The ground DSD spectra used for calibration and validation were collected by a Parsivel2 (Particle Size and Velocity) optical disdrometer. The working principle, strengths, and limitations of the Parsivel2 have already been discussed in great depth in previous studies and will not be part of this study . For example, the Parsivel is susceptible to errors in the lower drop diameter range, which can affect the DSD shape and number concentrations. However, no efforts have been made to try to correct for these issues within the context of this study. The raw DSD data consist of particle counts across 32 nonuniformly spaced diameter classes ranging from 0 to 25 mm with a sampling resolution of 30 s. From the raw DSD, integrated quantities such as rainfall rate (R) and radar equivalent reflectivity factor (Z) can be derived . The disdrometer measurements were used to fit gamma DSD models and derive constrained relations between μ and Λ parameters at different temporal resolutions, which is necessary for retrieving DSDs from polarimetric radar measurements. At the same time, the disdrometer measurements were also used to (indirectly) validate the radar retrievals and study their consistency over time and across different events.

Similarly to , preprocessing is applied to the disdrometer data.

Only the liquid type of precipitation was considered for further analysis. All DSDs with observations above the 22nd diameter class (drop diameters greater than 7 mm) were discarded, since they correspond to mixed or solid precipitation.

The radar data used to perform the DSD retrievals were collected by TU Delft's polarimetric S-band (λ=9.1 cm) FMCW radar TARA (Transportable Atmospheric RAdar; ) in Cabauw, the Netherlands. TARA was collocated with additional sensors. This included a Parsivel disdrometer (see , Fig. 1) provided by the Leibniz Institute for Tropospheric Research (TROPOS). For this experiment, the radar antenna elevation angle of TARA was fixed at 45∘ with constant azimuth. The collected polarimetric radar observables included the reflectivity factor at horizontal polarization (Zhh) and differential reflectivity (Zdr) at 200 m of height (corresponding to the minimum range of TARA). The full specifications of TARA during the ACCEPT campaign are given in Table 1 of .

In order to make the radar data comparable with the disdrometer data, all Zhh and Zdr measurements were down-sampled over successive 30 s sampling intervals. The radar and disdrometer data were then synchronized by determining the time shift that maximized the correlation coefficient between Zhh Parsivel and Zhh TARA.

Concerning the calibration of Zhh and Zdr, noise measurements were performed every day to account for possible variations in range, especially at the beginning and end of the IF filter. Before the start of the campaign, the calibration of Zdr was verified using vertical profiling of drizzle and very light rain. The resulting histograms showed a mean offset of -0.11 dB with a standard deviation of 0.05 dB. Consequently, an offset of +0.11 dB was added to the measured Zdr for the whole ACCEPT campaign. For the calibration of Zhh, the transmit power was stored in the dataset, and there was a near-field correction for the non-full-overlap of the transmit and receive antenna beams using the method described in . However, an end-to-end calibration for Zhh was missing.

A total of seven rain events over the whole measurement campaign were selected for further analysis. The criteria used to select events were as follows.

Each event must consist of predominantly stratiform rain and exhibit a well-defined melting layer signal in the radar data.

Table  presents a summary of the duration, rain intensity, and mass-weighted mean diameter (based on the disdrometer data) for each of the seven selected events. As can be seen, most of the events last between 120 and 150 min. The longest on 3 November is slightly longer than 4 h. The low rain intensity and mass-weighted mean drop diameter values confirm that the selected events are mostly comprised of light to moderate stratiform rain. This makes sense given the criteria used to select the events and the fact that the ACCEPT campaign took place in October–November in the Netherlands at a time when heavy convective events are rare.

For illustration purposes, one of the seven events (E2, 11 October 2014) is plotted in Fig. . As can be seen, this event mostly consists of stratiform rain with a moderate intensity of approximately 1.8 mm h-1 and a total duration of approximately 3 h between 10:30 and 13:45 UTC, including a short break between 12:45 and 12:55 UTC according to disdrometer observations on the ground (Fig. ). The mass-weighted mean diameter is 1.1 mm, which is typical for light stratiform rain and small raindrop sizes. Event 2 was chosen because it has a relatively stable, well-defined melting layer around 2 km height as shown by the enhanced values of Zhh and Zdr in Fig.  at the top and bottom, respectively. The event also has a relatively low horizontal wind speed, which makes it easier to compare the radar retrievals aloft with the disdrometer measurements on the ground.

The model used to approximate raindrop size distributions (DSDs) in this paper is the gamma distribution proposed by : 1N(D)=N0Dμe-ΛD=NTΛμ+1DμΓ(μ+1)e-ΛD, where N(D) is the raindrop size distribution in mm-1 mm-3, μ is the shape parameter (unitless), Λ is the slope parameter (mm-1), N0 is the intercept parameter (mm-1μ m-3), and NT is the total number concentration (m-3). The advantage of NT over N0 is that its unit does not depend on μ . For convenience, the gamma model is reformulated in terms of the mass-weighted mean diameter Dm (mm) and the generalized intercept parameter Nw (mm-1 m-3) to 2N(D)=Nwf(μ)DDmμe-(4+μ)DDm, where f(μ), Nw, and Dm are given by 3f(μ)=644(μ+4)(μ+4)Γ(μ+4),4Nw=44πρwLWCDm4,5Dm=4+μΛ.

In the equations above, LWC denotes the liquid water content (in g m-3), and ρw is the density of liquid water (10-3 g mm-3).

It should be mentioned that even though the gamma distribution is the most popular and widely accepted model for representing DSDs in the literature, several studies have questioned its adequacy , setting criteria and proposing different tools to check the gamma hypothesis on a case-by-case basis.

The best parameters (μ, Dm, and Nw) for describing the DSDs measured by the disdrometer are obtained by using normalized parameterization of the gamma DSD model based on Dm (ratio of fourth- to third-order moment). To estimate μ, we first calculate Dm and Nw (directly from the measured DSD spectra). The value of μ is determined by testing all possible values of μ between -3 and 15 and choosing the one that minimizes the cost function (CF, Eq. ). Finally, we derive Λ through its relationship with Dm and μ (Eq. ): 6CF=∑i=322∣log⁡10(Nobs(Di))-log⁡10(N(Di∣μ))∣, where Di is the center of the ith diameter class in the Parsivel disdrometer and Nobs (Di) represents the volumetric size distribution measurements for each diameter class. Note that the index i ranges from 3 to 22 because the first two diameter classes in the Parsivel are always zero and the diameter classes above 22 correspond to particles that are too large to be associated with rain.

When an empirical relation between shape and scale parameters is used the gamma model is often called constrained gamma. Note that the term “constrained gamma” denotes a gamma DSD model in which the shape and rate parameters are linked by a deterministic function. Mathematically, this is equivalent to reducing the number of free parameters from three to two, which is convenient in radar-based DSD retrievals. However, the uncertainty related to estimating μ and Λ based on observed DSD spectra remains. Hence, the constrained gamma DSD model and all its associated moments still remain stochastic in nature.

Numerous studies have used and proposed constrained relationships between these two DSD parameters. The most common models are based on second-order polynomial fits, firstly introduced by . Since then, several other studies have proposed updated polynomial μ–Λ relationships based on either seasonal or regional criteria . Polynomial models between μ and Λ were also proposed for DSD retrievals using microwave link measurements . In this study, μ–Λ relationships are modeled using a slightly different power-law model: 7Λ=α(μ+3)β, with two coefficients α and β as given in Eq. ().

The power-law model above was chosen mainly for mathematical reasons since it ensures that Λ remains positive across all scales and avoids the problem of having to choose between a first-, second-, or third-order polynomial. The power-law model is also easier to justify than a parabola from a physical and mathematical point in light of the scale invariance of DSDs under proper normalization, as pointed out by previous researchers . However, for the sake of completeness, we also examined the polynomial model during our study and concluded that it did not make a big difference from a practical point of view (i.e., it has similar goodness of fit over the considered range of μ values). Nevertheless, we decided to use the power-law model in this study since it is more appropriate than a polynomial from a theoretical point of view.

Note that the goal of this study is not to question the validity of previous μ–Λ relationships nor optimize the parameters behind them (which depend on the dataset used) but to take a closer look at the sensitivity of the obtained fits to various underlying assumptions. Critical aspects that were investigated include whether the μ–Λ relation remains stable with respect to different sampling resolutions, drop number concentrations, types of stratiform rain events, or the validity of the gamma DSD hypothesis itself. At the same time, one has to keep in mind that the limitation of the Parsivel in terms of the detection of small droplets might lead to overestimated Dm and μ values, since the width of the distribution will be underestimated.

Because the gamma DSD model involves three parameters, three different radar measurements representative of three weighted moments of the DSD are required to retrieve the DSD in a given radar resolution volume. The retrieval method used in this paper is described in . It involves a combination of reflectivity factor at horizontal polarization (Zhh), differential reflectivity (Zdr), and a empirical relationship between the DSD shape parameter (μ) and slope parameter (Λ), commonly referred to as a μ–Λ relationship. The main steps of the retrieval method can be summarized as follows.

Impose a μ–Λ relationship Λ=g(μ) based on nearby disdrometer observations or literature values. In our case, a power-law relationship is used: 8Λ=0.514(μ+3)1.339, where the prefactor and exponent were determined by combining all the data from all seven events in Table .

In the following, we analyze the variations of the μ–Λ relationships from one event to another. For this, a filter was applied identical to , and only the cases which satisfied the gamma model hypothesis were considered. The adequacy of the gamma model was assessed based on a combination of a Kolmogorov–Smirnov goodness-of-fit test and Kullback–Leibler divergence. In total, approximately 40 % of the DSDs passed the tests and were accepted. On an event-to-event basis, that number varies between 36 % and 45 %.

In order to investigate and visualize possible differences between events, all seven events were plotted using different colors in Fig. . The overall relationships by were added for comparison. As can be seen in Fig. , most of the event-specific μ–Λ relations stay relatively close to the overall relation, except for events 2 and 6 for which larger deviations for higher values of μ (i.e., μ>8) are visible. For event 6, the differences can be explained by the limited range of μ, with most values remaining between 3 and 5 and only a single observation falling between 5 and 15. This limited range of variability significantly affects the reliability of the estimated μ–Λ relationship, especially for values smaller than 3 and larger than 5. For event 2, the differences can be explained by the presence of a few outliers in the upper-right part of the scatter plot, corresponding to DSDs with low number concentrations and high sampling uncertainties.

For each selected event, the sample sizes, the fitted power-law parameters α and β, and their percentage relative differences against the overall relation are presented in Table . The relative errors of the parameters depend on the characteristics of each event, with event 1 being the closest to the overall relation and event 6 exhibiting the largest differences. In order to have a more complete picture of each event, the correlation coefficient between μ and Λ as well as the root mean square deviation (RMSD) between μ and Λ points of each event and the overall relationship were calculated and are presented in Table . Even though event 6 has the weakest correlation coefficient, it has the lowest RMSD mainly due to its small sample size (the smallest in the event list) and the way the data are concentrated close to the fitted line. Event 1 shows the strongest relation between μ and Λ, while at the same time event 2 has the highest RMSD because of its outliers in the upper-right part of the scatter plot.

The event-specific and overall μ–Λ relations are clearly different from previously proposed relations by . For a fixed μ value, the overall μ–Λ relation for the seven selected events predicts higher Λ values compared with the ones by . This can be explained by the fact that Λ is inversely proportional to the mass-weighted mean diameter and that the relations were derived under different climatological conditions in Oklahoma in the US, where convective rain events with larger raindrops are more common than in the Netherlands.

Although the overall relationship might not necessarily be optimal for each individual event, our results show that it still provides a fairly good approximation of the average μ–Λ relationship across all seven considered events. Also, one has to keep in mind that the low sample sizes and limited ranges for μ make it practically impossible to derive reliable and representative μ–Λ relations for each individual event. To avoid sampling issues such as those encountered in event 6 and increase the robustness of our results, all remaining sensitivity analyses and retrievals were therefore conducted using the overall μ–Λ relation.

One crucial factor that could affect the μ–Λ relationship is the gamma DSD assumption. To investigate this issue, we temporarily added back all DSDs that were excluded from the previous analysis because they did not conform to the gamma model according to the criteria set by . For each event, we recalculated the individual μ–Λ relationship and compared the new results to the ones obtained using only the DSDs that satisfied the gamma assumption. In six out of seven cases, the inclusion of the non-gamma cases resulted in larger α and smaller β values. However, these changes were not reflected visually in the μ–Λ scatter plot as the two opposite changes compensate for each other. Therefore, apart from slightly changing the parameter values, the gamma hypothesis does not appear to have a strong effect on the overall μ–Λ relation. Also, the changes to α (0.518 from 0.514) and β (1.328 from 1.339) were rather small and not statistically significant. The fact that the overall μ–Λ relation is rather stable with respect to the gamma DSD hypothesis is an interesting result, especially given the fact that there are large differences in sample sizes between non-gamma (1829) and gamma DSDs (652).

Using the overall relationship from Sect. 4.1 as a reference, the influence of the number concentration on the μ–Λ relationship was investigated. It would be interesting to investigate whether the events for which the DSD is predominantly number-controlled lead to more or less stable μ–Λ relationships than events with size-controlled DSDs. Three different NT thresholds corresponding to different percentiles of NT (25 %, 50 % and 75 %) were applied, and only the DSDs with number concentrations above these thresholds were considered. In Fig. , the three derived μ–Λ relations obtained after applying the NT filters are shown against the overall relation (no filter). As the NT threshold is increased from 225 to 300 and 390 m-3 (Figs. b–d), the μ–Λ relation remains relatively stable for lower μ values, gradually getting closer to the one proposed by , especially for higher values of the shape parameter (μ>7). This can be partly explained by the fact that, on average, higher NT values correspond to higher rainfall intensities and larger drop diameters. Also, the average mass-weighted mean diameter increases by approximately 10 % as we increase the threshold on NT. This may not represent a big change, but it can be enough to slightly affect the μ–Λ relation. However, we believe the main reason the μ–Λ relation changes with increasing NT is sampling uncertainty. Indeed, our dataset predominantly features stratiform rain events with low rainfall intensities, low number concentrations, and relatively low and constant mass-weighted mean diameters (see Table ). As we apply higher thresholds on NT, the DSD samples that only contain a small number of drops and are associated with a higher sampling uncertainty get removed. Consequently, the remaining DSDs with higher number concentrations tend to be associated with lower sampling uncertainties, which leads to more reliable μ–Λ estimates. Moreover, it is worth pointing out that because of the way μ is estimated through the cost function in Eq. (), the error distribution of μ tends to be positively skewed. On average, we are therefore more likely to overestimate μ and underestimate the spread of the DSD rather than the opposite. Since μ and Λ values are positively correlated through their relation with Dm in Eq. (), any overestimated μ value automatically results in an overestimated Λ value (to compensate and get the correct Dm). Consequently, as we increase the NT threshold, sampling errors get reduced and the positively skewed outliers with high μ and Λ values progressively disappear. This removes more and more points on the upper side of the μ–Λ curve, pushing the new relation down towards the one proposed by . Regarding the sensitivity of the α and β parameters describing the μ–Λ relationship, our analyses show that they exhibit an opposite behavior, increasing and decreasing, respectively, as we increase the threshold on NT. The latter can be attributed to a gradual flattening of the relationship and increase of the intercept parameter. Note that another similar approach to reduce the uncertainty in the estimated μ–Λ relationship without applying a threshold on NT could be to consider temporal aggregation intervals longer than 30 s. However, this would significantly reduce the amount of data available for analysis.

In the following, the DSD data corresponding to the seven selected events were resampled at four different temporal resolutions of 30, 60, 240, and 480 s to investigate the sensitivity of the μ–Λ relationship to the choice of the temporal resolution. Similarly to before, only the resampled DSDs which satisfied the gamma hypothesis were kept for analysis. Figure  shows that the overall μ–Λ relationship remains very stable, regardless of the considered sampling resolution. Table  shows more details about the fitted power-law parameters α and β at each resolution, including their percentage relative differences against the overall relation at 30 s. We can see that the relative error affecting the parameters slightly increases as the temporal resolution is reduced. The latter can be attributed to the lower number of samples available for fitting the parameters. Apart from these obvious sampling effects, the choice of the temporal aggregation scale seems to have very little effect on the overall μ–Λ relationship, which remains rather stable across multiple aggregation timescales.

Note that as we decrease the temporal resolution, the mean values of μ and Λ (Fig. ) also decrease. This means that there is a progressive transition from peaked DSDs at higher sampling resolutions to broader, more widespread DSDs at lower resolutions. Decreasing the sampling resolution therefore causes the μ and Λ values to shift toward the bottom-left part of the scatter plot. However, while the points shift, they remain remarkably close to the initial μ–Λ curve derived at the highest temporal resolution of 30 s. The fact that the μ and Λ values change with resolution but that the overall relation between them is preserved across scales suggests that there is a fundamental physical link between certain moments of the DSD, such as the spread and the mean. Also, this relation seems to be quite robust regardless of whether the gamma assumption is valid or not and is only slightly affected by NT. In steady rainfall conditions, it should therefore be possible to use the same μ–Λ relationship for DSD retrievals across multiple temporal scales. This is of high importance given the fact that μ–Λ relations are often used to retrieve DSDs from radar observations, which have different sampling volumes and levels of aggregation than disdrometer data. Moreover, the use of a μ–Λ relationship may still be justified from a physical point of view, even if the underlying DSDs do not strictly comply with the gamma distribution hypothesis. Obviously, the fact that we have selected relatively similar stratiform events with low rainfall intensities and low temporal variability is a crucial factor here since it means that by resampling, we do not significantly change the properties of the DSDs or mix together different rainfall regimes. By contrast, larger differences in μ–Λ relationships can be expected for mixed-type rainfall events with multiple and rapid alternations between stratiform and convective rain.

On the other hand, there is still substantial controversy in the literature around the reason why μ–Λ relations exist in the first place and why certain DSD parameters are linked to each other. One justification could be that the effective number of parameters needed to describe most DSDs is probably fewer than three. In other words, under proper normalization, all DSDs look rather similar to each other. For example, introduced a single DSD normalization technique based on one reference moment (usually the rain rate). Later, and proposed a more general normalization technique based on two reference moments (usually the third and sixth moments). The existence of a μ–Λ relationship may just be the consequence of such scaling laws. In their study, have also argued that data filtering can have a strong influence on the relation itself, leading to spurious links between μ and Λ. However, this is not the case in our study. On the contrary, our results show that when events with similar characteristics are chosen, the overall μ–Λ relationship can be rather stable, barely depending on the different filters applied to the data (e.g., inclusion or exclusion of non-gamma DSDs or minimum threshold for Zhh and Zdr). Other studies have pointed out that the constraints linking μ and Λ during parameter fitting can lead to correlated errors between estimated gamma DSD parameters and biased relationships . Indeed, because of the way we fit μ and Λ through Dm (see Sect. 3.1, DSD model), the parameters end up being positively correlated with each other. In other words, if μ is overestimated, Λ will also be overestimated because it has to compensate for the bias in μ. To address this, proposed a σ'–Dm relationship, wherein Dm is the mass-weighted mean diameter and σ' a new mass spectrum standard deviation, defined and constructed to be statistically independent of Dm. Even though their approach seems to lead to smaller biases, our results show that it is also possible to derive reliable μ–Λ relationships without defining a new σ, simply by excluding the non-gamma DSDs cases and carefully filtering out DSDs with very low NT values.

Given the systematic underestimation of the reflectivity factor in TARA, a bias correction was applied before proceeding with the DSD retrievals. Indeed, the bias correction was considered essential to get more reliable results, especially for NT. Since the NT retrievals require the reflectivity to be converted from logarithmic (dB) to linear scale (mm6 m-3), a multiplicative adjustment factor known as the G/R ratio (i.e., the ratio of the sum of Parsivel to TARA reflectivity values) was used to bias-correct the TARA measurements, treating the disdrometer observations as the reference truth. The value of the G/R ratio was 4.52, which confirmed the large calibration bias of TARA. To address the bias, all TARA reflectivity values (in linear scale) were multiplied by 4.52 and the new DSD parameters were retrieved. As expected, the first two DSD parameters μ and Dm were completely unaffected by the bias adjustment, as they only depend on Zdr (see Sect. 3.4, DSD retrieval method). Figure , on the other hand, shows that NT retrievals were substantially improved, and the bias decreased from 276 to 89 m-3. Despite the lower bias, we can see that large uncertainties remain in the retrieved NT values, as highlighted by the large scatter and frequent outliers.

In the following, a small additional bias adjustment was applied to Zdr to try to account for the large difference in sampling volumes between the TARA radar and the Parsivel disdrometer. This second adjustment is conceptually different from the one applied to Zhh, which was primarily due to calibration issues. Contrarily to Zhh, the differential reflectivity Zdr of TARA is assumed to be well-calibrated. Therefore, the differences in mean and standard deviation are primarily attributed to differences in scale, height, and measurement principles. Note that this scale bias also applies to Zhh. However, for Zhh, the effect is masked by the large calibration bias and the two cannot be separated.

According to Fig.  (bottom), the average Zdr values measured by TARA are 0.03 dB larger than the ones from the Parsivel disdrometer; this makes sense given that the radar sees a larger measurement volume, which makes it more likely to contain at least a few larger drops. Even though a 0.03 dB difference seems small, such a bias can have a significant effect on the DSD retrievals given that the majority of Zdr values are rather small (e.g., between 0.1 and 0.2 dB). A 0.03 dB bias in Zdr therefore represents a relative error of 15 %–30 %.

Figure  shows the retrieved DSD parameters after correcting for the scale bias. We see a reduction of the bias affecting μ and Dm, which are directly linked to Zdr. The bias affecting μ is halved from 2.11 to 1.12, and the bias affecting Dm is reduced from -0.09 to -0.02 mm. The correlation coefficient remains relatively stable, regardless of the scale correction. Despite the improvements for μ and Dm, the NT retrievals remain problematic, with a low correlation coefficient of 0.12 (compared to 0.17 without scale bias correction) and moderate bias of -32 m-3 (compared to 89 m-3 without correction). Also, the average NT value increased significantly from 261 to 382 m-3 (+46 %), which highlights the large sensitivity of NT to changes in the differential reflectivity.

The results presented in the previous sections have shown that, unlike μ and Dm, the uncertainty surrounding the NT retrievals tends to be much larger. This can be explained by the fact that NT is the last parameter to be retrieved in Eq. (), which makes it more susceptible to error propagation and accumulation during the first steps of the retrieval procedure. Errors in retrieved NT values can be due to the retrieval method itself (e.g., the assumed μ–Λ relation and gamma DSD model), biased radar observations (e.g., calibration errors in Zhh or/and Zdr), or additional biases due to differences in measurement scale, height, and measurement principle between radars and disdrometers. Considering the fact that the events used in this study mainly consist of weak or light stratiform rain, the errors and uncertainty affecting the measured Zdr values are very likely to play an important role.

The scatter plot of retrieved NT values versus disdrometer data in Fig.  (bottom) shows a low correlation coefficient and a significant underestimation from TARA, mainly due to the huge bias in Zhh (6.44 dBZ). However, it is worth noticing that even after applying a calibration bias correction to Zhh, there was no substantial improvement in terms of the NT retrievals (Fig. ). Even though the bias in NT was reduced (89 m-3 compared to 276 m-3), the scatter increased and the correlation coefficient remained low (0.17). The scale correction for Zdr results in even worse agreement (correlation coefficient 0.12; Fig. , bottom). In general, two distinct groups of data points with drastically different error properties can be seen. For the first, the retrieved NT values are severely overestimated compared to the Parsivel disdrometer by up to 1 order of magnitude. For the second group, the retrieved NT values are up to 10 times lower than the disdrometer values.

The conclusion is that there are two different types of combinations of Zhh–Zdr that result in unreliable NT retrievals. The first group is comprised of low Zdr values compared to Zhh, which results in overestimated NT values. These are all the pairs of Zhh–Zdr in the lower-right part of Fig. . Since Zdr is low, the only way to get a high reflectivity is by increasing NT. The second group consists of relatively high Zdr values compared to Zhh, which leads to underestimated NT values. These points correspond to the top-left part of Fig. . Since Zdr is large, the only way to get a low Zhh is to decrease NT. Together, these two different types of outliers are responsible for the large scatter observed in retrieved NT values.

Each retrieval has its own uncertainty and error characteristic, depending on the pair of Zhh–Zdr. For example, the scale correction has different impacts on the different subgroups. Even though there is a general increase in NT to compensate for the new reduced value of Zdr, the aforementioned correction had a significant impact on the subgroup, which corresponds to the points that are overestimated by TARA and negligible for the ones that are underestimated.

A possible way to reduce the uncertainty affecting the NT retrievals and thereby avoid large errors is to filter out all potentially problematic combinations of Zhh–Zdr. In the following, a filter which aims to control the uncertainty in NT by removing certain Zhh–Zdr combinations that are difficult to handle is applied. Note that these “outliers” in the Zhh–Zdr space are not necessarily wrong. They are just problematic in the sense that they can potentially result in very large errors in terms of retrieved NT. The applied filter is two-dimensional depending on both Zhh and Zdr values since the uncertainty derives from their combination. A power-law model was used to fit the radar observables Zhh and Zdr after calibration and scale bias correction, respectively. Based on that model, an upper and lower curve defining the limits of acceptable Zhh and Zdr pairs is obtained by adding or subtracting a given tolerance from Zhh as in Fig. . For illustration purposes ±6 dB was selected, but several other options (i.e., 2, 4, and 8 dB) were examined as well. Table  lists all options together with their corresponding performances for μ, Dm, and NT. We see that by removing certain points beyond the lower and upper limits in the Zhh–Zdr space, it is possible to improve the correlation between the observed and retrieved μ, Dm, and NT values while keeping a similar bias. For μ and Dm, the best tolerance (in terms of correlation) seems to be ±2 and ±4 dB. However, these are rather strict, which means that a large fraction of the data points would have to be discarded (i.e., 56 % and 23 %, respectively) for a modest gain in performance. For the NT retrievals, the optimal tolerance appears to be ±6 dB, which discards less than 9 % of the data but still manages to increase the correlation (0.12 to 0.24) and decrease the absolute value of the bias (-32 to 10 m-3). Note that, contrarily to μ and Dm, filtering out more data points does not necessarily increase the performance in terms of the NT retrievals. Figure  shows the final radar DSD retrieval results after applying a filter with a tolerance of ±6 dB.