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

161). Assumption 2H: the instantaneous location of all scatterers is statistically uniform inthe illuminated volume; DLH (pp. 158 and 161). Assumption 3H: the mean motion of allscatterers is statistically homogeneous in the illuminated volume; that is, U = U ,Vi= V , and Wi= W (DLH, p. 163). Assumption 4H: the instantaneous reflectivity ofall scatterers is statistically homogeneous in the illuminated volume; DLH (pp. 158 and 161).Assumption 5H: turbulent motion of all scatterers inside the illuminated volume isstatistically homogeneous and isotropic; DLH (p.163 and Sec. 4). Following this assumption,2the turbulence intensity was characterized in DLH and HDC by σt=2w = u 2 = v2shown in DHL (sec. 5.3), σtis related to the spectral width in the Doppler method, therefore,2the measured value in Eq. (19) is w (DHL, p. 169). Assumption 6H: specific functionalform of CF or spectrum for the reflectivity ∆ niof scatterers in the illuminated volume; e.g.,the Gaussian CF with the correlation lengths ρ chand ρ czin the horizontal and verticaldirections (DLH, sec. 4), a power law of the Kolmogorov type with specified parameters(DHL, sec. 5), or another. Assumption 7H: the vertical correlation length ρczis much smallerthan the range resolution σr. Assumption 8H: specific horizontal correlation length ρch; e.g.,ρch> ρcz(DLH, p. 170), or another.Relations between STARS and HAD. To relate the second order CF and SF, one can apply therstandard set of assumptions about the received signals E( xak,, t); the assumptions arepresented and discussed by Ishimaru (1997, sec. 4-9). Let us consider the in-phase and therrquadrature components I ( xak,, t)and Qx (ak ,, t)in Eq. (1) as two statistically stationary andrindependent Gaussian random processes, and the phase of the signals E( xak,, t)to beruniformly distributed over 2 π . Let us further consider the joint distribution of I ( xak,, t),r r rr rQx (ak ,, t),I( xak,+∆ xmk, t+τ ), and Qx (ak ,+ ∆ xmk, t+τ ) to be Gaussian as well, and theantenna centers to be close to each other. The relevant consequences from these assumptionscan be reproduced from Ishimaru (1997, sec. 4-9) in our notations as follows:r r rS( xak ,, t) = S( xak ,+∆ xmk, t+ τ ) = S(22)r r r r rEx (ak ,, tEx ) (ak ,, t) = Ex (ak ,, tEx ) (ak ,+∆ xmk, t+ τ ) ≈0(23)Using Eqs. (2), (3), (14), (22), and (23), one can derive the following relation between thesecond order CF and SF for received signals (Praskovsky et al., 2003e):r2D( xmk, τ) 2⎡r∆ = 1 − D( ∆xmk, τ)⎤(24)⎣⎦This equation provides a “bridge” between SF and CF-based SA techniques, e.g., directrelations between HAD and STARS. It follows from Eqs. (4) and (5) that:r r r rd0( ∆ xmk ) = D( ∆xmk ,0), d1( ∆ xmk ) = [ ∂D( ∆xmk, τ)∂τ]τ = 0r 2 r 2 r 2 r (25)22 d2 ( ∆ xmk ) = ⎡⎣∂ D( ∆xmk , τ) ∂ τ ⎤⎦ , 2 d (0autoxa, k) = ⎡∂ Dauto( xa,k, τ)∂τ⎤τ= ⎣ ⎦τ=0Combining Eqs. (24), (25), and (15) - (20), one can estimate coefficients d 0, d , d , and 1 2dautousing the cross and auto CF as follows (the estimates are denoted by the tilde):2d% r0( x ) 2⎡rmk1 C( x ,0) ⎤r r∆ = − ∆mk= 2{ 1−exp[ −2 c0( ∆ xmk )]}= d0( ∆xmk)(26)⎣⎦2d% r1( x ) 2 ⎡rmkC( xmk , τ) τ⎤r r r∆ =− ∂ ∆ ∂ = 4 c1( ∆xmk )exp[ −2 c0( ∆ xmk )]= d1( ∆xmk) (27)⎣⎦τ = 0i. As465