Abstract

It was recently demonstrated that magnitudes of the power-normalized cross-correlation functions of complex amplitudes in neighboring range bins are identical to the fractional contributions made by radar coherent backscatter in the direction of propagation to the total backscattered power in rain and snow. Here, a theoretical framework is presented for calculating the noise associated with estimates of these normalized cross correlations. This noise is identical to the statistical uncertainties in . Radar signals consist of two components: the usual incoherent backscatter often modeled by a Gaussian process and a coherent component modeled for the purposes of these calculations by a phasor C of fixed magnitude that rotates at a constant angular velocity ωC. Using the representation of the cross-correlation function as the average over the real part of the phasor dot products, it is found that the noise in this function comes from the dot products of C with the incoherent-scatter phasors in each range bin as well as the dot product between the two incoherent phasors. Furthermore, as long as ωC ≠ 0 and the number of statistically independent realizations (samples) k is sufficiently large, the noise is represented well by a normal distribution with mean 0 and with a variance that goes as 1/(2k). It is then shown that as the magnitude of C increases it acts to suppress the variance of . A formula is derived that gives the standard deviation of as a function of the number of statistically independent samples in the observation and the observed value of . Two examples, one in rain and the other in snow, are also presented.

1. Introduction

Recent results (Jameson and Kostinski 2010a,b) show that precipitation can produce radar coherent backscatter (Bragg scatter) when the streamers of precipitation are spaced on scales that resonate with the radar wavelength. In particular, Jameson and Kostinski (2010b) show that the cross-correlation function ρ12 between simultaneous complex amplitudes in neighboring range bins normalized by the geometric mean of their respective powers is the mean fractional coherent contribution to the total backscattered power: ; Z1 and Z2 are the mean backscattered powers (i.e., mean squared amplitudes of the backscattered waves) in the first and second range bin, respectively. In this brief note, the statistics of the noise in measurements of in the direction of wave propagation derived using range cross correlation are computed. The uncertainties (variances) of are then calculated.

To be specific, because the power backscattered to a radar most generally has incoherent and coherent components, at any given instant one can assume, therefore, that the phasors in each of the two range bins can be written as E1 = C + r1 and E2 = C + r2, where C is the common coherent component. This common C is the reason that ρ12 ≠ 0, as discussed in Jameson and Kostinski (2010b). Here, r represents the incoherent phasor component.

For the moment, if one lets E1 = a1 + jb1 and E2 = a2 + jb2, where j is the imaginary number, and use , one has that ρ12 = 〈a1a2 + b1b2〉 = 〈ℜ(E1 · E*2)〉 so that

 
formula

where Φ is the angle between E1 and E2, the asterisk denotes complex conjugation, ℜ denotes the real part of a complex number, and 〈〉 denotes ensemble (or time) averaging. The final step in (1) is possible because |E1|, |E2|, and Φ are statistically independent.

First, let us consider the case in which C = 0. Then the correlation would be between independent sample volumes having only random components in common at any instant. Angle Φ would then be equally likely to assume any value in [0, 2π] so that 〈cos(Φ)〉 → 0 as would .

When C is not zero, however,

 
formula

where t denotes the index of a sequence of samples. As a consequence,

 
formula

where is the “noise” given by

 
formula

Note that (3) is identical to (6) in Jameson and Kostinski (2010b) except for the additional noise term. Note also in using (3) that the variance of is equivalent to the variance of the observed .

Because r1 and r2 are random vectors, all of the dot products within the parentheses in (4) approach the null as the number of independent samples (realizations) k increases so that ρ12/(Z1Z2)1/2 is an unbiased estimator of . Because most measurements contain only a finite number of independent samples, however, two questions arise: How fast does the variance of (and, therefore, of ) decrease with increasing k? What role does C (or ) play in this variance? In that regard, it is reasonable to guess that when , where is that due solely to purely incoherent scatter [i.e., the r1 · r*2 term in (4)], which, in turn, depends upon k. On the other hand, as , N → 0 so that one anticipates that the variance is likely to be of the form

 
formula

where f is a function such that .

When , it is straightforward to show that σ2(k)Inc = 1/(2k).That is, when C is 0 and one uses (4), it follows that and the mean is null. On the other hand, the mean , since the mean cos2 is 0.5 and the mean |r1|2 and |r2|2 are Z1 and Z2, respectively. Hence, for k samples, the estimate of the sample variance will be σ2(k)Inc = Z1Z2/(2k), which, when normalized by Z1Z2, yields 1/(2k). This represents the maximum possible variance or uncertainty in .

However, defined by (4) involves sums of normal product distributions (i.e., sums of modified Bessel functions of the second kind). Therefore, an exact determination of the distribution of would then require convolutions of six of those functions. Since parametric forms of such convolutions do not exist, we, instead, derive the approximate distributions of using Monte Carlo calculations described next.

2. Calculations

In radar meteorology, the so-called I and Q components refer to the sine and cosine contributions to the complex phasor [i.e., the a and b, respectively, in (1)]. For incoherent scatter, the particles act independently and randomly so that I and Q are statistically independent and Gaussian distributed as is discussed in any book on radar meteorology. This is the assumption here as well for the I and Q components of r1 and r2. Vector C is modeled as a fixed phasor rotating at a constant angular velocity ωC [consistent with the requirement in the derivations of radar coherent backscatter in Jameson and Kostinski (2010a,b)] contained within both neighboring range bins [as required in Jameson and Kostinski (2010b)]. Hence, the noise in the normalized cross-correlation function (identical to the noise in ) can be calculated as the residual:

 
formula

where the averaging is over k independent realizations. For each independent sample of k realizations,

 
formula

while

 
formula

where C(t) simply indicates that the constant-magnitude C is rotating from realization to realization. To determine the frequency distributions of and its properties (mean and variance), (7) was computed 100 000 times for each number of independent samples k from the set {5, 10, 20, 40, 80, 140, 200, 500}.

The Is and Qs are drawn from Gaussian distributions of statistically independent random numbers having mean 0 and unit variance. (Changing the variance is equivalent to changing the mean powers, which, because of normalization and the definition of , apparently does not affect the results as verified through calculations.) The magnitude of C is varied to produce different . The results are presented next.

3. Results

Let one first consider the distributions of for k = 10, 40, 140, and 500 as illustrated in Fig. 1. Beyond k = 10, these distributions are represented well by normal fits with mean 0. Further inspection shows that for k ≥ 20 the normal distribution is valid. Only when k < ∼20 does the distribution become more betalike with a very slight negative bias. It turns out that this is not important, however, because calculations show (see below) that the variance of the noise then becomes so large when k < 30–40 that ρ12 ceases to be a very useful estimator of . Hence, this normal distribution approximation is used. It is still necessary to compute the variances of (or equivalently the variances of ) using simulations, however.

Fig. 1.

The frequency distributions of the noise calculated using the complex amplitude cross-correlation functions between neighboring range bins normalized by the geometric mean of the powers in the two bins. As long as the number of independent samples k exceeds about 20 (a necessity for accuracy anyway as discussed in the text), the distributions are well represented by 0-mean Gaussian distributions as shown by the k = 500 example (solid line: curve with highest peak).

Fig. 1.

The frequency distributions of the noise calculated using the complex amplitude cross-correlation functions between neighboring range bins normalized by the geometric mean of the powers in the two bins. As long as the number of independent samples k exceeds about 20 (a necessity for accuracy anyway as discussed in the text), the distributions are well represented by 0-mean Gaussian distributions as shown by the k = 500 example (solid line: curve with highest peak).

Using these calculated variances, one finds that, even down to k = 5, the variances apparently go as 1/k. This is illustrated in Fig. 2 for several different . Note that the example fit corresponding to goes exactly as 1/k. Also note that there is a dependence on as well.

Fig. 2.

The standard deviation of , denoted as , as a function of the number of independent samples k used to determine the mean noise. They reveal a k−1/2 dependence as illustrated for the coherent fractional contribution to the total power, (long dash–short dashed line and gray solid line: bottom two curves).

Fig. 2.

The standard deviation of , denoted as , as a function of the number of independent samples k used to determine the mean noise. They reveal a k−1/2 dependence as illustrated for the coherent fractional contribution to the total power, (long dash–short dashed line and gray solid line: bottom two curves).

To be specific, in Fig. 3 it is found that

 
formula

The 1/(2k) coefficient is as expected (derived above), thus giving credence to the calculations. The larger the observed is, however, the smaller is . This happens because, as increases, coherency more and more dominates the total signal so that signal fluctuations from the incoherent scatterers become less and less important.

Fig. 3.

The formula relating the variance of the measured to k and . As discussed in the text, this formula is generally applicable for 5 ≤ k ≤ 500.

Fig. 3.

The formula relating the variance of the measured to k and . As discussed in the text, this formula is generally applicable for 5 ≤ k ≤ 500.

Equation (9) is found to work extremely well over all of these Monte Carlo results. For example, for and k = 5, (9) yields as compared with the Monte Carlo result of 0.1013. At the other extreme, for and k = 500, (9) yields 0.000 331 3 as compared with the Monte Carlo result of 0.000 331 3. The first example represents a worst-case scenario because and k are both small. Hence, the conclusion is that (9) adequately describes the uncertainty associated with estimates of derived using ρ12 over the range of likely meteorological values.

To illustrate this result, Fig. 4 shows an example from snow using data already described in detail in Jameson and Kostinski (2010b). For these data, the mean number of statistically independent samples over the 1000 1-ms samples was determined to be 67. Using that number and the observed , the ±1σ levels are denoted by the shaded region encompassing the observed . It is clear that the statistical reliability and the reality of for this set of observations are irrefutable. It also means that the observed fluctuations are likely real reflections of the transient spatial–temporal nature of the sources of the radar coherent backscatter.

Fig. 4.

An illustration of the application of (9) to some high-resolution (30 m) snow data as discussed in the text. All of the values of are statistically reliable.

Fig. 4.

An illustration of the application of (9) to some high-resolution (30 m) snow data as discussed in the text. All of the values of are statistically reliable.

For completeness, an example in rain is also shown (Fig. 5). As for the snow, these data have been described elsewhere in detail. The essential point here again is that the observed are real and that the variability is likely a true reflection of the spatial–temporal variability of the coherency found in rain.

Fig. 5.

As in Fig. 4, but for lower-resolution (150 m) rain as discussed in the text. As for the snow, all of the values of are statistically reliable.

Fig. 5.

As in Fig. 4, but for lower-resolution (150 m) rain as discussed in the text. As for the snow, all of the values of are statistically reliable.

4. Concluding statements

Based upon the understanding of the nature of radar coherent backscatter found in Jameson and Kostinski (2010a,b), the radar signal is modeled as a combination of a constant-amplitude but rotating phasor vector common to two neighboring sample volumes embedded within the ensemble of otherwise incoherent scatterers described using uncorrelated Gaussian I and Q phasor components. Monte Carlo simulations are used to generate simultaneous radar backscatter complex amplitudes in the two neighboring bins over a wide range of the number of independent samples k and fractional coherent backscatter contribution to the total power .

Using the average of the real part of the dot product between the phasors in the two sample volumes as a geometrical representation of the cross-correlation function, it is found that the noise component of the backscattered power-normalized complex amplitude cross-correlation function between neighboring bins stems from correlations of the coherent component with the incoherent components as well as between the incoherent components in the neighboring bins. Because a theoretical derivation for the distribution of is not possible, Monte Carlo simulations are used to generate simultaneous radar backscatter complex amplitudes in the two neighboring bins over a wide range of the number of independent samples k and fractional coherent backscatter contribution to the total power . It is shown that the noise is unbiased (0 mean) and that its distribution can be represented well as a normal distribution at least for the k required to yield meaningful estimates of . The results also show a simple expression revealing a 1/(2k) dependence of the variance of as well as a simple dependence on the difference of from unity. Hence, for a given k, the larger the is, the smaller are the standard deviation and relative error. This relation appears to be valid for 5 ≤ k ≤ 500.

This result makes physical sense because, as increases, the incoherent contribution to the cross-correlation function decreases in a manner analogous to what happens for the temporal autocorrelation function at one bin [see (9b) in Jameson and Kostinski 2010c]. Because in this context the very meaning of coherence is for signals to “cohere,” fluctuations are very small to nonexistent. Hence, incoherent scatter must produce the maximum possible variance in ρ12 so that as its contribution decreases so too do the uncertainties associated with estimates of .

Acknowledgments

This work was supported by the National Science Foundation under Grant ATM08–04440.

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Footnotes

Corresponding author address: A. R. Jameson, 5625 N. 32nd St., Arlington, VA 22207-1560. Email: arjatrjhsci@verizon.net