Effects of radar beamwidth and scatterer anisotropy on
multiple-frequency range imaging using VHF atmospheric radar
Jenn-Shyong Chen1, Jun-ichi Furumoto2, and Takuji Nakamura2
1Department of Computer and Communication Engineering, Chienkuo Technology Univer-sity, Taiwan
2Research Institute for Sustainable Humanosphere, Kyoto University, Japan
Submitted to Radio Science
(May, 2010
)Third Version
Manuscript no.: 2009RS004267R
Corresponding author: Jenn-Shyong Chen
Add: Department of Computer and Communication Engineering, Chienkuo Technology University,
No. 1, Jieshou N. Rd., Changhua 500, Taiwan E-mail: [email protected] Tel : +886-4-7111111 ext 2304, 3397 Fax: +886-4-7111163 Mobile Phone: +886-9-28128935 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
Effects of radar beamwidth and scatterer anisotropy on
multiple-frequency range imaging using VHF atmospheric radar
Jenn-Shyong Chen1, Jun-ichi Furumoto2, and Takuji Nakamura2
1Department of Computer and Communication Engineering, Chienkuo Technology Univer-sity, Taiwan
2Research Institute for Sustainable Humanosphere, Kyoto University, Japan Abstract
Benefiting from the changeable antenna array size and flexible radar beam direction of the Middle and Upper atmosphere (MU) radar system (34.85oN,
136.11oE), the effects of radar beamwidth and scatterer anisotropy on the performance of multiple-frequency range imaging (RIM) were examined in addition to numerical simulation. First, nine transmitter/receiver modes were employed to reveal that a wider radar beam gives a larger phase bias in the RIM processing. Based on this, layer positions and layer thicknesses were estimated from the imaged powers of various radar beamwidths after corrections of phase bias and range-weighting function effect. Statistical examination showed that the imaged layer structure was thicker for a larger radar beamwidth and this feature became more evident at a higher altitude, thereby demonstrating the influence of radar beamwidth on the practical performance of RIM. Second, the scatterer anisotropy in the layer structure was examined by means of a vertical and three oblique radar beams (5o, 10o, and 15o north), which were transmitted in company with the RIM technique. The vertical beam observed some single-layer and double-layer structures that were not always detected by the oblique beams, indicating the existence of anisotropic scatterers in the layers. In addition, a
comparison of layer positions between the vertical and oblique radar beams showed that anisotropic characteristics of the upper and lower layers of a double-layer
structure can be different, demonstrating one more capability of RIM for investigating fine-scale features of the atmospheric layer structures.
Keywords: Range imaging, Radar beamwidth, Scatterer anisotropy, Layer position, Layer thickness 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66
1. Introduction
Range imaging (RIM), using the advantages of frequency diversity, has been applied successfully to the mesosphere-stratosphere-troposphere (MST) radar for resolving fine layer structures in the radar volume for more than one decade [Palmer
et al., 1999] (also called frequency interferometry imaging (FII) by Luce et al.
[2001]). Plenty of observations with the RIM technique have been exhibited [e.g.,
Chilson et al., 2001; Palmer et al., 2001; Chilson et al., 2003; Yu and Brown, 2004; Luce et al., 2006, 2007, and 2008; Chen et al., 2008b, 2009a, and 2009b],
demonstrating the advantage of using RIM in atmospheric studies.
According to the RIM algorithms, cross-correlation functions (CCFs) of various frequency pairs are required [Palmer et al., 1999; Luce et al., 2001, 2006]. There have been many theoretical works on the CCF of a frequency pair after the pioneering study made by Kudeki and Stitt [1987], and some parameters like radar beamwidth, range-weighting function, anisotropy of irregularities (or aspect sensitivity), multiple layers, and so on, have been demonstrated to affect the magnitude and phase of the CCF to various degrees [e.g., Franke, 1990; Liu and Pan, 1993; Chu and Chen, 1995;
Chen et al., 1997; Luce et al., 2000]. Consequently, we can expect that the outcome of
RIM is also associated with these parameters. The range-weighting function has been considered for correcting the imaged power [Luce et al., 2001]; however, the effects of radar beamwidth and scatterer anisotropy on RIM were commonly neglected although they were also addressed in the literature. A major reason of this ignorance is that scatterer anisotropy (or aspect sensitivity) is thought to exist commonly in the atmospheric structures and it can narrow the effective width of the radar beam. Besides, it is difficult to verify these effects because a fixed radar beamwidth was used in previous observations. To master the application of RIM, these ignored effects on practical outcomes of RIM are worth examining. This is the main object of this study.
Numerical simulation is the first and necessary step for our study. We made use of the two-frequency models and analytical expressions given by Liu and Pan [1993] and Chen and Chu [2001] for estimates of the CCF, although other simulation
schemes also exist [e.g., Holdsworth and Reid, 1995; Muschinski et al., 1999]. All the parameters concerned have been included in the models employed. The influence of radar beamwidth, scatterer anisotropy, and layer altitude on the performance of RIM is shown in section 2.
For demonstrating the effect of radar beamwidth on RIM in practical
observations, various radar beamwidths are required. This requirement is difficult to achieve for most of the MST radars worldwide. However, the Middle and Upper atmosphere radar (MU radar), located in Shigaraki, Japan (34.85oN, 136.11oE), is one 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100 102 104
having such ability. As will be illustrated in section 3, twenty-five antenna groups in the whole array can be combined arbitrarily for transmission and reception, providing various radar beamwidths for observations. It deserves a notice that in the literature,
Hassenpflug et al. [2003] employed the old MU radar system to investigate the
consistency of horizontal correlation lengths of refractive index irregularities measured by different combinations of transmitter/receiver antennas (Tx/Rx mode hereafter), and with a method based on Full Correlation Analysis (FCA). They verified the effect of radar beamwidth on FCA. In this paper, we show the effect of radar beamwidth on RIM. To this end, it is necessary to calibrate the phase bias and the range-weighting function for different Tx/Rx modes first. We found that the calibrated phase bias varies with radar beamwidth positively. This is an interesting and valuable finding for the usage of RIM; the details are addressed in section 3.
After calibrating the RIM data properly, a comparison of the layer positions and layer thicknesses obtained from different radar beamwidths is possible, which is capable of indicating the effect of radar beamwidth on the outcome of RIM. This is discussed in section 4.
In addition, we also examined the RIM data obtained from vertical and oblique radar beams to study anisotropic characteristics of the scatterers in the layer
structures. In the literature, several theoretical and observational works for the two-frequency technique with oblique radar beams can be found [e.g., Palmer et al., 1992;
Liu and Pan, 1993; Chu and Chen, 1995; Luce et al., 2000]. For RIM, however,
observation is the practical way to verify the variation of layer structure with radar beam direction. In this study, we carried out an experiment for this purpose; the result is shown in section 5. Finally, the conclusions are given in section 6.
2. RIM Simulations for radar beamwidth and scatterer anisotropy A description of the CCF of a frequency pair for multiple layers is given in appendix A, which is an extract from Chen and Chu [2001]. Note that the analytical expression (A5) is valid only for a vertical radar beam, and, by assuming that the multiple layers in the radar volume are Gaussian-shaped and not correlative. Numerical simulation of RIM can be achieved by substituting the CCFs of various frequency pairs estimated with (A5) into the inversion algorithms such as the Capon and Fourier methods [Palmer et al., 1999]. To discuss simply and shortly the issues that are concerned, we did not consider the influence of signal-to-noise ratio (SNR). However, readers should bear in mind that lower SNR makes the imaging worse. In addition, quantitative variations in the imaged results are dependent on the radar and layer parameters given in the simulation; therefore, our numerical simulations only 106 108 110 112 114 116 118 120 122 124 126 128 130 132 134 136 138 140 142
provide qualitative features of the performance of RIM.
First, Figure 1a displays the imaged layer varying with radar beamwidth, where the results of Fourier and Capon methods are shown for comparison. The imaging range step was 1 m, and this value was employed for all numerical simulations and observational analysis in this study. In the work of Figure 1a, the radar beam was set to be vertical, and five frequencies46.00, 46.25, 46.50, 46.75, and 47.00
MHzwere used with a 1-s pulse length and a 3.6o half-power beamwidth. The layer was located at the central height (5.075 km) of the radar volume (i.e., zero position in ordinate), with a thickness of 5 m (i.e., the standard deviation of a
Gaussian-shaped layer), and isotropic irregularities with 3-m correlation length were assumed in the layer. The use of 3-m correlation length is based on half-wavelength backscattering mechanism of radar echoes as the frequencies of 4647 MHz are used in the simulation.
In Figure 1a, it is apparent that the Fourier method yields a much thicker layer that ascends gradually with beamwidth. In comparison, the Capon method can retrieve the layer structure better although it outputs a layer structure that becomes more diffusive and has higher mean position at a larger beamwidth. In fact, a quantitative
investigation can find that the Fourier-deduced layer thickness also increases with beamwidth but it does not vary as apparently as the Capon-deduced layer thickness (not shown). In view of this, the effect of radar beamwidth on the RIM-deduced layer is evident. Such effect may smear out fine layer structures, as illustrated in Figure 1b. As shown, the two layers, given with the same contributing weights and at positions of -12.5 m and 12.5 m, respectively, cannot be resolved by the Fourier method; even the Capon method is unable to separate the two layers at radar beamwidths larger than ~5o.
In addition to radar beamwidth, the layer’s height is also essential for the performance of RIM. This is demonstrated in Figures 1c and 1d, where a radar beamwidth of 3.6o was given in calculation. Again, the Capon method offers a better imaging. However, as seen in Figure 1c, the Capon-deduced layer structure gets wide apparently along the layer height, and in Figure 1d, the two layers cannot be identified for the layers located at heights above ~20 km.
Deterioration of layer imaging, originating from broader radar beam and higher layer altitude, can be mitigated when the irregularities become more anisotropic. Some simulation cases resulting from the Capon method are shown in Figure 2, where horizontal isotropy was assumed, and, vertical and horizontal correlation lengths were 3 m and 30 m, respectively. Except for correlation lengths, the radar and layer
parameters used here are the same as those of Figure 1. The topmost panels of Figures 2a and 2b illustrate that the layers can be retrieved clearly at the height of ~5 km for 144 146 148 150 152 154 156 158 160 162 164 166 168 170 172 174 176 178 180
any of the beamwidths given, which improves greatly the results shown in the third row of Figure 1a and 1b, respectively. The imaged layer structures, however, are deteriorated again at higher altitudes and only a very narrow beam is capable of separating the two layers, as illustrated in the second and third rows of Figures 2a and 2b. A more detailed variation of the imaged layer structures with height is shown in Figure 2c (the bottommost panels). As seen, for a radar with beamwidth of 3.6o, the two layers can be resolved up to the height of ~30 km (left panel), which is better than the isotropic condition shown in Figure 1d; nevertheless, the situation becomes worse again for a radar with beamwidth of 7o (right panel), where the workable altitude for resolving the two-layer structure is less than 20 km.
Distribution of the imaged power simulated above will be somewhat different when the layer parameters and radar parameters are changed. For example, given a larger distance between the two layers and/or using seven or nine frequencies within a larger frequency range can improve the recognition of the two-layer structure in Figures 1 and 2. In the present MU radar system, however, only five frequencies between 46 MHz and 47 MHz are available for practical observations.
In general, the features disclosed from the above simulations are within the prediction of the theoretical works addressed in the literature, and they can be
explained as follows. A horizontal layer structure covers a larger range interval in the radar volume for a wider beamwidth and/or at a higher altitude. In the horizontal layer, the scatterers situated at off-zenith directions have the ranges mostly farther than the mean layer height, and they may contribute to the echoes received by a radar with finite beamwidth and consequently smear the range power distribution imaged by RIM, causing a larger layer thickness and a higher mean layer position. As the anisotropy of the scatterers increases, the echoes returning from larger off-zenith directions will decrease quickly. As a result, the RIM-deduced layer will be closer to the modeled layer. In any event, it may be troublesome for RIM to examine the fine-scale structures at altitudes like the mesosphere and ionosphere, unless the radar beam is extremely narrow or the radar scatterers are highly anisotropic.
3. Experimental configurations and calibration results
3.1 Experimental configurations
Figure 3 displays the antenna array configuration of the MU radar. The whole antenna array is partitioned into twenty-five antenna groups, and each antenna is equipped with a transmitter-receiver module. Software can combine the received signals of one to twenty five antenna groups for an output, and different combinations up to twenty five can be assigned simultaneously for reception, which diversifies the 182 184 186 188 190 192 194 196 198 200 202 204 206 208 210 212 214 216
usage of the radar.
Two RIM experiments were carried out for the purposes of this study; some relevant radar parameters are listed in Table 1. The first experiment employed three different combinations of antenna groups for transmission (full antenna array, seven and three antenna groups around array center, which are denoted as Txfull, Tx7, and Tx3, respectively). The three combinations of antenna groups yield different radar beamwidths, and were operated alternately with an experimental cycle about 4.5 min. Moreover, the same three combinations of antenna groups were also used for
reception (denoted by Rxfull, Rx7, and Rx3, respectively) during each transmission of beamwidth. As a result, there are nine Tx/Rx modes in the experiment, with
beamwidths between 3.67o (for Txfull/Rxfull) and 9.74o (for Tx3/Rx3).
The second RIM experiment employed the whole antenna array for transmission and reception. In addition to the vertical direction, the radar beam was also tilted northward to the zeniths of 5o, 10o, and 15o, respectively. The four beam directions were proceeded alternately with a cycle about 5 min.
Note that in the aforementioned experiments, each transmitting mode or radar beam direction was conducted for at least one minute before switching to the next transmitting mode or radar beam direction, and the data set collected each time (1024 samplings) was used for an estimate of the CCF. We will suppose that for the radar beamwidth smaller than 10o, a data set with such long period meets the assumption of enough number of scatterers in the radar volume. In other words, the estimated CCF can reveal relevant statistics of the scatterers in the radar volume. It is mentionable here that in this study, all the investigations into the layer parameters are based on statistical views.
3.2 Calibrations for various RIM experiments
Different radar parameters and antenna array configurations employed for RIM may result in different phase biases and range-weighting functions, so it is necessary to carry out calibrations for various RIM data first. We used the calibration approach proposed by Chen and Zecha [2009a], which can offer more parameters for correcting the imaged power. These parameters are: (1) optimal phase bias, (2) optimal (or effective) standard deviation (z) of a Gaussian-shaped range-weighting function, and (3) a curve for yielding the value of z at a specific SNR. See appendix B for a brief description of the calibration approach used.
The left two columns of Figure 4 show the histograms of the optimal phase biases and z values for the nine Tx/Rx modes in the first experiment. In this work, the data below ~10 km and with SNR>0.125 (~-9 dB) were used. It is obvious that each histogram of the optimal phase biases has a peak, whose location can be 218 220 222 224 226 228 230 232 234 236 238 240 242 244 246 248 250 252 254
regarded as the likely phase bias (bias) of RIM. An important feature can be observed: bias varies with Tx/Rx mode. For the Txfull/Rxfull mode, bias was about 125o; for the Tx3/Rx3 mode, however, bias rose to about 170o. An evident tendency is that a larger phase bias corresponds to a broader radar beam. Notice that the radar beamwidths of the Txfull/Rxfull, Tx7/Rx7, and Tx3/Rx3 modes given in panels (a), (e), and (i) are the two-way half-power beamwidths, while others are mean half-power beamwidths; for example, the beamwidth of the Txfull/Rx7 mode was estimated as
(3.67o+6.98o)/2≒5.33o.
The dependence of phase bias of RIM on radar beamwidth, like the interpretation given for the simulation results, is owing to the horizontal layer structure covering a larger range interval in a wider radar beamwidth (refer to the final paragraph of section 2). A wider radar beamwidth, as demonstrated in Figures 1a and 1b, yields a RIM layer with higher position than the given one. To compensate this deviation of layer position, therefore, a larger range correction (i.e., a longer time delay) is
necessary for a wider radar beamwidth in the calibration processing, which leads to a larger phase bias.
It deserves a mention that in our previous works, the existence of likely phase bias was attributed to the time delay of the signal in the radar system. If such attribution is valid, the actual time delay of the signal in the radar system should be shorter slightly than the one indicated by a finite beamwidth, according to the dependence of phase bias on radar beamwidth.
Regarding the distribution of the z values, the peak location was about 70 m for all Tx/Rx modes, which represents approximately the theoretical standard deviation of the Gaussian range-weighting function. Note that the spread of z toward larger values was owing to the data having lower SNR. Also, the minor peak (z 150 m) of the Tx3/Rx3 mode arose from a large amount of low-SNR data. If a larger SNR threshold is given for the calibration, it can be seen that the distribution of z (and phase bias) concentrate more around the main peak location (not shown).
The right column of Figure 4 reveals a dependent relationship between z and SNR. As demonstrated by Chen and Zecha [2009a], such a relationship comes from pursuing the optimal continuity of the imaged powers at the boundaries of the sampling gates for widespread SNR in observations, and a fitting curve can be obtained for determining the approximate values of z at different SNRs. Despite slight differences in the fitting parameters, the nine fitting curves are essentially similar. It should be mentioned here that the fitting curve extending to low SNR data is beneficial to give an effective value of z in correcting the imaged powers around the boundaries of the sampling gates when SNR is low.
To verify the phase bias obtained above, we inspected further the phases of the 256 258 260 262 264 266 268 270 272 274 276 278 280 282 284 286 288 290 292
two-frequency CCFs (termed frequency-domain-interferometry phase hereafter and abbreviated as FDI phase). This is illustrated in Figure 5, where the distributions of FDI phases for the frequency pairs with frequency separations of 1000 kHz, 500 kHz, and 250 kHz are displayed. Only the two modes of Txfull/Rxfull and Tx3/Rx3, whose beamwidth difference is the largest among the nine Tx/Rx modes, are shown. A detailed discussion on the meaning of such distributions has been addressed in previous works [e.g., Chen, 2004]. In a word, the peak location of the distribution can indicate the bias of FDI phase. A careful inspection can find a slight discrepancy between these distributions: the differences between the mean peak locations of the two Tx/Rx modes were about 20o and 15o, respectively, for the frequency separations of 500 kHz and 250 kHz. For the frequency separation of 1000 kHz, however, it is difficult to determine the difference in mean peak location between the two Tx/Rx modes because of their shallower curvatures. In conclusion, the influence of radar beamwidth on the FDI phase indeed exists, which supports the calibration results shown in Figure 4.
The same calibration process was also made for the second RIM experiment (one vertical beam and three oblique beams). The results showed that the likely phase biases and z values of the three oblique beams were almost the same as those of the vertical beam although the oblique beams observed more data with lower SNR (not shown). In view of this, the likely phase bias and z did not vary with radar beam direction in the second experiment.
4. Dependence of layer thickness and layer position on radar beamwidth The numerical simulations in section 2 have shown the variations of layer thickness and layer position with both radar beamwidth and altitude. We verify this with some practical observations in this section.
Figure 6 displays the height-time intensity imaged by Capon’s method for the Txfull/Rxfull mode. The calibration parameters shown in Figure 4 have been employed for correcting the imaged power. As displayed, there were many thin and stable layer structures above the height of ~5 km. By contrast, broader and variable
layers/turbulent structures were observed in the lower troposphere (<~4 km). Note that the vertical strong patches above 7 km, occurring at ~10.5 UT and ~11.0 UT, were the radar echoes reflected from aircraft, which should be ignored. Quantitative comparisons of layer parameters (position and thickness) between various Tx/Rx modes can reveal the effect of radar beamwidth on RIM more clearly; some
comparisons are exhibited in Figures 7-9, in which the layer position was estimated with the contour-based approach used by Chen et al. [2008a], and the layer thickness 294 296 298 300 302 304 306 308 310 312 314 316 318 320 322 324 326 328 330
was defined as the standard deviation of a Gaussian curve that was used to fit the imaged powers around the estimated layer position. This approach can determine the positions and thicknesses of the multiple layers within the imaged range, but we considered only the single-layer situation for diminishing possible unreliability in comparison. Moreover, only the data with SNR>0.5 (-3 dB) were adopted. We have examined the results obtained from data having higher SNR (e.g., >0 dB and 6 dB) and found that the resultant characteristics were similar to Figures 7-9 except that fewer estimates were obtained. This suggests that the features revealed in Figures 7-9 are not caused mainly by low SNR. In other words, radar beamwidth indeed plays a role in the RIM-deduced layer parameters. The details are addressed as follows.
Figure 7a displays the differences in layer position between the three receiving modes used with the full-array transmission, in which the distribution curve in each range bin was self-normalized. Note that for a clearer inspection, each range bin in Figures 7-9 contains the data of two range gates in observations. Figure 7a shows that the position differences centered on zero, and the profiling curve (thick solid line), which indicates the difference between the numbers of positive and negative position differences (positive minus negative) and is presented as a percentage of the total number in each range bin (see upper abscissa for the scale), also varied around zero. In view of this, the layer positions estimated from various receiving modes were sta-tistically close after corrections of phase bias and range weighting function effect. On the other hand, Figure 7b exhibits a difference in layer thickness, which shows that statistically, the layer thickness estimated from the receiving mode, Rx3, was the largest, and that estimated from the receiving mode, Rxfull, was the smallest. Such a feature is more evident at higher altitudes, as indicated by the profiling curve of num-ber difference. These consequences point out clearly an observable role of radar beamwidth in the performance of RIM. The same examination was also made for the two transmitting modes of Tx7 and Tx3, and it gave similar results to the above (not presented).
We may wonder about the consequence if a specific set of calibrated parameters is applied to other receiving modes. For example, using the calibrated parameters of Rx-full for the other two receiving modes, we obtained the products shown in Figure 8. In Figure 8a, it is seen that the Rx3 (Rx7) mode yields the layer having the altitude appar-ently higher than the Rx7 and Rxfull (Rxfull) modes; this is consistent with the simulation results of using various radar beamwidths. On the other hand, the features seen in Fig-ure 8b are similar to those of FigFig-ure 7b, except that the thickness differences shown in Figure 8b are smaller in statistics.
In Figure 7, the differences in effective beamwidth between the three Tx/Rx modes are not very large because of the same transmitting beam (Txfull) used in obser-332 334 336 338 340 342 344 346 348 350 352 354 356 358 360 362 364 366 368
vations. In Figure 9, the comparison of layer parameters between the three Tx/Rx modes, Txfull/Rxfull, Tx7/Rx7, and Tx3/Rx3, are shown. The two-way 6-dB beamwidths of the three Tx/Rx modes are about 3.67o, 6.98o, and 9.74o, respectively. As seen in Figure 9a, the profiling curves show that the estimated positions are statistically close; however, the distributions of position differences are more divergent than in Figure 7a, especially worse in the lower troposphere. One cause of this phenomenon could be that the three Tx/Rx modes were conducted alternately and each mode operated for about one minute, the layer position might change greatly during several minutes, es-pecially in the lower troposphere where more variable layers or turbulent structures were observed (see Figure 6). On the other hand, Figure 9b discloses again the influ-ence of radar beamwidth on the estimated layer thickness. Comparing with Figure 7b, the thickness differences shown in Figure 9b are larger (notice the scales of the ab-scissa in the two figures are different), demonstrating the effect of radar beamwidth more clearly.
5. RIM of oblique radar beams
In this section, we show the RIM data collected from the vertical and three oblique beams in the second experiment. Figure 10 displays a portion of the imaged powers, where the ordinates of the maps of the oblique radar beams have been corrected to coincide with the vertical beam. As seen, the vertical beam observed the clearest layer structures in the range direction. We discuss the following three features:
(1) Two quasi double-layer structures were observed between 18 UT and 20 UT: the first one descended from ~8.7 km to ~8.4 km, and the second one was located between ~7.8 km and ~8.0 km. It can be seen that with the increase of tilt angle of the radar beam, the imaged layer structures become more diffusive and the double-layer structures cannot be seen clearly.
(2) The layer structure descending from ~8.4 km to ~8.0 km between 18 UT and 20 UT, as observed by the vertical beam, faded out in the observations of using oblique beams.
(3) The major structures seen by 10o- and 15o-oblique beams were similar. The above three features seem to be associated with anisotropic characteristic (or aspect sensitivity) of the scatterers in the layer structures. Estimate of layer position from the imaged power can aid in the investigation of these imaged structures, as shown in Figure 11, where the layer positions estimated from the vertical and oblique radar beams are compared.
The double-layer structure is known to be the final stage of the dynamic process 370 372 374 376 378 380 382 384 386 388 390 392 394 396 398 400 402 404
of Kelvin-Helmholtz instability [Browning and Watkins, 1970; Worthington and
Thomas, 1997], and the model of turbulent layers presented by Woodman and Chu
[1989] can be one of the models to illustrate turbulent characteristics of a double-layer structure; that is, the observed two layers arise from sharp gradients of refractive index that are anisotropic, but between them the gradient of refractive index is much gentler and so the irregularities are more isotropic. As a result, the radar echoes from the region between the two layers are much weaker. One likely example is the double-layer structure descending from ~8.7 km to ~8.4 km in Figure 10, and a comparison of the positions estimated from different radar beam directions can be seen in the slanted rectangular box in Figure 11. As seen, although the imaged powers of the oblique radar beams are much weaker, the double layers can also be identified sometimes by the oblique radar beams. That indicates that the scatterer anisotropy in the two layers is not so extremely high that the layers are still observable for the three oblique radar beams. Note that only the data with SNR>0.5 are displayed in Figure 11.
The other double-layer structure, located between ~7.8 km and ~8.0 km, is somewhat different from the first one, as indicated in the region of the lower
rectangular box in Figure 11. The upper layer can be determined by all radar beams although the layer positions obtained from the oblique radar beams are slightly higher than from the vertical radar beam. On the other hand, the lower layer cannot be detected by the oblique radar beams. In view of this, the upper layer may be close to isotropic, but the lower layer is extremely anisotropic. Nevertheless, we notice that the imaged powers of the lower layer obtained from the vertical radar beam are not very strong and clear, this can also lead to the result that the layer cannot be determined definitely.
The second feature, a single layer descending from ~8.4 km to ~8.0 km between 18 UT and 20 UT, is demonstrated in the elliptical region of Figure 11. This layer was observed clearly by the vertical radar beam but not detected by the oblique radar beams, as indicated by the absence of layer position in the elliptical region. In view of this, this layer could be quite anisotropic. As for the third featuresimilar structures were observed by the two oblique radar beams with off-zenith angles of 10o and 15o, it indicates that in this observation, the scatterers detected by a radar beam tilted to a zenith angle larger than ~10o are almost isotropic.
In addition to the aforementioned three features, the estimated layer thickness was found to increase as the zenith angle of the radar beam direction increased, as demonstrated by the distributions of layer thicknesses and their mean profiling curves shown in Figure 12. Such characteristic has been predicted by the two-frequency expressions for oblique radar beams [Liu and Pan, 1993]. The profiling curve in Figure 12 also displays an increase of layer thickness with the sampling height/range, 406 408 410 412 414 416 418 420 422 424 426 428 430 432 434 436 438 440 442
which is more evident for a radar beam tilted to a larger zenith angle. The dependence of layer thickness on the zenith angle of the radar beam direction, and on
height/range, can be explained as follows: a horizontal layer structure covers a larger range interval in the tilted radar volume, causing a wider distribution of the imaged powers in the range direction, and such a feature is more evident at a higher altitude where the radar volume is larger.
To verify the observations shown in Figures 10-12, we have applied the analytical expressions derived by Liu and Pan [1993] to numerical simulations of single and double-layer structures, with different anisotropic scatterers in the layers and for various oblique radar beams. However, the expressions given by Liu and Pan [1993] were derived only for a single layer in the radar volume. To deal with multiple layers, we have to follow the assumption used by Chen and Chu [2001]; that is, the layers in the radar volume are not correlative so that the respective CCFs of the layers can be summed. Consequently, the expression (9) given by Liu and Pan [1993] can be employed. In use of the expression, the coordinates of transmitter and receiver were the same (x0=0), and the time lag () was set equal to zero; besides, the radar
parameters used in section 2 were employed again.
Figure 13 exhibits three simulation cases. The first case, shown in the panel (a), simulates a highly anisotropic layer located at the central height (5.075 km) of the range gate, with the vertical and horizontal correlation lengths of 3 m (lz) and 15 m
(lt), respectively. We can see that as the zenith angle of the radar beam direction
increases, the imaged layer structure broadens and its location descends. However, the echo power obtained in the simulation drops very quickly along the zenith angle of the radar beam direction (see the solid curve and refer to the right ordinate). For example, for a radar beam tilted to the zenith angle of 5o, the simulated echo power is about -20 dB relative to the vertical radar beam (0 dB), indicating that the layer can hardly be observed by a largely tilted radar beam. This case can illustrate the layer located in the elliptical region of Figure 11.
The second simulation is shown in Figure 13b, where two layers with slightly anisotropic scatterers (lz=3m and lt=4.5m) were given at positions of -30 m and 30 m.
As shown, the two imaged layers are more diffusive and wider for a radar beam tilted to a larger zenith angle. The simulated echo power also decreases with the zenith angle of the radar beam direction but does not reduce to a level as low as for the first case. Therefore, the two layers could be detected by off-vertical radar beams but with lower intensity. This result may describe, to some degree, the feature of the layers observed in the slanted rectangular region of Figure 11.
Figure 13c illustrates the third simulation, where two layers with different degrees of scatterer anisotropy were given. It is shown clearly that the highly 444 446 448 450 452 454 456 458 460 462 464 466 468 470 472 474 476 478 480
anisotropic layer (lz=3m and lt=15m), located at the lower position, fades out rapidly
along the zenith angle of the radar beam direction. By contrast, the isotropic layer (lz=
lt= 3 m), located at the higher position, dominates gradually the imaged layer structure
and acquires a broader thickness when the radar beam is tilted to a larger zenith angle, although the simulated echo power decreases accordingly. This picture may give a description of the layers observed in the rectangular region of Figure 11.
The above three simulations for oblique radar beams yield some qualitative evidences for the observations. Real atmospheric conditions, however, should be more complicated and noise is also crucial to imaging techniques. More simulations can be achieved but showing them here is beyond the scope of this study.
In this section, we have first applied the product of RIM to reveal the anisotropic characteristic of layer structure. Such manner of investigation is different from the existing methods; e. g., a comparison of echo powers between vertical and oblique radar beams that can only show an averaged outcome within a range-gate interval. Considering that isotropic and anisotropic layers/irregularities can exist closely or simultaneously within the same range-gate interval, the comparison of echo powers between differently oblique radar beams may not be enough for such case.
6. Conclusions
In this study, the effects of radar beamwidth and scatterer anisotropy on the performance of RIM were examined by numerical simulation and from practical observation. Observations with the MU radar in Japan were carried out by transmitting radar waves and receiving the echoes with different radar beamwidths and beam directions. It was shown that the phase bias in the RIM processing varies with beamwidth, that is, a larger beamwidth leads to a larger phase bias. Based on this, we have estimated the layer position and layer thickness more properly for various radar beams that were transmitted by the same radar system. Consequently, the effects of radar beamwidth on the performance of RIM can be investigated in addition to numerical simulation. Statistical examination demonstrated that a broader radar beam gives a broader layer thickness, which is more evident at higher altitudes. Moreover, an oblique radar beam also yields a broader layer thickness. All these observational results are consistent with previous theoretical predictions.
This study then demonstrated an application of RIM to the scatterer anisotropy or aspect sensitivity characteristic of the layer structure by means of vertical and oblique radar beams. Comparisons of the imaged powers and layer positions between vertical and oblique radar beams have disclosed different levels of scatterer anisotropy in the imaged layer structures that existed closely or simultaneously in a range gate. Such a 482 484 486 488 490 492 494 496 498 500 502 504 506 508 510 512 514 516 518
fine-scale examination is different from the existing methods having a range-gate resolution, and can be carried out for more observations in the future.
Appendix A: Two-frequency model and expressions
Assuming that there are N layers in the radar volume and the distribution of the echoing scatterers in each layer is Gaussian function, the output voltages of a receiver at two different transmitting frequencies can be written as
, ... , ... 2 22 2 21 1 2 1 12 2 11 1 1 n n n n v w v w v w V v w v w v w V (A1)
where vij is the output voltage of a receiver for a single layer in the radar volume, and
the first and second subscripts in vij denote the radar frequency and the layer,
respec-tively. Variables w1, w2, …and wn are the weights of the layers contributing to the
radar returns, which are related to the intensities of the refractive index fluctuations in the layers. With (A1), a two-frequency cross-correlation function can be expressed as
),
(
)
...
)(
...
(
N 1 N 1 * 2 1 N 1 * 2 1 2 * 2 22 2 21 1 1 12 2 11 1 * 2 1m
n
v
v
w
w
v
v
w
V
w
v
w
v
w
v
w
v
w
v
w
V
V
n m m n m n n n n n n n n n
(A2)Assuming further that the radar echoes generated from the nth and the mth layers are not correlative, then the term <v1nv*2m> in (A2) can be eliminated. To obtain an analyt-ical expression from (A2), the following form of v is used [Doviak and Zrnic’, 1984;
Franke, 1990]:
r
r
r
r
d
r
kr
j
z
z
t
n
f
W
C
v
s s l l
v(
)
2(
)
(
,
)
exp[
(
2)
2/
2
2]
exp(
2
)
, (A3)where W(r) is the range weighting function, fθ2(r) is the two-way beam pattern
func-tion. The Gaussian term, exp[-(z-zl)2/2σl2], is the distribution function of the scatterers
in the layer, in which zl and σl represent layer position and layer thickness,
respec-520 522 524 526 528 530 532 534 536 538 540 542 544 546 548
tively, and z is the vertical component of the spatial vector r. Δn is the refractive index fluctuation responsible for the radar echo, k is the wavenumber of the transmitted radar wave, rs is the range between Δn and the radar receiver, C is a constant related
to radar parameters. Note that the origin of the coordinate system in (A3) is the center of the radar volume. If the radar beam is horizontally symmetric and pointed verti-cally, and the matched filter is used for reception, W(r) and fθ2(r) can be expressed
ap-proximately as [Franke, 1990]
,
)
2
exp(
)
(
)
(
,
)
4
exp(
)
(
)
(
2 2 2 2 2 2 t zf
f
z
z
W
W
r
r
(A4)where σz=0.35cτ/2, c is the light speed and τ is the pulse length, ρ2=x2+y2, x and y are
the two horizontal components of the vector r, σt =21/2hθ6/3.33, θ6 is the 6-dB angular
width of the function fθ2(ρ) and h is the central height of the radar volume. With (A3) and (A4), (A2) becomes
, e e e e e 2 N 1 ) / 4 2 ( / 2 2 4 / ) 2 ( / 2 4 0 3 2 * 2 1 2 2 0 4 2 2 2 2 2 1 2 2 n z kz kz j z k A k k z r n y x y x n n l n l n r n l n l n r n r n z n l n l n A w A A a a z C V V (A5) where zln and σln represent the position and thickness of the nth layer, respectively, and
, / 1 8 / 1 , 4 / ) / 4 / ( , 4 / ) / 4 / ( , / ) 2 ( , / / 1 , / / 1 , / 2 / 1 / 1 2 2 0 2 1 0 2 0 2 2 2 2 z r z y y y y x x x x y y x x l z r l A l a a b A l a a b A z k k b z k j a z k j a n n n n (A6)
where σx and σy are the second moments of the radar beam in x and y directions,
re-spectively. lx, ly, and lz are, respectively, the spatial correlation lengths of the refractive
550 552 554 556 558 560 562 564 566 568 570 572
irregularities in x, y, and z directions, α and β are the coefficients related to the wavenumber power spectrum of refractive irregularities (α=1 and β=2 for the 3-di-mensional Gaussian spectrum; α=2.5lz-1.4 and β=1.5 for the approximation form of the
3-dimensional –11/3 power law spectrum [Chen et al., 1997]). (A5) can be normal-ized by <|V1|2>1/2<|V2|2>1/2 to obtain the coherence function. In the literature, Luce et
al. [1999] also carried out a derivation like the above. Moreover, the expression (A5)
can also be obtained by simplifying the general cross-correlation function given by
Liu and Pan [1993].
Appendix B: RIM calibration approach
Two parameters can be obtained from the calibration of RIM proposed by Chen
and Zecha [2009a]: range (time) delay of the radar signal and SNR-dependent
standard deviation (z) of the Gaussian range-weighting function. The following estimator has been used:
N 1 i 1i 2i 2i 1i N 1 i 1i 2i 2 2i 1i ) P P 2 -P P ( P P ) P -(P ERR (B1)where P1i and P2i are two sets of the imaged powers estimated around the boundary between two adjacent range gates, in the same height interval (namely, the upper boundary of the lower gate and the lower boundary of the upper gate). Giving various range (time) delays and z values in calculating the two sets of imaged powers will result in different values of ERR, but there exits a pair of range (time) delay and z that make ERR smallest, which are the optimal values enabling the two sets of imaged powers closest. Under such condition, it is supposed that the imaged powers of two adjacent range gates are mostly continuous at the common edge. For the reason of convenience, the estimated range (time) delay is transformed into a phase angle with the scale of a range gate (or pulse length) to 360o, which is termed “optimal phase bias” in the text.
It deserves a mention that z is theoretically not related to SNR but dependent on the pulse shape and filter bandwidth employed. The SNR-dependent z is due to the SNR-dependent performance of inversion algorithms such as Capon’s method. To mitigate discrepancy in the effects of SNR on the imaged powers of two adjacent range gates, and, to acquire the optimal continuity between the imaged powers at range gate boundary, we can use a SNR-dependent z. Although the upper and lower parts of a range gate usually suffer slightly different values of z according to the bration results, it has been demonstrated to be not a noticeable drawback of the cali-574 576 578 580 582 584 586 588 590 592 594 596 598 600 602 604 606 608
bration approach. Acknowledgements
This work was supported by the National Science Council of ROC (Taiwan) through grants NSC95-2111-M-270-001-MY3 and NSC98-2111-M-270-001, and also supported by the International Collaborative Research Program of MU radar (Ref. No.: 20MU-A39). The MU radar is operated by the Research Institute for Sustainable Humanosphere, Kyoto University, Japan. We would like to thank Prof. Mamoru Yamamoto especially for encouraging the radar experiment.
610 612 614 616
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Figure Captions
Figure 1. RIM simulation for isotropic layer. Both horizontal and vertical correlation lengths of the scatterers are 3 m, and the standard deviation of the Gaussian-shaped layer is 5 m. (a) Variation of the imaged layer structure with radar beamwidth, the central height of the range gate is 5.075 km. (b) Same as (a) but for a two-layer structure. (c) Variation of the imaged layer structure with layer height for the radar beamwidth of 3.6o. (d) Same as (c) but for a two-layer structure. The contributing weights of the layers in the two-layer structure are the same.
Figure 2. RIM simulation for anisotropic layer. Horizontal and vertical correlation lengths of the scatterers are 30 m and 3 m, respectively, and the standard deviation of the Gaussian-shaped layer is 5 m. (a) Variation of the imaged layer structure with radar beamwidth for the range gates at 5.075 km, 20.075 km, and 85.075 km, respectively. (b) Same as (a) but for a two-layer structure. (c) Variation of the imaged layer structure with layer height for the radar beamwidths of 3.6o (left) and 7o (right). The contributing weights of the layers in the two-layer structure are the same.
Figure 3. Antenna array configuration of the MU radar. The two antenna subarrays embraced by thick solid and dashed lines, respectively, were used in the first experiment in addition to the full antenna array. In the second experiment, only the full antenna array was employed.
Figure 4. Histograms of the optimal phase biases (left column) and z values (middle column) for various Tx/Rx modes. The phase and z bins are 10o and 5 m,
respectively. Vertical lines at left column indicate the phase bias of 125o, and the two numbers given in each panel are mean phase bias and radar beamwidth, respectively. Right column: scatter plot of z versus SNR, in which the solid curve is the fitting result with the four constants given in each panel [Chen and Zecha, 2009a].
Figure 5. FDI phase distributions at frequency separations (Δf) of 1000 kHz, 500 kHz, and 250 kHz for the Txfull/Rxfull and Tx3/Rx3 modes, respectively. Phase bin is 10o.
Figure 6. Range (height)-imaged powers of the Txfull/Rxfull mode.
Figure 7. Comparisons of layer parameters between different receiving modes. Full antenna array was used for transmission (Txfull). (a) Distribution of layer position differences. The distribution curve in each range bin is self-normalized, and the profiling curve (thick solid line) shows the difference between the numbers of positive and negative position differences (positive minus negative), presented as a percentage of the total number in each range bin (see upper abscissa for the scale). (b) Same as (a), but shows the difference in layer thickness.
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Figure 8. Same as Figure 7, but results from the calibrated parameters of full-array reception (Rxfull).
Figure 9. Same as Figure 7, but compares the three modes of Txfull/Rxfull, Tx7/Rx7, and Tx3/Rx3.
Figure 10. Range (height)-imaged powers of four radar beam directions: (a) vertical, (b) 5o north, (c) 10o north, and (d) 15o north. The Txfull/Rxfull mode was employed. Vertical thick blue lines indicate the time gap (~4 min) between data files, and horizontal dashed lines separate the 150-m range gates.
Figure 11. Comparisons of the layer positions estimated from vertical and three oblique radar beams (5o, 10o, 15o north).
Figure 12. Histograms of layer thicknesses observed by variously tilted radar beams. The distribution curve in each range bin is self-normalized, and the solid profiling curve shows the mean layer thickness varying with height/range.
Figure 13. RIM simulation for oblique radar beam and using the Capon method, in which the cross-correlation function derived by Liu and Pan [1993] was employed. Beamwidth: 3.6o. Standard deviation of the Gaussian-shaped layer: 5 m. Central height of range gate: 5.075 km. Right ordinate shows the scale for the normalized echo power (solid curve). (a) Single layer at position of 0 m. (b) and (c): two layers at positions of 30 m and -30 m, and with the same contributing weights. Horizontal and vertical correlation lengths (lt and lz ) of the scatterers in (b) are the same, but
different in (c). 750 752 754 756 758 760 762 764 766 768
