The detection of solar signatures in polar volume data of weather radars is described in . In the method a reflectivity signal which originates from a continuous microwave source is searched along radials in the operational scan data. Figure  shows an example where four radars observe the solar signal simultaneously close to the spring equinox. As a radar signal processor usually corrects the received echoes for the range dependence and the atmospheric attenuation, the “reflectivity” signals from the sun increase as a function of range. The received solar spectral power at the antenna feed PH,V for the horizontal and vertical polarizations (per MHz in dBm) can be calculated from the reflectivity signature as a function of range ZH,V(r) in dBZ: PH,V=ZH,V(r)-20log⁡10r-2ar-CH,V-10log⁡10Δf, where CH,V is the radar constant in dB according to for horizontal and vertical polarizations respectively, a is the one-way gaseous attenuation in dB km-1, and Δf is the receiver 3 dB bandwidth in MHz, assumed to be the same for both polarization channels. In the case of a proper solar signal, the power P is constant along the range. Depending on the hardware of the radar, the volume coverage pattern, the season, and the latitude of the radar, several tens of sun hits are found per day. Uncorrected reflectivity data (i.e. noise subtracted but ground clutter filtering not applied) are used for this analysis, as especially time-domain Doppler clutter filters can attenuate the solar signal by several dBs. The solar signal may also have contamination caused by ground clutter and precipitation. This can be circumvented by discarding data below 1∘ elevation and using data from long ranges (e.g. > 100 km) only.

The sun hits have a symmetric distribution around the sun position which is slightly wider in azimuth than in elevation. A typical example is shown in the left panel of Fig. . The larger width in azimuth is caused by the integration while the antenna is scanning, but is also influenced by the averaging window . The distribution of linear powers is well approximated by a Gaussian form; hence the power PH,V for the horizontal and vertical polarizations in dB can be written as follows: PH,V(x,y)≡ax⋅x2+ay⋅y2+bx⋅x+by⋅y+c, where the coordinates x and y are defined as follows: x=(ϕread-ϕsun)⋅cos⁡θsuny=θread-θsun, where ϕ and θ denote azimuth and elevation, “read” refers to the angle reading of the radar antenna, and “sun” refers to the calculated sun position. The observed azimuth differences are multiplied by cos⁡θsun to make them invariant to the elevation e.g.p. 516. Equation () is linear in the parameters ax to c, and thus the sun data can easily be fitted to this equation by the least squares method. Parameters ax and ay are related to the widths of the distributions of the x and y values, parameters bx and by to the elevation and azimuth biases, and parameter c to the peak power, i.e. when the antenna is pointing exactly at the sun. Note that these parameters need to be defined separately for the horizontal and vertical polarizations. We assume that the biases and widths are independent of the pointing angles, and that the microwave centre of the sun is close to the centre of the optical disk. The elevation width Δθ, the azimuth width Δϕ, the elevation bias Bθ, the azimuth bias Bϕ, and the power when the antenna is pointing directly to the sun, P^sun, can be calculated from the linear parameters : Δϕ,θ2=-40log⁡102ax,y≈-12ax,yBϕ,θ=-bx,y2ax,yP^H,V=c-bx24ax-by24ay. The widths are obtained from Eq. () when the corresponding parameter ax,y is negative.

As data from different elevations are analysed together, the solar elevation needs to be corrected for the effects of refraction. We use the analytical formulas for the atmospheric refraction of radiowaves, which are consistent with the commonly used k-model or 4/3-model . As the solar signal traverses all atmosphere, the expected value of k is less than 4/3, which is valid close to the surface. Hence we use k=5/4, which fits best to the model calculations and radar observations according to . To avoid severe refraction, data below 1∘ elevation are discarded.

The modelling of the Zdr signatures depends on which quantities are calculated in the radar signal processor. In case the horizontal reflectivity factor ZH and the vertical reflectivity factor ZV are both available separately, the analysis described above is repeated for both polarizations. An estimate of the solar Zdr is obtained from the difference of the horizontal and vertical solar powers, PH-PV, by applying Eq. (): Zdr≡ZH-ZV=PH-PV+(CH-CV). In addition one obtains the pointing difference of the horizontal and vertical polarizations, and it is also possible to determine how the widths compare to each other.

If the quantities available from the signal processor are ZH and Zdr, the vertical reflectivity can be calculated as ZH-Zdr, and the procedure outlined above can be followed. In the FMI system the signal processor uses the same radar constant for both polarizations. Then, in fact, the quantity provided by the signal processor is the difference of powers Pdr=PH-PV instead of Zdr calculated from the reflectivity factors. Also in this case the analysis can be carried out as above, using the same radar constant for H and V processing when applying Eq. (). The CH-CV factor is in this case included in the system Zdr bias, and taken into account when the system bias is subtracted (see Sect. ).

It is also possible to do the analysis directly to Zdr, following the procedure of . If we expand Eq. () using the expressions for the horizontal and vertical powers in Eq. () and equal the coefficients of the resulting equation with those of the two-dimensional polynomial for Z^dr, which is Zdr≡a¯x⋅x2+a¯y⋅y2+b¯x⋅x+b¯y⋅y+c¯, and apply Eqs. () to (), we arrive at the following equations: a¯x=ax,h-ax,v≈-12⋅(1Δϕ,h2-1Δϕ,v2)a¯y=ay,h-ay,v≈-12⋅(1Δθ,h2-1Δθ,v2)b¯x=bx,h-bx,v=-2⋅(ax,hBϕ,h-ax,vBϕ,v)≡-2a¯xB¯ϕb¯y=by,h-by,v=-2⋅(ay,hBθ,h-ay,vBθ,v)≡-2a¯yB¯θZ^dr=c¯-b¯x24a¯x-b¯y24a¯y, where B¯ϕ is the azimuth bias, B¯θ is the elevation bias, and Z^dr is the differential reflectivity at the extreme point. The curvature equations, (Eqs. ) and (), reveal that the curvatures in elevation and azimuth may have the same or opposite signs. Hence the Zdr surface is either an elliptic or a hyperbolic, i.e. a saddle surface. The pointing equations (Eqs. () and ()) indicate that whenever the H and V pointing directions agree, Zdr is also pointing to the same direction. This is easily seen, e.g. if Eq. () is solved for the bias: B¯ϕ=ax,hBϕ,h-ax,vBϕ,vax,h-ax,v. When the curvatures (i.e. widths) of the horizontal and vertical polarizations agree, either in azimuth or elevation, the Zdr curvature is zero, and the pointing is undefined. Expansion of the right hand side of Eq. () shows that, when a pointing difference between the two polarizations is present, Z^dr does not evaluate to ZH-ZV as defined in Eg. () but there are in addition terms proportional to the pointing difference. Hence the direct fitting to Zdr implies that the pointing differences are assumed to be small.

The right panel of Fig.  shows a scatter plot of the differential reflectivity, based on the same data as the reflectivity in the left panel. The distribution looks very different from that of the reflectivity, as the curvature has opposite signs in the elevation and azimuth directions. The distribution has the form of a saddle surface. In the vertical (elevation) axis the values are negative at the edges indicating that the V lobe is wider than the H lobe, whereas in the azimuth direction the opposite is true, in agreement with the antenna measurements shown in Table  and Eqs. () and (). The saddle surface form is different from the symmetric form seen in the Trappes radar data . In the Trappes case the surface has a minimum in the solar direction, indicating that the horizontal beam is wider than the vertical beam in all directions.

Calibration of differential reflectivity using polarimetric properties of rain has first been demonstrated by . Differential reflectivity at vertical incidence is intrinsically zero for raindrops; hence the measured Zdr is an estimate of the system bias Zdrbias, which is a sum of the differential biases in reception, ΔRdr, and transmission, ΔTdr: Zdrbias=ΔRdr+ΔTdr. Here ΔRdr and ΔTdr consist of differences of waveguide and other losses and the antenna gain between the horizontal and vertical polarization channels, and ΔTdr is in addition affected by the differences in the transmitted power between the channels.

For Zdr bias estimation, all FMI polarimetric radars scan 360∘ in azimuth with vertically pointing antenna every 15 min. A full azimuth scan is performed to reduce the effects of the orientation of the scatterers or possible asymmetries caused by the radome and the water on it. There is a transient feature during the first few kilometres from the radar, because the horizontal and vertical receiver channels (i.e. the T/R limiters) return to their normal operation in a different way after the transmission pulse. The range of this effect is radar dependent and varies from 2 to 7 km for the FMI radars. Because of transient effect, Zdr bias is analysed from altitudes where the transient has died out. Figure  shows an example of the Zdr bias analysis. In the analysis an average of Zdr values of the sweep over 360∘ in azimuth is calculated for each altitude level defined by the 125 m range gate used in these measurements. As a quality measure, only data with cross correlation coefficient ρHV>0.9 and SNR > 5 dB are accepted for the analysis, and data beyond two standard deviations from the mean are discarded. If more than 80 % of the 360∘ azimuth sector is covered by good quality data points, the mean Zdr bias is calculated for that altitude level. This last requirement is used to mitigate the potential Zdr bias caused by the orientation symmetry of the scatterers or by other disturbances during the measurement. This procedure is repeated for every vertical scan, and the final result is calculated as a daily mean of these quality controlled measurements.

The climatological melting layer height in Finland during the summer is about 3 km, which means that most of the data used for the analysis are obtained from solid precipitation. Snow is not spherical, but the assumption of zero intrinsic Zdr is justified because of the random motion of the snow, and the averaging in the azimuth. For the case in Fig.  the upper edge of the melting layer is at about 2 km altitude. All curves are continuous and smooth across that altitude.

The quality control of the solar hit data is necessary to get results of good quality. , and used a two-stage approach for ensuring that rain and clutter do not contaminate the hit data. The first stage is to use data from far ranges only, e.g. 100 km , which guarantees that clutter is not included, and the hit average is made of points above the rain in most cases. In addition standard selection methods, such as removing data points with high standard deviations, were applied. The second stage was to do the fitting of data to the model twice. After the first fit, data points too far (e.g. 1 dB) from the fitted curve are removed and a second fit is carried out on the remaining points. This method works efficiently when a small number of outliers appears in the data, unless they are strong or far from the solar direction so that the first fit is too far from the truth. The results are further improved if one assumes that the antenna is already well pointed and uses only hits close to that direction. This is an efficient method for avoiding using signal from RLANs (Radio Local Area Network), which produce signatures resembling those of the sun. The use of these methods guarantees that good results are obtained in a majority of cases. As the antenna pointing or the calibration levels are not adjusted on a daily basis based on these results, occasional bad results cause no trouble. describe a partially similar set of methods which offer the same functionality for the improvements of the quality of solar hits.

The number of sun hits and the statistical accuracy can be increased by also using data from closer ranges. Then the probability that the data are contaminated by rain or clutter increases, and one has to devise a method for removing the contaminated data prior to calculating the solar hit power. One possible method is a two-stage estimation, in which the first estimate of the solar hit power is calculated from the full data set, and this power together with an estimate of the width of the distribution is used to remove non-solar data. This is illustrated in Fig. , which shows a simulated solar hit case with rain at ranges less than 75 km. The figure shows that the mean of data between the 50 km and the 250 km ranges (blue) is much higher than the median of the same data (red), which in turn is close to the mean of data beyond the 200 km range (green). If the last is taken as an unbiased estimator of the solar power, the conclusion is that the median is a much better estimator than the mean when all data beyond 50 km is used. The median represents the solar power as long as clearly more than half of the points are genuine solar hit points, but will of course be more biased when the amount of contaminated points increases. Evidently, the power estimate based on far ranges only is even better but less precise due to lower number of samples used and less robust against outliers. The estimates of the width of the distribution confirm the above. If the standard deviation is used to determine the width of the distribution, the estimate is biased by precipitation contamination, as indicated in the figure. The median absolute deviation (MAD) gives a width estimate which is close to the width estimated from far ranges only.

Figure  shows the result of applying the methods to 1 month of solar hit data from the Anjalankoski radar. Instead of fixed range limits, as in Fig. , altitude limits are used, which enables us to use data from several elevation angles together. The figure illustrates the probability that any of the estimators produce a biased estimate of the solar hit power. The reference power is obtained by calculating the mean hit power above 8 km, which is free of rain contamination for the data used in the study. Figure  shows that filtering by mean produces biased estimates even if the start altitude is put to 6 km, and that the number of biased estimates is significant for lower start altitudes.

The overall best performance is obtained if the filtering window is first estimated using data from high altitudes and the final estimate of the mean is calculated using data also from the lower altitude. Then a biased estimate is obtained only in very few cases. The median estimator, although nearly as good, produces biased estimates of the power in some cases. One can reduce the computational load for real-time analysis of large data sets, e.g. for the analysis of all European solar hit data within the OPERA (Operational Programme for Exchange of Weather Radar Information) data centre by fixing the width of the filtering window instead of estimating it for each solar hit. A recommended value is at least 3 times the standard distribution deviation of the solar hit data, amounting in the FMI case to about 2 dB. Fixing the window width is possible because the statistics of the solar hit data do not vary from case to case, as pointed out by .

The effect of the improved filtering is also noticeable in the fitted parameters. For the case in Fig. , the number of good solar hits increases by about 10 % and the root mean square error of the fit decreases by about 3 % when the two better filtering methods are used instead of the simple mean filtering (blue bar). This holds when the data are used starting from 2 km upwards. The improvement is case dependent and depends to a great extent on the number of rainy days in the data set. In the analysis we have discarded all solar hits with standard deviation greater than 2.5 dB, which is about 3–4 times the standard deviation of genuine solar hits. Without this additional screening, the improvement when using the two better methods would be much greater.

The filtering of the differential reflectivity Zdr is not as straightforward as the filtering of the reflectivity, because Zdr in rain and from the sun do not deviate significantly from each other. The data contaminated by rain can be removed prior to averaging by using the reflectivity data as a proxy to indicate which data points are based on sun and which on rain. Hence the estimates for Zdr are calculated as the mean and standard deviation of the Zdr profile by applying the same range indices as had been used for the reflectivity data.

There is a number of different methods to estimate the Zdr bias from the solar hits. , when solving Eq. () for the Zdr, first estimated the widths using a larger data set and fixed the widths in the fitting to improve the stability of the fit. This is one of several methods available for the estimation of the Zdr bias. An obvious second choice is to do a full 5-parameter fit, such as recommended for reflectivity data . It is also possible to do 3-parameter fitting by fixing the pointing to that obtained from the reflectivity fit, and thus fitting for the power and the two width values only. Noting that the Zdr bias would be constant over the solar disk for fully matching antenna beams, one can take a mean or a median of all Zdr hits. And, finally, it is possible to analyse the reflectivity channels separately, either by a 5- or 3-parameter fit, and get the estimate by differencing the power.

Figure  compares Zdr estimates by five methods presented above for 1 month of data. In the following we assess the methods by their ability to estimate correctly the value of Zdr at direct solar pointing. One can readily notice that the results obtained by using the mean or the median are clearly different from those obtained by the fitting methods. This is not surprising, because the Zdr field seen in Fig.  is not constant. Taking a mean might work for a saddle surface if the averaging is restricted to a small area around the solar direction, but would not work if the Zdr surface has a clear minimum or maximum at the solar direction, such as for the case shown in . Hence these methods are not recommendable.

The three fitting methods give comparable results in the FMI case, where the widths of the two polarizations differ. If the azimuth and elevation widths are close to each other, the direct fitting to Zdr becomes more unstable, because the errors of the power and the widths are correlated. And if widths in either elevation or azimuth (or both) agree, the surface degenerates into a plane in that dimension, and the fitting to Zdr is ill posed as an inverse problem and not at all possible. Hence a direct fit to the Zdr is not guaranteed to work for all radars. A separate 5-par fit to both PH and PV is instead a safe method which works in all cases, and therefore recommended. The Zdr result is then obtained as the difference of the determined powers. If the solar SNR is low, a 5-par fit may not converge. One can then improve the stability of the fitting by fixing the widths to theoretical values or values based on large amount of data, as recommended by .

The additional benefit of doing the fitting to both polarization channels separately is that one can determine the angle between the pointing directions of the two polarizations. The two upper panels of Fig.  show statistics of the elevation and azimuth pointing results of the H and V polarizations for 1 month of data for the Anjalankoski (ANJ) radar. The analysis confirms that the H and V beams are well aligned. The pointing difference between ZH and ZV is less than 0.02∘ for ANJ. A similar analysis of all the radars confirms that for most other radars in the network the pointing difference is less than 0.01∘. These pointing angle differences are similar to those reported earlier by for the German network.

The lower panels compare the width values of the two polarizations. There is a clear difference in the image widths so that for the V polarization the image is wider in elevation and for the H polarization in the azimuth. This is a result of the antenna design, and the results are typical for the whole network and are consistent with the width values given in Table .

Figure  shows Zdr results from five polarimetric FMI radars, based on separate 5-par fitting to PH and PV, as recommended above. As the measured quantities in the FMI system are ZH and Zdr, ZV has been calculated as their difference. The Zdr results are differences of the fitted horizontal and vertical powers, without subtracting the system Zdr bias. Figure  shows that the bias varies from radar to radar and that the standard deviation ranges from 0.03 to 0.05 dB. This is an indication of the stability of the radar system itself and of the analysis method. The standard deviations are significantly lower than the 0.2 dB reported by , probably due to the better rain rejection in the estimation of the hits. The standard deviations are also somewhat lower than the 0.05 dB reported by , which were obtained as differences of median power values instead of fitted powers.

Figure  shows Zdr bias values based on the zenith scan measurements for the same radars as used in Fig. . The results are not available for all days, because the analysis requires precipitation above the radar. Yet the bias has been determined for more than half of the days. It is seen that the Zdr bias varies from radar to radar but that the bias is stable as indicated by the standard deviation of the data. Comparison with Fig.  shows that standard deviations are slightly higher than those for the solar data.

Both the solar and the zenith methods are based on using an object with intrinsic zero differential reflectivity. Yet the results are not directly comparable, because the solar results depend on the receiver biases only, whereas the zenith results depend both on the receiver and the transmitter sides. One could, in principle, use the known loss and gain biases to correct the observations. Any significant deviation of the corrected value from zero would indicate an error in the loss and gain figures. In systems where both polarizations are processed using a single radar constant, one needs to include the CH-CV factor, discussed in Sect. , in the estimation.

In many operational systems, the zenith Zdr is used as the system bias which is subtracted from all Zdr measurements, including the solar ones. The system bias is a sum of transmitter and receiver biases (Eq. ); hence the receiver biases cancel out and the result should amount to -ΔTdr, when the system bias is subtracted. The possibly existing CH-CV factor also cancels out in the operation. Comparison of the sum of the differential transmitter losses and antenna gains given in Table  with the difference of the solar measurements in Fig.  and zenith measurements in Fig.  shows that the two estimates are within 0.1 dB for four radars and about 0.5 dB for one radar. Noting the stability of the results from both the solar and zenith methods and assuming that the antenna gains do not change over time, the most obvious explanation is that the transmitter losses are not correct. The fact that the transmitter loss values might be incorrect does not affect the accuracy and usability of the Zdr measurements in precipitation, because the zenith scan measurement takes all loss factors into account, and the bias determined from it corrects for all possible errors in the loss or gain difference values.

The interpretation of the solar Zdr measurements depends on how often the system bias is updated. Usually the system bias is updated manually once the zenith observations indicate that the existing value is no more valid. In this case the solar method is useful for monitoring the receiver side, even though the solar Zdr with system bias subtraction amounts to -ΔTdr. Clearly, if something changes in the receiver chain, the solar Zdr changes readily. If something changes in the transmitter chain, excluding the antenna, no effect is seen in the solar Zdr until the system bias is updated. The effect of such a change in the system bias is seen in Fig.  in which the Zdr values make a downward step on 21 August. If the system bias was updated in near real time on a daily basis, the solar Zdr would actually follow the changes on the transmitter side. Daily setting of the system bias is not usual, and not even possible, because it requires the presence of precipitation.