## Abstract

It has long been recognized that path-integrated attenuation (PIA) can be used to improve precipitation estimates from high-frequency weather radar data. One approach that provides an estimate of this quantity from airborne or spaceborne radar data is the surface reference technique (SRT), which uses measurements of the surface cross section in the presence and absence of precipitation. Measurements from the dual-frequency precipitation radar (DPR) on the Global Precipitation Measurement (GPM) satellite afford the first opportunity to test the method for spaceborne radar data at Ka band as well as for the Ku-band–Ka-band combination.

The study begins by reviewing the basis of the single- and dual-frequency SRT. As the performance of the method is closely tied to the behavior of the normalized radar cross section (NRCS or σ^{0}) of the surface, the statistics of σ^{0} derived from DPR measurements are given as a function of incidence angle and frequency for ocean and land backgrounds over a 1-month period. Several independent estimates of the PIA, formed by means of different surface reference datasets, can be used to test the consistency of the method since, in the absence of error, the estimates should be identical. Along with theoretical considerations, the comparisons provide an initial assessment of the performance of the single- and dual-frequency SRT for the DPR. The study finds that the dual-frequency SRT can provide improvement in the accuracy of path attenuation estimates relative to the single-frequency method, particularly at Ku band.

## 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)

Calculation of the NRCS (σ^{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 be the apparent NRCS in a raining area at incidence angle *θ*, measured with respect to the local normal to the surface, and let be the reference NRCS at the same angle, formed from data taken in rain-free areas. An estimate of the two-way path-integrated attenuation (dB) is given by

where the angular brackets on the first term is used to indicate that the rain-free reference value is usually an average in space or time. In terms of the specific attenuation [*k* (dB km^{−1})] along the radar beam at angle *θ*, the two-way path-integrated attenuation (PIA) is defined as

where the integral is taken along the main lobe of the antenna in the radar range direction from the radar, *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)

A dual-frequency version of the SRT follows from an application of (1) to data from radars operating at frequencies *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

Letting δσ^{0} = σ^{0}(*f*_{1}) − σ^{0}(*f*_{2}), (3) can be written in the form

where

Equation (3′) shows that the only difference between the single- and dual-frequency versions of the method is that, in the latter case, the differential normalized surface cross section is used instead of the surface cross section itself; specifically, a subtraction of the differential σ^{0} in rain from the rain-free differential σ^{0} yields an estimate of differential path attenuation.

In most dual-frequency retrieval algorithms (e.g., Seto et al. 2013), what is needed is not δ*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

For each channel of the DPR (Ku band, Ka band, and KaHS) up to five single-frequency estimates of PIA can be generated corresponding to different surface reference estimates. A similar statement holds for the dual-frequency results derived from (5). What is needed is, in some sense, a best estimate. In the following discussion, the procedure is outlined only for the single-frequency estimates but the results apply equally well to δ*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)

where the first term on the right-hand side is the *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

which is usually approximated by the spatial or temporal sample variance of the reference data. From these individual estimates, we wish to obtain an effective or “best” PIA. We assume it can be written in the form

where the weights *w*_{j} are such that .

Assuming that the individual estimates are statistically independent, then the variance of (8) is

To minimize the variance in (9), we use the method of Lagrangian multipliers, which yields

where

The effective PIA is then

If we define the reliability RL as the ratio of the PIA to its standard deviation, then for the *j*th estimate

To apply this definition to the effective PIA estimate, we define RL_{eff} as

Computing RL_{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

Equations (10), (12), and (16) define, respectively, the weights, effective PIA, and effective reliability that are generated by the algorithm for the Ku-band, Ka-band, and KaHS single-frequency SRT results. Also provided in the output are the individual PIA estimates (PIAalt), where the first two elements of the array correspond to the forward/backward along-track estimates, the third and fourth elements to the forward/backward cross-track estimates, and the fifth and sixth elements to the standard and wet surface/light rain temporal estimates. As pointed out earlier, the temporal estimates are presently not being computed, while the cross-track estimates are computed only over ocean. From (5), the path attenuations at Ku band and Ka band are derived from the dual-frequency SRT. Similar quantities to those described above are derived for these estimates as well.

## 3. Errors in the methods

As indicated in the previous section, a first-order estimate of the error in the single-frequency SRT estimate of path-integrated attenuation is given by

that is, to first order, the variance in the attenuation estimate is equal to the variance of the rain-free normalized surface cross section. A similar approximation can be used for the dual-frequency estimates:

Note that if , a condition that approximately holds for the DPR data, then the right-hand side of (18) reduces to , where *ρ* is the correlation coefficient of so that the variance of the difference tends to zero as the surface cross sections become perfectly correlated. An equivalent way to state this is that the differential cross section is perfectly stable outside the raining area so that, in the presence of rain, any deviation from this constant value can be attributed to atmospheric attenuation.

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.

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.

Although the results in Fig. 3 are informative, they are not directly related to the accuracy of the path attenuation estimates. One measure of the reliability of the PIA is the ratio of the mean to the standard deviation of the estimate, RL ≡ *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

Equations (19) can be viewed as an approximate measure of the reliability of the Ku-band, Ka-band, and differential path attenuation estimates. It is important to note from (5) that if the *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

Using the results in Fig. 3 and assuming *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.

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.

## 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.

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.

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.

### 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.

### 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.

There are several measures of consistency between the path attenuation estimates. Over ocean, four estimates of path attenuation are usually generated so correlation coefficients and RMS deviations between the six pairs can be computed. Over land, only two estimates are produced, corresponding to the forward/backward along-track reference data, so that only a single correlation coefficient and RMS deviation are available. Another kind of RMS error can be defined as follows (Meneghini et al. 2012):

where the quantities on the right-hand side are given by (6), (8), and (10). The index *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:

Using (22), RMS_{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.

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.

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.

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

*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:.

## Footnotes

This article is included in the Precipitation Retrieval Algorithms for GPM special collection.

^{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).