## ABSTRACT

The NOAA W-band radar was deployed on a P-3 aircraft during a study of storm fronts off the U.S. West Coast in 2015 in the second CalWater (CalWater-2) field program. This paper presents an analysis of measured equivalent radar reflectivity factor *Z*_{em} profiles to estimate the path-averaged precipitation rate and profiles of precipitation microphysics. Several approaches are explored using information derived from attenuation of *Z*_{em} as a result of absorption and scattering by raindrops. The first approach uses the observed decrease of *Z*_{em} with range below the aircraft to estimate column mean precipitation rates. A hybrid approach that combines *Z*_{em} in light rain and attenuation in stronger rain performed best. The second approach estimates path-integrated attenuation (PIA) via the difference in measured and calculated normalized radar cross sections (NRCS_{m} and NRCS_{c}, respectively) retrieved from the ocean surface. The retrieved rain rates are compared to estimates from two other systems on the P-3: a Stepped Frequency Microwave Radiometer (SFMR) and a Wide-Swath Radar Altimeter (WSRA). The W-band radar gives reasonable values for rain rates in the range 0–10 mm h^{−1} with an uncertainty on the order of 1 mm h^{−1}. Mean profiles of *Z*_{em}, raindrop Doppler velocity, attenuation, and precipitation rate in bins of rain rate are also computed. A method for correcting measured profiles of *Z*_{em} for attenuation to estimate profiles of nonattenuated profiles of *Z*_{e} is examined. Good results are obtained by referencing the surface boundary condition to the NRCS values of PIA. Limitations of the methods are discussed.

## 1. Introduction

Precipitation is one of the most difficult and confounding meteorological variables to measure accurately and to sample sufficiently for meaningful averages. Most applications (e.g., hydrology, oceanic salinity budgets, global energy balances, soil moisture analysis) require grid-averaged precipitation rates. Because of the greatly patchy nature of precipitation, undersampling makes the use of surface-based conventional rain gauges problematic. Ground-based scanning radars and satelliteborne radars can greatly improve sampling, but they introduce a host of accuracy issues (e.g., Lee and Zawadzki 2006; Haynes et al. 2009). Two common issues with radar-based methods are the absolute calibration of the radar and the variation of radar–rain retrieval relationships with precipitation microphysics (Steiner et al. 2004; Lee and Zawadzki 2006). Conventional rain gauges have biases associated with wind effects on collection efficiency that are geometry dependent (Ciach 2003); gauges typically provide accumulations, and estimates of rain rate from gauge data often have poor time resolution. Disdrometers, which measure the raindrop size distribution (DSD), offer a superior surface characterization of precipitation microphysics because both the rain rate *R* and the equivalent radar reflectivity factor *Z*_{e} can be computed from the observations.

While precipitation reaching the surface is the overarching variable in many weather applications, precipitation formation processes are a critical research topic. Observational research into cloud–precipitation microphysical relationships has been dominated by airborne in situ DSD and ground-based millimeter-wavelength Doppler radar–observing systems (Kollias et al. 2007). The advent of DSD and Doppler spectrum moment techniques (Frisch et al. 1995, 1998) increased the utility of remote sensing methods, which have subsequently expanded to a variety of approaches [including multiwavelength, multi-Doppler peak, clear-air vs drop scattering modes; for more information see Tridon et al. (2013), Tridon and Battaglia (2015), and Williams (2016)]. Airborne (Galloway et al. 1999) and satelliteborne (Haynes et al. 2009) millimeter-wavelength radars have greatly expanded the scope of radars to investigate the spatial distribution and vertical structure of precipitating cloud systems. Multiwavelength methods have a rich history of application with surface-based systems (Firda et al. 1999; Williams 2012), but applications with airborne systems (e.g., Tian et al. 2007) are relatively rare—principally because of engineering constraints associated with matching beamwidth and performance characteristics at different wavelengths when space and weight conflict with aerodynamic performance. Millimeter-wavelength radars have the major advantage that they are sensitive to both clouds and precipitation and, because of their smaller size, are ideal for airborne platforms. The advantages of multiple wavelength techniques are moot if you have only one radar (e.g., TRMM, https://trmm.gsfc.nasa.gov/overview_dir/pr.html; and *CloudSat*, http://cloudsat.atmos.colostate.edu/instrument). Thus, we are motivated to squeeze as much information as we can from a single wavelength system.

One important weakness of moment-based methods in estimating precipitation rate and microphysics is that the zeroth, first, and second moments of the radar-reflectivity-weighted Doppler velocity spectrum are essentially the sixth–eighth moments of the DSD for the Rayleigh-type scattering [see (9) in Frisch et al. 1995]. Thus, radar moment methods may poorly constrain rain-rate retrieval, which is essentially the 3.67th moment of the DSD. An independent constraint of one or more of the lower-order DSD moments could improve radar rain-rate retrievals. In this paper we use radar attenuation as a constraint to estimate mean multiparameter profiles and time series of layer-averaged rain rates below an airborne W-band Doppler radar. The observations we are using are from the NOAA Physical Science Division (PSD) W-band radar (Moran et al. 2012) deployed on a NOAA P-3 aircraft for seven flights during the second CalWater (CalWater-2; CalWater-2015) field program off the U.S. West Coast in 2015 (Ralph et al. 2016).

This paper presents an analysis of processing measured equivalent radar reflectivity factor profiles *Z*_{em} to estimate precipitation rate using the observed decrease of *Z*_{em} with range below the aircraft. The rain rate is approximately proportional to the attenuation coefficient in rain (i.e., the slope of the reflectivity profile, assuming a prevalence of rain attenuation over changes of nonattenuated reflectivity *Z*_{e}) as described in Matrosov (2007). We find that the two-tiered rain-rate retrieval method of Chandra et al. (2015), where large rain rates are computed from the attenuation of uncalibrated *Z*_{em} and light rain rates (less than approximately 1 mm h^{−1}) are estimated from calibrated reflectivity using a *Z*_{e}–*R* parameterization, is an effective procedure. The Chandra et al. (2015) two-tiered methodology was based on ground-based vertically pointing Ka-band radar observations where attenuation estimates fell below measurement uncertainties for light rain rates.

A second but related method to estimate rain rate uses the measured normalized radar cross section NRCS_{m} retrieved from the return of the ocean surface. Since NRCS is fairly well characterized as a function of wind speed and angle relative to nadir (Li et al. 2005), the calculated normalized radar cross section NRCS_{c} is independent of radar attenuation. Thus, the difference between the measured and calculated NRCS represents the total column attenuation, which is also known as the path-integrated attenuation (PIA). As with the reflectivity gradient rain-rate method, the estimated PIA is related to rain rate such that PIA yields an estimate of the total column average rain rate below the aircraft (Meneghini et al. 1983). The total rain-rate estimates retrieved from the W-band radar measurements are compared to estimates from two other systems on the P-3: a Stepped Frequency Microwave Radiometer (SFMR; Uhlhorn et al. 2007) and a Wide-Swath Radar Altimeter (WSRA; Walsh et al. 2014).

In addition to estimating the layer-averaged rain rate, we wish to investigate the use of profiles of radar-derived parameters to retrieve information about precipitation microphysics. Here we explore the *value added* by attenuation observations. The approach centers on averages of radar profiles of the first three spectral moments plus attenuation in bins of rain rate. We have applied the Hitschfeld–Bordan inversion technique of Iguchi and Meneghini (1994) to retrieve profiles of unattenuated reflectivity using the surface return (NRCS) as the reference. This yielded profiles of *Z*_{e} consistent with values extrapolated to the aircraft altitude. Note that the surface reference (i.e., NRCS) and reflectivity gradient approaches have been used with the spaceborne W-band radar aboard *CloudSat* (e.g., Haynes et al. 2009; Matrosov 2011).

It is not practical to install an expensive radar on an aircraft just to measure rain rate. Our purpose here is to evaluate the use of attenuation to improve information extraction from a millimeter-wavelength radar being used for cloud/precipitation research. Some caveats to consider: If rain rate is derived from the vertical gradient of measured reflectivity, then inhomogeneity in the vertical distribution of raindrops and cloud absorption will compromise the results. For rain rate derived from NRCS, uncertainty in the NRCS model and the 10-m wind speed retrieval are likely the largest source of error. Corrections for atmospheric gaseous absorption affect both approaches, but the errors are small if temperature/humidity profiles are available. The correction of measured reflectivity profiles relies on accurate NRCS measurement, the determination of the clear-air NRCS, and a specification of the attenuation-to-reflectivity relationship (which is somewhat sensitive to precipitation microphysics). NRCS can be used to provide an absolute calibration check of the radar reflectivity.

## 2. Experimental details

### a. CalWater-2

The CalWater-2015 (Ralph et al. 2016) field deployment off the U.S. West Coast included NOAA’s flagship Research Vessel *Ronald H. Brown* (*RHB*), as well as a P-3 and G-IV aircraft. The U.S. DOE–sponsored Atmospheric Radiation Measurement (ARM) Cloud Aerosol Precipitation Experiment (ACAPEX) campaign provided the DOE ARM Mobile Facility 2 (AMF2) observing system, mounted on the NOAA vessel, as well as the DOE G-1 aircraft and support for aerosol and microphysics sensors at the coast. The NASA ER-2 aircraft flew several missions as well with remote sensors tailored partly for validation of a prototype space-based sensor being tested on the International Space Station. The California Department of Water Resources (DWR)-sponsored statewide extreme precipitation network, tailored to observe landfalling atmospheric rivers (Ralph et al. 2016), was a foundation of the experiment. The observation period was January through March 2015. Here we discuss measurements taken on the NOAA P-3 aircraft.

### b. P-3 measurements

NOAA’s P-3 aircraft are equipped with a unique array of scientific instrumentation, radars, and recording systems for both in situ and remote sensing measurements of the atmosphere, Earth, and its environment (http://www.omao.noaa.gov/learn/aircraft-operations/aircraft/lockheed-wp-3d-orion). In situ sensors provided flight-level meteorological and navigation information. The P-3 also deployed 80 Vaisala RD-94 dropsondes during the period in the region near 37°N, 127°W.

Rain-rate values were estimated from two systems on the P-3: the SFMR and the WSRA. These estimate rain rate averaged over altitude below the aircraft. The WSRA has 80 narrow beams spread over ±30° in the cross-track direction. It uses a subset of ±14° to estimate the sea surface mean square slope (mss) and the path-integrated attenuation at its Ku-band 16-GHz operating frequency. The nadir returned power is normalized to a constant mss and altitude. The maximum normalized returned power over a significant time interval is assumed to be rain free, and any decrease from that value is attributed to rain attenuation. Because the WSRA determines NRCS and mss independently by scanning incidence angles, it yields a fundamentally more unbiased measurement of the rain rate. It is also weakly attenuated compared to the W band, which reduces the resolution but allows for useful observations from greater altitudes and larger rain rates. Walsh et al. (2014) discuss the algorithm in detail and compare the results with the SFMR on the same aircraft and the National Mosaic and Multi-Sensor Quantitative Precipitation Estimation (QPE) system (Zhang et al. 2011; Lakshmanan et al. 2006) archive product from the NEXRAD network measurements. The SFMR passive technique for extracting the rain rate from its six C-band frequencies (4.74, 5.31, 5.57, 6.02, 6.69, 7.09 GHz) is more complex and has long been under development on NOAA aircraft (Black and Swift 1984; Uhlhorn et al. 2007; Walsh et al. 2014). Klotz and Uhlhorn (2014) detail the evolution of the technique and its present status, which produced the results compared with the W-band rain rates in this paper.

The observations we are focusing on are from the NOAA PSD W-band radar deployed on the P-3 for seven flights between 27 January and 9 February 2015^{1}.

The radar is described in depth by Moran et al. (2012). The initial deployments were ship based (Moran et al. 2012; Ghate et al. 2014) but aircraft deployments began in 2013 (Fairall et al. 2014).

### c. Radar configuration and calibration

The W-band radar operated in one Doppler spectra mode with a focus on measuring rain below the aircraft. Doppler spectra were recorded to disk every 0.3 s, and the first three moments (i.e., zeroth, first, and second) were calculated to estimate reflectivity, mean Doppler velocity, and Doppler velocity spectrum width. Pertinent radar operating parameters are listed in Table 1. Note that the first range gate was set to 489 m below the aircraft to avoid destroying the receiver from strong surface returns when the aircraft was below 500-m altitude.

For distributed targets within a radar resolution volume, the measured raw reflectivity factor *Z*_{em_raw}(*h*) (mm^{6} m^{−3}) at range (km) is related to the received power via

where is a function of the complex refractive index of water at the radar operating wavelength =3.17 mm, and is the radar calibration constant that incorporates all radar internal gains and losses plus other corrections discussed below. After calibrating the PSD W-band radar antenna at an antenna range, absolute reflectivity accuracy is expected to be approximately ±1 dB. In we include a correction factor that accounts for the additional loss of 1.6 dB from the radome in the belly of the aircraft plus a correction determined by calibration against NRCS in clear conditions. The NRCS calibration flight was done on 9 February 2015, which was dominated by clear skies. The P-3 dropped 17 sondes; the mean 10-m wind speed was 4.7 m s^{−1} with a standard deviation of 0.5 m s^{−1}. We determined a final correction to the radar constant, equivalent to +0.2 ± 1.0 dB, to force observed NRCS to agree with values in Li et al. (2005) at an incidence angle of 12° (at this angle NRCS has minimal wind speed dependence). The uncertainty in this correction is a combination of scatter in measured NRCS at 12° and variability of specified values in Li et al. (2005). An uncertainty of 1 dB translates to an uncertainty in NRCS-derived rain rate of about 0.32 mm h^{−1} (weakly wind speed dependent) for an aircraft altitude of 2.5 km.

Because of attenuation, *Z*_{e} is given as

where γ_{total} is the total specific attenuation (dB km^{−1}) at range (km) of length (km) and is composed of specific attenuation from oxygen γ_{o}, water vapor γ_{vapor}, cloud γ_{cloud}, and precipitation γ_{rain}. The total specific attenuation can be expressed as

For the remainder of this paper we will use the notation that *Z*_{em} is defined as the raw reflectivity corrected by the combined absorption of water vapor and oxygen (),

where dB*Z*_{em} = 10 log_{10}(*Z*_{em}) and *G*_{υ} is the gaseous attenuation, which is defined as

The factor of 2 in this equation arises because the radar has a two-way path. If we assume the cloud attenuation can be neglected, then changes in *Z*_{em} can be related to rain attenuation. A stratus cloud with a liquid water content of 0.1 g m^{−3} would have an attenuation of approximately 0.4 dB km^{−1}—roughly comparable to rain with a rate of 0.5 mm h^{−1} (e.g., Matrosov 2009), representing a bias in rain-rate estimates based on attenuation.

Values of *γ*_{υ} were obtained using the atmospheric absorption methods from the International Telecommunication Union (www.itu.int/dms_pubrec/itu-r/rec/p/R-REC-P.676-3-199708-S!!PDF-E.pdf). For CalWater-2 we computed *G*_{υ} =2.2 dB for *h* = 2.5 km using the mean water vapor, temperature, and pressure profiles from 19 CalWater-2 sondes dropped in the observation region by the NOAA G-IV on 5 February 2015 (precipitable water path of 2.2 cm from the surface to and an aircraft altitude of 2.5 km). The term *G*_{υ}(*h*) was 0.07 dB at the first radar range gate and 2.20 dB at the surface. The variability (standard deviation associated with the variability of the sonde profiles) of *G*_{υ}(*h*) was ±0.18 dB at the surface and ±0.12 dB at 1-km altitude.

## 3. Radar–precipitation relationships

### a. Processing for surface cross section

With regard to surface returns, at near-nadir incidence the W-band radar observes a strong spike in measured *Z*_{em} from the ocean surface (Li et al. 2006) that is referred to as the ocean scattering cross section (where *δR* is the radar range gate thickness). The measured reflectivity factor is converted to NRCS_{m} using

where 180 is a conversion factor converting reflectivity factor from mm^{6} m^{−3} to m^{3}. At W band and with 25-m range resolution, the first term on the right-hand side of (6) is 137.9.

### b. Rain profile retrievals using reflectivity gradient

After correcting for attenuation, the simplest *R* (mm h^{−1}) retrievals are based on *Z*_{e}*–R* power-law relationships of the form

The *Z*_{e}*–R* relationships are estimated in several ways, for example, fitting observed *Z*_{e} versus surface-based rain measurements or using airborne or ground-based measurements of the rain DSD to compute nonattenuated values of *Z*_{e} and *R*. The rain rate can be expressed as a function of *Z*_{e} by inverting (7).

Note that (7) is poorly posed for retrieving rain rate at W band, partly because of attenuation and partly because *Z*_{e} at W band includes both the Rayleigh scattering regime for small raindrops and the non-Rayleigh scattering regime for raindrops greater than about 0.8 mm in diameter. Because of non-Rayleigh scattering, the changes in nonattenuated reflectivity at W band with increasing rain rate are not that pronounced as at lower frequencies.

The relationship between rain-specific attenuation and rain rate can be expressed with a power law of the form

Some estimated coefficients from previous studies are given in Table 2. Given the data scatter in the γ_{rain}–*R* correspondence, (8) can be assumed to be linear with *b*_{γ} = 1 (Matrosov 2007). The bootstrap values given in Table 2 are obtained from relationships based on NRCS rain rates and observed attenuation and reflectivity, that is, solely determined by CalWater-2 W-band observations. The CalWater P-3 values are computed from a Droplet Measurement Technologies Precipitation Imaging Probe (PIP), which sizes drops in 62 equally spaced bins from 0.10 to 6.2 mm in diameter. The Matrosov (2010), bootstrap, and P-3 PIP are considered the most representative for these observations, so they were used to compute the averages.

The linearized mean relationship between the attenuation coefficient γ_{rain} and rain rate is written as

where *c* = 1.11 km dB^{−1} mm h^{−1} as estimated from T-matrix modeling using DSDs collected with the Joss–Waldvogel disdrometer during a Hydrometeorology Testbed (HMT) field project in California (Matrosov 2010). The term on the right, *k*(*z*), is a dimensionless correction factor accounting for an increase in raindrop fall velocity as the air density *ρ*_{a} (kg m^{−3}) decreases with height *z* above the surface (Foote and duToit 1969). The factor 1.1 is , where kg m^{−3} is the density of air at the surface. For this paper a fixed density correction of 1.04 was used, which corresponded to an altitude of 1.0 km and CalWater-2 atmospheric conditions.

Since attenuation coefficients are usually specified in decibels per kilometer, they can be related to the vertical gradient of measured dB*Z*_{em} as

where the first term on the right-hand side describes changes of nonattenuated reflectivity dB*Z*_{e}, If the vertical gradient of dB*Z*_{e} is small compared to that resulting from attenuation, then the rain attenuation can be computed from the slope of dB*Z*_{em} versus altitude as follows:

where each term in (11) is height dependent. Equation (11) provides the estimate of the mean attenuation coefficient under the assumption that the vertical changes of nonattenuated reflectivity are small compared to changes of observed reflectivity as a result of attenuation (i.e., the gradient of dB*Z*_{e} is much smaller than the gradient of dB*Z*_{em}). This equation can be applied to different segments of the entire rain layer by calculating a mean gradient within the segments of the layer. Note also that the Doppler velocity threshold (Chandra et al. 2015) provides a way of choosing segments/layers where the gradient approach provides sensible results.

### c. Path-integrated rain retrievals

The radar backscatter from the sea surface allows another method to compute the path-averaged rain rate from the total attenuation from the aircraft to the surface. For our purposes here we restrict the analysis to nadir-pointing profiles only, so at a given wavelength the NRCS_{c} is a function of wind speed only,

where *U*_{10} is the wind speed at a height 10 m above the ocean, *F*(0)^{2} = 0.32 is the Fresnel reflection coefficient at 20°C for seawater at W band at normal incidence and mss is the mean squared slope of the surface waves. Thus, the difference between the measured NRCS_{m} and the value, NRCS_{c}, gives a PIA defined as

The NRCS-based rain rate, *R _{nc}*, can be computed as

where *h*_{s} is the range to the surface. The advantage of (14) is that it does not require near-uniform vertical profiles of rain but gives the mean rain rate between the aircraft and the surface. The disadvantage is that it requires a specification of NRCS_{c}; NRCS_{m} is computed from *Z*_{em} at the surface as per (6). The model for NRCS_{c} we are using is

### d. Profiles of Z_{e} versus Z_{em}

A considerable amount of work in the literature concerns retrieving the true (i.e., nonattenuated) *Z*_{e} profile from the radar-observed profile. The simplest approach is to combine (10) with a specification of attenuation in terms of *Z*_{e} as in (8), where

In this case, Iguchi and Meneghini (1994) show that the Hitschfeld and Bordan (1954) relation can be expressed as

where

and

Note that the integral *S*(*h*) is in terms of *Z*_{em}, so the right-hand side of (17) is solely in terms of measured quantities.

The use of (17) is known to be problematical at W band because the absorption is large and the retrieval is referenced to the value of *Z*_{em} nearest to the aircraft. In our case, we must also extrapolate the *Z*_{em} profile from the first range gate (489 m below the aircraft) up to the aircraft to compute *S*(*h*). For this paper, we have used NRCS_{m} to reference the profile at the surface (Iguchi and Meneghini 1994),

where *S*(*h*_{s}) is the value of *S*(*h*) at the surface and

This approach has two advantages: the correction is most accurate where it is largest (near the surface) and it does not depend on the extrapolation of the *Z*_{em} profile above the first range gate. The second point follows because

which depends only on the profile between *h* and the surface.

## 4. Processing and analysis

Only one flight (1900–2100 UTC 5 February 2015) yielded significant “stratiform” rain that is suitable for our analysis. Here we use the term *stratiform* to describe wide-scale, weakly convective precipitation associated with midlatitude frontal regions (referred to as atmospheric rivers). We are not using it to refer to broad areas of precipitation in outflow regions from deep tropical convection. The flight on 7 February 2015 had significant rainfall, which is suitable for applying the NRCS approach, but it is too patchy to be able to claim relative vertical homogeneity (i.e., the presence of uniform rain everywhere in a layer from the aircraft altitude to the surface). On 5 February 2015, the aircraft was flying below a large region of precipitating clouds (i.e., it was not in cloud). In some periods there were low-level “scud” clouds below the aircraft with tops around 0.5 km. Radar measurements from the NOAA ship *Ronald H. Brown* indicated cloud tops at 7-km altitude with a freezing-level bright band at about 3-km altitude. Photographs taken from the P-3 and a visible satellite image can be found online (ftp://ftp1.esrl.noaa.gov/psd3/cruises/CALWATER_2015/P3/P3_02_05_2015/Pics_Plans_PPTs/Photos/).

An example of radar *Z*_{em} profile measurements is shown in Fig. 1. The P-3 location during the flight is shown in Fig. 2 with color-coded indications of 10-m wind speed from the SFMR. Measured and parameterized values of NRCS are shown in Fig. 3a with resultant rain rate in Fig. 3b. In Fig. 4 we show rain-rate estimates (smoothed to a 1-min time resolution) from NRCS, the SFMR, and the WSRA for the entire 3-h period. The WSRA has been biased corrected for slow variations in the transmit power. Some elements of the phased array antenna were not operating correctly and the problem was intermittent. The comparison between NRCS and SFMR retrievals is better but still not good for lighter rain rates. The agreement is better at rain rates greater than about 2 mm h^{−1}. The correlation coefficient between NRCS and SFMR rain rates is 0.81, while for NRCS–WSRA rain rates it is 0.71.

### a. Processing methods

We have examined several methods for estimating the rain rate from the measured reflectivity profiles from two points of view: 1) time series of layer-averaged rain rate computed from each profile of *Z*_{em} and 2) profiles of radar variables averaged in bins of rain rate. The time series methods are as follows:

Compute a linear regression for each observation of dB

*Z*_{e}versus*h*of the form The rain rate is then estimated from this slope using (9). The intercept, dB*Z*_{ei}, is reflectivity at the aircraft height (*h*= 0), which is an estimate of the unattenuated dB*Z*_{e}(valid when rain is observed in the first range gate and assumes that rain is present in the whole layer from the aircraft altitude to the surface).Compute a layer-averaged attenuation from the difference in reflectivity at two range gates as (dB

*Z*_{em1}*−*dB*Z*_{em2})*/*(*h*1*− h*2) and get an estimate of rain rate using (9). This estimate is somewhat akin to the NRCS estimate but does not depend on a surface backscatter model. Here we have used range gates at altitudes of*h*_{1}= 0.67 km (altitude = 1.83 km) and*h*_{2}= 2.3 km (altitude = 0.20 km).We have also used a hybrid approach following Chandra et al. (2015) where

*Z*–*R*and attenuation retrievals are combined. The procedure is to compute an estimate of*R*from reflectivity at a selected range gate,*Z*_{em3}, via (7). If the measured Doppler velocity at*h*_{3}is less than a threshold, then it replaces*R*with the value from method 2 (above). Here we used*h*_{3}= 0.67 km and a Doppler velocity threshold of −3.0 m s^{−1}. Chandra et al. (2015) used a Doppler threshold of −5 m s^{−1}for a Ka-band radar. In the same conditions, a Doppler velocity observed at W band is about −3.5 m s^{−1}(see Firda et al. 1999 or Fig. 5 in Tian et al. 2007). Tridon and Battaglia (2015) note that the reflectivity departs significantly from the Rayleigh limit at fall velocities of 3.3 and 4.9 m s^{−1}for W and Ka bands, respectively. The slightly more conservative threshold (−3.0 m s^{−1}) we used reflects the stronger absorption at W band.

The bin-averaged methods use the rain rate from NRCS. We average the first three spectral moments and *d*(dB*Z*_{e})*/dh* in bins of rain rate at every range gate. The mean attenuation profile for each rain-rate bin is −0.5* < *d*(*<*dB*Z*_{em}*>*)*/dh>*, where *dh* = 0.025 km. Thus, we obtain profiles of mean *Z*_{em}, Doppler velocity, and attenuation at different rain rates.

### b. Rain-rate time series

An important issue to solve is how to treat the nonideal nature of the nonattenuated reflectivity profiles in the processing (Matrosov 2009). Examples of three types of dB*Z*_{e} profiles are shown in Fig. 5. A glance at Fig. 1 shows periods when there is no rain at aircraft flight level or the first observable range gate (e.g., the period 2021–2026 UTC). Thus, a vertical derivative will indicate negative attenuation near the first range that has precipitation (see the red profile in Fig. 5). The blue line in Fig. 5 shows a profile where only rain occupies the region above 1 km. The black line in Fig. 5 shows a case with significant return throughout the profile.

We have examined rain-rate estimates using methods 1 and 2 smoothed to a 1-min time resolution. These are pure rain dB*Z*_{e} gradient–based approaches. Both methods can produce negative rain rates and substantial overestimates of the rain rate when the precipitation below the aircraft is vertically inhomogeneous (i.e., rain is not present everywhere below the aircraft). One simple check to avoid the worse cases is to require the gradient be positive or to require that the dB*Z*_{em} at the first usable range gate has measurable rain and that the dB*Z*_{em} at that range gate exceeds the value of dB*Z*_{em} near the surface. For values that do not meet the criteria, we set the rain rate = 0. We found that method 3 (hybrid) is superior to methods 1 and 2, yielding a more accurate mean and higher correlation with *R*_{nc} (see Table 3). Figure 6 shows the rain-rate time series with method 3 and the NRCS-based method.

### c. Bin-averaged profiles

Figure 7 shows profiles of mean dB*Z*_{em} and vertical Doppler velocity *W* averaged in bins of rain rate as determined by the NRCS method for the 3-h period on 5 February 2015. For each rain-rate interval, the means were computed for profiles where SNR > −10 dB, which is the minimum value yielding reliable estimates of Doppler velocity. The six rain-rate intervals used in the remainder of this paper are [0–0.25, 0.25–0.7, 0.7–1.5, 1.5–3, 3–6, and 6–13] mm h^{−1}. The average rain rates for each interval are listed in the legend of Fig. 7. The fraction of profiles that exceed the SNR minimum *f*_{q} is given in Table 4. The SFMR- and WSRA-based rain rates were too noisy and uncertain to use as an index for bin averaging. The measured Doppler vertical velocity is corrected for the pitch component of aircraft motion relative to the air via

The residual given by (23) should be the mean Doppler velocity of the precipitation. Here *SOG* is the aircraft speed over ground (between 100 and 140 m s^{−1}); COG is the aircraft course over ground, pitch is the aircraft pitch angle; *ψ* is aircraft heading; *U*_{w} and Dir are the wind speed (taken from the SFMR) and wind direction (taken from the P-3 flight-level data), respectively. Equation (23) is derived from the corrections in Fairall et al. (2014) when aircraft roll = 0 [equivalent to (7) in Heymsfield 1989]. A 1° pitch on the P-3 will change the observed Doppler velocity about 2 m s^{−1}, so the accuracy of the navigation system and the relative alignment of the radar are important. To evaluate this we compared the mean Doppler shift of the *surface* return with the correction using (23) as a function of pitch for the calibration flight on 9 February 2015. Since the aircraft is flying multiple headings, we can assume the mean Doppler velocity as a result of the surface current is 0 [see Haimov and Rodi (2013) for a more sophisticated example applied over land]. We applied a 0.5° correction to the P-3 pitch data to get zero mean Doppler at a pitch angle of 0. In the past (Fairall et al. 2014) we have done a similar pitch calibration using Doppler measurements from nonprecipitating clouds (which have negligible sedimentation velocity) in the first range gates below the aircraft. However, in CalWater-2 we did not have sufficient periods of nonprecipitation clouds just below the aircraft.

The following are some factors to note in Fig. 7:

The mean reflectivity value near the surface for the maximum rain rate is −16 dB

*Z*, which is greater than, but close to, the radar noise level (−24 dB*Z*; see Fig. 5).The slopes at lower rain rates are confined to the upper part of the profile and are actually larger than the slopes for intermediate rain rates. This likely indicates inhomogeneous profiles with most of the rain confined within 1 km below the aircraft. Thus, attenuation deduced from this regime is not reliable (i.e., gradients of nonattenuated reflectivities are not small compared to the gradients resulting from attenuation).

The dB

*Z*_{em}values at the top of the measured profiles for the two largest rain-rate bins are about the same. The slope for the highest rain rate shows much more attenuation, so a lot of the signal has been lost between the aircraft and the first range gate (about 10 dB).Mean Doppler velocities are between −3 and −4.5 m s

^{−1}and roughly become more negative with increasing rain rate. Doppler velocity is not the same as drop sedimentation velocity. The W-band radar is less sensitive (relative to the Rayleigh scattering regime) to droplets larger than about 1 mm. Thus, smaller drops with lower fall velocities are more heavily weighted than for radars at longer wavelengths. A Doppler velocity of [−3.0 to −4.5] m s^{−1}at W band corresponds to about [−4 to −7.5] m s^{−1}at X band (see Fig. 5 in Tian et al. 2007). X-band Doppler velocities are similar to pure Rayleigh Doppler velocities for rain rates less than about 20 mm h^{−1}. Rayleigh velocities of [−4 to −7.5] m s^{−1}correspond to a gamma distribution of droplets with [0.6 to 1.5]-mm mode diameter (Steiner et al. 2004), which is typical for light rain (0.1–5 mm h^{−1}). Thus, the observed mean W-band Doppler velocities are consistent with our NRCS rain-rate estimates.

### d. Attenuation corrections of observed reflectivity

To compute (18) to correct the measured dB*Z*_{em} for attenuation, we must integrate along the entire propagation path from the aircraft to the surface. However, the radar’s first range gate is 0.5 km below the aircraft. Thus, we need to fill in the dB*Z*_{em} profile from the aircraft out to the first range gate. We have done this by fitting a linear regression to the mean dB*Z*_{em} profile starting at 1.83-km altitude and ending at 1.33-km altitude. Then, using the slope and intercept of the fit to the profile, we extrapolate dB*Z*_{em} values in 19 additional range gates between range gate 1 and the aircraft altitude. This is a total of 169 range gates going from the surface to the aircraft altitude. This is illustrated in Fig. 8, where mean dB*Z*_{em} are shown for the six selected rain-rate intervals used in Fig. 7. The extrapolated portions of the profiles are shown as dotted symbols. In Fig. 8 (and subsequent figures) the measured reflectivity dB*Z*_{em} is weighted by *f*_{q} to represent the average measured reflectivity within the rain-rate interval (not just the reflectivity while it is raining). The weight form is calculated by multiplying *Z*_{em} by *f*_{q}.

We have used the average measured unweighted reflectivity profiles as inputs into (20) and then the retrieved values are multiplied by *f*_{q} to yield “true” weighted dB*Z*_{e} profiles, which are shown as dashed lines in Fig. 8. We experimented with different parameters in the *γ*–*Z*_{e} relationship given in (16) but settled on *α*_{γ} = 0.050 and *β*_{γ} = 1.0 based on the reflectivity intercept values. The near-surface part of the corrected reflectivity profile is not sensitive to the coefficients because it is set by PIA. The relative effects of PIA versus the *γ*–*Z*_{e} relationship can be gauged by noting the two terms in the denominator of (20). At the surface the *q*[*S*(*h*_{s}) − *S*(*h*)] term is zero, yielding

The sensitivity to PIA disappears at heights where .

One measure of the profile retrieval’s effectiveness can be evaluated at the top of the profile (*h* = 0), where we expect the retrieved values to be the same as the observed (unattenuated) values. Thus, we expect the correction at *h* = 0 to be small,

Values for 10^{−β*PIA/10}, *qS*(*h*_{s}), and are given in Table 4 using *α*_{γ} = 0.050 and *β*_{γ} = 1.0. Also, values of derived from the direct fit from the P-3 PIP (*α*_{γ} = 058, *β*_{γ} = 0.85) and Matrosov (*α*_{γ} = 0.033, *β*_{γ} = 0.97) are given in Table 4. The bootstrap values yield negligible corrections for the heavier rain rates, while the PIP and Matrosov values yield corrections on the order of 2–3 dB.

Two things to notice from Fig. 8 are that the corrected profiles are not perfectly vertically homogeneous. The profile for the lowest rain-rate bin, for which the correction has no effect, is the most inhomogeneous. At rain rates of 1.0 mm h^{−1} and above the profiles are reasonably vertically homogeneous, except at the highest rain rate below 0.5-km altitude. The second thing to notice is that the *Z*_{e} values are quite close to the values (*Z*_{ei}) extrapolated to the height of the aircraft from the measured profiles. This suggests the consistency between precipitation retrievals and values of the reflectivity, the *γ*–*Z*_{e} relationship, and NRCS. The differences in the extrapolated reflectivities and the measured values at 500 m below the aircraft are consistent with the results shown in Figs. 1 and 2 of Hogan et al. (2003). Figure 1 in Hogan et al. shows estimates of unattenuated reflectivities as a function of rain rate; their Fig. 2b shows estimates of reflectivity after attenuation through 500 m at these same rain rates. The differences increase from 2 to 7 dB as the rain rate increases from 1 to 8 mm h^{−1}. For the same rain rates, the differences values at 500 m below the aircraft with extrapolated values in Fig. 8 increase from 2 to 8 dB.

### e. Evaluation of Z_{e}–R, γ–R, and γ–Z_{e} relationships at W band

It is well known that, because of the complexity and variability of rain microphysics, there are no universal *Z*_{e}*–R, γ*–*R*, and *γ*–*Z*_{e} relationships. Table 2 gives examples of variability for the W band. To determine relationships for a given set of observations *Z*_{e}*–R* and *γ*–*Z*_{e} relationships requires unattenuated *Z*_{e} values, which is problematical at W band, where attenuation is substantial. We have two sources of unattenuated *Z*_{e}: estimates at the aircraft altitude obtained by extrapolating the profiles of dB*Z*_{em} (the intercept occurs at *h* = 0, so it is unaffected by attenuation) and estimates of dB*Z*_{e} obtained by applying (20) to the observed profiles. This retrieval requires an assumed *γ*–*Z*_{e} relationship. We used the dB*Z*_{e} intercepts to determine the *Z*_{e}*–R, γ*–*R*, and *γ*–*Z*_{e} parameters and then verified that the retrieved *Z*_{e} were reasonably well fit.

Figure 9 shows the mean *Z*_{e} values (upper panel) and the mean attenuation (lower panel) at three altitudes as a function of the bin-average rain rate. We selected three levels from the retrieval plus the intercept because that is at most as many degrees of freedom in the profiles. We did not use near-surface profiles because of the anomalous behavior of the highest rain-rate profile below 0.5-km altitude. In the case of reflectivity, the value extrapolated from the uncorrected reflectivity (*Z*_{ei}) is shown as the square. In Fig. 10 we show the *γ*_{R} values from Fig. 9b plotted against the *Z*_{e} values from Fig. 9a. In Fig. 10 we can see residual attenuation (on the order of 1 dB km^{−1}) as *Z*_{e} approaches 0. The *γ*–*Z*_{e} parameters used in the correction are shown to be a good fit to the data. The fits shown in these graphs are given as the bootstrap values in Table 2.

### f. Profiles of rain rate

Finally, we present the mean rain-rate profiles obtained by averaging the profile of the vertical derivative of dB*Z*_{em} in rain-rate bins. While individual 0.3-s dB*Z*_{em} profiles yield a noisy derivative profile, when averaged the results are reasonably smooth. We then use (9) and (11) to compute the profiles of the rain rate (Fig. 11a) with as discussed in section 3b. If we had used the bootstrap parameters, then *R* would be 15% lower. We have multiplied the rain rate by the fraction of bins with detectable rain to yield an actual rain rate including the dry periods. Figure 11b shows rain-rate profiles retrieved via the *R–Z*_{e} relationship [(7)] with the corrected reflectivities—a clear illustration of why the hybrid method works. The bias in the attenuation-derived rain rates is apparent compared to the *R–Z*_{e}-derived values, particularly at low rain rates. Using the bootstrap parameters improves the agreement for higher rain rates.

### g. Summary rain-rate statistics

In Table 3 we compare simple statistics for the different rain-rate estimates. We have added one estimate that is independent of the W-band radar, Rain2—the mean of the WSRA and SFMR rain rates. Rain2 has the same mean rain rate as rain from NRCS. The grand mean rain rate across all methods is 1.12 ± 0.17 mm h^{−1}; the mean while raining is 2.18 ± 0.21 mm h^{−1}. The hybrid method and SFMR have the highest correlations with *R*_{nc}. Note that the hybrid method, which depends on the observed reflectivity at 0.5 and 2.0 km below the aircraft and on the Doppler return at 0.5 km below the aircraft, is independent of *R*_{nc}, which depends on the reflectivity of the ocean surface.

## 5. Discussion and conclusions

In this paper we examined several approaches to estimating rain-rate time series, profiles, and statistics using radar reflectivity. The data are from the PSD W-band Doppler radar deployed on a NOAA P-3 aircraft during the CalWater-2 field program. Our primary goal was to investigate the use of the radar signal attenuation to improve estimates of rain rate and precipitation microphysical parameters below the aircraft (observation altitude was 2.5 km). The analysis is limited to 3 h from a flight in wide-scale frontal precipitation on 5 February 2015. In principle, profiles of rain rate can be computed from the profile of attenuation if cloud absorption and vertical inhomogeneity are negligible. However, individual profiles (3-Hz acquisition rate) may be quite vertically inhomogeneous because of the patchy nature of precipitation—this leads to noisy vertical derivatives. We used 1-min averages to smooth out some of the inhomogeneity.

The relationship of attenuation coefficient to the rain rate was found to be near linear and quite robust with good comparisons of our observations with several others in the literature. At rain rates near 1 mm h^{−1} and below, the observed attenuation coefficient levels off, which is a possible consequence of cloud attenuation. The relationships of reflectivity factor with rain rate or attenuation coefficient were also robust. Our W-band radar measurements, after correction for attenuation, were a good fit to an assumed power law of (Fig. 10) and compared reasonably with fits to computations using measured DSDs (see Table 2).

The NRCS method provided the most consistent estimate of layer-averaged rain rate from the W band, though, unlike the reflectivity gradient methods, it is applicable when the surface backscatter in the absence of rain is well characterized. Uncertainty in the rain–attenuation relationship makes the rain-rate values uncertain by about 15%. Uncertainty in the water vapor or cloud attenuation and in the NRCS wind speed parameterization contributes an additional rain-rate uncertainty of about ±0.5 mm h^{−1}. Without a standard we are unable to discern which of the rain estimates is superior. The most obvious difference between NRCS, WSRA, and SFMR-based methods is in the behavior in lighter rainfall. The SMFR tends to be spikey with very little light rain, while the WSRA shows higher values in light rain. The WSRA-based estimates are expected to be unbiased, but because of weaker attenuation they are also expected to have more trouble resolving light rain. The three other methods (i.e., methods 1–3) for estimating rainfall from W-band dB*Z*_{em} profiles/gradients were not as effective as the NRCS method. The two gradient methods were unreliable in inhomogeneous rain distributions when rain is not present at all altitudes below the aircraft. The hybrid method of Chandra et al. (2015), which combines an *R–Z*_{em} retrieval in light rain and an *R–γ* gradient-based retrieval in heavier rain was better with a 0.79 correlation with *R*_{nc}.

Compositing , Doppler velocity, or in bins of rain rate (Figs. 7, 8, and 11) yields very clean profiles. At lower rain rates there are anomalous gradients of dB*Z*_{e} in the upper heights, presumably because light rain is occurring at higher altitudes but is not reaching the surface. The profiles of mean gradient-derived rain rate in bins of NRCS rain rate (Fig. 11a) are smooth and the values are consistent at higher rain rates (but about 10% higher than NRCS-based rain-rate estimates). The use of the NRCS-referenced Iguchi and Meneghini (1994) method to reconstruct unattenuated profiles of dB*Z*_{e} from the composited observed (attenuated values) was robust. The retrieved profiles were consistent with estimates of unattenuated reflectivity from linear regressions of *Z*_{em} against range. The observed relationship of the corrected reflectivity with measured attenuation was consistent with the relationship assumed in the retrieval.

For research in cloud physics and cloud turbulence from mobile platforms (aircraft or ships), a W-band Doppler radar is likely the optimal choice. The combination of small size, sensitivity, narrow beamwidth, and reasonable (relatively) cost make it feasible to find installation space on crowded research aircraft and to stabilize pitch and roll motions on ships. For airborne systems a narrow beamwidth is necessary to minimize the broadening of the Doppler width (0.45 m s^{−1} broadening for the NOAA W band on the P-3 aircraft). For the measurement of rain rate with a single radar, a longer wavelength radar with less attenuation is likely preferable—depending on the scientific target. However, for space, weight, and sensitivity reasons, the W band might be preferred for ice cloud/precipitation studies from a high-altitude aircraft, such as the NCAR High-Performance Instrumented Airborne Platform for Environmental Research (HIAPER; (Vivekanandan et al. 2015). Clearly multiple wavelength approaches are preferable, but there are engineering challenges. Installing a Ku-band system with the same beamwidth (0.7°) as the NOAA W band would mean finding a place for a 1.5-m-diameter antenna.

## Acknowledgments

The authors thank the NOAA Aircraft Operations Office of Marine and Aviation Operations flight crew and support team for the making the airborne W-band flights possible. We also thank Marty Ralph, Ryan Spackman, and the rest of the CalWater science team. The comments from two reviewers were extremely helpful. Sergey Matrosov was funded in part by the NASA Project NNX16AQ36G. Christopher Williams was funded in part by the NASA Precipitation Measurement Mission (PMM) NNX13AF89G.

## REFERENCES

*Second Conf. on Radar Meteorology*, Zurich, Switzerland, Amer. Meteor. Soc., 433–438.

_{α}-band Doppler radar and a microwave radiometer

*Centimeter and Millimeter Wavelength Radars in Meteorology*. Lhermitte Publications, 550 pp.

_{a}-W dual-wavelength Doppler spectral ratio technique

## Footnotes

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

^{1}

Aircraft deployments include Tropical Storm Karen, Hurricane Patricia, and CalWater-2. Raw and processed data for the PSD observations can be found online (at ftp://ftp1.esrl.noaa.gov/psd3/cruises/CALWATER_2015/). Dropsonde profiles as Matlab.mat files from the G-IV are available (ftp://ftp1.esrl.noaa.gov/psd3/cruises/CALWATER_2015/G4/data/) as are the W-band radar data [ftp://ftp1.esrl.noaa.gov/psd3/cruises/CALWATER_2015/P3/Wband/; with the moment files in netCDF format in the *mom* directory, and the P-3 navigation and flight-level data in .txt files (which includes the SFMR) are in the *Aircraft* directory].