Proceedings with Extended Abstracts (single PDF file) - Radio ...

d%r( x ) ⎡ rC( x , ) ⎤ 4 ⎡ r rc ( x ) c ( x ) ⎤ rexp 2 c ( x )[ ]2 2 2 22∆mk=− ∂ ∆mkτ ∂ τ =2∆mk−1∆mk−0∆⎣⎦mkτ = 0⎣⎦22 2( πγ ) ( )4 4 δ 2 2 α 4 4 πγ 2 2 α2 2r r r= d ( ∆ x ) + 256 ∆x t u D exp −4∆x Dmk mk mk mkd% r( x ) ⎡ rC ( x , ) ⎤ r r4 c ( x ) d ( x )(29)222auto a, k=−⎢∂auto a, kτ ∂ τ =auto a, k=auto a,k⎣⎥⎦τ= 0where d , d , d , and 0 1 2dautoare given by Eqs. (8), (9), (11), and (12).COMPARISON OF THE TECHNIQUESA good agreement between STARS and CF-based SA techniques FCA and HAD inmeasuring the mean horizontal winds U , V , and intensity of the vertical turbulentvelocity2w(28)was found for the NCAR Multiple Antenna profiler and the Middle and UpperAtmosphere radar, respectively (Praskovsky et al., 2003c, d, e). The agreement seems to beexpected because STARS is related to the CF-based techniques; see the previous section.However, the relations between the SF and CF do not indicate the equivalence betweenSTARS and HAD (or any other) CF-based technique. Below we show that the CF and SFbasedSA techniques are conceptually different in spite of being formally related to eachother for a particular case of the second order functions.CF can be applied only to the globally statistically stationary random processes. Realphysical processes are almost never globally stationary while practically any process can besafely considered as being the locally statistically stationary; e.g., Tatarskii (1971), andMonin and Yaglom (1975). The basic STARS Assumption 1S about a local stationarity ismuch less restrictive than the basic HAD Assumption 1H about a global stationarity. Anotherimportant feature of SF is the presence of a small parameter τ → 0 . The small parameteralways significantly simplifies a physical task by both leading to the asymptotically exactsolutions, and requiring a smaller number of less restrictive assumptions; e.g., Migdal (1977).Indeed, the STARS Assumptions 2S and 3S for estimating the mean horizontal winds are lessrestrictive than the HAD Assumptions 2H - 4H. Furthermore, STARS requires only twoadditional Assumptions 4S and 5S for deriving operational equations for turbulence2 2 2characteristics w , u , v , and uv while HAD requires four much restrictive2additional Assumptions 5H - 8H for estimating the only turbulence characteristic w .CF characterizes fluctuations of a random process at all scales but mainly at the large onesof the order of the process's integral time scale T cor. Fluctuations at large scales are heavilyaffected by external conditions, and the functional form of CF can never be universal; e.g.,Townsend (1956, secs. 1.8, 1.9). In particular, the Gaussian functions (15) and (16) in theHAD technique are merely good approximations near the peak values of the auto and crossCF; in practice the rigorous validity of Eqs. (15) and (16) over a wide range of τ would bemore an exception than a rule. SF characterizes fluctuations at small scales τ