the dispersion of angles is small (not crossing the discontinuity), 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 radians, i.e. a right angleRight angle
In geometry and trigonometry, a right angle is an angle that bisects the angle formed by two halves of a straight line. More precisely, if a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles...
. The final factor adjusts this figure upwards so that it produces the double-pass result of
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 σθ 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.
A variant might be to weight each wind direction observation by the wind speed at that time.