## 1. Introduction

Radar reflectivity factor *Z* is most commonly calculated from noise-subtracted signal power measurements (Ulaby et al. 1982, 492–495; Doviak and Zrnić 1993; Skolik 1990, 23.1–33; McDonough and Whalen 1995). At low signal-to-noise ratio (SNR), this technique requires an independent estimation of the noise component of the signal-plus-noise measurements. This is typically accomplished by shifting the measurement to a signal-free region in time (Ivić et al. 2013, 2014b) or frequency (Hildebrand and Sekhon 1974; Siggia and Passarelli 2004; Urkowitz and Nespor 1992). However, the noise component cannot always be estimated with sufficient accuracy to prevent significant additional errors in the signal estimate. For example, if the noise samples are gathered prior to each transmit pulse, then multitrip echoes can contaminate the noise estimate. Even in the absence of multitrip echo contamination, the transmitted pulse leakage can desensitize receiver amplifiers such that the receiver gain is a function of range. If the noise samples are collected offset in frequency, then the noise estimate may be biased as result of uncompensated variation in the receiver frequency response. The noise power also depends on the receiver noise figure and the radiometric brightness temperature of the scene, so measuring the noise power corresponding to an internal matched load or temporal averaging “noise” gates when the scene brightness temperature varies can lead to significant errors (Ivić et al. 2013, 2014b). A better method is to average the noise power in several range gates past the farthest scatterers. To ensure signal-free gates, the pulse repetition interval (PRI) needs to be increased beyond what would be necessary to sample the maximum signal range. However, this reduces the folding velocity and the number of samples available for signal measurement at a given temporal averaging interval, consequently, degrading sensitivity. Furthermore, it is often desirable to average many hundreds or even thousands of samples to boost sensitivity, thereby extending the signal power measurement range 10–20 dB below the receiver noise floor. As a result, a significant portion of the measurements are made well below unity SNR, where to avoid large errors in the estimated signal power, the required noise component measurement accuracy can be difficult to achieve. For example, when integrating 2000 pulses, signals that are 10 dB below the noise level can be detected with a high probability of detection (>90%) at a false alarm rate of 0.01. However, at this signal level, a ±0.25-dB error in the estimated noise component introduces approximately +2 to −4 dB error in the calculated signal power.

The coherent power (CP) technique is an alternative method for measuring the radar received signal power without the need for estimating the noise power. The key to this technique is correlating pairs of signal-plus-noise samples in which the signal components are correlated but the noise components are independent. This can be implemented with dual-polarized radars operating in simultaneous transmit and receive (STAR) mode (Seliga and Bringi 1976, Zrnić 1996) by computing the lag-0 cross-correlation magnitude, also known as copolar power (Keranen and Chandrasekar 2014), or using time-delayed CP by computing the lag-1 autocorrelation magnitude. This time-delayed version of CP, normalized to the signal power, has been used for censoring (Bell et al. 2013), and the combination of dual-polarization and the time-delayed version of CP has been shown to improve signal detection (Ivić et al. 2014a, 2009b).

The polarimetric method depends on the fact that the vertically and horizontally polarized copolar backscatter from precipitation is typically highly correlated, while the noise components of the vertically and horizontally polarized receiver channels are independent. The time-delayed version of the CP technique relies on the target fading much more slowly than the pulse-pair spacing, so the signal components of such closely spaced pulses are highly correlated but the noise components are not. The pulse-to-pulse phase difference of the signal components has a mean offset that depends on the mean Doppler velocity, but as long as the PRI is much shorter than the inverse of the spectrum width, this phase difference will be approximately constant. However, the noise component phase is uniformly distributed over

Previously reported applications of CP have been used exclusively to improve signal detection or censoring (Ivić et al. 2009b; Bell et al. 2013; Ivić et al. 2014a; Keranen and Chandrasekar 2014). This paper proposes the use of CP as an alternative to the commonly used noise-subtracted power (NSP) technique for measuring signal power, particularly at low SNR, when the estimation and subtraction of the noise component is necessary with the NSP method. The CP technique is compared to the conventional NSP method when both techniques are properly implemented; that is, when the noise component estimate accuracy is much better than that of the signal-plus-noise measurements and when CP pulse-pair spacing is much smaller than the inverse of the signal bandwidth. The analysis presented in this paper shows that these techniques have similar potential accuracy and detection probabilities with some variation depending on the desired probability of false alarm. However, the important benefit of the CP technique is that it can maintain this benchmark accuracy at low SNR when the noise-subtracted power technique potentially degrades as a result of errors in the noise component estimate. This paper focuses on the time-delayed version of CP, but some of the results can be applied to the dual-polarized CP method.

In section 2 the conventional NSP technique is reviewed. In section 3, the CP estimate probability density function, bias, and standard deviation are characterized, and the probability of false alarm (PFA) and probability of detection (PD) of the CP and NSP techniques are compared for various threshold levels and SNRs. In section 4 a low SNR CP bias correction procedure is described, and in section 5 a correction for bias caused by finite pulse-to-pulse signal correlation is outlined. In section 6 analytical and simulation results are validated with cloud and precipitation data collected with the University of Wyoming (UWyo) airborne Ka-band precipitation radar (KPR).

## 2. Noise-subtracted power

*N*

_{n}noise samples can be expressed in terms of the standard deviation

*I*and

*Q*noise samples as

*N*signal-plus-noise power samples is

*N*

_{I}independent signal-plus-noise power samples is a chi-squared distribution of 2

*N*

_{I}degrees of freedom (Ulaby et al. 1982, 486–487; Mead 2016; Ivić et al. 2009b, appendix A)

^{1}However, at high SNR,

*R*is the signal-plus-noise autocorrelation, and

_{p}) for a given threshold (

*T*) can be expressed using the cumulative distribution function (CDF) for a chi-squared distribution with

*2N*

_{I}degrees of freedom [Abramowitz and Stegun 1964, their Eq. (26.4.1); cf. Wolfram 2012a] denoted by

*D*[Mead 2016, rearranged form of Eq. (40)]:

*threshold factor, T*

_{f}, such that

*N*

_{n}≫

*N*, and the noise component estimate is unbiased). The chi-squared distribution converges to the Gaussian distribution with an increasing number of independent samples averaged; so, for large

*=*0), the probability of detection is the probability of false alarm (PFA):

The Gaussian approximation might be easier to use than the inverse *D* function, but chi-squared distribution is slow to converge to Gaussian; therefore, Eq. (14) should be used only if a large number of samples are averaged or for calculating thresholds that correspond to relatively high false alarm rates. The relationship between *T*_{f} and PFA and the differences based on the use of the chi-squared and Gaussian pdfs are illustrated in Fig. 2. The Gaussian pdf approximation is shown as a red dashed line and the black lines are for the chi-squared pdf, both with perfect noise subtraction. The Gaussian approximation underestimates the required threshold level and results in a higher-than-desired PFA.

NSP technique probability of false alarm vs thresholding evaluated using the chi-squared pdf (black lines) and based on the Gaussian pdf approximation (red dashed line), both with perfect noise subtraction. Chi-squared distribution is slow to converge to Gaussian, so the approximations should be used only for large *N* or high PFA rates.

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

NSP technique probability of false alarm vs thresholding evaluated using the chi-squared pdf (black lines) and based on the Gaussian pdf approximation (red dashed line), both with perfect noise subtraction. Chi-squared distribution is slow to converge to Gaussian, so the approximations should be used only for large *N* or high PFA rates.

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

NSP technique probability of false alarm vs thresholding evaluated using the chi-squared pdf (black lines) and based on the Gaussian pdf approximation (red dashed line), both with perfect noise subtraction. Chi-squared distribution is slow to converge to Gaussian, so the approximations should be used only for large *N* or high PFA rates.

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

## 3. Coherent power

*I*and

*Q*signal and noise components as

*m ≪ N*, the superscript asterisk (*) denotes a complex conjugate, and the circumflex (^) denotes the estimated value. An integer

### a. Noise-only coherent power

*I*

_{n}and

*Q*

_{n}noise samples are independent, zero-mean Gaussian random variables with a standard deviation

*N*, based on the central limit theorem (Papoulis 1984, p. 194), the real and imaginary components of the average in Eq. (17) converge toward independent Gaussian random variables, with zero mean and

*N*, is Eq. (32) in Lee et al. (1994):

*N*), as illustrated in Fig. 3. Furthermore, the Rayleigh pdf is easier and more efficient to use than Eq. (19) for the evaluation of

*N*(Jentschura and Lötstedt 2011).

CP estimate (*N* samples of noise (SNR = 0). Equation (19) from Lee et al. (1994) (solid black lines), and Rayleigh approximation (dashed red lines).

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

CP estimate (*N* samples of noise (SNR = 0). Equation (19) from Lee et al. (1994) (solid black lines), and Rayleigh approximation (dashed red lines).

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

CP estimate (*N* samples of noise (SNR = 0). Equation (19) from Lee et al. (1994) (solid black lines), and Rayleigh approximation (dashed red lines).

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

*N*= 10 and

*N*= 100 overestimates the mean by about 1% (0.04 dB) and 0.1% (0.004 dB) and underestimates the standard deviation by 5% and 0.5%, respectively.

*C*(

*T*),

*N*= 10 samples and a 10% intended false alarm rate, the threshold calculated with Eq. (23) yields ~10.03% PFA

_{cp};

*N*= 50 samples and 1% intended PFA

_{cp}yields ~1.1%; and

*N*= 100 samples and 0.1% PFA

_{cp}yields 0.106%.

CP probability of false alarm (PFA_{cp}) vs threshold factor (*T*_{f}) for *N* = 10, 50, and 100 samples averaged. Values calculated using the Lee et al. (1994) pdf of Eq. (19) (solid black lines) and those based on the Rayleigh approximation (dashed red line).

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

CP probability of false alarm (PFA_{cp}) vs threshold factor (*T*_{f}) for *N* = 10, 50, and 100 samples averaged. Values calculated using the Lee et al. (1994) pdf of Eq. (19) (solid black lines) and those based on the Rayleigh approximation (dashed red line).

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

CP probability of false alarm (PFA_{cp}) vs threshold factor (*T*_{f}) for *N* = 10, 50, and 100 samples averaged. Values calculated using the Lee et al. (1994) pdf of Eq. (19) (solid black lines) and those based on the Rayleigh approximation (dashed red line).

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

### b. Signal-plus-noise coherent power

*N*,

*N*, the multiplication in Eq. (24) can be carried out and the real and imaginary components combined as follows:

*N =*1000). The Rayleigh approximation of the zero SNR CP pdf and the Gaussian pdfs for increasing

Simulated CP (red) and NSP (black) estimate pdf for *N* = 1000. Gaussian distribution thick green lines) and Rayleigh pdf (thick dashed blue line) are shown to illustrate that the CP pdf can be approximated as Rayleigh for *N*. CP estimate also converges toward the chi-squared (Gaussian for large *N*) power estimate pdf with increasing SNR and near-unity

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

Simulated CP (red) and NSP (black) estimate pdf for *N* = 1000. Gaussian distribution thick green lines) and Rayleigh pdf (thick dashed blue line) are shown to illustrate that the CP pdf can be approximated as Rayleigh for *N*. CP estimate also converges toward the chi-squared (Gaussian for large *N*) power estimate pdf with increasing SNR and near-unity

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

Simulated CP (red) and NSP (black) estimate pdf for *N* = 1000. Gaussian distribution thick green lines) and Rayleigh pdf (thick dashed blue line) are shown to illustrate that the CP pdf can be approximated as Rayleigh for *N*. CP estimate also converges toward the chi-squared (Gaussian for large *N*) power estimate pdf with increasing SNR and near-unity

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

*N*and

*k*= 0, and

*k*> 0 and

Note that for *k* = 0, *I*_{k}(0) = 1, and for *k* > 0, *I*_{k}(0) = 0 , so the Beckmann distribution [Eq. (33)] reduces to Rayleigh [Eq. (18)] when *k* = 0 to 4) in the infinite sum of Eq. (33) is sufficient to limit errors in the calculation of the first two moments to <1% (for *N* and *N*_{I} > 50). For

Since the Beckmann distribution is valid only for large *N* and numeric evaluation of the pdf, Eq. (33), and its moments is difficult, especially for large arguments, simulations were used to find a practical equation for the mean of the CP estimate in the transition region: *N*, and the partially correlated signal component (

*N=*

*I*and

*Q*terms are independent Gaussian random variables with variances

*I*and

*Q*terms is allowed here because the sum of Gaussians is also a Gaussian random variable. It is also convenient for the modeling of a desired pulse-pair correlation; note that the same randomly generated numbers are used in both the first and second correlated terms (

*mT*

_{s}correlations, while all other signal spectrum shapes are lost, this simple model is adequate to simulate independent weather radar pulse-pairs samples.

*P*

_{n}plus the noiselike uncorrelated portion of the pulse-pair sample,

*N*,

*N*= 10, 50, and 100 as a function of

*B*

_{f}[Eq. (39)] were optimized to minimize the maximum deviation from simulated

*B*

_{f}over

*N*> 10, and

*N*> 10. At high

*N*.

CP bias factor as a function of coherent signal component-to-noise ratio and number of samples averaged. Simulation results (red) and Eq. (39) approximation (black); *N* = 100 (solid line), *N* = 50 (dots), and *N* = 10 (dashes).

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

CP bias factor as a function of coherent signal component-to-noise ratio and number of samples averaged. Simulation results (red) and Eq. (39) approximation (black); *N* = 100 (solid line), *N* = 50 (dots), and *N* = 10 (dashes).

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

CP bias factor as a function of coherent signal component-to-noise ratio and number of samples averaged. Simulation results (red) and Eq. (39) approximation (black); *N* = 100 (solid line), *N* = 50 (dots), and *N* = 10 (dashes).

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

*N*> 50, evaluated for

The CP estimated standard deviation (*N* > 10 is within 7% and using the Beckmann/Gaussian pdf–based estimate mean for *N* > 50 is within 2% of the simulated

### c. PD comparison of CP and NSP techniques

The CP probability of detection *N* (*N* = 1000) and

NSP (solid black) and CP technique (dashed red) PD as a function of *N*_{n} ≫ *N*) noise subtraction is assumed in the power estimates and

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

NSP (solid black) and CP technique (dashed red) PD as a function of *N*_{n} ≫ *N*) noise subtraction is assumed in the power estimates and

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

NSP (solid black) and CP technique (dashed red) PD as a function of *N*_{n} ≫ *N*) noise subtraction is assumed in the power estimates and

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

## 4. Coherent power bias at low SNR

The signal-free CP bias is *N* [Eq. (20)]. With increasing SNR and *N*, this bias diminishes and the CP estimate converges to the pulse-pair signal component covariance magnitude [Eq. (40)]. The noise bias with increasing SNR was evaluated using the simulations assuming unity pulse-pair signal correlation magnitude

Simulated ratio of CP to signal power as a function of *N* = 1000 (solid line), 100 (dots), and 10 (dashes). Shown is a 0.35 dB bias at 3 dB

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

Simulated ratio of CP to signal power as a function of *N* = 1000 (solid line), 100 (dots), and 10 (dashes). Shown is a 0.35 dB bias at 3 dB

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

Simulated ratio of CP to signal power as a function of *N* = 1000 (solid line), 100 (dots), and 10 (dashes). Shown is a 0.35 dB bias at 3 dB

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

It usually makes sense to remove bias from data, but in this case there is rarely a practical benefit. CP bias is significant only for SNRs below low threshold levels (corresponding to about 10% or greater false alarm rates), but more importantly CP estimates at these low

*N*) as SNR increases. We therefore again used the simulations to characterize the relationship between CBR and SBR as a function of

*N*> 50:

The comparison of simulated and the Eq. (44) model relationship between SBR and CBR is shown in Fig. 9. Note that CP bias is only about 10% (0.4 dB) when SBR is 3 dB (

Simulated CBR vs SBR (black lines), for SBR ranging from 0 to 4, along with the approximate formula relating CBR to SBR [Eq. (44)] (red lines) for *N* = 1000 (solid), 100 (dots), and 10 (dashes). Unit slope line is drawn as the unbiased reference. Horizontal distance between this reference line and CBR is the bias.

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

Simulated CBR vs SBR (black lines), for SBR ranging from 0 to 4, along with the approximate formula relating CBR to SBR [Eq. (44)] (red lines) for *N* = 1000 (solid), 100 (dots), and 10 (dashes). Unit slope line is drawn as the unbiased reference. Horizontal distance between this reference line and CBR is the bias.

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

Simulated CBR vs SBR (black lines), for SBR ranging from 0 to 4, along with the approximate formula relating CBR to SBR [Eq. (44)] (red lines) for *N* = 1000 (solid), 100 (dots), and 10 (dashes). Unit slope line is drawn as the unbiased reference. Horizontal distance between this reference line and CBR is the bias.

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

## 5. Coherent power bias as a result of spectrum width

^{2}The CP bias associated with

*N*and SNR, is the corresponding correlation coefficient,

*m*= 1 is illustrated in Fig. 10.

High SNR and unit *m* CP bias (dB), as a function of normalized spectrum width. Equation

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

High SNR and unit *m* CP bias (dB), as a function of normalized spectrum width. Equation

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

High SNR and unit *m* CP bias (dB), as a function of normalized spectrum width. Equation

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

This method, using lag-50- and lag-150-*μ*s pulse spacing, was tested on cloud and precipitation data collected with the UWyo KPR radar, and the results agree within 0.1 dB with the bias estimated by directly computing the ratio of the measured noise-subtracted power and CP as described in section 6.

## 6. Experimental results

The radar reflectivity factor was estimated from data collected using the conventional NSP and CP techniques with KPR, on board the University of Wyoming King Air 200T research aircraft (Fig. 11). KPR is a compact, dual-beam, solid-state Doppler radar that operates from a standard Particle Measurement Systems (PMS) canister. A 10-W peak-power solid-state amplifier is used to transmit pulses of a chirped (frequency modulated) waveform, immediately followed by an offset frequency short pulse for close-range measurements. The received signals from both pulse segments are simultaneously measured with a wide bandwidth front-end and dual-channel back-end receiver sections. KPR can operate in a variety of modes, including pulse-pair and full spectrum processing. When using a single up- or down-looking antenna, constant PRI is employed. When simultaneous upward and downward pulse-pair measurements are required, the system can be configured to alternately transmit pairs of pulses to the up- and down-looking antennas, as illustrated in Fig. 12. The real-time computed radar data products include radar reflectivity factor and Doppler velocity moments from pulse pairs. A summary of the KPR system’s parameters is listed in Table 1.

UWyo KPR is a compact, dual-beam Doppler radar that operates from a standard PMS probe canister.

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

UWyo KPR is a compact, dual-beam Doppler radar that operates from a standard PMS probe canister.

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

UWyo KPR is a compact, dual-beam Doppler radar that operates from a standard PMS probe canister.

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

KPR interleaved up-/down-looking antenna pulsing sequence. For every *N* power samples collected per averaging interval, *N*/2 pulse pairs are available for processing at lag-1 and lag-3 spacing. Lag-1 and lag-3 pulse pairs are averaged to obtain the same number of pulse-pair noise samples (*N*) as NSP, but the lag-3 pulse-pair phase is first divided by 3 to match the lag-1 phase.

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

KPR interleaved up-/down-looking antenna pulsing sequence. For every *N* power samples collected per averaging interval, *N*/2 pulse pairs are available for processing at lag-1 and lag-3 spacing. Lag-1 and lag-3 pulse pairs are averaged to obtain the same number of pulse-pair noise samples (*N*) as NSP, but the lag-3 pulse-pair phase is first divided by 3 to match the lag-1 phase.

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

KPR interleaved up-/down-looking antenna pulsing sequence. For every *N* power samples collected per averaging interval, *N*/2 pulse pairs are available for processing at lag-1 and lag-3 spacing. Lag-1 and lag-3 pulse pairs are averaged to obtain the same number of pulse-pair noise samples (*N*) as NSP, but the lag-3 pulse-pair phase is first divided by 3 to match the lag-1 phase.

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

KPR key system parameters.

The dataset presented in Figs. 13–19 was collected on 4 September 2016 in southeastern Wyoming, about 50 km east of Rawlins. The aircraft was flying through convective cells approximately 4 km above ground and ~2.2 km above the melting layer in −11°C flight-level temperature. The radar PRF was a constant 20 kHz, transmitting alternating pairs of pulses to the up- and down-looking antennas as shown in Fig. 12. The resulting lag-1 and lag-3 pulse-pair spacings were 50 and 150 *μ*s, respectively. Each pulse contained a Tukey tapered 2.5-*μ*s-long 5-MHz bandwidth chirped pulse, immediately followed by a similarly tapered 0.25-*μ*s short RF pulse at an offset frequency. The resulting range resolution of the compressed chirp and short radio frequency (RF) pulses was approximately 35 m. The up- and down-pointed beams and short and compressed chirped pulse datasets were combined such that the short-pulse data were used to about 800 m from the aircraft and the chirped data, with better sensitivity, beyond. Each noise-subtracted power and coherent power reflectivity point was calculated by averaging 2000 samples (0.2-s integration time; five profiles per second per beam). The signal components of the samples were partially correlated as a result of the high PRF, so the equivalent number of independent high-SNR signal samples was only about 200.^{3} To obtain the same (2000) number of pulse-pair samples per CP reflectivity estimate, the lag-3 pulse-pair phasor phase was divided by 3 and then the lag-1 and lag-3 phasors were averaged.

CP reflectivity image, without thresholding or bias correction. CP bias mean and standard deviation, in the upper-right corner test region, agree with Eqs. (17) and (18).

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

CP reflectivity image, without thresholding or bias correction. CP bias mean and standard deviation, in the upper-right corner test region, agree with Eqs. (17) and (18).

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

CP reflectivity image, without thresholding or bias correction. CP bias mean and standard deviation, in the upper-right corner test region, agree with Eqs. (17) and (18).

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

CP SNR (dB).

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

CP SNR (dB).

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

CP SNR (dB).

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

Noise-bias-corrected CP reflectivity image, with thresholding that corresponds to 50% false alarm rate using Eq. (20). False alarm rate of the signal-free region inside the black rectangle in the upper-right corner is 0.499, confirming the correct thresholding. PD in the small upper-left rectangle is 0.821, compared to 0.870 in the same region in the NSP image of Fig. 16.

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

Noise-bias-corrected CP reflectivity image, with thresholding that corresponds to 50% false alarm rate using Eq. (20). False alarm rate of the signal-free region inside the black rectangle in the upper-right corner is 0.499, confirming the correct thresholding. PD in the small upper-left rectangle is 0.821, compared to 0.870 in the same region in the NSP image of Fig. 16.

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

Noise-bias-corrected CP reflectivity image, with thresholding that corresponds to 50% false alarm rate using Eq. (20). False alarm rate of the signal-free region inside the black rectangle in the upper-right corner is 0.499, confirming the correct thresholding. PD in the small upper-left rectangle is 0.821, compared to 0.870 in the same region in the NSP image of Fig. 16.

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

NSP reflectivity, without thresholding (*T*_{f} = 0; 50% PFA_{P}). False alarm rate of a signal-free region inside the black rectangle in the upper-right corner is 0.498 (50% expected). PD in the small upper-left rectangle is 0.870 compared to 0.821 in the same region in the CP reflectivity image of Fig. 15, confirming that the power technique is slightly more sensitive to weak signals than CP when the PFA is high (at low threshold) in Fig. 7.

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

NSP reflectivity, without thresholding (*T*_{f} = 0; 50% PFA_{P}). False alarm rate of a signal-free region inside the black rectangle in the upper-right corner is 0.498 (50% expected). PD in the small upper-left rectangle is 0.870 compared to 0.821 in the same region in the CP reflectivity image of Fig. 15, confirming that the power technique is slightly more sensitive to weak signals than CP when the PFA is high (at low threshold) in Fig. 7.

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

NSP reflectivity, without thresholding (*T*_{f} = 0; 50% PFA_{P}). False alarm rate of a signal-free region inside the black rectangle in the upper-right corner is 0.498 (50% expected). PD in the small upper-left rectangle is 0.870 compared to 0.821 in the same region in the CP reflectivity image of Fig. 15, confirming that the power technique is slightly more sensitive to weak signals than CP when the PFA is high (at low threshold) in Fig. 7.

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

CP reflectivity (noise bias corrected), with thresholding set to yield 0.1% false alarm rate using Eq. (20). False alarm rate of the signal-free region inside the rectangle in the upper-right corner was evaluated to be 0.0011 (0.0013 in the power image of Fig. 18). PD in the small upper-left rectangle is 0.906 compared to 0.850 in the same region using NSP in Fig. 18, confirming higher CP sensitivity to weak signals than the power technique at lower false alarm rates, as indicated in Fig. 7, in spite of the approximately 0.5-dB lower CP SNR as a result of nonzero spectrum width (Fig. 19).

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

CP reflectivity (noise bias corrected), with thresholding set to yield 0.1% false alarm rate using Eq. (20). False alarm rate of the signal-free region inside the rectangle in the upper-right corner was evaluated to be 0.0011 (0.0013 in the power image of Fig. 18). PD in the small upper-left rectangle is 0.906 compared to 0.850 in the same region using NSP in Fig. 18, confirming higher CP sensitivity to weak signals than the power technique at lower false alarm rates, as indicated in Fig. 7, in spite of the approximately 0.5-dB lower CP SNR as a result of nonzero spectrum width (Fig. 19).

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

CP reflectivity (noise bias corrected), with thresholding set to yield 0.1% false alarm rate using Eq. (20). False alarm rate of the signal-free region inside the rectangle in the upper-right corner was evaluated to be 0.0011 (0.0013 in the power image of Fig. 18). PD in the small upper-left rectangle is 0.906 compared to 0.850 in the same region using NSP in Fig. 18, confirming higher CP sensitivity to weak signals than the power technique at lower false alarm rates, as indicated in Fig. 7, in spite of the approximately 0.5-dB lower CP SNR as a result of nonzero spectrum width (Fig. 19).

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

NSP reflectivity, with thresholding set to yield a 0.1% false alarm rate using Eq. (14). False alarm rate of a signal-free region inside the black rectangle in the upper-right corner was evaluated to be 0.0013 (0.0011 in the CP image of Fig. 17). PD in the small upper-left rectangle is 0.850 compared to 0.906 in the same region in the CP image of Fig. 17.

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

NSP reflectivity, with thresholding set to yield a 0.1% false alarm rate using Eq. (14). False alarm rate of a signal-free region inside the black rectangle in the upper-right corner was evaluated to be 0.0013 (0.0011 in the CP image of Fig. 17). PD in the small upper-left rectangle is 0.850 compared to 0.906 in the same region in the CP image of Fig. 17.

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

NSP reflectivity, with thresholding set to yield a 0.1% false alarm rate using Eq. (14). False alarm rate of a signal-free region inside the black rectangle in the upper-right corner was evaluated to be 0.0013 (0.0011 in the CP image of Fig. 17). PD in the small upper-left rectangle is 0.850 compared to 0.906 in the same region in the CP image of Fig. 17.

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

Lag-1/lag-3 estimated CP negative bias and loss of SNR caused by aircraft motion and turbulence using Eq. (49). This agrees to less than 0.1 dB with the bias calculated directly by taking the ratio of NSP and CP.

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

Lag-1/lag-3 estimated CP negative bias and loss of SNR caused by aircraft motion and turbulence using Eq. (49). This agrees to less than 0.1 dB with the bias calculated directly by taking the ratio of NSP and CP.

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

Lag-1/lag-3 estimated CP negative bias and loss of SNR caused by aircraft motion and turbulence using Eq. (49). This agrees to less than 0.1 dB with the bias calculated directly by taking the ratio of NSP and CP.

Citation: Journal of Atmospheric and Oceanic Technology 35, 1; 10.1175/JTECH-D-17-0058.1

The average of 2000 signal-plus-noise power samples yields a zero SNR measurement standard deviation of about 16.5 dB below the mean noise power. Thus, even with a relatively high (for a research weather radar) threshold factor of 2.5 (4. dB and ~1% PFA), the detection threshold is about 12.5 dB below the mean noise floor. To maintain close to this measurement precision and sensitivity using the NSP technique, the noise component was estimated by averaging the last 20 (signal free) range gates of each averaged ray, totaling 40 000 samples, to estimate the noise component of each signal-plus-noise measurement. The resulting noise component estimate standard deviation is

The accuracy of the noise component estimate can be worse however because of the bias caused by receiver gain variation in range or radar echo. Noise estimate uncertainties as high as 2 dB have been cited by Keranen and Chandrasekar (2014) with operational weather radars. This is likely excessive with most research radars, but nevertheless estimating the noise component to about 0.02-dB uncertainty is not a trivial requirement. This makes the CP technique particularly attractive with KPR, and without the need for signal-free range gates for noise subtraction, PRF can be increased to maximize sensitivity. The staggered PRI mode shown in Fig. 12 has the added benefit that second-trip echo is uncorrelated in the CP estimate and therefore the associated bias is reduced with averaging according to Eq. (40).

The CP reflectivity image, without thresholding or bias correction, is shown in Fig. 13 and the corresponding SNR in Fig. 14. In Fig. 13 the zero SNR bias is clearly visible outside the cloud and precipitation regions. The mean and standard deviation of the CP signal-free noise floor in Eqs. (20) and (21) for large *N* are confirmed in the upper-right signal-free rectangular region. The mean measured CP bias to

The 50% false alarm rate CP with bias correction and NSP images are shown in Figs. 15 and 16, respectively. The correct thresholding (50% PFA) in both images is confirmed by the PFA in the signal-free regions in the upper-right corner (0.499 CP and 0.501 NSP PFA; ~0.002 standard deviation). The slightly better sensitivity of the NSP technique at low threshold levels (high PFA), indicated in Fig. 7, is confirmed by a slightly higher power PD (0.87; ~0.017 standard deviation) than the detection rate of CP (0.82; ~0.017 standard deviation) in the upper-left corner test region. When the threshold level is increased, so the PFA is close to 0.1%, in Figs. 17 and 18, the CP detection rate (0.91; ~0.011 standard deviation) is slightly higher than the power PD (0.85; ~0.011 standard deviation), in a test region at the 4-km range above the aircraft and in the left part of the images, as predicted by the analytical and simulation results plotted in Fig. 7.

The last image presented, in Fig. 19, compares the multilag [using Eq. (49)] to the direct NSP-to-CP ratio method of estimating CP bias due to spectrum width. In this dataset, the multilag method agrees with the direct NSP-to-CP ratio to within ~0.1 dB.

## 7. Conclusions

The CP and NSP techniques were shown to have similar sensitivity and accuracy to measure weather radar signals when properly implemented. The fact that the CP technique does not require an independent noise estimate can be very useful, particularly with radars that average a large number of samples and consequently extend the measurement dynamic range well below unity SNR, where the required noise component measurement accuracy is increasingly difficult to achieve. On the other hand, the CP technique requires the pulse-pair spacing to be much less than the inverse of the signal bandwidth to ensure close to unity signal component correlation. In addition, some thresholding should be applied to mask low SNR CP bias. Properly collected CP data are actually slightly more sensitive to weak signals than the power technique when the threshold level is set high (low false alarm rate), but the more important benefit of the CP technique is its inherent noise-canceling property.

The coherent power estimate probability density function for a high number of samples (*N* and *N*_{I}) has been identified to be the Beckmann distribution, which converges to the Rayleigh pdf in the special case when the pulse-to-pulse signal correlation or SNR is zero, and to Gaussian pdf at high SNR and

Data collected with the University of Wyoming airborne Ka-band precipitation radar (KPR) confirm the analytical and simulation results of zero SNR CP residual noise and the comparison between CP and NSP techniques. The CP residual noise floor mean and the standard deviation of Eqs. (20) and (21) for a large number of samples are in close agreement with calculations using Eq. (19) (Lee et al. 1994), simulations. and measurements, while it was found that the expressions of Eq. (A10) in Keranen and Chandrasekar (2014) are slightly biased estimates. Equation (23), for computing the required threshold level to achieve a specific false alarm rate, has also been confirmed. The CP signal power measurements agree to within 0.1 dB with the NSP measurements after correcting the effects of spectrum width using the multilag technique outlined in this paper. The combination of analytical, simulation, and experimental results confirm that CP can be a more accurate alternative to the NSP technique at low SNR and when the noise component cannot be estimated at an accuracy that is considerably better than that of the signal-plus-noise measurements.

## Acknowledgments

The authors thank Dr. James Mead of ProSensing, Inc., for the helpful discussions and suggestions related to the analyses presented herein, and the AMS reviewers for their thorough review and numerous constructive suggestions. The development of KPR and the data collection effort were funded by NASA EPSCoR Grant NNX13AN09A to the University of Wyoming.

## APPENDIX

### Statistics of the Real and Imaginary Components of Pulse-Pair Average

*mT*

_{s}pulse-pairs. In this appendix the mean and variance of the real and imaginary components (

the signal–signal terms are

the signal–noise terms are

the noise–signal terms are

the noise–noise terms are

#### a. Noise–noise terms

Each product term has a variance of *N* real and 2*N* imaginary terms,

#### b. Signal–noise and noise–signal terms

Similarly,

#### c. Signal–signal terms

*N*

_{I}≫ m) is the equivalent number of independent samples, while the imaginary parts cancel, so

The cross terms are identical, so

The noncoherent portion of the signal is just like noise, so

#### d. Mean and variance of the pulse-pair sum terms

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^{1}

Noise samples collected from successive range gates can be partially correlated when the data system sampling frequency is faster than the effective receiver (noise) bandwidth.

^{3}

The measured normalized spectrum width of this dataset is approximately 2.5 m s^{−1} / 84 m s^{−1} = 0.03, and from Doviak and Zrnić (1993, p. 128) *N* and