Two important assumptions were usually made: 1) The statistics follow Gaussian distributions; 2) The integral length scale of the flux can be estimated by the length scales of the single turbulent quantities e.g., the vertical wind and the temperature.
For turbulence probes the results of the error calculations had two consequences: 1) The statistical errors of turbulent fluxes measured in the convective boundary layer (CBL) are around 100 per cent; 2) To decrease these errors a large distance (airborne systems) or fetch (ground stations like towers etc.) has to be recorded. Normally this consumes too much time to disregard the instationarity of the CBL.
During the field experiment LITFASS 1998 above a heterogeneous site 60 km southeast of Berlin the airborne measurement systems Helipod and Do 128 (both Technical University of Braunschweig) flew simultaneous missions in the CBL. The Helipod is an autonomous probe attached to a rope under a helicopter and operates at 40 m/s airspeed. The Do 128 is a twin-propellered research aircraft that travels at 60 m/s. To meet the ratio of airspeeds the two systems flew two different square-shaped flight patterns - one with 10 km legs, the other with 15 km legs - simultaneously around a common central area. Within this central area, several ground-stations, a 99 m meteorological tower, and a wind profiler (with RASS) completed the experimental arrangement.
As a result the area-averaged turbulent fluxes of heat, humidity, and momentum measured by the Do 128 and the Helipod were nearly identical. The discrepancy was of the order of a few W/m^2. The statistical errors as calculated with the usual method were much larger: several 10 W/m^2. Since this was systematical, a closer look at the procedure seemed to be appropriate.
First we calculated the statistical contribution of the turbulent properties wind, humidity and temperature under stable and unstable thermal stratification. A simple Chi-Square test shows that the assumption of Gaussian distributions is not indicated. The histograms of the turbulent properties (named the real distribution) usually show a distribution with finite kurtosis and skewness. These distribution density functions were then approximate by three-parameter IDB distributions (Hjorth distribution). Special cases of this distribution type are the Weibull, Rayleigh, and the exponential distribution. The turbulent fluxes calculated using the real, the IDB, and the Gaussian distribution were about the same. The further analysis shows the influence of the non-Gaussian distributions on the calculation of the statistical errors.
The integral scale is the most dominant (and physically the most descriptive) quantity in the flux-error calculation. As many authors reported the integral scale of the turbulent flux itself is in praxis sometimes hard to calculate since the integral of the associated correlation function is not clear-cut. This is due to the 'wild' behavior of some measured correlation functions that i.e. passes zero several times or never at all.
This is the reason why most authors substitute the integral scale of the turbulent flux by an estimate that is built using the integral scales of the single turbulent quantities (e.g. the vertical wind and the temperature for the sensible heat flux) whose auto-correlation functions behave much more reasonably.
For the measurements during LITFASS 1998 we show: The calculation of the flux errors leads to values that are smaller by a factor of 2 to 10 when considering two items: 1) No assumption of Gaussian distributions; 2) No approximation for the integral scale of the fluxes but 'direct' calculation. The resulting flux errors now fit very well into the overall picture of fluxes measured with different systems at ground level and various heights within the CBL.
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