## 1. Introduction and background

A desirable goal in scatterometry is to extend the estimation of the near-surface wind vector over ocean to raining areas. For frequencies at X band (10 GHz) and higher, path attenuation caused by precipitation typically results in underestimates in the normalized radar cross section (NRCS or *σ*^{0}) of the surface leading to errors in wind estimates.

Several attenuation correction methods have been proposed and analyzed (Draper and Long 2004a,b; Stiles et al. 2010). In the Draper and Long method the geophysical model function, GMF, which relates *σ*^{0} to the wind speed and direction, is modified by a scalar multiplicative term *α*_{r} that accounts for the two-way path attenuation caused by precipitation. The GMF is also modified by an additive term *σ*_{e} that is used to account for two mechanisms: volume scattering by rain at the range gate(s) at which *σ*^{0} is estimated and a change from the rain-free *σ*^{0} caused by modifications in the ocean wave spectrum generated by the raindrop splash effect (Moore et al. 1983; Bliven et al. 1997; Weissman et al. 2002; Stiles and Yueh 2002; Contreras et al. 2003; Draper and 2004a,b; Contreras and Plant 2006; Weissman and Bourassa 2008; Bliven and Giovanangeli 1993; Tournadre and Quilfen 2003). Both unknowns are parameterized in terms of the near-surface rainfall rate *R*. The authors report that the method generally works well when applied to SeaWinds scatterometer data on the QuikSCAT satellite using rain rates from the TRMM Precipitation Radar.

As the High-Altitude Imaging Wind and Rain Airborne Profiler (HIWRAP) is able to measure backscattered power from precipitation along the column, and in particular returns from rain just above the surface, the volume scattering contribution to *σ*^{0} can be estimated. The second mechanism associated with the raindrop splash effect on *σ*^{0} is more difficult to assess as it depends on rain rate, the polar and azimuthal incidence angles and wind speed (Contreras and Plant 2006). While the HIWRAP data can be used to estimate rainfall rate just above the surface, which affords the potential to estimate the drop splash effect and wind speed, this will not be pursued in this paper.

Stiles et al. (2010) describe a method for retrieving winds in the presence of rain from a proposed dual-frequency C-/Ku-band scatterometer using an artificial neural network (ANN) approach. The additional information from the dual-frequency data is sufficient to estimate wind speed and direction as well as path attenuation corrections. Simulation results show good retrieval accuracy for winds up to 50 m s^{−1} even in the presence of heavy rain. While the present study does not use a neural network, the relative invariance of *δσ*^{0} to wind speed suggest the possibility that the neural network is able to exploit this property in the retrievals.

The HIWRAP airborne instrument (Li et al. 2016) uses frequencies similar to those of the Dual-Frequency Precipitation Radar (DPR) on the GPM satellite (Hou et al. 2014; Kojima et al. 2012) with a scan geometry similar to the SeaWinds scatterometer which flew aboard the QuikSCAT satellite with a data record from July 1999 to November 2009 (Jones et al. 1982; Wentz and Smith 1999). The HIWRAP system design parameters are given in Table 1. The minimum detectable reflectivity factors shown in the table ensure that rain rates below 1 mm h^{−1} can be detected at both frequencies; the upper limit of rain detectability is determined by the magnitude of the path attenuation and the system noise. The minimum surface detection capability is a function of path attenuation, system noise and the intrinsic value of *σ*^{0} which, over ocean, depends on incidence angle and wind speed and direction.

HIWRAP system parameters.

In the context of recently developed airborne scatterometers, it should be noted that Rodriguez et al. (2018) provide a comprehensive description of the theory, instrument design and data analysis from the airborne Ka-band scatterometer, DopplerScatt.

The attenuation-correction approach described below is a modification of a procedure used to estimate path attenuations in dual-frequency airborne and spaceborne radar data (Meneghini et al. 2012, 2015). Changes in the method are made to account for the higher incidence angle of HIWRAP, which introduces larger variations in the rain-free *σ*^{0}, and to account for the change from cross-track to conical scanning geometry.

The data analyzed in this paper were acquired during the NOAA Sensing Hazards with Operational Unmanned Technology field campaign (SHOUT-2016; https://uas.noaa.gov/Program/projects/shout). The campaign ran from 24 August to 10 October 2016, during which time HIWRAP was flown on the Global Hawk unmanned aircraft within and in the vicinity of Tropical Cyclones/Hurricanes Gaston (two flights), Hermine (one flight), Erika (one flight), and Matthew (three flights).

The organization of the paper is as follows. The attenuation correction method is described in section 2. Alternative estimates related to path attenuation, but independent of surface scattering properties, are discussed in section 3 followed by results from the HIWRAP data in section 4. Section 5 gives a preliminary assessment of the influences of the choice of rain–no-rain threshold, rain volume scattering, and the raindrop splash effect on *σ*^{0}.

## 2. Description of the basic method

Under rain-free conditions, the normalized surface cross sections at Ku and Ka band tend to be well correlated so that the data, in [*σ*^{0}(Ku), *σ*^{0}(Ka)] space, line up closely about a linear regression line where the primary variations in the data, over ocean, depend on wind speed and direction. In the presence of rain, the Ku- and Ka-band signals are attenuated so that the data tend to fall below the rain-free regression line. Moreover, because the attenuation at Ka band is about 6 times larger than that at Ku band, the apparent surface cross sections at Ka band decreases about 6 times faster than at Ku band. The combined changes from wind speed/wind direction and path attenuation lead to scatterplots, in the presence of rain, with an approximate slope of 6 in [*σ*^{0}(Ku), *σ*^{0}(Ka)] space with variations about this line determined by changes in wind speed/direction and deviations from the nominal slope. Deviations in the slope can be caused by variations in the raindrop size distributions along the column, contributions to the attenuation from cloud liquid water, water vapor, oxygen, and mixed-phase precipitation, and changes in the surface cross sections caused raindrop splash effects.

The results in Fig. 1 show the behavior of *σ*^{0}(Ku) and −*δσ*^{0}, *σ*^{0}(Ku) − *σ*^{0}(Ka), from the 30° conically scanning channel during a 30-min period over an area of the ocean with rain present. [Throughout the paper *δX* is used to denote a frequency difference in *X*, that is, *δX* = *X*(*f*_{1}) − *X*(*f*_{2}), where *f*_{1} > *f*_{2}]. With the Global Hawk altitude of approximately 18 km and a conical scan angle of 30°, the swath width is approximately 21 km. (The swath for the outer cone angle of 40° is about 30 km.) With an airspeed of approximately 200 m s^{−1}, the 30-min data segment depicted in the figure represents a flight leg of about 360 km. In the top plot, rain is depicted in black where rain is judged to be present at a particular field of view (FOV) when the measured radar reflectivity factor, at Ku band, exceeds 10 dB*Z* over 10 range gates along the column, excluding those ranges where antenna sidelobe surface clutter is significant. On the left are shown *σ*^{0}(Ku) measurements where data from the forward and backward portion of the scan are shown in the top and bottom panels respectively. A pair of images are shown on the right during the same time period but for −*δσ*^{0} = *σ*^{0}(Ku) − *σ*^{0}(Ka). Although the rain areas usually can be identified in the *σ*^{0}(Ku) images as a reduction in magnitude relative to the surrounding areas, the identification is often masked by variations in *σ*^{0}(Ku) with wind speed and direction. Of course, since the instrument is designed for the estimation of wind, this sensitivity is to be expected. For the −*δσ*^{0} plots on the right, the variations in rain-free areas are small so that the identification of rain areas, coincident with a spike in −*δσ*^{0}, are quite clear. This relative invariance of *δσ*^{0} to wind speed and direction makes it a good candidate for correcting the data for attenuation effects.

Scatterplots of *σ*^{0}(Ku) versus *σ*^{0}(Ka) are shown in Fig. 2 under rain-free (left) and raining (right) conditions for measurements made during Tropical Cyclone Hermine off the coast of Florida on 1 September 2016. Note that the rain-free data approximately line up along the 1:1 line while the data under raining conditions yield, in this case, a regression slope of 6.

*σ*

^{0}(Ku),

*σ*

^{0}(Ka)] under rain-free and raining conditions suggests a straightforward way to estimate path attenuations

*A*at the two frequencies. As a preliminary step, a linear regression through the rain-free data yields coefficients

*α*and

*β*of the regression

*σ*

^{0}(NR, Ka) =

*α*+

*βσ*

^{0}(NR, Ku), where “NR” denotes the no-rain data. From the data taken in rain, a second linear regression is performed, giving

*σ*

^{0}(

*R*, Ka) =

*p*+

*rσ*

^{0}(

*R*, Ku). For a given measurement in rain, denoted by

*r*, as determined by the regression slope through the rain data, until it crosses the rain-free regression line. The point of intersection gives estimates of the attenuation-corrected NRCS [

*σ*

^{0}(Ku) and

*σ*

^{0}(Ka)] and the Ku and Ka path attenuations [

*A*(Ku) and

*A*(Ka)]:

It is important to note that the attenuation estimates, but not the NRCS estimates, are independent of radar calibration errors since the attenuations remain the same if all data are shifted by arbitrary amounts along the *σ*^{0}(Ka) and *σ*^{0}(Ku) axes. Stated differently, *A*(Ku) and *A*(Ka), given by (3) and (4), are unaffected if all (rain and rain-free) *σ*^{0}(Ka) are replaced by *σ*^{0}(Ka) + *c*_{Ka} and all *σ*^{0}(Ku) are replaced by *σ*^{0}(Ku) + *c*_{Ku}, where *c*_{Ka} and *c*_{Ku} are arbitrary constants.

Results from the attenuation correction of the data displayed in Fig. 2 are shown in Fig. 4 represented in terms of the probability density functions (pdfs) of the rain-free *σ*^{0} (solid line) and the in-rain *σ*^{0} both before (dashed line) and after (dotted line) correction. The Ku-band and Ka-band results are shown in the left- and right-hand panels, respectively.

Implicit in the method is the assumption that the ratio of *A*(Ka) and *A*(Ku) is a constant given by the slope *r*. Analysis of disdrometer data (Liao et al. 2014) shows, however, that this is an approximation and that the ratio depends on the raindrop size distribution. Indeed, analysis of the HIWRAP data in half-hour segments shows changes in the slope not only from segment to segment but between data taken by the inner and outer conical scans. The slopes of the rain and rain-free data, moreover, depend not only on the segment of data that is analyzed but also on the choice of threshold separating the rain and rain-free data. These issues will be addressed in section 5.

## 3. Alternative estimates of path attenuation

In this section we examine two methods that can be used to assess the accuracy of the dual-frequency *σ*^{0} technique described above. In the first, an expression for the difference in path attenuations along two directions is expressed in terms of measured radar reflectivity factors. This approach does not yield the path attenuation but only the difference in attenuations at either Ku or Ka band along two directions and can be considered a type of constraint. In the second method, the measured dual-frequency ratio, using data just above the surface, is used to approximate the differential path attenuation. Unlike the method described in section 2, both are independent of surface scattering properties. To distinguish the two differences, the notation Δ*X* is used to denote a difference in the variable *X* along two paths at a fixed frequency while *δX* is used to denote a frequency difference in *X* where both measurements are along the same path from the radar to the surface.

### a. Geometrical path attenuation difference

*Z*and

*Z*

_{m}, respectively, where

*Z*

_{m}is directly proportional to the radar return power and where the difference between

*Z*and

*Z*

_{m}(dB) is equal to the path attenuation

*A*(dB) to range

*r*:

*Z*,

*Z*

_{m}, and

*A*on radar range

*r*, frequency

*f*, and the azimuthal and polar angles

*ϕ*and

*θ*are made explicit. As a consequence of the conical scanning geometry and system design, a scattering volume just above the surface viewed along angles (

*ϕ*,

*θ*) in the forward cone will be seen at a later time in the backward cone at angles (

*π*−

*ϕ*,

*θ*). For this backward view, an equation, analogous to (7), can be written

*Z*, particularly in stratiform rain conditions. With this assumption, the right-hand sides of (7) and (8) can be equated giving

*ϕ*,

*θ*) and (

*π*−

*ϕ*,

*θ*), we assume that approximately the same volume/surface are seen by the outer cone along the directions (

*ψ*,

*θ*′) and (

*π*−

*ψ*,

*θ*′), where

*ψ*and

*θ*′ denote the azimuthal and polar angles of the outer cone and where the relationship between angles

*ϕ*and

*ψ*is tan

*θ*′sin

*ψ*= tan

*θ*sin

*ϕ*. Taking the pairwise difference in path attenuations between all combinations yields 6 equations of the type given by (9). To simplify the equations, let

*A*

_{1}and

*Z*

_{m1}be associated with the path attenuation and measured near-surface radar reflectivity factor along the direction (

*ϕ*,

*θ*),

*A*

_{2}, and

*Z*

_{m2}with (

*π*−

*ϕ*,

*θ*),

*A*

_{3}, and

*Z*

_{m3}with (

*ψ*,

*θ*′) and

*A*

_{4}, and

*Z*

_{m4}with (

*π*−

*ψ*,

*θ*′). The 6 constraint equations, for each frequency, then become

*Z*

_{m}(

*i*,

*j*) is the difference between

*Z*

_{m}just above the surface along direction

*i*and

*Z*

_{m}just above the surface along direction

*j*,

*Z*

_{mi}−

*Z*

_{mj}, while Δ

*A*(

*i*,

*j*) is the difference in path attenuations along directions

*i*and

*j*,

*A*

_{i}−

*A*

_{j}. As the

*Z*

_{m}are directly measured quantities and as the attenuations can be estimated from (3) and (4), (11) provides a way to test the consistency of the

*δσ*

^{0}method.

### b. Measured dual-frequency ratio

A recent study of the measured dual-frequency ratio and its use in estimating the differential path attenuation through rain is given by Durden (2018). Earlier work using somewhat similar approaches can be found in Eccles and Mueller (1971) and Meneghini et al. (1992).

_{m}, is defined in a similar manner except that the measured radar reflectivities are used rather than the true, that is, attenuation-corrected values. Recalling that

*δA*=

*A*(Ka) −

*A*(Ku), it follows from the definitions of DFR and DFR

_{m}and (7) that

*r*

_{NS}is used to indicate that the quantities on the right are to be evaluated just above the surface. DFR

_{m}(

*r*

_{NS}) is a measurable quantity whereas DFR(

*r*

_{NS}) is not. If the raindrop size distribution is parameterized by a gamma distribution, the DFR is a function of the size and shape parameters but is independent of the particle number concentration (Liao and Meneghini 2019; Meneghini et al. 1992); for the (Ku, Ka) frequency pair DFR is small and negative for small values of the median mass diameter

*D*

_{m}but increases monotonically with

*D*

_{m}after reaching a minimum around 1 mm, where the exact value depends on the shape parameter

*μ*and the temperature of the particles. If the DFR term is neglected so that the estimate of

*δA*is taken simply to be DFR

_{m}(

*r*

_{NS}) then the estimate of differential path attenuation will tend to be slightly negatively biased for small

*D*

_{m}and positively biased for larger values of

*D*

_{m}. An additional issue with this estimate is that DFR

_{m}will be biased if the relative calibration errors of

*Z*

_{m}(Ku) and

*Z*

_{m}(Ka) are nonzero. This error can be mitigated if the average value of DFR

_{m}near the storm top is subtracted from the estimate. The reason for this is that, in the absence of calibration errors, the DFR near the storm top should be close to zero because the particle sizes are generally small. In the presence of calibration errors, this term will mitigate the bias.

*r*

_{ST}, then the estimate of differential path attenuation used in this paper becomes

_{m}near the storm top. For the HIWRAP data that have been processed, we have obtained a value of 1.25 dB which is used in the results below.

## 4. Results

As noted earlier, in the inner swath, each field of view at and just above the surface is viewed along 4 directions: the forward-looking inner swath (FI), backward-looking inner swath (BI), forward-looking outer swath (FO), and backward-looking outer swath (BO) (Fig. 5). Associated with each of these directions are the dual-frequency measured normalized surface cross sections and the measured dual-frequency radar reflectivity factors taken just above the surface. From these data, estimates of the differential path attenuation *δA*_{σ} from (5) and *δA*_{DFR} from (15) are computed and compared.

To put the results, shown below, into context, the measured Ku-band radar reflectivity data from 1 September 2016 for the time period 1550–1627 UTC are shown in Fig. 6. The four panels, beginning from the top, consist of range profiles of *Z*_{m} from the forward-looking (*ψ* = 0) outer swath, backward-looking (*ψ* = *π*) outer swath, forward-looking inner swath (*ϕ* = 0), and backward-looking (*ϕ* = *π*) inner swath. Note that these are the four views of the rain and surface as seen in the plane defined by the aircraft flight direction and nadir. A second set of plots for Ka band for the same time period and viewing parameters is shown in Fig. 7.

In both sets of images the peak surface return, seen as an intense band of reflectivity, is taken as the origin so that distances are measured from this peak. As the aircraft pitch is positive, the distances between the aircraft and surface are larger in the forward view than that in the aft view. A “bright band,” indicating a region of melting hydrometeors, is present in most of the Ku band and some of the Ka-band images at a range of about 5 km from the surface. Surface clutter, arising from the nadir surface return along an antenna sidelobe is clearly seen in the rain-free portions of the Ku-band data: for the outer swath, the clutter is occurs at ranges centered about 4.5 km above the surface while for the inner swath, the clutter can be seen at ranges of about 2 km from the main surface return. The surface clutter is much less intense at Ka band but can be seen in the outer swath images. While clutter correction procedures are available (e.g., Kubota et al. 2016), such correction was not performed on the HIWRAP data as the clutter has a negligible effect on the methods used here as the reflectivity data are taken well below 2 km.

Two-dimensional histograms between *δA*_{σ} and *δA*_{DFR} using the 1550–1627 UTC 1 September 2016 data (a segment of which is shown in Figs. 6 and 7), are given in Fig. 8. As noted above, the comparisons are done along four directions. In general, the correlation coefficient is relatively high with *ρ* ~ 0.9 along with regression lines, shown in black, with slopes ranging from 0.96 to 1.12. However, the *δA*_{DFR} estimates, as derived from (15), tend to be on the order of 0.5–1 dB larger than those derived from the *δσ*^{0} data. Although an error analysis is beyond the scope of the paper, it is worth noting that both retrievals are affected by mismatched beamwidths at the two frequencies and sampling fluctuations caused by the finite number of independent samples used in estimating *Z*_{m} and *σ*^{0}. A second example, comparing *δA*_{σ} and *δA*_{DFR} for the same day but for time period 1103–1206 UTC, is shown in Fig. 9.

The other validation method based on path attenuation differences along two directions generally produced rather poor correlations, particularly with the Ku-band data. The reason for this appears to arise from the fact that the larger beamwidth at Ku band prevents the selection of ranges close to the surface because of the influence of the surface return. In particular, as shown by Table 1, the ratio of Ku-band to Ka-band beamwidths is about 3. Using a range gate of 0.075 km we find that at Ka band, the range gate must be offset from the gate of the peak surface return by at least two gates at 30° incidence and at least three at 40° incidence. At Ku band the gate offset must be at least five gates for the inner swath and at least eight gates for the outer swath.

Because of the required offsets from the surface at Ku band it is difficult to satisfy the assumption that the scattering volumes from the two directions are nearly identical. On the other hand, relatively good correlations are found when the comparisons are restricted to Ka band and in particular to the inner swath Ka-band data where gates close to the surface can be chosen without surface contamination. Two examples of these comparisons are given in Fig. 10.

The scatterplots in Fig. 10 show a correlation coefficient of about 0.9 with relatively small offset in the means of the two estimates. Since Δ*A* represents a difference of path attenuations along the forward- and backward-look angles, errors in path attenuation that are constant between the two estimates are not detected. However, errors, such as those that might be caused by the splash effect or volume scattering, that depend on the magnitude of the path attenuation or rain rate, could appear as a nonlinearity in the scatterplots. Although this nonlinearity is not evident in the results of Figs. 8–10, we investigate in the next section how these effects might influence the path attenuation as well as the attenuation-corrected NRCS estimates.

## 5. Discussion

There are several issues associated with the path attenuation and the corresponding attenuation-corrected NRCS estimates given by (1) and (2). First is the sensitivity of the estimates to the regression parameters and in particular to the slope, *r*, obtained by a linear regression, in the [*σ*^{0}(Ku), *σ*^{0}(Ka)] plane, through data measured in the presence of rain. While the coefficients remain fairly constant for the rain-free data, the slope *r* depends on the rain–no-rain threshold that is chosen and on the segment of data that is selected. One reason for this sensitivity can be understood by noting that in light rain, the NRCS is more sensitive to changes in surface wind speed and direction than on path attenuation, the former of which tends to follow along the 1:1 line. Changing the rain–no-rain threshold to allow more light rain cases in the raining category serves to decrease *r*; smaller *r* values will, in turn, increase the estimates of both Ku- and Ka-band path attenuations. One way to mitigate the problem is to raise the threshold, eliminating the very light rain cases, and then determine *r* from the filtered data.

An example of this procedure is shown in Fig. 11 using data from 1 September 2016 over the time period 1550–1628 UTC. For the top plots, the rain–no-rain threshold at Ku band was chosen to be 15 dB*Z* over 10 gates; that is, if the measured *Z*(Ku) exceeds 15 dB*Z* in at least 10 range gates along the path, rain is judged to be present. With this choice, the regression slope *r* through the rain data is 7. Results for Ku-band (left) and Ka-band (right) probability density functions for no-rain, rain (uncorrected) and rain (attenuation-corrected) are shown at the top of Fig. 11. (It should be noted that for the example shown in Figs. 2 and 4, which uses the same dataset, the 10 dB*Z* rain–no-rain threshold yields a slope of 6.) If the rain–no-rain threshold is increased from 15 to 20 dB*Z*, the number of raining pixels is reduced by 49% from 14 840 to 7561, yielding a slope of 10.2. Results for this regression slope are shown in the bottom plots of Fig. 11. Note that the rain-free pdfs are unchanged from the top to bottom plots. What does change is the pdfs of the attenuation-corrected data (dotted lines) where an increase in the slope *r* decreases the width of the distributions primarily by reducing the number of higher *σ*^{0} values. (It should be noted that the pdfs of the rain-corrected data change if the light rain cases are reintroduced once the slope is determined from the filtered dataset. In particular, for the results shown in Fig. 11, the light rain rate cases, those between 10 and 15 dB*Z* for the top plots and those between 10 and 20 dB*Z* for the bottom plots, are discarded and therefore not included in the rain-corrected pdfs.)

A second issue is the influence of volume scattering from rain at the surface range gate(s) where the NRCS is computed. An examination of Fig. 6 shows that for the HIWRAP antennas and system parameters, the volume scattering at Ku band usually can be ignored for both the inner and outer conical scans since the surface return is significantly higher than the rain return except in the presence of heavy rain. At Ka band, however, this assumption is not always warranted particularly in the presence of moderate to heavy rain and at the higher incidence angle of the outer scan. In particular, Fig. 7 shows that the Ka-band surface return at a number of fields of view is comparable to the rain return just above the surface.

*ε*

_{Ka}that represents the volume scattering contribution, then replacing

*σ*

^{0}is equal to the sum of the measured

*σ*

^{0}and the path attenuation, it follows that, in the presence of volume scattering at Ka band, the modified attenuation-corrected NRCS,

*σ*

^{0}, by

*β*≈ 1), in the attenuation-corrected estimates of

*σ*

^{0}(Ka) and

*σ*

^{0}(Ku).

To get an idea of the magnitude of the effect, assume that *β* = 1, *r* = 6, and *ε*_{Ka} = 3 (i.e., the rain-free data regression line has a slope of 1 in the [*σ*^{0}(Ku), *σ*^{0}(Ka)] plane, *A*(Ka)/*A*(Ku) = 6 and the rain and surface contributions are equal). These values give a reduction in the Ka-band and Ku-band path attenuation estimates of 3.6 and 0.6 dB, respectively, and a negative bias of 0.6 dB in the estimated *σ*^{0} at both frequencies. It is worth noting that an approximate correction for the volume scattering effect is straightforward if the rain returns just above the surface are used along with an approximation for the fractional rain volume at the surface gates. Under these assumptions, computations show that the changes in the NRCS rain data are usually negligible, with the exception of cases of strong Ka-band attenuation, particularly at the higher incidence angle (40°). (Note that the results shown in Figs. 2, 4, and 11 include this rain volume correction.) For example, the standard deviation in NRCS(Ka) at 30°, under rain conditions, for the data shown in the right panel of Fig. 2, increases from 6.5 dB without correction to 6.67 dB with correction; at 40°, the standard deviation increases from 5.71 without correction to 6.05 with correction. The changes at Ku band are smaller while the slope of the regression lines through the rain data remains virtually unchanged.

If we assume that the splash effect changes the Ku- and Ka-band surface cross sections by equal amounts (*ε*_{Ka} = *ε*_{Ku} = *ε*) we find, with *β* = 1, that the path attenuation estimates from (20) and (21) are the same as they would be without the splash effect [*A*′(Ka) = *A*(Ka), *A*′(Ku) = *A*(Ku)]. However, (22) and (23) show that the modified NRCS are offset by the same amount as the measured NRCS; that is, the second terms on the right-hand sides of (22) and (23) are equal to *ε*. In other words, in this case, the measured NRCS, that are assumed to be affected both by attenuation and the splash effect, are corrected for attenuation but are not corrected for the splash effect.

These effects can be understood with reference to the diagram in Fig. 3, where a perturbation by *ε* in the unperturbed NRCS values, that is, *r*, from the unperturbed and perturbed points to the rain-free regression line are the same, but produces an offset *ε* in the estimated NCRS values. The diagram also shows that offsets in the measured NRCS along the rain regression line of slope *r* have the opposite effect in the sense that, for offsets of this type, the method gives the correct values for the NRCS but incorrect values for the path attenuations.

## 6. Summary

Estimation of the wind vector derived from the radar surface cross section in the presence of rain is an important issue in scatterometry. In this paper, a method of path attenuation correction at Ku and Ka bands has been presented based on the relative insensitivity of *δσ*^{0}, *σ*^{0}(Ka) − *σ*^{0}(Ku), to changes in wind speed and direction. Comparisons to path attenuations derived from methods independent of surface scattering properties generally show good agreement with the results from the *δσ*^{0} approach.

While the path attenuation estimates are independent of radar calibration errors, they depend on the regression slope *r*, which in turn depends on the choice of rain–no-rain threshold as well as the data segment that is chosen. Further work is needed to fully assess this error source and its impact on the retrieval. Other error sources include volume scattering and raindrop splash effects. The former source of error can be estimated from the capability of HIWRAP to measure the rain as well as the surface return; the problem is more serious at Ka band than at Ku band and a simple, approximate correction method is available as long as the surface returns are larger than those from the rain. While the splash effect is not evident in comparisons with surface-independent methods of path attenuation estimation, the effect remains a potential source of significant error particularly at higher incidence angles, higher rain rates and lower wind speeds. However, if the splash effect perturbs the measured cross sections by the same amount at Ku and Ka bands, the path attenuations derived by the *δσ*^{0} method appear be relatively independent of this effect though not the NRCS values themselves.

## Acknowledgments

The HIWRAP instrument was developed by Goddard Space Flight Center under NASA’s Instrument Incubator Program. The NOAA program provided support for HIWRAP during 2015-2017 SHOUT flights. We wish to thank Matthew McLinden and Lihua Li of Code 555 GSFC for supporting HIWRAP development and deployment and for their work on data processing and analysis.

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