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#### Vector vs. Scalar Averaging of Wind Data

The wind is described as having both a direction and a magnitude (speed), and it is therefore a vector quantity.  Although the wind is a vector quantity, the wind direction and speed can be treated separately as scalar values.  In collecting wind data, samples are typically collected at a high frequency and then averaged over a time period of a few minutes to an hour.  Depending on the application and the instrumentation, the data may be vector averaged, scalar averaged or averaged using both techniques.

In scalar averaging wind data, instruments such as a cup or propeller anemometer and a wind vane are used to make independent measurements of the wind speed and direction.  The instruments are sampled at regular intervals and simple arithmetic averages of the outputs are calculated over the averaging period.

In vector averaging, either the orthogonal components of the wind are measured directly with a wind instrument or the speed and direction are measured with an anemometer and a wind vane and then they are used to derive the orthogonal components.  To obtain the vector-averaged speed and direction, the components are summed and vector averaged at the end of the averaging time.

During periods of moderate to high wind speeds, the difference between vector and scalar averages will be small.  In the case of wind speed, vector-averaged speeds will never be larger than the scalar-averaged values and will generally be lower.  Larger differences will occur with greater wind direction variance, which typically occurs at lower wind speeds (below about 2 meters per second).  As an extreme example, suppose we had a constant wind from the north at 5 meters per second for 5 minutes followed by a constant wind of 5 meters per second from the south for 5 minutes.  If we calculated both the vector and scalar averages for the 10-minute period, the vector-averaged speed would be zero, whereas the scalar-averaged speed would be 5 meters per second.  In most real-world situations, the wind direction variability is much less than this extreme example.  In moderate wind speeds, say above about 5 meters per second, the wind direction standard deviation (sigma theta) is typically about 5 to 10 degrees, and the difference between the scalar and vector averages of wind speed will generally be within one to two tenths of a meter per second.

Shown below are some actual wind speed data from a 10-m meteorological tower equipped with a cup anemometer and wind vane: Both vector- and scalar-averaged speeds were recorded. The vector- and scalar-averaged speeds were obtained with the same anemometer and wind vane, but the averages were derived using the different vector and scalar averaging techniques. These data were averaged over 15-minute periods. Also plotted on the figure below are the corresponding measurements of sigma theta. As shown in this figure, there is very little difference between the vector- and scalar averaged speeds above about 5 meters per second, although the vector-averaged speeds are slightly lower. Sigma theta values during these periods of higher speeds generally are below about 10 degrees. At lower speeds, larger departures between the vector and scalar speeds occur and the corresponding sigma theta values become large.

In the case of wind direction, there will also be differences between vector and scalar averages, particularly at lower speeds. Vector averaging, in effect, weights the direction for speed whereas the scalar-averaged direction is independent of speed. Typically, the difference between vector- and scalar-averaged wind directions will be within a few degrees.

Sodar systems inherently measure the wind using vector averages.  That is, they measure the wind components and then combine the component measurements to form a wind vector at selected averaging intervals. Unlike cup or propeller anemometers and wind vanes, sodar systems do not measure the wind speed and wind direction independently.

Some industries, such as the wind energy industry, have traditionally used scalar-averaged wind data obtained from anemometers on towers. One might argue that vector-averaged wind speeds may be more representative of the actual wind power available to a wind turbine. Nevertheless, in principle, if it were desired, empirical or theoretical conversion factors could be developed to estimate the scalar speed from the sodar vector-averaged speed. Ideally, wind direction standard deviation (sigma theta) measurements from the sodar would be used to accomplish this, but sigma-theta measurements from sodar systems are generally not very accurate. This is because sodar systems use multiple beams to make the wind measurements, and these do not occur at the same point in space and time. As an alternative approach, using the sodar standard deviation of vertical speed (sigma-w) will probably provide the most reliable method. Sigma-w and sigma-theta are generally well correlated, and thus sigma-w can be a surrogate measurement of sigma-theta, which is directly related to the difference between scalar- and vector-averaged speeds.