Estimates of the frequency of surfaces exceeding threshold values of PNR have been obtained by exploiting the climatology gathered by the polar-orbiting nadir-pointing CloudSat W-band radar and simulating the WIVERN returns at slant incidence angle. The method, proposed by and refined in , accounts for the additional path integrated attenuation and the reduction in surface normalized backscattering cross sections when considering the slanted viewing geometry of the WIVERN radar. Figure  shows the cumulative distribution functions (CDF) of the surface peaks (and equivalently of PNR) for land and ocean surfaces. Since ocean surfaces are far less bright than land surfaces at WIVERN incidence angles, a lower percentage of sea than land surface profiles exceeds any given threshold. For instance, more than 36 % and 99 % of the surface peaks exceed -5 dBZ (i.e. PNR=10 dB) for ocean and land surfaces, respectively. Two factors can actually decrease the number of surface returns effectively useful for the calibration:

the presence of low clouds and precipitation that could perturb the shape of the reflectivity and of the Doppler velocity profile (discussed later in Sect. );

The Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) global digital elevation model (DEM) (https://asterweb.jpl.nasa.gov/GDEM.asp; last access: 2 June 2023), with a resolution of 1′′×1′′ (i.e. 30.9m×30.9m at the Equator), has been used to examine the validity of the flat surface assumption by evaluating the variability of the elevation in areas comparable to that swept by the WIVERN radar footprint with different integration lengths. Boxes with latitudinal extent of 18′′ and longitudinal extents of 18, 36, 72, 90 and 180′′ were considered, which corresponds roughly to from no integration to 5 km integration length. The standard deviations of the elevation, σelev, within each box were then calculated for the entire global data set. Figure  shows the variation of the cumulative land fraction with σelev, for the five box sizes adopted. The respective values of σelev for increasing box size, given in the form (50th percentile, 70th percentile, 90th percentile), are (7.0 m, 12.0 m, 37.3 m), (8.0 m, 14.4 m, 47.1 m), (9.4 m, 17.9 m, 59.5 m), (10.0 m, 19.2 m, 63.5 m), and (12.0 m, 23.6 m, 74.9 m). It is clear that, for a given percentile, the value of σelev increases as the box size increases. These findings suggest that roughly half of the land surfaces have a variability in elevation less than 10 m within characteristic WIVERN averaging areas. Such surfaces will likely be useful for the calibration methods I–II (Sects.  and ), but a dedicated study to properly assess the impact of surface elevation variability is needed.

A final consideration is that, generally, ocean surfaces will not appear to have zero Doppler velocities, but they will have biases of the order of less than 1 ms-1. This is due to the interplay between waves and currents . Here it is assumed that corrections for such effects can be performed based on auxiliary information or that they will average out when considering views from different directions.

As illustrated in Fig. , an elevation mispointing bias moves the apparent motion of the ARC position inside the WIVERN antenna pattern along the elevation direction (y axis), e.g. from the blue to the red line. Note that uncertainties in the atmospheric refraction between different models are expected to be of the order of 1 arcsec (i.e. lower than 5 µrad) and are therefore neglected in this study. Correspondingly, the actual power measured by the ARC, derived by using Friis formula with an ARC gain of 20 dB, changes because of the difference in the antenna pattern. Such change depends on the specific position of the overpass (e.g. on the minimum distance of the boresight position to the ARC) and on the details of the antenna pattern. In order to simulate the capability of the ARC measurement to identify and quantify the mispointing, different possible boresight ground tracks have been simulated (like the blue and the red lines in Fig. a). For each ARC position at a given elevation distance θ‾, the ARC signal is simulated and compared with the returns sampled at different elevation distances θ‾+Δθ (with a maximum shift Δθ of ±1000 µrad and sampled every 5 µrad).

The mean square distances (MSD) of simulated ARC received signals at position θ‾ and θ‾+Δθ sampled at different time tk with k=1,…,Nt is computed as 3MSD(θ‾,θ‾+Δθ)=∑k=1Nt(PARCθ‾(tk)-PARCθ‾+Δθ(tk))2Nt-1000µrad≤Δθ≤1000µrad. All power signals below -80 dBm in the summation have been excluded in order to make sure there will be no effective impact from ARC receiver noise. A 0.5 and 1 dB noisiness is introduced in the antenna pattern to account for uncertainties in the antenna pattern. A total of 400 different realizations of the antenna pattern are used to compute the MSD in Eq. () so that a distribution of MSD can be derived for each pair (θ‾,θ‾+Δθ).

The almost symmetric shape of the antenna pattern causes the similarity between two signals sampled at opposite elevation angles with respect to the boresight (i.e. with θ‾=±Δθ). Two examples of the 50th (line) and 5–95th percentiles (shading) of the MSD pdfs (probability density functions) are shown in Fig.  for θ‾=160 µrad (Fig. a) and θ‾=480 µrad (Fig. b). As expected, the minimum is found in relation to a shift between the pairs Δθ equal to 0 µrad, but the interplay between the width of the pdf and the prominence of the local minimum introduces uncertainties in the determination of the mispointing.

When considering an overpass close to the ARC (e.g. θ‾=160 µrad; Fig. a), the same MSDs are encountered on a vast interval of Δθ, ranging inside the first antenna main lobe in a symmetric way, i.e. with θ between -θ‾-Δθ‾ and θ‾+Δθ‾ with Δθ‾=250 µrad for this specific example. When considering ARC positions farther away from the boresight, a second local minimum may form with regard to Δθ=-2⋅θ‾ (e.g. for θ‾=480 µrad, the second local minimum is at around Δθ=-960 µrad; Fig. b).

Figure a shows the uncertainty in the elevation mispointing determination as a function of the elevation distance of the ARC from the WIVERN antenna boresight. It is determined by computing for which elevation mispointing the upper 95th percentile, corresponding to the minimum of the MSD (level identified by the red line in Fig. ), exceeds the lower 5th percentile in the adjacent mispointing angles. The black and the red shading represent the result obtained considering an antenna pattern uncertainty of 0.5 and 1.0 dB, respectively. Results show that uncertainty is maximized in correspondence to an overpass of the boresight exactly over the ARC.

Given the fact that there will be more than one overpass each month (see Fig. ), it seems realistic to bring this uncertainty down to less than 50 µrad.

On the other hand, a mispointing in azimuth does not move the apparent motion of the ARC position inside the WIVERN antenna pattern (Fig. ), but only translates the received power at the ARC in time. The transmission time and flight time of the radar pulses (∼2.2 ms) are well known; excess path lengths in the atmosphere are expected to be less than a few metres, and thus delays are expected to be of the order of less than 0.01 µs, negligible in this context . The procedure followed for the elevation mispointing is replicated introducing a shift in azimuth, Δϕ. In this case, 4MSD(θ‾,Δϕ)=∑k=1Nt(PARCθ‾(tk)-PARCθ‾(tk+Δϕ/Ωa))2Nt-1000µrad≤Δϕ≤1000µrad. As before, pdfs of MSD are computed and an estimate of the uncertainty in the azimuthal mispointing is derived based on percentiles. The right panel of Fig.  shows that the closer the overpass is to the ARC, the lower is the uncertainty in the azimuthal mispointing determination.

The previous methodology requires the ARC to be positioned in a location within a few kilometres (a few thousand µrad) from the radar boresight location at the ground. To estimate the number of useful calibration points as a function of ARC locations, the WIVERN orbit and boresight positions have been propagated for 50 d. Although the satellite ground track has a repeat cycle of 5 d, the boresight will not trace the same path after this period, and thus different regions will be observed within the swath. The simulation rationale consists of selecting a distribution of ARC locations over the region of interest (Fig. a) and counting the number of overpasses within a given footprint. A 1 km spacing in latitude and longitude has been selected to generate the ARC distribution over the region, whereas a time step of 0.5 ms (equal to 250 m along the scan track) has been selected to guarantee a good sampling of the scan track. The simulation has been repeated considering the footprint corresponding to 1, 3, 5 and 10 times the antenna beamwidth.

Figure  shows the average results for a 10 d period obtained from the 50 d simulation. Figure a displays the number of overpasses over the selected ARCs for a 10× beamwidth footprint. The image shows several hotspots located at specific latitudes and longitudes, while an optimal longitude-independent cluster of hotspots (red line) exists at around 79∘ of latitude. Since the enhanced number of overpasses at such locations is generated by the intersections occurring at the lower border of the swath, their positions are around 400 km south of the satellite ground track when it reaches its highest latitude. Three 50km×50km regions have been selected around some of the hotspots at latitudes 45, 66.5 and 78.7∘. Figure b shows the histogram of the overpasses within these regions, considering the 10× beamwidth footprint. As expected, a greater number of overpasses occur when moving toward higher latitudes.

Table  summarizes the results when taking different beamwidths. Clearly, when considering the 10× beamwidth footprint (i.e. within roughly 6000 µrad), a sensible number of overpasses (from at least more than 13 at 45∘ latitude to more than 53 at 78.7∘ latitude for a 10 d period) is possible over sites whose latitudinal position is properly selected.

To address these questions the data set produced in that combines the CloudSat reflectivity observations and the ECMWF winds has been exploited. Since CloudSat is orbiting in a sun-synchronous polar orbit like the one foreseen for WIVERN, the winds sampled in the ascending and descending orbits have the same statistical variability expected for WIVERN.

Figure  shows an example of simulation of WIVERN measurements from a portion of CloudSat orbit through a northeastern Atlantic widespread low which brought a significant amount of precipitation and high winds over Ireland on 19 February 2007. The CloudSat reflectivity curtain (Fig. a) is combined with the corresponding ECMWF wind reanalysis (Fig. b) to simulate the slant reflectivity (Fig. c) and Doppler velocity (Fig. d) curtains that would be observed by WIVERN at side view (ϕ=90∘). Because of the slant incidence angle, the WIVERN surface reflectivity is much lower than that of CloudSat apart from over land (see land/sea flag at the top of a). In Fig. d, the black contour highlights the areas where the WIVERN Doppler velocity accuracy would be better than 2 ms-1.

Our method comprises the following steps:

The data set has been divided in ascending (A) and descending (D) orbits for latitudes between -65 and 65∘.

The pdfs of in-cloud horizontal winds retrieved by WIVERN when looking sideways are shown in Fig. a for A/D orbits (diamond/asterisk markers) and for different height ranges as indicated in the legend. The same plot is repeated on the right side considering all-sky condition winds. In the latter condition, A and D wind distributions looks pretty much identical. However, when considering only in-cloud winds, the two distributions take different shapes, suggesting the existence of a diurnal cycle (A and D winds are sampled 12 h apart) affecting in-cloud winds. This result makes the option of identifying azimuthal mispointing only by using WIVERN ascending and descending measurements challenging because the pdfs of A/D in-cloud winds are intrinsically different, and therefore large biases in the winds (typically of the order of 1 ms-1, i.e. ϕ≈200 µrad) are needed to see a neat separation between the two pdfs for accumulation times of at least 10–15 d (not shown).

On the other hand, when considering the ECMWF as a reference, Fig.  demonstrates that the histogram of the differences for the side winds has a standard deviation of the order of 3.66 ms-1 with an average of about 80 000 winds per day. In this case, only WIVERN's wind measurements characterized by an SNR higher than 0 dB have been taken into account. This large amount of winds demonstrates that the error in the estimate of the azimuthal bias will become negligible (≤0.2 ms-1) already after few minutes. The real limit becomes, in this case, the assumption that the reference ECMWF winds are unbiased. The validity of such an assumption applies to global averages over few days with an upper limit for the bias of circa 0.3 ms-1. Note that the same reasoning can be applied to the forward/backward line of sight winds and therefore to the elevation biases.