Benefiting from the changeable antenna array size and flexible radar beam direction of the middle and upper (MU) atmosphere radar (34.85°N, 136.11°E), the effects of radar beam width and scatterer anisotropy on the performance of multiple-frequency range imaging (RIM) were examined in addition to numerical simulation. First, nine transmitter/receiver modes were employed to reveal that a wider radar beam gives a larger phase bias in the RIM processing. Based on this, layer positions and layer thicknesses were estimated from the imaged powers of various radar beam widths after corrections of phase bias and range-weighting function effect. Statistical examination showed that the imaged layer structure was thicker for a larger radar beam width and this feature became more evident at a higher altitude, thereby demonstrating the influence of radar beam width on the practical performance of RIM. Second, the scatterer anisotropy in the layer structure was examined by means of a vertical and three oblique radar beams (5°, 10°, and 15° north), which were transmitted in company with the RIM technique. The vertical beam observed some single-layer and double-layer structures that were not always detected by the oblique beams, indicating the existence of anisotropic scatterers in the layers. In addition, a comparison of layer positions between the vertical and oblique radar beams showed that anisotropic characteristics of the upper and lower layers of a double-layer structure can be different, demonstrating one more capability of RIM for investigating fine-scale features of the atmospheric layer structures.
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 According to the RIM algorithms, cross-correlation functions (CCFs) of various frequency pairs are required [Palmer et al., 1999; Luce et al., 2001, 2006]. There have been many theoretical works on the CCF of a frequency pair after the pioneering study made by Kudeki and Stitt , and some parameters like radar beam width, range-weighting function, anisotropy of irregularities (or aspect sensitivity), multiple layers, and so on, have been demonstrated to affect the magnitude and phase of the CCF to various degrees [e.g., Franke, 1990; Liu and Pan, 1993; Chu and Chen, 1995; Chen et al., 1997; Luce et al., 2000]. Consequently, we can expect that the outcome of RIM is also associated with these parameters. The range-weighting function has been considered for correcting the imaged power [Luce et al., 2001]; however, the effects of radar beam width and scatterer anisotropy on RIM were commonly neglected although they were also addressed in the literature. A major reason of this ignorance is that scatterer anisotropy (or aspect sensitivity) is thought to exist commonly in the atmospheric structures and it can narrow the effective width of the radar beam. Besides, it is difficult to verify these effects because a fixed radar beam width was used in previous observations. To master the application of RIM, these ignored effects on practical outcomes of RIM are worth examining. This is the main object of this study.
 Numerical simulation is the first and necessary step for our study. We made use of the two-frequency models and analytical expressions given by Liu and Pan  and Chen and Chu  for estimates of the CCF, although other simulation schemes also exist [e.g., Holdsworth and Reid, 1995; Muschinski et al., 1999]. All the parameters concerned have been included in the models employed. The influence of radar beam width, scatterer anisotropy, and layer altitude on the performance of RIM is shown in section 2.
 For demonstrating the effect of radar beam width on RIM in practical observations, various radar beam widths are required. This requirement is difficult to achieve for most of the MST radars worldwide. However, the middle and upper (MU) atmosphere radar, located in Shigaraki, Japan (34.85°N, 136.11°E), is one having such ability. As will be illustrated in section 3, twenty-five antenna groups in the whole array can be combined arbitrarily for transmission and reception, providing various radar beam widths for observations. It deserves a notice that in the literature, Hassenpflug et al.  employed the old MU radar system to investigate the consistency of horizontal correlation lengths of refractive index irregularities measured by different combinations of transmitter/receiver antennas (Tx/Rx mode hereafter), and with a method based on Full Correlation Analysis (FCA). They verified the effect of radar beam width on FCA. In this paper, we show the effect of radar beam width on RIM. To this end, it is necessary to calibrate the phase bias and the range-weighting function for different Tx/Rx modes first. We found that the calibrated phase bias varies with radar beam width positively. This is an interesting and valuable finding for the usage of RIM; the details are addressed in section 3.
 After calibrating the RIM data properly, a comparison of the layer positions and layer thicknesses obtained from different radar beam widths is possible, which is capable of indicating the effect of radar beam width on the outcome of RIM. This is discussed in section 4.
 In addition, we also examined the RIM data obtained from vertical and oblique radar beams to study anisotropic characteristics of the scatterers in the layer structures. In the literature, several theoretical and observational works for the two-frequency technique with oblique radar beams can be found [e.g., Palmer et al., 1992; Liu and Pan, 1993; Chu and Chen, 1995; Luce et al., 2000]. For RIM, however, observation is the practical way to verify the variation of layer structure with radar beam direction. In this study, we carried out an experiment for this purpose; the result is shown in section 5. Finally, the conclusions are given in section 6.
2. RIM Simulations for Radar Beam Width and Scatterer Anisotropy
 A description of the CCF of a frequency pair for multiple layers is given in Appendix A, which is an extract from Chen and Chu . Note that the analytical expression (A5) is valid only for a vertical radar beam, and, by assuming that the multiple layers in the radar volume are Gaussian-shaped and not correlative. Numerical simulation of RIM can be achieved by substituting the CCFs of various frequency pairs estimated with (A5) into the inversion algorithms such as the Capon and Fourier methods [Palmer et al., 1999]. To discuss simply and shortly the issues that are concerned, we did not consider the influence of signal-to-noise ratio (SNR). However, readers should bear in mind that lower SNR makes the imaging worse. In addition, quantitative variations in the imaged results are dependent on the radar and layer parameters given in the simulation; therefore, our numerical simulations only provide qualitative features of the performance of RIM.
 First, Figure 1a displays the imaged layer varying with radar beam width, where the results of Fourier and Capon methods are shown for comparison. The imaging range step was 1 m, and this value was employed for all numerical simulations and observational analysis in this study. In the work of Figure 1a, the radar beam was set to be vertical, and five frequencies (46.00, 46.25, 46.50, 46.75, and 47.00 MHz) were used with a 1-μs pulse length and a 3.6° half-power beam width. The layer was located at the central height (5.075 km) of the radar volume (i.e., zero position in ordinate), with a thickness of 5 m (i.e., the standard deviation of a Gaussian-shaped layer), and isotropic irregularities with 3-m correlation length were assumed in the layer. The use of 3-m correlation length is based on half-wavelength backscattering mechanism of radar echoes as the frequencies of 46–47 MHz are used in the simulation.
 In Figure 1a, it is apparent that the Fourier method yields a much thicker layer that ascends gradually with beam width. In comparison, the Capon method can retrieve the layer structure better although it outputs a layer structure that becomes more diffusive and has higher mean position at a larger beam width. In fact, a quantitative investigation can find that the Fourier-deduced layer thickness also increases with beam width but it does not vary as apparently as the Capon-deduced layer thickness (not shown). In view of this, the effect of radar beam width on the RIM-deduced layer is evident. Such effect may smear out fine layer structures, as illustrated in Figure 1b. As shown, the two layers, given with the same contributing weights and at positions of −12.5 m and 12.5 m, respectively, cannot be resolved by the Fourier method; even the Capon method is unable to separate the two layers at radar beam widths larger than ∼5°.
 In addition to radar beam width, the layer's height is also essential for the performance of RIM. This is demonstrated in Figures 1c and 1d, where a radar beam width of 3.6° was given in calculation. Again, the Capon method offers a better imaging. However, as seen in Figure 1c, the Capon-deduced layer structure gets wide apparently along the layer height, and in Figure 1d, the two layers cannot be identified for the layers located at heights above ∼20 km.
 Deterioration of layer imaging, originating from broader radar beam and higher layer altitude, can be mitigated when the irregularities become more anisotropic. Some simulation cases resulting from the Capon method are shown in Figure 2, where horizontal isotropy was assumed, and, vertical and horizontal correlation lengths were 3 m and 30 m, respectively. Except for correlation lengths, the radar and layer parameters used here are the same as those of Figure 1. Figures 2a (top) and 2b (top) illustrate that the layers can be retrieved clearly at the height of ∼5 km for any of the beam widths given, which improves greatly the results shown in Figures 1a (bottom) and 1b (bottom), respectively. The imaged layer structures, however, are deteriorated again at higher altitudes and only a very narrow beam is capable of separating the two layers, as illustrated in Figures 2a (middle) and 2a (bottom) and Figures 2b (middle) and 2b (bottom). A more detailed variation of the imaged layer structures with height is shown in Figure 2c. As seen, for a radar with beam width of 3.6° (Figure 2c, left), the two layers can be resolved up to the height of ∼30 km, which is better than the isotropic condition shown in Figure 1d; nevertheless, the situation becomes worse again for a radar with beam width of 7° (Figure 2c, right), where the workable altitude for resolving the two-layer structure is less than 20 km.
 Distribution of the imaged power simulated above will be somewhat different when the layer parameters and radar parameters are changed. For example, given a larger distance between the two layers and/or using seven or nine frequencies within a larger frequency range can improve the recognition of the two-layer structure in Figures 1 and 2. In the present MU radar system, however, only five frequencies between 46 MHz and 47 MHz are available for practical observations.
 In general, the features disclosed from the above simulations are within the prediction of the theoretical works addressed in the literature, and they can be explained as follows. A horizontal layer structure covers a larger range interval in the radar volume for a wider beam width and/or at a higher altitude. In the horizontal layer, the scatterers situated at off-zenith directions have the ranges mostly farther than the mean layer height, and they may contribute to the echoes received by a radar with finite beam width and consequently smear the range power distribution imaged by RIM, causing a larger layer thickness and a higher mean layer position. As the anisotropy of the scatterers increases, the echoes returning from larger off-zenith directions will decrease quickly. As a result, the RIM-deduced layer will be closer to the modeled layer. In any event, it may be troublesome for RIM to examine the fine-scale structures at altitudes like the mesosphere and ionosphere, unless the radar beam is extremely narrow or the radar scatterers are highly anisotropic.
3. Experimental Configurations and Calibration Results
3.1. Experimental Configurations
Figure 3 displays the antenna array configuration of the MU radar. The whole antenna array is partitioned into twenty-five antenna groups, and each antenna is equipped with a transmitter-receiver module. Software can combine the received signals of one to twenty five antenna groups for an output, and different combinations up to twenty five can be assigned simultaneously for reception, which diversifies the usage of the radar.
 Two RIM experiments were carried out for the purposes of this study. The first experiment employed three different combinations of antenna groups for transmission: full antenna array, seven and three antenna groups around array center, which are denoted as Txfull, Tx7, and Tx3, respectively. The three combinations of antenna groups yield different radar beam widths, and were operated alternately with an experimental cycle about 4.5 min. Moreover, the same three combinations of antenna groups were also used for reception (denoted by Rxfull, Rx7, and Rx3, respectively) during each transmitting mode. As a result, there are nine Tx/Rx modes in the experiment, with beam widths between 3.67° (for Txfull/Rxfull) and 9.74° (for Tx3/Rx3).
 The second RIM experiment employed the whole antenna array for transmission and reception. In addition to the vertical direction, the radar beam was also tilted northward to the zeniths of 5°, 10°, and 15°, respectively. The four beam directions were proceeded alternately with a cycle about 5 min.
 Note that in the aforementioned experiments, each transmitting mode or radar beam direction was conducted for at least one minute before switching to the next transmitting mode or radar beam direction, and the data set collected each time (1024 samplings) was used for an estimate of the CCF. We will suppose that for the radar beam width smaller than 10°, a data set with such long period meets the assumption of enough number of scatterers in the radar volume. In other words, the estimated CCF can reveal relevant statistics of the scatterers in the radar volume. In this study, all the investigations into the layer parameters are based on statistical views.
3.2. Calibrations for Various RIM Experiments
 Different radar parameters and antenna array configurations employed for RIM may result in different phase biases and range-weighting functions, so it is necessary to carry out calibrations for various RIM data first. We used the calibration approach proposed by Chen and Zecha , which can offer more information for correcting the imaged power: (1) optimal phase bias, (2) optimal (or effective) standard deviation (σz) of a Gaussian-shaped range-weighting function, and (3) a curve for yielding the value of σz at a specific SNR. See Appendix B for a brief description of the calibration approach used.
Figures 4 (left) and 4 (middle) show the histograms of the optimal phase biases and σz values for the nine Tx/Rx modes in the first experiment. In this work, the data below ∼10 km and with SNR > 0.125 (∼−9 dB) were used. It is obvious that each histogram of the optimal phase biases has a peak, whose location can be regarded as the likely phase bias (ϕbias) of RIM. An important feature can be observed: ϕbias varies with Tx/Rx mode. For the Txfull/Rxfull mode, ϕbias was about 125°; for the Tx3/Rx3 mode, however, ϕbias rose to about 170°. An evident tendency is that a larger phase bias corresponds to a broader radar beam. Notice that the radar beam widths of the Txfull/Rxfull, Tx7/Rx7, and Tx3/Rx3 modes given in Figures 4a, 4e, and 4i are the two-way half-power beam widths, while others are mean half-power beam widths; for example, the beam width of the Txfull/Rx7 mode was estimated as (3.67°+6.98°)/2≒5.33°.
 The dependence of phase bias of RIM on radar beam width, like the interpretation given for the simulation results, is owing to the horizontal layer structure covering a larger range interval in a wider radar beam width (refer to the final paragraph of section 2). A wider radar beam width, as demonstrated in Figures 1a and 1b, yields a RIM layer with higher position than the given one. To compensate this deviation of layer position, therefore, a larger range correction (i.e., a longer time delay) is necessary for a wider radar beam width in the calibration processing, which leads to a larger phase bias.
 It deserves a mention that in our previous works, the existence of likely phase bias was attributed to the time delay of the signal in the radar system. If such attribution is valid, the actual time delay of the signal in the radar system should be shorter slightly than the one indicated by a finite beam width, according to the dependence of phase bias on radar beam width.
 Regarding the distribution of the σz values, the peak location was about 70 m for all Tx/Rx modes, which represents approximately the theoretical standard deviation of the Gaussian range-weighting function. Note that the spread of σz toward larger values was owing to the data having lower SNR. Also, the minor peak (σz ≈ 150 m) of the Tx3/Rx3 mode arose from a large amount of low-SNR data. If a larger SNR threshold is given for the calibration, it can be seen that the distribution of σz (and phase bias) concentrate more around the main peak location (not shown).
Figure 4 (right) reveals a dependent relationship between σz and SNR. As demonstrated by Chen and Zecha , such a relationship comes from pursuing the optimal continuity of the imaged powers at the boundaries of the sampling range gates for widespread SNR in observations, and a fitting curve can be obtained for determining the approximate values of σz at different SNRs. Despite slight differences in the fitting parameters, the nine fitting curves are essentially similar. It should be mentioned here that the fitting curve extending to low SNR data is beneficial to give an effective value of σz in correcting the imaged powers around the boundaries of the sampling range gates when SNR is low.
 To verify the phase bias obtained above, we inspected further the phases of the two-frequency CCFs (termed frequency domain-interferometry phase hereafter and abbreviated as FDI phase). This is illustrated in Figure 5, where the distributions of FDI phases for the frequency pairs with frequency separations of 1000 kHz, 500 kHz, and 250 kHz are displayed. Only the two modes of Txfull/Rxfull and Tx3/Rx3, whose beam width difference is the largest among the nine Tx/Rx modes, are shown. A detailed discussion on the meaning of such distributions has been addressed in previous works [e.g., Chen, 2004]. In a word, the peak location of the distribution can indicate the bias of FDI phase. A careful inspection can find a slight discrepancy between these distributions: the differences between the mean peak locations of the two Tx/Rx modes were about 20° and 15°, respectively, for the frequency separations of 500 kHz and 250 kHz. For the frequency separation of 1000 kHz, however, it is difficult to determine the difference in mean peak location between the two Tx/Rx modes because of their shallower curvatures. In conclusion, the influence of radar beam width on the FDI phase indeed exists, which supports the calibration results shown in Figure 4.
 The same calibration process was also made for the second RIM experiment (one vertical beam and three oblique beams). The results showed that the likely phase biases and σz values of the three oblique beams were almost the same as those of the vertical beam although the oblique beams observed more data with lower SNR (not shown). In view of this, the likely phase bias and σz did not vary with radar beam direction in the second experiment.
4. Dependence of Layer Thickness and Layer Position on Radar Beam Width
 The numerical simulations in section 2 have shown the variations of layer thickness and layer position with both radar beam width and altitude. We verify this with some practical observations in this section.
Figure 6 displays the height-time intensity imaged by Capon's method for the Txfull/Rxfull mode. The calibration parameters shown in Figure 4 have been employed for correcting the imaged power. As displayed, there were many thin and stable layer structures above the height of ∼5 km. By contrast, broader and variable layers/turbulent structures were observed in the lower troposphere (<∼4 km). Note that the vertical strong patches above 7 km, occurring at ∼10.5 UT and ∼11.0 UT, were the radar echoes reflected from aircraft, which should be ignored. Quantitative comparisons of layer parameters (position and thickness) between various Tx/Rx modes can reveal the effect of radar beam width on RIM more clearly; some comparisons are exhibited in Figures 7–9, in which the layer position was estimated with the contour-based approach used by Chen et al. [2008a], and the layer thickness was defined as the standard deviation of a Gaussian curve that was used to fit the imaged powers around the estimated layer position. This approach can determine the positions and thicknesses of the multiple layers within the imaged range, but we considered only the single-layer situation for diminishing possible unreliability in comparison. Moreover, only the data with SNR > 0.5 (−3 dB) were adopted. We have examined the results obtained from data having higher SNR (e.g., > 0 dB and 6 dB) and found that the resultant characteristics were similar to Figures 7–9 except that fewer estimates were obtained. This suggests that the features revealed in Figures 7–9 are not caused mainly by low SNR. In other words, radar beam width indeed plays a role in the RIM-deduced layer parameters. The details are addressed as follows.
Figure 7a displays the differences in layer position between the three receiving modes used with the full-array transmission, in which the distribution curve in each range bin was self-normalized. Note that for a clearer inspection, each range bin in Figures 7–9 contains the data of two range gates in observations. Figure 7a shows that the position differences center on zero, and the profiling curves (thick solid lines), which indicate the difference between the numbers of positive and negative position differences (positive minus negative) and is presented as a percentage of the total number in each range bin (see upper abscissa for the scale), also vary around zero. In view of this, the layer positions estimated from various receiving modes were statistically close after corrections of phase bias and range weighting function effect. On the other hand, Figure 7b exhibits a difference in layer thickness, which shows that statistically, the layer thickness estimated from the receiving mode, Rx3, was the largest, and that estimated from the receiving mode, Rxfull, was the smallest. Such a feature is more evident at higher altitudes, as indicated by the profiling curve of number difference. These consequences point out clearly an observable role of radar beam width in the performance of RIM. The same examination was also made for the two transmitting modes of Tx7 and Tx3, and it gave similar results to the above (not presented).
 We may wonder about the consequence if a specific set of calibrated parameters is applied to other receiving modes. For example, using the calibrated parameters of Rxfull for the other two receiving modes, we obtained the products shown in Figure 8. In Figure 8a, it is seen that the Rx3 (Rx7) mode yields the layer having the altitude apparently higher than the Rx7 and Rxfull (Rxfull) modes; this is consistent with the simulation results of using various radar beam widths. On the other hand, the features seen in Figure 8b are similar to those of Figure 7b, except that the thickness differences shown in Figure 8b are smaller in statistics.
 In Figure 7, the differences in effective beam width between the three Tx/Rx modes are not very large because of the same transmitting beam (Txfull) used in observations. In Figure 9, the comparison of layer parameters between the three Tx/Rx modes, Txfull/Rxfull, Tx7/Rx7, and Tx3/Rx3, are shown. The two-way 6-dB beam widths of the three Tx/Rx modes are about 3.67°, 6.98°, and 9.74°, respectively. As seen in Figure 9a, the profiling curves show that the estimated positions are statistically close; however, the distributions of position differences are more divergent than in Figure 7a, especially worse in the lower troposphere. One cause of this phenomenon could be that the three Tx/Rx modes were conducted alternately and each mode operated for about one minute, the layer position might change greatly during several minutes, especially in the lower troposphere where more variable layers or turbulent structures were observed (see Figure 6). On the other hand, Figure 9b discloses again the influence of radar beam width on the estimated layer thickness. Comparing with Figure 7b, the thickness differences shown in Figure 9b are larger (notice the scales of the abscissa in Figures 7 and 9 are different), demonstrating the effect of radar beam width more clearly.
5. RIM of Oblique Radar Beams
 In this section, we show the RIM data collected from the vertical and three oblique beams in the second experiment. Figure 10 displays a portion of the imaged powers, where the ordinates of the maps of the oblique radar beams have been corrected to coincide with the vertical beam. As seen, the vertical beam observed the clearest layer structures in the range direction. We discuss the following three features:
 1. Two quasi double-layer structures were observed between 18 UT and 20 UT: the first one descended from ∼8.7 km to ∼8.4 km, and the second one was located between ∼7.8 km and ∼8.0 km. It can be seen that with the increase of tilt angle of the radar beam, the imaged layer structures become more diffusive and the double-layer structures cannot be seen clearly.
 2. The layer structure descending from ∼8.4 km to ∼8.0 km between 18 UT and 20 UT, as observed by the vertical beam, faded out in the observations of using oblique beams.
 3. The major structures seen by 10°- and 15°-oblique beams were similar.
 The above three features seem to be associated with anisotropic characteristic (or aspect sensitivity) of the scatterers in the layer structures. Estimate of layer position from the imaged power can aid in the investigation of these imaged structures, as shown in Figure 11, where the layer positions estimated from the vertical and oblique radar beams are compared.
 The double-layer structure is known to be the final stage of the dynamic process of Kelvin-Helmholtz instability [Browning and Watkins, 1970; Worthington and Thomas, 1997], and the model of turbulent layers presented by Woodman and Chu  can be one of the models to illustrate turbulent characteristics of a double-layer structure; that is, the observed two layers arise from sharp gradients of refractive index that are anisotropic, but between them the gradient of refractive index is much gentler and so the irregularities are more isotropic. As a result, the radar echoes from the region between the two layers are much weaker. One likely example is the double-layer structure descending from ∼8.7 km to ∼8.4 km in Figure 10, and a comparison of the positions estimated from different radar beam directions can be seen in the slanted rectangular box in Figure 11. As seen, although the imaged powers of the oblique radar beams are much weaker, the double layers can also be identified sometimes by the oblique radar beams. That indicates that the scatterer anisotropy in the two layers is not so extremely high that the layers are still observable for the three oblique radar beams. Note that only the data with SNR > 0.5 are displayed in Figure 11.
 The other double-layer structure, located between ∼7.8 km and ∼8.0 km, is somewhat different from the first one, as indicated in the region of the lower rectangular box in Figure 11. The upper layer can be determined by all radar beams although the layer positions obtained from the oblique radar beams are slightly higher than from the vertical radar beam. On the other hand, the lower layer cannot be detected by the oblique radar beams. In view of this, the upper layer may be close to isotropic, but the lower layer is extremely anisotropic. Nevertheless, we notice that the imaged powers of the lower layer obtained from the vertical radar beam are not very strong and clear, this can also lead to the result that the layer cannot be determined definitely.
 The second feature, a single layer descending from ∼8.4 km to ∼8.0 km between 18 UT and 20 UT, is demonstrated in the elliptical region of Figure 11. This layer was observed clearly by the vertical radar beam but not detected by the oblique radar beams, as indicated by the absence of layer position in the elliptical region. In view of this, this layer could be quite anisotropic. As for the third feature, similar structures were observed by the two oblique radar beams with off-zenith angles of 10° and 15°, it indicates that in this observation, the scatterers detected by a radar beam tilted to a zenith angle larger than ∼10° are almost isotropic.
 In addition to the aforementioned three features, the estimated layer thickness was found to increase as the zenith angle of the radar beam direction increased, as demonstrated by the distributions of layer thicknesses and their mean profiling curves shown in Figure 12. Such characteristic has been predicted by the two-frequency expressions for oblique radar beams [Liu and Pan, 1993]. The profiling curve in Figure 12 also displays an increase of layer thickness with the sampling height/range, which is more evident for a radar beam tilted to a larger zenith angle. The dependence of layer thickness on the zenith angle of the radar beam direction, and on height/range, can be explained as follows: a horizontal layer structure covers a larger range interval in the tilted radar volume, causing a wider distribution of the imaged powers in the range direction, and such a feature is more evident at a higher altitude where the radar volume is larger.
 To verify the observations shown in Figures 10–12, we have applied the analytical expressions derived by Liu and Pan  to numerical simulations of single and double-layer structures, with different anisotropic scatterers in the layers and for various oblique radar beams. However, the expressions given by Liu and Pan  were derived only for a single layer in the radar volume. To deal with multiple layers, we have to follow the assumption used by Chen and Chu ; that is, the layers in the radar volume are not correlative so that the respective CCFs of the layers can be summed. Consequently, the expression (9) given by Liu and Pan  can be employed. In use of the expression, the coordinates of transmitter and receiver were the same (x0 = 0), and the time lag (τ) was set equal to zero; besides, the radar parameters used in section 2 were employed again.
Figure 13 exhibits three simulation cases. The first case, shown in Figure 13a, simulates a highly anisotropic layer located at the central height (5.075 km) of the range gate, with the vertical and horizontal correlation lengths of 3 m (lz) and 15 m (lt), respectively. We can see that as the zenith angle of the radar beam direction increases, the imaged layer structure broadens and its location descends. However, the echo power obtained in the simulation drops very quickly along the zenith angle of the radar beam direction (see the solid curve and refer to the right ordinate). For example, for a radar beam tilted to the zenith angle of 5°, the simulated echo power is about −20 dB relative to the vertical radar beam (0 dB), indicating that the layer can hardly be observed by a largely tilted radar beam. This case can illustrate the layer located in the elliptical region of Figure 11.
 The second simulation is shown in Figure 13b, where two layers with slightly anisotropic scatterers (lz = 3 m and lt = 4.5 m) were given at positions of −30 m and 30 m. As shown, the two imaged layers are more diffusive and wider for a radar beam tilted to a larger zenith angle. The simulated echo power also decreases with the zenith angle of the radar beam direction but does not reduce to a level as low as for the first case. Therefore, the two layers could be detected by off-vertical radar beams but with lower intensity. This result may describe, to some degree, the feature of the layers observed in the slanted rectangular region of Figure 11.
Figure 13c illustrates the third simulation, where two layers with different degrees of scatterer anisotropy were given. It is shown clearly that the highly anisotropic layer (lz = 3 m and lt = 15 m), located at the lower position, fades out rapidly along the zenith angle of the radar beam direction. By contrast, the isotropic layer (lz = lt = 3 m), located at the higher position, dominates gradually the imaged layer structure and acquires a broader thickness when the radar beam is tilted to a larger zenith angle, although the simulated echo power decreases accordingly. This picture may give a description of the layers observed in the rectangular region of Figure 11.
 The above three simulations for oblique radar beams yield some qualitative evidences for the observations. Real atmospheric conditions, however, should be more complicated and noise is also crucial to imaging techniques. More simulations can be achieved but showing them here is beyond the scope of this study.
 In this section, we have first applied the product of RIM to reveal the anisotropic characteristic of layer structure. Such manner of investigation is different from the existing methods; e.g., a comparison of echo powers between vertical and oblique radar beams that can only show an averaged outcome within a range-gate interval. Considering that isotropic and anisotropic layers/irregularities can exist closely or simultaneously within the same range-gate interval, the comparison of echo powers between differently oblique radar beams may not be enough for such case.
 In this study, the effects of radar beam width and scatterer anisotropy on the performance of RIM were examined by numerical simulation and from practical observation. Observations with the MU radar in Japan were carried out by transmitting radar waves and receiving the echoes with different radar beam widths and beam directions. It is shown that the phase bias in the RIM processing varies with beam width, that is, a larger beam width leads to a larger phase bias. Based on this, we have estimated the layer position and layer thickness more properly for various radar beams that were transmitted by the same radar system. Consequently, the effects of radar beam width on the performance of RIM can be investigated in addition to numerical simulation. Statistical examination demonstrated that a broader radar beam gives a broader layer thickness, which is more evident at higher altitudes. Moreover, an oblique radar beam also yields a broader layer thickness. All these observational results are consistent with previous theoretical predictions.
 This study then demonstrated an application of RIM to the scatterer anisotropy or aspect sensitivity characteristic of the layer structure by means of vertical and oblique radar beams. Comparisons of the imaged powers and layer positions between vertical and oblique radar beams have disclosed different levels of scatterer anisotropy in the imaged layer structures that existed closely or simultaneously in a range gate. Such a fine-scale examination is different from the existing methods having a range-gate resolution, and can be carried out for more observations in the future.
Appendix A: Two-Frequency Model and Expressions
 Assuming that there are N layers in the radar volume and the distribution of the echoing scatterers in each layer is Gaussian function, the output voltages of a receiver at two different transmitting frequencies can be written as
where vij is the output voltage of a receiver for a single layer in the radar volume, and the first and second subscripts in vij denote the radar frequency and the layer, respectively. Variables w1, w2, …and wn are the weights of the layers contributing to the radar returns, which are related to the intensities of the refractive index fluctuations in the layers. With (A1), a two-frequency cross-correlation function can be expressed as
 Assuming further that the radar echoes generated from the nth and the mth layers are not correlative, then the term 〈v1nv*2m〉 in (A2) can be eliminated. To obtain an analytical expression from (A2), the following form of v can be used [Doviak and Zrnic', 1984; Franke, 1990]:
where W(r) is the range weighting function, fθ2(r) is the two-way beam pattern function. The Gaussian term, exp[−(z–zl)2/2σl2], is the distribution function of the scatterers in the layer, in which zl and σl represent layer position and layer thickness, respectively, and z is the vertical component of the spatial vector r. Δn is the refractive index fluctuation responsible for the radar echo, k is the wave number of the transmitted radar wave, rs is the range between Δn and the radar receiver, C is a constant related to radar parameters. Note that the origin of the coordinate system in (A3) is the center of the radar volume. If the radar beam is horizontally symmetric and pointed vertically, and the matched filter is used for reception, W(r) and fθ2(r) can be expressed approximately as [Franke, 1990]
where σz = 0.35cτ/2, c is the light speed and τ is the pulse length, ρ2 = x2 + y2, x and y are the two horizontal components of the vector r, σt = 21/2hθ6/3.33, θ6 is the 6-dB angular width of the function fθ2(ρ) and h is the central height of the radar volume. With (A3) and (A4), (A2) becomes
where zln and σln represent the position and thickness of the nth layer, respectively, and
where σx and σy are the second moments of the radar beam in x and y directions, respectively. lx, ly, and lz are, respectively, the spatial correlation lengths of the refractive irregularities in x, y, and z directions, α and β are the coefficients related to the wave number power spectrum of refractive irregularities (α = 1 and β = 2 for the three-dimensional Gaussian spectrum; α = 2.5lz−1.4 and β = 1.5 for the approximation form of the three-dimensional −11/3 power law spectrum [Chen et al., 1997]). (A5) can be normalized by 〈|V1|2〉1/2 〈|V2|2〉1/2 to obtain the coherence function. In the literature, Luce et al.  also carried out a derivation like the above. Moreover, the expression (A5) can also be obtained by simplifying the general cross-correlation function given by Liu and Pan .
Appendix B: RIM Calibration Approach
 Two parameters can be obtained from the calibration of RIM proposed by Chen and Zecha : range (time) delay of the radar signal and SNR-dependent standard deviation (σz) of the Gaussian range-weighting function. The following estimator has been used:
where P1i and P2i are two sets of the imaged powers estimated around the boundary between two adjacent range gates, in the same height interval (namely, the upper boundary of the lower gate and the lower boundary of the upper gate). Giving various range (time) delays and σz values in calculating the two sets of imaged powers will result in different values of ERR, but there exits a pair of range (time) delay and σz that make ERR smallest, which are the optimal values enabling the two sets of imaged powers closest. Under such condition, it is supposed that the imaged powers of two adjacent range gates are mostly continuous at the common edge. For the reason of convenience, the estimated range (time) delay is transformed into a phase angle with the scale of a range gate (or pulse length) to 360°, which is termed “optimal phase bias” in the text.
 It deserves a mention that σz is theoretically not related to SNR but dependent on the pulse shape and filter bandwidth employed. The SNR-dependent σz is due to the SNR-dependent performance of inversion algorithms such as Capon's method. To mitigate discrepancy in the effects of SNR on the imaged powers of two adjacent range gates, and, to acquire the optimal continuity between the imaged powers at range gate boundary, we can use a SNR-dependent σz. Although the upper and lower parts of a range gate usually suffer slightly different values of σz according to the calibration results, it has been demonstrated to be not a noticeable drawback of the calibration approach.
 This work was supported by the National Science Council of ROC (Taiwan) through grants NSC98-2111-M-270-001 and NSC99-2111-M-270-001-MY2, and also supported by the International Collaborative Research Program of MU radar (20MU-A39). The MU radar is operated by the Research Institute for Sustainable Humanosphere, Kyoto University, Japan. We would like to thank Mamoru Yamamoto especially for encouraging the radar experiment.