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

2 3D ( , ) ( ) ( ) ˆ ( ) ˆp∆ xmk τ = ap ∆ xmk + bp ∆ xmk τ + cp ∆ xmkτ + O( τ )r r (2)D x τ d x ˆ τ O τ +p p 1p, auto(k, a, ) =p, auto( a,k) + ( )where ˆ τ = τ / δt,and δ t is the inter-sample time interval. Equations for the coefficients a ,b , pcp,and dp,autoat p = 2 and 4 are presented in PP while only those at p = 2 are consideredbelow. The key assumption for deriving Eqs. (2) is the following: the characteristics of eachscatterer xi( t ), yi(), t zi(), t Ui(), t Vi(), t Wi(), t and ∆ ni( t),i = 1, 2, ..., M, are locally statisticallystationary random processes. Following this assumption, the instantaneous velocity of eachscatterer can be presented as a sum of the mean and turbulent components:{ Ui( t), Vi( t), Wi( t)} = { Ui , Vi , Wi } + { ui( t), vi( t), wi( t)}(3)It is also assumed that the scatterers have a statistically uniform spatial distribution in thehorizontal directions inside the illuminated volume, and that their mean motion is statisticallyhomogeneous, that is: Ui= U and V i= V for i = 1, 2, ..., M. These assumptions aresufficient for deriving equations for estimating the mean horizontal velocities U and Vfrom coefficients a pand bp.To derive equations for estimating turbulence characteristicsfrom coefficients c pand dp, auto,it is further assumed that turbulence is statisticallyp phomogeneous ( w = w ,iuv= uv for i = 1, 2, ..., M, and j = 0, 1, ..., p), andj p−j j p−ji ithe integral scale of wi( t ) is smaller thanσ ror/andσ h. Hereafter σrand σhdenote the radarrange resolution and the linear width of the transmitted beam. The equation for c ( )2∆x rmkwaspreviously presented in a simplified form [PP, Eq. (62)] where intensity of the horizontalturbulent velocity was omitted. The explicit expression for the coefficient is as follows:r⎡2( 2 2 2 4 r2) 2 ( 2 2c)2( x )w γ U V 8π γ xmk Umk u ⎤∆ + ∆ +mkmk2 2r = 32πδt⎢ + −⎥ (4)2 2 2 4 41 −a2( ∆xmk)/2 ⎢ λ α D α D⎥⎢⎣⎥⎦where [PP, Eq. (60)]: ( r2a ) 2 1 exp ( 4)2 2 2 22x ⎡r∆mk= − − πγ ∆xmkαD⎤⎢⎣⎥(5)⎦Here λ is the radar wavelength; D and γ are the transmitting antenna dimensions and antenna2 2factor; α = 1 + ( σh/ σa), and σais the receiver field of view linear width. The values Umkand umk( t ) are the mean and turbulent components for the projection of the instantaneoushorizontal velocity { Ut ( ), Vt ( ),0} of a scattering medium along the baseline ∆xrmk. To deriveEq. (4), it is further assumed that the integral scale of u ( t ) is approximately equal to, orlarger than σh. It follows from Eq. (4) that:r2 2 2 2 2 2 2 2 2d2, auto( xa,k) = 32πδt ⎡ w λ + γ ( U + V ) αD⎤⎢⎣⎥(6)⎦2This equation relates the intensity of the vertical turbulent velocity w to the ``measurable''coefficient d2,autoin Eq. (2) for the auto SF. Combining Eqs. (4) - (6) with the standardrelation between the instantaneous values Umk( t ) , Ut (), and Vt () for a baseline{ ∆x, ∆ y ,0} (Doviak and Zrnic, 1993, sec. 9.3), one can derive the following expression:mkmk2 2( ) 2( ) ( )u + U ∆ x + uv + U V ∆x ∆ y + v + V ∆y2 2 2 2mk mk mk mk2 2( )2∆ xmk +∆y rmk ⎡ r c2( ∆x) ⎤mk=2 r2 ⎢d2, auto( xa,k) − r ⎥16ln [ 1 −a2( ∆xmk) / 2]δ t ⎣1 −a2( ∆xmk) / 2 ⎦mkp(7)303