Extended Application of a Novel Phase Calibration Approach of ...

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JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY

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Extended Application of a Novel Phase Calibration Approach of Multiple-Frequency Range Imaging to the Chung-Li and MU VHF Radars JENN-SHYONG CHEN Department of Computer and Communication Engineering, Chienkuo Technology University, Changhua, Taiwan

CHING-LUN SU AND YEN-HSYANG CHU Institute of Space Science, National Central University, Jhongli, Taiwan

GERNOT HASSENPFLUG* Research Institute for Sustainable Humanosphere, Kyoto University, Uji, Japan

MARIUS ZECHA Leibniz-Institut fu¨r Atmospha¨renphysik, Ku¨hlungsborn, Germany (Manuscript received 11 February 2009, in final form 16 June 2009) ABSTRACT Multiple-frequency range imaging (RIM), designed to improve the range resolution of radar echo distribution, is now available for the recently upgraded Chung-Li VHF radar (24.98N, 121.18E). To complete the RIM technique of this radar, a novel phase calibration approach, proposed initially for the Ostsee Wind (OSWIN) VHF radar, was employed to examine the effects of phase bias and the range-weighting function on the received radar echoes. The estimated phase bias indicated a time delay of ;1.83 ms for the signal in the radar system. In contrast, such a time delay is more difficult to determine from the phase distribution of twofrequency cross-correlation functions. The same calibration approach was also applied successfully to the middle and upper atmosphere (MU) radar (34.858N, 136.118E) and revealed a time delay of ;0.33 ms for the radar parameters employed. These calibration results for various radars demonstrate the general usability of the proposed calibration approach. With the high-resolution performance of RIM, some small-scale Kelvin– Helmholtz (KH) billows, double-layer structures, and plumelike structures in the troposphere that cannot be seen in height–time intensity plots have been recognized in present observations. The billows and double layers were found to be closely related to strong vertical wind shear and small Richardson number, supporting the hypothesis of a dynamic process of KH instability. On the other hand, the plumelike structures were observed to grow out of a wavy layer and could be attributed to saturation and breaking of gravity waves. These fine structures have shown some remarkable features resolved by the RIM method applied to VHF radars in the lower atmosphere.

1. Introduction Range imaging (RIM) with multiple transmitting frequencies has been successfully used with VHF/UHF atmospheric radars, which improves the range resolu* Current affiliation: National Institute of Information and Communications Technology, Tokyo, Japan. Corresponding author address: Jenn-Shyong Chen, Department of Computer and Communication Engineering, Chienkuo Technology University, No. 1, Jieshou N. Rd., Changhua 500, Taiwan. E-mail: [email protected] DOI: 10.1175/2009JTECHA1295.1 Ó 2009 American Meteorological Society

tion of radar echo distribution greatly (the sampling step of RIM can be as small as 1 m). Many studies have been carried out with the help of high-resolution range imaging of the atmospheric structures: for example, Kelvin–Helmholtz instability (KHI) (Chilson et al. 2003; Luce et al. 2007), high-resolution wind profiling (Yu and Brown 2004), and vertical velocity bias associated with KHI (Chen et al. 2008). In the literature, RIM is also known as frequency interferometric imaging (FII) (Luce et al. 2001). RIM is based on some inversion algorithms that were initially applied to multiple-receiver coherent radar

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NOTES AND CORRESPONDENCE

imaging (CRI) to resolve multiple echo centers in the radar volume (Hysell 1996; Palmer et al. 1998; Yu et al. 2000). The success of applying the RIM technique to the atmosphere is also based on precise phase calibration of the radar echoes. Therefore, some approaches have been employed for the phase calibration in the RIM processing (Kilburn et al. 1995; Fernandez et al. 2005). Recently, we also proposed a calibration approach for the OSWIN VHF radar (54.18N, 11.88E) (Chen and Zecha 2009, hereafter CZ). The proposed approach is not only practical for the phase calibration of RIM, but also improves the correction of the range-weighting function effect on the imaged powers. To validate the general usability of the proposed calibration approach, we applied this approach to the Chung-Li VHF radar (24.98N, 121.18E; Taiwan), which has been upgraded recently, and the middle and upper atmosphere (MU) radar (34.858N, 136.118E; Japan). The first range imaging of the Chung-Li radar is achieved after appropriate phase and range-weighting corrections. In this note, we give a brief description of the proposed calibration approach in section 2. In section 3, we present some calibration results of the Chung-Li and the MU radars. The first RIM processing carried out with the Chung-Li radar is displayed, and, moreover, some remarkable KH billows, double-layer structures, and plumelike structures observed by the MU radar are exhibited and examined. Conclusions are given in section 4.

2. Instruments and methodologies The Chung-Li and MU radar systems are introduced briefly in the appendix, along with the equations of RIM used in this study. The issues of phase bias and rangeweighting effect are also described briefly. The calibration approach proposed by CZ is based on a simple hypothesis: the difference between the imaged powers around the edge of two adjacent range gates should be very small after optimal (or effective) range-weighting correction and phase compensation. Accordingly, the following estimator has been utilized to calculate the difference of the imaged powers: N (P1i P2i )2 P1i P 5 2 1 2i , ERR 5 P1i P2i P1i i51 i51 P2i N

å

å

(1)

where P1i and P2i are the two sets of imaged powers estimated at the same heights above ground, around the edge of two adjacent range gates; N is the number of the imaged powers. ERR varies with the parameter pair of phase compensation and range-weighting function given in cal-

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culation: ERR will be zero when P1i 5 P2i. If P1i 6¼ P2i, however, ERR becomes larger as the difference between P1i and P2i increases. Therefore, we can find the optimal correction for the imaged powers according to the minimum ERR. Since the phase imbalances of different frequency pairs are supposed to be mainly owing to the time delay of the signal in the radar system and the time delay corresponds to a range delay, for convenience we can delay the range of the sampling gate [variable r in (A3)] in calculating the imaged powers. To compare with the phase bias obtained from the FDI phase distribution (FDI phase: the phase of the cross-correlation function between two echoes of a frequency pair), however, the range delay is transformed into a phase angle by regarding the change of phase within a range-gate interval as 3608. Therefore, in practical calculation we use the term (phase bias/ 3608) 3 (gate interval) (m) for the range delay or (phase bias/3608) 3 (pulse length) (ms) for the time delay, in which the ‘‘phase bias’’ is the variable of phase angle. Note that a so-defined phase bias is not limited to the phase interval between 08 and 3608, but varies from zero to any positive number to reveal the optimal time delay in the radar system. The choice of lower and upper bounds of phase bias in a calculation depends on the radar system. Moreover, the range of the sz values (standard deviation of a Gaussian range-weighting function; see appendix) for calibration can be given according to the radar pulse length employed. For example, a range of 50–300 m (100–600 m) was chosen for a 1-ms (2-ms) pulse length in our present calculation. The computational step size of phase bias was 108, and the step size of sz was 5 m (10 m) for 1-ms (2-ms) pulse length. Owing to inevitable noise, such obtained optimal phase biases (and sz) for different gate pairs at different times are sometimes not exactly the same. Therefore, the likely phase bias (and sz) is revealed from the histogram of the optimal phase biases (and sz). Readers can refer to CZ for a detailed description of the practical calibration procedure.

3. Calibration results Table 1 lists the RIM experiments examined. Cases 1–4 were carried out with the Chung-Li radar, and case 5 was observed with the MU radar.

a. Chung-Li radar Figure 1 shows the distributions of the optimal phase biases and optimal sz values of the lower 30 gate pairs for case 1; each distribution contains 100 estimates. As seen, the phase biases of the lower gate pairs were more concentrated, where their mean signal-to-noise ratios (SNRs)

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TABLE 1. Radar parameters used in RIM experiments. Cases 1–4: Chung-Li radar and case 5: MU radar. Cases 2–4 are carried out alternately, and each mode is ;110 s. The sampling time resolution is ;0.256 s for cases 1–4 and ;0.512 s for case 5.

Case 1 2 3 4 5 a b c

Start–end time/date

Pulse shapea

Rx filter bandwidth (kHz)

Pulse code

Range resolution (m)

Sampling step, zo (m)b

Frequency setc

1700–2200 UTC 30 Mar 2008 0250–0420 UTC 11 Apr 2008 0250–0420 UTC 11 Apr 2008 0250–0420 UTC 11 Apr 2008 2200 UTC 8 Feb– 0700 UTC 10 Feb 2006

S

500

No

300

300, 1800

fa

G

250

No

300

300, 1800

fa

S

250

No

300

300, 1800

fa

G

500

No

300

300, 1800

fa

S

1000

16-bit Spano code

150

150, 3000

fb

G: Gaussian, S: square. zo: the start of sampling height. fa 5 [51.75, 51.875, 52.0, 52.125, 52.25] MHz (Df 5 0.125 MHz); fb 5 [46.0, 46.25, 46.5, 46.75, 47.0] MHz (Df 5 0.250 MHz).

were larger, too. Rough inspection can tell that the likely phase bias was ;3208. On the other hand, the phase biases of the upper gate pairs were dispersive and their mean SNRs were small; usually the results at low SNR are less reliable. As for the optimal sz shown in the right panel, the distributions of the lower gate pairs centered on ;170 m, which can be regarded as the likely value of sz that is close to the theoretical one. Nevertheless, we find that lower SNR leads to a larger and more dispersive value of sz, as observed clearly at the upper gate pairs. In theory, sz is not related to SNR; however, the echoes at low SNR are statistically more random, causing the value of sz to be also SNR dependent. To obtain smoother imaged powers at gate boundaries, therefore, a value of sz adaptive to SNR can be employed. The proposal of the SNR-dependent sz is one main innovation of the proposed calibration approach. The distributions of the optimal phase biases (and sz) of different gate pairs can be merged to form a more reliable histogram, as shown in Fig. 2. Figures 2a–d display the results of cases 1–4, respectively. As seen, the peak locations of the four histograms are close (Fig. 2, left); that is, between 3208 and 3508. Taking 3308 as the representative phase bias, this indicates a time delay of ;1.83 ms (3308/3608 3 2 ms). As for the histogram of sz, we can find differences in the peak locations of the four cases (right panels): ;170, ;260, ;220, and ;210 m for cases 1–4, respectively, which indicate approximately the theoretical value of sz in the absence of noise. According to the pulse shape and filter bandwidth employed in the experiment, the Gaussian range-weighting function for case 1 is narrowest, and that for case 2 is broadest. In view of this, the likely values of sz indicated in Fig. 2 are reasonable. It should be mentioned that all the characteristics of the sz histogram shown here are

similar to those of the OSWIN VHF radar demonstrated by CZ. Moreover, the values of optimal sz (not the likely sz) are also SNR-dependent, and Eq. (3), from CZ, can be employed again to find the mean relationship between optimal sz and SNR. The above calibration suggests a time delay of ;1.83 ms for the Chung-Li radar system. It is expected that such a time delay can also be determined from the FDI phase distribution. Theoretically, the distribution of FDI phases of each gate is a quasi-Gaussian shape with the peak located at the phase angle predicted for the center of the radar volume; such a shape is related to the range-weighting function as well as the intensity distribution of the radar main beam. However, the unbiased peak location occurs only when no phase imbalance exists between echoes at two different carrier frequencies. As presented in CZ, the peak location of the FDI phase distribution can be estimated in advance according to the radar parameters used. For our present experiments, the extents of FDI phases and their peak locations are summarized in Table 2 for different frequency separations Df. Note that the phase extents and peak locations of Df 5 125 kHz (250 kHz) and 375 kHz (750 kHz) are different for the four grouped gates; for Df 5 250 kHz (500 kHz) the FDI phases dwell in different phase extents at odd and even gates. As for Df 5 500 kHz (1000 kHz) the FDI phases always lie between 08 and 3608 with the peak at 1808. Taking case 1 as an example, Fig. 3 exhibits the FDI phases observed, where Fig. 3a–d display the respective FDI phase distributions of the grouped gates for four frequency pairs. Other frequency pairs result in similar features to those displayed here as long as their frequency separations are the same; for brevity, they are not shown here. Referring to Table 2, an inspection of

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FIG. 1. Distributions of (left) optimal phase biases and (right) optimal sz of different gate pairs. Each distribution is self-normalized.

the distributions in Fig. 3 reveals that the differences (phase imbalance) of observed and predicted peak locations are variable: they are about 2458, 1608 (odd gates) and 22208 (even gates), 808 and 22758 and 808 and 708, 21358 and 21358 and 21258 and 2208 for the frequency pairs Df 5 500, 250, 125, and 375 kHz, respectively. Such a set of phase imbalances is peculiar, different from that of the OSWIN radar, and thus needs further clarification. In fact, the negative values of phase imbalances are due to phase ambiguity. Therefore, we conclude that the phase imbalances are about 3158, 1608, 808, and 2308 for Df 5 500, 250, 125, and 375 kHz, respectively. Such a quasi-linear variation in phase imbalances corresponds to a time delay of ;1.78 ms, which is close to the ;1.83 ms revealed by the proposed calibration approach.

The above examination suggests again that we should be careful sometimes in using the FDI phase distribution for calibration; phase ambiguity may occur when the time delay of the signal in the radar system is large. In comparison and to its advantage, the calibration approach as used does not encounter this problem of phase ambiguity. Moreover, one may wonder if the time delay is so large that it could be ;1.83 1 2n ms, where n is zero or a positive integer. In the case of n 5 1 or 2, the time delay is ;2.83 or ;4.83 ms, corresponding to a phase imbalance of ;6908 (3308 1 3608) or ;10508 (3308 1 2 3 3608) for Df 5 500 kHz. We examined this and concluded that such large time delays can be discarded, as demonstrated in Fig. 4. In Fig. 4, we cannot find representative peaks of phase biases (the peaks at the leftmost edge of

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FIG. 2. Histograms of (left) optimal phase biases and (right) optimal sz. (a)–(d) Cases 1–4 listed in Table 1, respectively. The phase bin is 108, and the sz bin is 10 m.

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NOTES AND CORRESPONDENCE TABLE 2. Chung-Li radar and MU radar (in brackets) for Df range from 500 to 125 kHz. 500 kHz

250 kHz

[1000 kHz] Gates 1, 5, 9, . . . 2, 6, 10, . . . 3, 7, 11, . . . 4, 8, 12, . . .

125 kHz

[500 kHz]

375 kHz

[250 kHz]

[750 kHz]

Extent (deg)

Peak (deg)

Extent (deg)

Peak (deg)

Extent (deg)

Peak (deg)

Extent (deg)

Peak (deg)

0–360 [0–360] 0–360 [0–360] 0–360 [0–360] 0–360 [0–360]

180 [180] 180 [180] 180 [180] 180 [180]

0–180 [0–180] 180–360 [180–360] 0–180 [0–180] 180–360 [180–360]

90 [90] 270 [270] 90 [90] 270 [270]

180–270 [0–90] 270–360 [90–180] 0–90 [180–270] 90–180 [270–360]

225 [45] 315 [135] 45 [225] 135 [315]

180–360–90 [0–270] 90–360 [180–360–180] 0–270 [180–360–90] 270–360–180 [90–360]

315 [135] 225 [45] 135 [315] 45 [225]

the abscissa are unreliable), and the optimal values of sz are not appropriate. Based on the above calibration results, one of the RIM observations carried out with the Chung-Li radar is shown in Fig. 5. Figure 5a discloses some echo layers at a range resolution of 300 m; these echo layers are range smeared. Figure 5b displays the imaged powers without calibration; as seen, we cannot resolve the echo layers correctly. In comparison, Fig. 5c shows that the range imaging is improved greatly after proper corrections of phase bias and range-weighting effects. As seen, thin

layered and turbulent/convective structures can be identified above and below the height of ;4 km, clearly indicating different atmospheric characteristics in the middle troposphere and in the boundary layer. In the literature, similar structures have been observed and addressed in detail with the MU radar (e.g., Luce et al. 2006, 2007).

b. MU radar The general usability of the proposed calibration approach was further tested using MU radar data, as

FIG. 3. For case 1: distributions of FDI phases of four frequency pairs with different frequency separations Df. The phase bin is 108.

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FIG. 4. As in Fig. 2, but the interval of phase bias of the abscissa is changed. The data are from case 1.

demonstrated in Fig. 6. Figure 6a indicates that the phase bias is ;1208, corresponding to a time delay of ;0.33 ms (1208/3608 3 1 ms). The likely value of sz is ;90 m, and the values of sz are also SNR dependent (not shown). Figures 6b–e show some FDI phase distributions for several frequency pairs; the other distributions, not shown, possess similar features to the panels here, depending on their frequency separations. Referring to the predicted peak locations listed in Table 2, the phase imbalances can be estimated from the mean peak locations of the humps, except for the panel of Df 5 1000 kHz: ;608, ;258, and ;908 for Df 5 500, 250, and 750 kHz, respectively. A quasi-linear relationship between the phase imbalance and the frequency separation is evident, although the FDI phase distribution of Df 5 1000 kHz does not show an observable hump. The cause of the concealed hump for Df 5 1000 kHz is currently unknown. Again, the proposed calibration approach can avoid the need to deal with such an indistinct phase distribution. Figure 7 exhibits a special RIM example from our observations. Note that the time starts from the right side of the abscissa, which makes it more convenient to compare with the wind field depicted later. As seen, the diagram of the received power shows nothing exciting

(Fig. 7a), and the Capon imaged powers without any calibration are range-biased severely (Fig. 7b). After appropriate calibration, however, some KH billows are revealed between 6.0 and 6.6 km, during 12.0–12.3 h (Fig. 7c). Before and after these KH billows are doublelayer structures. Note that the observations were interrupted for tens of seconds each 4-min interval for data storage, making the imaged power vary abruptly sometimes (e.g., at 1200 UTC). The close relationship between double-layer structures and KH billows has been addressed in detail (Browning and Watkins 1970; Worthington and Thomas 1997) and has also been demonstrated by simulations (Fritts et al. 2003, 2006). Vertical wind shear is the most important factor responsible for KH billows, and this can be recognized from Fig. 7d, where the horizontal wind field, derived from full-correlation analysis (FCA), is shown. As seen, significant vertical wind shear existed between 6.0 and 6.6 km. An estimate of mean Richardson number during the period displayed here is about 0.1 ; 1.0, by taking the values of 1024 ; 1023 rad22 s22 for the square of the assumed Brunt–Va¨isa¨la¨ frequency (a typical range reported in the literature). This magnitude of Ri is around the threshold (0.25) required for the occurrence of active KH billows. Recently, Luce et al.

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FIG. 5. A portion of the observations of case 1. (a) Received radar echoes with a range resolution of 300 m. (b) Range imaging without phase and range-weighting corrections. (c) Range imaging with adaptable sz (varying with SNR) and phase correction (3308). Sampling step of range is 2 m in the imaging process and time resolution is ;30 s.

(2008) reported some KH billows above the jet stream and attributed it to the trigger of an inertia–gravity wave generated by the jet stream. Different from that case, the KH billows shown in Fig. 7 are far away from the core of the jet stream and persist only for a short period; in view of this, it seems that our case here is mainly related to the local wind shear. According to the authors’ knowledge, the case shown in Fig. 7 could be the clearest observation so far of the relationship between KH billows, double-layer structures, and vertical wind shear demonstrated by RIM with a VHF radar. Figure 7c also discloses a double-layer structure around the height of 6.8 km. In fact, such a structure appears quite often in our present MU radar data. In the literature, the model of turbulent layers addressed by Woodman and

Chu (1989) can be one interpretation of the double-layer structure observed; that is, highly anisotropic turbulence located at both edges of a turbulent layer can reflect/ scatter the radar echoes much stronger than the refractive index irregularities inside the main and more wellmixed turbulent layer. In addition to this model, we suspect that small-scale KH billows may occur very often and intermittently and could be the main mechanism of double-layer structures observed within a small vertical range. Unfortunately, even the RIM used with VHF radars cannot resolve such small KH billows yet; only the consequent double-layer structure is recognizable. One more novel observation with the MU radar is shown in Fig. 8. The received powers with 150-m resolution do not reveal any notable features (Fig. 8a). After

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FIG. 6. For case 5 (MU radar). (a) As in Fig. 2. (b)–(e) Distributions of FDI phases of four frequency pairs with different frequency separations Df. The phase bin is 108.

range imaging, however, two thin layers are disclosed around the heights of 5.2 and 6.2 km. (Note again that the observations were interrupted for tens of seconds each 4-min interval, causing abrupt change at times in the imaged power between data files.) The lower layer is wave modulated and its shape is slightly tilted due to vertical wind shear, which has been examined for bias of vertical velocity by Chen et al. (2008). Existence of atmospheric waves can be revealed from the vertical wind field shown in Fig. 8c. As shown, vertical velocities below the height of ;5.5 km indicate a wave activity having a period of ;4 min, especially between 20.1 and 20.4 h. However, the wave activity stopped at the height of ;5.5 km, indicating a possible critical level at

;5.5 km for this wave. An absorbing phenomenon is the plumelike structures between the two thin layers, which seem to grow out of the lower wavy layer. We suspect that these plumelike structures arose from saturation and breaking of gravity waves. This has shown a remarkable feature observed only by the RIM of a VHF radar in the lower atmosphere so far.

4. Conclusions An extended employment of a novel calibration approach for range imaging (RIM), which was initially applied to the OSWIN VHF radar recently, was made in

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FIG. 7. For case 5 (MU radar). (a) Received radar echoes with a range resolution of 150 m. (b) Ranging imaging without phase and range-weighting corrections. (c) Range imaging with adaptable sz (varying with SNR) and phase correction (1208). Sampling step of range is 1 m in the imaging process and time resolution is ;32 s. (d) Horizontal wind field.

this study for both the Chung-Li and the MU VHF radars. The first RIM experiment with the Chung-Li radar was also carried out successfully. Our studies have suggested that the major cause of the phase biases of RIM could be the same for the three

radar systems mentioned: time delay of the signal in the system, which leads to shift in FDI phase. Moreover, it is revealed that the calibration using the FDI phase distributions of different frequency pairs could be more difficult when the time delay in the radar system is large,

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FIG. 8. For case 5 (MU radar). (a) Received radar echoes with a range resolution of 150 m. (b) Range imaging with adaptable sz (varying with SNR) and phase correction (1208). Sampling step of range is 1 m in the imaging process and time resolution is ;32 s. (c) Vertical winds: gray (black) indicate upward (downward) velocities.

as happened in the case of the Chung-Li radar. By comparison, the proposed calibration approach is capable of indicating the value of time delay without the problem of phase ambiguity. With the high-resolution performance of RIM, a double-layer structure accompanied by some KH billows has been observed clearly with the MU radar, and their formation is closely related to significant vertical wind shear and small Richardson number, supporting the hypothesis of a dynamic process of KH instability. Moreover, some plumelike structures associated with a wavy layer are first reported in this note and are suspected to be the consequence of saturation and breaking of gravity waves. Based on these initial demonstrations, the RIM technique used with VHF/UHF pulsed

radars indeed aids in the physical interpretation of atmospheric phenomena by displaying high-resolution radar echoes in the range direction. More scientific studies on the atmosphere can thus be expected in the future in connection with microstructures and dynamics of the atmosphere. Acknowledgments. This work was supported by the National Science Council of ROC (Taiwan) under Grants NSC95-2111-M-270-001-MY3 and NSC98-2111M-270-001. The Chung-Li radar is maintained by the Institute of Space Science, National Central University, ROC (Taiwan), and the MU radar is operated by the Research Institute for Sustainable Humanosphere, Kyoto University, Japan.

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APPENDIX Radar Systems and Range Imaging a. Radar systems Figure A1 shows the configurations of the main antenna arrays of the Chung-Li and MU VHF radars. The ChungLi radar is operated at a central frequency of 52 MHz, in which the ST array is used mainly for observations of the lower atmosphere (;2–20 km) and the IONO array is for the ionosphere. The ST and IONO arrays consist of three subarrays formed by 8 3 8 and 4 3 8 four-element linear Yagi antennas, which are connected to three identical but independent transmitter–receiver modules. The upgraded oscillator can currently generate carrier frequencies from 51.75 to 52.25 MHz for RIM experiments (0.5-MHz imaging bandwidth so far; a larger imaging bandwidth is possible but it needs further testing). Moreover, radar pulse shape can be Gaussian or rectangular, and various filter bandwidths are available. [Currently, the source for more radar characteristics is via e-mail at [email protected]. edu.tw or from http://sdbweb.ss.ncu.edu.tw/VHFradar/ ChungLiVHF_home.htm (presented only in Chinese).] The MU radar, operated at a central frequency of 46.5 MHz and 1-MHz imaging bandwidth from 46 to 47 MHz, possesses an array consisting of 475 crossedYagi antennas, and each antenna is equipped with an independent but identical type of transmitter–receiver module. The radar array is divided into 25 independent groups with 19 crossed-Yagi antennas each. Each antenna group has its own digital receiver for the combined signal. Furthermore, software can combine the outputs of these receivers arbitrarily. For our present study, the combined output of the full antenna array was employed. Readers can refer to Hassenpflug et al. (2008) for a detailed description of the MU radar system.

b. Range imaging Capon’s method (Capon 1969) has been demonstrated to be excellent for RIM (Palmer et al. 1999; Luce et al. 2001; Chilson et al. 2003), and one set of the Capon equations can be simply expressed as P(r) 5

1 eH R1 e

(A1)

in which 2

R11 6R 6 21 R 56 6 4 Rn1

R12 R22 Rn2

3 R1n R2n 7 7 7 .. 7 5 . Rnn

(A2)

FIG. A1. Configurations of the Chung-Li and MU VHF radars.

and e 5 [ej2k1 r , ej2k2 r , . . . , ej2kn r ]T ,

(A3)

where P(r) is the imaged power at range r; the superscripts H and 21 in (A1) and T in (A3) represent the Hermitian, inverse, and transpose operators, respectively, and kn is the wavenumber of the nth carrier frequency. The term Rmn is the nonnormalized cross-correlation function of the signals calculated at zero time lag for a pair of transmitting frequencies; the phase of Rmn is termed frequency-domain-interferometry (FDI) phase in this paper. Equations (A1)–(A3) present the processing without Doppler frequency sorting of the echoes. More algorithms used with RIM and their comparisons can be found in the literature (e.g., Luce et al. 2006). The phase of Rmn may be biased owing to the phase imbalance between the two echoes of a frequency pair. Such phase bias/imbalance creates offsets in the imaged power location and/or smears the image, making true

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physical phenomena like KH billows hard to see or creating the impression of false phenomena, and thus need to be compensated. To find suitable phase compensation is essential. According to our previous studies (Chen 2004; CZ), the reason for phase biases of different frequency pairs is suspected to be mainly the time delay of the signal in the radar system. The exact cause of such time delay is unknown, but possibly originates from many parts of the radar system: response times of the elements, mixing and filtering of the echoes, propagation delay of the signals in the cable lines/circuits, among other causes. In addition to phase bias, the range-weighting effect on P(r) may also be considerable, especially toward the edges of the gates, and should be removed. This can be accomplished approximately by using the inverse of the range-weighting function W2(r) 5 exp(2r 2/sz2) (Franke 1990; Luce et al. 2001). Theoretically sz can be estimated from the radar pulse shape and filter bandwidth; however, considering the ubiquitous noises embedded in the echoes, the use of a noise-dependent sz has been recommended by CZ to improve the continuity of the imaged powers at gate boundaries. REFERENCES Browning, K. A., and C. D. Watkins, 1970: Observations of clear air turbulence by high power radar. Nature, 227, 260–263. Capon, J., 1969: High-resolution frequency-wavenumber spectrum analysis. Proc. IEEE, 57, 1408–1419. Chen, J.-S., 2004: On the phase bias of multiple-frequency radar returns of mesosphere-stratosphere-troposphere radar. Radio Sci., 39, RS5013, doi:10.1029/2003RS002885. ——, 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. ——, G. Hassenpflug, and M. Yamamoto, 2008: Tilted refractiveindex 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. Chilson, P. B., T.-Y. Yu, R. G. Strauch, A. Muschinski, and R. D. Palmer, 2003: Implementation of range imaging on the Platteville 915-MHz troposphere profiler. J. Atmos. Oceanic Technol., 20, 987–996. Fernandez, J. R., R. D. Palmer, P. B. Chilson, I. Ha¨ggstro¨m, and M. T. Rietveld, 2005: Range imaging observations of PMSE using the EISCAT VHF radar: Phase calibration and first results. Ann. Geophys., 23, 207–220.

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