Radar beam- and range-weighting effects on three-dimensional radar
imaging for the atmosphere
Jenn-Shyong Chen and Chun-Hua Chen
Department of Computer and Communication Engineering, Chienkuo Technology University, Taiwan
Paper submitted to Radio Science
Corresponding author: Jenn-Shyong Chen
Add.: Department of Computer and Communication Engineering,
Chienkuo Technology University, No. 1, Jieshou N. Rd., Changhua City 500, Taiwan E-mail: [email protected] Tel : +886–4–7111111 ext 2300 Mobile Phone: +886–9–28128935 Fax: +886–4–7111163 2 4 6 8 10 12 14 16 18 20 22
Radar beam- and range-weighting effects on three-dimensional
radar imaging for the atmosphere
Jenn-Shyong Chen and Chun-Hua Chen
Department of Computer and Communication Engineering, Chienkuo Technology University,
Taiwan
Abstract
Multiple-receiver and multiple-frequency radar imaging techniques used with VHF atmospheric radar can improve, respectively, angular and range resolutions of the atmospheric irregularity structure inside the radar volume. In addition to the imaging method itself, calibration of the imaging is also crucial to yield a better visualization of the irregularity structure. In this paper, the three-dimensional radar imaging using multiple receivers and multiple frequencies simultaneously is
demonstrated on the basis of numerical simulation with the Capon method. More than previous works did, radar beam and range weighting effects on the imaging were examined. Beam weighting effect on angular brightness distribution of the scattering region is apparent, but it gives different impacts on the range imaging as
accompanying with the range weighting effect: the higher the range position of the scattering region is, the smaller the synthetic influence of both weighting effects will 24 26 28 30 32 34 36 38 40 42
be. Beam weighting effect also causes a range-shortening effect for a localized scattering region, which should be considered in interpreting the shift of range brightness distribution with angular location of the target. Moreover, it is
demonstrated that adaptable beam width is effectual to recover a two-blob structure and a wavy layer. An experimental case was also investigated to demonstrate the need of correcting beam weighting effect using adaptable beam width; meanwhile, it suggests that an adaptable range weighting function could be helpful to a further amendment of the three-dimensional imaging.
1. Introduction
Atmospheric radar imaging using multiple-receiver and multiple-frequency techniques has been introduced to VHF atmospheric radar for years to make an attempt on reconstructing atmospheric refractivity structure in the radar volume. The two techniques can improve, respectively, angular and range resolutions of the refractivity structure, which are termed, respectively, coherent radar imaging (CRI) [Woodman, 1997; Palmer et al., 1998] and range imaging (RIM) [Palmer et al., 1999] or frequency-domain interferometric imaging (FII) [Luce et al., 2001]. Furthermore, three-dimensional (3-D) imaging, using multiple receivers and multiple frequencies 44 46 48 50 52 54 56 58 60 62
simultaneously, has also been developed to retrieve the 3-D atmospheric structure [Yu
and Palmer, 2001]. Following these pioneering studies, a number of applications to
the atmosphere have been carried out [e.g., Hysell et al., 2004; Chau and Hysell, 2004; Palmer et al., 2006; Chen et al., 2007; Chen et al.; 2008; Yu et al., 2010].
The radar imaging is a process with inversion algorithm such as Fourier, Capon, and Maximum Entropy methods [Yu and Palmer, 2001], yielding the estimate so-called power density or brightness function that is related to fluctuations of refractive index in the space. The brightness estimates, however, are known to be weighted by radar beam (or antenna pattern) and range weighting functions of the radar system. Removals of both weighting effects with theoretical weighting functions usually result in bias or unrealistic features; in view of this, adjustable weighting functions have been suggested for RIM [Chen and Zecha, 2009] and CRI [Chen and Fuumoto, 2011], respectively, to yield more plausible scattering structures.
One purpose of this paper is to show a simulation study of 3-D imaging for the scattering structures, based on the instruction given by Yu and Palmer [2001] but with a simplified simulation model. Moreover, radar beam and range weighting effects on the 3-D imaging are considered. The radar beam weighting effect was not examined explicitly in the CRI simulation works made by Yu and Palmer [2001]. Recently,
Chen and Furumoto [2011] proposed an approach of using adaptable beam width to
64 66 68 70 72 74 76 78 80
observational CRI data, in which the adaptable beam width was derived from the CRI data of several beam directions and was demonstrated to vary with the
transmission/reception configuration as well as signal-to-noise ratio of the data. Nevertheless, the study is short of simulation illustration. Radar beam weighting effect on RIM was also demonstrated in the literature [Chen et al., 2010], but we discuss it in more detail for the 3-D imaging. Finally, an observational case was investigated to show the effectiveness of correction of radar beam and range
weighting effects on the 3-D imaging. Virtually, the works exhibited in this paper can be regarded as extended and supplementary researches of previous studies.
The mathematical expressions of 3-D radar imaging for the atmosphere are given briefly in section 2. Radar scattering model for simulation is described in section 3. Section 4 gives the simulation results, and section 5 demonstrates an observational case as well as some perspectives of correction of radar beam and range weighting effects in an actual situation. Conclusions are stated in section 5.
2. Three-dimensional radar imaging
A detailed tuition of the 3-D radar imaging for the atmosphere has been given by 82 84 86 88 90 92 94 96 98 100
Yu and Palmer [2001]. In this section, a short description is made in brief notation.
Giving N spatially separated receivers at locations of D1, D2,…, Dn and transmit-ting M carrier frequencies at f1, f2,…, fM from pulse to pulse, the signals from these
re-ceivers and carrier frequencies are first expressed as a column vector:
T MN M M N N t S t S t S t S t S t S t S t S t S (t) )] ( ) ( ) ( ) ( ... ) ( ) ( ) ( ) ( ) ( [ 2 1 2 22 21 1 12 11 S , (1)
where the superscript T represents the transpose, and Sij(t)is the signal from carrier
frequency i and receiver j. The cross-correlation function of the signal S(t) is needed in the imaging process:
MM
N
M
M
M
H
V
V
V
V
V
V
t
t
V
V
V
)
(
)(
2
1
2
22
21
1
12
11
S
S
V
, (2)where the superscript H represents the Hermitian (conjugate transpose) operator. V is a MN×MN matrix (termed visibility matrix), and Vpq is a N×N matrix consisting of
102 104 106 108 110 112 114 116
cross-correlation functions between receivers for the frequency pair of p and q. For example,
*
1
1
*
12
1
*
11
1
*
1
12
*
12
12
*
11
12
*
1
11
*
12
11
*
11
11
11
N
N
N
N
N
N
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
V
, (3a),
*
1
2
*
12
2
*
11
2
*
1
22
*
12
22
*
11
22
*
1
21
*
12
21
*
11
21
21
N
N
N
N
N
N
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
V
(3b)where asterisk indicates the conjugate of a complex number. Note that the time vari-able t has been omitted from (3) for notational simplicity. Vpq is the estimate of
visibil-ity function without time lag; that is, imaging for different Doppler frequencies of the signals is not considered here.
The imaging processing is to retrieve the so-called brightness distribution via the visibility matrix V, defined as
118 120 122 124 126 128
Vw w a a H , R W , R B ) ( 1 ) ( 2 (4)
where a=[sinθsinφ, sinθcosφ, cosθ], which is the angular direction at zenith angle θ and azimuth angle φ. The variable R is the range. W(a, R) is the weighting function for the echoes in the radar volume, formed by radar beam (antenna pattern) and range weighting function. Matrix w is the weighting vector for retrieving the brightness dis-tribution, given as T k R k j k R k j k R k j k R k j k R k j k R k j k R k j k R k j k R k j N M M M M M M N N e e e e e e e e e ] [ ) 2 ( ) 2 ( ) 2 ( ) 2 ( ) 2 ( ) 2 ( ) 2 ( ) 2 ( ) 2 ( 2 1 2 2 2 2 2 1 2 2 1 1 2 1 1 1 1 1 D a D a D a D a D a D a D a D a D a w , (5)
where ki is the wave number of carrier frequency i. Substituting (5) into (4) yields the
so-called Fourier brightness that has a coarse resolution. To enhance the imaging res-olution, the Capon method is one of the choices:
e V e a a 2 1 1 ) ( 1 ) ( H , R W , R B , (6) 130 132 134 136 138 140 142 144 146
where e has the same form as (5). Eqs. (4) and (6) indicate clearly that brightness esti-mate is weighted by the weighting function W(a, R). Removal of W(a, R) from the brightness estimate is thus necessary to reconstruct 3-D atmospheric structure.
3. Simulation model
The output voltage of a receiver for the echoes backscattered from the refractiv-ity irregularities in the radar volume can be given briefly as
dv
kR
j
R
W
R
L
S
v(
a
,
)
(
a
,
)
exp(
2
)
, (7)where L(a, R) represents the 3-D distribution of scatterers, W(a, R) denotes the 3-D weighting function, k is the wave number of carrier frequency, R is the range, and
vector a indicates angular direction.
v denotes the radar volume of integration. Thecross-correlation function between signals from the ith receiver with mth carrier fre-quency and the jth receiver with nth carrier frefre-quency can be written as
148 150 152 154 156 158 160 162 164
v
)
exp(
)
,
(
)
,
(
)
,
,
,
(
k
k
S
S
* vL
2
R
W
2
R
j
d
R
D
i mD
j n
mi nj
a
a
, (8a) j n i m k k kR aD aD 2 . (8b)Eq. (8) is basically similar to that given by Yu and Palmer [2001], only the temporal lag between signals is not considered here and the phase has opposite sign. L2(a, R)
gives the part of brightness function/spectrum, and Δk = kn - km.
For simplicity, L2(a, R) and W2(a, R) are expressed by Gaussian functions in both
angular and range/height directions:
]
2
)
(
exp[
]
2
)
(
exp[
]
2
)
(
exp[
)
,
(
)
,
(
2 2 2 2 2 2 2 2 t t l yo y xo x oz
z
z
L
R
L
a
a
, (9))
2
exp(
)
2
exp(
)
2
exp(
)
,
(
2 2 2 2 2 2 2 b y b x rR
R
W
a
, (10)where zo, xo, and yo represent, respectively, mean altitude, off-zenith angular
loca-tions of the scattering region center in zonal and meridional direcloca-tions, and l and t
are standard deviations of the scattering region in the vertical and angular directions. z 166 168 170 172 174 176 178 180 182
is the variable of height. r is given theoretically as 0.35c /2, where c is the speed of
the light and is the pulse length. b is beam width. Note that L(a, R) is assumed to be
symmetric in angle with respective to its central location (xo, yo), and W(a, R) is
symmetric in angle with respective to zenith.
In our simulation, the scatterers are located within a horizontal region, and they are confined to a horizontal layer when t gets very large. With this scattering model,
the 3-D distribution of scatterers, L(a, R), becomes a function of height, as expressed in (9). A schematic model is depicted in Fig. 1. Our simulation model is different from the dynamic simulator proposed by Cheong et al. [2008]. The dynamic simulator assumes thousands of scatterers moving with the three-dimensional wind field and so can produce sample-to-sample time series data. By contrast, our simulation model is stationary and can only give the output which is equivalent to the average of time se-ries data and so cannot be used for some studies like spectral-based measurements. In spite of this limitation, the simulation model used here is sufficient for the present study.
The cross-correlation function in (8) is estimated for each set of receivers and carrier frequencies by giving suitable integration grid, for example, 1 m in range and 0.1o in angular plane. The obtained cross-correlation functions are then used with (2)
and (4), or (6) to retrieve the brightness distribution. It should be mentioned that the 184 186 188 190 192 194 196 198 200 202
integration grid size may cause a difference between computational results [Cheong et
al., 2008], but the difference is hardly observed in the present simulation model when
the integration grid is smaller than 10 m in range and 0.5o in angle. It is surly that the
computation accuracy also depends on the size and shape of the scattering structure model; this part, however, is not the main subject of the study. In the following sec-tions, the main subject: radar beam and range weighting effects on 3-D radar imaging, is discussed.
4. Simulation results
The degree of weighting effect on the 3-D brightness may depend on a variety of factors, including the characteristics of scattering structure (location, size, shape, position…) and the radar parameters (beam width, pulse length and its shape, filtering bandwidth…). In this section, some selected cases are exhibited to convey the thought of 3-D weighting effect. In section 4.1, we illustrate the weighting effect for three factors: radar beam width, range and angular locations of the target. The scattering model of a horizontal layer or a single localized region was assumed. In section 4.2, a two-blob structure was examined and mitigation of the 3-D weighting effect was made. The third case, as shown in section 4.3, is a wavy layer in the radar volume, 204 206 208 210 212 214 216 218 220 222
which is a common situation in the radar remote sensing of the atmosphere.
4.1 Horizontal layer and single scattering region 4.1.1 Variation with range position of the target
First, a horizontal layer and a localized scattering region shown in Figure 1 were investigated with our 3-D imaging algorithm, respectively. Also shown in Figure 1 is the wavy layer which will be examined in section 4.3. The scatterers in the localized region are Gaussian-distributed in the vertical and angular directions; in the layer, however, the scatterers are only confined in the vertical direction with a Gaussian function. Simulation parameters are listed in Table 1, and only the Capon result are shown in this paper for its higher resolution than the Fourier method.
Figure 2 displays the imaging results, in which the brightness values are exhibited in angular (CRI) and range (RIM) direction, respectively. In showing the 1-D RIM profile, the brightness values at equal-range surface are summed. Note that the hori-zontal layer and the localized region are given at some specific heights with respec-tive to the central height of the range gate, as indicated by the horizontal dotted line in the plot of RIM profile. The horizontal dashed line presents the range position of the maximum point in the model structure (solid curve). For the horizontal layer model, the dotted and dashed lines are overlapped.
In Figure 2(a), five maps of 2-D imaging at different ranges are exhibited in the 224 226 228 230 232 234 236 238 240 242
left column for the case of a horizontal layer at height of -120 m, and the second col-umn shows the corresponding RIM profile (dashed curve). Shown in the third and fourth columns are the RIM profiles for the layers located at heights of 0 m and 120 m, respectively, but their 2-D imaging maps are not displayed for their features are very similar to the maps in the left column. Figure 2(a) displays that the 2-D imaging structure is localized in angular direction even the simulation model is a horizontal layer. Obviously, the localization in angular direction arises from the radar beam weighting effect. On the other hand, the RIM profile shows a layer structure with range position higher than the altitude of the model layer (solid curve), but the differ-ence in peak location between the retrieved and the original layers gets smaller as the layer structure is located at a higher altitude. Such feature of RIM profile is a syn-thetic result of radar beam and range weighting effects, and it can be explained as fol-lows. The radar beam gives not only angular weighting effect, but also its finite beam width that will bias the RIM layer to a higher range position, as demonstrated by
Chen et al. [2010]; on the other hand, the range weighting function centralizes the
RIM layer to the central position of the range gate. As a result, the two effects are ei-ther constructive or destructive, depending on the layer being at the lower or upper parts of the range gate. For a layer just around the central height of the range gate (the third column), however, the range weighting effect on the peak location of the re-244 246 248 250 252 254 256 258 260
trieved RIM profile is small, and so the bias of the retrieved RIM profile arises mainly from the finite beam width.
The features observed in Figure 2(a) vary a bit when a localized scattering re-gion at off-vertical location, not a horizontal layer, is given for 3-D imaging. As seen in Figure 2(b), the 2-D imaging maps reveal that the brightness center strays from the given center (asterisk) and deflects to the zenith, which arises obviously from the beam weighting effect. As for the RIM output, the peak location of the retrieved RIM profile also depends on the altitude of the scattering region, but the bias of peak loca-tion of the retrieved RIM profile is obviously smaller than that of a horizontal layer, and even more, the case of zl=120 m has a RIM profile lower than the model structure.
In view of this, the range weighting effect seems over the effect of finite beam width for this case. For a localized scattering region, however, we have to consider another effect in addition to the range weighting effect: the beam weighting effect deflects the brightness distribution to the zenith and so shortens the range position of the scatter-ing region, which lowers the range position of the retrieved RIM profile. These con-siderations can figure out the range variation of the retrieved RIM profiles in Fig. 2(b). It is of interest to see that the three effects range weighting, finite beam width, and beam weighting are almost balanced off for the case of zl=0 m in this simulation
model. 262 264 266 268 270 272 274 276 278 280
4.1.2 Variation with radar beam width
Figure 2 reveals the inevitable effects of radar beam and range weighting func-tion on 3-D imaging (or CRI and RIM). To demonstrate such effects more expressly, Figure 3 shows a simulation with beam width of 7o; the other parameters are the same
as those listed in Table 1. For the case of a horizontal layer shown in Figure 3(a), it is seen that a larger beam width causes a larger deviation of position to the retrieved RIM profile, as referring to Figure 2(a). As the layer becomes localized in angular di-rection, the beam width effect on the RIM output is reduced again, as indicated in Fig-ure 3(b) where the difference in peak location between the model and the retrieved RIM profile gets smaller. It is noticed that in Fig. 3(b) the retrieved RIM profile for the scattering region at height of 120 m is very close to the model, indicating that the beam width effect on RIM is almost balanced off by the range and beam weighting ef-fects around this height.
Compared with Figure 2(b), the 2-D imaging maps in Figure 3(b) shows that the beam weighting effect on the 2-D imaging is smaller for that the brightness center is closer to the given center (asterisk). The same feature also occurs in the simulation case with a horizontal layer, as shown in Figure 3(a), where the 2-D brightness distri-bution is wider than that in Figure 2(a) and so is closer to the horizontal layer model. One more noticeable feature observed in Figure 3(a) is that the maximum of the 282 284 286 288 290 292 294 296 298 300
2-D brightness distribution at range position of -80 m or higher is not at zenith. Such phenomenon can be explained with the Gaussian layer model shown in Figure 4. In Figure 4, the dotted curve indicates the equal-range surface for the scattering echoes. In the radar volume, surface 1 is above the central height of the layer. The scattering points A and C are closer to the central height of the Gaussian layer, and so their scat-tering echoes are stronger than that at point B. As a result, maximum brightness does not appear at zenith. By contrary, surface 2 is below the central height of the layer. The scattering point b is closer to the central height of the layer, and so the scattering echoes are stronger than that at points a and c, leading to a maximum at zenith.
4.1.3 Variation with angular location of the target
Figures 2 and 3 illustrate the weighting effect at different range positions of the target, as well as different beam widths. One more inspection regarding the angular location of the target is displayed in Figure 5, where three localized regions are lo-cated at zenith angles of 1o, 5o and 9o, respectively. Beam width is 3o and altitude of
the target is -120 m; the other parameters are listed as the localized region in Table 1. In the 2-D imaging, it is apparent that the bias of the brightness center gets larger when the target is located at the place farther from the zenith; this feature is attributed to the Gaussian-shaped beam weighting function and has been addressed in Chen et
al. [2008] with a simulated radar beam pattern. For the 1-D RIM output, the three
fac-302 304 306 308 310 312 314 316 318 320
tors: range weighting, finite beam width, and beam weighting are all needed to inter-pret the variation of the RIM profile, just like the discussion for Fig. 2 (b).
For the target at zenith of 1o, the beam weighting effect on shortening the range
of the RIM profile is small and so the range weighting function and finite beam width shift the RIM profile to a higher range position. With the increase of the zenith angle of the target, however, the range-shortening effect arising from the beam weighting effect gets more and more crucial and ultimately overrides the range-extending effects of range weighting and finite beam width. As a result, the retrieved RIM profile is bi-ased to the range position lower than the model, as demonstrated in the situation of 9o
zenith angle.
4.2 Multiple scattering regions
One of the capabilities of multiple-receiver and multiple-frequency techniques is to distinguish multiple targets in the radar volume. This capability was examined and shown in Figure 6. Two scattering regions are located at (z, , )= (-40 m, 5o, 45o) and
(40 m, 5o, 180o), respectively. Each scattering region is modeled by a Gaussian
distribution with standard deviations of 12.5 m and 2.5o, respectively, in the vertical
and angular directions. Radar beam width is 3o, and the other simulation parameters
are the same as those listed in the right column of Table 1.
In Figure 6, the retrieved RIM profile shown in the right column presents that the 322 324 326 328 330 332 334 336 338 340
two scattering regions can be identified in range but recognition of the two scattering regions is poor. The 2-D brightness distributions at different range positions, as shown in the left column, can also reveal two scattering regions, but displacement of bright-ness distribution arising from beam weighting effect is evident. It will be shown later (Figure 7) that the 2-D brightness distribution obtained by adding the brightness val-ues together along the range direction cannot separate the angular locations of the two scattering regions if the beam weighting effect is not removed. It is possible, however, to improve the recognition of the two scattering regions with other imaging algo-rithms such as Maximum entropy method, as demonstrated by Yu and Palmer [2001]. We do not examine this issue repeatedly here because our goal is orientated towards the beam weighting effect on the imaging.
Figure 7 displays some 3-D views of CRI and RIM brightness distributions corrected with various beam widths and a fixed range weighting function. The concept of adaptable beam width proposed by Chen and Furumoto [2011] was also tested. Because of without consideration of noise in the simulation, the cubic equation form given by Chen and Furumoto [2011] was attempted here:
2 1 0 c c m b , (11)
where is the beam width, which is a function of zenith angle . Taking m=3, 342 344 346 348 350 352 354 356 358 360
c1=0.0025, and c2=2.800 as an example, the corrected result is shown in the
bottom-right plot of Figure 7. Note that (11) is just an attempt of fitting to recover the simu-lated brightness distribution and so it is not the only choice; any expression that is able to recover the model structure well can be employed. In the analysis of true data, the expression of adaptable beam width is associated with SNR and antenna configu-ration used in observation, as demonstrated by Chen and Furumoto [2011] for CRI. This issue will be discussed in section 5 with an experimental case.
In Fig. 7, the CRI brightness, exhibited with contour lines on the zonal-merid-ional plane, was obtained by adding the brightness values together along the range di-rection. On the other hand, two kinds of RIM outputs are displayed: one of them is like that in Figure 6, as shown on the range-meridional plane; the other is summed from the four brightness values at equal range that are estimated around the angular centers (+) of the 3-D imaged brightness, as shown on the range-zonal plane. The re-spective RIM profiles shown on the range-zonal plane yields a mean RIM profile around each angular center (+), which can suppress the smearing effect or interference from the directions farther from the angular center (+) and so provide a better range resolution of the localized region. A similar processing has been made in the 3-D data processing and for the RIM layer around zenith direction [Yu et al., 2010].
For comparison, the two black spots in Figure 7 represent the central locations of 362 364 366 368 370 372 374 376 378
the model regions, and their coordinates projecting on angular plane and range direc-tion are indicated by the red-dashed lines. The RIM and CRI outputs are discussed as below:
4.2.1 RIM output
It is shown that the retrieved RIM profiles (blue curves) on the range-zonal plane are very close to the two scattering models, the difference in the peak location of the retrieved RIM profile depends on the beam width used with the correction of beam weighting effect. In this case, correction with beam width of 4o or adaptable beam
width leads to a better result. As for the retrieved RIM profiles (blue curves) on the range-meridional plane, they are basically similar to that in Figure 6 but the correction with smaller beam width results in smaller contrast between summit and valley, making the recognition of the two peaks worse. By contrast, correction of using adaptable beam width yields a better recognition of the two peaks in the retrieved RIM profile: the contrast between summit and valley is larger.
Note that a fixed range weighting function was used in this case for correction of range weighting effect. We will show in the observation that this may not be the best scheme to retrieve the RIM profile. Using an adaptable range weighting function could be helpful to improve the RIM profile further.
380 382 384 386 388 390 392 394 396 398
4.2.2 CRI output
First, the two angular centers (+) are evidently different from the models (*) without correction of beam weighting effect, and the contoured 2-D brightness distribution displays only one center around zenith (see the up-left plot of Figure 7). After correction of beam weighting effect, however, the two angular centers (+) separate from each other and get close to the models (*); meanwhile, two contouring centers also appear in the 2-D brightness distribution and get close to the models except for the use of 3.3o beam width. The brightness value resulting from the use of
3.3o beam width is severely over-corrected, leading to some peak values at the edges
of the 2-D brightness distribution. In fact, over-correction at the edges of the 2-D brightness distribution has occurred since the use of 4.5o beam width.
In view of the above situation, the use of a fixed beam width cannot yield correct echo centers and meanwhile recover the brightness distribution properly. To obtain a more acceptable output, correction of brightness values with adaptable beam width, as expressed in (11), could be one of the solutions. The corrected 2-D brightness
distribution is exhibited in the bottom-right plot of Figure 7. As shown, not only the angular centers (+) of the two scattering regions are very close to the models (*), but also the 2-D brightness distribution displays a befitting result.
4.3 A wavy layer in the radar volume
400 402 404 406 408 410 412 414 416 418
Considering that slanted layers, which are usually associated with wave activities, often exist in the radar volume, a wavy layer was examined in addition. For
simplicity, the wavy layer is supposed to be homogeneous in the meridional direction and Gaussian-distributed in the vertical direction, and with wavelength of 2hotan10o in
the zonal direction and amplitude of 160 m, where ho is the altitude of the radar
volume center. The height of the wavy layer relative to ho is expressed as
o o o o 10 tan 2 ) 10 tan ( 2 sin 160 ) ( h h x x h (12)
where x is the location variable in the zonal direction. The simulation parameters are similar to the horizontal layer in Table 1, but the beam width is 3o.
Figure 8 shows the simulation results, where the contoured brightness
distributions of five slices at range positions of -160m, -80m, 0m, 80m, and 160m are displayed. The thick solid curve indicates the range position of the layer center. In the upper plot, it is seen that the contour centers move basically with the range position of the layer center, which indicates the usability of the 3-D imaging. Of course the brightness distribution suffers beam weighting effects so that the brightness values around both edges of zonal dimension are too small to disclose the layer structure. If a fixed beam width, say 3.3o, is used to correct the brightness value, the brightness
420 422 424 426 428 430 432 434 436 438
values will be over-corrected severely around the edges of imaging, as illustrated in the middle plot; this is obviously not appropriate. On the other hand, the bottommost plot shows an attempt of recovering the brightness distribution with (11), and m=2.4. The corrected brightness distribution exhibits some qausi-aligned structures in meridional direction, which in fact can be clarified with the wavy layer model. In the contour map at range position of zero, three meridional-aligned structures around zonal angle of 0o and both zonal edges can be found, which indicate the approximate
locations of the wavy layer over there. Similar meridional-aligned structures also appear at range positions of -80 m and 80 m. As for the contour maps at range positions of 160 m and -160 m, near the locations of summit and valley of the wavy layer, only one meridional-aligned structure is seen. In view of this, correction of weighting effect indeed works here although the modified brightness distribution can still be improved to match with the wavy layer model.
Note that the value given to m is slightly different from that used for Figure 7 (m=3), which is in the cause of pursuing a feature closer to the layer model. We have attributed it to the more complex structure of the layer model, approximate form of (11), and some numerical errors. We should bear in mind that in practical data analysis, a suitable expression for mitigation of the weighting effect should be obtained from the data themselves. The mitigation processing made here is just to 440 442 444 446 448 450 452 454 456
demonstrate the possibility of recovering the brightness distribution with some suitable expressions.
5. Observation demonstration
An experimental case, observed by the MU radar in Japan, was examined and shown in this section to support the necessity of mitigation of the weighting effect. The whole antenna array was used for transmission and nineteen receivers were set for reception. Five frequencies, 46.00, 46.25, 46.50, 46.75, and 47.00 MHz, were transmitted with pulse length of 1 s. The effective radar beam width of this transmis-sion/reception mode was ~3o under the condition of noise-free [Chen and Furumoto,
2011].
Figure 9 shows the original imaging, where a single scattering region is found to be around the zenith and at range position of ~-40 m. The 3-D views of CRI and RIM brightness distributions corrected with various beam widths and range weighting functions are exhibited in Figure 10, like that shown in Figure 7. The black spot indicates the central location of the scattering region found in the original imaging (shown in the upper-left plot). The RIM curve displayed on the range-meridional plane was obtained by summing the brightness values at equal-range surface; on the other hand, the RIM curve shown on the range-zonal plane was estimated only with 458 460 462 464 466 468 470 472 474 476
the four brightness values estimated at the same range and around the angular centers (+), providing a mean output of RIM around each angular center. As discussed in Figure 7, the RIM profile shown on the range-zonal plane suffers less smearing effect or interference from the directions farther from the angular center (+) and so is supposed to provide a better range resolution of the localized region.
First, correction with 3o beam width and a theoretical range weighting function
leads to over-corrected 2-D brightness distribution and RIM profile at both angular and range boundaries, as shown in the upper-right plot, although the scattering center, indicated by the symbol of plus, can still be identified. By using the angular- and SNR-dependent adaptable beam width suggested by Chen and Furumoto [2011],
0 4 2 0 3 10 b b b SNR c c , (13)
where c31.4751, c4-9.7430, and SNR>-10 dB, another contour center appears at the
corner of ~(-5o, -5o) in the 2-D brightness distribution (middle-left plot), and
moreover, the brightness is not corrected excessively. However, the RIM profiles are still corrected excessively at range boundaries. In view of this, the following equation of r was used for an adaptable range weighting function to make an attempt to
mitigate the over-corrected RIM profile: 478 480 482 484 486 488 490 492 494 496
max 2 r r ro ro r , (14)
where ro = 0.35c/2=52.5 m, c is the speed of the light and is the pulse length. In
this case, is 1 s and rmax=100 m. As a result, more proper RIM profiles can be
obtained, as seen in the middle-right plot. Note that Eq. (14) is just for the purpose of demonstrating the thinking of using an adaptable range weighting function along the range. A reliable expression for the experimental data, unfortunately, cannot be achieved with the present observation. In the literature, an approach to finding a SNR-dependent range weighting function was proposed by Chen and Zecha [2009], which is based on the point of improving the continuity of the RIM profile around the range gate boundaries, not derived from the concept of adaptable range weighting function along the range. In view of this, a range weighting function adaptive to range is worthy of pursuing in the future for improving the accuracy of radar imaging.
In spite of the workable thought of an adaptable range weighting function, we found that the correction of range weighting effect may not be crucial in this case, as demonstrated in the bottommost plot of Figure 10, where the range weighting effect was ignored but the 2-D brightness distribution was hardly influenced by this ignorance. In any case, the importance of an adaptable range weighting function to 498 500 502 504 506 508 510 512 514
radar imaging needs a further examination with suitable experimental data.
6. Conclusions
With multiple receivers and multiple frequencies, numerical examination of three-dimensional (3-D) radar imaging for the atmosphere was made. Radar beam and range weighting effects on the 3-D imaging have been investigated, providing ex-tended and supplementary works of previous studies.
We have illustrated the weighting effect for different beam widths, range and an-gular locations of the target. A horizontal layer and a localized scattering region were examined, respectively. It has been shown that for a horizontal layer, beam weighting effect localizes the angular brightness distribution apparently, and finite beam width biases the range brightness distribution to a higher range position. For a localized scattering region, the beam weighting effect deflects the angular brightness distribu-tion to the zenith, and the severity of the deflecdistribu-tion gets greater for the scattering re-gion located at a larger zenith angle. In the interpretation of the retrieved range bright-ness distribution for a localized scattering region, however, we have to consider the range-shortening effect arising from the beam weighting effect. It has been shown that the peak location of the retrieved range brightness can be higher or lower than the model, depending on the location of the target and the beam width; such feature can 516 518 520 522 524 526 528 530 532 534
be attributed to the competition between the effects of range weighting function and radar beam as well as its finite beam width. To mitigate these inherent angular and range weighting effects, we also demonstrate that adaptable beam width is capable of recovering the brightness distribution to some degree and then gives a more proper lo-cation of the model structure; a two-blob structure and a wavy layer have been exam-ined to validate this.
In addition to the numerical investigation, an experimental case has been studied to verify the effectiveness of employing adaptable beam width. It was shown that a single-center scattering structure turned into a double-center scattering structure, and the angular brightness distribution still retained a proper feature to yield reliable angu-lar centers. On the other hand, correction of range weighting effect yielded brightness values that were corrected excessively around range edges. In view of this, an adapt-able range weighting function has been attempted to reduce the over-correction, but a proper expression associated with the experimental data cannot be obtained with the present radar samples; specific radar parameters and experimental processing is needed for this issue.
The shift in brightness distribution due to weighting effect may not be fatal sometimes for derivation and/or interpretation of some atmospheric phenomena, but it deserves to know the details and limitations of radar imaging applied to the atmos-536 538 540 542 544 546 548 550 552 554
phere and this paper contributes some thoughts to this area. Once the weighting effect is mitigated properly, the atmospheric parameters derived from the corrected bright-ness distribution will be more reliable.
Acknowledgements
This work was supported by the National Science Council of ROC (Taiwan), project no. NSC99-2111-M-270-001-MY2. The authors would like to thank the colleagues of MU radar for providing experimental data for the study.
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Table Captions
Table 1. Simulation parameters for the 3-D imaging in Figures 2 and 3. East and
north directions have positive angles.
Figure Captions
Figure 1. A schematic description of the scattering models for simulation.
Figure 2. Three-dimensional radar imaging resulting from the Capon method. The
imaged values are normalized for each plot. (a) and (b) result from a horizontal layer and a horizontally localized region, respectively. Radar beam width is 3o and
beam direction is vertical. Left column: 2-D brightness distributions at different range positions for a target at height of -120 m (with respective to the central height of the radar volume), where the symbol “*” indicates the angular center of the model structure. Columns 2-4: 1-D RIM profiles for a target at heights of -120 m, 0 m, and 120 m, respectively. Solid and dashed curves are the model and the re-trieved RIM profiles, respectively. The horizontal dotted and dashed lines indicate, respectively, the central height and range of the model structure.
Figure 3. The same as in Fig. 2, but with beam width of 7o.
Figure 4. A horizontal layer structure in the radar volume. The scattering points (A, B,
C) and (a, b, c) are located at surfaces 1 and 2, respectively. The surface is defined
by equal-range points. 616 618 620 622 624 626 628 630 632 634
Figure 5. Presented like Figure 2(b), but for a target at different angular locations.
Beam width is 3o and altitude of the target is -120 m.
Figure 6. Three-dimensional radar imaging for a two-blob structure. Left: 2-D
brightness distribution at different range positions. Right: 1-D RIM profile. Radar beam width is 3o. Remaining descriptions of curve and symbol are the same as
Figure 2.
Figure 7. A 3-D view of the three-dimensional radar imaging for a two-blob
struc-ture. The brightness values are modified with different beam widths (b in the
rec-tangular boxes) and a fixed Gaussian range weighting function (r=105 m). Black
spots indicate the central locations of the two scattering regions in the space, and their coordinates projecting on the horizontal and vertical planes are indicated by red-dashed lines. The symbol “+” shows the angular center of the 3-D brightness distribution, and respective 1-D RIM profiles estimated around the two angular centers (+) are displayed on the range-zonal plane, where the black-solid curve gives the original model structure and the blue-solid curve presents the retrieved 1-D RIM profile. Another kind of 1-1-D RIM profile, summed from the brightness val-ues of equal-range surface, is displayed on the range-meridional plane. All the pro-filing curves are rescaled for comparison of peak locations. Shown on the zonal-meridional plane is the 2-D brightness distribution, presented with contour lines. 636 638 640 642 644 646 648 650 652
Figure 8. A 3-D view of the three-dimensional radar imaging for a wavy layer. The
brightness values are modified with different beam widths (b in the rectangular
boxes) and a fixed Gaussian range weighting function (r=105 m).Thick solid
curve represents the range position of the layer center. The contoured brightness distributions of five slices at range positions of -160m, -80m, 0m, 80m, and 160m are displayed. Black spot indicates the radar volume center, and its coordinates projecting on the horizontal and vertical planes are indicated by red-dashed lines.
Figure 9. Three-dimensional radar imaging for an experimental case, presented like
Figure 6.
Figure 10. A 3-D view of the radar imaging for the experimental case shown in
Figure 9, presented like Figure 7. The solid curves shown on different vertical planes are the retrieved RIM profiles, as defined in Figure 7 for the blue curves. 654 656 658 660 662 664 666
