M 



E D(B m ) 



sinO^ 



dd 



M 



E D(dj 



cos 8 



dQ 



(26) 



With O determined by Equation 26, moments m 1 , m 2 , and « 2 can be computed 

 from Equations 22, 24, and 25, respectively. 



Kuik, van Vledder, and Holthuijsen (1988) define a measure of directional 

 spread (herein called circular width) o as 



o = (2 - 2/w,) 1/2 



(27) 



a measure of asymmetry of a directional distribution {circular skewness) y as 



1 1 3/2 



- - -m. 



2 2 2 



(28) 



and a measure of the flatness of a directional distribution (circular kurtosis) 6 as 

 6 -8m, +2m, 



6 = 



(2 -Imtf 



(29) 



Quartile Parameters 



Two parameters that are modestly more intuitive than the corresponding circu- 

 lar parameters, and are also useful for characterizing spread and asymmetry in di- 

 rectional distribution function are the quartile spread A6 and quartile asymmetry 

 A used by Long and Oltman-Shay (1991). The concept is based on the fact that 

 any directional distribution function integrates to unity such that an integral from 

 the direction of minimum energy m (where m mln is the discrete direction index 



at which minimum energy occurs) to any arbitrary angle creates a function 

 7(0 m - m ) that increases monotonically from zero to an upper limit of unity. 



The directions at which this integral (interpolated as necessary from discrete data) 

 has the values -, -, and - are the first quartile, median, and third quartile direc- 



4 2 4 



tions of the directional distribution, respectively. Differences among these direc- 

 tions then provide information about the spread and asymmetry of the distribution. 



16 



Chapter 4 Characterizing Parameters 



