This paper demonstrates the multiple-frequency range imaging (RIM) which was implemented recently on the OSWIN VHF atmospheric radar (54.1°N, 11.8°E), Germany. A simple but practical phase calibration method is introduced. We validate the RIM technique and the proposed calibration method successfully by examining various radar experiments with different pulse lengths, mono and coded pulses, evenly and unevenly spaced frequencies, and receiver filter bandwidths. The proposed calibration method not only mitigates the phase imbalance between the echoes received at different transmitting frequencies, but also provides a likely value of standard deviation (σz) of the Gaussian range-weighting function for correcting the range-weighting effect. Moreover, it is found that σz can be adaptive to signal-to-noise ratio when it is employed in practice; this procedure improves the continuity of the imaged powers of RIM around the boundaries of range gates, and an empirical expression has been proposed for this. With the improved power distribution around gate boundaries, we can obtain more available estimates of layer altitudes and exhibit the pass of the layer through gate boundaries clearly. Two observations are shown to demonstrate the maturation of the RIM technique used with the radar: convective cells and double-layer structures. These atmospheric structures cannot be seen clearly in the original presentation of signal-to-noise ratio (or height-time intensity) of the radar echoes having 150-m or 300-m range resolution.
 Transmitting multiple frequencies to observe the atmosphere may become a basic capability of the modern mesosphere-stratosphere-troposphere (MST) radars. This observational technique starts with the frequency domain interferometry (FDI) using two slightly different frequencies [Kudeki and Stitt, 1987; Franke, 1990], takes wing after some sophisticated signal-processing algorithms such as the Capon method [Capon, 1969; Palmer et al., 1999] and the maximum entropy, etc., were introduced to deal with the echoes received at several carrier frequencies, and is now known as range imaging (RIM) [Palmer et al., 1999] or frequency interferometric imaging (FII) [Luce et al., 2001] in MST radar community.
 The sophisticated algorithms used with RIM/FII are inversion processes, which were initially introduced to multiple-receiver coherent radar imaging (CRI) to determine multiple echo centers (angle of arrival or direction of arrival) in the radar volume [Palmer et al., 1999]. With similar processing, these algorithms enable us to recognize multiple irregularity layers in the radar volume and then give estimates of the altitudes and thicknesses of the layers. Consequently, the range resolution of echo distribution (or brightness distribution) is improved greatly. With the information of layer altitude, layer thickness, and echo distribution, plenty of atmospheric studies have been carried out; for example, Kelvin-Helmholtz instability (KHI) [Chilson et al., 2003; Luce et al., 2007a], high-resolution wind profiling [Yu and Brown, 2004], fine-scale layer structures [Chilson et al., 2001; Palmer et al., 2001; Luce et al., 2006, 2007b], and vertical velocity bias associated with KHI [Chen et al., 2008].
 Recently, the OSWIN VHF radar (54.1°N, 11.8°E; Germany) has established the RIM/FII technique for the observations of the atmosphere. To validate the capability of the RIM/FII technique used with this radar is thus one of the purposes of this study. We starts with the essential work-phase calibration in the RIM processing. Precise phase calibration can improve the continuity of the imaged powers at the boundaries of range gates, making the locations of the irregularity layers in the radar volume more reliable.
 In the literature, several approaches have been employed for the phase calibration in RIM/FII; for example, reading the peak location of the FDI phase distribution (FDI phase is defined as the phase of the cross-correlation function of two echoes received by a pair of frequencies [Kilburn et al., 1995; Brown and Fraser, 1996]), measuring the initial phases of the carrier frequencies from the signals that are leaked from the transmitted signals back to the receiver by an ultrasonic delay line [Palmer et al., 2001; Chilson et al., 2001], applying the relationship between echo power and FDI phase [Chen, 2004], utilizing genetic algorithms [Fernandez et al., 2005]. Among them the use of FDI phase distribution is simpler. Nevertheless, the shape of FDI phase distribution is related to some radar parameters, and usually, it needs enough radar echoes to obtain a reliable distribution in statistics; the latter could be a problem at times. For example, the mesosphere summer echoes (MSE), occurring in summer season at midlatitudes, are striking VHF/UHF radar echoes that are related to variations of temperature and water vapor in the mesosphere, and can be observed for studying the waves/dynamics of the atmosphere around the mesopause height. However, MSEs are sometimes weak and limited in time and height intervals. In view of this, in this paper we propose a new method of phase calibration. In addition to the phase calibration, an improved correction of the range-weighting function effect on the imaged power can also be achieved.
 A brief description of the OSWIN VHF radar and the RIM method is given in section 2. In section 3, we detail the process of phase calibration with the proposed method and demonstrate several RIM experiments to validate the calibration method. Two initial RIM observations are shown in section 4. Finally, conclusions are stated in section 5.
2. Instrument and Range-Imaging Technique
 The OSWIN VHF radar, primarily operated at a central frequency of 53.5 MHz, can now generate the carrier frequencies from 53.25 MHz to 53.75 MHz with the smallest frequency step of 1 kHz. The carrier frequencies are generated with the same oscillator and can be changed from pulse to pulse to meet the basic requirement of RIM, that is, the echoes with various carrier frequencies are received almost simultaneously. Radar pulse shape can be Gaussian or rectangular and complementary codes are usually employed to raise signal-to-noise ratio (SNR). For more descriptions of the radar configuration/characteristics, browse the Web site http://www.iap-kborn.de or refer to Latteck et al. .
 Among the inversion algorithms used with RIM/FII, the Capon method [Capon, 1969] is robust and handy [Palmer et al., 1999; Luce et al., 2001]. Without considering Doppler frequency sorting of the echoes, the Capon equations can be simply expressed as
where P(r) is the imaged power at the range r after range-scanning processing. The superscripts H and −1 in (1a) and T in (1c) represent, respectively, the Hermitian, inverse, and transposition operators. kn is the wave number of the nth carrier frequency. Rmn is the nonnormalized cross-correlation function of the signals calculated at zero-time lag for a pair of frequencies. Rmn can be simply written as
where 〈 . 〉 means ensemble average. rm and rn indicate the ranges of the scatterers, and ϕm and ϕn are the phase terms associated with the system responses to different transmitting frequencies. After performing the ensemble average, the phase angle of (1d) is named FDI phase here. The phase terms ϕm and ϕn are not necessarily the same, causing inaccurate range distribution of P(r). Many factors can contribute to the phase imbalance ϕm − ϕn; for example, the initial phases at different carrier frequencies, the time delay of the signal in the media and radar system, and so forth. The major factors of the phase imbalance may vary from system to system and are difficult to identify exactly; however, it is possible to correct the phase imbalance by compensating suitable phase angle in Rmn. To find such suitable phase angle is thus essential and a variety of approaches have been attempted, as mentioned in the introduction. There are two noticeable facts here: first, ϕm − ϕn is the phase difference between the two signals, so absolute measurements of ϕm and ϕn are not necessary in the calibration; second, Rmn is not normalized, so P(r) can indicate the echo distribution in range.
 In addition to compensating the phase imbalance, the range-weighting effect on P(r) should be removed. This can be done approximately by using the inverse of the range-weighting function W2(r) = exp(−r2/σz2) [Franke, 1990; Luce et al., 2001], where σz is the parameter needed to determine appropriately for various radar experiments.
 After the above corrections, P(r) may possess several distinct brightness maxima, indicating multiple layers in the radar volume. The resolvable number of layers in the radar volume depends on many factors, e.g., the frequency spacing and the maximum frequency separation, the distances between the layers and the layer thickness, the affection of SNR on some inversion algorithms like the Capon method, and so forth. As a start, we use five frequencies between 53.25 MHz and 53.75 MHz in the present RIM experiments although transmitting more frequencies are possible. The maximum frequency separation is 0.5 MHz and we transmit the pulse length ≥ 2 μs. The use of the above radar parameters is partly subject to the capabilities/limitations of the present radar system and for the reason of protecting the radar system.
 We carried out several radar experiments for the troposphere and lower stratosphere with the RIM technique. Table 1 lists some important radar parameters employed for the experiments. Sampling time was ∼0.2 s for these experiments, and 128 or 256 data points were used for one estimate of Rmn with (1d), resulting in the time resolutions of about 30 s to 1 min. It should be mentioned that some experiments were carried out with two or three observational modes alternately for multiple purposes; therefore the real time resolutions of the RIM results were double for the cases 6 and 8 (∼2 min), and triple for the cases 1–4 (∼3 min).
Table 1. Radar Parameters Used With the RIM Experimentsa
 As well known, the shape of FDI phase distribution at a range gate is closely related to the weighted radar volume formed by range weighting function and radar beam both. Range weighting function results from the receiver filtering process and can be approximated by a Gaussian form with the maximum at the center of the range gate [Doviak and Zrnic', 1984]. On the other hand, the radar beam can be modeled by a two-dimensional Gaussian function on the plane perpendicular to the range direction and with the maximum at the beam centerline. There is a convolution of the beam shape function with the atmospheric echoes, causing the weighting of the echoes to be greatest at the center of the radar volume and decreases gradually outwards. As a result, the echoes returning from the irregularities closer to the center of the radar volume usually dominate the received power, causing the measured FDI phase to deflect to the phase angle at the center of the radar volume. After collecting enough echoing situations, it is expected to obtain a quasi-Gaussian distribution of FDI phases with the peak at the phase angle predicted for the center of the radar volume, provided there is no phase imbalance between the echoes received at different carrier frequencies. Such expectation is valid even for the echoes accompanied with noise (white noise). In theory, ensemble average in (1d) would eliminate the affection of noise on the FDI phase, and for pure noise the FDI phases are randomly, uniformly distributed, and should not be related to the range weighting function or radar beam shape.
 The peak location of the FDI phase distribution at a range gate can be estimated in advance when suitable radar parameters are employed (e.g., frequency separation, pulse width, sampling time delay, sampling step, etc.). For example, with pulse width 2 μs, sampling step 2 μs, and sampling time delay 8 μs (multiple of 2 μs), Table 2 summarizes the extents of FDI phases of the grouped range gates as well as their peak locations at different frequency separations (Δf). As indicated, for Δf = 500 kHz the FDI phases of all range gates inhabit between 0° and 360°, with a peak at 180°. However, for Δf = 250 kHz the FDI phases of odd and even range gates distribute over different phase extents and peak at different phase angles. As for Δf = 125 kHz and Δf = 375 kHz, the FDI phases of the four grouped gates dwell in different phase extents and also peak at different phase angles. Such regularly distributed FDI phases make it possible to find the phase imbalance in (1d), as exhibited in Figure 1.
Table 2. FDI Phase Distribution and Its Peak Location Within the Range of a Gatea
Figures 1a–1d display the FDI phase distributions of four frequency pairs of the case 1 for different groups of gates. The results of other frequency pairs unshown are similar to the displayed ones having the same frequency separation. To avoid unexpected affection of the noise, however, we used a SNR threshold, 0.125 (∼−9 dB), in adopting the FDI phase. Referring to Table 2, we can find that the peak locations of the distributions deflect to right slightly. For example, the frequency pair with Δf = 250 kHz possesses two expected peak locations at 90° and 270° (see Table 2), but the observed distributions show two mean peaks at ∼120° and ∼300°. Apparently, there is a shift of ∼30° in the observed distributions, that is, the phase imbalance is ∼30°. With similar inspection, the phase imbalances are ∼60°, ∼45°, and ∼15°, respectively, for Δf = 500 kHz, 375 kHz, and 125 kHz. Accordingly, there is a linear relationship between the phase imbalance and the frequency separation.
Figures 1e–1g show the full FDI phase distributions of all frequency pairs of the cases 1 (except for Δf = 500 kHz). For saving pages, we display the integrated distributions of 40 gates. For the situations of Δf = 250 kHz and Δf = 125 kHz (Figures 1e and 1f, respectively), their respective humps in the distributions can be identified definitely, and it is clear that the shift of the peak location depends only on the frequency separation, not on the absolute values of the two frequencies. As for the situation of Δf = 375 kHz (Figure 1g), it is difficult to distinguish the 4 peak locations shown in the Figure 1d. This is because the FDI phase distributions overlap largely and peak at different phase locations, causing the merged distribution smearing. In conclusion, the phase imbalance depends only on the frequency separation, which is similar to that examined by Chen  for the Chung_Li VHF radar. This feature could be attributed to the group delay of the signals in the media and the time delay from the data processing in the radar system. To compensate these phase imbalances, therefore, we can simply consider a time delay (or range error) in the RIM analysis. The time delay is about (60°/360°) × 2μs = 0.333 μs for the case here, corresponding to a range error of ∼50 m.
Figures 2a–2c show the cases 2–4. For saving pages, only four of the distributions are presented for each case. The unshown distributions are similar to the demonstrated ones having the same frequency separation. Generally speaking, it is more difficult to determine the phase imbalances here. This is partly because the pulse shapes and Rx filter bandwidths employed in the cases 2–4 result in shallower range weighting functions, causing the contrast between the peaks and the valleys in the FDI phase distributions smaller and less observable, especially for the observations with large frequency separations (e.g., Δf = 500 kHz and Δf = 375 kHz). This points out one possible defect of using the FDI phase distribution for phase calibration when the experimental radar parameters produce a shallow range weighting function and/or the amount of the available echoes are not enough to reveal the range-weighting function shape. The following calibration method can avoid such inconvenience.
3.2. A New Calibration Method of FDI Phase Imbalance
 Our new calibration method can find two desired parameters: the time delay for phase compensation and the standard deviation (σz) of the Gaussian range weighting function for range-weighting correction. The key thought/hypothesis is simple, that is, the imaged powers of RIM around the edge of two adjacent range gates should be very close after optimal range-weighting correction and phase compensation (by giving a corresponding time delay/range error in RIM). Based on this thought, the estimator given below can be employed to estimate the difference between the two sets of imaged powers, P1 and P2, around the edge of two adjacent range gates:
where N is the number of the imaged powers. If P1 = P2, then ERR = 0. If P1 ≠ P2, then ERR > 0 and ERR gets larger when the difference between P1and P2 is larger. Given various values of time delay/range error and σz to estimate P1 and P2, it is able to find the smallest value of ERR. The time delay/range error and σz with the smallest ERR can be regarded as the optimal values in correcting the imaged powers. To compare with the phase imbalance obtained from the FDI phase distribution, however, the time delay/range error can be transformed into phase angle by regarding the change of FDI phase within a range-gate interval as 360°. Therefore, in practice we use the term, (phase bias/360°) × pulse length, for the time delay, in which the “phase bias” is the variable given in the calculation. Such obtained optimal phase biases (and σz) from different pairs of range gates at different times would not be the same value due to some intrinsic uncertainties of the echoes, the inevitable noises, and the SNR-dependent result of the Capon method. However, a histogram of the optimal phase biases (and σz) is able to indicate a likely value of phase bias (and σz). We will see this later.
 Note that, with (2), we need the imaged powers outside of the nominal range gate; however, it is not necessary to use all of the imaged powers. For example, for the 2-μs pulse length with 300-m range resolution, P1 can be the imaged powers between 120 m and 180 m of the lower gate, and P2 can be the imaged powers between −180 m and −120 m of the upper gate. If the scanning range of each gate is from −180 m to 180 m with 2-m scanning step, P1 contains the last 31 imaged powers of the lower gate and P2 contains the first 31 imaged powers of the upper gate. The scanning step can be smaller, say, 1 m, then P1 will be the last 61 values of the lower gate and P2 contains the first 61 values of the upper gate. Smaller scanning step consumes more calculating time but does not lead to observably different results according to our experience. Although one can employ the imaged powers within a larger range extent around the gate boundary, the range-weighting correction at the places far away the center of the range gate may be excessive because the Gaussian range-weighting function is only an approximate form within the range gate interval. In view of this, we suggest using the imaged powers between −30 m and 30 m, centering on the height of gate boundary.
3.2.1. Mono Pulses: Cases 1–4
 The cases 1–4 were examined again with the proposed calibration method, as shown in Figure 3. We summarize three important features as follows:
 1. The four histograms of the optimal phase biases exhibit similar shapes, and peak almost at the same phase angle ∼70°, corresponding to a time delay of ∼0.389 μs (Figure 3, left). This result is in good agreement with that obtained from the FDI phase distributions of the case 1 (∼0.333 μs).
 2. The distributions of σz peak at about 150 m, 250 m, 190 m, and 190 m, respectively, which can be regarded as the likely values of σz of their respective Gaussian range-weighting functions (Figure 3, right). The difference in the likely values of σz between the four cases arises from different pulse shapes and Rx filter bandwidths employed (see Table 1). The experiment of the case 1 utilized square pulse with the Rx filter bandwidth of 317 KHz, producing a narrowest range weighting function compared with the other three experiments. On the other hand, the Gaussian pulse with the Rx filter bandwidth of 159 KHz in the case 2 results in a broadest range weighting function. Therefore, the likely values of σz revealed in Figure 3 make sense.
 3. The dispersion of phase biases and σz in the histograms is related to low SNR. See Figure 4 for further explanation.
 In Figure 4, the left four panels show the two-dimensional histograms of the optimal phase biases and σz, in which the number of each bin is indicated by gray intensity. Darker pixel represents larger number and the gray intensity is self-normalized. As indicated by the darkest pixels, the likely locations of phase biases of the four cases are almost the same (around 70°), but the likely values of σz are different. In the middle panels, we can see that the optimal phase biases get more dispersive at lower SNR. This can be attributed to the noises which degrade the echoes and introduce statistical uncertainty into the estimated phase biases. The SNR here is the mean estimated from the sampled signals of two adjacent gates. In the right four panels, one can see the general relationship: a lower SNR correlates with a larger σz. The fitting curves result from the equation
where SNRmin is the threshold of SNR for fitting, and the value of −10 dB was used in our studies. The constants a, b, c, and d are indicated in each panel. (3) is a more appropriate curve we have found so far for our experimental data. There may be various expressions for different radar experiments, and simpler formulas could also exist.
Figure 4 indicates that it would make the imaged powers at gate boundaries more continuous if we use the values of σz and phase bias adaptive to SNR in correcting the imaged powers of each gate. However, this is not practical. The optimal σz and phase bias are estimated from a pair of range gates, say, the first and the second range gates, and the second and the third range gates. Different pairs of range gates may result in different optimal σz and phase bias. Consequently, there may be not consistent values of σz and phase bias for the second range gate. In practice, the imaged power and its variations in range are small at low SNR, often making the phase calibration doubtful. Considering this, we can simply adopt the phase angle of the peak location revealed from the histogram of the optimal phase biases to correct all data. On the other hand, it would be good to use the σz adaptive to SNR to improve the continuity of the imaged powers around gate boundaries, especially in low-SNR condition. The fitting curve (3) can be employed to produce a suitable value of σz at a given SNR. Considering three consecutive range gates again, practical procedures are as follows:
 1. Estimate the mean SNRs of the first and the second gates, and the second and third gates, respectively. Let the results be SNR1 and SNR2.
 2. Calculate the values of σz from (3) with SNR1 and SNR2. Let the results be σz1 and σz2.
 3. Use σz1 for the first gate and the lower half part of the second gate, but use σz2 for the lower half part of the third gate and the upper half part of the second gate.
 With the above three steps, the upper and lower parts of the second/middle range gate usually suffer slightly different values of σz, but this would smooth the imaged powers at both boundaries of the second/middle gate. Repeating the above procedure, the continuity of the imaged powers at the boundaries of the third and the fourth gates, the fourth and the fifth gates, and so on, can be improved. It should be mentioned here that σz is related to the radar parameters used, which does not depend on SNR in theory. The feature of SNR-dependent optimal σz is just due to the inevitable noises in the echoes. Considering other factors may also vary the imaged power more or less, the above procedure of range-weighting correction is certainly not very perfect. Nevertheless, we will see that it indeed works in making the imaged powers more continuous at gate boundaries and is also beneficial for locating the imaged power centers or layer altitudes around gate boundaries (see below).
Figure 5 demonstrates one example of our studies. As seen, the SNR in Figure 5a shows some descending layer structures with severe smear in range. By contrast, range imaging exhibits the layer features much well (Figures 5b and 5c). However, there is a difference between the two outputs of range imaging—the calibration processing results in more continuous imaged powers at gate boundaries (Figure 5c) than that without any correction (Figure 5b). A close-up of the brightness profiles at the time of 12:06 UT is shown in Figure 5d. As seen, there are steep changes in the uncorrected profile (black) at the heights of 6.9 km and 7.2 km (namely, at gate boundaries). After a range-weighting correction with σz = 150 m and phase compensation, the profile (blue) peaks at gate boundaries, which is due to overmodification. To mitigate the steeply changed or overmodified imaged powers at gate boundaries, therefore, the range-weighting correction with SNR-dependent σz can be employed, as indicated by the red curve.
 As mentioned, such smoothed powers at gate boundaries are beneficial for locating the layer altitudes. The locating results are shown in Figures 5e and 5f, where the layer altitudes were estimated with the contour method proposed by Chen et al.  for including the multiple-layer cases. In Figure 5e, the layer altitudes estimated with 150-m σz and adaptive σz, respectively, are close; however, the use of 150-m σz produces fewer available estimates of layer altitudes, causing many discontinuities at gate boundaries, for example, the layer beginning at ∼6.5 km and descending to ∼4.5 km at ∼12:00 UT. Such discontinuity of layer altitude is due to overmodification of the imaged powers. With a more appropriate value of σz for this case, e.g., 250 m according to Figure 2b, we obtain the results shown in Figure 1g. As seen, both layer altitudes are in good agreement. Based on this, 250 m has been an appropriate value of σz for this case to correct the imaged powers as well as to locate the layer altitudes. To be superior, adaptive σz can be employed.
 Note that although the value of σz, 150 m, is not appropriate for the case here, it is more suitable for the case 1. In any case, using adaptive σz is suggested. To remind readers here again, the use of adaptive σz is to mitigate the steep change of the imaged powers at gate boundaries; the value of σz is theoretically not related to SNR, and its likely value can be found with our new calibration method, namely, the peak location in the histogram of the optimal σz (Figure 3).
3.2.2. Coded Pulses: Case 5
 Since coded pulses are usually utilized in radar observations to raise SNR, to validate the RIM experiment with coded pulses is essential for practical applications. One of the experimental results is shown in Figure 6, in which the coded pulses with 8-bit complementary codes were transmitted.
Figure 6a displays only some distributions of the FDI phases for saving pages; the unshown distributions are analog to these distributions, depending on the frequency separation Δf. The features of these distributions are similar to the case 1, and according to Table 2, we can find the phase imbalance of each frequency separation easily. There is a linear dependence of phase imbalance on frequency separation except for the situation of Δf = 375 kHz: ∼80°, ∼40°, and ∼20° for Δf = 500 kHz, 250 kHz, and 125 kHz, respectively, corresponding to a time delay of ∼0.444 μs. Such time delay can also be found with the proposed calibration method, as indicated in Figures 6b and 6c. As shown, the likely value of phase bias is ∼90°, which is equal to the time delay of ∼0.5 μs, very close to the value 0.444 μs, revealed from the FDI phase distributions. Other features of the optimal phase biases and σz are also similar to those of the cases 1–4. More experiments have been carried out with 2- and 4-bit coded pulses, and their calibration results are alike (not shown). In view of these, the RIM technique with coded pulses is indeed workable for our radar system.
3.2.3. The 4-μs Pulse Length: Case 6
 A longer pulse length was employed in the case 6. Sampling range resolution was 600 m but the frequency separation was a half of that in the cases 1–5. Figure 7 shows the results. In Figure 7a, the linear relationship between phase imbalance and frequency separation is not recognizable. By comparison, Figure 7b indicates clearly that the likely value of phase bias is around −60° and the values of the optimal σz gather at ∼520 m. The likely phase bias is different from those of the cases 1–5. The exact causes of such difference are unknown but could be associated with the radar parameters used. This reminds us that phase calibration is indeed necessary for the RIM experiments with different radar parameters. In addition, the optimal values of σz are much larger here than those of the cases 1–5, which is due to the longer pulse length (larger range resolution) employed. The dependence of optimal phase bias and σz on SNR is also apparent, as shown in Figure 7c, where we have used a curve to fit the relationship between σz and SNR.
3.2.4. Unevenly Spaced Frequencies: Case 7
 The case 7 is an experiment with unevenly spaced frequencies, which was carried out to validate the proposed calibration method further. Some of the integrated FDI phase distributions are displayed in Figure 8a. Without any surprise, the distributive aspects of Δf = 500 kHz and Δf = 250 kHz are similar to the case 1. However, the uses of Δf = 100 kHz and Δf = 50 kHz result in, respectively, five and ten peaks in the histograms. This makes sense because using Δf = 100 kHz (50 kHz) causes different FDI phase extents within five (ten) consecutive range-gate intervals; each range-gate interval occupies 72° (36°). As for the situation of Δf = 200 kHz, the FDI phases also dwell in different extents within five consecutive range-gate intervals, but the phase span of each range-gate interval is 144°. Owing to considerable overlap of the FDI phase distributions, it is difficult to distinguish the respective peaks in the integrated histogram of Δf = 200 kHz; however, we can find the expected peaks by inspecting the respective histograms of all range gates, just as the situation of Δf = 375 kHz shown in Figure 1. The above features also exist in the FDI phase distributions of other frequency pairs of this experiment. Inspecting all FDI phase distributions, we can find again the linear relationship between the phase imbalance and the frequency separation.
Figures 8b and 8c illustrate the calibration with the proposed method. As seen, the scenarios are the same as those of the cases 1–5. Figure 8d displays the SNR and the imaged power after calibration. As seen, the RIM method performs very well. A detailed examination of advantages and drawbacks of using unevenly spaced frequencies in practical observation is an interesting issue but is not a main purpose of this study. We leave it aside for the moment.
4. More Observations
 Two observations are briefly shown here to demonstrate further the performance of the RIM technique used with the OSWIN radar. Figure 9 displays a portion of the observations in the case 5. The original SNR is featureless (Figure 9a), but the imaged powers (Figure 9b) reveal some double-layer structures at the heights of ∼4.5 km and ∼5.5 km. Similar structure was also observed by Luce et al.  with the MU radar in Japan. Double-layer structure is though to be related to KHI effect [Browning and Watkins, 1970; Worthington and Thomas, 1997] and so is in connection with vertical wind shear. Figure 9c illustrates the horizontal wind field derived from the full-correlation analysis (FCA) with the radar echoes, and Figure 9d shows the profiles of mean zonal and meridional velocities. As seen, the meridional component possesses more observable variations as well as larger vertical wind shear. If these double-layer structures are indeed in connection with the vertical wind shear observed, the meridional wind may be more crucial. Unfortunately, there are no temperature measurements nearby during the radar experiment and so it is difficult to examine this case further with the Richardson number or other parameters. It is thus suggested that radiosonde or lidar observation can be carried out along with the RIM experiment in the future.
Figure 10 shows an observation of convective condition (case 8). As seen, the original SNR reveals the convective structures roughly, but the imaged powers show three convective cells more clearly around the times of the beginning, 1100 UT, and the end of the observations, respectively. However, only the middle one was recorded completely. Also plotted are vertical velocities, indicated by short-vertical line segments with black and red colors. It is interesting to see that stronger radar echoes appeared on the edges of the convective cells. This is not surprising because of the extremely different characteristics between the air parcels inside and outside the convective cells, which causes severe variations in refractive index around the boundaries of the convective cells. Moreover, the vertical wind field combined with the imaged powers provides us a clearer observation of the atmospheric motion. For example, we can see large updrafts (∼0.5 m/s maximum) in the middle of the convective cell appearing around the time of 1100 UT, and observe downdrafts just on both sides of the updrafts. It should be mentioned that this case was carried out for other purposes with the receivers aligned in zonal, so the complete horizontal wind is not available.
 There are two main purposes/contributions in this study: (1) validate the RIM technique implemented recently on the OSWIN VHF radar; (2) propose a simple but practical phase calibration method for the RIM processing, providing the following advantages/achievements:
 1. Reveal the likely phase bias/time delay of the echoes without difficulty so far for various pulse lengths, pulse shapes, mono and coded pulses, and receiver filter bandwidths.
 2. Can be applied to the RIM experiments with evenly and unevenly spaced frequencies.
 3. Determine the value of σz associated with the range weighting function. For practical application, σz can be adaptive to SNR in smoothing the imaged powers at gate boundaries, which is beneficial for locating the layer altitudes around gate boundaries.
 Two observations have been shown to demonstrate the robust performance of the RIM method used with the radar, that is, double-layer structures, and convective cells with strong radar echoes on the edges of the cells and severe change of vertical velocities in the cells. Being the first attempt of using the RIM method on the OSWIN VHF radar and focusing on the calibration purpose, however, this study has not carried out a detailed investigation into the RIM data for scientific messages. Further studies/experiments are considering. Especially, it is expected in the near future that the RIM technique, combined with coherent radar imaging (CRI) and lidar measurements, can be applied to the midlatitude mesosphere summer echoes (MSE), detected overhead by the OSWIN VHF radar, and the polar mesosphere summer echoes (PMSE), observed in the polar region. Studies of wave activities, microphysics, and dynamics of the MSE/PMSE can thus be improved with fine echo imaging (having the range resolution of several meters only) as well as from precise location of layer altitude.
 This work was supported by the National Science Council of ROC (Taiwan) under grant NSC95-2111-M-270-001-MY3. The OSWIN VHF radar is maintained by the Leibniz-Institut für Atmosphärenphysik (IAP) at Kühlungsborn, Germany. The authors would like to thank the three anonymous referees for their valuable comments/suggestions on this study.