a. Formulation

For oversampled signals the received waveform is a convolution of the reflectivity profile and the transmit pulse. It can be shown that the correlation of the range samples is a function of the transmitted pulse shape and can be expressed as (following Bringi and Chandrasekar 2001)

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where R_υ (t, t + Δt) is the correlation function of range samples at the receiver output, h(t) is the convolution of the transmit pulse {U_tr(t)} and the impulse response function of the receiver filter {g(t)}, C is a constant, and f_η(t; Δt) is the normalized time-correlated scattering cross section of particles. Torres and Zrnic (2003a) proposed transforming the correlated range samples into independent samples using the covariance matrix information of the range samples.

The eigen-decomposition method is used to diagonalize the positive definite covariance matrix. Let 𝗖_v be the normalized Hermitian covariance matrix of range samples; then, its eigen-decomposition can be expressed in terms of a unitary matrix 𝗨 and a diagonal matrix Λ as follows (Golub and Loan 1996): columns of 𝗨 are the eigenvectors of 𝗖_v, and the diagonal elements of Λ (λ_i) are the eigenvalues of the covariance matrix,

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Formulation of the transformation matrix 𝗪 is shown below, where 𝗗 is a diagonal matrix with elements {λ^−1/2_i}.

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The whitened (or uncorrelated) data vector X_h is obtained by a linear transformation comprising V_h (oversampled), the vector containing correlated data, and the transformation matrix 𝗪 as

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Both X_h and V_h are column vectors of size L, where L is the oversampling factor.

The CSU-CHILL radar has two independent transmitters and a slight mismatch in the transmitted waveforms, which results in slightly different covariance properties. Section 2b shows the transmit waveforms and their correlation characteristics. This presents a different scenario than the results presented by Torres and Zrnic (2003b), where the studies are based on the assumption of identical waveforms being transmitted in H and V polarization. The following discussion analyzes how a mismatch between transmitted pulses of different polarizations influences the copolar correlation and consequently the estimation accuracy of polarimetric parameters.

The cross-correlation matrix of the oversampled H and V signals can be expressed as

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Following Eq. (4), the above equation for whitened vectors can be rewritten as

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Using the previous equation and recognizing that 𝗖^os_v,h = E(V_v,hV_v,h*^T), the estimators for power can be expressed as

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Therefore, the copolar correlation of whitened data can be estimated as

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The cross-correlation in range–time between oversampled signals in two polarizations (𝗖^os_vh) is the product of copolar correlation at lag zero (ρ^os_co) and the cross-correlation of the transmit pulses (𝗥_vh). Using this and normalizing the numerator and denominator by the mean power of the H and V signals, Eq. (8) can be rewritten as

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If the transmitted pulses are identical, the autocorrelation of the range signals in any polarization would equal the range–time cross-correlation between the H and V signals. Otherwise, the numerator and denominator in Eq. (9) would be unequal. Therefore, in the presence of a mismatch between the correlation functions of the H and V transmit pulse, a diagonalization scheme based on Eq. (3) may not be successful. This would bias the copolar correlation of the whitened signals, which will consequently affect the estimation quality of the polarimetric parameters.

b. Implementation

The CSU-CHILL radar can transmit short (0.33 μs) and long (1 μs) pulses. Presently, its oversampling capabilities are limited to 3 MHz. Figure 1 shows a block diagram of the CSU-CHILL radar’s receiver chain. Directional couplers are inserted between the transmitter and circulators to obtain high-quality samples of each transmit pulse. Next, the waveforms are downconverted to 10 MHz (Brunkow 1999). The 10-MHz intermediate frequency (IF) signal is digitized at 40 MHz and a low-pass filter is used to perform quadrature detection and produce I and Q samples. Figure 2 shows the characteristics of the CSU-CHILL transmit pulse in H and V polarization. Figure 3 shows the magnitude and phase of the correlation functions of the H and V polarization signal. The existence of a differential phase pattern between the phase patterns of the transmitted waveforms (Fig. 2) was verified by observing multiple realizations of the waveforms at the H and V receiver output.

Figure 4 shows the normalized power spectrum of the transmit pulses that are respectively filtered by a 1- and 3-MHz bandwidth filter. It can be seen that even though the transmitted pulse is limited to 1 MHz, there are high-frequency components several tens of decibels below which that otherwise would be attenuated by a matched filter. Observation of the variability in the received signal within intervals as small as the subpulse period enables better generation of independent samples from the correlated data.

a. Case I: Simulation study

Figure 5 shows a mean reflectivity profile observed by the CSU-CHILL radar. This was used to simulate dual-polarized signals using samples of the H and V transmitted pulse (Fig. 2) using an oversampling factor of 5. Fifty time series were generated for each range bin, with each realization containing 128 samples. A spectral domain–based algorithm, as described in Chandrasekar et al. (1986), is used for the simulation. Two different finite-impulse response (FIR) filters with bandwidths of 1 and 5 MHz are used to simulate signals at the receiver output. Power estimates from the oversampled/whitened data are obtained using the following algorithm:

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where V_h,υ refer to returns signals in H/V polarization and L and N are the oversampling factor and length of time series, respectively.

The whitening transformation matrix is computed by using the magnitude of the correlation functions. Computation of the transformation matrix from a real correlation matrix has also been suggested by Ivic et al. (2003). Range correlation coefficients up to L – 1 lags were used to compute the whitening transformation matrix.

The standard deviation of reflectivity estimates obtained from oversampled (only averaged) and whitened (averaged after whitening) data are shown in Fig. 6. The figure compares the accuracy of reflectivity estimates using the following methods: (a) normal sampling using a narrow receiver filter, (b) oversampling using a wideband receiver filter, and (c) subsequent whitening of the oversampled signals obtained using both matched and wideband receiver configurations. It must be observed that the wideband receiver provides estimates with the smallest error. Increasing the receiver bandwidth causes the noise power to increase, whereas the improvement obtained from an increased number of independent samples offsets the effect of noise enhancement (Torres 2001). Therefore, it would be desirable to find an optimum bandwidth for best estimation quality. However, this paper does not focus on that aspect.

The following equations are used to estimate the differential reflectivity (Z_dr), differential phase (ϕ_dp), and magnitude of the copolar coefficient (ρ_co):

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Figures 7a and 7b show the standard deviation of Z_dr and ϕ_dp estimates obtained from the simulation. It should be noted that the degree of improvement in estimation accuracy for Z_dr and ϕ_dp after whitening is less when compared to the reflectivity estimates. It is known that a wideband receiver increases the noise power, which affects the copolar correlation to some extent. However, nonidentical transmitted pulses are another major factor that affects the copolar correlation of whitened data. A comparison between the whitening-based polarimetric parameter estimators using different receiver configurations shows this. However, the influence of noise boost is not as significant as the impact of the mismatch between transmit pulses. Figure 8 shows the estimates of copolar correlation obtained from simulated data and the estimated copolar coefficient obtained from Eq. (9) using the measurements of H and V transmitted pulses. This figure also shows the estimates of ρ_co obtained after whitening from another simulation based on the same input profile (Fig. 5). It used the H transmitted pulse to generate time series data for both polarizations. The result shows that there is almost no bias in ρ_co estimates when identical pulses are used in the transmission, and the impact of noise boost resulting from whitening is minimal. A significant drop in copolar correlation can be observed when the mismatch is introduced. It is worth noting that the results from the simulation are in agreement with estimates based on Eq. (9).

b. Case II: Data analysis

Figure 9a shows a range profile of reflectivity, copolar correlation, and SNR as observed by the CSU-CHILL radar. The data were collected using an oversampling factor of 3 and a 1-μs pulse. A wideband receiver filter with 3-MHz bandwidth was used. The collected time series data were whitened using the same transformation as described earlier in the paper. Figure 9b shows the bias in copolar correlation following whitening. Figure 9c shows the standard deviation of the estimates for various mean parameters obtained from whitened range oversamples. A significant improvement in the standard deviation of reflectivity and mean velocity estimates must be noted. It must be observed that even under such high SNR conditions there is not any visible improvement in the standard deviation of ϕ_dp estimates. There is occasional improvement in the measurement accuracy of Z_dr estimates, but they are not consistent.