6.3.1.1 SPECTRAL PROCESSING

LCE long-time-dwell clutter data were acquired with a stationary antenna over 70-s data recording intervals called experiments. Each LCE experiment involved the recording of 80 I and Q sample pairs per pulse repetition interval (PRI) at 2- MHz sampling frequency (i.e., 75-m range gate spacing), leading to a 6-km total recorded range swath. Clutter experiments were usually taken in sequential groups covering the various LCE polarization combinations. Phase One long-time-dwell clutter data were acquired similarly to LCE data. Table 6.3 provides the specific Phase One clutter data acquisition and spectral processing parameters applicable to the Phase One spectral results shown herein from Katahdin Hill.

All the LCE clutter power spectra shown subsequently were computed directly as fast Fourier transforms (FFTs) of the sampled temporal pulse-by-pulse return, including the dc component. A four-term Blackman-Harris window function was utilized, with highest sidelobe level at −92 dB [37]. Each temporal record of 30,720 pulses (first 61.44 s of each 70-s experiment) was divided into contiguous groups of 5,120 samples; a 1,024-point complex FFT was generated for each group by utilizing every fifth pulse; and the amplitudes of the resultant set of FFTs were arithmetically averaged together in each

Doppler cell to provide the spectrum illustrated. Thus each LCE spectrum shown is the result of averaging six individual spectra (each from a 1,024-point FFT) from an overall record of 1.024-min duration, using an effective 10-ms PRI and an effective 100-Hz PRF. The CPI for each FFT was 10.24 s.

Table 4.10 in Chapter 4 indicates that the typical correlation time for windblown trees at L-band is ∼ 1 s (see also Table 6.3). Thus the CPI utilized in LCE spectral formation, usually being ∼ 10 times the correlation time, is more than adequate to allow the random process to fully develop. The Phase One clutter spectra shown were computed similarly to the LCE spectra. Table 6.3 provides the particular Phase One spectral processing parameters utilized in generating the Phase One clutter spectra from Katahdin Hill. Table 6.3 includes both the CPI (i.e., time dwell per FFT) used and the typical correlation time of windblown trees for each Phase One frequency band, as specified by Table 4.10. These numbers indicate that the CPI covers many correlation periods at each of the five Phase One radar frequencies.

6.3.1.2 SYSTEM STABILITY

For a steady target, the spectral processing of either the Phase One or LCE radar yields a very narrow spectrum containing only dc power at zero-Doppler velocity. Figure 6.4 shows results from such steady targets. Figure 6.4(a) shows the measured LCE clutter spectrum from a desert terrain cell under very still (0 mph) wind conditions. Figures 6.4(b) and (c) show measured Phase One spectra from a large municipal water tower at L- and X-bands, respectively; in each case, clutter from trees in the same cell as the water tower just begins to broaden the spectrum near the base of the water-tower dc spike. Figure 6.4(d) shows the measured Phase One VHF spectrum from a cell containing tall grass; the windblown motion of the grass is indiscernible in this VHF measurement. In all these results, the width of the dc spectral component from the steady target is essentially the limit of spectral resolution provided by the Blackman-Harris window function, which is cleanly maintained over the full spectral dynamic range of the radar down to the system noise level (i.e., 71 to 77 dB down for Phase One, 80 dB down for LCE). The window function sidelobes occur below the noise level of either system.

6.3.1.3 SPECTRAL NORMALIZATION

Chapter 6 shows measured clutter spectra normalized to compare with analytic representations of clutter spectral density. The first step in spectral normalization is to convert the FFT output from power per spectral resolution cell to power/(m/s) so as to be directly comparable with analytic spectral shapes defining continuous density functions. This conversion is performed by dividing each point in the FFT velocity spectrum by the width of the Doppler velocity resolution cell Av. The Doppler velocity resolution cell is wider than the sampling interval by a factor equal to the equivalent noise bandwidth ENBW of the window function. Thus Δv = (λ/2) ·(PRF/N) ·ENBW, where N is the number of points in the FFT. For the four-term Blackman-Harris window used here, ENBW = 2.004 [37]. The second step in spectral normalization is to bring the power in the measured spectrum to unity for comparison with analytic spectral shapes for which the integral over the entire velocity domain is unity. In some circumstances, total power in the spectrum is brought to unity by dividing each spectral point by total spectral power. In other circumstances, where concern is only with the ac spectral shape and not the particular amount of dc power present, the ac power in the spectrum is brought to unity by dividing each spectral point by total spectral power times 1/(r + 1).

6.3.1.4 SPECTRAL POWER

Total, dc, and ac spectral power were computed in the time domain for the Phase One and LCE spectral results. Total power over the temporal record of each FFT is 1/N times the sum of the squares of the I and Q samples of received power. The dc power over the same temporal record is the square of 1/N times the sum of the I samples plus the square of 1/N times the sum of the Q samples (i.e., coherent sum). The ac power over the same temporal record is total power minus dc power. These computations were performed over each FFT contributing to each spectrum. The final time-domain quantities for total, dc, and ac spectral power are each means of the resulting set of values, one value per FFT, applicable to each spectrum.

The dc power obtained by summing coherently over a temporal record of backscatter from windblown trees depends on the length of the record (the CPI) over which the summation is performed. If the random process were well behaved (i.e., stationary), the coherent sum would converge and be largely independent of record length for lengths much greater than the correlation period of the process. However, windblown clutter backscatter records are not rigorously stationary. The statistics of the process sometimes appear to be characterized as intervals of stability separated by abrupt transitions from one stable state to another. Such abrupt changes can be caused, for example, by large tree limbs suddenly shifting position. Because the coherent sum does not always converge, the ratio of dc/ac power computed in the time domain is dependent on CPI duration.

Table 6.4 provides two examples of the dc/ac ratio obtained as a function of CPI duration for two LCE long-time-dwell backscatter measurements from cells containing windblown trees, one taken under very light wind conditions and the other under very windy conditions. The correlation times for these two experiments are estimated to be ∼ 4 s for the light-air data and ∼ 1 s for the windy data. The clutter spectra from these two measurements are discussed in Section 6.3.2.1. It is evident that a significant dc component exists in the light-air data in Table 6.4 and very little dc power exists in the windy data. The results in the table may be interpreted in terms of decreasing sampling bandwidth of the zero-velocity Doppler filter with increasing CPI, where sampling interval bandwidth is given by λ/(2 × CPI). Because no weighting is employed in these time-domain computations, the filter has a sin x/x response.

In the frequency domain, the zero-th Doppler bin can contain both a singular dc power component existing as a discrete delta function and ac power existing as a continuous density function. The ac power in the zero-th Doppler bin decreases with increasing resolution (i.e., with increasing CPI duration), whereas the singular dc component is theoretically independent of resolution. However, in all the measured clutter spectra shown here, normalization (division) of power/cell by Δv was performed for all cells, including the zero-Doppler cell. First-order credibility checks of the reasonableness of dc/ac ratios provided herein for spectra containing strong dc components (i.e., strong dc spikes, for example, in desert or cropland or at VHF) should therefore be performed by first decreasing the height of the spike by |Δv| dB before estimating the resultant dc/ac ratio. Note that the peak spectral level at zero Doppler (before or after division by Δv) is not itself used in any direct way in Chapter 6 to normalize measured spectral data (measured spectra are never simply aligned by peak level).

One reason that the zero-Doppler cell is not normalized differently from others is to maintain the relative shape of the raw FFT before normalization (whatever is done to one cell is done to all cells). Another reason is that the power in the zero-Doppler cell of windblown foliage spectra is often not dominated by a singular dc (i.e., delta function) component, in which circumstances the continuous power in the zero-Doppler cell requires normalization by Δv similarly to all other Doppler cells. Excess quasi-dc power in Doppler resolution cells near the zero-Doppler cell is included as dc power in computing the dc/ac power ratio used as spectral modeling information in spectra where excess quasi-dc power exists. The dc/ac ratio in which the excess quasi-dc component is included in the singular dc term is obtained in the frequency domain by best-fitting the spectral tail at high Doppler velocities with an exponential ac shape function. The excess quasi-dc power is that which exists above the approximating exponential in cells of very low Doppler velocity close to zero Doppler. The fitting process is largely independent of spectral resolution as long as the resolution is adequate to define the exponential spectral tail. As a result, the dc/ac ratio used herein, which is that required for the ac exponential shape function to match the measured spectral tail both in relative shape and in absolute level, is also largely independent of CPI duration and spectral resolution. This fitting process is discussed in Section 6.4.2.

6.3.2.1 VARIATIONS WITH WIND SPEED

Figures 6.5–6.7 are examples of LCE-measured windblown forest clutter spectra under windy, breezy, and light-air conditions, respectively. The data in these figures are normalized to show ac spectral shape P_ac(v) plotted against a logarithmic Doppler velocity axis similar to the modeled curves of Figure 6.1. Each figure compares the measured ac spectral shape with several exponential shape functions of various values of shape parameter β. In Figure 6.5, the measured data follow the exponential curve of shape factor β = 5.2 remarkably closely over the full spectral dynamic range shown. This match of measured ac spectral shape with exponential is among the best in the Lincoln Laboratory database, although other examples exist both of LCE and Phase One windy-day clutter spectra with equally good fits to exponential. Furthermore, the spectrum of Figure 6.5 is among the widest measured; its shape factor β = 5.2 is the basis of the “worst case, windy” specification in Table 6.1 and Figure 6.1.

Also shown in Figure 6.5 is a narrower Gaussian spectral shape function. The particular Gaussian shape shown corresponds to Barlow’s [7] much-referenced 20-dB dynamic range historical measurement (see Section 6.6.1.1). It is evident in Figure 6.5 that the overall rate of decay in the LCE data is much more exponential than Gaussian in character. Thus these LCE data support the general consensus of agreement subsequently arrived at [10–20, 27, 28] of spectral tails wider than Gaussian in windblown clutter spectra. Li [19] explains that tails wider than Gaussian are theoretically required by branches and leaves in oscillatory—as opposed to merely translational—motion.

The LCE spectral data of Figures 6.6 and 6.7 indicate that measured ac spectral shapes remain reasonably well represented by exponential shape functions under less windy conditions, with increasing values of exponential shape factor with decreasing winds (i.e., β = 8 for breezy conditions, β = 12 for light-air conditions). The results of Figures 6.6 and 6.7 are representative of many similar spectra measured in other cells on other days. Recall that normalization to P_ac(v) requires raising the P_tot(v) spectrum by 10 log₁₀ (r + 1) decibels on the vertical ordinate. The value of r applicable in Figures 6.5, 6.6, and 6.7 is 0.7, 18.9, and 29.8 dB, respectively. It is evident in Figures 6.6 and 6.7 that the measured data begin to depart from the approximating exponentials for v < 0.2 m/s as they begin to rise into the quasi-dc region, contributing to the large values of r in these data. As the amount of dc spectral power increases, less spectral dynamic range is left for measuring the ac power in the spectral tail, indicated by the rapidly rising effective system noise levels with respect to P_ac(v) as wind speed decreases and dc/ac ratio increases in Figures 6.5–6.7.

As discussed later (Sections 6.6.1 and 6.6.2), power-law spectral tails plot as straight lines in plots of 10 log₁₀P vs log₁₀v such as those of Figures 6.5–6.7. Thus a power law of shape n = 3 (30 dB/decade) fits the data of Figure 6.6 reasonably well down to the P_ac = −20-dB level.

However, this n = 3 power law cannot be extrapolated to lower levels. The local power-law (local slope tangent to the data curve) rate of decay in Figure 6.6 strongly increases (to n = 6 or 7) at the lower spectral power levels in Figure 6.6. Likewise, at first consideration, an n = 4 power law (40 dB/decade) might be thought to be a reasonable match to the measured data of Figure 6.7. The apparent goodness of this straight-line fit to the data in Figure 6.7 is heightened by the data beginning to rise above the exponential in the low-Doppler quasi-dc region v < 0.2 m/s and by the data flaring away from the exponential toward the noise level as they become limited in signal-to-noise (S/N) ratio when approaching to within 10 dB of the noise floor at higher Doppler velocities around 1 m/s. However, these two effects tend to obscure a more fundamental exponential-like rate of decay (increasing local tangent slope with decreasing power level) as shown in the region 0.2 < v < 0.7 m/s in Figure 6.7, and the upper-level power-law rate of decay can no more be extrapolated to lower levels in the light-air data of Figure 6.7 than in the windier data of Figures 6.5 and 6.6. To do so would lead to physical implausibility as the upper-level light-air power law would extrapolate to lower-level ac power levels (e.g., P_ac = −60 dB) exceeding in spectral width those measured under windy conditions at the same lower levels.

Not all the Phase One and LCE clutter spectra measured under breezy and windy conditions are as closely exponential as the measured spectrum in Figure 6.5. However, like that spectrum, they all demonstrate increasing downward curvature (convex from above) with increasing Doppler velocity and decreasing power level as their most characteristic general feature in plots of 10 log P vs log v such as that of Figure 6.5. The main reason the exponential form is used herein for modeling clutter spectral shapes is that it, too, when plotted as 10 log P vs log v, possesses this increasingly downward curving shape (see Figure 6.1) while remaining wider than Gaussian as required by the measured data (see Figure 6.5). In contrast, the spectral tails of power-law functions do not have increasing downward curvature on 10 log P vs log v axes but plot linearly (i.e., extrapolate rapidly to excessive spectral width) on such axes. This matter is further discussed in Sections 6.6.1 and 6.6.2.

Most of the Phase One- and LCE-measured clutter spectra are not completely and precisely representable by any simple analytic function over their full spectral ranges. Many of these measured spectra are somewhat wider than exponential (i.e., are concave from above in 10 log P vs v plots), but they are almost always much narrower than power law (i.e., are convex from above in 10 log P vs log v plots). In such circumstances, absorbing excess quasi-dc power in the dc term usually gives the exponential model the flexibility to match the relative shapes and absolute levels of the measured spectra over extensive spectral tail regions.

Gale Force Winds (Scaled Estimate). Figure 6.8 shows a different set of three LCE-measured windblown-forest clutter spectra under light-air, breezy, and windy conditions, displayed as 10 log P_tot vs v. In contrast to the light-air, breezy, and windy spectra of Figures 6.5–6.7 (which come from three different range cells) the spectra shown in Figure 6.8 are from the same 7-km range cell on three different measurement days. These Figure 6.8 spectra clearly indicate that spectral extent from a given range cell increases strongly with increasing wind speed. In approximate measure, the data of Figure 6.8 indicate similar-sized steps of increasing spectral width for wind speeds increasing by approximate factors of 3 (from 1–2 to 6–7 mph, and from 6–7 to 18–20 mph). This observation implies that spectral width increases approximately linearly with the logarithm of wind speed [28, 38]. Such data are the basis of Eq. (6.3) specifying spectral width as a function of wind speed in the clutter model of Section 6.2. In Figure 6.8, the maximum spectral extent in the data 70 dB down from their zero-Doppler peaks is ∼ 1, 2, and 3 m/s for the light-air, breezy, and windy spectra, respectively. In these results, as ac clutter power increases and spreads out with increasing wind speed, dc clutter power decreases, as indicated by dc/ac ratios r of 0.1, −1.5, and −4.5 dB for the light-air, breezy, and windy spectra, respectively.

Figure 6.9 shows the same three spectra of Figure 6.8, now displayed as 10 log P_tot vs log v. Figure 6.9 also shows a scaled extrapolation to higher wind speeds by a further factor of 3, that is, from the 18–20 mph of the “windy” spectrum to 54–60 mph gale force wind speeds. This estimate was obtained by finite-difference extrapolation of the light-air, breezy, and windy Doppler velocities, say v_a, v_b, and v_c, to v_d, the estimated gale force Doppler velocity, at multiple spectral power levels, assuming constant factors-of-3 increases in wind speed throughout. It is evident in the figure that the gale force spectral estimate is well modeled by an exponential curve of shape factor β = 4.3; this is the basis of the “typical” gale force exponential ac shape parameter specification β = 4.3 in the clutter model of Section 6.2. Increasing gale force β in this model from its typical specification based on the scaled estimate shown in Figure 6.9 to a worst-case specification of β = 3.8 brings it into very close agreement (in terms of gross spectral extent at the −14-dB level) with the only known measurements of windblown clutter under actual gale wind conditions, namely, the very early measurements of Goldstein [9] that are further discussed in Section 6.6.3.5.

6.3.2.2 INVARIANCE WITH RADAR FREQUENCY

The idea that spectral extents of windblown ground clutter Doppler-velocity spectra are in large measure invariant with radar frequency, or equivalently, that spectral widths in Doppler-frequency spectra are approximately proportional to radar frequency, has been discussed in the technical literature of the subject since the early days of radar development [6, 8, 9]. For example, early work in comparing spectral widths with radar frequency conducted at the MIT Radiation Laboratory during World War II by Herbert Goldstein and others is summarized by Goldstein’s conclusion that “The widths of the [Doppler-frequency] spectra … increase with wind speed and … appear to be essentially proportional to [radar] frequency” [8, 9]. This idea remains true in the LCE and Phase One spectral results, as indicated in Figures 6.10 and 6.11—Figure 6.10 shows VHF, L-, and X-band Doppler-velocity forest spectra under windy conditions; Figure 6.11 shows UHF, L-, and S-band forest spectra under breezy conditions.

However, the early results were mainly in the 1- to 10-cm range of wavelengths and, by today’s standards, over very limited spectral dynamic ranges (∼ 20 dB). Results such as those of Figures 6.10 and 6.11 extend the idea of frequency invariance of windblown clutter Doppler-velocity ac spectral shape over very much greater spectral dynamic ranges (> 60 dB) and to very much longer radar wavelengths [from X-band (λ = 3.3 cm) to VHF (1.8 m)]. These results are rather surprising, since the dominant, wavelength-sized scatterers at VHF (large branches, limbs) are presumably different than those at X-band (leaves, twigs). As will be shown, much more dc power exists in VHF windblown clutter spectra than at higher radar frequencies, for one reason because the VHF energy partially penetrates the foliage to reach the underlying stationary tree trunks and ground surface. As a result, the ac spectral power at VHF is measured at lower levels in the available spectral dynamic range. Still, over very many spectral measurements of windblown trees at VHF and UHF, ac spectral spreading caused by internal motion in windblown clutter generally exists at VHF and UHF at lower absolute levels of P_tot(v) but roughly equivalently in the relative shape and extent of P_ac(v) to that observed in the higher, L, S, and X microwave bands.

In Figure 6.10, the VHF and X-band spectra were measured by the Phase One radar at Katahdin Hill under windy conditions on two different days in April at 2.8-km range. The L-band spectrum was measured by the LCE radar at Wachusett Mt. on 11 September at 6-km range. The Figure 6.11 spectra were all measured by the Phase One radar at Katahdin Hill also at 2.8-km range under breezy conditions in late April or early May. In general measure, the three windy-day spectra of Figure 6.10 are essentially identical in overall ac spectral shape, as are the three breezy-day spectra of Figure 6.11. Of course, temporal (minute-to-minute, hour-to-hour) and spatial (cell-to-cell, site-to-site) variability exist in LCE- and Phase One-measured clutter spectra under nominally similar wind conditions, and not all such measurements overlay one another as exactly as those shown in Figures 6.10 and 6.11. Concerning variability, even in the results of Figures 6.10 and 6.11 for which the same nominal range and azimuth apply, the spatial cells still encompass different overlapping ground areas due to the different azimuth beamwidths. Also, “… there are the usual uncertainties [because of the lack of] … simultaneity of the measurements in time” [9].

In considering possible means by which variations with radar frequency might be introduced in clutter velocity spectra, amplitude fluctuations caused by scatterer rotation and the wig-wag shadowing of background leaves by leaves in the foreground have been discussed [14, 19, 29, 39] as possible mechanisms that might complicate clutter spectra over and above phase fluctuations caused by the scatterer velocity distribution. However, one theoretical model exists [14] that incorporates scatterer rotational and shadowing effects and still provides radar frequency-independent clutter Doppler-velocity spectral shapes (i.e., “to a first approximation, the spectrum … depends only on the product λf” [14]). It is not suggested here that if multifrequency spectra could somehow be measured simultaneously from exactly the same spatial assemblage of windblown foliage, fine-scaled specific differences would not be observed in ac spectral shape with radar frequency. However, in looking across all the LCE- and Phase One-measured spectral data and the variations that exist therein, no significant trend is observed in ac Doppler-velocity spectral shape with radar frequency, VHF to X-band, as opposed, for example, to the strong trend seen in ac spectral shape with wind speed.

6.3.2.3 INVARIANCE WITH POLARIZATION

The LCE and Phase One clutter spectral data indicate that clutter spectral shape from windblown vegetation is largely independent of radar polarization, to the extent that this can be determined in non-simultaneous measurements. Figure 6.12 shows one set of three sequential LCE measurements of windblown treed-cell ground clutter spectra at VV-, HH-, and HV-polarizations obtained at approximately 2-min intervals, which are of essentially identical spectral shape. The spectral artifact in the HV-pol. spectrum of Figure 6.12 at ∼ 2.8 m/s is probably a bird. Many other LCE and Phase One VV- and HH-pol. spectra were compared from common cells selected from other sites and experiments [21]. These usually showed little or no variation in spectral shape with polarization. When variations of spectral shape with polarization did occur, such variations were usually relatively random with little evidence for the existence of any strong general effect on spectral shape with polarization.

Other investigators have also found little effect on windblown clutter spectral shape with polarization. For example, Kapitanov et al. [13] observed that “The spectra of [X-band] echo signals from forest for different polarizations [vertical and circular] … are on the average similar.” Unpublished X-band windblown clutter spectral results obtained at Lincoln Laboratory by Ewell [20] at VV-, HH-, and circular polarizations in shrubby desert terrain also indicated no consistent differences in the spectral shapes obtained at the various polarizations.

6.3.2.4 TEMPORAL VARIATION

All LCE clutter spectra in Chapter 6 are averages of six individual 1,024-point FFTs, each formed from a 10.24-s duration temporal backscatter record. Windy-day wind conditions frequently vary considerably from one 10-s interval to the next. Such variability of windy-day conditions within a treed resolution cell from one 10.24-s interval to the next often leads to significant variability in successive individual 10.24-s dwell FFTs [21]. However, the results of Figures 6.13 and 6.14 indicate that over longer periods of 60- to 80-s, windy-day treed-cell clutter spectra can be expected to become more stationary. Figure 6.13 shows LCE treed-cell clutter spectra for three repeated experiments on a windy day, each of which is formed from an overall temporal record of 61.44 s (100-Hz PRF, 1024-point FFTs, 6 FFTs averaged). The shape of the spectrum from the first experiment is essentially identically replicated by the shape of the spectrum from a following experiment begun 65 min later, suggesting that enough averaging of wind variations occurs within 1 min to lead to some degree of convergence in average spectral shape. But wind is an extremely nonstationary dynamic random process with complex short- and long-term variation. For example, on the windy/gusty day on which the data of Figure 6.13 were collected, the gusts happened to die down over the 70-s interval covering the second experiment (begun 9 min after the first), and the spectrum formed from that data is indeed considerably narrower than the other two.

Figure 6.14 shows Phase One L-band clutter spectra for three repeated breezy-day experiments for very nearly the same treed cell at Wachusett Mt. for which the LCE data of Figure 6.13 apply. Each of these Phase One spectra is formed from a temporal record of 81.9-s duration (125-Hz PRF, 2048-point FFTs, 5 FFTs averaged). The spectrum from the first experiment is nearly identical to that of the second experiment, begun 4 min later. The spectrum from the third experiment, begun 9 min after the first, is somewhat narrower. In a similar set of five sequential experiments begun 15 min before those of Figure 6.14, the range of variability of spectral shape was similar. The slightly changing average wind conditions within 81.9-s intervals over 4- or 5-min periods resulted in only very small changes in the measured spectra, such as shown in Figure 6.14. Such results indicate that the range of variability in clutter spectra formed by averaging over 60- to 80-s data intervals under nominally similar wind conditions generally is quite low, compared with the more variable individual FFTs formed from 10- to 16-s data intervals during the same period.

6.3.2.5 EFFECTS OF SITE/SEASON/TREE SPECIES/CELL SIZE

It is not difficult to find Phase One and LCE spectra of essentially identical exponential spectral shape—two are shown in Figure 6.15. The LCE spectrum (a) measured on September 10 (leaves on deciduous trees) at Wachusett Mt. at 7.9-km range, HH-pol., 150-m range resolution, and 2° depression angle essentially overlays and replicates the Phase One spectrum (b) which was measured on May 3 (leaves not yet emerged) at Katahdin Hill at 2.4-km range, VV-pol., 15-m range resolution, and 0.5° depression angle. These two measurements were obtained with different radar receivers, and the two spectra were produced using different data reduction and processing software. Thus commonality can exist in spectral shape, in large measure because of the large cells and large degree of spatial averaging involved, despite a host of underlying differences including measurement instrumentation and parameters (range, cell size, illumination angle, polarization), site, and time of year.

The Phase One and LCE radars mimic long-range ground-based surveillance radars in the relatively long ranges, large resolution cell sizes, and low illumination angles of their measurements. Cell sizes, typically several hundred meters on a side, are large enough to encompass a large spatial ensemble of scatterers (many trees) as well as variable local wind currents within the cell. Because of the complexity of the scattering ensemble and nonuniform winds within such large cells and the temporal and spatial variability of the ensemble from cell to cell over large numbers of cells, it is difficult to discern significant site-to-site differences or significant trends with season and/or tree species in the Phase One and LCE spectral data. Other, more fine-scaled investigations, both historical [13–16] and recent [40–42], involving small illumination spot sizes (e.g., 1.8-m diameter [42]) on individual trees at short ranges (e.g., 30 m [42], 50 m [16]), provide results showing variation on treed-cell spectral shape with the type or species of trees. However, such small differences are largely absorbed within the general ranges of statistical variability in the Phase One and LCE measurements and are thus of limited consequence for the longer ranges and larger cells of surveillance radars.

From Phase One Katahdin Hill measurements acquired from a specific set of forested cells once a week over a nine-month period [21], L-band spectra were examined for seasonal variations in three wind regimes, viz., calm to light-air conditions, light-air to breezy, and windy. These results marginally showed that in each wind regime, spectral widths 60 to 70 dB down were only very slightly wider by no more than ≈0.5 m/s for summer measurements (leaves on deciduous trees) than for winter measurements (leaves off deciduous trees). An early study of Phase One spectra involved eight forested sites, four in western Canada and four in the eastern U.S. For the western Canadian sites, the dominant tree species were aspen and spruce. For the eastern U.S. sites, the forest was mixed (oak, beech, maple, hemlock, pine). Measured Doppler-velocity spectra generally showed no significant major differences in shape or extent from one forested measurement site to another, either within each group or from group to group. No significant discernible difference has been observed in the shapes of Phase One spectra from cells of 150-m range resolution compared with cells of 15-m range resolution.

6.3.3.1 VARIATION WITH WIND SPEED

Forested ground clutter cells contain many scatterers. Each scatterer is positioned randomly within the cell and hence produces an elemental scattered signal of random relative phase with respect to the other scatterers. Some of the scatterers, such as leaves and smaller branches, move in the wind, producing fluctuating signals with time-varying phases. Other scatterers, such as tree trunks and larger limbs, are more stationary, producing steady signals of fixed phase. The total clutter signal is the sum of all the elemental backscattered signals, both steady and fluctuating. At high wind speeds, most of the foliage is in motion, and the ratio r of dc to ac power in the clutter spectrum is relatively low. In such circumstances, and in the higher microwave bands where little foliage penetration occurs, the steady component can become vanishingly small, whereupon Eq. (6.1) simplifies to P_tot(v) ≅ P_ac(v). Goldstein correctly anticipated, however, that “As the wind velocity decreases, … the steady-to-random ratio [i.e., r] would be expected to increase” [9]. Thus under light winds, a large proportion r/(r + 1) of the clutter power is at dc.

Even so, the small proportion of clutter power 1/(r + 1) that, under light winds, remains at ac can still troublesomely interfere with desired target signals. Therefore, it is necessary that a windblown clutter spectral model quantify the dc/ac ratio r expected from forested or other types of vegetated cells as a function of wind speed.

Figure 6.16 shows an LCE-measured clutter spectrum from a treed cell under very light wind conditions. The most striking characteristic of this light winds spectrum is its extreme narrowness, with spectral spreading occurring only at relatively low power levels and to relatively small extent in Doppler. This spectrum contains a large steady or dc component, and since the spectral density decays smoothly and continuously (albeit rapidly) away from the peak zero-Doppler level, it also contains high levels of quasi-dc power at very low but non-zero Doppler velocities. This spectrum may be approximately modeled utilizing a value of r = 29.8 dB, in which excess quasi-dc power is included in the dc term, and an exponential shape function for the spectral tail of shape parameter β = 12. The spectrum of Figure 6.16 is generally representative of many similarly narrow LCE and Phase One clutter spectra measured in other treed cells and on other light wind days.

Figure 8.18 An LCE windblown forest clutter spectrum measured under light wind conditions.

The particular value of dc/ac ratio r applicable to any given treed clutter cell is highly variable. Such variability is illustrated in the results shown in Figure 6.17, in which ratios of dc to ac spectral power obtained from LCE clutter measurements from many treed cells are shown as a function of wind speed. There is some difficulty in precise specification of wind speed in such results since anemometer measurements usually provide only a one-point-in-space indication of wind conditions for the total test area. Over and above the inherent variability in these data, Figure 6.17 indicates a strong trend such that the ratio of dc to ac spectral power in dB decreases approximately linearly with the logarithm of wind speed w.

8.3.3.2 VARIATION WITH RADAR FREQUENCY

Whether a scatterer in a forested clutter cell is classified as stationary or in motion depends on the radar wavelength. A back-and-forth scatterer motion of 3 cm would produce a steady signal of essentially fixed phase at VHF, but a fluctuating signal passing through all possible phases at X-band. Furthermore, at VHF and UHF, significant energy penetrates the foliage to scatter from the stationary underlying ground surface, whereas at X-band little or no energy reaches the ground. Again, Goldstein correctly anticipated that, because of such effects, the dc to ac ratio in windblown clutter spectra “… should therefore decrease with [decreasing] wavelength” [9].

Figure 6.18 shows a Phase One-measured spectrum at VHF from a treed cell under windy conditions (same spectrum as shown in Figure 6.10). It is evident in Figure 6.18 that a large dc component exists in this VHF clutter signal such that the ratio of dc to ac power in the spectrum is 14.8 dB. In contrast to the large dc component in the light winds spectrum of Figure 6.16, in which significant quasi-dc spectral power also occurs, in the lower frequency VHF spectrum of Figure 6.18 the dc component exists largely as a discrete delta function at the spectral resolution of the processing in the zero-Doppler bin. Much smaller dc components occurred in measured spectra at higher radar frequencies from the same forested cell under similarly windy conditions. Although there is a large dc component in the VHF spectrum, it also contains a significant amount of ac power of considerable spectral extent. This VHF spectrum comes from a single FFT of 61.44-s CPI (i.e., no averaging), this relatively long CPI being required to provide adequate spectral resolution at this relatively low radar frequency. The VHF spectrum of Figure 6.18 is representative of many other Phase One-measured VHF clutter spectra from other forested cells and on other windy days.

Figure 6.19 shows ratios of dc to ac spectral power vs radar frequency, VHF to X-band, obtained from Phase One clutter measurements under windy conditions at three forested sites. The solid line in the figure joins the median positions of each in-band cluster of data points; the dashed line joins the median positions of the bounding VHF and X-band clusters only. These lines indicate, over and above the inherent variability in the data, a strong trend such that the dc to ac ratio in dB decreases approximately linearly with the logarithm of the radar carrier frequency f_o. Thus at X-band in Figure 6.19, virtually all the spectral power is ac, whereas at VHF the ac power occurs at levels 15 to 25 dB below the dc power. The information shown in Figures 6.17 and 6.19 substantiates the early expectations [9] in these matters. Such information was used to develop the empirical relationship given by Eq. (6.4), which relates dc to ac ratio in windblown clutter spectra with wind speed and radar frequency.

6.4.3.1 DESERT

Ground clutter spatial resolution cells in which some of the backscattered power comes from stationary scattering elements such as the underlying terrain surface itself or large fixed discrete objects (water towers, rock faces) provide correspondingly larger values of dc/ac ratio r in the resulting clutter spectra. In the ground clutter spectral measurements conducted by the LCE radar at its desert site, portions of barren desert floor were visible between the sparse desert bushes (greasewood, creosote) typically of heights of 3 or 4 ft. Figure 6.22 illustrates a typical LCE clutter spectrum measured at this desert site under windy conditions (wind speed ≅ 20 mph). It is evident that this desert clutter spectrum contains a large dc component at zero-Doppler velocity, the result of backscatter from the stationary desert floor.

The ratio of dc/ac power computed directly in the time domain for the desert spectrum of Figure 6.22 is 24.1 dB. Normalization of measured spectral data to P_tot(v) involves division of power per resolution cell by the width of the cell Δv. For the data of Figure 6.22, Δv equals −16.22 dB, so this procedure raises the power in each cell by 16.22 dB. For spectra like that of this figure with a large dc component, the power in the zero-Doppler cell is not distributed over Δv but exists as a singular dc component. Reducing the zero-Doppler peak by 16.2 dB allows more straightforward interpretation of r = 24.1 dB as simply the ratio of the reduced zero-Doppler peak to the total ac power (i.e., the power level of the step function existing in the unit interval −0.5 < v < +0.5 m/s and containing the same total ac spectral power as the measured spectrum).

The desert/windy clutter spectrum of Figure 6.22 contains extensive ac spectral spreading at lower power levels. The low-level ac spectral spreading in these data is due to wind-induced motion of the desert foliage. Other LCE-measured desert spectra from similar cells under light or calm wind conditions showed little or no spectral spreading over as much as 80 dB of spectral dynamic range [see Figure 6.4(a)]. Unlike the forest/light wind spectrum of Figure 6.16, which also contains a large dc component but in which the spectral power decays rapidly but broadens continuously away from the zero-Doppler peak, the dc component in the desert/windy spectrum of Figure 6.22 exists more as a discrete delta function at the spectral resolution of the window function over the higher levels of spectral power.

Figure 6.23 shows the ac spectral tail region of the desert/windy spectrum of Figure 6.22 at higher Doppler velocities as 10 log P_ac(v) vs log v. Also shown in Figure 6.23 is the forest/windy spectrum previously shown in Figure 6.5. Both measured spectra in Figure 6.23 are normalized to P_ac(v). The 24-dB dc/ac ratio in the desert spectrum (a) of Figure 6.23 results in an effective 24-dB loss of sensitivity (i.e., higher noise level) compared with the forest spectrum (b) in measuring ac spectral shape. Otherwise, the ac spectral shape of the desert/windy spectrum in Figure 6.23 almost exactly overlays that of the forest/windy spectrum over their common interval of available spectral dynamic range.

It was previously shown that the exponential shape factor β = 5.2 provides an excellent match to the forest/windy spectrum of Figure 6.23. On the basis of these data, the windblown clutter ac spectral shape that applies for densely forest-vegetated clutter cells under windy conditions also applies for more open desert-vegetated cells. That is, these data suggest that the ac spectral shape function caused by windblown vegetation in a cell may be, at least to a first-order approximation, somewhat independent of the type of vegetation in the cell. However, this equivalence does not extend to include dc/ac ratio; the dc to ac ratio of clutter spectral power is much higher in partially open desert than in forest terrain.

Knolls, Utah. Figure 6.24 shows X-band desert clutter spectra measured by the Phase One radar at two western U. S. desert measurement sites—Booker Mt., Nevada, and Knolls, Utah—in summer season. At both sites these spectra were obtained as eight-gate averages, 15 FFTs per gate (2,048-point FFTs, PRF = 500 Hz, hor. pol., gate width = pulse length = 150 m). The Booker Mt. spectrum of Figure 6.24(a) was measured from barren level mud flats under calm wind conditions and light rain at 12- to 13.2-km range and 2.3° grazing angle. This Booker Mt. spectrum shows essentially no spectral spreading beyond the window-function resolution over a 60-dB spectral dynamic range (except for the small residual spur indicated at the base of the dc-spike 56 dB down), and again evidences the field capability of the Phase One radar for making X-band clutter spectral measurements at low Doppler frequencies [compare with Figure 6.4(c)].

In contrast to no spectral spreading at Booker Mt. in Figure 6.24(a), Figures 6.24(b) and (c) show an X-band desert spectrum from Knolls with significant spectral spreading. The Knolls spectrum was measured from very level terrain containing sparse, low, desert scrub vegetation under 11- to 16-mph wind conditions at 2.5- to 3.7-km range and 0.5° grazing angle. This X-band desert spectrum, although containing a significant dc component, contains less dc power than the LCE L-band Nevada desert spectrum of Figure 6.22. As a result, more ac spectral dynamic range is available for defining the shape of the X-band desert spectrum in Figure 6.24(c) than was available for defining the shape of the L-band desert spectrum in Figure 6.23. This X-band Knolls desert spectrum [Figures 6.24(b), (c)] is very similar to other Phase One and LCE windblown clutter spectra observed at other radar frequencies and from other types of vegetation.

The basic similarities of the Knolls spectrum to other Phase One- and LCE-measured spectra are in the increasing downward curvature with increasing Doppler velocity and decreasing power level displayed in Figure 6.24(c) on 10 log P vs log v axes. This downward curvature also characterizes the exponential shape factor. Thus through a broad central region of increasing downward curvature in the data of Figure 6.24(c), the exponential shape factor β = 11.2 overlays the data and captures its general shape. At upper levels, the data rise somewhat above the approximating exponential in the quasi-dc region |v| < 0.21 m/s; this excess quasi-dc power is included as dc power in the dc/ac ratio of 5.2 dB ascribed in modeling this spectrum. The slight broadening beyond exponential at very low levels in Figure 6.24(c) as the measured data approach to within 10 dB of the noise level is partially the result of S/N limitations in this region. In contrast to the exponential shape factor generally capturing the shape of this spectrum, power-law rates of decay [Figure 6.24(c)] increase from n ≅ 2 (20 dB/decade) at upper levels to n ≅ 4 (i.e., 40 dB/decade) at intermediate levels to n > 6 (> 60 dB/decade) at the lowest levels; in other words, no single power law comes close to capturing the general shape of the X-band desert spectrum shown in Figures 6.24(b) and (c).

8.4.3.2 CROPLAND

The Phase One radar acquired clutter measurement data at the farmland site of Beulah, N. Dakota, during the first two weeks of June [44]. The open agricultural fields in the measurement sectors around Beulah were mostly in wheat; in early June, the wheat was not yet very high. Trees occurred at ∼1% to 3% incidence by area, for the most part at low-lying elevations along creekbeds. The primary Beulah measurement swath examined comprised 16 contiguous 150-m gates from 4.5- to 6.75-km range. Over this swath, the terrain was well illuminated by the Phase One radar at a depression angle of ∼ 0.4°. Windblown wheatland clutter spectra were generated for all 16 primary-swath gates at vertical polarization and pulsewidth in all five Phase One frequency bands. In each band, 2048-point FFTs were generated and averaged to form the spectrum for each gate. The effective PRF and number of FFTs averaged in each band were at X-band, 500-Hz PRF, 15 FFTs; at S-band, 167-Hz PRF, 5 FFTs; at L-band, 100-Hz PRF, 3 FFTs; at UHF, 33.3-Hz PRF, 1 FFT; and at VHF, 12.5-Hz PRF, 1 FFT. Considerable gate-to-gate variation occurred in the resultant clutter spectra.

Six particular gates (10–12 and 14–16) at longer ranges in the primary swath were selected as under strong illumination and essentially containing pure wheat fields within the azimuth main beamwidth in all bands. Spectra from these gates were averaged together in each band to provide a generalized indication of pure windblown wheatland clutter spectra.

Figure 6.25 shows the resultant generalized clutter spectra from pure windblown wheatland in all five Phase One frequency bands. These measurements were obtained on four different days under relatively windy conditions. The results show significant spectral spreading in each of the four upper bands (X, S, L, and UHF) to either side of a strong dc component. Spectral spreading in these bands extends to power levels 60 to 70 dB below zero-Doppler peaks and to corresponding Doppler velocities generally < ∼1 m/s. In contrast, the VHF wheatland spectrum shows absolutely no spectral spreading to a level 65 dB down under 12-mph winds. This complete lack of VHF spectral spreading is due to a combination of (a) a high degree of penetration of the VHF radiation through the wheat stalks to the underlying stationary ground, and (b) the amplitude of the 12-mph windblown motion of the wheat stems being < 1λ at VHF (< 1.7 m). Note that the UHF wheatland spectrum, which shows the greatest degree of spreading of any of the four upper-band spectra to a maximum Doppler velocity of ∼1.25 m/s, was measured under much windier conditions (27 mph).

The ac spectral spreading in the four upper-band wheatland clutter spectra of Figure 6.25 is highly exponential in shape, as indicated by the relatively linear decay displayed on the 10 log P vs v axes used in Figure 6.25. The three ac spectral shapes at S-, L-, and UHFbands are remarkably exponential, and in fact are reasonably well-fitted by the exponential shape factor β = 9, which is not out of line with the values of exponential shape parameter appropriate for windblown forest at similar wind speeds (see Table 6.1). Each spectrum also shows a strong dc component. However, the X-band spectrum in Figure 6.25 shows a much decreased dc component, such that much of the pure dc power in the lower bands has spread into a quasi-dc region |v| ≤ ∼ 0.25 m/s at X-band. The dc to ac ratios in these spectra for X-, S-, L-band, and UHF are: 4, 17, 21, and 17 dB respectively.

Beiseker, Alberta. Another agricultural site visited by the Phase One radar in summer season was Beiseker, Alberta (see Section 3.4.1.4.2). The Phase One summer visit to Beiseker was in August when the crops (mostly wheat and other grains) were mature and thus higher than for the June visit to Beulah. Figure 6.26 shows an S-band summer Beiseker cropland clutter spectrum measured as a 16- gate average of 150-m range gates at vertical polarization (5 FFTs per gate, 2048-point FFTs, PRF = 500 Hz). The PRF of 500 Hz used to make this spectrum was higher than the 167 Hz generally used at S-band, resulting in somewhat reduced spectral resolution. This Beiseker spectrum is from measurements at 10- to 12.4-km range under 7- to 12-mph winds. The measurement geometry resulted in illumination at a depression angle of ∼0.4°. The 2.4-km Beiseker measurement sector was almost entirely in mature wheat, although it was somewhat broken in places and dissected with a stream bed, resulting in small secondary incidences of herbaceous rangeland and minor scrub/brush.

The 16-gate, mature-wheat, Beiseker S-band spectrum of Figure 6.26 is well represented as exponential, although containing less dc power than the 6-gate, young-wheat, Beulah S-band spectrum included in Figure 6.25. The exponential shape factor that best represents the Beiseker spectral shape is β = 13, greater than the β = 9 Beulah value as a result of the lighter winds at Beiseker. Power-law rates of decay shown in Figure 6.26(b) increase from n ≅ 3 (30 dB per decade) at intermediate levels to n ≅ 5 (50 dB per decade) at lower levels. The dc/ac ratio in which excess quasi-dc power is included as dc power to enable the β = 13 shape factor to fit the spectral tail in Figure 6.26 is 12.2 dB.

The Beiseker 16-gate spectrum of Figure 6.26 is essentially identical in all respects to a corresponding Beulah wheatland 16-gate spectrum³⁵ (not shown). That is, although detailed gate-to-gate differences exist in these data caused by variations in topography, crops, and wind speeds, when averaged over broad 2.4-km swaths these results indicate that exponential spectral spreading generally occurs in open cropland terrain without much site-specific variation.

de Loor’s Results. Much earlier measurements of clutter spectra from windblown crops (including wheat) made with a noncoherent X-band system were reported by de Loor, Jurriens, and Gravesteijn [45]. The one example shown by de Loor et al. is an ac spectrum for full-grown wheat under 5- to 7-mph winds of ∼ 40-dB spectral dynamic range. This spectrum appears to be highly exponential and is very similar in shape to the Phase One-measured Beulah pure-wheat spectra of Figure 6.25. Spectral widths at the −3- and −10-dB levels as a function of wind speed for winds up to 22 mph are also provided by de Loor et al. in composite plots covering four different crop types (wheat, alfalfa, sugar beets, potatoes). The maximal extent points in these results fall in closely with the predictions of the current windblown clutter exponential model at corresponding −10- and −25-dB levels under windy conditions (see Figure 6.1), after taking into account that de Loor’s results are noncoherent (i.e., wider spectra by a factor of ∼1.4, see Section 6.6.1.1). However, de Loor et al. “could not conclusively [show] significant [spectral] differences between different crop types.” Detailed comparisons of Phase One spectral results with those of de Loor et al. reinforce Phase One indications suggesting no large differences in ac spectral shape or extent at microwave frequencies between windblown tree spectra and windblown crop spectra, or indeed, between windblown crops at different stages of maturity, from different sites, and of different crop types, under windy conditions.

6.4.3.3 RANGELAND

Clutter spectral results were obtained for rangeland terrain in which patches of trees (45-ft high aspen) and patches of shrubs (15-ft high willow) occurred over open areas of herbaceous (grassy) rangeland [24, 25]. Five-frequency spectra were obtained, each averaged over 76 contiguous 15-m range gates, in which a number of FFTs per gate were also averaged. A significant dc component existed in these spectra (e.g., of dc/ac ratio as much as 15 to 20 dB), largely the result of geometrically visible open ground occurring between the patches of trees and shrubs. “Shoulders” were evident in the spectral shapes at the onset of the ac spreading just at the base of the dc spikes [cf. Figures 6.22, 6.24(b), 6.25]. Such shoulders in spectral shape have been associated with the natural resonant frequency of the trees (see Section 6.6.1.3). Beyond these shoulders, the rates of decay over the ac spectral tail regions in the rangeland spectra were highly exponential. See [25] for plots of these rangeland spectra, as well as other considerations involved in their modeling including their dc/ac ratios.

6.4.4.1 PRELIMINARY ANALYSIS

The input clutter power within one pulse repetition interval T entering the single delay-line canceller is given by

where f_p = 1/T is the PRF and P_tot(f) is the total clutter spectral power density as given by Eq. (6.1) (transformed from Doppler velocity v to Doppler frequency f). Since f_p must be much greater than the clutter spectral extent of P_tot(f) for successful MTI operation, the limits of integration in the preceding integral can be further expanded to ± ∞ without practical consequence on P_ic, whereupon

The equivalence of the infinite integral expression to unity in this equation is required by the definition of P_tot(f) in Eq. (6.1). The frequency response function in the power domain for a delay line of time delay T is given [1] by

Thus the residual clutter power after cancellation is given by

In this equation, the sine function in |H(f)|² may be replaced by its small-angle approximation (i.e., its argument), again because f_p = 1/T greatly exceeds the practical spectral extent of P_tot(f). The result of this substitution is

The total clutter spectral power density P_tot(f) is given by Eq. (6.1) as

where r is the ratio of dc to ac power in the clutter spectrum and P_ac(f) is the ac spectral shape factor. Substituting this expression for P_tot(f) into the above equation for P_oc and recognizing that the term involving integration over the delta function term vanishes, the general result is that the output clutter power from a single delay-line canceller is given by

(6.6)

The MTI improvement factor I of the single delay-line canceller is given [1,42] by

where , the average gain of the canceller, can be shown [1] to equal 2. Therefore, I of the single delay-line canceller is given by

where P_oc is given by Eq. (6.6).

6.4.4.2 EXPONENTIAL CLUTTER

Clutter distributed exponentially in Doppler frequency f is given by Eq. (6.2) as

where β = (λ/2)β and β is as specified in Eq. (6.3) or Table 6.1. It follows that

Using the standard result , the preceding equation reduces to

Representing I of the single delay-line canceller in exponentially distributed clutter as I_β, it follows that

(6.7)

where v_p = (λ/2)f_p. Skolnik has extended Eq. (6.7) to apply to n delay line cancellers in cascade (see [1], p. 158), as:

(6.8)

6.4.4.3 GAUSSIAN CLUTTER

Clutter distributed Gaussianly in Doppler frequency f is given by Eq. (6.17) as

where g’ = (λ/2)²g. It follows that

Using the standard result the preceding equation reduces

If I of the single delay-line canceller in Gaussianly distributed clutter is represented as I_g, it follows that

(6.9)

Except for the inclusion of the term (r+1), which explicitly shows the effect on I_g of nonzero dc/ac spectral power ratio r, this expression for I_g is otherwise identical to that provided elsewhere [1,42]. Both Eqs. (6.7) and (6.9) reduce to

(6.10)

where σ is the standard deviation in the respective spectrum [i.e., in Gaussian-distributed clutter, ; in exponentially distributed clutter, .

6.4.4.4 NUMERICAL EXAMPLES

It is evident from Eqs. (6.7), (6.9), and (6.10) that I obtained in Gaussian clutter exceeds that obtained in exponential clutter by a decibel amount given by

(6.11)

assuming that the radar parameters λ and f_p and the clutter dc/ac ratio r are the same in both cases. As a numerical example, consider Barlow’s [7] much-referenced Gaussian clutter spectrum under windy (20-mph) conditions for which g = 20 (see Section 6.6.1.1). The value of β from the exponential clutter model of Section 6.2 for similar windy (15- to 30-mph) conditions is specified in Table 6.1 as β = 5.7. Applying Eq. (6.11) using these clutter parameters shows that the single delay-line canceller improvement factor obtained in g = 20 Gaussian clutter is 3.9 dB greater than that obtained in β = 5.7 exponential clutter, independent of radar frequency and processing-specific parameters such as pulse repetition frequency. That is, the fact that exponential clutter spreads the clutter in Doppler beyond that of the more usually assumed Gaussian clutter results in a 3.9 dB loss in improvement factor.

As a further numerical example, the values of I_β obtained by an X-band radar (i.e., λ = 3 cm) of f_p = 1000 Hz operating in exponential clutter are shown in Table 6.5 as computed from Eq. (6.7). The exponential clutter for which Table 6.5 applies is that of the model of Eqs. (6.1), (6.2), (6.4) and Table 6.1 under various wind conditions.

(i) visibility curves: the inverse of I_c (i.e., SIR_i required to leave SIR_o = 0 dB) as a function of the Doppler frequency f for given N;

(i) improvement factor plots: I_c as a function of N, for given Doppler frequency f.

6.5.2.1 X-BAND RESULTS (1-D)

The X-band results to follow compare optimum MTI filter performance as given by Eq. (6.14) for corresponding Gaussian and exponential windblown ground clutter spectral shapes under breezy wind conditions. The model utilized for clutter power spectral density (PSD) is that specified in Section 6.2, particularly by Eq. (6.1) which specifies P_tot(v) in terms of dc/ac ratio r and the ac shape factor P_ac(v). The exponential spectral shape is modeled as specified in Section 6.2.1, particularly by Eq. (6.2) which specifies exponential spectral shape in terms of the exponential shape parameter β, β being specified as a function of wind speed (e.g., breezy, windy) in Table 6.1 and Figure 6.1. The Gaussian spectral shape is modeled as specified by Eq. (6.17) in Section 6.6.1.1, where the Gaussian shape parameter g is that required to give the same one-lag temporal autocorrelation coefficient ρ as that of the exponential spectrum under given wind conditions. This procedure is one way of normalizing the Gaussian and exponential shapes to be of equivalent extent in Doppler as a function of wind speed, the autocorrelation function being the inverse Fourier transform of the clutter power spectrum. The dc/ac ratio r in the clutter spectrum is modeled as specified in Section 6.2.2, particular by Eq. (6.4) which models r as a function of wind speed w and radar carrier frequency f_o (see also Figure 6.2). In the results to follow, f_o = 9.2 GHz (X-band), PRF = 1/T = 500 Hz, and CNR = 50 dB.

Figure 6.27 shows two visibility curves, corresponding to exponential and Gaussian clutter spectra under breezy conditions with dc/ac ratio r = unity, and for N = 16 coherently integrated pulses. For the exponential spectrum, β = 8 s/m; for both spectra, ρ = 0.9908, and r = 0 dB. Each curve shows the SIR_i necessary to obtain P_D= 0.8 and P_FA = 10⁻⁶ vs normalized Doppler separation fT. Both curves decay from large required values of SIR_i at low values of fT, where clutter strength is high, to bottom out at a much lower floor value of SIR_i ≅ −61 dB at higher values of fT, the floor level being determined by the thermal noise limit. The curve for the Gaussian PSD decays much faster than the curve for the exponential PSD. That is, the MTI filter operating under the standard assumption of Gaussian clutter spectral shape significantly outperforms the MTI filter operating under the more realistic assumption of exponential clutter spectral shape. The loss in detection performance due to the exponential PSD is given by the signal power increase required by the disturbance with exponential PSD to match the performance of the disturbance with Gaussian PSD. It is evident in Figure 6.27 that this loss in detection performance caused by the exponential PSD is significant over a wide range in fT (i.e., for 0.05 < fT < 0.3) and at its maximum becomes as large as 15 dB. Corresponding results under windy conditions (not shown here) show larger differences in performance between Gaussian and exponential PSDs, with significant detection loss due to the exponential PSD spanning 0.07 < fT < 0.4, and becoming as large as 18 dB [32]. It is apparent that the predicted performance of the MTI processor is very sensitive to the assumed shape of the clutter Doppler spectrum.

Figures 6.28 and 6.29 show improvement factor plots under assumptions of exponential and Gaussian clutter spectra, respectively (i.e., corresponding to the visibility curves of Figure 6.27). As in Figure 6.27, wind conditions for the results of Figures 6.28 and 6.29 are breezy. Comparison between the two figures indicates that, for a normalized target Doppler frequency of fT = 0.2, (i.e., well within the main clutter spectral region), to obtain I_c = −50 dB in the presence of an exponential spectrum requires the coherent integration of N = 10 pulses, while integration of only 6–7 pulses is sufficient for a Gaussian spectrum. In passing from breezy to windy conditions, the achievable improvement factor significantly degrades (i.e., under the assumption of an exponential PSD and close to zero Doppler at fT = 0.1, the loss in improvement factor in passing from breezy to windy is on the order of 9 dB [32]).

6.5.2.2 L-BAND RESULTS (1-D)

The L-band results that follow parallel the X-band results of the previous subsection, except that the comparison of optimum MTI filter performance [Eq. (6.14)] is now extended to include power-law—as well as Gaussian and exponential—windblown ground clutter spectral shapes. The power-law PSD is modeled as specified by Eq. (6.18) in Section 6.6.1.2; i.e., as parameterized by the power-law exponent n which typically takes on values of n = 3 or 4. Equivalent spectral extent for given wind conditions among the three spectral shapes of Gaussian, exponential, and power-law is now specified in these L-band results in terms of equal −3-dB frequency bandwidth in the three shapes, rather than specifying equal temporal correlation coefficients as in the X-band results. A further distinction is that the L-band results are provided for “windy” (as opposed to “breezy”) wind conditions, with wind speed specified at 30 mph. In the L-band results to follow, f_o = 1.23 GHz, PRF = 500 Hz, and CNR = 70 dB (as opposed to 50 dB in the X-band results).

These parameters are appropriate for an L-band ground-based surveillance radar.

Figure 6.30 shows L-band visibility curves for Gaussian, exponential, and power-law clutter spectral shapes under windy conditions, for integration of N = 32 coherent pulses. For the exponential clutter PSD, the shape parameter for windy conditions is given by β = 5.7 (see Table 6.1). The Gaussian and power-law clutter PSDs have shape parameters providing equivalent −3 dB spectral width to that of the exponential PSD. The dc/ac ratio is as specified by Eq. (6.4) for w = 30 mph and f_o = 1.23 GHz. As in the X-band results, the curve for the Gaussian-shaped clutter PSD decays much faster in Figure 6.30 than that for the more realistic exponentially-shaped clutter PSD. In the region fT < 0.1, performance prediction based on the exponential model requires SIR_i up to 22 dB greater than that specified by the Gaussian model. In the region fT > 0.1, both Gaussian and exponential PSDs are below the thermal noise level, which determines the required SCR_i ≅ − 84 dB in this region. The n = 3 power-law clutter PSD has a much higher spectral tail than the Gaussian or exponential PSDs (see Figures 6.44 and 6.55), resulting in much greater SCR_i being required over the complete range 0 < fT < 0.5 to ensure the same detection performance. Performance prediction based on the n = 3 power-law clutter PSD is excessively pessimistic, compared to that based on the more realistic exponential clutter PSD. The n = 4 power-law prediction in Figure 6.30 is less pessimistic but still results in significant performance mismatch.

Figure 6.31 shows L-band improvement factor plots for clutter of exponential spectral shape under windy conditions (i.e., corresponding to the exponential curve of Figure 6.30) but here showing variability with number of integrated pulses N; the results are parameterized by fT. It is apparent that for high target Doppler frequencies, a small number of pulses is sufficient to yield the best performance achievable, namely, I_c|_MAX = CNR|_dB + 10 log₁₀(N); whereas for lower Doppler frequencies, larger numbers of pulses are necessary to converge to the same condition. Figure 6.32 compares improvement factor plots at fT = 0.5 and fT = 1/16, assuming corresponding Gaussian, exponential, and n = 3 power-law clutter PSDs. Unlike I_c for the Gaussian and exponential clutter PSDs, I_c for the power-law clutter PSD does not converge to the ideal performance, even at high Doppler frequencies.

(i) visibility curves: inverse of I_c as a function of the Doppler frequency f for given N (with constant N × K = 64 and θ = 0);

(ii) improvement factor plots: I_c as a function of N (with constant N × K = 64 and θ = 0), for given Doppler frequency f.

6.5.3.1 X-BAND RESULTS (2-D)

The X-band results to follow compare STAP system performance (i.e., 2-D case) for corresponding Gaussian and exponential windblown ground clutter spectral shapes under breezy conditions. The same methodology for modeling the clutter spectra and the same values of radar parameters are utilized as specified at the beginning of Section 6.5.2.1 in the 1-D X-band case. Figures 6.33 and 6.34 show visibility plots for exponential and Gaussian clutter PSDs, respectively, under breezy conditions with dc/ac ratio r = unity. Various curves are shown in these two figures for different combinations of N and K, thus exchanging temporal DOF for spatial DOF. The optimum number of pulses is on the order of N = ∼16, which provides the lowest SIR_i value needed to achieve assigned values of P_D for fixed values of P_FA. The undulations of the curve for N = 64 are caused by the two antenna sidelobes in this degenerate case of K = 1. Such undulations appear when the number of spatial DOF is so low as to allow little cancellation to be possible. Comparison between the results of Figures 6.33 and 6.34 reveals that the detection loss incurred by the airborne MTI processor in this 2-D example under the realistic assumption of an exponential clutter PSD as opposed to the more common assumption of a Gaussian clutter PSD can be as much as 10 dB.

Figures 6.35 and 6.36 show improvement factor plots assuming exponential and Gaussian clutter PSDs, respectively, corresponding to the visibility curves of Figures 6.33 and 6.34. These figures indicate that for target Doppler frequencies well within the main clutter spectral region, i.e., for fT = 0.1 or 0.2, the MTI performance assuming Gaussian clutter significantly exceeds that assuming exponential clutter by amounts on the order of 8 to 10 dB for various numbers N of pulses integrated. In addition, the improvement factor I_c is much less sensitive to the number of temporal DOF N with the exponential spectrum than with the Gaussian spectrum, indicating that selection of the optimum ratio of temporal to spatial DOF depends on the assumed shape of the clutter spectrum. As in the 1-D case, in passing from breezy to windy conditions, the achievable improvement factor significantly degrades (i.e., under the assumption of an exponential PSD, and close to zero-Doppler at fT = 0.1, the loss in 2-D improvement factor in passing from breezy to windy is ≈10 dB [32]).

6.5.3.2 L-BAND RESULTS (2-D)

The 2-D L-band results to follow compare STAP system performance for different assumed shapes of clutter spectra using the same methodology for modeling the clutter spectra and using the same values of radar parameters as specified at the beginning of Section 6.5.2.2 for the 1-D L-band case. Figures 6.37, 6.38, and 6.39 show 2-D L-band visibility plots for exponential, Gaussian, and n = 3 power-law clutter PSDs, respectively, under windy conditions and for various numbers N of coherently integrated pulses as processed by the optimum space-time processor. In all three figures, the undulations in the curve for N = 64 are caused by effects of the two antenna sidelobes in this single-antenna-element case, as mentioned in the previous subsection. For the Gaussian and exponential clutter PSDs, the optimum number of pulses to provide the lowest SIR_i needed to achieve assigned values of P_D for fixed values of P_FA is N = 8. In contrast, for the n = 3 power-law clutter PSD, the optimum number of pulses is N = 32.

Figure 6.40 compares the Gaussian, exponential, and n = 3 power-law visibility plots for N = 8 and N = 32 together in one figure. At N = 32, the results for Gaussian and exponential clutter still show residuals of the space-time cancelled sidelobes. It is apparent in these results that a significant loss in performance (i.e., as much as 10 dB) occurs over a wide range in target Doppler frequencies (i.e., 0 < fT < 0.3) when using the realistic exponential spectrum compared to the more commonly used Gaussian spectrum. However, the additional loss in performance that occurs in utilizing an n = 3 power-law clutter spectrum is generally very much greater (i.e., as much as 30 dB) and extends over all available Doppler frequencies (i.e., 0 < fT < 0.5). That is, the predictions based on the power-law model require a very much higher SIR_i at the system input to obtain the same detection performance as with the Gaussian or exponential models, and the impact of the space-time cancellation is very much lower for the power-law model due to the very slow decay of the power-law spectral tails. This slow decay also masks the effects of the antenna sidelobes at K = 2 using the power-law model, such effects being clearly evident in the corresponding results using the other two models.

Figure 6.41 shows 2-D L-band improvement factor plots for windy exponential clutter PSD corresponding to the visibility plots for exponential clutter shown in Figures 6.37 and 6.40. Since clutter PSD modeled to be of exponential shape most closely matches the measured spectral clutter data as discussed elsewhere throughout Chapter 6 (e.g., see Figures 6.5, 6.21, 6.54), the modeled results for I_c under windy conditions shown in Figure 6.41 are the best estimate of actual L-band clutter cancellation performance to be expected. The data indicate that the optimum number of integrated pulses N to maximize I_c for the various parameterized values of target Doppler frequency fT shown is usually such that N ≅K, in agreement with approximate theoretical expectations discussed elsewhere [48, 49] in this regard.

Figure 6.42 compares 2-D L-band improvement factor plots for windy Gaussian, exponential, and n = 3 power-law clutter PSDs at two values of fT, viz., fT = 1/32 and 1/2. At fT = 1/32 (i.e., well within the main region of clutter Doppler extent and, in fact, quite close to zero-Doppler), the prediction based on Gaussian clutter significantly overestimates I_c (by 13 dB for N = 8 pulses) and the prediction based on n = 3 power-law clutter significantly underestimates I_c (by 10 dB at N = 8 pulses) compared to the more realistic prediction based on exponential clutter PSD. At fT = 1/2 (i.e., at the highest target Doppler frequency), the predictions for I_c based on the Gaussian and exponential models are essentially identical since the clutter for both these models is well below thermal noise at such high Doppler, with the result that the performance for these two models at fT = 1/2 is dependent only on thermal noise. In contrast, the estimated clutter cancellation performance using the n = 3 power-law clutter PSD model at fT = 1/2 is much poorer (i.e., is far too pessimistic, by 25 dB for N = 8 pulses) since the power-law clutter spectral tail is still well above thermal noise even at such high Doppler.