The representation of deep convection at cloud-resolving and global climate model scales (CRMs and GCMs) remains a serious challenge (Lin et al., 2006; Jakob, 2010). Part of the challenge can be attributed to the lack of comprehensive observations of dynamics and microphysics in these vigorous cloud systems (Ferrier, 1994; Milbrandt and Yau, 2005; Mrowiec et al., 2012; Donner et al., 2016). In particular, cloud dynamical insights may provide necessary guidance for improving these simulations at convection allowing scales and act as a basis for improving convective parameterizations at GCM scales (Lang et al., 2007; Wu et al., 2009; Nicol et al., 2015).

Despite the importance of vertical velocity measurements in deep convection, such measurements are difficult to acquire. Aircraft penetration of convective clouds offers the most direct method to measure these vertical air motions (Lenschow, 1976). However, practical hazards and operational costs have resulted in a valuable, but limited, dataset (e.g., Byers and Braham, 1948; LeMone and Zipser, 1980; Donner et al., 2001). Recent studies using profiling Doppler radars have suggested an ability to retrieve vertical velocities in convective clouds with an uncertainty of the order of 1–2 m s-1, thus offering a viable substitute for in situ aircraft measurements (Jorgensen and LeMone, 1989; Cifelli and Rutledge, 1994; May and Rajopadhyaya, 1999; Williams, 2012; Heymsfield et al., 2010; Giangrande et al., 2013a; Kumar et al., 2015; Giangrande et al., 2016). Furthermore, profiling radars provide a high degree of detail of convective clouds in time and height, and can sample even the most intense convective cores. However, profiling radars potentially have a limited role for direct (rather than statistical) cloud model constraint due to their narrow view of these large, three-dimensional systems.

Scanning Doppler radars and the use of multi-Doppler retrieval techniques may help overcome known in situ aircraft and profiling radar sampling limitations. In addition to improved spatial representation of deep convection, this measurement approach offers an ability to document the three-dimensional structure of updrafts and downdrafts. Unfortunately, multi-Doppler retrieval applications are not straightforward. First, the number of radars in the network and their respective locations has a direct impact on the retrieval quality. Second, distributed Doppler radar networks, including mobile radar deployments, are not widely available or standardized. Operational radar networks often provide inadequate coverage throughout the depth of deep convective clouds, particularly at cloud top, necessary to constrain traditional vertical velocity retrievals at higher altitudes. Finally, since retrieval methodologies evolve, it is often difficult to establish a consensus pick among the many versions of multi-Doppler retrieval techniques that have been proposed.

In simpler terms, these retrieval techniques can be categorized as either “iterative” or “simultaneous”, based on their treatment of mass continuity. Iterative techniques solve the integral mass continuity equation throughout the column, contingent on known vertical velocity boundary conditions at the bottom level (i.e., upwards integration) and/or top level (i.e., downwards integration) (e.g., O'Brien, 1970; Ray et al., 1980; Protat and Zawadzki, 1999). This requires an estimate of horizontal wind divergence at each level made in a previous step, hence the non-simultaneity of iterative techniques (Dowell and Shapiro, 2003; Potvin et al., 2012a). By their nature, iterative upwards/downwards integration techniques propagate information in one direction, thus errors in horizontal wind divergence accumulate throughout the column, which in turn leads to larger errors in vertical velocity (e.g., Ray et al., 1980).

In contrast, simultaneous techniques treat mass continuity similar to other analysis constraints by inserting it directly into the cost function. This avoids accumulation of errors throughout the column since mass continuity is analyzed everywhere simultaneously. Moreover, these techniques are known to mitigate retrieval instabilities in poorly constrained regions like the dual-Doppler radar baseline (Bousquet and Chong, 1998; Dowell and Shapiro, 2003). Since simultaneous techniques are by definition 3DVAR techniques, we will refer to them as such throughout the remainder of this study. The 3DVAR approach has previously been shown to provide more accurate dual-Doppler retrievals than traditional techniques in observing system simulation experiments (OSSEs; Gao et al., 1999; Potvin et al., 2012a).

Several studies have investigated multi-Doppler wind retrieval uncertainties by identifying or utilizing (i) the importance of Doppler radar measurement errors and beam geometry (e.g., Doviak et al., 1976; Nelson and Brown, 1987; Matejka and Bartels, 1998; Bousquet et al., 2008), (ii) the influence of radar data objective analysis (e.g., Clark et al., 1980; Gal-Chen, 1982; Testud and Chong, 1983; Chong et al., 1983; Given and Ray, 1994; Majcen et al., 2008; Shapiro et al., 2010; Collis et al., 2010), and (iii) OSSEs (e.g., Fanyou and Jietai, 1994; Gao et al., 1999; Liou and Chang, 2009; Potvin et al., 2012b; Potvin and Wicker, 2012). However, few studies have compared practical retrieval performance to other independent air motion estimates from aircraft or ground-based profiling radars (e.g., Collis et al., 2013; Newsom et al., 2014). Collis et al. (2013) recently compared iterative dual-Doppler wind retrievals in tropical convection with those from a collocated dual-frequency wind profiler. However, because of the suboptimal location of the wind profiler near the dual-Doppler baseline, several assumptions had to be made before evaluating the two datasets.

While 3DVAR wind retrievals have been studied using OSSEs, an implementation, verification and sensitivity analysis using independent datasets from actual observations is noticeably missing. The US Department of Energy (DOE) Atmospheric Radiation Measurement (ARM) program provides an excellent opportunity to investigate the benefits and relevant issues associated with multi-Doppler wind retrievals over the Southern Great Plains (SGP) in Oklahoma (Mather and Voyles, 2012). During the Midlatitude Continental Convective Clouds Experiment (MC3E), a joint field campaign between the DOE ARM program and the National Aeronautics and Space Administration (NASA) Global Precipitation Measurement (GPM) mission Ground Validation (GV) program (Jensen et al., 2016), the SGP site collected unique datasets from a distributed scanning Doppler radar network and radar wind profilers (RWPs) for several deep convective systems. These 3-D wind fields are strongly desired to analyze structures and characteristics of convective events (e.g., Liu et al., 2015; Donner et al., 2016). This study applies the 3DVAR radar wind retrieval to the MC3E deep convective events over the ARM SGP site and presents optimization of constraint weights used in the cost function. These 3DVAR retrievals are validated using data from the RWPs and compared with an iterative upwards integration technique.

This paper is organized as follows. A description of the dataset and radar data processing is presented in Sect. 2. Section 3 provides background information for 3DVAR wind retrievals from multiple scanning Doppler radars. Retrieval sensitivity and a method for producing physically sound wind fields is discussed in Sect. 4. Sections 5 and 6 presents 3DVAR wind retrieval results in the context of how they compare with those from independent collocated radar wind profilers as well as with an iterative upwards integration method. Section 7 is reserved for summary and concluding remarks.

The MC3E took place during April–June 2011 in northern Oklahoma and surrounding states. A total of five events from MC3E were analyzed for this study and are listed in Table 1. These events represent a variety of warm-season convection over Oklahoma, including nocturnal elevated convection (25 April 2011), widespread stratiform precipitation with embedded convection (11 May 2011), mesoscale convective system (MCS) and associated squall line (20 May 2011), and isolated severe supercell thunderstorms (23–24 May 2011). The approximate time frame defining each event reflects profiling radar observations recorded at the SGP Central Facility (CF). The multi-Doppler radar wind retrieval in this study utilized plan position indicator (PPI) measurements operated by three X-band radars and one C-band radar from the ARM scanning precipitation radar network and one NEXRAD WSR-88D S-band radar. The ARM RWPs were used to evaluate these retrievals (Atmospheric Radiation Measurement (ARM) Climate Research Facility, 2011a, b, c). The environmental background wind fields are obtained from the ARM Merged Sounding value-added product that combines the observations from radiosonde soundings at the SGP CF (available every 3 h during MC3E), microwave radiometers, surface meteorological instruments, and European Centre for Medium Range Weather Forecasts (ECMWF) model output to produce a dataset at 1 min intervals and at 266 altitude levels (ARM, 1996).

Results presented in this study capitalize on the physical constraints of radial velocity observations (Jo), anelastic mass continuity (Jc), surface impermeability (Jp), background wind field (Jb), and spatial smoothness (Js). Assuming that u=[u1,…,un], v=[v1,…,vn], and w=[w1,…,wn] are the eastward, northward, and vertical wind components on the n-point analysis grid, respectively, we have the cost function J(u,v,w)=Jo+Jc+Jp+Jb+Js. The optimal wind field solution is at the (global) minimum of J which implies that the gradient of J with respect to u, v, and w vanishes. For applications requiring large-scale (e.g., 1×106 variables) nonlinear cost functional minimization, it is often necessary to use an iterative conjugate-gradient algorithm (Navon and Legler, 1987). In Gao et al. (1999), where a similar cost function and conjugate-gradient minimization algorithm were used, u and v were found to be well recovered within the first 50 minimization iterations; however, w lacked both coherency and strength until 200+ iterations. We use these values as a reference point for the minimum number of iterations required to minimize Eq. (2).

Typically, the constraint weight matrices (Λ) found in Eqs. (5), (8), and (9) are treated as adjustable parameters (λ), controlling the degree to which each constraint influences the final solution. In essence, the values prescribed to each weight are often determined through trial and error (e.g., Gao et al., 1999). Fundamentally, there exists a range of values for each weight that produces a physically sound wind field. A thorough sensitivity analysis could be used to determine this parameter space, but this is often ignored because studies typically consider theoretical wind retrieval performance by comparing it to a known truth field (e.g., model output in an OSSE). The weights optimized to minimize the residual error between the retrieved and truth wind fields are then adopted (e.g., Gao et al., 1999; Potvin et al., 2012a). For applications involving real radar datasets where no truth field is available, one must consider (i) determining the parameter space which produces physically sound wind fields and (ii) characterizing the solution spread within the parameter space determined by (i).

This section addresses these two points through an extensive sensitivity analysis within the experimental domain indicated by the dashed blue box in Fig. 1. This domain has the same 250 m grid spacing and vertical extent as the larger domain, but covers a smaller horizontal area of 20 km × 20 km. Utilizing a smaller domain for the sensitivity analysis reduces processing time and allows for the isolation of specific cloud type regimes (e.g., convective versus stratiform). Since convective air motion retrievals are the primary interest of this study, the sensitivity analysis was done during a time when intense convection filled the experimental domain on 23 May 2011, using scanning radar observations valid between 22:36 and 22:43 UTC.

Point (i) is addressed by answering the following two questions. The first is, how well does the wind retrieval satisfy radial velocity observations? The second is, how well does the wind retrieval satisfy anelastic mass continuity? The second question is particularly important in the context of numerical modeling and convective parameterizations.

The wind retrieval is said to satisfy the radial velocity observations of one or more radars if the RMSD between the retrieval and observation is within the uncertainty estimate of the observations themselves. Since it is impractical to account for all sources of error inherent in mapped radial velocity observations, we establish a range of uncertainty and require the RMSD to be within this range. We employed radial velocity measurement error of approximately 0.5 m s-1, which is a common value for regions of low signal-to-noise ratio (∼ 20 dB; Fang et al., 2004) and larger Doppler spectrum width (Doviak and Zrnić, 1993; Bringi and Chandrasekar, 2001). The additional uncertainty introduced when mapping irregular radial velocity data to a regular grid is estimated to be of the order of 1 m s-1. Therefore, we consider the wind field to satisfy radial velocity observations if it produces a RMSD with one or more radars within 0.5–1.5 m s-1, computed over the entire analysis domain. To determine the degree to which the wind field satisfies anelastic mass continuity, following Shapiro et al. (2009), we define the normalized mass continuity residual (NMCR) as NMCRi=Di2wiρiair∂ρiair∂z2+∂ui∂x2+∂vi∂y2+∂wi∂z2-1∀i=1,…,n, where Di is given by Eq. (6). As NMCR approaches zero, anelastic mass continuity becomes perfectly satisfied. However, this is not necessarily desirable since this condition is not exactly satisfied in nature, and even if it were, discretization errors would prevent precise satisfaction of Eq. (6). Therefore, we propose a range for NMCR, averaged over the entire analysis domain, between 1 and 10 %, whereby anelastic mass continuity is said to be adequately satisfied.

The response of CSAPR-I7 ṽr RMSD and NMCR to perturbing multiple constraint weights is analyzed. The results are shown in Fig. 4. We first discuss the impact of the continuity and background weights (Fig. 4a–b). What is immediately evident in Fig. 4a is the strong dependence of ṽr RMSD on λb, with little to no dependence on λc. Even with only a factor of 2 increase in λb, the wind retrieval diverges substantially from the radial velocity observations and converges towards the background wind field. As λb→0.5, CSAPR-I7 ṽr RMSD approaches the specified upper limit of 1.5 m s-1. This is important to note since 3DVAR retrievals have been found to be relatively insensitive to minor changes (e.g., not orders of magnitude) in other constraint weights (e.g., Gao et al., 1999; Potvin et al., 2012a). However, in Fig. 4b, λb has a decreased effect on the degree to which the wind retrieval satisfies mass continuity. As expected, this is primarily controlled by λc, not only within the continuity-background parameter space but also in the continuity-smoothness parameter space shown in Fig. 4c–f. NMCR is particularly sensitive to λc when λc < 250. Outside of this range, NMCR is generally more stable with respect to λc and NMCR is typically less than 20 %. However, as seen in the right column of Fig. 4, in order to obtain NMCR ≤ 5 %, λc must generally be 500 or larger.

Unlike the continuity-background sensitivity analysis, both RMSD and NMCR metrics appear highly unstable in certain regions of the continuity-smoothness parameter spaces investigated in Fig. 4c–f. For λsu and λsv, which control the degree of smoothing of the horizontal wind components in Eq. (9), CSAPR-I7 ṽr RMSD becomes unstable as these two weights approach values of 400 and larger. A similar phenomenon occurs for NMCR in Fig. 4d. These highly unstable regions of the parameter space are likely the result of nonlinear effects introduced by the squared second-order partial derivatives defined in Js and should be avoided altogether. For values of λsu and λsv below approximately 100, CSAPR-I7 ṽ RMSD is within 1.5 m s-1 and relatively stable. However, the parameter space in which this holds true gradually shrinks as λc increases towards 1000. Mass continuity is also adequately satisfied for λsu=λsv < 100 and λc > 250, with NMCR typically less than 10 %. Results for λsw are similar to those of λsu and λsv except for one aspect. Since λsw controls the degree of smoothing of the vertical wind component in Js, it has little influence on CSAPR-I7 ṽr RMSD since the vertical wind component is generally not well sampled by scanning radars. This manifests itself in Fig. 4e, which shows CSAPR-I7 ṽr RMSD to have much less dependence on λsw compared to λsu and λsv.

Each panel in Fig. 4 contains over 2000 wind field realizations, each of which was concurrently saved. Therefore, we compute the 3DVAR vertical velocity solution spread from these thousands of realizations, allowing us to address point (ii) above. The ranges of the optimized λ values derived from this sensitivity analysis and λ values used for the retrieval case studies are recorded in Table 3. It is found that within the range of constraint weights defined in the analysis column of Table 3, the vertical velocity solution spread is relatively narrow at 1.5 m s-1. This provides a form of uncertainty estimate for the 3DVAR wind retrievals presented in this study. It follows that we expect the 3DVAR vertical velocity retrievals to be relatively stable over a large range of constraint weights, with an uncertainty estimate of the order of 1–2 m s-1.

This section investigates the benefits of the 3DVAR approach for convective events as compared to an iterative upwards integration technique. First, a similar sensitivity analysis was performed for the iterative upwards integration technique, the results of which are shown in Fig. 5. Similar to the 3DVAR results in Fig. 4a, CSAPR-I7 ṽr RMSD is highly dependent on λb and less so on λc. However, for continuity-background parameter spaces where λc>5 or λo>0.5, there are sharp increases in ṽr RMSD exceeding approximately 2.5 m s-1, well outside the 1.5 m s-1 upper limit. In fact, the parameter space in which ṽr RMSD is below 1.5 m s-1 is very small (approximately λc<1 and λo<0.1), and when looked at together with the normalized mass continuity residual (NMCR), no continuity-background parameter space exists in which both metrics are reasonably satisfied for an iterative upwards integration technique. It is worth noting that as λc is increased, NMCR appears to approach an asymptote around a value between 10 and 15 %. This indicates that even in the parameter space where radial velocity observations are effectively ignored (e.g., ṽr RMSD greater than 3 m s-1), iterative upwards integration techniques still have difficulty properly satisfying mass continuity. Results are also poor for the continuity-smoothness sensitivity analysis, in particular they were more unstable, and therefore they are not shown.

Next, we compare the actual wind fields retrieved by both techniques to determine if the difference found in the sensitivity analysis impacts the wind retrievals. The RMSD of radial velocity and the NMCR are estimated from the 3DVAR and an iterative upward integration method for the five cases at times when a strong convective region passed over the SGP CF for a 20 km × 20 km domain centered around the SGP CF, and the result is listed in Table 4. The 3DVAR technique provides lower NMCR and radial velocity RMSD values than the upward integration technique for the five cases. We note that the radial velocity RMSD values for the upward integration technique from the 25 April and 11 May cases are very low, close to the values found with 3DVAR techniques. These two events are nocturnal elevated convection (25 April) and widespread stratiform precipitation with embedded convection (11 May), respectively. Both cases included narrow or weaker convective regions, and exhibit propagation speeds that are slower than in the remaining MC3E cases featuring isolated severe convective cells and organized MCS events. The results suggest that the upward integration technique is comparable to the 3DVAR approach for the two cases where the mass continuity would be satisfied, whereas the 3DVAR technique demonstrates an advantage for the severe convective events.

Detailed comparisons of the retrieved wind fields from the two techniques are performed using the squall line event on 20 May 2011 at 10:40 UTC. This event featured the largest areas of strong convection for this MC3E dataset and substantial surface wind convergence ahead of the convective line that was well sampled by the scanning radar network. Strong wind convergence at or near the surface was indirectly observed around 10:40 UTC by UAZR-C1 as strong upwards motion lasting close to 5 min. The upwards integration technique used here is ideal for surface-driven events since the horizontal wind divergence profile should be well defined, particularly near the lower boundary. Figure 6 presents the wind retrievals from these two retrieval techniques within the 20 km × 20 km area surrounding the CF. Both techniques retrieve similar wind convergence patterns and upwards motion at 1 km a.g.l., with the iterative upwards integral technique retrieving a slightly stronger convergence line near the surface and therefore enhanced upwards air motion near the surface. Both methods also satisfy radial velocity observations below 2 km a.g.l., as shown in Fig. 6f. As expected, there is a large discrepancy between the two retrieval techniques when it comes to satisfying mass continuity. At each analysis height in Fig. 6g, the 3DVAR retrieval is adequately satisfying mass continuity, with NMCR < 10 % at each height and NMCR = 5 % over the entire domain. For the iterative upwards integration retrieval, NMCR never gets below 30 % at any given height, and over the entire domain NMCR = 52 %. Differences in the vertical velocity field between these techniques becomes more pronounced with increasing altitude due in part to the iterative upwards integration retrieval not adequately satisfying mass continuity throughout the column. At 8 km a.g.l., the vertical velocity fields no longer exhibit similar spatial patterns or intensities.

The accumulation of differences with height is most evident in Fig. 6f. For the iterative upwards integration retrieval, CSAPR-I7 ṽr RMSD quickly grows larger than 2 m s-1 at heights above 5 km a.g.l., whereas the 3DVAR method is able to satisfy CSAPR-I7 radial velocity observations at almost all analysis levels. As a result, the large spread in vertical velocity differences between the two retrieval techniques shown in Fig. 6h primarily comes from analysis levels above 5 km a.g.l. Over the entire analysis domain, vertical velocity MBD, MAD, and RMSD are all large at 1.3, 5.8, and 7.7 m s-1, respectively. Below 5 km a.g.l., the MBD, MAD, and RMSD decrease substantially to 0.5 m s-1 (38 %), 2.6 m s-1 (45 %), and 3.9 m s-1 (51 %), respectively. Nonetheless, these still represent large differences in the wind fields retrieved by each technique. That the radar coverage near ground level is not sufficient for these retrievals (particularly, as viewed from the KVNX radar, Fig. 2b) could be an explanation for the unsatisfactory mass continuity behavior in the iterative upwards integral technique.

This study addresses the utilization of the ARM SGP scanning radar network in retrieving robust air motion estimates in five deep convective cloud events observed during MC3E, with the inclusion of surrounding NEXRAD WSR-88D assets. The ARM SGP site promises to provide a unique convective air motion dataset moving forward due to its continuous operation. Plans are currently being made for a second C-band scanning dual-polarization Doppler radar to be placed due south of the SGP CF, which would provide an additional constraint for these 3DVAR retrievals in the future.

First, this study optimized constraint weights of the cost function used in the 3DVAR multi-Doppler wind retrieval. The sensitivity analysis of the constraint weights suggests that the 3DVAR vertical velocity retrievals are relatively stable over a large range of constraint weights, with an uncertainty estimate of the order of 1–2 m s-1. The retrieved updrafts from 3DVAR retrievals are compared with an iterative upward integration technique and independent ARM wind profiler observations. Our comparisons in terms of the constraint weight sensitivity testing indicated that there is a large constraint weight parameter space in which 3DVAR retrievals are able to simultaneously satisfy radial velocity observations and mass continuity and give stable solutions for vertical velocity retrieval. This is in contrast to the iterative upwards integration technique that has difficulty properly satisfying mass continuity. These comparisons for the five MC3E cases suggested that the 3DVAR technique can produce smaller errors in updraft retrievals than the iterative upward integration technique. Particular emphasis for this improvement was on the severe convection events that included large areas of strong convection.

Additional focus on the squall line case on 20 May 2011 revealed that the two techniques retrieved similar vertical velocity spatial patterns, including a large region of upwards motion associated with the surface convergence zone. However, the magnitudes of the vertical velocities between the two methods were considerably different, with MAD and RMSD of the order of 3 and 4 m s-1, respectively. These large differences were likely caused by the iterative upwards integration technique inadequately satisfying mass continuity. In particular, the mean NMCR was 52 % for the iterative upwards integration retrieval compared to 5 % for the 3DVAR retrieval, and the radial velocity RMSD quickly grew larger than 2 m s-1 above 5 km a.g.l. for the iterative upwards integration retrieval.

The 3DVAR retrieval was also evaluated in terms of how well it behaved when compared to collocated column measurements of radar reflectivity (for alignment) and retrieved vertical velocity from wind profilers. Time–height comparisons showed good visual agreement between reflectivity measurements, which was reinforced by correlations greater than 0.8 at most heights, MBD near -1.5 dB, and MAD typically less than 3 dB. The spatial and temporal characteristics of 3DVAR vertical velocity retrievals were also generally in good agreement with those from the wind profilers. Prominent updraft and downdraft features retrieved by the UAZRs were repeatedly observed in the 3DVAR dataset. Time–height comparisons showed reasonable agreement for most events analyzed, with moderate correlations of the order of 0.5, MBD less than 0.5 m s-1, MAD within 1 m s-1, and RMSD generally less than 1.5 m s-1. In the context of deep convective drafts, where velocities can easily exceed 15 m s-1, these differences are arguably negligible.

One omission for this sensitivity analysis was that the advection and time evolution of convective clouds during the typical PPI volume scan window was not considered (e.g., Gamache et al., 1995; Protat and Zawadzki 1999; Collis et al., 2013). A radar PPI scan from the NEXRAD and ARM radars during MC3E generally took 6–7 min to complete, allowing for substantial cloud movement and evolution in faster-moving deep convective events. In this regard, the co-gridding of the network radars cannot represent an actual snapshot of the 3-D convective structure, ultimately limiting the ability for this 3DVAR approach to satisfy the mass continuity equation (e.g., Clark et al., 1980; Gal-Chen, 1982). A further analysis using radar forward simulator and high-resolution (∼ 0.5 km) and frequent (every 20 s) model output will be needed to better address these issues and other potential source of errors (e.g., PPI strategy, radar beam width, sensitivity) in future retrievals.