Accuracy Evaluation of Doppler Velocity on a Spaceborne Weather Radar through a Random Signal Simulation

Satoru Kobayashi Communications Research Laboratory, Koganei, Tokyo, Japan

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Hiroshi Kumagai Communications Research Laboratory, Koganei, Tokyo, Japan

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Toshio Iguchi Communications Research Laboratory, Koganei, Tokyo, Japan

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Abstract

Digitized signals for a spaceborne weather radar are generated to inspect a Doppler signal processor, in which the digital signals are converted to analog through an arbitrary wave generator. A conventional Rice harmonic analysis is applied to include fluctuations of Fourier coefficients explicitly in a time series rather than in an ensemble. The accuracy of Doppler velocity is studied through this simulation for a mean estimator of contiguous pulse-pair operation in a space mission, characterized by a low signal-to-noise ratio (SNR) and a short coherence time. A linear perturbation formula is shown to deviate from the simulation as the SNR decreases and the pulse-pair interval increases. Furthermore, a theoretical limit in measurement accuracy is derived, beyond which the correlation signal is to be practically regarded as white noise, losing the physical meaning of measurement.

Corresponding author address: Hiroshi Kumagai, Communication Research Laboratory, 4-2-1 Nukii-kita Koganei, Tokyo, Japan. Email: kumagai@crl.go.jp

Abstract

Digitized signals for a spaceborne weather radar are generated to inspect a Doppler signal processor, in which the digital signals are converted to analog through an arbitrary wave generator. A conventional Rice harmonic analysis is applied to include fluctuations of Fourier coefficients explicitly in a time series rather than in an ensemble. The accuracy of Doppler velocity is studied through this simulation for a mean estimator of contiguous pulse-pair operation in a space mission, characterized by a low signal-to-noise ratio (SNR) and a short coherence time. A linear perturbation formula is shown to deviate from the simulation as the SNR decreases and the pulse-pair interval increases. Furthermore, a theoretical limit in measurement accuracy is derived, beyond which the correlation signal is to be practically regarded as white noise, losing the physical meaning of measurement.

Corresponding author address: Hiroshi Kumagai, Communication Research Laboratory, 4-2-1 Nukii-kita Koganei, Tokyo, Japan. Email: kumagai@crl.go.jp

1. Introduction

The feasibility of spaceborne weather Doppler radars has been conceptually studied by Meneghini and Kozu (1990) and Amayenc et al. (1993), who discuss both the pulse-pair method based on covariance function and the digital Fourier transform (DFT) method. In view of optimal estimators, the former method should be avoided because its variance of mean frequency is far from the Cramer–Rao bound for a broad spectrum and a long sampling interval, especially for a high signal-to-noise ratio (SNR) as described by Zrnić (1979a) and Dias and Leitao (2000). However, for a space mission, this method has been considered to be best for the following reasons: First, compact implementation in a satellite requires simple complexity of signal processing. Second, the pulse-pair method is of higher potential than the DFT method for SNR < 0 dBZ, typically measured during a space mission. Furthermore, for such a low SNR, the variance of mean frequency (velocity) is not as far from the Cramer–Rao bound as it is for high SNR (Doviak and Zrnić 1993; Zrnić 1979a). Based on these advantages, a practical design of a spaceborne cloud profiling radar was proposed by the European Space Agency as a part of the Earth, Clouds, Aerosols and Radiation Explorer (EarthCARE) mission (Harris and Battrick 2001). In this mission, the fairing sizes of available launchers confine the diameter of a radar antenna to 2.5 m, which will give a coherence time of Tc ≈ 100 μs at a supposed altitude of around 450 km from the earth's surface. Kobayashi et al. (2002), independently from the EarthCARE, studied pulse-pair operations from space on the basis of a linear perturbation theory. It concluded that a promising operation in regard to accuracy is a polarization diversity method (Doviak and Sirmans 1973), with a pulse-pair interval of Ts = 60 μs and a pulse repetition interval of Tpri = 222 μs (4.5 kHz). However, in this mode of operation the second signal of paired pulses coming back from the altitude of 9 km suffers from strong interference due to the ground clutter of the preceding first pulse. This is an adverse effect since one purpose of Doppler mode of operation in the EarthCARE mission is detection of cirrus existing over the altitude of 8 km. A substitute mode of operation is adoption of contiguous pulse-pair operation with Ts = Tpri = 100–125 μs, sacrificing the unambiguous range and the precise correction against a misaligned beam. It is, however, noted in this mode of operation that the pulse-pair interval Ts = 100–125 μs larger than the coherence time Tc ≈ 100 μs along with low SNR (≲0 dB) may cause large scatters about an estimated mean value, in some cases, breaking the linearity of the perturbation theory based on small scatters about the mean value (Zrnić 1977). Thus, the validity of the perturbation theory must be confirmed through numerical simulation for the critical regime defined by TsTc, and SNR ≲0 dB, prior to implementation.

For this purpose, two kinds of numerical simulations can be considered, one of which is particle simulation, and the other is random signal simulation. Taneli et al. (2001) recently took the former simulation for a space mission along with the moderate regime defined by Ts < Tc and SNR ≳0, rather than the critical regime. Their results showed that the perturbation theory gave good agreement with the particle simulation for their concerned region. The random signal simulation, on the other side, has been performed mainly for ground-based parameters by a number of authors (e.g., Sirmans and Bumgarner 1975; Zrnić 1979a,b; and Dias and Leitao 2000). Advantages in this simulation are its short calculation time and easy handling of parameters such as spectral width of Doppler velocity συ, or coherence time Tc, and pulse-pair interval Ts.

The Communications Research Laboratory (CRL) in Japan is fabricating a pulse-pair signal processor to evaluate the feasibility of spaceborne Doppler radars. A block diagram of the processor is depicted in Fig. 1, in which digitized signals are input to an arbitrary waveform generator (AWG) to convert the signals to analog. Thereafter, noise generated in a radio frequency (RF)–intermediate frequency (IF) unit merges with the signals from the AWG, being in-phase and quadature (IQ)-demodulated. Finally, the demodulated IQ signals enter an analog-to-digital (A/D) board to calculate the mean Doppler velocity and standard deviation for a given mean estimator. In order to operate this processor in real time, it is necessary to develop a prompt generator of simulated signals. Furthermore, these signals should be as close to real signals as possible. Experiments with this processor will be reported separately in the future. In this paper, a generation of random signals will be proposed so that the characteristic of contiguous pulse-pair operation during a space mission is well reflected. Thereafter the accuracy of Doppler velocity in a mean estimator will be evaluated through the simulation.

2. Random signal generation

Random signals on contiguous pulse-pair Doppler operation have been simulated using a harmonic analysis, referred to as the Rice method (Rice 1944), in the works of Sirmans and Bumgarner (1975) and Zrnić (1975, 1979a,b). The Rice method is presented as follows. To a given power spectrum S(ω), the complex current J(t) (voltage) at a time t ∈ [0, T] for IQ detection can be represented in terms of a discrete Fourier series:
i1520-0426-20-6-944-e1
in which ωn has been defined as
ωnπnT−1
and the complex coefficient cn is decomposed to the real coefficients an and bn as
cnanibn
Here an and bn are Gaussian random variables satisfying the following relations:
i1520-0426-20-6-944-e3
In these equations, 〈x〉 represents the mean value of x, and δnm designates the Cronecker delta. In short, Eqs. (3)–(5) can be summarized as
c*mcnδnmσ2n
In these equations, σ2n is related to the power spectrum S(ω) of the signal through
σ2n−1Sωω,
in which Δω = 2πT−1 is an sampling interval of discrete frequency.
Sirmans and Bumgarner (1975) assumed the phase of cn (i.e., arg cn) as a random variable, while fixing the amplitude of cn at a constant defined in Eq. (8) for each frequency component n within a train of signals:
c*mcnδnmσ2n
On the other hand, Zrnić (1975) assumed an exponential distribution to |cn|2, which can be shown to be equivalent to the Rice method defined in Eqs. (1)–(7). Relating to these previous works, the following is to be mentioned: Suppose that the accuracy of a Doppler mean estimator is represented as a function of the power spectrum or its correlation function as illustrated in section 6.4-5 of Doviak and Zrnić (1993); then, as far as the same power spectrum is used, both the simulations can be proven to give the same accuracies despite the qualitative difference in random signals.

However, for inspection of the IQ signal processor, similarity of simulated signals to the experimental is substantial. We should note for the Rice method that both the amplitude and phase of cn are fixed for a given n within a time train of T, and the distribution of cn never fluctuates within T. Instead, the randomness and fluctuation of cn is introduced by considering an ensemble of such trains to give the required power spectrum. Consequently, even though the original Rice method suffices for simulating the accuracy of velocity for the Doppler mean estimator mentioned above, the fluctuation of cn had better be included explicitly in a time series, for which the following method can be considered.

First, a set of currents Ji(t) (i = 1, 2, …) is generated independently by the Rice method. Second, these Ji(t) are connected smoothly through Gaussian-like envelope functions along with mutual overlaps, as ideally shown in Fig. 2a. However, simulating this model consumes a lot of time due to a large number of currents Ji(t). To simplify the problem, Fig. 2b shall be considered in place of Fig. 2a. In the concerned time region, three random currents, J0(t), J1(t), and J2(t), are generated from a power spectrum S(ω), and trapezoidal envelope functions instead of the Gaussian-like are introduced, being constituted of two parts: transient envelope regions of time duration Tlt extending in [−(Tlt + Tls), −Tls] and [0, Tlt], and the steady envelope regions in between. In the first transient region [−(Tlt + Tls), −Tls], the random signal J1(t) starts growing with an envelope function B(t) and reaches the constant amplitude of unity, holding in the steady region [−Tls, 0]. In the second transient region [0, Tlt], the signal J1(t) decays with an envelope function A(t), and, instead, the other random signal J2(t) grows again with the envelope function B(t). For macroscopically homogeneous clouds, we can assume that the 〈J*(t1)J(t2)〉 is determined uniquely by the correlation function r(t) corresponding to a power spectrum S(ω) during a radar scan, namely
Jt1Jt2rt2t1
for arbitrary t2 and t1 within a radar scan. In the steady region, Eq. (9) is automatically satisfied. In the transient region [0, Tlt], the total signal can be written in the form
JtAtJ1tBtJ2t
and the time correlation can be represented as
Jt1Jt2At1At2Bt1Bt2rt2t1
However, Eq. (9) indicates that the functions A(t) and B(t) must satisfy
At1At2Bt1Bt2
Trivially, rigorous solutions of A(t) and B(t) cannot exist except for constants. Thus, Eq. (12) should be assumed to be nearly equal to unity as an approximation. Furthermore, simple forms are assumed as follows:
i1520-0426-20-6-944-e13
In order to calculate Eq. (12), the time t2 is set at
t2t1Ts
in which Ts designates a pulse-pair interval. Then Eq. (12) can be written as a function of the first pulse time t1 along with the parameters Ts, Tlt, and Tls:
i1520-0426-20-6-944-e15
Since the transition duration Tlt and the steady duration Tls are parameters within which signal correlation must hold, we can consider that these parameters are at least the length of coherence time Tc. In this paper, the steady duration Tls is assumed to be Tls = 2Tc ≈ 200 μs so that we can continuously evaluate the accuracy beyond the pulse interval Ts ≈ 120 μs, as required in the EarthCARE mission. Furthermore, from numerical calculations for Eq. (15), the transient durations of Tlt = 2Tc and Tlt = 10Tc are found to be adopted as a good parameter up to Ts ≈ 50 μs and Ts ≈ 160 μs, respectively.

3. Accuracy of Doppler velocity in a space mission

The evaluation of accuracy in Doppler velocity is performed for contiguous pulse-pair operation. Suppose that a radar antenna of 2.5-m diameter with a beamwidth of 0.089° aims toward the nadir direction from a platform moving at an altitude of 450 km at the velocity of υpl = 7.64 km s−1, then the total spectral width of Doppler velocity can be considered to be decomposed (Kobayashi 2002; Kobayashi et al. 2002):
σ2υσ2dopσ2cldσ2win
In Eq. (16), σcld and σwin designate the spectral widths due to the distributions of cloud falling velocity weighted by backscattering cross sections and wind velocity, respectively, which are assumed to be σcld = 0.5 m s−1 (Gossard et al. 1997) and σwin = 1 m s−1 (Amayenc et al. 1993). The σdop denotes the spectral width reflecting the combined effect of Doppler fading and vertical wind shears (Kobayashi 2002). Ignoring the wind shears, we can calculate σdop = 3.72 m s−1. Hence Eq. (16) yields the total velocity width of συ = 3.85 m s−1, and the corresponding coherence time is given by
Tc2υ−1μ
where k is the radiation wavenumber for 95.04 GHz. Converting the spectral width of Doppler velocity to the frequency width through the equation σω = 2υ, the IQ current J(t) can be simulated as in section 2. A mean estimator over an along-track integration of distance d is presented for contiguous pulse pairs in the form (Zrnić 1977)
i1520-0426-20-6-944-e18
in which N designates the total number of pulse pairs in the distance d with the platform velocity υpl, given by
NdυplTs
Finally, the estimated mean Doppler velocity is derived from
υkTs−1RTs
and the standard deviation (accuracy) of υ, referred to as std dev (υ), can be calculated through iterations of the simulation.

Simulation results of std dev (υ) for d = 1 km, which is required by the EarthCARE mission (Harris and Battrick 2001), are plotted in Fig. 3 for the parameters Tlt = 10Tc and Tls = 2Tc. The ordinate and abscissa represent the std dev (υ) and the pulse-pair interval Ts, respectively. The solid lines are generated from the averages of 10 000 iterations for the SNRs from −15 to +10 dB, on which the superimposed noises have been assumed to be white. The broken lines are calculated by a perturbation formula in Doviak and Zrnić (1993). Good agreements are seen for Ts ≲ 130 μs with higher SNRs, as expected from the postulation of the perturbation theory. In addition, simulations for the other parameters, Tlt = 2Tc and Tls = 2Tc, were also performed, the result of which was almost identical to that shown in Fig. 3; for Ts < 50 μs, especially, the agreement was excellent.

Notice that all the solid lines in Fig. 3 converge to the line marked “Noise” in the region of Ts > 200 μs, which can be explained as follows. As the SNR decreases and Ts increases, the Doppler correlation signal is expected to behave like white noise. The ensemble average of υ, referred to as 〈υ〉, cannot be uniquely defined rigorously for white noise, because the value of 〈υ〉 strongly fluctuates from one ensemble to another. In such a situation, however, once 〈υ〉 is determined for a given ensemble, which is always possible, std dev (υ) can be formally calculated as shown below. Without losing generality, we can assume 〈υ〉 = 0, or equiva-lently 〈θ〉 = 0, along with the phase θ = argR(Ts) in the IQ space. Then θ distributes uniformly in [−π, π] for the white noise, and the standard deviation of θ can be calculated:
θ−0.5π.
We thus obtain the expression of the Noise line as
υθkTsT−1sμ
This Noise line indicates that accuracies near it have little physical meaning in view of measurements.

4. Conclusions and discussion

Signals for a spaceborne weather radar have been simulated as discussed in section 2. This simulation reflects the fluctuation of the complex coefficient cn explicitly in a time series (i.e., during a radar scan), which, on the other hand, appears in an ensemble in the original Rice method. This feature of the former method suits for the experiment of Fig. 1, because the real-time processing for the fluctuating signal is substantial. However, it is noted that for the parameters satisfying Eq. (15), such as Tlt = 10Tc and Tls = 2Tc, the method in this paper gives the same time correlation and power spectrum as the original Rice method does. Thus, as far as the accuracy of the mean estimator of Eq. (18) is concerned, both the methods should give identical results. In fact, the simulation of Fig. 3 showed excellent agreement with that of the original Rice method as shown in Fig. 4.

In Figs. 3 and 4 the emphasis is placed on deviations from the perturbation theory on low SNRs (≤0 dB) and/or Ts longer than a coherence time Tc = 93 μs. Especially at Ts ≳ 150 μs large deviations occur even for relatively high SNRs (5 and 10 dB), because the signals lose the coherence, approaching noise. Obviously, monotonously increasing degradations of the accuracy on the perturbation theory with increasing Ts and decreasing SNR, seen in Figs. 3 and 4, are unrealistic, but they should be suppressed by nonlinear effect, as the simulation results show. This nonlinearity can also be expected from the following consideration. Suppose that an ensemble of R(Ts), defined in Eq. (18), has scatters of ΔI and ΔQ in the IQ space about a mean value of 〈ReR(Ts)〉 = Ime, and 〈ImR(Ts)〉 = 0, that is, 〈θ〉 = 0. Since it is intractable to know the exact expression of std dev (θ), we can roughly represent as
θQIme
which can be linearized for |ΔQ| ≪ 1:
θQIme
Equation (24) is related to the perturbation theory in previous works, and is also the origin of the divergence for ΔQ/Ime → ∞ as Ts → ∞. On the other hand, the nonlinear expression Eq. (23) will never diverge, as ΔQ/Ime → ∞. Here the exact asymptotic form of std dev (θ) in a large Ts is again unknown, but the form in the extreme limit of large Ts has been already given exactly in Eq. (21). Additionally, it should be mentioned that the degradation of velocity accuracy with short Ts ≈ 10 μs is caused by reduction in phase resolution for the small Ts but not related to increase in the the value of std dev (θ) (Kobayashi et al. 2002).

The EarthCARE (Harris and Battrick 2001) requires Doppler measurement to an accuracy of 1 m s−1 over an along-track integration of 1 km for clouds of −20 dBZ, corresponding to SNR = −5 dB under the parameters of this paper. Our simulations indicate that this requirement can be achieved for Ts ≲ 120 μs around which the perturbation theory is effective with a little deviation from the values of simulations, as shown in Figs. 3 and 4, thus also supporting linear analyses in Kobayashi et al. (2002).

Throughout this paper, the superimposed noises have been assumed to be white. However, when short Ts ≈ 10 μs is concerned, this assumption cannot always be guaranteed, as in the hardware experiment in Fig. 1; therefore, colored noise, which may have significant impact should be taken into account.

REFERENCES

  • Amayenc, P., Testud J. , and Marzoug M. , 1993: Proposal for a spaceborne dual-beam rain radar with Doppler capability. J. Atmos. Oceanic Technol., 10 , 262276.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Dias, J. M. B., and Leitao J. M. N. , 2000: Nonparametric estimation of mean Doppler and spectral width. IEEE Trans. Geosci. Remote Sens., 38 , 271282.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Doviak, R. J., and Sirmans D. , 1973: Doppler radar with polarization diversity. J. Atmos. Sci., 30 , 737738.

  • Doviak, R. J., and Zrnić D. S. , 1993: Doppler Radar and Weather Observations. 2d ed. Academic Press, 562 pp.

  • Gossard, E. E., Snider J. B. , Clothiaux E. E. , Martner B. , Gibson J. S. , Kropfli R. A. , and Frisch A. S. , 1997: The potential of 8-mm radars for remotely sensing cloud drop size distributions. J. Atmos. Oceanic Technol., 14 , 7687.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Harris, R. A., and Battrick B. , Eds.,. . 2001: EarthCARE—Earth, Clouds, Aerosols and Radiation Explorer. ESA Publications Division, 130 pp.

    • Search Google Scholar
    • Export Citation
  • Kobayashi, S., 2002: A unified formalism of incoherent, quasi-coherent, and coherent correlation signals on pulse-pair Doppler operation for a cloud-profiling radar: Aiming for a space mission. J. Atmos. Oceanic Technol., 19 , 443456.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kobayashi, S., Kumagai H. , and Kuroiwa H. , 2002: A proposal of pulse-pair operation on a spaceborne cloud-profiling radar in the W band. J. Atmos. Oceanic Technol., 19 , 12941306.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Meneghini, R., and Kozu T. , 1990: Spaceborne Weather Radar. Artech House, 199 pp.

  • Rice, S. O., 1944: Mathematical analysis of random noise. Bell Syst. Tech. J., 23 , 282332.

  • Sirmans, D., and Bumgarner B. , 1975: Numerical comparison of five mean frequency estimators. J. Appl. Meteor., 14 , 9911003.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Taneli, S., Im E. , Durden S. L. , Facheris L. , Giuli D. , Haddad Z. S. , and Smith E. A. , 2001: Spaceborne radar measurements of vertical rainfall velocity: The non-uniform beam filling considerations. Proc. 2001 Int. Geoscience and Remote Sensing Symp., Vol. II, Sydney, Australia, IEEE, 679–681.

    • Search Google Scholar
    • Export Citation
  • Zrnić, D. S., 1975: Simulation of weatherlike Doppler spectra and signals. J. Appl. Meteor., 14 , 619620.

  • Zrnić, D. S., 1977: Spectral moment estimates from correlated pulse pairs. IEEE Trans. Aerosp. Electron. Syst., AES-13 , 344354.

  • Zrnić, D. S., 1979a: Estimation of spectral moment for weather echoes. IEEE Trans. Geosci. Electron., GE-17 , 113128.

  • Zrnić, D. S., 1979b: Spectrum width estimates for weather echoes. IEEE Trans. Aerosp. Electron. Syst., AES-15 , 613619.

Fig. 1.
Fig. 1.

A block diagram of a pulse-pair signal processor. Digitized simulated signals are converted to analog by an arbitrary waveform generator (AWG). Noise generated in an RF–IF unit merges with the signals from the AWG

Citation: Journal of Atmospheric and Oceanic Technology 20, 6; 10.1175/1520-0426(2003)020<0944:AEODVO>2.0.CO;2

Fig. 2.
Fig. 2.

Schematic diagrams of envelope functions. (a) A set of random signals Ji(t) is generated independently by the Rice method. These Ji(t) are connected through Gaussian-like envelope functions to give the fluctuation of the complex coefficient cn explicitly in a time series. (b) A simplified model. Trapezoidal envelope functions, which are constituted of the steady envelope regions and the transient envelope regions, are adopted

Citation: Journal of Atmospheric and Oceanic Technology 20, 6; 10.1175/1520-0426(2003)020<0944:AEODVO>2.0.CO;2

Fig. 3.
Fig. 3.

Accuracy in Doppler velocity vs pulse-pair interval Ts for contiguous pulse-pair operation. The along-track integration and the spectral width of velocity are set at d = 1 km and συ = 3.85 m s−1 (Tc = 93 μs), respectively. The simulations are performed over 10 000 times, as described in section 2, for the parameters Tlt = 10Tc and Tls = 2Tc. The noises superimposed on the data SNR = −15 to +10 dB have been assumed to be white. The broken lines are calculated by a perturbation formula of Doviak and Zrnić (1993)

Citation: Journal of Atmospheric and Oceanic Technology 20, 6; 10.1175/1520-0426(2003)020<0944:AEODVO>2.0.CO;2

Fig. 4.
Fig. 4.

Accuracy in Doppler velocity vs pulse-pair interval Ts for contiguous pulse-pair operation. The simulations, performed by the original Rice method, are otherwise the same as in Fig. 3. Good agreement with Fig. 3 indicates that the method of this paper (Fig. 3) gives the same correlation and power spectrum as the original Rice method, despite a qualitative difference in random signals

Citation: Journal of Atmospheric and Oceanic Technology 20, 6; 10.1175/1520-0426(2003)020<0944:AEODVO>2.0.CO;2

Save
  • Amayenc, P., Testud J. , and Marzoug M. , 1993: Proposal for a spaceborne dual-beam rain radar with Doppler capability. J. Atmos. Oceanic Technol., 10 , 262276.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Dias, J. M. B., and Leitao J. M. N. , 2000: Nonparametric estimation of mean Doppler and spectral width. IEEE Trans. Geosci. Remote Sens., 38 , 271282.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Doviak, R. J., and Sirmans D. , 1973: Doppler radar with polarization diversity. J. Atmos. Sci., 30 , 737738.

  • Doviak, R. J., and Zrnić D. S. , 1993: Doppler Radar and Weather Observations. 2d ed. Academic Press, 562 pp.

  • Gossard, E. E., Snider J. B. , Clothiaux E. E. , Martner B. , Gibson J. S. , Kropfli R. A. , and Frisch A. S. , 1997: The potential of 8-mm radars for remotely sensing cloud drop size distributions. J. Atmos. Oceanic Technol., 14 , 7687.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Harris, R. A., and Battrick B. , Eds.,. . 2001: EarthCARE—Earth, Clouds, Aerosols and Radiation Explorer. ESA Publications Division, 130 pp.

    • Search Google Scholar
    • Export Citation
  • Kobayashi, S., 2002: A unified formalism of incoherent, quasi-coherent, and coherent correlation signals on pulse-pair Doppler operation for a cloud-profiling radar: Aiming for a space mission. J. Atmos. Oceanic Technol., 19 , 443456.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kobayashi, S., Kumagai H. , and Kuroiwa H. , 2002: A proposal of pulse-pair operation on a spaceborne cloud-profiling radar in the W band. J. Atmos. Oceanic Technol., 19 , 12941306.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Meneghini, R., and Kozu T. , 1990: Spaceborne Weather Radar. Artech House, 199 pp.

  • Rice, S. O., 1944: Mathematical analysis of random noise. Bell Syst. Tech. J., 23 , 282332.

  • Sirmans, D., and Bumgarner B. , 1975: Numerical comparison of five mean frequency estimators. J. Appl. Meteor., 14 , 9911003.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Taneli, S., Im E. , Durden S. L. , Facheris L. , Giuli D. , Haddad Z. S. , and Smith E. A. , 2001: Spaceborne radar measurements of vertical rainfall velocity: The non-uniform beam filling considerations. Proc. 2001 Int. Geoscience and Remote Sensing Symp., Vol. II, Sydney, Australia, IEEE, 679–681.

    • Search Google Scholar
    • Export Citation
  • Zrnić, D. S., 1975: Simulation of weatherlike Doppler spectra and signals. J. Appl. Meteor., 14 , 619620.

  • Zrnić, D. S., 1977: Spectral moment estimates from correlated pulse pairs. IEEE Trans. Aerosp. Electron. Syst., AES-13 , 344354.

  • Zrnić, D. S., 1979a: Estimation of spectral moment for weather echoes. IEEE Trans. Geosci. Electron., GE-17 , 113128.

  • Zrnić, D. S., 1979b: Spectrum width estimates for weather echoes. IEEE Trans. Aerosp. Electron. Syst., AES-15 , 613619.

  • Fig. 1.

    A block diagram of a pulse-pair signal processor. Digitized simulated signals are converted to analog by an arbitrary waveform generator (AWG). Noise generated in an RF–IF unit merges with the signals from the AWG

  • Fig. 2.

    Schematic diagrams of envelope functions. (a) A set of random signals Ji(t) is generated independently by the Rice method. These Ji(t) are connected through Gaussian-like envelope functions to give the fluctuation of the complex coefficient cn explicitly in a time series. (b) A simplified model. Trapezoidal envelope functions, which are constituted of the steady envelope regions and the transient envelope regions, are adopted

  • Fig. 3.

    Accuracy in Doppler velocity vs pulse-pair interval Ts for contiguous pulse-pair operation. The along-track integration and the spectral width of velocity are set at d = 1 km and συ = 3.85 m s−1 (Tc = 93 μs), respectively. The simulations are performed over 10 000 times, as described in section 2, for the parameters Tlt = 10Tc and Tls = 2Tc. The noises superimposed on the data SNR = −15 to +10 dB have been assumed to be white. The broken lines are calculated by a perturbation formula of Doviak and Zrnić (1993)

  • Fig. 4.

    Accuracy in Doppler velocity vs pulse-pair interval Ts for contiguous pulse-pair operation. The simulations, performed by the original Rice method, are otherwise the same as in Fig. 3. Good agreement with Fig. 3 indicates that the method of this paper (Fig. 3) gives the same correlation and power spectrum as the original Rice method, despite a qualitative difference in random signals

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