Measurement of Range-Weighting Function for Range Imaging of VHF Atmospheric Radars Using Range Oversampling

Measurement of Range-Weighting Function for Range Imaging of

VHF Atmospheric Radars Using Range Oversampling

Jenn-Shyong Chen1_{, Ching-Lun Su}2_{, Yen-Hsyang Chu}2_{, Ruey-Ming Kuong}3_, and Jun-ichi Furumoto4

1_{Department of Information and Network Communications, Chienkuo Technology University,}

Taiwan

2_{Institute of Space Science, National Central University, Taiwan} 3_{Chung-Shan Institute of Science and Technology, Taiwan}

4_{Research Institute for Sustainable Humanosphere, Kyoto University, Japan}

Manuscript submitted to

Corresponding author: Jenn-Shyong Chen E-mail: [email protected] Tel: +886-4-7111111 ext. 2300 Fax: +886-4-7111163 Mobile: +886-9-28128935 2 4 6 8 10 12 14 16 18 20 22 24

Measurement of Range-Weighting Function for Range Imaging of

VHF Atmospheric Radars Using Range Oversampling

Jenn-Shyong Chen1_{, Ching-Lun Su}2_{, Yen-Hsyang Chu}2_{, Ruey-Ming Kuong}3_, and Jun-ichi Furumoto4

1_{Department of Information and Network Communications, Chienkuo Technology University,}

Taiwan

2_{Institute of Space Science, National Central University, Taiwan} 3_{Chung-Shan Institute of Science and Technology, Taiwan}

4_{Research Institute for Sustainable Humanosphere, Kyoto University, Japan}

0BAbstract

Multi-frequency range imaging (RIM) used with the atmospheric radars at ultra and very high frequency (VHF) bands is capable of retrieving the power distribution of the backscattered radar echoes in the range direction, with some inversion algo-rithms such as the Capon method. The retrieved power distribution, however, is weighted by the range-weighting function (RWF). Modification of the retrieved power distribution with a theoretical RWF may cause over-correction around the edge of the sampling gate. In view of this, an effective RWF that is in a Gaussian form and varies with signal-to-noise ratio (SNR) of radar echoes has been proposed to mitigate the range-weighting effect and thereby enhance the continuity of the power distribu-tion at gate boundaries. Based on the previously proposed concept, an improved ap-proach utilizing the range-oversampled signals is addressed in this article to inspect the range-weighting effects at different range locations. The shape of the Gaussian RWF for describing the range-weighting effect was found to vary with off-center range location in addition to the SNR of radar echoes, that is, the effective RWF for the RIM was SNR- and range-dependent. The use of SNR- and range-dependent RWF can be of help to improve the range imaging to some degree at the range location out-side the range extent of a sampling gate defined by the pulse length. To verify the pro-posed approach, several radar experiments were carried out with the Chung-Li (24.9o_{N, 121.1}o_{E) and MU (34.85}o_{N, 136.11}o_{E) VHF atmospheric radars.}

1. Introduction

The range-weighting function (RWF), defined by the convolution of transmitter pulse envelope and receiver filter impulse response, gives the weighted contribution of individual scatterers in the range direction of the radar volume. For an ideal rectangular pulse envelope and infinite filter bandwidth, the RWF is a rectangular shape with range extent of c/2, where  is the pulse length and c is the speed of the 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62

light. In practice, however, the filter bandwidth is finite and the transmitter pulse envelope can be Gaussian or usually not perfectly rectangular, leading to a stretched-out, non-rectangular RWF. Consequently, the sampled signal contains the contribution of scatterers outside the ideal range extent c/2. For the VHF atmospheric radar, the non-rectangular RWF is usually modeled by a Gaussian function, and then the range resolution of the sampled signal is defined from the modeled RWF, i.e., 0.35 c/2 [Doviak and Zrnić, 1984].

In addition to the transmitter pulse envelope and receiver filter, the digital signal processing of echo samples along the range-time dimension can also affect the RWF [Torres and Curtis, 2012]. One example is the whitening transformation on the range-oversampled signals of weather radar. The whitening transformation can improve the estimates of spectral moments (e.g., signal power, mean Doppler velocity, Doppler spectrum width) [Torres and Zrnić, 2003], but degrades the range resolution of the sampled signals. To improve the range resolution, Yu et al. [2006] proposed a resolution enhancement technique (termed RETRO) using the Capon method (a method of linearly constrained minimum variance) for the range-oversampled signals; nevertheless, the Capon method may add additional range-weighting effect to the echoes due to its adaptive range-weighting pattern in the imaging process.

The Capon method was developed by Capon (1969), which is an inversion algorithm using multi-channel signals, and has been applied to VHF atmo-spheric radar to enhance the angular and range resolutions of the scattering structure [Palmer et al., 1998, 1999; Yu et al., 2001]. To enhance the angular resolution, the echoes received by multiple receivers are used, which is termed coherent radar imag-ing (CRI); similarly, the echoes with different transmitter frequencies are employed in the inversion algorithm to improve the range resolution, and is termed range imaging (RIM) or frequency interferometric imaging [Luce et al., 2001]. RIM and CRI give the estimate of so-called power density or brightness that can repre-sent the fluctuations of refractive index in the space. Plenty of studies associated with CRI and RIM have been carried out [e.g., Yu et al., 2001; Chilson et al., 2002; Yu and Brown, 2004; Cheong et al., 2004; Palmer et al., 2006; Chen et al., 2007; Luce et al., 2008; Chen et al., 2008; Hassenpflug et al., 2008].

Among the studies of RIM and CRI, Chen and Zecha [2009] and Chen et al. [2009] raised the concept of effective RWF for RIM; following this, Chen and Furu-moto [2011] proposed the effective beam-weighting function (BWF) for CRI. These works were carried out under the consideration that the estimated brightness is weighted by the RWF and BWF; a further examination on this subject was made by Chen et al. [2011] for the three-dimensional radar imaging. In these studies, the effec-tive RWF was obtained on the basis of a Gaussian function, and its shape was a func-64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100

tion of signal characteristics, e.g., signal-to-noise ratio (SNR). The effective BWF was also derived from the use of a Gaussian function; however, its shape was dependent on off-beam angle in addition to SNR.

The SNR-dependent characteristic of the effective RWF and effective BWF resulted partly from the performance of the inversion algorithm, e.g., the Capon method, which is also SNR-dependent. The issue of SNR was also discussed in the whitening transformation [Torres and Zrnić, 2003] and the RETRO technique [Yu et al., 2006]. On the other hand, the dependence of the effective BWF on off-beam angle was attributed to the Gaussian form used in computation. Namely, the Gaussian function used for the angular intensity of radar beam is only an approximate form, which is suitable within the region around the main beam axis, but deviates gradually from the true shape as the off-beam angle increases. Another factor causing the angular-dependent effective BWF could be the extra weighting pattern from the imaging method, which is adaptive to the echo distribution in the scattering volume [Yu et al., 2006].

In view of the dependence of the effective BWF on off-beam angle, we consider that the effective RWF used for RIM may vary with range. To verify this assertion is one goal of this study. Another goal of this study is to develop an approach for the measurement of RWF from the observed RIM data, similar to that made by Chen and Furumoto for CRI [2011]. In the work carried out by Chen and Furumoto [2011], multiple beam directions were employed to reveal the variation of effective BWF with off-beam angle, and the estimator of minimum mean square error was utilized to determine the optimal similarity between the brightness in the overlapped angular regions of adjacent beam directions. In this paper, we propose an analogous experimental setup to verify the range-dependent effective RWF by using the range oversampling technique. The similarity between the brightness in the overlapped range extents of adjacent sampling gates was examined to ascertain if the shape of the effective RWF changes with range or not.

This paper is organized as follows. Section 2 details the methodology of obtaining the effective RWF from the range-oversampled signals. Section 3 describes the experimental setup. The multi-frequency and range oversampling techniques of the Chung-Li VHF radar and the Middle and Upper (MU) atmosphere VHF radar are introduced. Section 4 exhibits some typical results and section 5 discusses more experimental results to complete the applicability of the approach. The conclusions are drawn in the final section. For the information about the RIM, see the Appendix or the related references listed in this paper.

2. Methodology 102 104 106 108 110 112 114 116 118 120 122 124 126 128 130 132 134 136 138

A schematic description of methodology and experiment is given in Figure 1. The panel (a) of Figure 1 describes the sampling gates with range extent of 300 m (corresponding to the range resolution of a 2-s rectangular pulse length with infinite receiver bandwidth) and the sampling range step of 50 m (corresponding to the sampling time step of ~0.33 s). Each solid box indicates a sampling gate with range extent of 300 m. The numbers given in and out of the sampling gate are the off-center range locations of the districts separated by dashed lines, which are the places we are searching for their respective range-weighting effects or the shapes of RWF. Notice that the districts confined by dashed lines are not sub-sampled gates but artificial partitions in the sampling gate for describing the methodology.

In the application of VHF atmospheric radar, a Gaussian RWF that is symmetric to the range center of the sampling gate is commonly used, as defined below:

)

2

exp(

)

(

r

r

W







where r is the range location with respect to the range center of the sampling gate, and r is the standard deviation of the Gaussian form. W2(r) represents the two-way RWF.

With expression (1) for the oversampled range gates in Figure 1(a), the locations of 25 m in the gate 1 (g1) and -25 m in the gate 2 (g2) are at the same height, and can be assumed to have the same range-weighting effect because they are the points symmetric to the range center of the sampling gate. Similarly, the locations of 75 m in g1 and -75 m in g4, the locations of 125 m in g1 and -125 m in g6, and so on, are at the same height and have the same range-weighting effect, respectively. Even more, the range-weighting effect at the location of the dashed line (i.e., the boundary between the two districts just below and above the dashed line) can be the same as that in another sampling gate, for example, the locations of 50 m in g1 and -50 m in g3, the locations of 100 m in g1 and -100 m in g5, and so on. Based on this, the two sets of brightness distributions around the locations of r and –r, respectively, in two matching sampling gates will be very close to each other after a recovering of brightness with the reciprocal of expression (1). This concept is analogous to that proposed by Chen and Furumoto (2011) for the determination of the effective BWF suitable for CRI. The role of the oversampled range gates in this study is equivalent to that of the part-overlapped radar beams conducted by Chen and Furumoto (2011).

In the practical process of RIM, the RWF suitable for recovering the brightness may change with SNR, as first examined by Chen and Zecha [2009]. For our present study, we applied again the estimator of mean square error used by Chen and Zecha 140 142 144 146 148 150 152 154 156 158 160 162 164 166 168 170 172 174

[2009]:    N 1 i _{1i 2i} 2 2i 1i P P ) P -(P ERR (2)

where P1i and P2i are two sets of brightness distributions estimated around the locations of r and –r, respectively, in two matching sampling gates. N is the number of the brightness values. A brief description of the estimation of P1i and P2i is given in the Appendix. The above estimator is zero when P1=P2. If P1P2, the value of ERR is positive and gets larger as the difference between P1 and P2 increases.

In computing, different values of range (time) delay and r are given to recover

the two brightness sets in (2) before ERR is estimated, yielding various values of ERR. The range (time) delay and r with the smallest ERR are then regarded as the

optimal values in correcting the brightness distribution. Here, the parameter of range (time) delay is thought to be the delay of the signal accumulated as the signal propagates through media and the radar system, which adds additional phase angles to the signals at different transmitter frequencies and must be compensated in the process of RIM. It will be shown later that the range (time) delay does not vary with the pair of sampling gates used, but r does.

Any two consecutive sampling gates (g2 and g3, g3 and g4, and so on) can provide an optimal value of r for the locations of 25 m and -25 m. With the same processing,

the gate pairs of g1 and g3, g2 and g4,…, can yield the optimal value of r for the

locations of 50 m; the gate pairs of g1 and g4, g2 and g5,…, can yield the optimal value of r for the locations of 75 m; and so on. The Gaussian RWF can extend to

the range locations farther from the center of the sampling gate, and therefore estimation can be made for a pair of sampling gates with a larger distance, e.g., g1 and g8 for 175 m, g1 and g9 for 200 m, g1 and g10 for 225 m, and so on. If the expression (1) can represent the true RWF well, the values of r estimated at various

range locations should be very close to each other. However, it will be shown later that so obtained r varies with the SNR of radar echoes and the gate pair used, thereby

making shapes of RWFs different at various range locations and SNRs.

The panel (b) of Figure 1 describes another experiment which uses oversampling: sampling gate with range extent of 150 m (corresponding to the range resolution of a 1-s rectangular pulse length with infinite receiver bandwidth) and sampling range step of 37.5 m (corresponding to the sampling time step of 0.25 s). In this case, the gate pairs of [(g1, g2), (g2, g3),…], [(g1, g3), (g2, g4),…], [(g1, g4), (g2, g5),…], …, are used to estimate the values of r at the off-center range locations of

18.75 m, 37.5 m, 56.25 m, …, respectively. 176 178 180 182 184 186 188 190 192 194 196 198 200 202 204 206 208 210 212

3. Experimental setup

Two VHF atmospheric radars were utilized to demonstrate our approach: the Chung-Li radar (24.9o_{N, 121.1}o_{E; Taiwan) and the MU radar (34.85}o_{N, 136.11}o _E; Japan). The beam directions were directed to vertical, and the beam widths of the two radars were 7.4o _{and 3.6}o_{, respectively.}

In the experiment using the Chung-Li radar, the sampling gates and the sampling range steps are sketched in Figure 1(a), which has been described in section 2. Several frequency sets were examined and the Capon method was employed in the process of RIM with an imaging step of 2 m. Experiments with rectangular and Gaussian pulse shapes, and with the filter bandwidths of 500 kHz and 1 MHz, were conducted, as listed in Table 1 for the cases I~V. Figure 1(b) describes the experiment of the MU radar, and the radar parameters are listed in Table 1 for the case VI. Only rectangular pulse shape with matched filter was examined for the MU radar, however, and imaging step was only 1 m in range. In any case, the effective RWF was determined based on the Gaussian form given in (1) for each pair of pulse shape and filter bandwidth.

In computing, 512 samples were used for an estimate of cross-correlation function between a pair of frequencies in the cases I, II, III and VI, and 256 samples in the cases IV and V. The time resolutions were thus ~65 s and ~46 s, respectively.

4. Results

As pointed out in section 2, the use of the expression (2) can provide two parameters: range (time) delay of the signal and r of the RWF. The range (time)

delay adds different phase angles to the signals at various transmitter frequencies. In order to facilitate inspection, we have transformed the range (time) delay to a phase angle with the equivalence of 300-m range delay (or 2-s time delay) to 360o_{, and} termed it as phase bias hereafter. In this way, we do not need to inspect the respective phase angles of the signals at various transmitter frequencies caused by the range (time) delay.

To yield more reliable results, 100 estimates from the estimator of (2) were assembled respectively for each set of gate pairs to show the statistical distributions of phase biases and r values at different range locations. The first three cases are

exhibited in Figure 2, in which the histograms of phase biases and z values at

different off-center range locations (r): 25 m, 50 m, and so on, are shown. Here, r

z 

  2 , where r is defined in the expression (1). In Figure 2, we can observe

that, no matter what r was, the peak of the phase bias distribution always located around 500o _{for the three cases. This means that the range (time) delay of the signal in} 214 216 218 220 222 224 226 228 230 232 234 236 238 240 242 244 246 248 250

the radar system was not related to the pulse shape and the filter bandwidth.

By contrast, the peak location (denoted as Pz hereafter) of the distribution of z

values varied with r, especially in the panels (a) and (b) when |r| was larger than 250 m. A detailed inspection find that Pz first decreased slightly with |r| and then increased with |r|. Such a variation is more prominent in the panel (b), where we can see that the smallest Pz occurred at r150 m; notice the same location of r for the smallest Pz in the panels (a) and (b). In the panel (c), however, the minimum Pz came about at r300 m, although the peak locations for various r were very close. From these results, we summarize that:

(i) The Gaussian form given in (1) cannot fully describe the range-weighting effect at different range locations, leading to a variation in the peak location of z

distribution with r.

(ii) The convolution of Gaussian pulse shape and impulse response of 500 kHz-bandwidth filter yields a moderate change in the range-weighting effect as compared with the other two cases. Therefore, the Gaussian form given in (1) can describe the RWF of the case III better than the other two cases. This can be indicated by the nearly constant peak location of z distribution in the case III.

In addition to the r-dependence, z for the Capon-RIM has been shown to vary with

SNR [Chen and Zecha, 2009]. Figure 3 displays the scatter relationships between z

and SNR at different r for the three cases shown in Figure 2. The relationships between z and SNR were very similar to that shown by Chen and Furumoto [2011]

for the study of beam-weighting effect on CRI. In view of this, it is expected that the approach used by Chen and Furumoto [2011] can also be applied to the present study to find a usable expression for describing the range-dependent scatter relationships between z and SNR. The process is as follows:

1) The scatter diagrams of SNR and z in Figure 3 indicate the hyperbola relation-ship, xy=Ao, where Ao is an undetermined value, x=SNR, and y=z; therefore, z=Ao/SNR. Moreover, it is found that minimum values of z and SNR are needed to establish the relationship between SNR and z further, and thus the following equation can be used:

zo o z σ SNR A σ    10 , (3)

where the minimum threshold of z is denoted as zo, and a minimum threshold of

-10 dB is chosen for SNR.

2) zo can be regarded as the value of z at infinite SNR, and can be obtained

approximately by extending the hyperbola relationship in Figure 3 to very high 252 254 256 258 260 262 264 266 268 270 272 274 276 278 280 282 284 286

SNR. Taking the case I as an example, if the values zo=[152, 150, 148, 147, 145,

145, 149, 151, 158, 166, 180, 200, 220, 242, 260] m are chosen initially for |r| =[25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300, 325, 350, 375] m, then the curve 2 8 . 2 1 r 120 c c σzo     (4)

is capable of depicting the relationship between zo and r at infinite SNR

approximately. Figure 4 shows the fitting result, with the constants c1≒2.223310-5 _{and c2≒147.3803. Expression (4) yields the values of }

z at

various r in the circumstance of infinite SNR, as indicated by the numbers given in each panel of Figure 3(a).

3) Ao is found to be different in the scatter diagrams of SNR and z at different r, and could be a function of zo. Comparing with that examined by Chen and

Furumoto for CRI [2011], zo is not mono-incremental, leading to a more complex

expression of Ao. After a trial of fitting, the following expression can be used to

describe the variation of Ao with zo:

0 1 2 2 3 3 a a a a A_o  _zo  _zo  _zo , (5)

where [a3, a2, a1, a0]=[ -0.6398, 286.3239, -42731.3180, 2127502.5079] for r  ~140 m, and [a3, a2, a1, a0]=[ 0.002632, -1.5745, 324.5014, -20909.8945] for r > ~140 m.

The fitting curves obtained from the expressions (3)-(5) are displayed in Figure 3(a). It should be pointed out that the expressions (3)-(5) are empirical and they are not unique. Other sets of expressions could also describe the relationship between z, r,

and SNR. This paper, however, is not intended to show all possibilities of the fitting, but to raise the concept of an adjustable RWF and to find usable expressions for recovering the RIM brightness appropriately. The same process can be applied to the cases II and III to determine their available expressions.

To take a look at the shape of the effective RWF determined by the adjustable z

in (3), a plot of some effective RWFs varying with SNR and r is shown in Figure 5. Taking the panel of r=150 m as an example, the range weight was ~0.8 at SNR=-5dB and decreased to ~0.5 at SNR=25 dB. However, the theoretical Gaussian RWF, shown by the curve with circles, gave a weight of only ~0.3. With the decrease of r, the weights of all RWFs at r got closer, as shown in the panels of r=50 m and r=5 m.

It is deserving of mention that the parameter r in (1) is 0.35c/2 for a rectangular

288 290 292 294 296 298 300 302 304 306 308 310 312 314 316 318 320 322 324

pulse shape with matched filter, as defined in previous studies [e.g., Franke, 1990]. For example, 0.35c/2 is 105 m for the case I, which is apparently smaller than the representative value of z at |r|200 m seen in Figure 2(a), namely, ~150 m. This is due to the definition z  2r. Dividing z by 2 is r, and accordingly, 150 m divided by 2 equals ~106 m, which is very close to the value of 105 m.

5. Discussion

5.1 Imaging corrected with different RWFs

With the expressions (3)-(5) to correct the imaging result, i.e., multiply the brightness of RIM with the reciprocal of the effective RWF, it is expected to recover the brightness value more suitably for the location outside the range extent defined by c/2. We discuss this using the results presented in Figure 6. The panel (a) of Figure 6 displays the height-time intensity of three non-overlapped sampling gates g55, g61, and g67 in the case I, and the panel (b) shows the RIM of the three sampling gates after correction with the expressions (3)-(5) for respective gates. The imaging process was applied between -150 m and 150 m in range for each gate. In the panel (b), layer or turbulent structure can be seen to vary with time and height. As the range imaging was performed between -450 m to 450 m only for the central gate of the three gates and without correction, the range-weighting effect was seen clearly, as shown in the panel (c), where only the structures around the range center were disclosed. When we corrected the brightness with the SNR-dependent RWF proposed by Chen and Zecha [2009], we obtained the result shown in the panel (d). Apparently, the brightness was over-corrected near the upper and lower edges of the imaging map. On the other hand, the correction using the expressions (3)-(5) can suppress the over-correction near the edges, as shown in the panel (e). In view of this, the expressions (3)-(5) indeed improves the imaging as compared with the uncorrected imaging (the panel (c)), although their performance may still be unsatisfactory near the upper and lower edges of the imaging map of this case. That means that there is still room left for the expressions (3)-(5) to enhance the imaging. As we have pointed out, the expressions (3)-(5) are not unique, other sets of expressions which may yield better imaging are possible. Another likely cause of unrecoverable brightness near the edges of the imaging map in Figure 6 is that the inversion method has its intrinsic limitation on retrieving the echoes at the locations far from the range center of a sampling gate, where the echoes are very weak due to severe suppression by the RWF.

Comparing with the case I, the Gaussian pulse shape and 500 kHz-bandwidth filter in the case III results in a smoother RWF that could be characterized nearly by a Gaussian function with constant standard deviation. It is thus expected that the SNR-dependent RWF proposed by Chen and Zecha [2009] is adequate for the case III. As 326 328 330 332 334 336 338 340 342 344 346 348 350 352 354 356 358 360 362

seen in Figure 7, the extended imaging for the central gate (the panel (c)) was very similar to the imaging of the three non-overlapped sampling gates (the panel (b)), although visible difference still existed around the edges of the imaging map.

5.2 RWFs of different frequency sets

In this subsection, different frequency steps were examined to verify the applicability of the approach proposed (namely, the cases IV and V in Table 1). This was executed in that the resolution/performance of RIM is related to frequency step in addition to frequency span. Compared with the sampling of temporal signals, the frequency step resembles the time step of sampling, and the frequency span is like the length of the temporal signals.

Figure 8 shows the histograms of phase biases and z values which are similar to those of Figure 2, but with the echoes of three, five, and seven frequencies in the case IV, respectively. The three frequencies were 51.75, 52.0, and 52.25 MHz, and the five frequencies were 51.75, 51.875, 52.0, 52.125, and 52.25 MHz. First, there was no apparent difference between the results of five and seven frequencies (the panels (b) and (c)). However, the phase bias was about 600o_{, which was larger than that (~500}o₎ for the cases I~III. Notice that the case IV and the cases I-III were carried out in summertime and wintertime, respectively. In view of this, it seems that variation in background temperature during different seasons, which can change the radar system characteristics, may alter the range (time) delay of the signal, causing a change of phase bias (Chen, 2004). On the other hand, the general features of the histograms of z values were similar to those of the case I.

As for the result of three-frequency set shown in Figure 8(a), the distribution of phase biases was getting scattered for |r|>250 m, and shifted severely for |r|=350 m. Besides, the distribution of z values was more dispersive than the other two cases. Such features can be attributed to a larger frequency step used for the imaging of the three-frequency set, as demonstrated in Figure 9. Figure 9 displays the grating lobe patterns of imaging with three- and five-frequency sets; the frequency steps were 250 kHz and 125 kHz, respectively. As seen, for the three-frequency set the layer structure appeared periodically at a distance of ~600 m. This means that the brightness used with the estimator of expression (2) became unreliable for |r|300 m. In view of this, it is expected that a smaller frequency step in the three-frequency set can improve the outcome of expression (2), for example, the frequency set of 51.875, 52.0, and 52.125 MHz. Figure 10 demonstrates this speculation. As seen, the peak locations of phase biases returned to ~600o _{at |r|=350 m, and all the distributions of phase biases and z} values were more reliable and close to those of five-frequency set. In fact, the computing deterioration caused by grating lobe pattern of imaging also occurs in the 364 366 368 370 372 374 376 378 380 382 384 386 388 390 392 394 396 398 400

approach using multiple receivers for measurement of beam-weighting function [Chen and Furumoto, 2011].

Finally, we considered equal and unequal frequency steps in the seven-frequency set, thereby conducted the experiment of case V for a comparison with the case IV, as listed in Table 1. It is found that there was no significant difference between the consequences of the cases IV and V (not shown).

5.3 RWF of MU VHF radar

To demonstrate the generality of the approach proposed in this study, the radar data collected by another radar system were examined. Figure 11 shows the results for the case VI, observed by the MU VHF radar. The experimental setup and the range-oversampling capability of the MU radar can refer to section 3 and Figure 1(b). As shown in Figure 11, the peak locations of phase biases (left panel) at different r were nearly constant, which were the same as those obtained during another winter season: 8-10 Feb, 2006 [Chen et al., 2009], indicating a very stable radar system. In the two right panels, we can observe that the peak locations of z values varied with SNR and r, and the scatter relationships between z and SNR at different r were also similar to that of the Chung-Li radar (Figure 3). Based on this, the approach proposed for the RIM to the measurement of the effective RWF works well on different radar systems.

6. Conclusions

The concept of adjustable or effective range weighting function (RWF) in the radar volume for the multi-frequency range imaging (RIM) of VHF atmospheric radar has been proposed in this paper, and an approach using the range-oversampled radar data for the measurement of effective RWF has also been addressed. It is shown that the shape of the effective RWF, which was assumed initially as a Gaussian form in the computation, depended not only on signal-to-noise ratio (SNR) but also on the off-center range location. The range-dependent characteristic of the effective RWF was more evident for the rectangular pulse shape with wider filter bandwidth, and could be attributed to the Gaussian form in the computation which cannot describe fully the range-weighting effect at various range locations. On the other hand, the dependence of the effective RWF on SNR could be associated with the inversion algorithm of RIM, e.g., the Capon method whose performance is also SNR-dependent.

Although the effective RWF was found to be SNR- and range-dependent, the shape of the effective RWF changed mainly with SNR within the range extent (c/2) of a sampling gate. In such circumstance, the brightness of the RIM within the range extent of a sampling gate can be corrected merely with the SNR-dependent effective 402 404 406 408 410 412 414 416 418 420 422 424 426 428 430 432 434 436 438

RWF. For the brightness located at the region farther from the sampling gate center, it could be over-corrected eventually by the SNR-dependent effective RWF. To suppress such overcorrection, the effective RWF adaptive to SNR and off-center range location has been recommended. A number of experiments with different frequency sets, pulse shapes, receiver filter bandwidths, equal and unequal frequency steps, and two radar systems have been conducted to demonstrate successfully the generality of the approach proposed.

Our approach to the measurement of effective RWF is based on a Gaussian form; nevertheless, it is possible to use other models for the RWF, which may not be as simple as the Gaussian function. Other models may need more parameters than the standard deviation of a Gaussian form for a complete description of the model, and the relationship between the parameters and SNR could be different from ours. To link up the effective RWF with the commonly used RWF, a Gaussian form for the effective RWF is a suitable choice.

The determination of the effective RWF can be of significance for some types of signal processing that may affect the RWF, for example, the whitening transformation on the weather radar echoes and the range imaging algorithm employed in this study. The RWF could be different from the original form defined by pulse shape and filter bandwidth. Therefore, the experimental approach to the measurement of the effective RWF is helpful for some types of signal processing/imaging applied to the atmospheric echoes of VHF radar. It is also expected to be of help to the three-dimensional imaging of VHF atmospheric radar, which needs considering the range-weighting and beam-range-weighting effects in reconstructing the three-dimensional scattering structure in the radar volume.

Acknowledgements

This work was supported by the National Science Council of ROC (Taiwan) through grants NSC99-2111-M-270-001-MY2 and NSC102-2111-M-270-001. The Chung-Li radar is maintained by the Institute of Space Science, National Central University, Taiwan. The MU radar is operated by the Research Institute for Sustainable Humanosphere (RISH), Kyoto University, Japan. The authors sincerely thank Prof. Toshitaka Tsuda for encouraging the experiment with the MU radar under the International Collaborative Research Program (Ref. No.: 23MU-A36).

Appendix: range imaging with multiple frequencies

Using a set of closely spaced transmitter frequencies for sequential radar pulses, the power distribution (or brightness distribution) in the range direction can be reconstructed with some inversion algorithms for the multi-frequency echoes; this is 440 442 444 446 448 450 452 454 456 458 460 462 464 466 468 470 472 474 476

well known as range imaging (RIM) [Palmer et al., 1999] or frequency interferometric imaging (FII) [Luce et al., 2001].

Among the inversion algorithms used with the RIM, the Capon method [Capon, 1969] is robust and handy. The inversion processing can be made in the time domain without considering Doppler frequency sorting of the echoes; it can also be made in the frequency domain [Palmer et al., 1999]. In the time domain, the Capon method can be expressed simply as

e R e 1 1 ) ( P r  _{H } , (1a)



























nn

n

n

n

n

R

R

R

R

R

R

R

R

R

...

.

. .

.

...

...

2

1

2

22

21

1

12

11

R

where P(r) is the so-called brightness at the range r. The superscripts H, -1, and T represent, respectively, the Hermitian, inverse, and transposition operators of the matrices. kn is the wavenumber of the n-th transmitter frequency. Rmn is the non-normalized cross-correlation function of two data sets calculated at zero-time lag for a pair of frequencies. Vector e is the “range steering vector”, obtained by resolving the constrained optimization problem [Luenberger, 1984]. For an introduction of the constrained optimization problem, see also Palmer et al. [1998].

In practical data analysis, we have to consider extra phase angles adding to the multi-frequency echoes. The extra phase angles adding to the echoes could arise from initial phases at different transmitter frequencies, the time delay of the signal propagating in the media and radar system, and so forth. For this problem, Chen [2004] suggested that the time delay of the signal could be the main cause if the radar system is stable, and Chen and Zecha [2009] proposed a self-calibration process to determine the time delay of the signal. The time delay of the signal leads to a range error and should be compensated in the range steering vector.

In addition to the time delay of the signal, the brightness is weighted by the range weighting function (RWF). The theoretical Gaussian RWF given by Franke [1990] 478 480 482 484 486 488 490 492 494 496 498 500 502 504 506

can be used to correct the range brightness, but an adjustable Gaussian RWF suitable for the RIM was proposed by Chen and Zecha [2009] to improve the continuity of the brightness at gate boundaries. Chen and Zecha [2009] showed that the adjustable Gaussian RWF was dependent on signal-to-noise ratio (SNR). In this paper, we further propose a form of adjustable Gaussian RWF that is not only related to SNR, but also to the range in the radar volume.

References

Capon, J., 1969: High-resolution frequency-wavenumber spectrum analysis. Proc. IEEE, 57, 1408-1418.

Chen, M.-Y., T.-Y. Yu, Y.-H. Chu, W. O. J. Brown, and S. A. Cohn, 2007: Application of Capon technique to mitigate bird contamination on a spaced antenna wind profiler. Radio Sci., 42, RS6005, doi:10.1029/2006RS003604. Chen, J.-S., 2004: On the phase biases of multiple-frequency radar returns of

mesosphere-stratosphere-troposphere radar. Radio Sci., 39, RS5013, doi:10.1029/2003RS002885.

Chen, J.-S., G. Hassenpflug, and M. Yamamoto, 2008: Tilted refractive-index layers possibly caused by Kelvin–Helmholtz instability and their effects on the mean vertical wind observed with multiple-receiver and multiple-frequency imaging techniques. Radio Sci., 43, RS4020, doi:10.1029/2007RS003816.

Chen, J.-S., P. Hoffmann, M. Zecha, and C.-H. Hsieh, 2008: Coherent radar imaging of mesosphere summer echoes: Influence of radar beam pattern and tilted structures on atmospheric echo center. Radio Sci., 43, RS1002, doi:10.1029/2006RS003593.

Chen, J.-S. and M. Zecha, 2009: Multiple-frequency range imaging using the OSWIN VHF radar: Phase calibration and first results. Radio Sci., 44, RS1010, doi:10.1029/2008RS003916.

Chen, J.-S., C.-L. Su, and Y.-H. Chu, G. Hassenpflug, and M. Zecha, 2009: Extended application of a novel phase calibration method of multiple-frequency range imag-ing to the Chung-Li and MU VHF radars. J. Atmos. and Oceanic Technol., 26, 2488-2500.

Chen, J.-S. and J. Furumoto, 2011: A novel approach to mitigation of radar beam weighting effect on coherent radar imaging using VHF atmospheric radar. IEEE Trans. Geosci. Remote Sens., 49, 3059-3070, doi:10.1109/TGRS.2011.2119374. Chen, J.-S., C.-H. Chen, and J. Furumoto, 2011: Radar beam- and range-weighting

effects on three-dimensional radar imaging for the atmosphere. Radio Sci., 46, RS6014, doi:10.1029/2011RS004715. 508 510 512 514 516 518 520 522 524 526 528 530 532 534 536 538 540 542

Cheong, B. L., M.W. Hoffman, R. D. Palmer, S. J. Frasier, and F. J. López-Dekker, 2004: Pulse pair beamforming and the effects of reflectivity field variation on imaging radars. Radio Sci., 39, RS3014, doi:10.1029/2002RS002843.

Chilson, P. B., T.-Y. Yu, R. D. Palmer, and S. Kirkwood, 2002: Aspect sensitivity measurements of polar mesosphere summer echoes using coherent radar imaging. Ann. Geophys., 20, 213–223.

Doviak, R. J., and D. S. Zrnić, 1984: Reflection and scattering formula for anisotropi-cally turbulent air. Radio Sci., 19, 325-336.

Franke, S. J., 1990: Pulse compression and frequency domain interferometry with a frequencyhopped MST radar. Radio Sci., 25, 565-574.

Hassenpflug, G., M. Yamamoto, H. Luce, and S. Fukao, 2008: Description and demonstration of the new Middle and Upper atmosphere Radar imaging system: 1-D, 2-D and 3-D imaging of troposphere and stratosphere. Radio Sci., vol. 43, RS2013, doi:10.1029/2006RS003603.

Luce, H., M. Yamamoto, S. Fukao, D. Hélal, and M. Crochet, 2001: A frequency radar interferometric imaging (FII) technique based on high-resolution methods. J. Atmos. Sol. Terr. Phys., 63, 221–234.

Luce, H., G. Hassenpflug, M. Yamamoto, S. Fukao, and K. Sato, 2008: High-resolution observations with MU radar of a KH instability triggered by an inertia-gravity wave in the upper part of a jet stream. J. Atmos. Sci., 65, 1711-1718. Luenberger, D. G., 1984: Linear and nonlinear programming, Addison-Wesley,

Reading, Mass.

Palmer, R. D., S. Gopalam, T.-Y. Yu, and S. Fukao, 1998: Coherent radar imaging using Capon’s method. Radio Sci., 33, 1585-1598, doi:10.1029/98RS02200. Palmer, R. D., T.-Y. Yu, and P. B. Chilson, 1999: Range imaging using frequency

diversity, Radio Sci., 34, 1485–1496.

Palmer, R. D., B. L. Cheong, M. W. Hoffman, S. J. Fraser, and F. J. López-Dekker, 2006: Observations of the small-scale variability of precipitation using an imaging radar. J. Atmos. Ocean. Technol., 22, 1122–1137.

Torres, S. M. and C. D. Curtis, 2012: The Impact of Signal Processing on the Range-Weighting Function for Weather Radars. J. Atmos. Oceanic Technol., 29, 796– 806.

Torres, S. M. and D. S. Zrnić, 2003: Whitening in Range to Improve Weather Radar Spectral Moment Estimates. Part I: Formulation and Simulation. J. Atmos. Oceanic Technol., 20, 1433–1448.

Yu, T.-Y. and R. D. Palmer, 2001: Atmospheric radar imaging using multiple-receiver and multiple-frequency techniques. Radio Sci., 36, 1493-1503, doi:10.1029/2000RS002622. 544 546 548 550 552 554 556 558 560 562 564 566 568 570 572 574 576 578 580

Yu, T.-Y., R. D. Palmer, and P. B. Chilson, 2001: An investigation of scattering mechanisms and dynamics in PMSE using coherent radar imaging. J. Atmos. So-lar Terr. Phys., 63, 1797–1810.

Yu, T.-Y., and W. O. J. Brown, 2004: High-resolution atmospheric profiling using combined spaced antenna and range imaging techniques. Radio Sci., 39, RS1011, doi:10.1029/2003RS002907.

Yu, T.-Y., G. Zhang, A. B. Chalamalasetti, R. J. Doviak, and D. Zrnić, 2006: Resolution Enhancement Technique Using Range Oversampling. J. Atmos. Oceanic Technol., 23, 228–240.

Table Captions

Table 1. Radar parameters for range oversampling and range imaging.

Figure Captions

Figure 1. Schematic plot of range-oversampling gates. (a) 300-m sampling gates

indicated by solid boxes. The sampling range step is 50 m. (b) 150-m sampling gates indicated by solid boxes. The sampling range step is 37.5 m. The numbers given in the districts separated by dashed lines are the off-center range locations of the district centers.

Figure 2. Histograms of phase biases and z values for three pairs of pulse shape and filter bandwidth: (a) s rectangular pulse with 500 kHz-bandwidth filter, (b) 2-s rectangular pulse with 1 MHz-bandwidth filter, and (c) 2-2-s Gaussian pulse with 500 kHz-bandwidth filter. r is the range location relative to the range center of the sampling gate.

Figure 3. Scatter diagrams of z versus SNR for the three pairs of pulse shape and filter bandwidth in Figure 2. In the panel (a), the curves result from the expressions (3)-(5), and the numbers given in each panel are the z values at infinite SNR, estimated with the expression (4).

Figure 4. Relationship between z at infinite SNR (denoted as z0) and off-center range location r. The fitting curve is from the expression (4).

Figure 5. The Gaussian RWF adaptive to signal-to-noise ratio and r. The theoretical RWF is shown by the curve with circles. In each panel, only the weights at the locations indicated by horizontal dashed lines can be used to correct the brightness of range imaging.

Figure 6. Range imaging of the case I, corrected with different range-weighting

functions. Three-gate and one-gate imaging plots are shown for comparison.

Figure 7. Range imaging of the case III, corrected with the range-weighting function

adaptive to SNR. Three-gate and one-gate imaging plots are shown for 582 584 586 588 590 592 594 596 598 600 602 604 606 608 610 612 614 616 618

comparison.

Figure 8. Same as Fig. 2, but for different frequency sets.

Figure 9. Grating lobe pattern of range imaging for different frequency steps. Figure 10. Same as Fig. 8(a), but the frequency step is 125 kHz.

Figure 11. For the MU VHF radar. (left two columns) Histograms of phase biases and

z values obtained from 1-s rectangular pulse shape with 1 MHz-bandwidth filter. (right column) Scatter diagrams of z versus SNR.

620 622 624 626 628