3. STATISTICS RELATED TO DIRECTIONAL DIFFERENCE
Linear statistics give information on the energy-based quantities such as wave height but
comparisons involving wave directions between the WIS hindcast information and the measured
information require circular statistics. Circular statistical quantities using the vector mean wave direction
were calculated for the coincident WIS and measurement locations for the 1990 Gulf of Mexico hindcast.
The mean directional difference between WIS and the measurement location was calculated for one
month of data (using one month of information helps to identify seasonal trends and identifies differences
in specific storms during the month). Monthly statistics give a measure of how one month's hindcast
information compares with what has been measured, and these statistics can be added to the monthly
statistics already developed for parameters like significant wave height and period. Concentration and
circular correlation, two other statistics using the monthly set of directional differences give more
information about how well the measured and hindcasted mean directions compare. The concentration
statistic measures the spread of the distribution. The circular correlation, similar in concept to the usual
linear correlation coefficient but applied to circular data, measures how the two distributions match. It is
important to look at all three of these statistics since one month could show a small mean directional
difference but could have low circular correlation and a concentration value indicating a distribution with a
lot of spread. The following paragraphs present the statistical equations and variables used in the tables
and graphs in section 4 on statistical results.
All wave directions for both WIS and the measurements are initially in meteorological convention.
These values are converted to polar coordinates for ease in dealing with trigonometric functions. The
directional difference, xi, is calculated for each hour that has a coincident directional measurement and
WIS value. The WIS direction and the measurement direction are assumed to be two unit vectors in polar
coordinates. Using the scalar and cross vector product on these two vectors enables calculation of a
directional difference (positive or negative). This difference was calculated so a positive difference
means that the measurement vector is more clockwise than the WIS wave direction vector (using polar
directions). An example of a directional difference calculation will help to clarify the results. If the
measured wave direction is 18.9 deg in meteorological convention (coming from NNE) and the WIS wave
direction is 57.9 deg in meteorological convention (from NE), the calculated difference in the statistical
program used for the statistics in this paper will be 39.0 deg. The mean directional difference is
calculated from the set of directional differences using the technique outlined in Bowers et al. (2000) to
calculate x , the mean directional difference for the month in question. Equations 1 through 5 below
show the steps in the calculation of x , the mean directional difference.
n
S = ∑ sin( xi )
(1)
i=1
n
C = ∑ cos( xi )
(2)
i =1
R = C2 + S2
(3)
R = R/n
(4)
x = tan -1 ( S / C)
(5)
R is called the resultant, and R is termed the mean resultant length since it is divided by n, the number
of coincident directions in the two distributions. The arc tangent function was calculated using the
FORTRAN atan2 function to assure proper signs and quadrants for the resulting x angle. The 95%
^
x sin -1 (1.96 / nR k )
(6)
Previous Page
