Estimation of Rain Rate from Airborne Doppler W-Band Radar in CalWater-21. 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 Ze 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 Zem to estimate precipitation rate using the observed decrease of Zem 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 Ze) 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 Zem and light rain rates (less than approximately 1 mm h−1) are estimated from calibrated reflectivity using a Ze–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 NRCSm 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 NRCSc 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 Ze 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.
This paper is organized as follows: experimental details are given in section 2, radar–precipitation relationships are discussed in section 3; processing methods and analysis results are described in section 4, and section 5 has discussion and conclusions.
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 20151.
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.
Specifications of the PSD W-band radar for CalWater-2 flights.
Values of γυ 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 Gυ =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 Gυ(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 Gυ(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
b. Rain profile retrievals using reflectivity gradient
Note that (7) is poorly posed for retrieving rain rate at W band, partly because of attenuation and partly because Ze 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.
Some estimated coefficients from previous studies are given in Table 2. Given the data scatter in the γrain–R correspondence, (8) can be assumed to be linear with bγ = 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.
Coefficients for rain-rate dependence of Ze [(7)] and γrain [(8)] at W band. Bootstrap refers to a relationship based on NRCS rain rates and observed attenuation and reflectivity. Lhermitte and Kollias values are computed from Marshall–Palmer DSD. Matrosov (2007, 2010) values are computed from disdrometer DSD measurements. P-3 PIP calculations are from the airborne in situ DSD measurements on 6 Feb 2015. More recent direct estimates (bold values) are used to compute an average and uncertainty. Values in the table correspond to Ze in mm6 m−3, γ in dB km−1, and R in mm h−1.
c. Path-integrated rain retrievals
d. Profiles of Ze versus Zem
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 (
An example of radar Zem 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.
Time–range cross section of reflectivity (dBZem) for (a) 1900, (b) 2000, and (c) 2100 UTC 5 Feb 2015. Vertical ordinate is height above the surface (altitude); horizontal ordinate is minutes for each hour (UTC). Surface return is apparent (bright red line at altitude near 0). Aircraft descended from 5 to 2.5 km in the beginning of the record. Banking maneuvers are visible as the short periods of extended range in the surface return (e.g., 20 h, 53 min). Note the period just after 19 h, 20 min when attenuation is so great there is no surface return.
Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0025.1
Flight path of the NOAA P-3 for 1900–2200 UTC 5 Feb 2015. Color of the path denotes 10-m wind speed (m s−1) from the SFMR. Satellite image for this day can be found online (
Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0025.1
(top) Sample time series of modeled NRCSc (blue) and NRCSm measured including attenuation (green) from 1900 UTC 5 Feb 2015 in CalWater-2. Note a few missing values just after 1935 UTC, when rain attenuation was sufficient to eliminate the surface return (you can see this as a notch in dBZem in Fig. 1 where the surface return disappears). At the end of the record there is no precipitation, so the blue and green lines coincide. (bottom) Precipitation from NRCSc–NRCSm.
Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0025.1
As in Fig. 3, except that a 1-min smoothed form of the rain rate is shown for the entire period 1900 through 2200 for NRCS (blue), SFMR (green), and WSRA (red).
Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0025.1
The peak NRCS rain rate in Fig. 4 is about 10 mm h−1, which is the approximate limit of the radar when flying at 2.5 km with 20 m s−1 10-m wind speed. This is because surface returns are no longer detectable for greater rain rates (e.g., a gap in the surface returns at 1920 UTC in Fig. 1).
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 Zem 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 dBZe versus h of the form
The rain rate is then estimated from this slope using (9). The intercept, dBZei, is reflectivity at the aircraft height (h = 0), which is an estimate of the unattenuated dBZe (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 (dBZem1−dBZem2)/(h1 − h2) 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 h1 = 0.67 km (altitude = 1.83 km) and h2 = 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, Zem3, via (7). If the measured Doppler velocity at h3 is less than a threshold, then it replaces R with the value from method 2 (above). Here we used h3 = 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.
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 dBZe 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.
Sample observed reflectivity profiles from 5 Feb 2015. Noise level of the radar (green line; in dBZem terms it increases with range from the radar). In the middle of the record with light precipitation from the surface up to 0.6 km (red line). Early in the record with no precipitation below 1 km (blue line). Later still with precipitation all the way to the surface (black line). The legend shows the time within the hour.
Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0025.1
We have examined rain-rate estimates using methods 1 and 2 smoothed to a 1-min time resolution. These are pure rain dBZe 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 dBZem at the first usable range gate has measurable rain and that the dBZem at that range gate exceeds the value of dBZem 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 Rnc (see Table 3). Figure 6 shows the rain-rate time series with method 3 and the NRCS-based method.
Comparison of mean rain rate (mm h−1) and correlation coefficients for the different methods using the 1-min time series. The mean while raining is computed by selecting periods where Rnc > 1 mm h−1. Rain2 refers to the mean of WSRA and SFMR rain rates.
Layer mean radar-derived rain-rate estimates from 5 Feb 2015. NRCS values (blue) are compared to the hybrid estimate (method 3) where R = Rmethod2 for Wc < −3.0 m s−1 and R =(Zem/15)0.91 for Wc > −3.0 m s−1. For method 2 the difference in reflectivity is computed between the two range gates at 1.83- and 0.20-km altitude; Zem and Wc are taken from the range gate at 1.83-km altitude.
Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0025.1
c. Bin-averaged profiles
Profiles of bin averages of (top) dBZem and (bottom) pitch-corrected Doppler vertical velocity for 3 h on 5 Feb 2015. Legend gives the mean rain rate (mm h−1) for the six rain-rate intervals of [0–0.25, 0.25–0.7, 0.7–1.5, 1.5–3, 3–6, and 6–13] mm h−1. Means were computed for profiles where SNR exceeded the threshold (−10 dB) for valid Doppler velocity estimates.
Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0025.1
Summary statistics of rain-rate-average reflectivity properties: fq is the fraction of valid returns in each rain-rate category; Zem(0.1) is the return-fraction weighted mean measured reflectivity at 0.1-km altitude, Zei is the weighted measured profile extrapolated to the aircraft altitude; Ze(0.1) and Ze (top) are retrieved reflectivity at 0.1-km altitude and aircraft height, respectively. The 10−β*PIA/10 is the first term in the denominator of (20), and q*S(hs) is the second term evaluated at aircraft altitude (h = 0). Term ΔdBZet is the total correction made via (20) to the reflectivity at aircraft altitude. The last two columns show the result using the direct γ–Ze fits from the P-3 PIP and Matrosov (2010), respectively, given in Table 2.
The following are some factors to note in Fig. 7:
- The mean reflectivity value near the surface for the maximum rain rate is −16 dBZ, which is greater than, but close to, the radar noise level (−24 dBZ; 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 dBZem 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 dBZem 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 dBZem profile from the aircraft out to the first range gate. We have done this by fitting a linear regression to the mean dBZem 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 dBZem 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 dBZem 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 dBZem is weighted by fq 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 Zem by fq.
Profiles of average dBZem and mean dBZe for the same rain-rate intervals used in Fig. 7. Average measured reflectivities are from Fig. 7a but have been scaled by fq (see Table 4). Average dBZem (solid lines). Portions of the profiles above 1.9 km that are the extrapolations using the slope and intercepts (dotted lines). The term dBZe was computed from dBZem using (20) with γR = 0.05*Zem1.0
Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0025.1
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 Ze values are quite close to the values (Zei) 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–Ze 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 Ze–R, γ–R, and γ–Ze relationships at W band
It is well known that, because of the complexity and variability of rain microphysics, there are no universal Ze–R, γ–R, and γ–Ze relationships. Table 2 gives examples of variability for the W band. To determine relationships for a given set of observations Ze–R and γ–Ze relationships requires unattenuated Ze values, which is problematical at W band, where attenuation is substantial. We have two sources of unattenuated Ze: estimates at the aircraft altitude obtained by extrapolating the profiles of dBZem (the intercept occurs at h = 0, so it is unaffected by attenuation) and estimates of dBZe obtained by applying (20) to the observed profiles. This retrieval requires an assumed γ–Ze relationship. We used the dBZe intercepts to determine the Ze–R, γ–R, and γ–Ze parameters and then verified that the retrieved Ze were reasonably well fit.
Figure 9 shows the mean Ze 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 (Zei) is shown as the square. In Fig. 10 we show the γR values from Fig. 9b plotted against the Ze values from Fig. 9a. In Fig. 10 we can see residual attenuation (on the order of 1 dB km−1) as Ze approaches 0. The γ–Ze 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.
Results from analysis of each mean profile using retrieved values of Ze (Fig. 8) and extrapolations of linear regression fits of the form 
Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0025.1
Results from analysis of profiles as per Fig. 9, except that attenuation is plotted directly against Ze.
Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0025.1
f. Profiles of rain rate
Finally, we present the mean rain-rate profiles obtained by averaging the profile of the vertical derivative of dBZem in rain-rate bins. While individual 0.3-s dBZem 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 
Rain-rate bin-averaged profiles of rain rate from the (a) averaged dBZem slope and (b) retrieved Ze values (Fig. 8) using the bootstrap Z–R relationship in Table 2.
Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0025.1
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 Rnc. 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 Rnc, 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 
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 dBZem 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–Zem retrieval in light rain and an R–γ gradient-based retrieval in heavier rain was better with a 0.79 correlation with Rnc.
Compositing 
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.
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.
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Aircraft deployments include Tropical Storm Karen, Hurricane Patricia, and CalWater-2. Raw and processed data for the PSD observations can be found online (at
