A DSD dataset is critical for establishing relationships between polarimetric radar variables for radar QPE algorithms. Disdrometers and rain gauges are usually deployed at the same meteorological site; although they sample the precipitation differently, their rainfall measurements in the same area should agree with each other. However, DSD-derived rainfall at six stations, directly calculated from the size–velocity pairs in Fig. 5 without any QC processing (denoted as RM), all presented unrealistically large values: maximum RM at LH, XJ, and HJ exceeded 200 mm that at DT exceeded 400 mm and that at WL and YH unbelievably exceeded 3×103 and 104 mm during typhoon Lekima. For convenient comparison of disdrometers with gauge rainfall series, RM is rewritten as 5RTM=RT+(RM-RT)/CTRM>RTRMRM≤RT, where RTM stands for the transformed rainfall value and RT stands for a rainfall threshold that is set a little larger than the maximum hourly gauge rainfall. CT is also manually set for each station, and CT partly indicates that RM is at least CT times higher than gauge rainfall. The RM part exceeding RT can shrink into a limited range interval, and RT and CT serve for comparing RM and DSD-derived rainfall after QC processing (denoted as RQC) in the same figure, as depicted in Fig. 6. Accordingly, CT of YH and WL in Fig. 6 is huge (800 and 500, ≥20 at the other stations). Meanwhile, DSD-derived maximum ZH, ZDR, KDP, and R exceeded 85 dBZ, 5.5 dB, 1500 ∘ km-1, and 15000 mm h-1, respectively (see Fig. 7a–c), and they are also abnormally larger than the final QC-processed counterparts (rectangles in Fig. 7a–c). If these unrealistic DSD-derived radar variables were directly utilized to establish the parameters of any radar QPE algorithm, an unrealistically overestimated radar rainfall field would be obtained. Afterward, the QC procedure in Sect. 2.3 is first imposed on the size–velocity pairs, and its performance and effectiveness are investigated through comparison with gauge rainfall recordings.

According to a visual comparison in Fig. 6, the severe overestimation of RM at all six stations is reduced after processing wind effects, and a better approximation is noticeable at XJ, HJ, LH, and DT in Fig. 6a–c and e, where the extra hail and graupel processing hardly change the residual differences. In contrast, the RQC time series at WL agrees well with its gauge rainfall recordings after the hail processing but is underestimated after extra graupel processing (see Fig. 6d). This implies that WL suffers from some solid particle contaminations. Still, these solid particles may melt and have similar size–velocity characteristics to raindrops, and their removal is responsible for the final underestimation of RQC at WL after QC processing. YH also suffered from solid particle contaminations. During its peak rainfall recording period between 16:00 and 22:00 UTC, 9 August 2019, RQC in Fig. 6e changes relatively less after the hail processing and still deviates largely from gauge rainfall recordings; conversely, RQC better approximates gauge rainfall after the graupel processing. This indicates that the terminal velocity of these filtered particles is more prone to graupel (not deduced by size). Section 3.2.1 further verifies the falling solid particles.

These residual solid particles could result in a false relationship between D0 and ZDR. As shown in Fig. 8a, the fitted curve uniformly passed through the scattergram, representing an excellent fitting relationship between D0 and ZDR. However, as mentioned above, these DSD-derived D0 and ZDR still suffer from some solid particle contaminations after processing the wind effects. Even after hail and graupel processing, the scattergram in Fig. 8b still presents a significant overfitted relationship between D0 and ZDR. The scatters with ZDR>2.5 dB are related to melting solid particles with D0 ranging from 1.5 to 4 mm, and some have raindrop-like sizes (<2 mm). Finally, DSD-derived radar variables constrained by ZDR<2.5 dB are assumed to be contributed by pure raindrops, and they are utilized to fit the D0–ZDR, LWC–KDP, and KDP–ZH relationships in Fig. 8c–e and the ZDR–ZH relationship in Eq. (2a) (see Fig. 8f): 6D0=0.2987×ZDR3-1.3229×ZDR2+2.1931×ZDR+0.3543,7LWC=2.0949×KDP0.6889,8KDP=1.5473×10-15×ZH8.8365. Combining these relationships and another relationship between the normalized concentration of raindrops (Nw, in mm-1 m-3), LWC, and the mean volume diameter of the drop size distribution (Dm, in mm) in Eqs. (9) and (10), 9Nw=44πρwLWCDm4,10Dm=4+μ3.67+μD0, where ρw is the water density (1 g cm-3), high-resolution DSD parameter fields can be derived from WZ-SPOL radar measurements.

The self-consistency can demonstrate the credibility of polarimetric radar measurements through scattergrams (Fig. 9). The scattergrams in Fig. 9b and d are obtained from all ZHC, ZDRC, and KDP measurements described in Fig. 11. The ZHPI approach (Bringi et al., 2001) with more constraints described in Gou et al. (2019a) effectively mitigates the attenuation effects on ZH and ZDR of the WZ-SPOL radar. The spatial fields of ZHM and ZDRM are not presented (they will not be used for the subsequent analysis), but it is noticeable that radar-measured ZHM, ZDRM, and KDP are not self-consistent before attenuation correction processing: it is obvious for ZHM>40 dBZ and KDP>1 ∘ km-1 that KDP–ZHM scatters deviates positively from the theoretical KDP–ZH curve (Eq. 8 as depicted in Fig. 8e), indicating that larger reflectivity values are anticipated for these KDP measurements. In addition, an overall deviation of ZDRM–ZHM distribution in Fig. 9c from the theoretical ZDR–ZH curve (the black curve stands for Eq. 2a as depicted in Fig. 8f) addresses a non-negligible negative ZDR bias before the differential attenuation correction. In contrast, the scattergram core areas in Fig. 9b and d (defined as log10(N)>1.6) exhibit more compact distribution along theoretical KDP–ZH and ZDR–ZH curves, demonstrating the effectiveness of the attenuation correction to enhance the self-consistency between ZHC, ZDRC, and KDP.

Radar measurements are feedback from drops in the air, but disdrometers collect DSD near the surface. In this sense, the comparison above also means that radar measurements tend to be more consistent with their surface counterparts after the correction. However, this does not mean that they completely agree; conversely, ZDRC still deviates largely from Z^DR when reflectivity exceeds 35 dBZ in Fig. 9d. In addition, the time series in Fig. 10 shows that extremely large DSD-derived ZH, ZDR, and KDP in Fig. 7 (time series not presented) have diminished, and they begin to approximate their radar-measured counterparts. The hail and graupel processing effectively improves the consistency between the gauge and disdrometer, as mentioned above; furthermore, DSD-derived ZH and KDP also simultaneously tend to better approximate radar-measured ZHC and KDP. Meanwhile, the residual differences between radar and DSD are still prominent in terms of ZDR, and larger DSD-derived ZDR than radar-measured ZDRC occurs at WL and YH, indicating that larger-sized drops are collected by WL and YH than the radar volume gates above.

Considering that Eq. (2a) is fitted based on S0, and RQC at WL agrees better with gauge rainfall if no graupel processing occurs, S0 can be refined: SI excludes large-sized drops by removing WL, SII further excludes large-sized drops from WL and YH, and SIII re-includes more large-sized drops by adding the size–velocity pairs removed by graupel processing at WL. In this way, three new ZDR–ZH relationships are re-established as 11aZ^DRr=3.477×10-4ZH(r)2.161DSD∈SI,11bZ^DRr=5.033×10-4ZH(r)2.0383DSD∈SII,11cZ^DRr=1.0652×10-4ZH(r)2.508DSD∈SIII. The further removal of the DT dataset from SII will change the ZDR–ZH relationship in Eq. (11b) very little (data not presented). Although there is an uncertainty that large-sized drops may source either from melting solid particles or the collision–coalescence, more large-sized drops in S0 and SIII make Eqs. (2a) and (11c) (higher Z^DR estimates) prone to the outcome of the dominant collision–coalescence process; conversely, more small-sized drops in SII and SIII make Eqs. (11a) and (11b) prone to dominant collision–breakup. Resultantly, Eqs. (11a) and (11b) exhibit smaller ZDR than that of Eqs. (2a) and (11c) for a given ZHC, which agrees well with the simulation result in Kumjian and Prat (2014). In Fig. 9d, radar-measured ZDRC tends to be more consistent with Eqs. (11a) and (11b) for a given ZHC than Eqs. (2a) and (11c) in the scattergram core area, and this ZDRC–ZHC scattergram reflects that the governing collision–breakup processes in radar volume gates restrain the drop size increase due to the coalescence–breakup balance, which means ZDRC does not grow similarly to coalescence-dominated volume gates.

The time series of vertical polarimetric radar measurement (Figs. 12–17), which is constructed with an altitude resolution of 100 m based on the technique in Zhang et al. (2005), is chosen to describe the microphysical evolutions upon each station; DSD-derived radar measurements in Sect. 3.1 assist in interpreting what occurred near the surface. The combination of radar and DSD can effectively explain the potential microphysical processes in the vertical gap between the air and the surface.

The freezing level (FL) is significant in the vertical measurements (see Figs. 12, 14, and 17), and its altitude is about 7 km: the layers with near-zero ZDRC measurements dominate above the FL, indicating the dominant dry snow aggregates (ZHC<30 dBZ); ρHV is relatively weaker (<0.98) below the FL, indicating the dominant mix-phase particles in the ML (near 6 km). In addition, the sustaining large KDP (>1 ∘ km-1) upon WL and HJ (Figs. 12 and 14) after 18:00 UTC, 9 August 2019, (after landfall) indicates the high concentration of solid particles above the FL. In addition, the significant upward extension of ZHC (>40 dBZ) and ZDRC (>1 dB) columns marked with black rectangles indicate the developing convective storms; the black ellipses indicate potential updrafts coupled with the storm; the blue ellipses indicate subsiding signatures of falling solid particles deducing from gradual decreasing heights of ρHV<0.98 over time. The convective storms are accompanied by abundant water content, as indicated by significant KDP (>0.5∘ km-1) columns extending upwards, which benefited the size increases of the falling solid particles. The microphysical processes of the solid particles differed at each station.

The WZ-SPOL radar initially measured similar ZDR but larger ZH and KDP compared with DSD at the WL station (rectangle 1 in Fig. 10) before the landfall of Lekima, and more concentrated hydrometeors aloft accompanying the updrafts compared to the surface in this process account for this phenomenon, which is verified in the first rectangle of Fig. 12a and c. Furthermore, two consecutive severe updrafts passed over WL, one from the outer eyewall and the other from the inner eyewall, causing the significant upward extension of ZHC, ZDRC, and KDP columns below the FL, as depicted in two black rectangles in Fig. 12. As illustrated in two black ellipses, some ice particles might ascend with the first updraft, then fall and melt in the time gap between two updrafts, with the signature of ρHV<0.98 reaching the lowest layer of 1.8 km; they instantly suffered from another size increase process confronting the second updraft (in the second ellipse) and then fell with the subsiding signals of ρHV and ZDRC (in the blue ellipse): ZDRC<0.5 dB was sustained when ρHV gradually transitioned from ρHV<0.84 around the FL to ρHV<0.98 on the lowest layer, indicating the existence of some near-spherical but mixed-phase particles during this falling process. These solid particles partly account for the larger DSD-derived ZDR near the surface than the WZ-SPOL radar (rectangle 2 in Fig. 10), but the coalescence of raindrops might also partly account for this DSD-derived larger ZDR.

Similar updrafts occurred upon the YH station (rectangle 3 in Fig. 10), and the WZ-SPOL radar measured similar KDP but weaker ZH and ZDR compared with DSD before the hail/graupel processing. Featuring with similar ZHC extending upwards upon the YH station, large ZDRC (>1.2 dB) and weak KDP (<0.2 ∘ km-1) accompanied the updrafts with ρHV<0.98 in the black ellipse of Fig. 13, indicating that dominant large-sized mixed-phase particles were developing around the ML. Then, in the blue ellipse, the subsiding signals of ρHV<0.84 formed in Fig. 13d after 16:30 UTC and tended to decrease their heights over time; finally, they transitioned to ρHV>0.98 on the top of the ZDRC (>2 dB) columns, attributing to the transformation of melting solid particles into big raindrops. Compared with surface DSD, the decrease in radar-measured ZHC and KDP reflects the reduction of LWC in the vertical gap; this LWC reduction did not contribute to the size increase of drops, because radar and DSD presented similar ZDR. Another possible explanation is that some LWC is absorbed by the falling solid particles, contributing to the filtered ZDR part in the hail/graupel processing.

Another solid particle falling occurred upon HJ, which is to the north of the landfall positions of Lekima. Even with the unnoticeable upward ZHC enhancement between 17:00 and 18:00 UTC, 9 August 2019, as depicted in Fig. 14a, the large ZDRC signals of the ML in Fig. 14b diminished due to the updraft, ZDRC and KDP both increased, and ρHV reduced steadily after 18:00 UTC above the FL in the black ellipses of Fig. 13b–d. Subsiding signals of ρHV<0.84 also emerged after 18:00 UTC, resulting in the ρHV reduction from 0.98 to 0.96 on the lowest layer. Conditioning VM by [0.5VB, 1.5VB] eliminated some size–velocity pairs of the solid particles at HJ, because solid precipitation particles have smaller terminal velocities than liquid particles. Conversely, the rising overestimation of RQC by reconditioning VM by [0.4VB, 1.5 VB] in Fig. 6b (the dotted green line) further verified this possibility.

These common characteristics feature in HJ, DT, LH, and XJ in Figs. 14–17: KDP (<0.5 ∘ km-1) above the FL indicated a lower concentration of ice particles upon DT, LH, and XJ than upon the other three sites, which refrains the size increase of falling solid particles through the aggregation process; the insignificant ZHC (<45 dBZ) and KDP (<0.2 ∘ km-1) extending upwards reflect the relatively low concentration of hydrometeors below the ML upon HJ, LH, and XJ, which refrains the further size increases of melting ice particles in the warm rain environment. The exceptions in ZDRC columns upon DT between 18:00 and 19:00 UTC in Fig. 15b were attributed to the falling melting ice particles upon an updraft with high LWC (KDP>1∘ in Fig. 15c); those in LH between 18:00 and 19:00 UTC were attributed to the sustaining weak updrafts (ZHC <45dBZ) but more concentrated ice particles above the FL. The deep ML (ρHV<0.98) also features these stations, and this signature even extends down to the lowest layer of LH and HJ with ZDRC>2 dB dwelling below the FL. In addition, most ice particles upon these four stations might have melted in the air before being collected by disdrometers near the surface, which effectively accounts for the small rainfall differences between disdrometers and rain gauges. However, the residual differences between radar and DSD are mainly related to the warm process of raindrops below the ML.

The deviation of ZDRC from Z^DR is a non-negligible phenomenon during landfall of Lekima: underestimated ZDRC in Fig. 11i–l compared with Z^DR in Fig. 11m–p emerged in areas with significant KDP in Fig. 11q–t, which simultaneously emerged around the GWS of YDM. Apparently, ZDRC cannot completely approximate Z^DR after correction; intrinsically, the microphysical composition issue, either dominant large-sized or small-sized raindrops filling in radar volume gates, resultantly determines final ZDRC–ZHC distribution. One typical radial range profile in Fig. 18 is detailed to clarify this phenomenon. The ellipse-surrounded storm area contributes the most attenuation and differential attenuation with maximum ΔZH =7.9 dBZ and ΔZDR = 0.645 dB, respectively, in Fig. 18a and b. Although the correction can result in enhanced consistency between ZH and KDP (see Sect. 3.1.2) and some ZDRC have indeed partly approximated well to Z^DR (outside the ellipse in Fig. 18b), the other ZDRC within the storm (in the ellipse) still have a residual ZDR bias of about -1 dB. Additionally, ρHV ranging from 0.99 to 1 (in Fig. 15c) further indicates the dominance of pure liquid precipitation; high LWC and Nw can be deduced from Eqs. (8) and (9) (KDP≈3.5 ∘ km-1; ΔΦDP ≈ 68.5∘ in Fig. 12c). ZH is a composite integral of hydrometeors with different sizes and number concentrations, and ZDR is sensitive to hydrometeor size; therefore, high concentrations of small-sized drops rather than large-sized drops contribute more to radar-measured ZHC in radar volume gates. This unique composition resultantly causes an overestimated Z^DR estimated by ZHC, or conversely, underestimated ZDRC compared with Z^DR.

The hydrometeor size sorting (HSS) partly accounts for the position inconsistency between ZHC and ZDRC, and it is significant in the rectangle-surrounded area of the inner eyewall, characterized by a maximum of ZDRC in Fig. 12i on the significant upwind gradients of KDP in Fig. 12q (Homeyer et al., 2021; Hu et al., 2020). Since ZHC in Fig. 12e and KDP in Fig. 12q are consistent with each other, the large Z^DR estimated by ZHC also horizontally deviates from the area with large ZDRC. Differential sedimentation of hydrometeors of various sizes is the intrinsic reason for HSS (Feng and Bell, 2019), which is significant in the outer eyewall. The higher LWC (>3 g m-3) features the outer eyewall as depicted in Fig. 16e; the area with large ZDRC (>2 dB) consists of dominant larger-sized drops with D0>1.8 mm in Fig. 16a, but relatively lower concentration with log10(Nw) > 4.4 in Fig. 16a than in its downwind area. Meanwhile, lower LWC (< 2 g m-3) features a cyclonical downwind area, but this area consists of dominant higher concentrated (log10(Nw) > 4.4) small-sized drops (D0<1.625 mm). However, HSS cannot account for the overall underestimation of ZDRC when pixels with large ZHC, ZDRC, and KDP coincide.

The collision process in warm rain has three probable colliding outcomes: bounce, coalescence, and breakup. In one volume gate, bounce cannot change raindrop size and concentration; coalescence boosts the size increase, but breakup increases the concentration. The existence of large raindrops with D0>1.8 mm around the GWS of YDM (in Fig. 19b and c) indeed back the occurrences of collision–coalescence processes, which corresponds to ZDRC (>1.8 dB in Fig. 11j and k). However, if the size increases contribute enough in one volume gate, ZDRC might have well-approximated Z^DR in the storm area and agree better with ZHC. In addition, raindrops cannot continue increasing their size; spontaneous breakup (Srivastava, 1971) or collision–breakup due to vertical wind shear (i.e., Deng et al., 2019) co-occurs during the falling process of drops:

i.

The first evidence comes from the radar-measured ZDRC–ZHC scattergram in Fig. 11d, and it tends to be more consistent with ZDR–ZH relationships dominated by small-sized drops related to the breakup, not large-sized drops related to the coalescence. This also agrees with the simulation results in Kumjian and Prat (2014).

The collision–coalescence indeed occurs, but the breakup balances the size increase. This is evident in the evolutions of D0 and Nw constrained by a given LWC, which is typical around the GWS of YDM. In Fig. 19c, g, and k, the identical LWC fill in the rectangle-surrounded and ellipse-surrounded regions; the latter exhibits larger D0 (>1.75 mm) but lower Nw with log10(Nw) <4.4; conversely, the former shows smaller D0 (<1.75 mm) but higher Nw with log10 (Nw) >4.4. Similar situations occurred in the two left columns in Fig. 19, and sparse large-sized D0 is only prominent in a small area (in ellipse and rectangle); high Nw but small D0 are features of the other parts of the typhoon. The LWC in one range gate will contribute not only to the size increase but also to the concentration, attributing to the balance between coalescence and breakup.

Combining the abovementioned observations, the overwhelming breakup of large-sized drops over coalescence firmly restrains the magnitudes of radar-measured ZDRC for a given ZHC, accounting for the noticeable deviation of ZDRC from Z^DR (in Fig. 11). Despite all this, collision–coalescence accompanied by the terrain-enhanced precipitation occurred when Lekima took high LWC (>2 g m-3) and passed over YDM, as depicted in Fig. 19f and g, resulting in an overall LWC reduction around the GWS of KCM (i.e., Fig. 19g to h). During this period, D0 and Nw simultaneously increased: D0 increased by about 0.5 mm from Fig. 19a to D0>1.5 mm in Fig. 19c; log10 (Nw) increased about 0.4–0.8 from Fig. 19i to k, and these enhancements coincided well with the GWS of YDM. The gradual but insignificant enhancement persisted around the GWS of KCM, including an LWC increase by about 1 g m-3 (i.e., Fig. 19e–h), a diameter transition from D0<1.25 mm to D0>1.5 mm (i.e., Fig. 19a–d), and growth of log10 (Nw) about 0.4 in sparse pixels (i.e., Fig. 19i–l), but this enhancement was relatively weaker than that around the GWS of YDM. This comparison indicates that extensive large-sized drops had formed and fallen around the GWS of YDM before Lekima moved to the north, which effectively accounts for the flood disasters. However, the utilization of radar-measured ZDRC may not derive accurate radar rainfall fields.

Utilizing the DSD dataset from S0, three primary radar rainfall rate relationships for R(ZH), R(KDP), and R(ZH, ZDR) are respectively established as 12aRZH=0.0544×ZH0.608,12bRKDP=45.0484×KDP0.7679,12cRZH,ZDR=0.0086×ZH0.9153ZDR-3.8606 based on the standard weighted least squares nonlinear fitting method and DSD-derived radar variables (depicted in Fig. 20). In addition, ZDRM, ZDRC, and Z^DR are integrated with ZH to exploit the impacts of the above-mentioned microphysical process on radar QPE algorithms. The pixel-to-pixel linear average accumulation scheme is utilized to retrieve radar 6 h rainfall fields for these radar QPE estimators and is then evaluated independently by comparing gauge 6 h rainfall measurements through the absolute normalized mean error (ENMA), root mean square error (ERMS), and correlation coefficient (ECC) as 13aENMA=∑i=1n|ri-gi|∑i=1ngi×100%13bERMS=1n∑i=1n(ri-gi)213cECC=∑i=1n(ri-r‾)(gi-g‾)∑i=1n(ri-r‾)2∑i=1n(gi-g‾)2, where ri and gi refer to radar rainfall estimates and gauge rainfall. The 6 h radar rainfall fields retrieved by R(ZHM), R(ZHC), R(KDP), R(ZHM, ZDRM), R(ZHC, ZDRC), and R(ZHC, Z^DR) are derived in Fig. 21, as well as the scattergram between radar rainfall estimates and gauge rainfall measurements, depicted in Fig. 22, to reveal their practical performances around the disaster area.

R(ZHM) presents lower rainfall estimates in Fig. 21a than the other radar rainfall estimators in Fig. 21b–f, although they have similar rainfall center shapes. In terms of statistical scores in Table 1, R(ZHM) does not perform the worst among all radar rainfall estimators. Its ERMS, ENMA, and ECC even outperform R(ZHM, ZDRM) by 57 %, 31.6 % and 7.9 %, and outperform R(ZHC, ZDRC) by 63.8 %, 34.9 % and 6 %, respectively. However, its underestimation can easily be perceived from the scatters in Fig. 22a when rainfall recordings exceed 100 mm in the center rainfall area. This phenomenon can be ascribed to the attenuation on ZHM caused by the highly concentrated hydrometeors in the storm during the landfall of Lekima.

In contrast, R(ZHC) in Fig. 21b presents higher rainfall estimates and R(ZHC) mainly overestimates, since more scatters are distributed above the diagonal line (y=x) as depicted in Fig. 22b, and its ECC outperforms that of R(ZHM) by 4.2 %, even with larger ENMA and ENMA scores. The overestimation of R(ZHC) in the rainfall center area conversely demonstrates the effectiveness of the attenuation correction based on the ZPHI approach, because the same R(ZH) relationship is utilized for the rainfall retrieval; the only difference is the replacement of ZHM with ZHC.

R(KDP) in Fig. 21c presents a similar rainfall field structure to R(ZHC). The scores of R(KDP) are just a little superior to that of R(ZHC) in Table 1, with its ERMS, ENMA, and ECC outperforming that of R(ZHC) by 3.1 %, 3.2 %, and 0.5 %, respectively. The scattergrams in Fig. 22b and c are also similar to each other, indicating that R(KDP) and R(ZHC) both overestimate, although R(KDP) is less overestimated when rainfall recordings are less than 100 mm. Their similar performances can be attributed to the consistency between radar-measured KDP and ZHC measurements as described in Sect. 3.1.2.

R(ZHM, ZDRM) and R(ZHC, ZDRC) in Fig. 21d and e both present significantly higher estimates in the rainfall center area than the others, which results in severe overestimation according to the scattergrams in Fig. 22d and e. Furthermore, R(ZHC, ZDRC) obtains the worst ERMS and ENMA scores of all radar rainfall estimators, and this can be explained based on the ZH-related and ZDR-related calculation items as demonstrated in Fig. 23: ZHC obtains much higher rainfall estimates through ZH-related items than ZHM. However, the calculation needs to be further adjusted through the ZDR-related item: the larger ZDR measurements correspond to fewer final rainfall estimates. A -0.5 dB ZDR bias could result in relatively less rainfall adjustment, according to Fig. 23. The attenuation effects on ZDRM make the corresponding rainfall calculation less adjusted, which can effectively account for the overestimation of R(ZHM, ZDRM). However, the correction cannot make ZDRC completely consistent with ZHC, but it is underestimated, as demonstrated in Sect. 3.1.2, which is related to the dynamic microphysical process described in Sect. 3.2.

The spatial texture of R(ZHC, Z^DR) in Fig. 21f presents slightly fewer rainfall estimates than R(ZHC) and R(KDP) in Fig. 21b and c, and the scattergram in Fig. 22f shows that R(ZHC, Z^DR) agrees better with the gauge rainfall than in Fig. 22b and c. R(ZHC, Z^DR) effectively reduces the overestimates and is obviously superior to R(ZHM, ZDRM) and R(ZHC, ZDRC). The ERMS/ENMA score of R(ZHC, Z^DR) is better than R(ZHC) and R(KDP), by 8.6%/5% and 5.7%/1.8%, respectively, although its ECC score is slightly worse by 0.2 % and 0.3 %. The superiority of R(ZHC, Z^DR) can also be apparently attributed to the incorporation of Z^DR. Z^DR is not a real radar measurement; it is directly estimated from ZHC from the theoretical DSD-derived ZDR–ZH relationship in Eq. (2a). Z^DR is naturally self-consistent with ZHC and KDP, since ZHC and KDP have agreed well with their DSD-derived counterparts regarding the KDP–ZH distributions and pixel-to-pixel comparisons in Sect. 3.1. The utilization of the DSD-derived ZDR–ZH relationship intrinsically assumes that composition in radar volume gates has a similar size and concentration to its surface counterparts; therefore, Z^DR can be seen as an equivalent radar variable. The replacement of ZDRC with Z^DR is also equivalent to imposing the surface raindrop size and concentration on radar measurements. The relatively larger Z^DR than ZDRC means a more significant adjustment can be performed for rainfall estimation using R(ZHC, Z^DR), according to Fig. 23, and this also indicates that the anticipated giant raindrops had fallen around the GWS of YDM. Except for the simultaneous D0 and Nw increases, the following alternative Z^DR indirectly verifies the dominant collision–coalescence around this area.

The consistency between radar and surface measurements is critical for radar QPE algorithms, but the microphysical process in the vertical gap between air and surface may worsen the practical performances of radar QPE. This is the case around the GWS of YDM: the primary outcome of the collision process transitions from a dominant breakup in the air to dominant coalescence near the surface due to the topographical enhancement. Using radar measurements on the lowest elevation angle to retrieve radar QPE implicitly assumes that they are representative of surface precipitation, but they are not in this situation; only R(ZH), R(KDP), and R(ZH, ZDR) relationships established based on the DSD dataset represent the feedback near the surface. Although R(ZHC, Z^DR) performs best, Z^DR can also be estimated by Eqs. (11a)–(11c). However, ZDRC changes little if a smaller or larger Z^DR estimated by Eqs. (11a)–(11c) is imposed in the correction procedure, as ZDRC–ZHC scattergrams shown in Fig. 24a–c. Furthermore, the corresponding three R(ZH, ZDR) relationships based on SI–SIII can be established as 14aRZH,ZDR=0.0088×ZH0.917ZDR-3.9203DSD∈SI,14bRZH,ZDR=0.0085×ZH0.9222ZDR-4.0371DSD∈SII,14cRZH,ZDR=0.0078×ZH0.9342ZDR-4.2321DSD∈SIII. The alternative utilization of SI–SIII slightly changes the parameters of R(ZH, ZDR) with minor rainfall rate differences estimated by Eqs. (12c), (14a)–(14c). However, the impacts on Z^DR are non-negligible, particularly for a given ZHC exceeding 35 dBZ, and smaller Z^DR means weaker adjustment for the ZHL-related item, as depicted in Fig. 23.

As in the analysis in Sect. 3.1.2, radar-measured ZDRC–ZHC in volume gates tend to be more consistent with SI and SII, because breakup overwhelms in the coalescence–breakup balance, so if breakup still dominates when these drops further fall on the ground, R(ZHC, Z^DR) estimated by Eqs. (14a) and (14b) should perform better than that estimated by Eq. (12c). However, in reality, their spatial fields in Fig. 25a and b and scattergrams in Fig. 26a and b conversely present a similar overestimation as R(ZHM, ZDRM) and R(ZHC, ZDRC), which contradict the anticipated results. Such a contradiction means Z^DR estimated by Eqs. (11a) and (11b) is not representative enough for surface precipitation. In contrast, R(ZHC, Z^DR) in Fig. 25c shows even lower rainfall estimates than that in Fig. 21f (obtained through Eq. 12c based on S0), which can also be seen by comparing the scattergrams in Figs. 26c and 22f. In addition, when large-sized drops are gradually excluded from the DSD dataset for Z^DR and R(ZHC, Z^DR), ECC changes little in Table 2, whereas ERMS and ENMA both exhibit a monotonic increasing tendency, implying the non-negligible contribution of large-sized drops around the GWS of YDM.

The dominant breakup in the air but dominant coalescence around the GWS of YDM can be ascribed to the overshooting of radar beams and the topographical enhancement. In this sense, the utilization of Z^DR instead of ZDRC equals a physical conversion of breakup-dominated outcome in one volume gate for a given ZHC into their coalescence-dominated counterparts in an average sense. In this conversion process, consistency between radar-measured ZHC and KDP in the air and the surface counterparts (DSD-derived KDP and ZH) has been achieved, as indicated in Sect. 3.1.2, demonstrating the mass conservation characteristics of falling drops. Therefore, radar-measured ZHC and KDP around the GWS of YDM may change insignificantly, which makes it conducive for R(ZHC, Z^DR) to obtain a better radar rainfall field.