# Whitening in Range to Improve Weather Radar Spectral Moment Estimates. Part I: Formulation and Simulation

Abstract: A method for estimation of spectral moments on pulsed weather radars is presented. This scheme operates on oversampled echoes in range; that is, samples of in-phase and quadrature-phase components are collected at a rate several times larger than the reciprocal of the transmitted pulse length. The spectral moments are estimated by suitably combining weighted averages of these oversampled signals in range with usual processing of samples (spaced at the pulse repetition time) at a fixed range location. The weights in range are derived from a whitening transformation; hence, the oversampled signals become uncorrelated and, consequently, the variance of the estimates decreases significantly. Because the estimate errors are inversely proportional to the volume scanning times, it follows that storms can be surveyed much faster than is possible with current processing methods, or equivalently, for the current volume scanning time, accuracy of the estimates can be greatly improved. This significant improvement is achievable at large signal-to-noise ratios.

## 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.

Did you find this useful? Give us your feedback

...read more

##### Citations

225 citations

### Cites background from "Whitening in Range to Improve Weath..."

...Oversampling and whitening of signals in range is a candidate for increasing the speed of volume coverage and reducing the errors of estimates (Torres and Zrnic 2003)....

[...]

81 citations

### Cites background from "Whitening in Range to Improve Weath..."

...Also, Torres and Zrnic (2003) proposed a technique that can significantly reduce statistical errors while maintaining the same level of current WSR-88D radar capabilities such as the scan rate....

[...]

78 citations

### Cites background or methods from "Whitening in Range to Improve Weath..."

...However, oversampling in range can be used to reduce the uncertainty of weather data without increasing update times (Torres and Zrnić 2003)....

[...]

...For example, range oversampling techniques (Torres and Zrnić 2003) use faster sampling rates at the radar receiver so that more samples are acquired in range without increasing the dwell times; range samples collected in this way can be decorrelated and used to reduce the variance of estimates via…...

[...]

...In addition, the accuracy of meteorological data can be improved by using range oversampling techniques (Torres and Zrnić 2003) or beam multiplexing (Yu et al. 2007)....

[...]

64 citations

60 citations

##### References

13,864 citations

13,734 citations

### "Whitening in Range to Improve Weath..." refers background in this paper

...Any H that satisfies (5) is called a square root of (Faddeev(R)CV and Faddeeva 1963) and is the inverse of a whitening transformation matrix, 21W 5 H , (6) which, if applied to the range samples, produces L uncorrelated random variables with identical power (Kay 1993)....

[...]

6,705 citations

### "Whitening in Range to Improve Weath..." refers methods in this paper

...Two such methods are the eigenvalue decomposition (Therrien 1992) and Cholesky decomposition, which is equivalent to Gram–Schmidt orthogonalization (Therrien 1992; Papoulis 1984)....

[...]

2,130 citations

1,887 citations

### "Whitening in Range to Improve Weath..." refers background in this paper

...That is, the range spanned by the power spectral density matches closely the range of eigenvalues (Johnson and Dudgeon 1993)....

[...]