Clutter from a Flat Surface

performed during the observation time to. Since many terms on the right-hand side of this equation are range-dependent, R_mc must generally be found graphically or by root-finding. Exceptions to this general rule, where this equation may be writ-ten in closed form for surface and to volume clutter, are given in Sections 3.3.5 and 3.4.6.

The graphical procedure is to plot separately the signal energy E from (1.25) and required energy I_{0 e}D_x(n) = (N0 + C_{0 e})D_x(n) as functions of target range R.

The detection requirement is met at ranges for which E  I_{0 e} + D_x(n) and the longest such range is R_mc. The root-finding procedure is to express E and I_{0 e}D_x(n) as separate functions of R, and create a computer program to find the largest R at which the two expressions are equal:

 

   

mc rootR e x

R  E R I R D n  (3.13)

where root_x[·] is the value of x at which the equality within the brackets is met.

3.3 DETECTION IN SURFACE CLUTTER

Surface clutter is backscatter from land or sea surfaces illuminated by the radar beam. Two geometrical models of surface clutter are described below: a simple, flat-Earth model, and a spherical-Earth model that is detailed further in Chapter 8.

3.3.1 Clutter from a Flat Surface

The geometry of surface clutter observed by a the mainlobe of a radar antenna at altitude h_r above a flat surface is shown in Figure 3.1. The clutter cell lies within

Radar Equations for Clutter and Jamming 63

the elliptical beam footprint defined by the beamwidths _e in elevation and _a in azimuth, reduced in each case by the beamshape loss L_p to account for the reduced two-way antenna gain of off-axis scatterers.¹ Sidelobe clutter is normally much smaller and insignificant, but is discussed in Section 3.6.

The width of the cell is set by the range R_c and the azimuth beamwidth _a. The depth (range dimension) is usually set by the range resolution _r:²

 

where _n is the width of the processed pulse and c is the velocity of light. As stated by Blake [3, p. 297], _n is the equivalent pulsewidth after processing. This is not the 3-dB width ₃ conventionally used, but rather a pulsewidth defined in [4, p.

342] in a way similar to the noise bandwidth of a network:

 

where a(t) is the pulse waveform voltage at the processor output and a_m is its peak value. The ratio n/3 varies from unity for a rectangular output pulse to approxi-mately 1.06 = 0.24 dB for the outputs of matched filters for rectangular pulses and for most pulse compression waveforms.

1 The beamshape loss used in equations for clutter is Lp = 1.33, as obtained for dense sampling by integration over the two-way beam pattern (5.1), and denoted by Lp0 (see Section 5.2.3). The addi-tional subscript 0 is used there to distinguish the dense-sampling value from the varying Lp applica-ble to targets for the general case that may include sparse sampling, as discussed in Chapter 5. The subscript 0 is omitted from equations in this chapter for compactness.

2 The resolution cell depth for CW radar is discussed in Section 3.3.6.

h_r

Figure 3.1 Surface clutter geometry.

Projected onto the surface, the depth of the beam footprint is increased by the cosecant of the grazing angle :

 

The beam footprint usually extends beyond the range resolution cell, but as  increases, and especially for narrowband waveforms, the range resolution cell may extend beyond the footprint. This leads to two alternative expressions for the sur-face area included in the resolution cell.

3.3.1.1 Pulsewidth-Limited Cell

This cell depth is limited by the pulsewidth when

cot (m)

The area contributing to the surface clutter for this case is³

 

The need to modify the half-power beamwidth in (3.18) by the factor L_p, now called beamshape loss, is recognized in early work on clutter [5, p. 483. Eq. (93)], but that factor is used inconsistently in subsequent work, perhaps because error from its omission is considered unimportant in comparison with other uncertain-ties in clutter modeling. Blake’s simple formulation [3, p. 26, Eq. (1.43)] omits this factor, but his more detailed discussion of clutter [3, p. 296, Eq. (7.7)] pre-sents an expression that invokes integration over the beam pattern.⁴ Other litera-ture uses a “two-way half-power beamwidth” defined as _a 2, which is 0.26 dB less than the correct value.

3 The beamwidth in (3.18) is assumed small enough that the small-angle approximation applies. Oth-erwise, a should be replaced by 2tan(a/2); the error in approximation is < 0.01 dB for a < 10.

4 That equation contains a typographical error in using the one-way voltage pattern f (,) rather than the two-way power pattern f (,)⁴.

Radar Equations for Clutter and Jamming 65

3.3.1.2 Beamwidth-Limited Cell

When the condition specified by (3.17) is not met, the surface clutter area is

 

2

2

= ^{c i a c i e}csc ^{c i}2^{a e}csc m

c i

p p p

R R R

A L L L

   

   (3.19)

For this beamwidth-limited case, the factor 1/L²p is the same as the integral in the weather-radar equation [6, p. 590, Eq. (5)], and in Blake’s detailed discussion pre-viously cited. The factor /4 = 1.05 dB, appearing in Blake [3, p. 296, (7.4)] and elsewhere, is inaccurate for actual antenna patterns, to which 1/L²_p = 2.48 dB ap-plies.

3.3.1.3 Unequal Transmitting and Receiving Beamwidths

The azimuth beamwidth used in calculating clutter area is the one-way, half-power beamwidth _a when the transmitting and receiving beam patterns are iden-tical. When they differ, an effective beamwidth may be substituted, calculated as

eff 2 2

2 _{at ar}

a

at ar

   

   ^(3.20)

It can be seen that _aeff = _at = _ar when the two are equal, approaching 2_a2 when a1 >> a2. For the latter case, the angular width of the clutter cell becomes

2 2

2_a L_p1.06_a . These relationships apply also to elevation beamwidth in (3.19) and in expressions derived in Section 3.4.1 for volume clutter.

3.3.1.4 Effect of Range Ambiguities

As discussed in Section 3.2, clutter energy originates not only from the resolution cell at the target range R but also in range ambiguities at intervals R_u within and beyond R. The range R_{c i} from the ith ambiguity is then used in (3.18) and (3.19) to find an area A_{c i} for each ambiguous clutter region, and (3.5) applied to find the total clutter energy. Multiple ambiguities may also occur for MPRF and HPRF in the beamwidth-limited case.