L L E(k

m=l n=l

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m , rPn)e-2ikmIR( n)-rol

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. E*(k m , rPn+nJe2ikmIR( n+nJ-rol

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(11.4.19)

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Therefore, ACF imaging is the summation over many images of ACF focusing. Because ACF focusing is a random function of correlation angle rPd when 1"0 is not at a target. Comparing (11.4.18) and (11.4.19), we note that the clutter is further suppressed by having one more averaging in (11.4.18) than in (11.4.19).

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11 ACF AND DETECTION OF' BURtED OBJECT

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Figure 11.4.2 Simulated image of larget.s embedded iu clutter by field imaging for circular

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Figure] 1.4.:J Simulated image of targeh ClIIOO:!rkod in duLter!.ly ACF

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SAn.

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4.3 Simulations of SAn Data fllJd ACF Processiug

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Figure 11.4.4 Simulaled image of SAR.

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Por linear SAH as shown in Fig. 11.4.1, a monost.a.t.ic radar moves along linear path at height H abo\'e the target region. The radar positiou u; Function of the anLenm:~ posil.ioIJ x as R(x) = xx - diJ + Hz. Therefore, the reccinxl signals can be written as

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L e2Ikln(:I'I-r~1 J~,,(k) + I: e2ikIR(z)-rnlff~(k)

/1=1

fl:=]

(11.4.20)

where dIe received sigllal is a function of wave lIUlnber k (frequency) and antenna position x. The field imaging is to correlate the received signal with a reference signal Ef.' and then sum over frequency I. m and azimutJull position x m . C'p(f() = and (11.4.2lb)

!..II the numerical simulations, the reference signal E() is chosen to be unity

L L:E(J':m.Xf/)EQ(k",re-2ik,nI'R(:l:n)-r..1 (11.4.21a)

m:l "=1

11 ACF AND DETECTION OF BURIED OBJECT

for simplicity. The correlation defined above has well defined peaks when f 0 approaches the locations of the scatterers. It was shown that in circular SAR, the angular correlation imaging can give clear images. On the other hand, in linear SAR, spatial correlation imaging gives fine cross-range resolution and frequency correlation imaging gives fine range resolution. In general, frequency spatial correlation will give better performance in correlation imaging for linear SAR. The frequency angular correlation function (FACF) imaging is defined by

Cr(fo ) =

L L L

m,=l n,=l m2=1

)e-2ik=1IR(xnl)~rol

n2=1,k d >K

(11.4.22) The memory effect is avoided in the summation by choosing kd > K. The magnitude of the difference of the two wave vectors kd is given by kd = [(k m1 cos <PI sin el - km2 cos <P2 sin e 2)2 + (k m1 sin <PI sin el - km2 sin <P2 sin e2)2 + (k m1 COSel - km2 COSe2) 2] 1/2

(11.4.23)

where the angular pairs (e l , <PI) and (e2, <P2) are the orientation angles for vectors R(xnJ-fo and R(xnJ-fo , respectively. In the numerical simulations of correlation focusing, kd is chosen to be larger than ko /16, i.e., kd > K = ~6' where k o is the center wave number. Equation (11.4.22) reduces to FCF imaging with n2 = nl and to ACF imaging by letting m2 = mI. The FCF imaging is then given by

C~FCF)(fo) =

n=l rn,=l

m2=1,k d >K

. E*(k

x )e2ik=2IR(xn)~rol m2' n

(11.4.24)

and ACF imaging is written as

Cr(fo ) =

m=l n,=l

L L n2=1,k >K L

. E*(k m , XnJe2ik~IR(xn2)-rol

(11.4.25)

For the numerical simulations of linear SAR imaging, 80,000 small particles are used to model clutter. They are randomly distributed in a layer region of 40>'0 x 40>'0 x 0.5>'0' Four targets are placed at positions of (10,14,0)>'0'

4.3 Simulations of SAR Data and ACF Processing

(10,28,0)>'0' (30,10,0)>'0' and (30,32,0)>'0' The dielectric constant of the targets and the scatterers are equal to (3.23 + iO.36). The back-scattering amplitudes for the targets and the particles are calculated based on Mie scattering. The receiver moves from X s = -d/2 to X s = d/2 with an increment of d/100. The horizontal position is Ys = d with d = 1732>'0 and the height is Zs = H = 1000>'0. The received signal is calculated over a frequency band 0.5io to 1.5io with an increment of O.Olio and 100 azimuthal positions with equal space. The simulated data is processed by the methods of field imaging and FACF imaging. The normalized results are shown in Fig. 11.4.5, and 11.4.6. Figure 11.4.5 shows field imaging of C2F(r o) = ICF(roW. The 4 targets are obscured by the background clutter. Figure 11.4.6 shows the results of FACF imaging with the memory effect avoided, which has lower clutter level than that in Fig. 11.4.5. The results of FCF imaging and ACF imaging are also shown in Fig. 11.4.7 and 11.4.8. We see that ACF imaging loses range resolution and FCF does not have good cross-range resolution. Therefore, FACF imaging gives better performance than ACF imaging and FCF imaging for linear SAR. To compare results quantitatively, we define the visibility of targets in clutter as a ratio of target signal and the average signal strength for the entire image covering the region. The visibility is calculated by

Max(II(ro)l)