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2 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 ⎦This linear equation relates three unknown values, the turbulence characteristicsand uv to “measurable” coefficients d 0, d , 2and dautoin Eqs. (4) and (5) for the secondordercross and auto SF. The characteristicsu2u2,(13),2v , and uv can be estimated uniquelyby applying Eq. (13) to the coefficients d 0( ∆x r mk), d ( ),2∆x r mkand dauto( x r a,k) for any threenon-parallel baselines ∆xr mkat ( m≠k)= 1, 2, …, N. Eqs. (2) - (5), (10), (12), and (13) are themajor operational equations for measuring the mean horizontal winds and the second-orderturbulence characteristics with the UCAR-STARS method.v2,464HAD: equations and assumptions. The non-dimensional second order cross CF for a pair ofreceivers with the phase centers x r ak ,and x r am ,, ( k ≠ m)= 1, 2, ..., N, can be defined as:* *C( ∆ x r mk, τ) = E( x r ak ,,) t E ( x r ak ,+∆ x r mk, t+τ) E( x r ak ,,) t E ( x rak ,,) t (14)rThe auto CF Cauto( xa,k, τ ) is a particular case of (14) at ∆xrmk= 0. Notations in this paperdiffer from those in DLH and HDC to match notations for SF. Following HDC, themagnitude of cross and auto CF can be presented in the form:r r r r 2 3C( ∆ x , ) exp ˆ ˆmkτ =⎣⎡−c0( ∆xmk ) −c1( ∆xmk ) τ −c2( ∆xmk) τ −O( τ ) ⎤⎦ (15)rr 2 3C ( ˆautoxa, k, τ ) = exp ⎡⎣−cauto( xa, k) τ −O( τ ) ⎤⎦(16)where the coefficients are given as follows:( r) 2 2 2 r 2 2 2c0 ∆ xmk= πγ ∆xmkαD(17)r2 2 r2 2c1( ∆ xmk ) =−8πγ ∆xmk Umkδt αD(18)r rc2 ( ∆ xmk ) = cauto ( xa,k)(19)r2 2 2 2 2 2 2 2 2cauto( xa,k) = 8πδt ⎡ w λ + γ ( U + V ) αD⎤⎢⎣⎥(20)⎦Combining Eqs. (17) and (18), one can get:U =− 4 ∆x r δ t c ( ∆x r ) c ( ∆xr )(21)( )[ 1 0 ]mk mk mk mkEquations (21), (19), and (20) relate the mean speed Umkand intensity of the vertical2turbulent velocity w to “measurable” coefficients c 0, c , c , and 1 2cautoin Eqs. (15) and(16) for the second-order cross and auto CF. The characteristics U , V , and2w can beestimated uniquely by applying Eqs. (14) - (16) and (19) - (21) to any two non-parallelbaselines ∆xr mkat ( m≠ k)= 1, 2, …, N.Eqs. (14) - (16) and (19) - (21) are the major operational equations for measuring themean horizontal winds and intensity of the vertical turbulent velocity with the HAD method.The assumptions that were adopted for deriving Eqs. (15) - (20) are not listed systematicallyin DLH but rather scattered throughout the paper. Below we try to systemize the assumptionsto the best of our ability as well as present them in the terms of those for STARS whenever ispossible.Assumption 1H: the characteristics of each scatterer xi( t ), yi(), t zi(), t Ui(), t Vi(), t Wi(), tand ∆ n ( t),i = 1, 2, ..., M, are globally statistically stationary random processes; DLH (p.i