5.4.3 FOREST TERRAIN

Much natural terrain is forested. The characteristics of land clutter from forested terrain are very different than from open terrain. Section 5.4.3 specifies the characteristics of forest land clutter based on 8,611 measured clutter histograms from 466 forest clutter patches. These measurements break down in two regimes of relief, high (terrain slopes > 2°) and low (terrain slopes < 2°), and in multiple depression angle regimes as shown in Table 5.24. These forest clutter measurements come from 27 different sites.

Because forest is much less supportive of multipath than is open terrain, particularly at high illumination angles, many of these forest measurements are much more measurements of relatively pure intrinsic σ° and are much less contaminated by dispersive propagation influences than are measurements in open terrain. To the extent that this is true, fewer measurements can establish trends in forest terrain than in farmland terrain. Nevertheless, the Phase One forest clutter measurements constitute 29% of all Phase One measurements in pure terrain. Thus, after farmland at 35%, forest stands as far and away the second most highly sampled terrain type among the eight pure terrain types of Chapter 5.

Classification of forest clutter patches was in three separate categories, viz., deciduous, coniferous, and mixed (see Table 2.1). However, the clutter measurements showed no significant variation among these classes. The results presented here are inclusive of all three classes.

Little seasonal variation was observed in the Phase One clutter measurements from forested terrain. One forested wilderness site—Brazeau, Alberta—was visited twice by the Phase One equipment, once in summer and once in winter. Typically, mean clutter strengths at

Brazeau between summer and winter seasons differed by only 1 or 2 dB with the summer measurements usually (but not always) being stronger (see Section 3.4.1.3.3 and Figure 3.24). The results presented in Section 5.4.3 are inclusive of all Phase One forest measurements independent of the season in which the measurement was made. As was provided for agricultural terrain, some interpretative discussion involving individual patch measurements continues to be provided concerning the uncertain outliers that occur in the following results for forest terrain.

It is interesting to compare the high- and low-relief forest mean clutter strength data to follow with the site-specific repeat-sector forest mean clutter strength characteristics from Blue Knob and Wachusett Mountain shown in Figures 3.3 and 3.4, respectively, in Chapter 3.

5.4.4 SHRUBLAND TERRAIN

Less discussion is provided for following terrain types in Section 5.4—shrubland, grassland, wetland, desert, mountains—than was provided for foregoing terrain types of urban, agricultural, and forest. The reason for this is to avoid redundant discussion—many of the points to be made have already been dealt with in previous discussions. Uncertain outliers continue to be indicated with open plot symbols for the terrain types that follow, although they are not discussed in the text.

The land cover classification system utilized provides rangeland as a major land cover category (see Table 2.1). Within rangeland, there are three specific subcategories—herbaceous rangeland, shrubby rangeland, and mixed herbaceous/shrubby rangeland. Hereafter, pure herbaceous rangeland clutter patches are referred to as grassland, and patches classified as shrubby or herbaceous/shrubby mixtures are referred to as shrubland. Forest, shrubland, and grassland thus comprise three land cover classes involving natural vegetation for which the height of the vegetation significantly decreases from forest to shrubland to grassland. As will be seen, this decrease in vegetation height results in significantly differing clutter characteristics from these three terrain types.

Table 5.29 shows the statistical populations underlying the shrubland clutter modeling information to follow. These populations are relatively low, involving only 1,030 measured clutter amplitude distributions from 51 patches. The high-relief shrubland patches come from three different sites, the low-relief shrubland patches from seven different sites.

5.4.5 GRASSLAND TERRAIN

Table 5.34 shows the statistical populations underlying the grassland clutter modeling information to follow. As with shrubland, the high-relief grassland populations remain small, but the low-relief grassland populations are significantly higher. High-relief grassland patches come from six different sites, the low-relief grassland patches from nine different sites.

5.4.6 WETLAND TERRAIN

Table 5.39 shows the statistical population underlying the wetland clutter modeling information to follow. This population comprises 447 measured clutter amplitude distributions from 17 clutter patches at three different sites. Many of these measurements occurred at Big Grass Marsh, Manitoba, which was visited by Phase One in winter weather (see Chapter 3, Section 3.4.1.5). All wetland clutter patches were of “level” landform classification.

Clutter modeling information for wetland is provided in Tables 5.40 and 5.41 and Figures 5.36 and 5.37. As shown in Figure 5.36, mean clutter strength for wetland is very weak—in fact, it is the weakest clutter strength in every band of any of the terrain types examined. The results of Figure 5.37 correspond quite closely with the Big Grass Marsh repeat sector results shown in Figure 3.8 in Chapter 3. Mean clutter strength is particularly weak at VHF and UHF in Figure 5.36, hovering in the neighborhood of −67 to −72 dB. Although very weak, these are accurate measured clutter strength numbers unaffected by Phase One sensitivity limitations and noise corruption (see Appendix 3.C for examples of accurate weak clutter computation at Big Grass Marsh). The wetland mean strength data show a strong trend of increasing strength with increasing frequency, UHF to X-band—obviously due to multipath propagation, and a weaker trend in most bands of increasing strength with increasing depression angle between 0° to 0.25° and 0.25° to 0.75°—the only depression angle regimes in which data exist for wetland. The fact that VHF mean clutter strengths are somewhat stronger than at UHF indicates that intrinsic σ° is significantly greater at VHF than UHF for wetland, a result previously seen in repeat sector data in Chapter 3.

Table 5.41 and Figure 5.37 indicate strong trends of decreasing spread in level wetland clutter amplitude distributions with increasing cell size A. In this respect, level wetland is like low-relief grassland (compare Figure 5.37 with Figure 5.35).

5.4.7 DESERT TERRAIN

Desert terrain does not explicitly appear in the land cover and landform terrain classification systems employed herein (see Section 2.2.3, Tables 2.1 and 2.2). A criterion utilized to specify desert clutter within the Phase One measurement database is that the desert clutter must come from two Phase One sites—Booker Mountain, Nevada, and Knolls, Utah—that were situated in typical western desert terrain (see Section 3.4.1.5). Booker Mountain was a high site; Knolls was a low site. At both sites, low-relief desert valleys were surrounded by steep terrain at much higher elevations. The low-relief valley floors were typically covered with sparse desert vegetation, but with occasional barren dry lake beds or salt flats. The surrounding high-relief terrain had occasional occurrences of patches of trees, principally on steep slopes at higher elevations. A further criterion for a clutter patch to be accepted as desert at either of these two sites is that, if low relief, its land cover classification be predominantly rangeland or barren; if high-relief, a component of trees was also allowed. Table 5.42 indicates that, for desert terrain thusly defined, there are available 2,884 measured Phase One clutter amplitude distributions from 279 desert clutter patches. These desert patches and histograms are approximately equally distributed between high-relief patches and low-relief patches. Repeat sector desert clutter measurements at Booker Mountain and Knolls were previously discussed in Chapter 3, Section 3.4.1.5 (in particular, see Figure 3.37).

5.4.8 MOUNTAINOUS TERRAIN

As with desert terrain, mountain terrain does not overtly appear in the terrain classification systems employed herein (see Tables 2.1 and 2.2). A criterion utilized to specify mountain clutter within the Phase One measurement database is that the mountain clutter must come from two Phase One sites—viz., Plateau Mountain and Waterton—that were situated in what is obviously extremely mountainous terrain in the Canadian Rocky Mountains. Plateau Mountain was a high site; Waterton was a low site; both sites are further described in Sections 2.4.2.1 and 3.4.1.2. Further criteria for a clutter patch to be accepted as mountainous at either of these two sites are that its landform classification must be ridged, moderately steep, or steep (see Table 2.2); and its land cover classification must be barren (e.g., perennial-snow-and-ice), rangeland, or forest (see Table 2.1). Table 5.50 indicates that, for mountain terrain thusly defined, there are available 1,653 measured Phase One clutter amplitude distributions from 133 clutter patches for which the terrain within the patch meets these criteria for being mountainous.

Clutter modeling information for mountains is given in Tables 5.51 and 5.52 and Figures 5.44 and 5.45. In Figure 5.44, mean clutter strengths for mountains are seen to be very strong, and, like high-relief forest (Figure 5.24), to show a significant trend of decreasing clutter strength with increasing frequency VHF to X-band (compare with the repeat sector mountain data of Figure 3.21; see also Figure 3.2). This trend in mountain terrain is due to increasing absorption of RF power from the incident radar wave by tree foliage with increasing frequency as discussed in Chapter 3, Section 3.4.1.2. As discussed there, when the mountain slopes are not forested, this trend disappears. Also, as in high-relief forest (Figure 5.24), there are significant trends with depression angle in most bands in the mountain data of Figure 5.44, but the trends are more complicated than those of increasing strength with increasing angle exhibited in forest. In fact, there is an inverse trend of decreasing strength with increasing angle in the mountain data of Figure 5.44 similar to the inverse trend shown in the high-relief desert data of Figure 5.38, which exists for the same reason in the mountain data as previously explained for the high-relief desert data. However, in the lower three bands in the mountain data of Figure 5.44 where depression angle effects are significant, the inverse trend is from the 1° to 2° depression angle regime (dark blue) through 2° to 4° (purple) to 4° to 6° (magenta) and > 6° (red); the trend does not include the 0° to 1° (cyan) data. In fact, looking closely at the mountain data in the lower three bands in Figure 5.44, an opposite, more common trend of increasing strength with increasing angle is seen from the −1° to 0° depression angle regime (dark green) to 0° to 1° (cyan) to 1° to 2° (dark blue). So the strongest mountain clutter occurs in all three lower bands in the 1° to 2° depression angle regime (dark blue), with mean strengths decreasing both with increasing depression angle (lower terrain) and decreasing depression angle (higher terrain) from there.

Modeling information for spreads in clutter amplitude distributions from mountain terrain is given in Table 5.52 and Figure 5.45. Spreads in mountain clutter amplitude distributions are small (although generally still significantly wider than Rayleigh statistics)—in fact, spreads in mountain clutter are the least of any terrain type (high-relief forest has the next lowest spread). In particular, the mountain spreads for large A (A ≅ 10⁶ m²) are nearer unity (i.e., more approaching Rayleigh) in Table 5.52 than any other terrain type (compare with high-relief forest, Table 5.26). These low spreads in mountain clutter amplitude distributions are due to the high terrain slopes and corresponding full illumination of the mountain clutter patches. As with most other terrain types, there exists a significant trend of decreasing spread with increasing cell size A in the mountain data of Table 5.52 and Figure 5.45.

5.5.1 MODEL VALIDATION

The Weibull clutter model has been in use at Lincoln Laboratory for a number of years in radar system studies involving clutter-limited operation against various airborne targets. These studies have involved radar systems ranging in frequency from VHF to X-band. Since the clutter model directly affects the outcomes of such studies, strong and ongoing interest exists in validating and improving the clutter model. In validating a clutter model, a radar engagement with a target is hypothesized to take place at one of the Phase One clutter measurement sites, and system performance is computed in two ways: (1) using the actual measured clutter at that site, and (2) using the clutter model to simulate clutter at that site. Successful validation requires that the performance measure computed using modeled clutter be acceptably similar to that computed using the actual clutter. Model validation encompasses many measurement sites.

For X-band tracking systems, the Weibull model generally works well in predicting the performance parameters employed in such studies. System performance is then generalized by computing the median performance parameter of the system over many sites. A simple spherical earth, constant-σ° clutter model cannot deliver the same general performance as the more in-depth Weibull model. That is, the constant σ° required in the simple model is system and performance-parameter dependent, whereas the Weibull statistical model is adaptive and resilient enough to work well for any clutter-limited system or performance parameter.

For VHF surveillance systems, radar performance parameters can be more difficult to predict due to propagation effects. The Weibull VHF model remains accurate in terrain of significant relief where propagation effects are not severe. However, in level open terrain, the accuracy of the Weibull model can, in some circumstances, be improved by adding deterministic computations of multipath propagation.

5.5.2 MODEL IMPROVEMENT

The clutter modeling information of Chapter 5 represents low-angle land clutter as a Weibull statistical process occurring over visible terrain regions. The Weibull distributions aggregate the widely varying returns received predominantly from an infinitude of spatially localized discrete sources. Studies have been conducted to shed further light on higher-order parametric effects that occur in low-angle land clutter beyond the level at which clutter modeling information is provided in Chapter 5. These studies include the following:

As indicated, each of these subjects has been mentioned or discussed elsewhere in this book.

Figure 4.3 (in Chapter 4) conveniently represents the structure of the Weibull statistical clutter model. Within this structure, what actually exists is an adaptive computer-resident modeling capability. That is, 59,804 measured clutter spatial amplitude distributions exist as computer files together with their terrain descriptive parameters. The clutter model is the template by which these data are sorted. Not only can general templates be provided for general terrain types such as are given in Sections 5.3 and 5.4, but also the template can be adjusted to provide more detailed results for terrain types highly specific to an individual user’s particular interests.

5.A.1 introduction

Chapter 5 provides empirical modeling information for low-angle radar land clutter spatial amplitude statistics in terms of Weibull probability distributions [1]. The purpose of this appendix is to provide information indicating why Weibull distributions were selected to model low-angle clutter in Chapter 5 and elsewhere in this book, as opposed to other possible distribution functions, including in particular the lognormal distribution and the K-distribution. The reasons for using Weibull distributions are as follows:

This appendix covers the following material. Section 5.A.2 further describes Weibull distributions. A number of plots are provided which give added insight into the behavior of Weibull distributions, and which make the clutter modeling information of Chapter 5—given in terms of Weibull coefficients—easier to work with. Section 5.A.3 numerically examines the shapes of the cumulative distribution function (cdf) of theoretical Weibull distributions as plotted against nonlinear Weibull and lognormal probability ordinates, and compares them with lognormal and K-distribution shapes plotted against the same ordinates. This provides understanding on how to interpret upwards and downwards curvature in cdf shapes of measured spatial clutter amplitude distributions when plotted against these ordinates, as to whether they are tending towards Weibull, lognormal, or K-shapes. Examination of the shapes in the extreme high tails is also included. Section 5.A.4 illustrates how Weibull distributions often fit measured low-angle clutter amplitude distributions better than other analytic distributions by comparing fits obtained with

Weibull, lognormal, and K-distributions to some measured Phase One X-band farmland clutter data.

5.A.2 WEIBULL DISTRIBUTIONS; a_w= 1, 2, 3, 4, 5

The Weibull probability density function (pdf) may be written

(5.A.1)

In Eq. (5.A.1), as elsewhere in this book, the random variable x represents clutter strength σ°F⁴, where F is the pattern propagation factor [2]. The quantity x = σ°F⁴ is in some places abbreviated as σ° in this appendix. Also, as represented elsewhere in this book, the Weibull shape parameter a_w is related to b in Eq. (5.A.1) as

(5.A.2)

The scale parameter c in Eq. (5.A.1) is related to the median value of clutter strength x₅₀ (or σ°₅₀) as

(5.A.3)

[cf. Eq. (2.B.18)]. The Weibull cumulative distribution function (cdf) follows from Eq. (5.A.1), as

(5.A.4)

The ratio of variance to the square of the mean for a Weibull distribution is given by

(5.A.5)

where Γ is the Gamma function, is the mean of x, and var(x) is the variance of x. The ratio of standard deviation-to-mean in a Weibull distribution, i.e., , which is much used in Chapter 5, is

(5.A.6)

The ratio of mean-to-median in a Weibull distribution is given by

(5.A.7)

Definitions, relationships, and other matters involving the Weibull distribution are also discussed in Appendix 2.B.

Figure 5.A.1 plots the shape of the Weibull pdf as given by Eq. (5.A.1) for four values of shape parameter a_w—namely, a_w = 0.666, 0.5, 1, and 2, following Sekine and Mao [1]. Table 5.A.1 shows the a_w shape regimes of Weibull distributions. It is apparent in Figure 5.A.1 that, for a_w < 1, the shapes are peaked at a positive modal value of x [for the Weibull distribution of Eq. (5.A.1), mode ], and the pdf decays from its modal peak to zero at both x = 0 and x = ∞. In contrast, for a_w >> 1, the shapes are not peaked.

For a_w = 1, the pdf is a negative exponential and decays from its maximum value of unity at x = 0 to reach zero at x = ∞. For a_w >1, the shapes are like the negative exponential in that they also decay from maximum values at x = 0 to reach zero at x = ∞. These shapes for a_w > 1 are called subexponential because their exponents b = 1/a_w are less than unity. Although similar to the negative exponential shape for a_w = 1, the subexponential shapes for a_w > 1 decay with increasing x from maximum values at x = 0 greater than unity, first at rates faster than the negative exponential, and then for large values of x at rates slower than the negative exponential. It is these slow rates of decay and long tails for large values of x that are of interest here in that they mirror the behavior of low-angle land clutter spatial amplitude distributions.

Thus interest is in the subexponential Weibull pdf’s for which a_w >1. Note that for simplicity, Figure 5.A.1 is plotted for c = 1 which means for —see Eqs. (5.A.1) and (5.A.3). This is done in the interest of making a simple plot, but the plot is not representative of realistic clutter amplitudes. In a number of results to follow in Sections 5.A.2 and 5.A.3, this appendix sets x₅₀ = 0.0001 (i.e., median clutter strength = −40 dB) and lets the Weibull shape parameter range subexponentially over the five values a_w = 1, 2, 3, 4, 5. These values of x₅₀ and a_w are representative of the values occurring for these parameters in many of the actual measured clutter spatial amplitude distributions shown in the main body of Chapter 5 and elsewhere in this book.

Thus Figure 5.A.2 plots the shape of the Weibull pdf as given by Eq. (5.A.1) for Weibull parameters x₅₀ = 0.0001 and a_w = 1, 2, 3, 4, 5. To accommodate these realistic parameters, logarithmic scales are used for both abscissa and ordinate (i.e., log-log paper). The median value x₅₀ = 0.0001 is shown at the extreme left end of the abscissa. Long decaying tails are seen over increasing values of x from the median value, of increasing length and decreasing rate of decay with increasing a_w. The label “Gaussian” on the a_w = 1 curve means that the in-phase (I) and quadrature (Q) clutter voltage components are each of zero-mean equal-variance Gaussian shape, which implies that the amplitude of the clutter voltage [I² + Q²]^1/2 is Rayleigh distributed, and the clutter power σ° = I² + Q² is exponentially distributed.

The distributions of Figure 5.A.2 are not similar to these actually observed or plotted in this book, when histograms of measured clutter spatial amplitude statistics are formed. The reason is that the clutter histograms as shown herein are formed of decibel values of clutter strength to keep them manageable because of the wide ranges of clutter power that are involved. Thus the histograms are formed in bins of width specified in decibels, typically in bins 1 dB wide (see Appendix 2.B). This constitutes a logarithmic transformation y = 10log₁₀x. Figure 5.A.3 shows the logarithmically transformed Weibull pdf of Eq. (5.A.1) such that p(x)dx = p(y)dy. Like Figure 5.A.2, Figure 5.A.3 continues to show results for x₅₀ = 0.0001 and a_w = 1, 2, 3, 4, 5. The pdf shapes in Figure 5.A.3 look like the typical shapes of the measured clutter histograms, examples of which are shown in a number of places in this book. These pdf shapes show a long, slowly diminishing, high-side tail of increasing extent and decreasing rate of decay with increasing value of a_w, which mimics the behavior of the measured histograms. All five pdf’s in Figure 5.A.3 are of bell shape, between narrow and high (a_w = 1) and wide and low (a_w = 5). The median value of 10log₁₀x₅₀ = y₅₀ = −40 dB is evident near the peaks of the bell shapes. Many of the measured histograms do not show much or any of the diminishing shape for low values of y to the left because the histograms become noise-limited for low values of y.

Figure 2.28 in Chapter 2 plots the shape of the Weibull cdf as given by Eq. (5.A.4) against a Weibull ordinate, also for x₅₀ = σ°₅₀ = 0.0001 and a_w = 1, 2, 3, 4, 5. The increasing extent of the high-side tail with increasing values of a_w is evident in Figure 2.28, as are the increasingly strong values of mean clutter strength in these distributions as a result of the high-side tails, and the increasingly large values of mean/median ratio with increasing a_w.

Figure 5.A.4 shows plots of the ratio of standard deviation-to-mean vs a_w, and of the ratio of mean-to-median vs a_w, as given by Eqs. (5.A.6) and (5.A.7), respectively. Note that these relationships are transcendental, involving the Gamma function. The plots are shown over the range 0 ≤ α_w ≤ 6, although principal interest here is in the subexponential range a_w > 1. It is apparent that both measures of Weibull spread in Figure 5.A.4—sd/mean and mean/median—increase very rapidly with increasing a_w; they are shown as decibel quantities in Figure 5.A.4 to accommodate their rapid increase. As indicated in Figure 5.A.4, when a_w = 1 the Weibull distribution degenerates to a simple negative exponential distribution for clutter power and power-like quantities including σ°, which means the in-phase and quadrature components of clutter voltage are distributed as zero-mean equal-variance Gaussians.

Extensive modeling information for spread in low-angle land clutter distributions is given in Chapter 5 in terms of tabulated and plotted values of both the actual measured quantities of spread in these distributions—namely, ratios of sd/mean and mean/median—and in terms of the modeling parameter used to capture these quantities—namely, the Weibull shape parameter a_w. The two plots in Figure 5.A.4 are useful in allowing quick, back-and-forth conversion between these transcendentally-related quantities.

The modeling information for spread in clutter amplitude distributions given in Chapter 5 is given in terms of linear regression between spread values for limiting cell sizes A = 10³ m² and A = 10⁶ m². The linear regression actually took place in the measured quantities of spread—sd/mean (dB) vs log₁₀A and mean/median (dB) vs log₁₀A. Many examples of the scatter plots of sd/mean (dB) vs log₁₀A and the resulting regression lines are shown in

Chapter 5; similar scatter plots and regression lines were formed of mean/median (dB) vs log₁₀A. The exact use of the modeling information for spread requires linear interpolation on log₁₀A of the provided underlying measured values of spread—sd/mean (dB) or mean/median (dB)—rather than a_w directly. However, as shown in the left-hand plot of Figure 5.A.4, the relationship between sd/mean (dB) and a_w is essentially linear for values of a_w such that a_w > 1.5. The value of a_w = 1.5 corresponds to a value of sd/mean = 1.90 dB. Almost all values of clutter spread given in Chapter 5 exceed a_w = 1.5 and/or sd/mean = 1.90 dB. Since the relationship between sd/mean (dB) and a_w is essentially linear for a_w > 1.5 and sd/mean > 1.90 dB, it does not matter whether the linear regression is done on a_w vs log₁₀A or sd/mean (dB) vs log₁₀A. By contrast, the right-hand plot in Figure 5.A.4 is less linear, so it matters more that, if modeling values of a_w are selected as derived from mean/median (dB), in this case the interpolation be of mean/median (dB) vs log₁₀A, not a_w vs log₁₀A.

5.A.3 COMPARISON OF WEIBULL, LOGNORMAL, AND K-DISTRIBUTIONS

This section discusses what observed curvature can imply in plots of cdf’s of clutter strength on nonlinear probability scales. The two nonlinear scales of interest are the Weibull scale (if the measured cdf is linear when plotted on a Weibull scale, the measured distribution is Weibull) and the lognormal scale (if the measured cdf is linear when plotted on a lognormal scale, the measured distribution is lognormal). All measured spatial clutter amplitude distributions from clutter patches were plotted on both Weibull and lognormal cumulative probability scales. The derivation of the plot axes in such Weibull and lognormal plots, the plotting of measured clutter distributions in the plots, and the regression analyses to obtain modeling information in the plots are all described in detail in Appendix 2.B. The question of interest here is, what does observed curvature in such plots imply about whether such clutter amplitude distributions are more accurately represented as Weibull, lognormal, or K-distributions?

Figure 5.A.5 is a notional diagram showing the kinds of curvature exhibited by Weibull, lognormal, and K-distributions when plotted as cdf’s using a nonlinear Weibull probability scale as the y-axis. As shown, a Weibull distribution plots as a straight line, a lognormal distribution plots with convex curvature as observed from above (i.e., is downward curving), and a K-distribution plots with concave curvature as observed from above (i.e., is upward curving). The plots of measured clutter cdf’s in overall measure tend more toward linearity on Weibull plots than they tend to be of strong general upwards or strong general downwards curvature, although examples of all three kinds of curvature occur in the data.

The notional diagrams of Figure 5.A.6 expand the discussion of Figure 5.A.5. Part (a) shows a Weibull distribution, which as required plots linearly against the Weibull y-axis used there; part (b) shows the same Weibull distribution which exhibits upward curvature (concave from above) when plotted against the lognormal y-axis used there. A lognormal distribution is also shown in (b) as the straight dashed line. It is apparent in (b) that a Weibull distribution has less high-end spread and more low-end spread than a lognormal distribution. Part (d) shows a lognormal distribution, which as required plots linearly against the lognormal y-axis used there; and part (c) shows the same lognormal distribution, which exhibits downward curvature (convex from above) when plotted against the Weibull y-axis used there. A Weibull distribution is also shown in (c) as the straight dashed line. It is apparent in (c) that a lognormal distribution has more high-end spread and less low-end spread than a Weibull distribution.

Two Clutter Patch Measurements. Figure 5.A.7 shows measured ground clutter amplitude histograms and cdf’s from two clutter patches, a Peace River patch in part (a) at the top, and a Magrath patch in part (b) at the bottom. At both top and bottom, the cdf (read on the left ordinate of each plot) is shown plotted against a Weibull y-axis in the left plot and against a lognormal y-axis in the right plot. At the top, the same Peace River histogram is repeated both to left and right, and likewise for the Magrath histogram at the bottom (the percent of samples per histogram bin is read on the right ordinate of each plot). In either part (a) or part (b), for any given value of σ° on the x-axis, the same value of cumulative probability P(σ°) is read from the (left plot) Weibull scale or the (right plot) lognormal scale. The presentation format used in each of the four plots in Figure 5.A.7 is similar to that used for Figures 5.2 and 5.3, and Figures 5.14 and 5.15; this format is described more fully in the discussions relating to these preceding figures in the main body of Chapter 5. For example, the vertical dotted lines show the positions of the 50-, 70-, 90-, and 99-percentiles in each distribution; and the vertical dashed line shows the position of the mean in the distribution.

The upper part of Figure 5.A.7 [i.e., part (a)] shows the histogram and cdf resulting from an L-band measurement of a clutter patch at Peace River, Alberta. This Peace River patch was of agricultural land cover with a secondary component of trees (∼10% incidence of trees) and was observed at 0° depression angle. It was situated beginning at 3.9 km from the radar, and extended 4.1 km in range and 79.5° in azimuth. In this measurement, 4% of the samples are at radar noise level (black). The measurement was conducted at 150-m pulse length and vertical polarization. The lower part of Figure 5.A.7 [i.e., part (b)] shows the histogram and cdf resulting from an X-band measurement of a clutter patch at Magrath, Alberta. This Magrath patch was also of agricultural land cover (but in this case with only ∼1% incidence of trees) and was observed at 1° depression angle. It was situated beginning at 5.7 km from the radar, and extended 6.2 km in range and 46° in azimuth. In this measurement, 57.3% of the samples are at radar noise level. This measurement was also conducted at 150-m pulse length, but at horizontal polarization.

The Peace River cdf in part (a) of Figure 5.A.7 is extremely linear as plotted against the Weibull y-axis to the left, and hence shows upwards curvature plotted against the lognormal y-axis to the right (cf. Figure 5.A.6, top). Therefore, this measured distribution is accurately modeled as a Weibull distribution of shape parameter a_w = 2.85 [this value of a_w obtained through a regression analysis least-mean-squares fit of the cdf shown to the left in (a) to a straight line; see Appendix 2.B]. Many measured clutter cdf’s, while not as linear as (a) (left) on a Weibull axis, show some degree of upwards curvature like (a) (right) on a lognormal axis. The histogram in the upper plots is indeed of similar shape to the theoretical Weibull pdf shown in Figure 5.A.3 for a_w = 3. The Magrath cdf in part (b) of Figure 5.A.7 is highly linear as plotted against the lognormal y-axis to the right, and hence shows downward curvature plotted against the Weibull y-axis to the left (cf. Figure 5.A.6, bottom). Therefore, this measured distribution is accurately modeled as a lognormal distribution with mean-to-median ratio ρ = 19.53 [also obtained through linear regression of the cdf shown to the right in (b), see Appendix 2.B]. Although not all measured clutter cdf’s are as linear on Weibull or lognormal axes as those shown in Figure 5.A.7 (a) (left) or (b) (right), considerably fewer measured clutter cdf’s show some degree of downwards curvature on a Weibull axis [like (b) (left)] than show some degree of upwards curvature on a lognormal axis [like (a) (right)]. That is, considerably more measured cdf’s tend towards Weibull curvature than towards lognormal curvature.

Table 5.A.2 shows additional statistical attributes relating to the Peace River and Magrath clutter measurements shown in Figure 5.A.7. The indicated measured values of sd/mean and mean/median of 5.5 and 11.1 dB, respectively, in the Peace River measured distribution compare relatively closely to the corresponding values of 6.0 and 11.5 dB that are applicable to an ideal analytic Weibull distribution of shape parameter a_w = 2.85. This close comparison confirms the Weibull fit to be a good approximation to the Peace River clutter distribution, as does the high value of the coefficient of determination obtained in this fit, equal to 0.9993 (see Table 5.A.2). Similarly, the indicated measured values of sd/mean and mean/median of 10.7 and 13.2 dB, respectively, for the Magrath measurement shown in Table 5.A.2 compare reasonably well to the corresponding value of 12.9 dB applicable to each in an ideal analytic lognormal distribution of shape parameter ρ = 19.53.

Note that, in a lognormal-like distribution [Table 5.A.2(b)], the values of sd/mean [theoretically equal to (ρ²−1)^1/2; see Appendix 2.B] and mean/median (theoretically equal to ρ) are nearly equal for ρ >> 1; whereas, in a Weibull-like distribution [Table 5.A.2(a)], the value of mean/median far exceeds the value of sd/mean (see Figure 5.A.4). As with the Weibull fit in Figure 5.A.7(a), the lognormal fit in Figure 5.A.7(b) is confirmed by the reasonable comparison between measured and analytic values of sd/mean and mean/median and by the high value of the coefficient of determination applicable to this fit, equal to 0.9955 (see Table 5.A.2). Attempting to fit the Peace River distribution as lognormal (upper-right) or the Magrath distribution as Weibull (lower-left) results in gross mismatches between measured and analytic values of sd/mean and mean/median, and significantly lower values of coefficient of determination.

K-Distribution. The Weibull distribution is defined in Appendix 2.B and in Section 5.A.2 of this appendix. The lognormal distribution is defined in Appendix 2.B. It now remains to define the K-distribution. The cumulative distribution function (cdf) of the K-distribution is given by [3–5]

(5.A.8)

where Γ is the Gamma function and K_v is the modified Bessel function of second kind of order v. The parameter v is the shape parameter of the K-distribution; 1/d is a scale parameter. The first two moments of the K-distribution are

(5.A.9)

and

(5.A.10)

The quantity d is given by

(5.A.11)

The K-distribution shape parameter v is given by

(5.A.12)

In this book, spread in distributions is often defined by the ratio of , where and . An important relationship for the K-distribution is

(5.A.13)

If spread in a Weibull distribution [from Eq. (5.A.5)] is equated with spread in a K-distribution [from Eq. (5.A.13)], the following equation is arrived at

(5.A.14)

that relates the Weibull shape parameter a_w with the K-distribution shape parameter v.

Comparison of Analytic cdf Shapes. This section graphically compares the shapes of Weibull, lognormal, and K-distributions. In what follows, numerical plots of theoretical Weibull, lognormal, and K-cdf’s are provided to show how the previously introduced ideas of cdf curvature as plotted on Weibull and lognormal ordinates (see the notional diagrams, Figures 5.A.5 and Figures 5.A.6) develop in actual numeric data.²⁷ The basis of the comparison is the same as used heretofore—namely, the median value x₅₀ of the distribution is always set to x₅₀ = 0.0001 (i.e., to −40 dB), and the spreads [i.e., ratios of or ] are those of Weibull distributions with values of shape parameter a_w = 1, 2, 3, 4, 5. Table 5.A.3 shows expressions for the shape parameters for the Weibull, lognormal, and K-distributions, and the relationship of each to spread as given by . Table 5.A.4 illustrates the distributions to be compared. The basic set of distributions are the Weibull distributions of x₅₀ = 0.0001 and a_w = 1, 2, 3, 4, 5. Table 5.A.4 shows the value of for each one of these distributions. Then Table 5.A.4 shows the five values of ρ for the five corresponding lognormal distributions with the same values of ; and similarly the five values of v for the five corresponding K-distributions.

Figure 5.A.8(a) shows the five basic Weibull distributions plotted against a Weibull y-axis.

Figure 5.A.8(b) shows the five corresponding lognormal distributions with values of ρ as given by Table 5.A.4 and of equal spread to the Weibull distributions, plotted against a lognormal y-axis. The lognormal distributions in Figure 5.A.8 are more tightly clustered than the Weibull distributions over the range shown because the lognormal spreads are more strongly affected by the lognormal high tails beyond the range plotted. That is, at high enough cumulative probability levels each lognormal distribution in Figure 5.A.8 eventually has a longer high tail than the corresponding Weibull distribution. This will be seen in what follows.

Figure 5.A.9 plots equivalent Weibull, lognormal, and K-distributions on Weibull y-axes for σ°₅₀ = −40 dB, and for the first three values of spread specified in Table 5.A.4. The upward and downward curvatures sketched in Figure 5.A.5 are evident in Figure 5.A.9. Note that the K-distribution corresponding to a_w = 1 (i.e., v = ∞) degenerates to negative exponential (or Rayleigh voltage) as does the Weibull for a_w = 1; and the K-distribution corresponding to a_w = 2 (i.e., v = 0.5) also is identically equal to the Weibull distribution of a_w = 2. For the a_w = 3 plot in Figure 5.A.9, the curvatures are as sketched in Figure 5.A.5.

Figure 5.A.10 shows the five basic Weibull and five basic lognormal distributions, each with σ°₅₀ = −40 dB and with values of sd/mean as specified in Table 5.A.4, in which each distribution is plotted against a Weibull y-axis. These plots show each Weibull distribution to be a straight line as required when plotted against a Weibull probability ordinate (see Appendix 2.B), and show each lognormal distribution to be of downwards curvature as indicated in the sketch of Figure 5.A.6(c) [compare also with the measured data in Figure 5.A.7(a) (left) and (b) (left)]. Figure 5.A.11 shows the same Weibull and lognormal distributions as in Figure 5.A.10, but here plotted against a lognormal y-axis. These plots show each lognormal distribution to be a straight line as required when plotted against a lognormal probability ordinate (see Appendix 2.B), and show each Weibull distribution to be of upwards curvature as indicated in the sketch of Figure 5.A.6(b) [compare also with the measured data in Figure 5.A.7(a) (right) and (b) (right)]. Considerably more low-angle measured clutter data tend to be Weibull-like or quasi-Weibull (i.e., displaying a significant degree of upwards curvature when plotted against lognormal ordinates, as shown by the Weibull plots in Figure 5.A.11), than appear to be lognormal-like or quasi-lognormal (i.e., displaying a significant degree of downwards curvature when plotted against Weibull ordinates, as shown by the lognormal plots in Figure 5.A.10).

In Figures 5.A.10 and Figures 5.A.11, the crossover points in the high-end distribution tails where the lognormal distribution starts to extend to higher values of σ° compared with the Weibull [see Figures 5.A.6(c) and (b)] only explicitly appear in the first two plots of each figure. Figure 5.A.12 shows the first four of the five basic Weibull and five basic lognormal distributions (see Table 5.A.4) plotted to very high values of cumulative probability to better show the behavior of the high-end tails. It is apparent in these plots that, at high enough probability levels, the lognormal distribution always reaches to higher values of σ° (i.e., always has a longer high-end tail) than the corresponding Weibull distribution, for a given ratio of sd/mean.

5.A.4 FITS TO MEASURED FARMLAND CLUTTER: WEIBULL VS LOGNORMAL VS K-

Many distributions have been proposed in the literature to model the amplitude pdf of spiky clutter; see, for example [1, 3–13]. This section provides results from a study (for which [14] is a summary) of two particular experiments of Phase One ground clutter that show that in these data Weibull statistics usually (but not without occasional exception) provide better fits to the clutter amplitude distributions than do lognormal or K-distributions.²⁸

Other results from this study are discussed in Chapter 4, Section 4.6.3, under the heading “Spatial Correlation.” In this study, the term clutter amplitude refers to the clutter voltage amplitude, that is, to the square root of the sum of the squares of the I and Q voltage components. In what follows, the empirical pdf is compared with the Weibull, lognormal, and K- pdf’s having the same first- and second-order moments, and with the Rayleigh pdf having the same variance. Additional details are available in [14], including expressions for the first- and second-order moments of the Weibull, lognormal, and K- pdf’s ([12, 15]; see also Appendix 2.B).

The two clutter experiments studied were of open farmland X-band clutter obtained at the clutter measurement site of Wolseley, Saskatchewan; one experiment is at VV-polarization, the other at HH-polarization. Both experiments are of Phase One survey data taken in slow scan mode (see Chapter 3) through one ∼90° azimuth sector in which 703 azimuth samples were collected per range cell. Each experiment contains four range intervals, and each range interval contains 316 range cells of 15-m sampling interval. Thus each range interval covers 4.74 km, and the four contiguous range intervals cover from 1 to 20 km. Figure 5.A.13 shows a three-dimensional clutter map covering the first range interval at VV-polarization; the spiky nature of the discrete farmland clutter is evident.

Figure 5.A.14 shows the histogram of the I component of the clutter voltage signal from the fourth range interval at VV-polarization—the histogram of the Q component is similar. Long, drawn-out, slowly diminishing tails are evident in the histogram of Figure 5.A.14. The histogram (or empirical pdf) of Figure 5.A.14 is compared in the figure with the Gaussian pdf having the same variance as the empirical pdf and zero mean (the very small, residual dc offset of each measured channel was estimated in each range interval and subtracted from the data). It is apparent that the I (and Q) components of spatially varying clutter amplitudes do not follow Gaussian distributions (i.e., the voltage amplitude is not Rayleigh, and the clutter strength distribution is not exponential). An X-band histogram and cumulative distribution of clutter strength σ°F⁴ from a clutter patch at this same Wolseley site were shown previously in Figure 5.15 of Chapter 5; these data of Figure 5.15 appear to be well modeled as a Weibull distribution. To be determined here is if this previous indication of Weibull is borne out in the following more analytically rigorous results.

Figure 5.A.15 plots clutter amplitude histograms for the second and fourth range intervals, both at HH-polarization, and compares them with Rayleigh, Weibull, lognormal, and K-distributions. In this figure, the measured pdf is compared with the Weibull, lognormal, and K- pdf’s having the same first and second order moments, and with the Rayleigh pdf having the same variance. For the second range interval (a), the data are very well fitted by the Weibull approximation over the full range of amplitudes shown. For the fourth interval (b), the data appear to generally lie between the Weibull and lognormal approximations.

Figure 5.A.16 shows an X-band clutter amplitude histogram obtained as temporal variations from a fixed cell containing windblown trees. These data were obtained at the Katahdin Hill measurement site; they constitute long-time-dwell data (see Chapter 3) in which 30,720 samples per cell were recorded at 500 Hz PRF (stationary antenna, VV-polarization; 15-m pulse length). These temporal data are shown here to indicate the profound differences in clutter statistics that result when the clutter is collected as spatial variations over many cells of open farmland, on the one hand (Figure 5.A.15), and as temporal variations from windblown foliage, on the other (Figure 5.A.16). The temporal data in Figure 5.A.16 very closely follow the Rayleigh distribution—the I and Q components of these data follow Gaussian pdf’s as shown in Figure 5.A.14. The theoretical Rayleigh pdf in Figure 5.A.16 which the empirical data so closely follow is normalized to have the same variance as the data.

For the Wolseley farmland spatial clutter data, Figure 5.A.17 shows the first six normalized moments as computed from the data and compares them to the theoretical higher moments of Weibull, lognormal, and K-distributions [12, 14, 15]. In these results, the parameters of the theoretical distributions are set by matching the first two moments of the theoretical distributions to the first two computed moments in the empirical clutter data. As in Figure 5.A.15, results are shown for the second and fourth range intervals at HH-polarization. For the second range interval (a), all four higher moments of orders 3, 4, 5, and 6 computed in the measured data match those of the Weibull distribution very closely—in fact, the measured moments actually overlay the Weibull theoretical moments in this plot. For the fourth range interval (b), the four higher moments computed in the measured data lie between the corresponding Weibull and lognormal moments—but somewhat more closely to Weibull. These results substantiate what initially appears to be true in the histograms of Figure 5.A.15. Note in Figure 5.A.17 that Weibull higher moments lying between lognormal and K-distribution higher moments shows in another way how Weibull distributions lie intermediate between lognormal and K-distributions in terms of degree of spread involved (compare with the figures showing distribution shape in Section 5.A.3).

Figure 5.A.18 plots the measured cdf’s, here for the third and fourth range intervals both at HH-polarization, on Weibull paper. The Weibull paper utilized is such that the abscissa is , and the ordinate is . In such a plot, a Weibull distribution for clutter amplitude plots as a straight line with slope given by the shape parameter of the Weibull distribution [14]. These matters are explained more fully in Appendix 2.B.

The empirical cdf for the third range interval is quite linear when plotted on the Weibull paper of Figure 5.A.18(a), indicating that a Weibull distribution accurately fits these data. The empirical cdf for the fourth range interval is also quite linear in Figure 5.A.18(b). In Figure 5.A.18, each measured clutter cumulative or cdf is compared with the method of moments (MoM) line representing the Weibull cdf with the same mean and mean square values as the measured data, and with the line obtained by linear least squares (LS) fitting to the data [16]. For each range interval, it is apparent in Figure 5.A.18 that the MoM and LS techniques provide very similar values for coefficients (i.e., slope, intercept) of Weibull distributions to match the measured cdf’s.

Note that the Weibull distributions shown in Figure 5.A.18 are for the clutter amplitude ; the shape parameter a_w for the corresponding cdf of clutter strength σ° is twice that obtained directly from Figure 5.A.18. Thus, the LS value of a_w applicable to the σ° cdf in both (a) and (b) of Figure 5.A.18 is a_w = 4.55.²⁹ This value compares closely with the value pertaining to the Wolseley data of Figure 5.15, viz., a_w = 4.7 (see Table 5.12), for which the applicable clutter patch straddles the first and second range intervals. In both parts (a) and

(b) of Figure 5.A.18, the regions where the data tails off at the left are the result of radar noise contamination and should be ignored.

A goodness-of-fit hypothesis test was applied to each of the eight sets of Wolseley X-band farmland clutter data—i.e., four range intervals, two polarizations. This test determines which distribution, Weibull, lognormal, or K-, best fits each set of data. Over their complete extents, measured low-angle clutter distributions almost never pass rigorous statistical hypothesis tests for belonging to any practicable theoretical distribution. Therefore, a modified Kolmogorov-Smirnoff test was employed such that only the data in the upper tail region was taken into account. The test is characterized by the parameter α that represents the probability of Type I error. Probability of Type I error is the probability of error that occurs in rejecting the null hypothesis when true. Here, the null hypothesis is that the measured (empirical) clutter distribution is equal to the hypothesized theoretical distribution under test. If this probability is low, the hypothesis that the empirical distribution is equal to the theoretical distribution should be rejected; if this probability is high, this same hypothesis should be accepted [17]. Results are shown in Table 5.A.5. The high probabilities under Weibull indicate that all eight data sets are reasonably well fitted by Weibull distributions in the region , some very closely. Only in three cases is the lognormal probability higher than Weibull, but the Weibull probability is still high in each of these three cases. Some of the lognormal probabilities are very low. All of the K-distribution probabilities are low or very low, indicating that none of the eight distributions are well fitted by K-distributions in their upper tails.

As a result of the various preceding analyses of the empirical Wolseley farmland spatial clutter amplitude distributions provided in Section 5.A.4, it is evident that, in the first and second range intervals, these clutter data are accurately represented and much better fitted by Weibull distributions than by lognormal or K-distributions. In the third and fourth range intervals, the empirical data show behavior intermediate between Weibull and lognormal.

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2. Nathanson, F. E., Reilly, J. P., Cohen, M. N. Radar Design Principles, 2nd ed. New York: McGraw-Hill, 1991.

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