## 1. Introduction

The demand for good spatial resolution from airborne and spaceborne weather radars under the constraint of antenna size leads to the use of higher frequencies. However, at frequencies at X band (10 GHz) and above, attenuation correction is usually needed to allow for the potential of accurate estimates of precipitation parameters. For a well-calibrated radar, the Hitschfeld–Bordan (HB) attenuation correction method (Hitschfeld and Bordan 1954) has been shown to be reasonably accurate for small to moderate path attenuations (Marzoug and Amayenc 1994; Iguchi and Meneghini 1994; Iguchi et al. 2000, 2009); however, at higher attenuations, the errors grow rapidly even in the presence of modest calibration errors or errors in the *k–Z* (specific attenuation–radar reflectivity factor) relation. The surface reference technique (SRT) can be used to complement the HB approach since the associated errors are nearly independent of the strength of the attenuation and primarily depend on the statistics of the normalized radar cross section (NRCS or σ^{0}) of the surface.

Although the basic method is well established (Meneghini et al. 1983, 2000), some refinements and alternative formulations have been proposed (Li et al. 2002, 2004; Durden et al. 2003, 2012; Meneghini et al. 2004; Takahashi et al. 2006; Seto and Iguchi 2007) as well as extensions to dual-frequency data (Meneghini et al. 1987; Tanelli et al. 2006; Meneghini et al. 2012). The method has been part of the operational processing of the TRMM and now GPM data and is used in the analysis of *CloudSat* data (L’Ecuyer and Stevens 2002; Mitrescu et al. 2010). The method has also been used in the analysis of airborne weather radar data (e.g., Liao and Meneghini 2005; Liao et al. 2008).

## 2. Surface reference technique

### a. Single-frequency method (SRT)

^{0}) of the surface follows the formulation given by Kozu (1995) that relates σ

^{0}to parameters of the radar and the radar return power from the surface. The equation for the estimate of path attenuation consists simply of a subtraction (dB) between the NRCS under rain-free and raining conditions. Let

*θ*, measured with respect to the local normal to the surface, and let

*k*(dB km

^{−1})] along the radar beam at angle

*θ*, the two-way path-integrated attenuation (PIA) is defined as

*r*= 0, to the surface,

*r*=

*r*

_{s}. Under nonuniform beamfilling (NUBF) conditions, greater care is needed in writing the equation for the PIA since different ray paths within the main lobe of the antenna generally correspond to different path attenuations (Nakamura 1991; Kozu and Iguchi 1996; Takahashi et al. 2006; Durden and Tanelli 2008; Tanelli et al. 2012; Meneghini and Liao 2013). For the discussion here, however, we assume that uniform beamfilling conditions exist in the main beam, at any range, in the plane orthogonal to the range direction.

An obvious way to compute reference data is to take, at each incidence angle, a running mean and standard deviation over a window of *N* fields of view (FOVs) under rain-free conditions, where *N* is nominally set to 8. When rain is encountered, the updated reference values are used to compute the rain-free sample mean and associated standard deviation. Reference data formed in this way are referred to as an along-track spatial reference.

A second type of reference can be derived from prior measurements of the NRCS at or near the rain area of interest. In particular, a latitude–longitude and incidence angle array can be defined to store the mean, standard deviation, and number of data points of σ^{0} taken under rain-free conditions. For this type of temporal reference, the first term on the right-hand side of (1) is obtained from a lookup table given the latitude–longitude and incidence angle information associated with the NRCS measurement at the raining field of view.

A third kind of reference is obtained by fitting a curve through σ^{0} data in the cross-track direction. [In both the TRMM PR and GPM dual-frequency precipitation radar (DPR), the radars scan in the cross-track direction, orthogonal to the direction of spacecraft motion.] In particular, over ocean, in the absence of attenuation, a good approximation to the variation in σ^{0} with angle is a quadratic. Thus, if the running mean of the along-track reference data at each angle is taken, with the associated standard deviation used as a weighting factor in the fit, then the coefficients of a quadratic curve can be found, which serves as the reference data across the scan [as was pointed out by Seto and Iguchi (2007), departures in the σ^{0} from quadratic variation can cause errors]. Using the TRMM PR data, it was found that two quadratic fits over the Ku swath—one in the inner swath and one in the outer swath—reduce this error. Over the narrower 125-km Ka band (and KaHS swaths^{1}), only a single quadratic is used. It should be noted that cross-track estimates are done only over ocean; an appropriate cross-track fitting function has not been found for over land.

Independent along-track and cross-track reference data can be obtained simply by processing the orbital data in reverse so that the reference data are taken after (backward going) the rain area is encountered rather than before (forward going). A schematic of the different types of spatial reference estimates is shown in Fig. 1.

In summary, the five types of reference data are forward/backward along track, forward/backward cross track, and temporal, each leading to a different, nearly independent PIA estimate. In addition, a modified temporal reference dataset has been proposed (Seto and Iguchi 2007), based on the fact that the NRCS over land is modified by the presence of light rain or the recent occurrence of rain and that these data can be used as a temporal reference. In the future, both types (conditioned and unconditioned on surface wetness) of temporal reference datasets will be prepared when sufficient DPR data are available.

It should be mentioned that at present, only three surface types are defined for the SRT: ocean, land, and coast, where the reference data value is matched to that surface type beneath the raining FOV. The use of a multicategory land classification has been proposed in order that spatial reference data are taken only from that class that matches the surface type under the raining FOV (Durden et al. 2012). Tests of this idea are presently being conducted. For this paper, however, only PIA estimates derived from the spatial reference datasets with the three surface types are presented.

### b. Dual-frequency method (DSRT)

*f*

_{1}and

*f*

_{2}. As is the case with the DPR, we assume that the measurements are made simultaneously using matched beams. Taking

*f*

_{1}to represent the Ka-band frequency (35.5 GHz) and

*f*

_{2}to represent the Ku-band frequency (13.6 GHz), then

*f*

_{1}>

*f*

_{2}and the true differential path attenuation, δ

*A*=

*A*(

*f*

_{1}) −

*A*(

*f*

_{2}), is always positive. An estimate for δ

*A*can be obtained from

^{0}= σ

^{0}(

*f*

_{1}) − σ

^{0}(

*f*

_{2}), (3) can be written in the form

^{0}in rain from the rain-free differential σ

^{0}yields an estimate of differential path attenuation.

*A*but

*A*(

*f*

_{1}) and

*A*(

*f*

_{2}). Following Meneghini et al. (2012), we assume that

*A*(

*f*

_{1}) and

*A*(

*f*

_{2}) are linearly related to δ

*A*; thus, if

*A*(

*f*

_{1}) = γ δ

*A*, then

*A*(

*f*

_{2}) = (γ − 1) δ

*A*. It is easy to show that γ =

*A*(

*f*

_{1})/[

*A*(

*f*

_{1}) −

*A*(

*f*

_{2})]. A more transparent way to express these relations is to introduce the ratio

*p*=

*A*(

*f*

_{1})/

*A*(

*f*

_{2}) so that

*A*(

*f*

_{1}) and

*A*(

*f*

_{2}) can be derived from δ

*A*and

*p*using the following equations:

### c. Effective PIA

*A*. To emphasize the fact that multiple estimates of path attenuation are available, (1) can be written as (omitting the overbar and

*θ*dependence for simplicity)

*j*th surface reference value and the second term is the apparent NRCS in rain. Associated with the

*j*th reference value is a variance, σ

_{j}

^{2}, where

*w*

_{j}are such that

*j*th estimate

_{eff}as

_{eff}requires a value for the standard deviation of the effective PIA. This can be found by noting that σ

_{eff}

^{2}= var(

*A*

_{eff}) so that

## 3. Errors in the methods

*ρ*is the correlation coefficient of

The above-mentioned equations show that an approximation for the accuracy of the SRT can be found from the statistics of the NRCS. Figure 2 shows the mean and standard deviation of the rain-free NRCS DPR data as a function of incidence angle for data in the inner swath, from −9° to 9°, acquired during August 2014. Note that the data are restricted to latitudes between 35°S and 35°N so that the DPR statistics can be compared with those from the TRMM PR.

(left) Mean and (right) standard deviation of rain-free σ^{0} over (top) ocean and (bottom) land as a function of incidence angle in the inner swath of the DPR (from −9° to 9°): DPR Ku band (red) and DPR Ka band (blue). TRMM Ku-band statistics (black) for August are shown for comparison. Data are taken from August 2014 over 35°S to 35°N to coincide with the TRMM measurements.

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

(left) Mean and (right) standard deviation of rain-free σ^{0} over (top) ocean and (bottom) land as a function of incidence angle in the inner swath of the DPR (from −9° to 9°): DPR Ku band (red) and DPR Ka band (blue). TRMM Ku-band statistics (black) for August are shown for comparison. Data are taken from August 2014 over 35°S to 35°N to coincide with the TRMM measurements.

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

(left) Mean and (right) standard deviation of rain-free σ^{0} over (top) ocean and (bottom) land as a function of incidence angle in the inner swath of the DPR (from −9° to 9°): DPR Ku band (red) and DPR Ka band (blue). TRMM Ku-band statistics (black) for August are shown for comparison. Data are taken from August 2014 over 35°S to 35°N to coincide with the TRMM measurements.

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

The left-hand panels of Fig. 2 show the mean NRCS as a function of incidence angle in the inner swath for ocean (top) and land (bottom). Results of the standard deviation in σ^{0}(Ku), σ^{0}(Ka) are shown in the right-hand panels. Note that in this inner swath (from −9° to 9°), the mean and standard deviation of σ^{0}(Ku) from the TRMM PR are quite close to those from the DPR Ku band. It is worth noting that this good agreement extends to the wider swath (from −18° to 18°) as well (as viewed by an observer on the spacecraft, facing in the direction of motion, negative angles are to the left).

The statistics critical to the SRT performance are shown in Fig. 3, where *ρ* for ocean (top) and land (bottom) is shown in the left-hand panels. The asymmetry in *ρ* with respect to nadir, particularly noticeable over ocean, is thought to be caused by surface sidelobe contamination. The right-hand panels show the standard deviations of σ^{0}(Ku), σ^{0}(Ka), and δσ^{0}. The smaller standard deviation in δσ^{0} relative to those in σ^{0}(Ku) and σ^{0}(Ka) indicates that the estimate of differential attenuation should have associated with it smaller variations than the single-frequency estimates. Moreover, the variations in the differential surface cross sections over ocean are nearly independent of incidence angle, whereas the single-frequency data show large variations with angle.

(top left) Correlation coefficient of [σ^{0}(Ku), σ^{0}(Ka)] over ocean; (bottom left) correlation coefficient over land; (top right) standard deviations of σ^{0}[Ku) (red), σ^{0}(Ka)] (blue), and [σ^{0}(Ka) − σ^{0}(Ku)] (black) over ocean; (bottom right) As in (top right), but over land.

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

(top left) Correlation coefficient of [σ^{0}(Ku), σ^{0}(Ka)] over ocean; (bottom left) correlation coefficient over land; (top right) standard deviations of σ^{0}[Ku) (red), σ^{0}(Ka)] (blue), and [σ^{0}(Ka) − σ^{0}(Ku)] (black) over ocean; (bottom right) As in (top right), but over land.

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

(top left) Correlation coefficient of [σ^{0}(Ku), σ^{0}(Ka)] over ocean; (bottom left) correlation coefficient over land; (top right) standard deviations of σ^{0}[Ku) (red), σ^{0}(Ka)] (blue), and [σ^{0}(Ka) − σ^{0}(Ku)] (black) over ocean; (bottom right) As in (top right), but over land.

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

*E*(

*A*)/std(

*A*). Letting

*A*(Ku) = 1 dB so that

*A*(Ka) =

*p*and δ

*A*= (

*p*− 1) and using the approximations in (17) and (18) gives

*p*=

*A*(Ka)/

*A*(Ku) is considered a constant, then the “reliability” of the dual-frequency-derived estimates of

*A*(Ku) and

*A*(Ka) are the same as the reliability of δ

*A*. Denoting the dual-frequency-derived estimates of path attenuation at Ku band and Ka band as

*A*

_{DF}(Ku) and

*A*

_{DF}(Ka), respectively, then, for

*p*constant

*p*= 6, (19) is easily evaluated. The results, in Fig. 4, show that, irrespective of whether a single- or dual-frequency method is used, the reliability is almost always higher over ocean than over land, reflecting the fact that the standard deviation of the NRCS is almost always lower over ocean than land at these incidence angles. Another feature of the results is that for a given incidence angle and surface type, the reliability of the differential estimate is always higher than those of the single-frequency and that this difference is particularly significant at Ku band. In other words, the Ku-band estimate of path attenuation is expected to be significantly more accurate if the dual-frequency version rather than the single-frequency version of the method is used. Use of the dual-frequency SRT at Ka band also improves the accuracy of the estimate for all incidence angles over land and ocean though the improvement is smaller. These results, however, assume that the

*A*(Ka)/

*A*(Ku) is constant, whereas analyses of raindrop size distribution data show that this ratio depends to some extent on the size distribution. Another consideration is the loss of the surface return at Ka band in heavy rain, which limits the applicability of the method. We will return to these issues later in the paper. Finally, it should be noted that the results in Fig. 4 scale with

*A*(Ku) so that if

*A*(Ku) is 2 rather than 1 dB, then all the results are scaled by a factor of 2; that is, the relative error of all the estimates decreases as the path attenuation increases.

Reliability (mean/standard deviation) of path attenuation estimates as a function of incidence angle for Ku band (red), Ka band (blue), and differential (black) estimates over (top) ocean and (bottom) land assuming *p* = 6 and a 1-dB path attenuation at Ku band.

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

Reliability (mean/standard deviation) of path attenuation estimates as a function of incidence angle for Ku band (red), Ka band (blue), and differential (black) estimates over (top) ocean and (bottom) land assuming *p* = 6 and a 1-dB path attenuation at Ku band.

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

Reliability (mean/standard deviation) of path attenuation estimates as a function of incidence angle for Ku band (red), Ka band (blue), and differential (black) estimates over (top) ocean and (bottom) land assuming *p* = 6 and a 1-dB path attenuation at Ku band.

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

Although Fig. 3 gives single values (one each for land and ocean) for the standard deviation of the NRCS at a given angle, these values represent spatial averages of the data from 35°S to 35°N. Shown in Fig. 5 are the standard deviations of σ^{0}(Ku) (top panel), σ^{0}(Ka) (middle panel), and [σ^{0}(Ka) − σ^{0}(Ku)] (bottom panel) on a 0.5° × 0.5° latitude–longitude grid for an incidence angle of 3° using rain-free DPR measurements taken during August 2014. Significant spatial variations are present over both land and ocean. Since the path attenuation estimates are made at the local level, the results in Figs. 3 and 4 provide only a first-order assessment of the expected performance of the SRT.

Maps of the standard deviation of the NRCS from DPR data taken during August 2014 at an incidence angle of 3° using a latitude × longitude grid of 0.5° × 0.5°. (top) Standard deviation of σ^{0}(Ku), (middle) standard deviation of σ^{0}(Ka), and (bottom) standard deviation of δσ^{0} = [σ^{0}(Ka) − σ^{0}(Ku)]. Note the change of scale for (bottom).

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

Maps of the standard deviation of the NRCS from DPR data taken during August 2014 at an incidence angle of 3° using a latitude × longitude grid of 0.5° × 0.5°. (top) Standard deviation of σ^{0}(Ku), (middle) standard deviation of σ^{0}(Ka), and (bottom) standard deviation of δσ^{0} = [σ^{0}(Ka) − σ^{0}(Ku)]. Note the change of scale for (bottom).

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

Maps of the standard deviation of the NRCS from DPR data taken during August 2014 at an incidence angle of 3° using a latitude × longitude grid of 0.5° × 0.5°. (top) Standard deviation of σ^{0}(Ku), (middle) standard deviation of σ^{0}(Ka), and (bottom) standard deviation of δσ^{0} = [σ^{0}(Ka) − σ^{0}(Ku)]. Note the change of scale for (bottom).

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

## 4. Results

### a. Overpass example

Before examining the statistical properties of the PIA estimates, it is instructive to present an overpass example. On 3 July 2014, the GPM satellite overflew Hurricane Arthur, located off the East Coast of the United States. Shown in Fig. 6 are the measured NRCS data from the Ku band (left panel) and Ka band (right panel) over a segment of the orbit. Note that the swath of the Ku-band radar is approximately 245 km, consisting of 49 fields of view over incidence angles from nadir to about ±18° (full swath), while the Ka-band swath is approximately 125 km, consisting of 25 FOVs over angles from nadir to about ±9° (inner swath). Note also that the Ka-band measurements include 24 interleaved fields of view (not shown) with a coarser range sampling (250 m) and higher sensitivity; data from these measurements are referred to as the KaHS (Kubota et al. 2014). Two features of the figures are worth noting: the clear difference at both frequencies between the land and ocean surface returns and the reduction in the apparent cross sections caused, in this case, by attenuation through the precipitation.

Measurements of NRCS from the GPM DPR on 3 Jul 2014, orbit 1957: (left) Ku band and (right) Ka band over Hurricane Arthur off the Atlantic Coast of the United States.

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

Measurements of NRCS from the GPM DPR on 3 Jul 2014, orbit 1957: (left) Ku band and (right) Ka band over Hurricane Arthur off the Atlantic Coast of the United States.

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

Measurements of NRCS from the GPM DPR on 3 Jul 2014, orbit 1957: (left) Ku band and (right) Ka band over Hurricane Arthur off the Atlantic Coast of the United States.

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

Estimates of the PIA from the single-frequency SRT are shown in Fig. 7 where, from left to right, the Ku-band, Ka-band, and KaHS results are displayed. Path attenuations at Ku band and Ka band derived from the dual-frequency SRT are shown in Fig. 8. It is important to note that as the DSRT can be applied only in the inner swath, where both Ku- and Ka-band data exist; for the angles outside the inner swath, the “dual frequency” PIA is set to the single-frequency estimate (i.e., Ku band). Although the dual-frequency estimates in the inner swath compare well with the single-frequency results of Fig. 7, the DSRT estimates tend to be less noisy, with fewer negative PIAs. On the other hand, an examination of the “reliability flag” for these data shows that in the most intense regions of the storm, within the inner swath, the Ka-band surface return is at or close to the receiver noise level. In these cases, Ka-band and dual-frequency estimates are produced but flagged to indicate that the PIA estimate is a lower bound and likely negatively biased. Indeed, a comparison between the single- and dual-frequency Ku-band path attenuations in these regions shows the Ku-band dual-frequency estimate to be consistently smaller than the Ku-band single-frequency estimate.

Single-frequency PIA (effective) estimates for (left) Ku band, (middle) Ka band, and (right) KaHS.

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

Single-frequency PIA (effective) estimates for (left) Ku band, (middle) Ka band, and (right) KaHS.

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

Single-frequency PIA (effective) estimates for (left) Ku band, (middle) Ka band, and (right) KaHS.

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

Dual-frequency-derived PIA (effective) estimates in the inner swath of the (left) Ku band and (right) Ka band. Note that the results in the outer swath of the Ku-band data are the same as the single-frequency results in Fig. 7.

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

Dual-frequency-derived PIA (effective) estimates in the inner swath of the (left) Ku band and (right) Ka band. Note that the results in the outer swath of the Ku-band data are the same as the single-frequency results in Fig. 7.

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

Dual-frequency-derived PIA (effective) estimates in the inner swath of the (left) Ku band and (right) Ka band. Note that the results in the outer swath of the Ku-band data are the same as the single-frequency results in Fig. 7.

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

The path attenuations shown in the previous figures are the effective PIAs. As shown in (12), this is obtained from the weighted sum of individual PIAs corresponding to different reference datasets. For example, the dual-frequency-derived Ku-band PIA in Fig. 8 was produced from a weighted sum of the four PIAs shown in Fig. 9, which, from left to right, correspond to estimates from the forward/backward along-track and the forward/backward cross-track reference data (the absence of cross-track PIA estimates over land in the two rightmost panels arises from the fact that cross-track estimates are made only over ocean). As noted earlier, temporal-based reference estimates have not yet been implemented in the algorithm but are expected to be by early 2016.

Dual-frequency Ku-band PIA estimates, in the inner swath, from four reference datasets. (from left to right) Forward and backward along track and forward and backward cross track. The PIA (effective) shown in Fig. 8 is the weighed sum of these data. Note that data in the outer swath are derived from single-frequency Ku-band data.

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

Dual-frequency Ku-band PIA estimates, in the inner swath, from four reference datasets. (from left to right) Forward and backward along track and forward and backward cross track. The PIA (effective) shown in Fig. 8 is the weighed sum of these data. Note that data in the outer swath are derived from single-frequency Ku-band data.

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

Dual-frequency Ku-band PIA estimates, in the inner swath, from four reference datasets. (from left to right) Forward and backward along track and forward and backward cross track. The PIA (effective) shown in Fig. 8 is the weighed sum of these data. Note that data in the outer swath are derived from single-frequency Ku-band data.

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

### b. Estimate of A(Ka)/A(Ku)

A value of *p* must be assumed or derived from the data in order to convert δ*A* to *A*(Ku) and *A*(Ka). In this paper, we assume that *p* = *A*(Ka)/*A*(Ku) = 6. This estimate was arrived at in two ways. In the first approach, measured raindrop size distributions were used to calculate the ratio of specific attenuation, *k* (dB km^{−1}), at Ka band and Ku band: *k*(Ka)/*k*(Ku). The second way to estimate *p* is by comparing *A*_{DF}(Ka) with *A*_{SF}(Ka). We focus on the latter approach here.

To explain the second procedure, note that the DSRT estimate at Ka band, *A*_{DF}(Ka), depends on *p*, while the single-frequency estimate, A_{SF}(Ka), does not. Thus, if we take the pairs [*A*_{SF}(Ka), *A*_{DF}(Ka)], filtering them to include only those pairs where both are considered “reliable” (where the PIA is much larger than its standard deviation), then that value of *p* that brings *A*_{DF}(Ka) into best agreement with *A*_{SF}(Ka), in an RMS sense, provides the value we seek. An example of this is shown in Fig. 10, where the data were taken at an incidence angle of −4.5° for land and ocean backgrounds. In the scatterplots shown, two values of *p* are used: for the left-hand plots *p* = 4, while in the right-hand plots *p* = 6. Note that for the *p* = 6 case, the regression slope is close to 1 for both land and ocean data. In contrast, the results for *p* = 4 show that the *A*_{DF}(Ka) are generally larger than *A*_{SF}(Ka). The *p* = 6 case also results in smaller RMS errors than other integer values of *p*. The results, moreover, are relatively insensitive to incidence angle. There is, however, some dependence on rain type with the regression slopes, where the values are smaller for convective rain than for stratiform rain. It should be noted that the rain classification algorithm is described in Awaka et al. (2009) for the single-frequency method and in Le and Chandrasekar (2013) for the dual-frequency method. For the results used in Fig. 10, we have used the dual-frequency rain classification results.

Scatterplots of *A*_{SF}(Ka) vs *A*_{DF}(Ka) at an incidence angle of −4.5°: (top left) ocean, *p* = 4; (top right) ocean, *p* = 6; (bottom left) land, *p* = 4; and (bottom right) land, *p* = 6. Data are separated into stratiform (blue) and convective (red) rain types using the results from the classification module of the DPR algorithm.

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

Scatterplots of *A*_{SF}(Ka) vs *A*_{DF}(Ka) at an incidence angle of −4.5°: (top left) ocean, *p* = 4; (top right) ocean, *p* = 6; (bottom left) land, *p* = 4; and (bottom right) land, *p* = 6. Data are separated into stratiform (blue) and convective (red) rain types using the results from the classification module of the DPR algorithm.

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

Scatterplots of *A*_{SF}(Ka) vs *A*_{DF}(Ka) at an incidence angle of −4.5°: (top left) ocean, *p* = 4; (top right) ocean, *p* = 6; (bottom left) land, *p* = 4; and (bottom right) land, *p* = 6. Data are separated into stratiform (blue) and convective (red) rain types using the results from the classification module of the DPR algorithm.

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

### c. Statistical results

Validation of the estimated SRT path attenuations is beyond the scope of the paper. Nevertheless, we can gain insight into the performance by the degree of consistency among the different PIA estimates. In the absence of error, the forward/backward along-track and cross-track estimates should yield the same result. When they do not, it points to an inconsistency and therefore error in the method. While this consistency check is useful, it is clear that perfect agreement among the PIAs does not imply an error-free estimate. For example, if rain perturbs the surface cross section away from the rain-free value, then the PIA will be biased even if the various methods are in perfect agreement. Another significant source of error that will not be detected by a consistency check is NUBF. Although this topic is also beyond the scope of the paper, it is worth noting that even if the SRT-derived path attenuation is exact in that it provides an error-free estimate of the difference between the NRCS outside and inside the rain, under NUBF this quantity is not the same as the path attenuation needed by the rain retrieval algorithm (Takahashi et al. 2006; Meneghini and Liao 2013). In the latter case, what is needed is the difference (dB) between the measured and true radar reflectivity factors just above the surface. Under uniform beamfilling conditions, where gradients in *Z* are allowed only along the radar range direction, the two quantities are equal; under NUBF conditions the two quantities generally will not be equal.

*j*denotes the method so that, over ocean

*j*would typically run from 1 to 4 and the path attenuation estimates would include those derived from the forward/backward along-track and forward/backward cross-track reference data. Equation (21) provides a weighted mean square deviation of the individual PIA estimates from the effective PIA for a single estimate of path attenuation. Assuming that rain is detected over

*N*fields of view over a particular background type

*B*at incidence angle

*θ*the following quantity is computed:

_{av}over ocean is shown in Table 1 as a function of incidence angle for the single- and dual-frequency estimates at Ku band [

*A*

_{SF}(Ku),

*A*

_{DF}(Ku)] and Ka band [

*A*

_{SF}(Ka),

*A*

_{DF}(Ka)]. As in all the data presented in the paper, the database is the set of measurements made during August 2014.

RMS_{av} ocean background, August 2014.

Comparisons between the single- and dual-frequency estimates of path attenuation at Ka band [*A*_{SF}(Ka) vs *A*_{DF}(Ka)] in Table 1 show that the RMS_{av} associated with the dual-frequency Ka-band estimates, A_{DF}(Ka), are 20%–43% smaller than the single-frequency Ka-band estimates, *A*_{SF}(Ka), where the degree of improvement depends on incidence angle. At Ku band, the improvement in *A*_{DF}(Ku) relative to that of *A*_{SF}(Ku) is more pronounced with RMS_{av} values for the dual-frequency estimates 81%–89% smaller than those of the single-frequency estimates.

Note that a comparison of the Ku- and Ka-band single-frequency results [*A*_{SF}(Ku) vs *A*_{SF}(Ka)] indicates that the absolute error of the estimate at Ku band is slightly smaller than at Ka band. However, as the path attenuation at Ka band is about a factor of 6 higher than at Ku band, the relative error at Ka band is about a factor of 5 smaller than that at Ku band. A comparison between the dual-frequency estimates at Ku and Ka band [*A*_{DF}(Ku) and *A*_{DF}(Ka)] shows that the absolute error in the *A*_{DF}(Ku) data is a factor of 6 smaller than that at *A*_{DF}(Ka) so that the relative errors of the two are the same. These properties follow from the fact that, for the DSRT both path attenuations are derived from the differential attenuation by simple scaling factors where *p* has been assumed to be equal to 6.

Table 2 shows the same set of results as Table 1 but for land background. Qualitatively, the results are similar to those over ocean. For land, however, the improvement in the dual- versus single-frequency-derived results is greater. In particular, the RMS error associated with the *A*_{DF}(Ka) data is 27%–62% smaller than that of the *A*_{SF}(Ka) data, while the RMS error of the *A*_{DF}(Ku) data is 89%–94% smaller than that of the *A*_{SF}(Ku) data.

RMS_{av} land background, August 2014.

Although these results indicate the potential for significant improvement in the accuracy of the dual-frequency estimates over those from the single-frequency SRT, we should again mention some of the disadvantages of the DSRT. One error source that is not highlighted by the RMS_{av} statistic is the fraction of data for which the Ka-band surface return is indistinguishable from the receiver noise power; that is, the surface return is fully attenuated by the precipitation. In these cases, the DSRT estimates of path attenuation at Ku band and Ka band are underestimated and can be considered as lower bounds.

To explore this issue in more detail, scatterplots of path attenuations derived from the forward (ordinate) and backward (abscissa) along-track reference datasets are shown in Fig. 11 for an incidence angle of 9° over ocean. It should be noted that all of the data have been plotted, reliable and unreliable, where the latter category includes negative path attenuations. The four panels correspond to along-track forward/backward comparisons for the *A*_{SF}(Ku) (top left), *A*_{SF}(Ka) (top right), *A*_{DF}(Ku) (bottom left), and *A*_{DF}(Ka) (bottom right). The points in black, comprising 99.2% of the data, represent those rain cases where the Ka-band (and Ku band) surface signal is at least 2 dB above the noise. For the remaining 0.8% of the data, shown in red, the Ka-band surface signal is less than 2 dB above the noise. With the exception of the top-left plot, which depends only on the Ku-band data, the loss of surface signal at Ka band affects all the results of Fig. 11 although it affects the dual-frequency (DSRT) results in the bottom two panels in a different way than the single-frequency, *A*_{SF}(Ka), results in the top-right plot.

Scatterplots of path attenuation from the forward along-track estimates (abscissa) vs the backward along-track estimates (ordinate) at an incidence angle of −9°. (top left) Ku-band PIA derived from SRT(Ku); (top right) Ka-band PIA derived from SRT(Ka); (bottom left) Ku-band PIA derived from DSRT; and (bottom right) Ka-band PIA derived from DSRT. The points in red represent cases where the surface SNR at Ka band is less than 2 dB [for clarity, the points in red in (bottom left) have been vertically displaced by 2 dB].

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

Scatterplots of path attenuation from the forward along-track estimates (abscissa) vs the backward along-track estimates (ordinate) at an incidence angle of −9°. (top left) Ku-band PIA derived from SRT(Ku); (top right) Ka-band PIA derived from SRT(Ka); (bottom left) Ku-band PIA derived from DSRT; and (bottom right) Ka-band PIA derived from DSRT. The points in red represent cases where the surface SNR at Ka band is less than 2 dB [for clarity, the points in red in (bottom left) have been vertically displaced by 2 dB].

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

Scatterplots of path attenuation from the forward along-track estimates (abscissa) vs the backward along-track estimates (ordinate) at an incidence angle of −9°. (top left) Ku-band PIA derived from SRT(Ku); (top right) Ka-band PIA derived from SRT(Ka); (bottom left) Ku-band PIA derived from DSRT; and (bottom right) Ka-band PIA derived from DSRT. The points in red represent cases where the surface SNR at Ka band is less than 2 dB [for clarity, the points in red in (bottom left) have been vertically displaced by 2 dB].

Citation: Journal of Atmospheric and Oceanic Technology 32, 12; 10.1175/JTECH-D-15-0044.1

In the case of the *A*_{SF}(Ka) estimates, if the reference value is constant, then *A*_{SF}(Ka) will also be constant if the Ka-band surface signal is lost since *A*_{SF}(Ka) will be equal to the difference between the reference value and the σ^{0} obtained with the surface power set equal to the noise power (NP). Denoting this lower bound by σ^{0}(Ka, NP), then *A*_{SF}(Ka) = σ^{0}(Ka, reference) − σ^{0}(Ka, NP) = constant. In contrast, the estimate of *δA* will be equal to the difference between the reference δσ^{0} and its value in the presence of precipitation: δ*A* = δσ^{0}(Reference) − δσ^{0}(Rain). When the Ka-band surface signal is fully attenuated, the last term on the right-hand side is δσ^{0}(Rain) = [σ^{0}(Ka, NP) − σ^{0}(Ku, Rain)] so that δ*A* = δσ^{0}(Reference) − [σ^{0}(Ka, NP) − σ^{0}(Ku, Rain)] or δ*A* = Constant + σ^{0}(Ku, Rain), where Constant = [δσ^{0}(Reference) − (σ^{0}(Ka, NP)]. Since σ^{0}(Ku) is almost always above the noise level, an increase in the Ku-band attenuation results in a decrease in σ^{0}(Ku). As a consequence, when the surface signal at Ka band is lost, the estimate of δ*A* will decrease as the path attenuation increases. This explains the larger excursions in the *A*_{DF}(Ka) (red) data (bottom-right panel) relative to the *A*_{SF}(Ka) (red) data (top-right panel) since once the Ka-band surface signal is lost, higher Ku-band attenuation actually results in smaller δ*A* and therefore smaller *A*_{DF}(Ku) and *A*_{DF}(Ka) [it should be mentioned that in almost all cases, the signal-to-noise ratio (SNR) at Ku band is greater than 2 dB even when the Ka-band surface signal is lost].

For the great majority of the data (points in black), where the surface signal is detected at both frequencies, the smaller amount of scatter in the DSRT data are taken to indicate a more stable estimate of path attenuation since it is relatively independent of the reference data used. For the results in Fig. 11, the agreement is with respect to the forward/backward along-track estimates, but comparisons among any of the four types of ocean reference data show similar behavior. In contrast, the results for the single-frequency Ku-band path attenuations (black points, top left) show much more scatter than the dual-frequency Ku-band path attenuations in the lower left, suggesting that the dual-frequency results are more accurate as long as the Ka-band surface return is greater than the noise level. This is consistent with the RMS_{av} results shown in Tables 1 and 2.

The fraction of raining data in which the Ka-band surface signal is lost is a function of incidence angle. This is a consequence of the fact that the surface cross section over both land and ocean decreases away from nadir incidence so that the nominal surface return power decreases as a function of incidence angle. This implies that the dynamic range—that is, the difference between the rain-free σ^{0} and the maximally attenuated σ^{0} (in other words, that σ^{0} corresponding to the value where the surface return power is equal to the receiver noise power)—decreases with incidence angle. Over ocean, the fraction of rain data for which the Ka-band surface return is less than 2 dB above the receiver noise is approximately 0.4% at nadir and 0.8% at 9°. Over land, the values are 0.75% at nadir and 2% at 9°.

## 5. Summary

Measurements of the normalized radar cross sections (NRCS) of the surface, σ^{0}(Ku) and σ^{0}(Ka), for the near-nadir incidence angles that comprise the inner swath (from −9° to 9°) of the DPR radar show relatively high correlation coefficients and relatively small values for the variance of σ^{0}(Ka) − σ^{0}(Ku). From the perspective of the surface reference technique, this implies that the rain-free differential σ^{0} provides a stable reference against which the differential path attenuation can be estimated. This stability can be quantified in part by comparing the different, nearly independent estimates of path attenuation generated by using different surface reference datasets. The results indicate that the dual-frequency SRT offers an improvement in accuracy over the single-frequency counterpart, particularly in the estimation of the Ku-band path attenuation.

The DSRT, however, has two drawbacks. The first is caused by the loss of the surface return at Ka band; this occurs at high rain rates where the Ka-band attenuation becomes larger than the difference between the rain-free NRCS and the receiver noise power. The second drawback arises from the errors incurred in estimating the path attenuations at Ku band and Ka band from the differential path attenuation. This conversion requires knowledge of the ratio *p* of the path attenuations at Ka band and Ku band. Although an assumed value of 6 provides good agreement between the Ka-band single- and dual-frequency estimates, studies using measured raindrop size distributions show that *p* is not constant but depends on the parameters of the size distribution. Whether improvements in the determination of *p* can be made with the measurements available is an open question.

Several refinements and upgrades to the algorithm are planned. The most important is implementation of the temporal reference. Although this will begin with a 0.5° grid resolution, an example of which is shown in Fig. 5, as additional data are acquired, the goal is to improve the resolution to 0.1°. The temporal reference dataset is expected to improve path attenuation estimates near and at land/water boundaries, including areas at or near coastlines, rivers, small islands and peninsulas, where spatial reference data are often sparse or missing. Implementation of multicategory land and wet soil/light rain reference datasets might further improve the estimates.

The best test of the method will come in determining whether the path attenuation estimates from the SRT or DSRT can offer accurate attenuation correction and the extent to which this can lead to improved estimates of the precipitation characteristics.

## Acknowledgments

We wish to thank members of the JAXA and NASA data processing teams for providing the data. This work is supported by Dr. Ramesh Kakar of NASA headquarters under NASA Grant NNH12ZDA001N-PMM.

## REFERENCES

Awaka, J., Iguchi T. , and Okamoto K. , 2009: TRMM PR standard algorithm 2A23 and its performance on bright-band detection.

,*J. Meteor. Soc. Japan***87A**, 31–52, doi:10.2151/jmsj.87A.31.Durden, S. L., and Tanelli S. , 2008: Predicted effects of nonuniform beam filling on GPM radar data.

,*IEEE Geosci. Remote Sens. Lett.***5**, 308–310, doi:10.1109/LGRS.2008.916068.Durden, S. L., Li L. , Im E. , and Yueh S. H. , 2003: A surface reference technique for airborne Doppler radar measurements in hurricanes.

,*J. Atmos. Oceanic Technol.***20**, 269–275, doi:10.1175/1520-0426(2003)020<0269:ASRTFA>2.0.CO;2.Durden, S. L., Tanelli S. , and Meneghini R. , 2012: Using surface classification to improve surface reference technique performance over land.

,*Indian J. Radio Space Phys.***41**, 403–410.Hitschfeld, W., and Bordan J. , 1954: Errors inherent in the radar measurement of rainfall at attenuating wavelengths.

,*J. Meteor.***11**, 58–67, doi:10.1175/1520-0469(1954)011<0058:EIITRM>2.0.CO;2.Iguchi, T., and Meneghini R. , 1994: Intercomparison of single-frequency methods for retrieving a vertical rain profile from airborne or spaceborne radar data.

,*J. Atmos. Oceanic Technol.***11**, 1507–1516, doi:10.1175/1520-0426(1994)011<1507:IOSFMF>2.0.CO;2.Iguchi, T., Kozu T. , Meneghini R. , Awaka J. , and Okamoto K. , 2000: Rain-profiling algorithm for the TRMM precipitation radar.

,*J. Appl. Meteor.***39**, 2038–2052, doi:10.1175/1520-0450(2001)040<2038:RPAFTT>2.0.CO;2.Iguchi, T., Kozu T. , Kwiatkowski J. , Meneghini R. , Awaka J. , and Okamoto K. , 2009: Uncertainties in the rain profiling algorithm for the TRMM Precipitation Radar.

,*J. Meteor. Soc. Japan***87A**, 1–30, doi:10.2151/jmsj.87A.1.Kozu, T., 1995: A generalized surface echo radar equation for down-looking pencil beam radar.

,*IEICE Trans. Commun.***E78-B**, 1245–1248.Kozu, T., and Iguchi T. , 1996: A preliminary study of non-uniform beam filling correction for spaceborne radar rainfall measurement.

,*IEICE Trans. Commun.***E79-B**, 763–769.Kubota, T., and Coauthors, 2014: Evaluation of precipitation estimates by at-launch codes of GPM/DPR algorithms using synthetic data from TRMM/PR observations.

,*IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens.***7**, 3931–3944, doi:10.1109/JSTARS.2014.2320960.Le, M., and Chandrasekar V. , 2013: Precipitation type classification method for dual-frequency precipitation radar (DPR) onboard the GPM.

,*IEEE Trans. Geosci. Remote Sens.***51**, 1784–1790, doi:10.1109/TGRS.2012.2205698.L’Ecuyer, T. S., and Stevens G. L. , 2002: An estimation-based precipitation retrieval algorithm for attenuating radars.

,*J. Appl. Meteor.***41**, 272–285, doi:10.1175/1520-0450(2002)041<0272:AEBPRA>2.0.CO;2.Li, L., Im E. , Durden S. L. , and Haddad Z. S. , 2002: A surface wind model–based method to estimate rain-induced radar path attenuation over ocean.

,*J. Atmos. Oceanic Technol.***19**, 658–672, doi:10.1175/1520-0426(2002)019<0658:ASWMBM>2.0.CO;2.Li, L., Im E. , Connor L. N. , and Chang P. S. , 2004: Retrieving ocean surface wind speed from the TRMM precipitation radar measurements.

,*IEEE Trans. Geosci. Remote Sens.***42**, 1271–1282, doi:10.1109/TGRS.2004.828924.Liao, L., and Meneghini R. , 2005: A study of air/space-borne dual-wavelength radar for estimates of rain profiles.

,*Adv. Atmos. Sci.***22**, 841–851, doi:10.1007/BF02918684.Liao, L., Meneghini R. , Tian L. , and Heymsfield G. M. , 2008: Retrieval of snow and rain from combined X- and W-band airborne radar measurements.

,*IEEE Trans. Geosci. Remote Sens.***46**, 1514–1524, doi:10.1109/TGRS.2008.916079.Marzoug, M., and Amayenc P. , 1994: A class of single- and dual-frequency algorithms for rain-rate profiling from a spaceborne radar. Part I: Principle and tests from numerical simulations.

,*J. Atmos. Oceanic Technol.***11**, 1480–1506, doi:10.1175/1520-0426(1994)011<1480:ACOSAD>2.0.CO;2.Meneghini, R., and Liao L. , 2013: Modified Hitschfeld–Bordan equations for attenuation-corrected radar rain reflectivity: Application to nonuniform beamfilling at off-nadir incidence.

,*J. Atmos. Oceanic Technol.***30**, 1149–1160, doi:10.1175/JTECH-D-12-00192.1.Meneghini, R., Eckerman J. , and Atlas D. , 1983: Determination of rain rate from a space-borne radar using measurements of total attenuation.

,*IEEE Trans. Geosci. Remote Sens.***GE-21**, 34–43, doi:10.1109/TGRS.1983.350528.Meneghini, R., Jones J. A. , and Gesell L. H. , 1987: Analysis of a dual-wavelength surface reference radar technique.

,*IEEE Trans. Geosci. Remote Sens.***GE-25**, 456–471, doi:10.1109/TGRS.1987.289857.Meneghini, R., Iguchi T. , Kozu T. , Liao L. , Okamoto K. , Jones J. A. , and Kwiatkowski J. , 2000: Use of the surface reference technique for path attenuation estimates from the TRMM precipitation radar.

,*J. Appl. Meteor.***39**, 2053–2070, doi:10.1175/1520-0450(2001)040<2053:UOTSRT>2.0.CO;2.Meneghini, R., Jones J. A. , Iguchi T. , Okamoto K. , and Kwiatkowski J. , 2004: A hybrid surface reference technique and its application to the TRMM precipitation radar.

,*J. Atmos. Oceanic Technol.***21**, 1645–1658, doi:10.1175/JTECH1664.1.Meneghini, R., Liao L. , Tanelli S. , and Durden S. L. , 2012: Assessment of the performance of a dual-frequency surface reference technique over ocean.

,*IEEE Trans. Geosci. Remote Sens.***50**, 2968–2977, doi:10.1109/TGRS.2011.2180727.Mitrescu, C., L’Ecuyer T. , Haynes J. , Miller S. , and Turk J. , 2010:

*CloudSat*precipitation profiling algorithm—Model description.,*J. Appl. Meteor. Climatol.***49**, 991–1003, doi:10.1175/2009JAMC2181.1.Nakamura, K., 1991: Biases of rain retrieval algorithms for spaceborne radar caused by nonuniformity of rain.

,*J. Atmos. Oceanic Technol.***8**, 363–373, doi:10.1175/1520-0426(1991)008<0363:BORRAF>2.0.CO;2.Seto, S., and Iguchi T. , 2007: Rainfall-induced changes in actual surface backscattering cross sections and effects on rain estimates by spaceborne precipitation radar.

,*J. Atmos. Oceanic Technol.***24**, 1693–1709, doi:10.1175/JTECH2088.1.Seto, S., Iguchi T. , and Oki T. , 2013: The basic performance of a precipitation retrieval algorithm for the Global Precipitation Measurement mission’s single/dual-frequency radar measurements.

,*IEEE Trans. Geosci. Remote Sens.***51**, 5239–5251, doi:10.1109/TGRS.2012.2231686.Takahashi, N., Hanado H. , and Iguchi T. , 2006: Estimation of path-integrated attenuation and its nonuniformity from TRMM/PR range profile data.

,*IEEE Trans. Geosci. Remote Sens.***44**, 3276–3283, doi:10.1109/TGRS.2006.876295.Tanelli, S., Durden S. L. , and Im E. , 2006: Simultaneous measurements of Ku and Ka-band sea surface cross sections by an airborne radar.

,*IEEE Geosci. Remote Sens. Lett.***3**, 359–363, doi:10.1109/LGRS.2006.872929.Tanelli, S., Sacco G. F. , Durden S. L. , and Haddad Z. S. , 2012: Impact of non-uniform beam filling on spaceborne cloud and precipitation radar retrieval algorithms.

*Remote Sensing of the Atmosphere, Clouds, and Precipitation IV*, T. Hayasaka, K. Nakamura, and E. Im, Eds., International Society for Optical Engineering (SPIE Proceedings, Vol. 8523), 852308, doi:10.1117/12.977375.

^{1}

Note also that the Ka-band measurements include 24 interleaved fields of view (not shown) with a coarser range sampling (250 m) and higher sensitivity; data from these measurements are referred to as the KaHS (Kubota et al. 2014).