Whitening in Range to Improve Weather Radar Spectral Moment Estimates. Part I: Formulation and Simulation
Summary (2 min read)
1. Introduction
- Doppler weather surveillance radars probe the atmosphere and retrieve spectral moments for each resolution volume in the surrounding space.
- With such processing, the variance reduction of averaged estimates is inversely proportional to the equivalent number of independent samples, which depends on the correlation between samples r and the total number of averaged samples (or pulses) M (Walker et al. 1980).
- The main advantage of this technique is that the whitening transformation is derived from a known correlation function.
2. Why whitening?
- Simple averaging, however, does not yield the best performance when the observations are correlated.
- Hence, it can be inferred that it is not the correlation between observations that limits the accuracy of a given estimator but the way those observations are used to compute the estimates.
- The authors suggest combining the whitening transformation of samples in range with autocovariance processing in sample time and thus improving the spectral moment estimates.
- The proposed processing increases the equivalent number of independent samples in a simple manner while the sacrifice in range resolution is minimal and the transmission bandwidth is not broadened.
- Even faster rates of volumetric data are required to determine the presence of transverse winds.
3. The whitening transformation
- For convenience, the contribution from the resolution volume to the received sampled complex voltage V(nTs) 5 I(nTs) 1 jQ(nTs) at a fixed time delay nTs can be decomposed into subcontributions s(lto, nTs) from L contiguous elemental shells or ‘‘slabs,’’ each ct/2L thick.
- The voltages s(l, n) are identically distributed, complex, Gaussian random variables, where the real and imaginary parts have variances s2, and the average power of s(l, n) is 5 2s2.
- Introducing pm into (3) produces ; this(R)rV needs to be done only once for a given pulse shape and receiver bandwidth.
- In general, the decomposition of the correlation matrix is not unique and many well-known methods can be applied to generate different whitening transformations.
- Two such methods are the eigenvalue decomposition (Therrien 1992) and Cholesky decomposition, which is equivalent to Gram–Schmidt orthogonalization (Therrien 1992; Papoulis 1984).
4. The noise enhancement effect
- The presence of noise is inherent in every radar system; therefore, it is necessary to analyze the performance of the whitening transformation, under noisy conditions.
- Note that an extra L factor should be added if comparing with the noise power in the classical processing.
- The trade-off between noise enhancement (radar sensitivity) and variance reduction makes the whitening transformation useful in cases of relatively large SNR.
- That is, the range spanned by the power spectral density matches closely the range of eigenvalues (Johnson and Dudgeon 1993).
- The analysis of these and other suboptimal techniques is a subject for further study.
5. Spectral moment estimators
- The estimation of spectral moments using a whitening transformation on oversampled data is performed in three steps.
- The performance of WTB estimators is compared with that achieved by the classical matched-filter-based (MFB) estimators and the estimators obtained from oversampled data and regular averaging.
- Figure 2 shows the normalized standard deviation of WTB, MFB, and OAB power (Fig. 2a), Doppler velocity (Fig. 2b), and spectrum width estimators (Fig. 2c) as a function of the SNR for the ideal case and a normalized spectrum width of 0.08.
- When compared with MFB (or OAB) estimates, WTB estimates exhibit a superior performance for large SNR.
6. Discussion
- Section 5 discussed the application of the whitening transformation to the estimation of spectral moments.
- That is, approximately L-times fewer samples are needed for WTB estimators to keep the same errors as the ones obtained without the aid of a whitening transformation.
- The constraint of uniform reflectivity is the principal assumption required to precompute the exact correlation of oversampled signals in range.
- It is understood that these idealized conditions will not be satisfied for all the resolution volumes in an operational environment, especially at the edge of precipitation cells, where very sharp gradients could exist.
- If noise dominates estimation accuracy, pulse compression has approximately an L2 edge in SNR over whitening.
7. Conclusions
- A method for estimation of Doppler spectral moments on pulsed weather radars was presented.
- As with the previous works, the whitening transformation is used in such a way that the equivalent number of independent samples equals the number of samples available for averaging and, consequently, the variance of the estimates decreases significantly.
- For SNRs larger than the SNRc, WTB estimates are preferred over classical estimates.
- Funding for this research was provided under NOAA-OU Cooperative Agreement NA17RJ1227.
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Citations
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References
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"Whitening in Range to Improve Weath..." refers methods in this paper
...With this decomposition, H 5 U*L1/2, and W is obtained as W 5 H21 5 L21/2UT, which is the Mahalanobis transformation (Tong 1995)....
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"Whitening in Range to Improve Weath..." refers background or methods in this paper
...The scheme exploits the idea of whitening to obtain independent samples, such as in the works of Dias and Leitão (1993), Schulz and Kostinski (1997), Koivunen and Kostinski (1999), Frehlich (1999), and Fjørtoft and Lopes (2001)....
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...Further, Koivunen and Kostinski (1999) explored practical aspects of the whitening transformation on estimation of signal power....
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...The procedure for implementing the whitening transformation follows Koivunen and Kostinski (1999) and is listed here for completeness (and reader convenience)....
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...Later, Frehlich (1993) improved Zrnić’s results and derived simplified expressions to test new estimators based on the ML approach....
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...For example, for a reflectivity gradient of 30 dB km21 [referred to as ‘‘extreme’’ in model 1 of Rogers (1971)], it can be shown that the variance reduction factor is approximately cut in half....
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