A Velocity Dealiasing Algorithm on Frequency Diversity Pulse-Pair for Future Geostationary Spaceborne Doppler Weather Radar

1 College of Electronic Engineering, Chengdu University of Information Technology, Chengdu 610225, China; lixh@cuit.edu.cn (X.L.); Qinzxiaxm@163.com (Z.Q.); hjx@cuit.edu.cn (J.H.); sunq224@163.com (Q.S.) 2 China Meteorological Administration, Key Laboratory of Atmospheric Sounding, Chengdu 610225, China 3 Meteorological Information and Technology Center, Chongqing Meteorological Bureau, Chongqing 401147, China; liiuzz@126.com * Correspondence: wczmail@foxmail.com; Tel.: +86-028-8596-7291


Introduction
Space-borne Precipitation Radar (SPR) is an active measurement radar used on a satellite platform that can observe rainfall characteristics in all weather conditions at all times with threedimensional (3D) detection in the range of the radar observation. SPR is the only effective tool that directly obtains global rainfall information [1]. The development of the SPR began in the 1980s. The first SPR was carried on the Tropical Rainfall Measuring Mission satellite, which was launched in 1997 [2]. The Dual Frequency Precipitation Radar (DPR) is working, carried by the Global Precipitation Measurement (GPM) that was launched in 2013 [3]. While studying the low-orbiting SPR, National Aeronautics and Space Administration (NASA) proposed the Next-Generation Radar (NEXRAD) in the space research project to deploy a weather radar in geostationary orbit for hurricane detection. In contrast to low-orbiting SPR, the Geostationary Spaceborne Doppler Weather Radar (GSDWR) not only provides high-frequency, wide-coverage rainfall detection, but also typhoon and strong storm monitoring near 1 h or higher frequencies. In addition, GSDWR not only measures precipitation fall speed, but also the movement of the horizontal wind field.
However, the GSDWR project faces many new challenges in the measurement of Doppler velocity, such as Doppler velocity ambiguity. Since the maximum unambiguous velocity of the short wavelength (Ka band) is relatively small, velocity ambiguity often occurs when measuring the speed of strong storms and hurricanes and other severe weather, which seriously affects the accuracy of the velocity measurement and causes large estimation error in the echo spectral moment estimation. The detection of strong storms and hurricanes is one of the main purposes of GSDWR. Therefore, we focused on the velocity ambiguity problem and created a method for velocity dealiasing for GSDWR to improve the efficiency, which is extraordinarily significant for GSDWR.
The most commonly used methods of velocity dealiasing include the staggered pulse repetition time (PRT) algorithm, dual pulse repetition frequency (PRF), batch processing, and polarization diversity pulse-pair (PDPP). The stagger PRT method was introduced by Sirmans to be used in a weather radar in 1976 [4,5], but because of non-uniform sampling, the signal and clutter spectrum change [6], so the use of this method has been limited for a long time. Of course, there is more research on dual PRF technology in recent years [7]. In 1997, Sachidananda and Zrnic put forward a method for solving distance ambiguity using the phase encoding of a SZ system [8]. This algorithm was validated by the National Center for Atmospheric Research (NCAR) and National Severe Storms Laboratory (NSSL). In 2006, Zhou [9] introduced batch processing to solve the weather radar data ambiguity problem and this method was able to separate overlapping echoes from time. In 1998, Pazmany proposed polarization diversity pulse-pair (PDPP) [10,11], which can effectively solve Doppler ambiguity but is only suitable for a dual polarization radar. However, each method has some disadvantages that require improvement. In this paper, we introduce a novel method for frequency diversity pulse-pair to extend the maximum unambiguous velocity while maintaining the accuracy of velocity estimation. The aim of this study was to provide a velocity dealiasing method for future GSDWR.
In this paper, the velocity ambiguity problem faced by GSDWR is firstly analyzed according to the relationship between satellite geometry and radial velocity measurement for hurricane observation. Then, a velocity dealiasing algorithm for frequency diversity pulse-pair (FDPP) is introduced and analyzed, which changes the interval time of the adjacent transmitting pulses to obtain better performance in the extension of the maximum unambiguous velocity and the velocity estimation accuracy. To evaluate the performance of FDPP used in GSDWR, a GSDWR echo model of frequency diversity is then built and simulated. The results show that this FDPP algorithm has better velocity estimation performance and can extend the maximum unambiguous velocity for GSDWR. Although the study is focused on spaceborne techniques, it can be also of interest to researchers working on improved Doppler radar velocity estimates, such as wind shear and mesocyclone storm detection [12].

GSDWR Introduction
Although the Geostationary Spaceborne Doppler Weather Radar (GSDWR) is still a conceptual radar [1,13,14], it has been studied in the United States and China and some of the related parameters of the radar system are shown in Table 1. The GSDWR is designed to work in geostationary orbit at an altitude of 36,000 km, with a maximum scan angle of 4°, and a spiral scanned antenna beam [15]. The scan model is shown in Figure 1a.
The geometric relationship between Geostationary Spaceborne Doppler Weather Radar (GSDWR) and the earth. The altitude of the geostationary orbit HOrb is 36,000 km and the scan angle θ is from 0° to 4°. (a) Spiral scan model; GSDWR performs the spiral scan from nadir to 4° and its horizontal resolution is from 12 km (nadir) to 14 km (4°); (b) Model of relationship between satellite and ground angle, describing the relationship between the scan angle θ, incidence angle α, and the grazing angle φGra; (c) Antenna scanning strategy, mainly used to describe the relationship between maximum ambiguous distance and PRF. The geometric relationship between the satellites and the earth is shown in Figure 1b, which can be expressed with Equations (1) and (2). The orbit height HOrb is 36,000 km, radar beam scan angle θ ranges from 0° to 4°, and the earth radius REar is 6370 km.
For the Doppler weather radar, the choice of PRF is crucial. Distance ambiguity cannot occur in the range of attention. It is necessary to consider the pulse generation, transmission time tolerance, and the switching time between the transmitting and receiving states. The correlation and independence between the echo signal samples and the Nyquist sampling rate also must be considered [16]. According to Figure 1c and Equation (3), where c is the speed of light, when the maximum detection distance H is 25 km and the scan angle range θ is 0° to 4°, the maximum PRF is 6 kHz.

Velocity Ambiguity of GSDWR
For the Geostationary satellite platform, we assumed that the speed of the satellite platform motion has no effect on the Doppler radial velocity component of GSDWR, so the Doppler radial velocity component observed by GSDWR can be computed as follows [13]: where u, v, and w are the velocity components of the 3D wind field, vT is the terminal velocity of the precipitation, and the angle θ and γ represent the incident angles of the beam in the east-west direction and the north-south direction, respectively, which can be expressed by Equation (5) [13].
where α and β represent the pointing angles in the north-south and the east-west direction, respectively; d is the distance between radar and meteorological target; and a is the earth's radius.
For simple evaluation of the relationship of radar radial velocity component with the horizontal wind field, we focused on the radial velocity component vrad from the composed horizontal wind speed VH from (u, v). From Equations (4) and (5), VH can be obtained by: (cos sin sin cos ) H rad Vv (6) The problems facing the Doppler weather radar include maximum unambiguous range and maximum unambiguous velocity, and these two values are both determined by the value of the PRF [17,18]. The two values, respectively, are expressed by: where c is the velocity of light and λ is the radar wavelength. Usually, in the spaceborne Doppler weather radar design, we firstly attempted to choose the correct PRF value to ensure there is no range ambiguity. To demonstrate the velocity ambiguity problem, a maximum range of 25 km for the range of precipitation observation was assumed. According to Equation (8), the PRF value could not be larger than 6 kHz; therefore, the maximum unambiguous velocity was 12.7 m/s. Taking the Saffir-Simpson Hurricane Wind Scale (SSHS) [19] as an example, the maximum horizontal wind speed is 59-69 m/s at SSHS-4 and ≥70 m/s at SSHS-5. Given the SSHS-5 horizontal velocity was 70 m/s, according to Equation (6), the relationship between the pointing angles (α and β) and the maximum radial velocity vrad can be obtained, as shown in Figure 2. From Figure 2, when SSHS-5 is observed by GSDWR, the radial velocity in the areas of pointing angles from 2° to 4° or −2° to −4° will be larger than the maximum unambiguous velocity of 12.7 m/s. If we added the terminal and vertical velocity components into the result of the radar radial velocity component, the area of the pointing angles will be further expanded and the problem of velocity ambiguity will become worse. As one of the main advantages of GSDWR is that it can observe and trace hurricanes and typhoons more frequently with its stationary observation of the earth, if the problem of velocity ambiguity cannot be resolved, the whole application of GSDWR will be affected.

FDPP Algorithm Principle
The Frequency Diversity Pulse-Pair (FDPP) algorithm is an improvement based on the stagger PRT algorithm [20,21]. Two pulses with different center frequencies of f1 and f2 are transmitted with a separation ΔT during the first PRT, and an inverting order pulse is transmitted with a separation ΔT during the next PRT. The pulse transmission mode of the FDPP algorithm is shown in Figure 3. Two central frequencies of f1 and f2 are in the pulse that is transmitted with an interval ΔT, but after time T, the order of the pulses is reversed and transmitted ( Figure 3). Figure 3. Illustration of a frequency diversity pulse-pair (FDPP) pulse transmission mode. Two pulses at different central frequencies of f1 and f2 are transmitted with a time interval ΔT. Generally speaking, ΔT is relatively small, only taking a few μs or dozens of μs. The combination of four pulses is a complete pulse repetition time. The difference between f1 and f2 is usually a few MHz or the value of radar band. PRT is the complete pulse repetition time.
In the receiver, the pulse-pair phase estimates of f1−f2 and f2−f1 are individually accumulated and recorded as ΔΦR1 and ΔΦR2, respectively. Then, the Doppler velocity vr is estimated from the sum of ΔΦR1 and ΔΦR2. The detailed deduction of vr is shown in Appendix A.
Because the sum of ΔΦR1 and ΔΦR2 is between [−π, π], the maximum unambiguous velocity expression is as shown in Equation (10).
The FDPP algorithm velocity estimate is same as the conventional pulse-pair algorithm by calculating the phase information through the correlation function. According to the correlation function previously defined [4], the variance in calculating ΔΦR1 and ΔΦR2 is same and can be expressed with Equation (11).
The spectrum width σv can be expressed as:

Analysis of FDPP Algorithm
In this section, the FDPP algorithm performance is theoretically analyzed with the algorithm above. In the calculation, f1 = 35.5 GHz, f2 = f1 + 10 MHz, ΔT = 10 μs, the theoretical maximum unambiguous velocity under these conditions is 105.5 m/s. Figure 4a shows the phase results of ΔΦR1 and ΔΦR2. From the figure, the phase directly calculated from Equation (9) appears folded and it limits the range of the maximum unambiguous velocity. If the phase is now directly used in Equation (9), the velocity value will not be as great as the theoretical maximum unambiguous velocity. We propose a phase modification method as shown in Equation (13), where k represents the number of phase folds, and here, k = 1. The velocity comparison between phase modification and no modification is shown in Figure 4c.   From analysing Equation (10), the maximum unambiguous velocity of FDPP is only related to ΔT and wavelength, which are independent of the PRF and completely solves the Doppler problem. However, the choice of PRF has a certain relationship with the accuracy of the velocity estimation. In addition, the accuracy of velocity estimation is influenced by the signal to noise ratio (SNR) and ΔT. When ΔT = 10 μs and the pulse width is 10 μs, the pulse central frequency f1 is 35.5 GHz, and f2 = f1 + 10 MHz. The relationship between the velocity estimation error and SNR is shown in Figure 5a. When the SNR is greater than −6 dB, the velocity estimation error is small, but when the SNR is less than −6 dB, the velocity estimation error increases with the decrease in the SNR. Similarly, Figure 5b shows the trend in the velocity estimation error with different SNR. When PRF is less than 9 kHz, the velocity estimation error is relatively small, and when the SNR is smaller, the estimation error is greater. Equation (10) shows that ΔT determines the maximum unambiguous velocity, but it is not as small as possible due to the accuracy of the velocity estimation. Using the same conditions above, at PRF = 4000 Hz, the velocity estimation error is evaluated with different ΔT, and the results are shown in Figure 5c. From the results, when ΔT is larger than 5 μs, the velocity estimation errors are small and the needs of GSDWR, whose pulse width will be 5-50 μs, can be met.  Maximum unambiguous velocity

Echo Simulation of GSDWR
Due to the limited conditions, GSDWR could not verify the algorithm or the whole machine performance through real experimental data. Related research used airborne radar or ground-based radar data to simulate space-borne weather radar echo signals [22]. The Earth Clouds, Aerosols, and Radiation Explorer (Earth CARE) satellite carried a spaceborne 94-GHz Doppler Cloud Profiling Radar (CPR), which included a synthetic echo signal from ground-based radar data, was used for algorithm validation and performance evaluation [23]. However, the echo signal simulation from the power spectrum, and generating frequency diversity time series from the spectrum, were not feasible in this paper. Therefore, the time domain echo signal model was used to simulate the GSDWR echo signal in this article. However, attenuation and weighting and other factors must also be considered. Firstly, the spatial distribution of the precipitation particles model was established by the rectangular grid method shown as Figure 6a, which ignored the influence of the curvature of the earth. The precipitation space was divided into several precipitation cells and all the particles in the precipitation cell were treated as a scattering unit. These precipitation particles had a certain relative motion [24]. According to the measured APR-2 data and Figure 6a, the spatial precipitation distribution model was established and the Gaussian distributed pseudo-random values were used to generate the particle relative velocity field in the precipitation cell [21]. The echo of the precipitation cell was calculated by the echo signal model as shown in Equation (14), and the precipitation particle relative parameters were determined, including position, velocity, antenna gain, and the path integral attenuation. The echo signal deduction model is shown in Appendix B.
where f0 is carrier frequency, u(t) is the complex envelope function of the signal, τi is the echo delay of the ith precipitation cell, which is determined by the radial distance and relative velocity of the precipitation target cell and radar, fd is the Doppler shift, and N is the number of the precipitation cell. However, the resolutions of APR-2 data and GSDWR data are different; a GSDWR data cell consists of 30 × 8 APR-2 data. Therefore, the antenna gain corresponding to the precipitation particles is different in different positions in a GSDWR beam element, so that the echo power is also different and must, therefore, be antenna weighted [25]. The weighting function expression is shown in Equation (15). To improve the detection ability of the spaceborne radar, the advanced pulse compression technology was used to improve the distance resolution, so the distance weighting was used. The distance weighting function uses the ambiguous function of the transmitting signal, as shown in Equation (17). The antenna weighted and distance weighted schemes are shown in Figure  6b.

Analysis of the Simulation Results
The GSDWR simulations were obtained using the environmental field data measured by the Kaband of the APR-2 weather radar. The APR-2 data were considered to eliminate factors such as aircraft attitude and jitter that may affect data quality. Firstly, according to the GSDWR parameters, FDPP model, and Equation (14), the APR-2 data of reflectivity and Doppler radial velocity were used to generate the echo signal. Then, the FFT transform was performed for the echo signal and the power spectrum was obtained. The GSDWR power spectrum was obtained by range and antenna weighting, and then the I/Q serial was obtained by inverse Fourier transform of the weighted power spectrum. Finally, the FDPP algorithm was used to estimate velocity and width. The flowchart of the simulation is shown in Figure 7a. The APR-2 echo reflectivity and velocity data are shown in Figure 7b. To quantitatively verify the velocity dealiasing effectiveness of the FDPP algorithm for GSDWR, the 30 (horizontal) × 216 (vertical) APR-2 data was used to constitute the radial data of GSDWR, as shown by the data range drawn in the red rectangle A as shown in Figure 7b.  Figure 8a, which shows that when the virtual velocity is greater than 8 m/s in the range of one to three kilometers, the velocity of the PPP estimation has folded, but the velocity of the FDPP estimation can accurately estimate the velocity value. To verify that the selection of PRF in the FDPP algorithm has no direct effect on the maximum unambiguity velocity, the PRF was adjusted to satisfy the maximum unambiguity distance. We adjusted the PRF to 1000 Hz, resulting in a maximum unambiguous velocity with PPP estimation algorithm of 2.1 m/s, and a maximum unambiguous velocity with FDPP estimation algorithm of 105.6 m/s. From Figure 8b, the velocity of the traditional PPP method always has ambiguous velocity, and then the FDPP estimation results are acceptable. Figure 9 shows the GSDWR radial velocity echo data according to Figure 7b and the same simulation conditions above. The actual horizon resolution is very poor, so horizontal slide processing was used to obtain more horizontal data for a better effect. Figure 9a displays the results obtained with the PPP method and Figure 9b provides the FDPP method results. Figure 9a shows the velocity ambiguity in the elliptic region, which occurred because the real velocity is greater than the maximum unambiguous velocity using the PPP method. Figure 9b shows the velocity value was estimated accurately in the ellipse region.

Performance Analysis of FDPP
The above analysis shows that the FDPP algorithm has a good velocity dealiasing effect. However, in actual applications, the precision of the velocity estimation needs to be considered. In the meteorological radar observation business, the precision of velocity estimation must be less than 1 m/s, and the GSDWR is proposed to be less than 0.5 m/s. Therefore, the velocity estimation bias of the FDPP algorithm under different SNR, PRF, and ΔT conditions was analyzed. Figure 10 shows the velocity estimation scatter and statistic histogram, where the red line represents the zero-deviation standard in scatter, and the histogram indicates the probability of velocity bias being less than 0.5 m/s. From the scatter figure, with the increase in SNR, the more the velocity data focus on the zero-deviation standard line. The histogram shows that the velocity estimation precision improves with the increase in SNR, which is also consistent with the previous theoretical analysis. Similarly, from the PRF analysis in Figure 11 and ΔT in Figure 12, almost no difference in velocity estimation accuracy was observed within the permitted range, which is also consistent with the previous theoretical analysis and Figure 5. Actually, when ΔT is 40 μs in Figure 11, the maximum unambiguous velocity is only about 26 m/s, and velocity ambiguity easily occurs for GSDWR. Figure 11. Velocity estimation bias and probability distribution of FDPP at different ΔT. The red line in the scatter plots represents the zero-deviation standard, and the histogram indicates the probability of velocity bias being less than 0.5 m/s. To construct the velocity spectrum width information in this simulation, Gaussian random velocity distribution was constructed based on APR-2 velocity data with a maximum velocity change of 4 m/s. The velocity and velocity spectral width results using the FDPP algorithm when the SNR was 10 dB, PRF was 4000 Hz, and ΔT was 10 μs are shown in Figure 13a. The maximum spectral width was also 4 m/s.
To further verify the validity of the velocity spectral width estimation, a QQPlot figure was used for verification analysis, as it can intuitively verify and analyze whether a group of data obeys or approaches a certain distribution characteristic. Figure 13b shows the distribution characteristics of the velocity spectrum width calculated under a variety of SNR conditions. Along with the increase in SNR, spectral width data were increasingly consistent with ideal distribution, which is also consistent with the spectral width distribution of the previous structure.  The red line represents the ideal distribution; the more data scattered closer to the red line, the closer the spectral width data is to the normal distribution.

Conclusions
The Spaceborne Weather Radar is an effective tool for global precipitation observation. However, Doppler velocity ambiguity and range ambiguity are urgent problems. The frequency diversity pulse-pair method outlined in this paper can effectively solve the Doppler velocity and distance ambiguity problems, which are limited by the radar PRF and wavelength, especially for millimeter wave radar. We firstly analyzed the principle and performance of the FDPP algorithm. From the principle analysis, the maximum unambiguous velocity can be determined by the radar wavelength and the interval time ΔT of the adjacent pulse-pair. ΔT has a certain range of choices, rather than an unlimited or large range. Then, a GSDWR echo signal model was established, which verified the velocity dealiasing algorithm. According to the results analysis, we chose ΔT as 10 μs in Ka band, which extended the maximum unambiguous velocity to 105 m/s, but it was impossible through staggered PRT within appropriate values of PRF. Additionally, the velocity estimation precision was also fully considered. As with other velocity estimation methods, SNR had a relatively large effect on velocity estimation accuracy. When the SNR value was greater than 10 dB, most of the estimation error was less than 0.5 m/s. Simultaneously, the choice of ΔT and PRF value had little effect on the velocity estimation accuracy within the appropriate range. So, according to the maximum detection distance and velocity, we could dynamically choose these two parameters, ΔT and PRF, during deployment. Compared with staggered PRT, the FDPP introduced in this paper has a larger velocity extension range and estimation performance, providing a better solution for Doppler's dilemma, especially for millimeter wave radar.
Author Contributions: C.W. and X.L. designed the experiments; C.W. performed the experiments and data analysis; Z.Q., F.L. and Q.S. designed the echo simulation; C.W. and X.L. analyzed the results; X.L. and J.H. supervised the work and provided comments on the results; C.W. wrote this paper. Approximately, we assume that k1 − k2 ≈ 0, k1 + k2 ≈ 2k1, and then:

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So, the velocity can be expressed as:

Appendix B
Assuming that the transmitted signal is represented by st(t), the echo signal from the precipitation target is denoted by sr(t), and τt is the echo delay time. According the radar meteorological from Doviak and Zrnić [26], echo signal can expressed as formula (A10). where G is antenna gain, θ and φ are the beam width in different directions, τ is the pulse width, λ is wavelength, As is the attenuation coefficient, and Z is reflectivity. Normally, the transmitted radar signal is a narrowband signal and to achieve pulse compression, the signal is used as a linear frequency modulation signal. The complex expression of the signal st(t) is written as: where f0 is the carrier frequency, u(t) is the complex envelope function and the expression is: