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 T_c ≈ 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 T_s = 60 μs and a pulse repetition interval of T_pri = 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 T_s = T_pri = 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 T_s = 100–125 μs larger than the coherence time T_c ≈ 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 T_s ≳ T_c, 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 T_s < T_c 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 σ_υ, or coherence time T_c, and pulse-pair interval T_s.

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:

View Expanded

in which ω_n has been defined as

ω_nπnT⁻¹

and the complex coefficient c_n is decomposed to the real coefficients a_n and b_n as

c_na_nib_n(2)

Here a_n and b_n are Gaussian random variables satisfying the following relations:

View Expanded

In these equations, 〈x〉 represents the mean value of x, and δ_nm designates the Cronecker delta. In short, Eqs. (3)–(5) can be summarized as

c^*_mc_nδ_nmσ²_n(6)

In these equations, σ²_n is related to the power spectrum S(ω) of the signal through

σ²_n⁻¹Sωω,(7)

in which Δω = 2πT⁻¹ is an sampling interval of discrete frequency.

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 c_n are fixed for a given n within a time train of T, and the distribution of c_n never fluctuates within T. Instead, the randomness and fluctuation of c_n 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 c_n had better be included explicitly in a time series, for which the following method can be considered.

First, a set of currents J_i(t) (i = 1, 2, …) is generated independently by the Rice method. Second, these J_i(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 J_i(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 T_lt extending in [−(T_lt + T_ls), −T_ls] and [0, T_lt], and the steady envelope regions in between. In the first transient region [−(T_lt + T_ls), −T_ls], 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 [−T_ls, 0]. In the second transient region [0, T_lt], 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(9)

for arbitrary t₂ and t₁ within a radar scan. In the steady region, Eq. (9) is automatically satisfied. In the transient region [0, T_lt], the total signal can be written in the form

JtAtJ₁tBtJ₂t(10)

and the time correlation can be represented as

Jt₁Jt₂At₁At₂Bt₁Bt₂rt₂t₁(11)

However, Eq. (9) indicates that the functions A(t) and B(t) must satisfy

At₁At₂Bt₁Bt₂(12)

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:

View Expanded

In order to calculate Eq. (12), the time t₂ is set at

t₂t₁T_s(14)

in which T_s designates a pulse-pair interval. Then Eq. (12) can be written as a function of the first pulse time t₁ along with the parameters T_s, T_lt, and T_ls:

View Expanded

Since the transition duration T_lt and the steady duration T_ls are parameters within which signal correlation must hold, we can consider that these parameters are at least the length of coherence time T_c. In this paper, the steady duration T_ls is assumed to be T_ls = 2T_c ≈ 200 μs so that we can continuously evaluate the accuracy beyond the pulse interval T_s ≈ 120 μs, as required in the EarthCARE mission. Furthermore, from numerical calculations for Eq. (15), the transient durations of T_lt = 2T_c and T_lt = 10T_c are found to be adopted as a good parameter up to T_s ≈ 50 μs and T_s ≈ 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 υ_pl = 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):

σ²_υσ²_dopσ²_cldσ²_win(16)

In Eq. (16), σ_cld and σ_win 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 σ_cld = 0.5 m s⁻¹ (Gossard et al. 1997) and σ_win = 1 m s⁻¹ (Amayenc et al. 1993). The σ_dop denotes the spectral width reflecting the combined effect of Doppler fading and vertical wind shears (Kobayashi 2002). Ignoring the wind shears, we can calculate σ_dop = 3.72 m s⁻¹. Hence Eq. (16) yields the total velocity width of σ_υ = 3.85 m s⁻¹, and the corresponding coherence time is given by

T_c2kσ_υ⁻¹μ(17)

where k is the radiation wavenumber for 95.04 GHz. Converting the spectral width of Doppler velocity to the frequency width through the equation σ_ω = 2kσ_υ, 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)

View Expanded

in which N designates the total number of pulse pairs in the distance d with the platform velocity υ_pl, given by

Ndυ_plT_s(19)

Finally, the estimated mean Doppler velocity is derived from

υkT_s⁻¹RT_s(20)

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 T_lt = 10T_c and T_ls = 2T_c. The ordinate and abscissa represent the std dev (υ) and the pulse-pair interval T_s, 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 T_s ≲ 130 μs with higher SNRs, as expected from the postulation of the perturbation theory. In addition, simulations for the other parameters, T_lt = 2T_c and T_ls = 2T_c, were also performed, the result of which was almost identical to that shown in Fig. 3; for T_s < 50 μs, especially, the agreement was excellent.

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 c_n 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 T_lt = 10T_c and T_ls = 2T_c, 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.

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 T_s ≲ 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 T_s ≈ 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.