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

which are adopted for deriving the equations. Detailed derivation of the equations anddiscussion of the assumptions can be found in PPa, PPb, and DLH, HDC, respectively.The transmitter of a SA profiling radar sends pulses of radio waves vertically upwardsinto the atmosphere and these are scattered by the refractive index irregularities to form amoving and changing diffraction pattern on the ground. Following PPa, the irregularities arereferred to as scatterers independent of their physical nature. Therefore, the scatterer isdefined as the refractive index irregularity scattering the transmitted waves of a specificfrequency, and it is a property of the atmosphere to which the radar is sensitive. Followingsuch a definition, each scatterer is characterized by its instantaneous locationrrx () t = { x (), t y (), t z ()}, t velocity W() t = { U (), t V(), t W()},t and reflectivity ∆ n ( t).i i i ii i i iHereafter t is time, i = 1, 2, ..., M, and M is the number of scatterers in the illuminatedvolume. The geophysical coordinate system with z axis directed upwards, x axis towards east,and y axis towards north is used hereafter; the values in the brackets { } denote the Cartesiancomponents of a vector.The magnitude and phase of the diffraction pattern is sampled with N ≥ 3 spatiallyseparated receiving antennas with the phase centers x rak ,where k = 1, 2, ..., N denotes thereceiver number. Each antenna provides a complex received signalr r rE( xak ,, t) = I( xak ,, t) + −1 Q( xak,, t)(1)where I and Q are the in-phase and quadrature components of the pure return from thescatterers with no noise or clutter. Without loss of generality, one can consider I( x r ak ,=I( x r ak ,= 0; hereafter the brackets denote the ensemble averages. Equations for puresignals can be used directly in practical measurements while noise can be taken into accountwhile calculating SF and CF; see PPa and HDC for details.i462UCAR-STARS: equations and assumptions. Consider a pair of receivers with the phasecenters x r ak ,and x r am ,, ( k ≠ m)= 1, 2, ..., N. The non-dimensional second order cross SF canbe defined as (Tatarskii, 1971, chap. 1A):r r r r 2 r r 2D( ∆ xmk, τ) = ⎡⎣Sx (ak ,, t) − Sx (ak ,+∆ xmk, t+ τ) ⎤⎦ ⎡Sx (ak ,, t) − Sx (ak ,, t)⎤⎣ ⎦(2)wherer 2 r 2 rSx (ak ,, t) = I( xak ,, t) + Q( xak,, t)(3)r r ris the instantaneous power of pure received signals; ∆ xmk = xam ,−xak,is a spatial separationbetween the antenna centers, and τ is a temporal separation between the signals. The auto SFrDauto( xa,k, τ ) is a particular case of (1) at ∆xrmk= 0. For any atmospheric profiling radar atτ → 0 and small enough ∆xrmk, SF (2) can be presented in the following form (PPa):r r r r 2 3D( ∆ x , ) ˆ ˆmkτ = d0( ∆ xmk ) + d1( ∆ xmk ) τ + d2( ∆ xmk) τ + O( τ )(4)rr 2 3D ( ˆautoxk, a, τ ) = dauto( xa, k) τ + O( τ )(5)where ˆ τ = τ / δt,and δ t is the inter-sample time interval. Eqs. (4) and (5) were derived inPPa using the only Assumption 1S: the characteristics of each scatterer xi( t ), yi(), t zi(), tUi(), t Vi(), t Wi(), t and ∆ ni( t),i = 1, 2, ..., M, are locally statistically stationary randomprocesses. The term “locally stationary” is used in the paper in the same sense as in a theoryof the fine-scale turbulence, e.g., Monin and Yaglom (1975, sec. 21). It stands for stationarityover a time period which is much smaller than the integral time scale of the random process.Following Assumption 1S, the instantaneous velocity of each scatterer can be presented as asum of the mean and turbulent components: