The frequency domain interferometry (FDI) technique uses two or more frequencies to measure the positions and thicknesses of the atmospheric thin layers embedded in the radar volume, in which the cross-correlation analyses of the radar echoes for the pairs of carrier frequencies are performed and the resultant amplitudes and phases (FDI phase) are both employed. However, in light of the possibility that the characteristics of radar system, mean refractivity gradient, and other factors that would significantly affect the FDI phase, calibration of the FDI phase is required to improve the measurement. In this study we employed three methods in measuring the phase bias in the FDI observation using the Chung-Li VHF radar; namely, (1) histogram of the FDI phases, (2) relationship between echo power and FDI phase, and (3) the FDI phase of aircraft. Both methods 1 and 2 are based on the range weighting effect on the radar echoes returned from the atmospheric scatterers; however, the first produced smaller FDI phase bias than the second. To examine such discrepancy in the results of methods 1 and 2, method 3 was exploited and provided more consistent values of phase biases with those of method 2. Considering that the radar echoes reflected from aircrafts are not related to uncertain conditions of the atmosphere such as mean reflectivity gradients and statistical characteristics, the results of methods 2 and 3 may be more reliable. Besides, the first two methods demonstrated that the FDI phase bias was quasi-linearly dependent on the separation of frequency pair, which not only consolidates the existence of the FDI phase bias but also indicates that a systematic phase compensation for the FDI analysis is possible. For example, considering 0.1-, 0.4-, and 0.8-μs time delays of signals for the returns of 1-, 2-, and 4-μs pulse lengths, respectively, the FDI phase biases can be removed effectively. Same methods and procedures can be applied to other radar systems.
 Mesosphere-stratosphere-troposphere (MST) radar is capable of detecting the atmospheric thin layers embedded in the radar volume by making use of multiple-frequency technique (Frequency Domain Interferometry/FDI technique thereafter). The simplest version of the FDI technique is using a pair of carrier frequencies transmitted alternately in consecutive radar pulses. After performing the normalized cross-correlation analysis of the radar echoes for the pair of carrier frequencies, the parameters of amplitude and phase (FDI phase thereafter) are obtained and then employed to estimate the position and thickness of a single layer in accordance with analytical expressions [Kudeki and Stitt, 1987; Franke, 1990; Liu and Pan, 1993]. For the condition of multiple layers in the radar volume, however, more frequencies are required. The FDI techniques using more than two frequencies also utilize the parameter pair of amplitude and phase, but different approaches such as Fourier, Capon, Maximum entropy, and so forth, are applied [e.g., Palmer et al., 1999; Luce et al., 2001].
 It is always suspected, however, that some phase terms resulted from radar system or other factors may exist in the FDI phase and vary with radar parameters, causing a bias in the FDI phase and thus leading to imperfect measurement of the atmospheric thin layer. In view of this, phase calibration is needed to improve the application of the FDI technique. The issue of phase bias in the FDI experiment has been noted and treated with different ways. Kilburn et al.  referred to the peak location of the histogram of FDI phase to estimate the phase bias in two-frequency observation. Similar method was also employed by Brown and Fraser  and Luce et al.  in their multiple–frequency observations. However, Chen and Chu  suggested that the relationship between echo power and FDI layer position can also be applied to reveal the FDI phase bias under the situation of single layer in the radar volume. In addition, Palmer et al.  and Chilson et al.  considered the initial phases of the carrier frequencies and measured them from the signals that were leaked from the transmitted signals back to the receiver by an ultrasonic delay line.
 Referring to the approaches mentioned above, we applied three methods in this study to measure the FDI phase bias for the Chung-Li VHF radar. They are, respectively, histogram of the FDI phase, relationship between echo power and FDI phase, and the FDI phase of aircraft. Note that the first two methods use the radar returns from the atmospheric scatterers and are based on the weighted radar volume formed by range weighting function and radar beam pattern both. Range weighting function results from the receiver filtering process and can be approximated by a Gaussian form when a specific receiver filter matched to the transmitted radar pulse is employed [Doviak and Zrnic, 1984]. However, the radar beam can be modeled by a two-dimensional Gaussian function that forms the size of the radar volume perpendicular to the direction of range. Consequently, the maximum weight will be at the center of the radar volume, and the echoes scattered/reflected from the targets situated at the place closer to the center of the radar volume dominate the returned echoes. This will cause the measured FDI phase to deflect to the phase angle corresponding to the center of the radar volume. It is thus expected to obtain a histogram of the FDI phase having the peak at the phase angle predicted for the center of the radar volume, provided there is no phase bias in the radar returns. Also, the echo power should have a maximum value at the peak location of the FDI phase histogram.
 In literature, Kilburn et al.  used the restriction of low FDI amplitude to construct the histogram of the FDI phase and pointed that the FDI phase bias may vary from case to case. In this study, however, the criterion of SNR was used. We indeed found systematic bias in the FDI phase. Moreover, phase calibration using the condition of a single layer in the radar volume, which was implemented by Chen and Chu , is one application of echo power. We will demonstrate that the condition of single layer can be released in using the relationship between echo power and FDI phase. As for the radar echoes reflected from aircrafts, they have been employed recently in measuring the system phase differences between receiving channels [Chen et al., 2002]. This application is based on the continuing movement of the aircraft aloft that results in continuous variation in the FDI phase with time. Chen et al.  have noted the phase bias in their FDI observations. We examined this problem in more detail in this study.
 This paper is organized as follows: the issue of FDI phase bias is described briefly in section 2. Typical observations and calibration results are stated in section 3. Section 4 is discussion, in which the experimental results conducted at different frequency pairs and radar parameters are presented. A systematic bias in the FDI phase can be seen from these experiments. The effect of mean refractivity gradient on the phase bias is also examined. Moreover, the aircraft echoes are applying to examining the issue of initial phase. Conclusions are offered in section 5.
2. FDI Phase Bias
 For simplicity, the radar echoes for two (or more) different transmitting frequencies can be written as
where Ai is the amplitude of radar return of the ith frequency, ki is the wave number corresponding to the ith frequency, ri is the distance between target and radar receiver, ϕi is a phase term governed by a number of factors such as the characteristics of the receiving channel, temperature effect on the radar system, transmitting frequency, aging of radar components, and so forth. After performing the cross-correlation analysis, 〈V1V2*〉, for the two radar returns in equation (1), we obtain a phase term expressed as
In normal FDI experiment the radar pulses with different carrier frequencies are transmitted almost at the same time so that the targets can be assumed to be constant in characteristics and their ranges are also treated as invariant during the transmission/reception of a frequency pair (or set). Therefore r1 = r2 = r, and equation (2) becomes
where θ is the so-called FDI phase in this paper. As mentioned above, many factors can contribute to the phase term ϕi, and thus ϕ1 − ϕ2. However, since same receiver is used in the FDI experiment, the factors of different receiving channels, temperature effect on the radar system, and aging of radar components can be ignored. One of the potential contributions to the term of ϕ1 − ϕ2 is the system response to different transmitting frequencies, for example, different initial phases. If ϕ1 − ϕ2 is known, r can be estimated accurately using the observed FDI phase and equation (3), in which k1 and k2 are given a priori. In practice, the range of the sampling gate has to be estimated a priori from the sampling delay time. Reversely, we can predict the FDI phase for a given r. It should be emphasized again that such work can be achieved only when the exact value ϕ1 − ϕ2 is known and the range of the sampling gate, which is calculated from the sampling delay time, is correct. Otherwise, the predicted FDI phase will be different from the observed one. To measure the difference between the observed and predicted FDI phases and to examine the potential factors of such difference are thus the main purposes of this study.
 The FDI coherence function is obtained from the normalized cross-correlation analysis defined as
where V1 and V2 are, respectively, two radar returns at different carrier frequencies, the angle brackets stand for ensemble average over a period of time, the asterisk denotes complex conjugate, ∣S12∣ and ϕ are the FDI amplitude and FDI phase, respectively. The radar parameters for the FDI experiments presented in this section were selected to result in 2π phase extent in a range bin. In addition, the matched filter was used to obtain the maximum signal-to-noise ratio of the radar returns. It should be mentioned that the two carrier frequencies were produced from two frequency synthesizers, respectively.
 In the methods of echo power and FDI phase histogram, the data with signal-to-noise ratio (SNR) larger than 2 were taken. The SNR is defined as “received radar power minus estimated noise power and then over the estimated noise power,” in which the numerator is the received echo power. The use of the SNR criterion is different from that considered by Kilburn et al. . In their study they considered that the echoes with the FDI amplitude smaller than some value, say, 0.3, were generated through volume scattering, and these echoes were adopted to construct the histogram of the FDI phases. Privately, we think that a threshold of SNR should also be used in the study of Kilburn et al.  although they did not indicate it in their paper, as referring to their high–time intensity plots. With the only criterion of SNR, we will show the restriction of small FDI amplitude can be released. In literature, Brown and Fraser  appeared to employ the criterion of the SNR, as we use here.
3.1. Typical Results
3.1.1. Histogram of the FDI Phases
Figure 1a is an example showing the histograms of the FDI phases for five consecutive sampling gates. The radar parameters employed for this experiment were set as follows. The pulse length was 2 μs, a pair of frequencies 51.75 and 52.25 MHz were transmitted alternately, the sampling time was 0.12 s, 512 raw data points were taken for the ensemble average in equation (4), and consequently the duration for an estimation of FDI coherence function was 61.44 s. Total period of observation was 67.5 hours. In Figure 1a, the phase bin of the histogram is five degrees. The FDI phase bias is estimated as follows. The histogram is shifted to left by five degrees each time, and negative FDI phase bins are moved to right by adding 360°. Then the mean value (first moment) of the FDI phase for the shifted histogram is calculated and the difference between the mean value and 180° is also estimated. Finally, the shift with the smallest difference between the mean value and 180° indicates the FDI phase bias. Note that, on the basis of the radar parameters employed in this experiment, the phase angle of 180° is the expected FDI phase corresponding to the central height of the sampling gate.
 One can observe from the samples shown in Figure 1a that the mean peak locations of the histograms are not in agreement with the expected one (180°) but bias systematically to the right by about 40° to 80°. This strongly suggests that some kind of systematic bias exists in the observed FDI phases. However, to obtain a more confident value of FDI phase bias, a histogram superposed from the histograms of several range gates is preferable. Such a superposed histogram is expected to meet the condition of volume scattering and so exhibit the Gaussian range weighting effect. Some results are displayed in Figure 1b. These experiments were carried out with the radar parameters listed in Table 1 for tens of hours, and the data within the height interval between 2.25 and 8.7 km were taken for the histograms. As seen, the mean peak locations of the histograms shift right by about 15°, 45°, and 40° from 180° (the expected phase angle) for the three pulse lengths, respectively. These results indicate that the FDI phase bias may vary with pulse length and frequency pair. Whether these phase biases are representative, more observations are in need of examination.
Table 1. Radar Parameters Employed in the Experiments Shown in Figures 1 and 2
Pulse Length, μs
Sampling time, s
Ensemble average time, s
Range interval for data, km
Transmitting frequency, MHz
Phase range in each range bin
3.1.2. Relationship Between Echo Power and FDI Phase
 To find the relationship between echo power and FDI phase, the echo powers in the histogram of the FDI phase can be averaged for respective phase bins. Note that the echo power defined here is the range-corrected one, namely, the one that is obtained by multiplying the echo power by the square of the central range of the sampling gate. Moreover, the echoes reflected from aircrafts have to be removed from the radar returns because they are usually much larger than those scattered/reflected from the atmospheric scatterers and can contaminate seriously the mean of echo power. Because the Chung-Li radar site is close to domestic and international airways, there are averagely several aircrafts flying through the viewing region of the radar within one hour. In view of this, we performed the following procedure to remove the aircraft echoes as completely as possible: several radar returns with extraordinarily large intensity compared to other returns in the whole ward were excluded first; second, for each phase bin the echo powers that are larger than 1.5 times of the standard deviation of the echo powers were also removed. As an example, Figure 2a shows the distributions of mean echo power over the FDI phase for the sampling gates shown in Figure 1a. As shown, the mean peak locations of the histograms deviate from the expected one (180°) by about 65° to 85°. This feature is very similar to that shown in Figure 1a except that the curves in Figure 2a are more spiked than those in Figure 1a. Presumably, the spikes in Figure 2a might be attributed to the remnant echoes of aircrafts, interference signals, or real strong returns from the atmospheric scatterers. Irrespective of these spikes, the distribution of echo power over the FDI phase is a quasi-Gaussian form, and this distribution is thought to be due to the Gaussian range weighting function.
 As has been done in Figure 1b, Figure 2b displays the diagrams superposed from the diagrams of several range gates for different pulse lengths and frequency pairs, which can provide a more representative FDI phase bias. As shown, the mean peak locations of the FDI phases deviate from the expected one by 35°, 80°, and 85°, respectively, for the pulse lengths of 1, 2, and 4 μs. Obviously, a similar feature is seen in Figures 1b and 2b; however, FDI phase histogram provides smaller phase bias than echo power-FDI phase relationship. To examine this inconsistency, we exploited one more method using the echoes of aircraft.
3.1.3. FDI Phase of Aircraft Echoes
 Both the FDI phase histogram and echo power-FDI phase relationship indicate a positive bias in the FDI phase; however, their biases are not in good agreement. To examine this inconsistency, we exploited the echoes of aircrafts to assist in the calibration.
 In applying the echoes of aircrafts, the coherent integration should be as few as possible because of fast movement of the aircraft. In view of this, no coherent integration was made in the observation. An example is shown in Figure 3a. The radar parameters employed for this example were that: pulse length, 2 μs; transmitting frequencies, 51.75 and 52.25 MHz; IPP, 2400 μs. According to the radar parameters used, the extent of the FDI phase for a range bin was 2π and the sampling time was 0.0048 s due to alternately transmitted radar pulses with two different carrier frequencies. Thirty points of raw data were averaged to compute the FDI coherence function, corresponding to a time resolution of 0.144 s in calculation. In Figure 3a (left two panels), the upper panel is the original FDI phase without phase compensation. As shown, striking phase traces, which are continuous in range gates but discontinuous at gate boundaries, are obviously seen. These striking phase traces indicate the movement of the aircraft in the range gates, in which the nearest range of the aircraft is about 11.25 km. Note the duplicated phase traces occurring in the upper and/or lower gates of the primary phase trace, which can be attributed to the range weighting function. Namely, the range weighting function causes two adjacent sampling volumes to overlap partly so that the target in the overlapped region may generate the echoes for two adjacent range samples. More important feature is the salient discontinuities of the primary phase trace appeared at gate boundaries. Apparently, such discontinuities resulted from some kind of systematic bias in the FDI phase. We found that such discontinuities can be almost taken away if a phase angle of −80° is compensated to the original FDI phases, as shown in the lower panel of Figure 3a. Referring to the middle panel of Figure 2b, the phase angle of −80° can also shift the distribution to center on the phase angle of 180°.
 Another example is shown in Figure 3b (right two panels), in which the pulse length of 4 μs and the frequency pair of 51.9 and 52.15 MHz were used. As displayed in the upper panel, the primary phase traces are not continuous at gate boundaries, which is very similar to that shown in Figure 3a. These discontinuities can be almost removed by compensating a phase angle of −70° to the original FDI phases. Again, the distribution in the right panel of Figure 2b, of which pulse length and frequency pair are the same to those used for Figure 3b, becomes almost symmetric to 180° when the phase angle of −70° is compensated to the original FDI phases.
 We also examined the data of 1-μs pulse length and decided the results unreliable because of abnormal behavior of FDI phase. Such abnormal behavior is likely due to the near sizes of the radar volume and the aircraft, or, other unknown factors.
3.2. More Observations
 We examined a number of FDI experiments conducted at the Chung-Li VHF radar in the years of 1998, 1999, and 2000 to find the systematic FDI phase bias using the first two methods (FDI phase histogram and echo power-FDI phase relationship); the results were summarized in Table 2. As shown, for respective pulse lengths the phase biases estimated from the same method are consistent with each other. Moreover, the phase biases for the pulse lengths of 2 and 4 μs are close if the same method is used; however, they are greater than those of 1-μs pulse length. Observable discrepancy exists between the phase biases estimated from FDI phase histogram and echo power-FDI phase relationship. Such a discrepancy is a puzzle to the authors at the moment.
Table 2. FDI Phase Biases Estimated From the FDI Phase Histogram and Echo Power-FDI Phase Relationshipa
Pulse Length 1 μs; FDI Phase, Echo Power Observational Period
Pulse Length 2 μs; FDI Phase, Echo Power Observational Period
Pulse Length 4 μs; FDI Phase, Echo Power Observational Period
The angles in the parentheses are the standard deviations of the FDI phases. The numbers indicated in the parentheses just after the observational periods are total hours. Refer to the Table 1 for the radar parameters used.
15° (57°), 35° (92°) 30 March 2000, 1700 LT to 31 March 2000, 0800 LT (15)
35° (81°), 50° (98°) 27 March 2000, 1230 LT to 29 March 2000, 1930 LT (55)
50° (92°), 75° (94°) 14 February 1998, 1400 LT to 17 February 1998, 1000 LT (68)
15° (83°), 35° (90°) 2 December 2000, 1000 LT to 4 December 2000, 1700 LT (55)
50° (91°), 85° (90°) 19 September 2000, 1330 LT to 11 October 2000, 1630 LT (60)
35° (88°), 85° (86°) 24 November 1999, 0900 LT to 30 November 1999, 1700 LT (56)
20° (80°), 30° (93°) 8 December 2000, 1000 LT to 10 December 2000, 1700 LT (55)
45° (95°), 75° (89°) 23 October 2000, 1000 LT to 24 October 2000, 1200 LT (26)
35° (100°), 70° (88°) 10 January 2000, 0950 LT to 14 January 2000, 1900 LT (19)
45° (93°), 75° (79°) 5 December 2000, 1200 LT to 6 December 2000, 1000 LT (22)
40° (90°), 85° (88°) 20 September 2000, 1330 LT to 12 October 2000, 1630 LT (50)
45° (82°), 80° (87°) 19 December 2000, 1100 LT to 22 December 2000, 0600 LT (67)
50° (93°), 85° (90°) 2 October 2000, 1710 LT to 13 October 2000, 1015 LT (93)
 In addition to the Table 2, tens of cases of the aircrafts observed with pulse lengths of 2 and 4 μs were investigated. As was done in Figure 3, the finding of the FDI phase bias is subjective. We gave different compensatory phases (an interval of five degrees) to the original FDI phases, and picked out the one leading to smoother variation in the FDI phase trace at gate boundaries. Since the determination is subjective, an uncertainty of about ten degrees of phase angle is possible. In this way the compensatory phase angles were found to be mostly between 50° to 90°, which are more consistent with those estimated with the echo power-FDI phase relationship.
 According to the experimental results shown in the previous section, the FDI phase bias was tens of degrees. However, other radar parameters may result in different FDI phase biases. For example, when a pair of frequencies are selected to produce π or π/2 FDI phase difference in a range bin, the distribution of the FDI phase will be very different from that of 2π-FDI phase difference in a range bin. Figure 4 shows two examples obtained with different frequency separations, in which the FDI data were taken with 2-μs pulse length for tens of hours and the frequency pairs were, respectively, 51.9 and 52.15 MHz, and 52 and 52.125 MHz for Figures 4a and 4b. Note that the two frequency pairs result in π and π/2 FDI phase differences in a range bin, respectively. As shown, double-peak and four-peak diagrams appear in Figures 4a and 4b, respectively. The double-peak feature is due to the fact that the FDI phases in odd and even range gates are situated in the phase ranges of 0-π and π-2π, respectively. Similarly, the four-peak feature is due to the different phase ranges of 0-π/2, π/2-π, π-3π/2, and 3π/2-2π for four consecutive range gates. In Figure 4a, the mean peak locations of the two phase groups are 120° and 295° respectively for the upper diagram, and are 130° and 300° respectively for the lower diagram. Because the unbiased peaks in Figure 4a are expected to locate at 90° and 270°, respectively, the biases for the two phase groups are 30° and 25° for the upper panel, but 40° and 30° for the lower panel. In analogous manner, the FDI phase biases in Figure 4b are around 15° for upper and lower panels both.
 A remarkable feature in Figure 4 is the quasi-linear increase of the FDI phase bias with frequency separation. To validate this, more observations were carried out with 1-, 2-, and 4-μs pulse lengths, and the frequency pairs producing π/2 and π FDI phase angles in a range bin were employed. Figure 5 is the summing-up diagram showing the means and standard deviations of the FDI phase biases. As indicated, for a specific pulse length the relationship between the FDI phase bias and frequency separation is nearly linear, especially the values estimated from the echo power-FDI phase relationship. Accordingly, it is strongly suggested that some systematic mechanisms are responsible for this linear feature.
 A number of factors may likely be responsible for the bias in the FDI phase, including the group delay of the signal in the media, the time delay resulted from the data processing in radar system, the initial phases of the transmitted frequencies, mean reflectivity gradient [e.g., Johnston et al., 2002], and so on. These potential factors are discussed in the following. First, mean reflectivity gradient gives additional weighting effect on the radar volume and can cause positive or negative bias in the FDI phase. For example, a negative reflectivity gradient combined with the range weighting function results in a lower range center for the radar volume [Johnston et al., 2002, Figure 8], leading to a negative shift in the peak of the FDI phase histogram for the observations displayed in Figures 1 and 2 (also in other experiments conducted for this study). Figure 6 discloses such effect. Figure 6a is the height-time intensity plot for the observation that has been discussed in Figure 1a. Note that the weak radar returns below ∼3 km are not reliable due to incomplete recovery of the instrument; however, radar returns are plentiful between 3 and 6 km. Figure 6b displays the histograms of echo powers by range gates as well as the profile of mean echo power (range compensated), in which the profile of mean echo power shows small reflectivity gradients (<5 dB/km). Figure 6c exhibits the profiles of the FDI phase offset (bias), standard deviation of the FDI phase, and the FDI coherence (the amplitude of equation (4)). As shown, the phase offsets derived from both methods (FDI phase histogram and echo power-FDI phase relationship) are mostly positive although several are close to zero. Moreover, both profiles of phase offsets vary coherently between the heights of 3.3 and 6.6 km, irrespective of their differences in value. The FDI coherences are around 0.5 and they look not to correlate with the phase offsets. On the contrary, echo power and phase offset are well correlated between the heights of 3.3 and 6.3 km, especially around the height of 5 km. We observed that the phase offsets decrease when the echo powers reduce with altitude, and vice versa. In view of this, mean reflectivity gradient seems to contribute a part of the FDI phase bias. If average of the phase biases is performed for all heights, however, this part of contribution may be diminished due to positive and negative shifts at different heights. Besides, we found two situations that are not in favor of the main role of mean reflectivity gradient in the FDI phase bias. First, the FDI phase biases for 2- and 4-μs pulse lengths are almost the same order, which does not adhere to the rule of pulse length-dependent bias on the range center. Note especially the case 3 of 2 μs and case 4 of 4 μs in Table 2 that were observed during almost same period of time, but their FDI phase biases are close. Second, we noted that mean reflectivity gradient has no effect on point target such as an aircraft [Johnston et al., 2002, Figure 4]. In view of the salient discontinuities of the FDI phases at gate boundaries for the aircraft echoes (Figure 3), the FDI phase bias that was not brought on through mean reflectivity gradient indeed appeared.
 In addition to mean refractivity gradient, other conditions in the atmosphere may also control the variation in the profile of phase offset. Note that we acquire the phase bias under the condition of volume scattering, which can exhibit the Gaussian range weighting effect. In a range bins, however, layer structure or local patches may deflect the mean FDI phase to some value for that the condition of volume scattering is not satisfied sufficiently. Accordingly, a changeful profile of the phase bias may be seen, as Figure 6c shows. In view of this, an average of the FDI phase histograms or echo power-FDI phase relationships for all range gates is more likely to meet the required condition and provide a more reliable value of phase bias. This is what we have done in this study.
 A puzzled feature in Figure 6c is that the phase biases derived from the FDI phase histogram are generally smaller than the other, which has been observed constantly and shown in the Table 2. In spite of this, the synchronous variation in the two profiles of phase bias indicates a similar part of phase bias measured by the two methods. To figure out the puzzle, further theoretical and simulated works are needed, or, employ other calibration methods and criterions. Since Kilburn et al.  used different criterion to find the FDI phase bias, it is worthy of seeing whether such obtained FDI phase bias is close to our results or not. We selected the data with the FDI amplitude of less than 0.3 to construct the histogram of the FDI phases again; moreover, the threshold of SNR > 1 was also given. The latter criterion is to reject noisy data, and the authors privately think that Kilburn et al.  should use this threshold of SNR according to their height-time intensity plots. Two examples are presented in Figure 7. As shown, the FDI phase histograms are similar sometimes (Figure 7a); but they may be quite different at other times (Figure 7b). An apparent shortcoming with the restriction of low FDI amplitude is that the number of data reduces greatly, causing a very spiked histogram of the FDI phases; even the Gaussian distribution form may disappear due to fewer data number (Figure 7b), which makes the estimated phase bias unreliable. Certainly, random distribution also occurs in the histogram of the FDI phases with only the SNR criterion when the data number is few; in such case, statistical meaning disappears. This is another consideration for that an average for all range gates is more appropriate for estimating the phase bias. Comparisons between the mean phase biases estimated with different methods and criterions are presented in Figure 8; data times are listed in the Table 2. The points indicating the phase biases are placed at their approximate time locations, which can provide a look at their variations with time. Note that there is only one data set used in 1998 and 1999 both. As shown in the upper three panels of Figure 8, the phase biases subject to low FDI amplitude are generally smaller than those obtained in this study; even their changes may not be coherent with our results (see the upper panel for 2-μs pulse length). Phase biases indeed changed with time, but they did not vary randomly. The time-coherent variation in the phase bias based on the SNR criterion is consistent with the altitude-coherent variation shown in Figure 6c. As for the slight change of phase bias with time, one cause could be the change of characteristics in radar system. Another potential cause is various conditions in the atmosphere, e.g., mean reflectivity gradient.
 Next, we turned to inspect the initial phase of the carrier frequency in the radar pulse. We considered two conditions in dealing with this issue: (1) the initial phase varies randomly from radar pulse to radar pulse, (2) the initial phase difference between two carrier frequencies in the radar pulses is constant. Condition 1 can be examined by means of the radar returns of the aircraft. Figure 9 is one example showing the phase of raw data for the aircraft in Figure 3a located in the range of 11.1–11.4 km. Observe that the phase varies smoothly with time that is consistent with the continuous movement of the aircraft aloft, the initial phases of the radar pulses are not random. Figure 9 is the result for the pulses with carrier frequency 1, and similar feature can also be seen for the radar returns with the other carrier frequency (not shown). This result also confirms the radar's capability of measuring Doppler shift by phase change of echoes over time.
 As for the condition 2, it can cause a systematic bias in the FDI phase and results in disconnected FDI phase trace at gate boundaries. For this part, we carried out an experiment for the aircraft with two same carrier frequencies that were transmitted, respectively, from two independent frequency synthesizers. Because of the use of same frequency, the first term 2(k2 − k1)r in (3) can be omitted so that the effect of time delay of the radar echoes is eliminated. We will expect the FDI phase is zero if the effects associated with the initial phases of the two same carrier frequencies in the radar pulses do not exist. Figure 10 shows the result. According to the information of Doppler shift and radar echo intensity, the aircraft was closest to the radar site at the time around 23 s and in the range of 2.7–3.0 km. One can observe that the FDI phases of the aircraft are indeed around zero; however, we can also see that before and after the time of ∼23 s the FDI phases shift, respectively, negatively and positively. Such slow variation in the FDI phase is due to the fast movement of the aircraft during the time of an IPP that leads to remarkable phase change in two successive radar returns. We have demonstrated this by numerical simulation (not shown here).
Figure 10 illustrates only the condition of two same frequencies produced by independent frequency synthesizers. For different frequencies, the initial phase problems may still exist due to the radar response to different transmitting frequencies. It may need other methods and instruments for verifying this response.
 Finally, the group delay of the signal in the media and the time delay resulted from the data processing in the radar system, which cause incorrect estimation of the range of the sampling gate, are likely important for the FDI phase bias. In case of this, we can estimate the time delay according to the quasi-linear relationship between phase bias and frequency separation shown in Figure 5. Referring to the results obtained from echo power-FDI phase relationship, the time delays are, respectively, around 0.1, 0.4, and 0.8 μs for the pulse lengths of 1, 2, and 4 μs. Taking these time delays into account in the FDI analyses, it is expected to eliminate the disconnection of the FDI phase or layer location at gate boundaries. For dual-frequency observation this expectation can be reached only under the condition of a signal layer or one target such as an airplane in the radar volume. In a real atmosphere, multiple layers, background scatterers [Chen and Chu, 2001], and so on, have made the verification difficult. Figure 11 shows an example for a try at this verification. This observation was implemented with 2-μs pulse length and a pair of frequencies, 51.75 and 52.25 MHz. Each range bin was 300 m. The upper panel of Figure 11 shows the range-time-intensity plot, in which a striking echo layer descending from the height of 7.2 to 6 km during the time interval between 1800 and 2000 UT can be seen. We would assume that this is the case of a single layer in the radar volume. The middle panel is the layer position derived from the FDI equation of single-layer model. As shown, although the position of the primary FDI layer generally agrees well with the echo layer, disconnections in the FDI layer position are apparently seen at gate boundaries. It is worthy of mentioning that we have estimated the layer position by various single-layer FDI equations existed in literature, namely, Kudeki and Stitt , Franke , Liu and Pan , and Chen et al. , but there is no much difference between these calculations. Now considering a time delay of 0.4 μs in determining the range of the sampling gate, or equally, the FDI phase is compensated by adding −72°, the FDI layer position at gate boundaries can become much smoother, as shown in the lowest panel of Figure 11. Discontinuity in the layer position at the range of 6.9 km around 1800 UT is still remarkable, which could be due to the multiple layers in the range gate. This example is consistent with our expectation; nevertheless, multiple-frequency (more than two) experiment will provide a more reliable verification for it can resolve the multiple layers in the range gate.
 Three methods were utilized in this study to measure the FDI phase bias for the Chung-Li VHF radar: (1) histogram of the FDI phase, (2) relationship between echo power and FDI phase, and (3) the FDI phase of aircraft. All the three methods suggest a positive bias in the FDI phase. However, the FDI phase bias derived from the first method is smaller than the other twos, which is one puzzled feature in this study. The data of 1-μs pulse length also resulted in smaller phase bias than 2- and 4-μs pulse lengths, which could be due to the inherent response of radar system to different pulse lengths, or other factors.
 We have tried to figure out the plausible causes of the FDI phase bias for the Chung-Li VHF radar, including mean reflectivity gradient, initial phase of the carrier frequency in the radar pulse, and time delay associated with the signal and the data processing in radar system. Mean refractivity gradient may contribute a part of the FDI phase bias, but a portion of the FDI phase bias that is not related to mean refractivity gradient is observable. Initial phase difference between the radar pulses with same carrier frequency was not found. For different carrier frequencies, however, further examinations are needed. It is difficult to conclude here which one is the final cause; two or more causes can be responsible for the observed phase bias. Irrespective of the causes, we have acquired some predictable values for the FDI phase bias observed by the Chung-Li radar. This is achieved by the quasi-linearly-dependent relationship between the FDI phase bias and frequency separation. In practical FDI analysis, the phase bias can be either compensated for each frequency pair or consider proper time delay in determining the range of the sampling gate. For the latter, the time delays are around 0.1, 0.4, and 0.8 μs, respectively, for 1-, 2-, and 4-μs pulse lengths, according to the results of method 2.
 It is worthy of pointing out that the radar echoes reflected from aircrafts are not related to uncertain conditions of the atmosphere such as mean reflectivity irregularity, SNR, and other statistical characteristics. In view of this, the aircraft echoes are worthy of using to assist us in calibrating the FDI phase bias, although it may fail for 1-μs pulse length.
 The bias in the FDI phase may change from system to system and from time to time, but same methods and procedures can be applied. Other calibration methods may be still required for different radar systems and radar parameters; this study contributes parts of the calibration.
 The author would like to express the greatest gratitude to Jürgen Röttger (Max-Planck-Institut für Aeronomie, Germany) and Yen-Hsyang Chu (Institute of Space Science-Center for Space and Remote Sensing Research, National Central University, Taiwan) for useful discussion on this paper. The author also thanks Ching-Huang Wu for assisting in the operation of the Chung-Li VHF radar. The comments from the reviewers are highly appreciated for that improve the materials and quality of this paper greatly. This paper was supported by the National Science Council of the Taiwan, under grants NSC91-2111-M-270-001 and NSC93-2111-M-270-001. The Chung-Li VHF radar is maintained by the National Central University, Taiwan.