1. Introduction
The Taylor (1938) frozenturbulence hypothesis is commonly used in physical science and engineering studies to infer the spatial distribution of flow variables from time series of those variables at fixed locations, study the statistical properties of turbulence, and mitigate analysis errors arising from the nonsimultaneity of data collections. According to this hypothesis, smallscale features in the velocity field can be idealized as patterns of unchanging form that are transported by a largerscale “wind stream.” The validity of this hypothesis is a topic of continued interest (Uddin et al. 1997; Burghelea et al. 2005; Dennis and Nickels 2008; Li et al. 2009; Castro et al. 2011; Higgins et al. 2012; Wilczek et al. 2014; Creutin et al. 2015). Not surprisingly, the scales of motion bear on the validity of the hypothesis, and the validity degrades for longer time intervals.
The patterntranslation components U and V used in a variety of radar, hydro, and mesoscalemeteorological applications have been estimated visually (Heymsfield 1978; Carbone 1982; Hildebrand and Mueller 1985; Austin 1987; Dowell and Bluestein 1997; Wurman et al. 2007), obtained as wind profiles from a numerical weather prediction model (Mittermaier et al. 2004), calculated by crosscorrelation analysis (Zawadzki 1973; Bellon and Zawadzki 1994; Fabry et al. 1994; Liu and Krajewski 1996; Anagnostou and Krajewski 1999; Gerstner and Heinemann 2008; Harrison et al. 2009; Nielsen et al. 2014; Thorndahl et al. 2014; Seo and Krajewski 2015), and derived from the minimization of a cost function in which (1.1), (1.2), or (1.3) is imposed as a weak constraint in a variational framework^{ 1 } (GalChen 1982; Liu et al. 2004; Shapiro et al. 2010a, hereafter SWP10a). SWP10a repurposed a singleDoppler velocity retrieval (SDVR) algorithm similar to the frozenturbulencebased SDVR algorithms of Tuttle and Foote (1990), Qiu and Xu (1992), Shapiro et al. (1995), Zhang and GalChen (1996), Liou (1999, 2007), Lazarus et al. (2001), Liou and Luo (2001), and especially the variational echo tracking procedure of Laroche and Zawadzki (1995). In repurposing SDVR as advection correction, SDVR “winds” are interpreted as patterntranslation components rather than air velocity components. Moreover, while the tracer may be of limited interest in SDVR (beyond its use in obtaining the “winds”), it is of primary interest in advection correction—it is the variable being analyzed. SWP10a speculated that, in many cases, a frozenturbulencebased SDVR procedure may work better as an advectioncorrection procedure than as a wind retrieval.
Unlike the expanded form of DR/Dt [which is (1.2)], the expanded form of (1.4) [see (2.2)] includes second derivatives with respect to space and time.
Attempts to mitigate analysis errors stemming from the nonsimultaneity of radar data collections often use advectioncorrection procedures in which U and V are treated as constant throughout the analysis domain or on each subdomain of a partitioned analysis domain. A notable exception is the SWP10a procedure, in which (1.3) is applied as a weak constraint along with smoothness constraints to obtain spatially varying U(x, y) and V(x, y) fields and an analyzed reflectivity field R(x, y, t) consistent with those fields and two time levels of input data. In a companion paper (Shapiro et al. 2010b, hereafter SWP10b), reflectivity data from a supercell storm were advection corrected to a time between the two data input times, and compared to data at that time. The spatially variable procedure produced analyses with smaller rootmeansquare errors and larger correlation coefficients than analyses obtained with any pair of constant U and V parameters. Additional tests in SWP10b focused on advection correcting υ _{ r } using U and V obtained from (i) the spatially variable reflectivitybased procedure, or (ii) a range of imposed constant U and V values. Although the reflectivitybased procedure yielded generally good results, the errors in υ _{ r } were slightly larger than the errors obtained using the best of the constant U and V analyses. SWP10b suggested that better results might be obtained using a υ _{ r }based procedure, that is, a procedure based on (1.4) instead of (1.3).
In this study we develop a frozenturbulencebased advectioncorrection procedure for the radial velocity field in which the retrieved pattern translation components are spatially variable. An important application is to the analysis of nonsimultaneously collected radial velocity data from two or more radars to common analysis times for use in multipleDoppler wind syntheses of squall lines, supercells, tornadoes, and other convective phenomena (Clark et al. 1980; GalChen 1982; Chong et al. 1983; Hildebrand and Mueller 1985; Smull and Houze 1987; Bousquet et al. 2007; Wurman et al. 2010; Wienhoff et al. 2018; Oue et al. 2019; and many other studies).
The paper is arranged as follows. In section 2 we introduce the cost function whose minimization underpins the analysis. The Euler–Lagrange equations arising from the minimization of this cost function are derived in sections 3 and 4. The numerical procedures used to solve the Euler–Lagrange equations are described in section 5. In section 6, analytical data are used to verify that the computer code for the analysis procedure is errorfree. Tests using data gathered by a mobile Xband Doppler radar, the Atmospheric Imaging Radar (AIR), of a band of heavy convective rainfall near El Reno, Oklahoma, on 4 September 2018, and a tornadic storm near Canadian, Texas, on 27 May 2015, are presented in section 7. Conclusions follow in section 8.
2. A cost function for advection correcting radial velocity data
As in SWP10a,b, we analyze data on a twodimensional (2D) Cartesian grid on which gradually varying “large scale” advection fields U = U(x, y) and V = V(x, y) advect smallerscale features in the data. The U and V fields are considered to be in a steady state. The analysis grid can be embedded on a surface of constant elevation angle [for data gathered in plan position indicator (PPI) mode] or on a surface of constant height [for data on constantaltitude PPI (CAPPI) surfaces]. The data on any analysis surface at any data input time are treated as if they were gathered instantaneously. A volume of data gathered as a sequence of surfaces can be analyzed sequentially, one surface at a time, with a single time stamp assigned for each surface. Some recently designed mobile Doppler radars can scan a single elevation angle in just a few seconds (e.g., French et al. 2014; Kurdzo et al. 2017; Wienhoff et al. 2018). However, operational radars such as those in the Weather Surveillance Radar1988 Doppler (WSR88D) network can take nearly 20 s to scan an elevation angle, even using a scan strategy designed for severe convective weather (Brown et al. 2005). In such cases, the peak displacement error on a PPI due to the assumed simultaneity of the data collection would be ~200 m in a storm translating at a speed of ~10 m s^{−1}. The significance of such an error may vary with the application, but should be minor if the scales of interest are ~1 km or larger.
The analysis domain is a rectangle with sides of length L _{ x } parallel to the x axis, and L _{ y } parallel to the y axis. The lower left corner of the domain is at the point (x _{0}, y _{0}). The analysis window extends from an initial time t _{ i } to an end time t _{ f }. Radial velocity data are incorporated into the analysis at three data input times: the initial time t _{ i }, the end time t _{ f }, and an intermediate time^{ 2 } t _{ m } (t _{ i } < t _{ m } < t _{ f }). The radial velocities are analyzed to all of the analysis (computational) times within the analysis window (Fig. 2). For notational simplicity, we omit the limits of integration on all integrals that span the full analysis domain and analysis window, as in the (upcoming) cost function.
The solution of (2.4) is α(x, y, t) = const (along trajectories).^{ 3 } We assign α = 0 to trajectories that enter or leave the domain at any time during the analysis window or are in a data void at any data input time, and α = 1 to trajectories on which data are available at all three data input times. Due to the “filling in” effect of the Cressman filter used to interpolate raw radar data to the Cartesian analysis grid, the data coverage at a data input time may be more extensive on the analysis grid than on the radar grid. However, since α(x, y, t) evolves as the numerical procedure iterates between evaluating Γ on trajectories, and solving the elliptic equations for U and V (section 5), the form of α(x, y, t) between data input times is not known until the procedure has converged. The κ∇U^{2} and κ∇V^{2} terms (∇ is the horizontal gradient operator, κ is a weight function) confer spatial smoothness on U and V. However, unlike the smoothness weight (β) in SWP10a,b, κ is not constant, but increases with radius as κ = Kr ^{2}, where K (which has units of m^{2} s^{−4}) is constant. The r ^{2} factor in κ allows the smoothness terms to keep pace with the radial growth of terms in the weakconstraint form of frozen turbulence associated with the factor r in Γ ≡ rυ _{ r } [an r ^{2} magnification factor for error in the weakconstraint form of frozen turbulence was noted by Matejka (2002)].
The Euler–Lagrange equations arising from (2.6)–(2.8) (sections 3 and 4) form a coupled set of differential equations for U, V, and Γ that will be solved iteratively (section 5).
In SWP10a, it was shown that many advectioncorrection procedures based on the frozenturbulence constraint are subject to solution nonuniqueness associated with temporal aliasing [e.g., the reflectivitybased procedures in SWP10a,b, GalChen (1982), Laroche and Zawadzki (1995), and Germann and Zawadzki (2002)]. A proof similar to that in SWP10a (not shown), establishes that the υ _{ r }based procedure in the present study is also subject to solution nonuniqueness. If one suspects that multiple cost function minima are likely (e.g., if there are wavy features in the data), the prospect of multiple solutions can be explored by using multiple first guesses for U and V, as in Shapiro et al. (2015). Use of a lowresolution “scaling guess,” as in Laroche and Zawadzki (1995) and Germann and Zawadzki (2002), may also be desirable.
3. Euler–Lagrange equation for Γ
The integrand in the east/west boundary integral in (3.9) is proportional to α and to the boundarynormal velocity component (U). Clearly, the integrand is zero on any east/west boundary point on which U = 0. However, the integrand also vanishes on any east/west boundary point on which U ≠ 0, since any trajectory launched at any analysis time from such a point is entering or leaving the domain (so α = 0). Thus, this boundary integral is identically zero. Similarly, it can be shown that the north/south boundary integral is identically zero.
The evaluation of the triple integral in (3.9) depends on the number n of data input times. The procedure corresponding to n = 3 is described in detail in this section. In appendix A we show that an n = 2 procedure yields meaningless results. A procedure based on arbitrary n (≥3) is outlined in appendix B, but its development is beyond the scope of this study.
For each trajectory, we solve (3.12)–(3.14) on subwindows 1 and 2, with subscripts 1 and 2 affixed to the solutions on the respective subwindows.
Although it should be clear from the context whether a time integral is evaluated at a fixed point or along a trajectory, we facilitate the distinction by using the symbol τ (with primes) to represent the dummy integration variable in the time integrals that are evaluated along trajectories, as in (3.16a) and (3.16b).
In arriving at (3.21a), (3.21b) and (3.22a), (3.22b), Cauchy’s formula for repeated integration (Herrmann 2011) was used to express the double integrals as single integrals.
One then gets C _{1} from (3.35), C _{2} from (3.37), A _{2} from (3.38), D _{1}, D _{2}, B _{1}, and B _{2} from (3.31)–(3.34), respectively, and Γ from (3.20a) and (3.20b).
4. Euler–Lagrange equations for U and V
The integrals in (4.8) and (4.9) are evaluated at fixed locations, not along trajectories.
5. Numerical procedures
As shown in appendix D, the solution of (5.1) and (5.2) subject to the Neumann conditions (4.2a), (4.2b) and (4.10a), (4.10b) for any outer iteration level (with Γ, DΓ/Dt, and Ψ known from the previous iteration) is unique. This simplifies the implementation of the SOR solver. However, this uniqueness result only pertains to the solution of (5.1) and (5.2) at any one outer iteration level. As discussed in section 2, uniqueness of the overall converged solution for U, V, and Γ cannot be ensured.
The analysis procedure is schematized in Fig. 2. During the (N + 1)th outer iteration, a fourthorder Runge–Kutta method (Press et al. 1992) is used to construct the patterntranslation trajectories x(t), y(t) from the U, V fields at the Nth iteration.^{ 7 } Trajectories are launched forward and backward in time from every analysis point at every analysis time. At each data input time, υ _{ r } data (really Γ) and divU are bilinearly interpolated to each trajectory. If data are missing on a trajectory at any data input time, we set α = 0 for that trajectory. On trajectories for which data are available at all data input times (α = 1), we calculate Γ from (3.20a) and (3.20b), DΓ/Dt from (3.19a) and (3.19b), and Ψ from (3.15a) and (3.15b). Since a trajectory passes through each analysis point at each analysis time, this calculation also updates Γ, DΓ/Dt, and Ψ at all analysis points at all analysis times. The updated variables are used to update the coefficients in (5.1) and (5.2). These equations are solved for U and V subject to boundary conditions (4.2a), (4.2b) and (4.10a), (4.10b) using SOR with an overrelaxation coefficient set to 1.95. If the maximum change in U or V from the Nth to (N + 1)th outer iteration levels is less than a prescribed tolerance (TolOUT = 0.001 m s^{−1}), the procedure is deemed to have converged; otherwise, the procedure returns to the trajectory calculation step. If the procedure fails to converge, the nonconvergence can be manifested as U, V values that oscillate (cycle) around unchanged values, or as continued growth of U, V to unrealistically large values, with concomitant development of large data voids as virtual particles quickly leave the domain. In the former case, if the oscillation is of sufficiently small amplitude, the procedure can still provide good estimates of the radial velocity field.
Although we have not developed a procedure to estimate appropriate values for K, an amplitude principle for K can be inferred from (2.3): if, for a dataset with radial velocity field υ _{ r1}(x, y, t), a smoothness constant K _{1} has been identified for which the retrieved U and V fields are deemed physically reasonable, then for a dataset with radial velocity field υ _{ r2}(x, y, t) having the same pattern as υ _{ r1} throughout the analysis window but an amplitude ε times that of υ _{ r1} (υ _{ r2} = ευ _{ r1}), the same U and V can be obtained using the smoothness constant K _{2} = ε ^{2} K _{1}. Although the allotherthingsbeingequal conditions for the validity of this principle can never be met in practice, the idea that larger values of K should be applied to datasets with larger radial velocity amplitudes may be useful.
6. Analytical data tests
The radial velocities are obtained from (6.5), with u and υ given by (6.7)–(6.9).
It can be shown that u and υ in the uniform translation test [i.e., given by (6.3) and (6.4) with constant U and V] satisfy the inviscid equations of motion exactly, but the u and υ fields in the solid body vortex test [i.e., given by (6.7)–(6.9) with U and V satisfying (6.6)] do not. However, since the analyses are not constrained by the dynamical underpinnings of the flow (other than the assumption that the frozenturbulence hypothesis is at least qualitatively valid), the failure of (6.7)–(6.9) to satisfy the equations of motion should not preclude their use in our tests. Of more concern, however, is that U, V fields satisfying (6.6) cannot satisfy all of the zeronormal gradient conditions (4.2a), (4.2b) and (4.10a), (4.10b). Thus, it will not be possible to retrieve U and V exactly in the solid body vortex test case. However, as we will see, we can retrieve U and V fields in good agreement with (6.6) away from computational boundaries.
In both sets of tests, we take A = 20 m s^{−1}, k = l = 2π/λ, and λ = 10 km. The analysis domain is a square with sides of length 30 km, and a gridpoint spacing of Δx = Δy = 250 m (121 × 121 points). The computational time step is Δt = 4 s. The analysis window is 6 min, with data supplied at 3 min intervals. In the uniform translation tests, data are generated from (6.3) and (6.4) with U = 5 m s^{−1} and V = 15 m s^{−1}. In the solid body vortex tests, data are generated using (6.7)–(6.9) with Ω = 3 × 10^{−4}s^{−1}. The velocity fields for these cases are shown in Fig. 3.
The U, V fields retrieved in the uniform translation tests were nearly constant and nearly equal to their true values for a range of values of the smoothness parameter. In tests run with K = 0.5, 1, 5, 10, 50, 100 m^{2} s^{−4}, the maximum error in U or V at any location was ≈0.5% of the true values, and the maximum error in υ _{ r } at any location or time was ≈0.12 m s^{−1}, about 0.34% of the true peak value of υ _{ r } (≈35 m s^{−1}).
We also ran a uniform translation test to explore the possibility of solution nonuniqueness. Following SWP10a and Shapiro et al. (2015), we note that if a tracer varies with wavelength λ in the y direction, and U _{true} and V _{true} are the true patterntranslation components, there are an infinite number of solutions U = U _{true}, V = V _{true} + Mλ/T (M is an integer, T is data input time interval) that satisfy the analysis constraints exactly. Associated with the first spurious modes M = ±1 (with T = 180 s, λ = 10 km, V _{true} = 15 m s^{−1}) are the aliased solutions V ≈ −40.56 m s^{−1} and V ≈ 70.56 m s^{−1}. To retrieve the −40.56 m s^{−1} aliased solution, we left the first guess for U at 0 m s^{−1}, but changed the first guess for V to −50 m s^{−1}. In a test with K = 1 m^{2} s^{−4}, the procedure converged with good precision to the aliased solution: U was nearly constant throughout the domain and close to its true value of 5 m s^{−1} (4.97 < U < 5.00 m s^{−1}), while V was nearly constant and close to the −40.56 m s^{−1} aliased solution (−40.45 < V < −40.42 m s^{−1}).
Results from a solid body vortex test in which K = 0.0002 m^{2} s^{−4} are shown in Fig. 4. As anticipated, the retrieved U, V fields are in qualitatively good agreement with the exact U, V fields in the interior of the domain, with larger errors—on the order of 1 m s^{−1}—in the vicinity of boundaries on which the true normal gradients are inconsistent with zeronormal gradient conditions (southern/northern boundaries for U, western/eastern boundaries for V). Despite ~1 m s^{−1} peak errors in U and V, the peak error in υ _{ r } at any point or time was only ≈0.15 m s^{−1}, or ≈0.75% of the true peak value of υ _{ r } which was ≈20 m s^{−1}. An additional experiment confirmed the amplitude principle for K described in section 5: doubling the amplitude of the radial velocity (taking A = 40 m s^{−1}) while increasing K by a factor of 4 (taking K = 0.0008 m^{2} s^{−4}) produced results that differed imperceptibly from those in the above experiment.
7. Tests using AIR data
The AIR is a mobile rapidscan Xband phased array radar, designed, maintained, and operated by the Advanced Radar Research Center (ARRC) at the University of Oklahoma (Isom et al. 2013; Kurdzo et al. 2017; Mahre et al. 2018; Griffin et al. 2019). We conducted advectioncorrection tests using AIR data of heavy convective rainfall in a band of thunderstorms during the afternoon of 4 September 2018 near El Reno, Oklahoma, and of a decaying EF2 rated tornado and its parent storm on 27 May 2015 near Canadian, Texas.
Here, X and Y are the analyzed and observed υ
_{
r
} fields, respectively, at the verification time, the summations are over the P analysis points on which both analyzed and observed υ
_{
r
} fields are available, and an overbar denotes an average over those P points; that is, for a variable ϕ,
To put the analysis results in context, we also ran a suite of control experiments:

Forward Persistence: The υ _{ r } field analyzed at the verification time t _{ υ } is the υ _{ r } field observed at the initial time t _{ i } . The velocity field is treated as stationary—no attempt is made to account for advection.

Backward Persistence: This run is similar to the Forward Persistence run, but with the υ _{ r } field analyzed at the verification time t _{ υ } taken as the υ _{ r } field observed at the intermediate time t _{ m } . Again, there is no advectioncorrection step.

Linear Time Interpolation: The υ _{ r } field is analyzed by linearly interpolating υ _{ r } data from the initial and intermediate times to the verification time. There is no advectioncorrection step.

Specified Constant U, V: The υ _{ r } field is advection corrected, but with U, V imposed as specified constants (U and V varied in 1 m s^{−1} increments from −15 to 15 m s^{−1}) rather than derived from (5.1) and (5.2). Of these 31^{2} = 961 experiments, the one yielding the smallest RMSE is referred to as the Best Constant U, V experiment.

Large K: The υ _{ r } field at the verification time is obtained from the full advection correction procedure, but with the smoothness constant K set to a value so large that the retrieved U, V fields are nearly constant. This run mimics the υ _{ r }based GalChen (1982) procedure.
In both the 4 September 2018 and 27 May 2015 deployments, the AIR was steered mechanically in azimuth, while transmitting a fan beam 20° wide in elevation and 1° wide in azimuth. For each pulse, data received by the AIR’s multielement antenna array were postprocessed using digital beamforming (Kurdzo et al. 2014). For both cases, the temporal resolution of the data was 10 s. The scan mode employed pulse compression with a 5.25μs pulse, yielding a native range resolution of 37.5 m (Kurdzo et al. 2014, 2017). AIR data were oversampled to 30 m in range and 0.5° resolution in both azimuth and elevation (factorof2 oversampling). Radial velocities from 27 May 2015 were dealiased using Solo II software (Oye et al. 1995), with questionable values removed manually. A groundclutter filter (Siggia and Passarelli 2004; Cho and Chornoboy 2005) was applied to the 4 September 2018 data. No dealiasing was needed for the 4 September 2018 case, and no groundclutter filter was needed for the 27 May 2015 case. More details about the 27 May 2015 dataset can be found in Griffin et al. (2019).
AIR data were interpolated to a 2D Cartesian analysis grid on a constant elevation angle surface with a uniform grid spacing (Δx = Δy = 250 m) using a Cressman filter (Cressman 1959) with 400 m radius of influence. Experiments with 4 September data were conducted on the 5.5° elevation surface because that surface had the most extensive data coverage. Experiments with 27 May data were conducted on the 4° elevation surface because that level was within a layer of good data coverage, tornado strength decreased with height above that surface, and tornado size decreased (albeit slightly) beneath that surface. We report on experiments for which data are input at ~2 min intervals, and the verification time is halfway between the initial and intermediate data input times. The radial velocity fields at the three data input times are shown for both cases in Fig. 5. The tornado appears in the 27 May data as a dipolelike feature in the radial velocity field centered on a point near x = −5 km, y = 3 km. It is strongest at the initial time. Even away from the tornado, the winds in the 27 May case are generally stronger and more variable than in the 4 September case.
For each dataset, we determined a threshold value of K at and beneath which the procedure did not converge, but subsequently increasing K by 0.001 m^{2} s^{−4} did yield convergence. The nonconvergence in these threshold experiments was associated with a smallamplitude “cycling” of U, V iterates around unchanging values, with only slight changes to υ _{ r } from one iteration level to the next.
Statistics from tests with the 4 September data are summarized in Table 1. The best of the advectioncorrection experiments had lower RMSE and higher CC than any of the nonadvectioncorrection experiments, with the RMSE in the two persistence experiments being more than twice as large as the RMSE in any of the advectioncorrection experiments. The RMSE generally decreased with decreasing K, and was lowest in the K threshold run (K = 0.001 m^{2} s^{−4}). Statistics from the Specified Constant U, V runs and a LargeK run are shown in Fig. 6. The RMSE in the LargeK run was similar to but somewhat larger than in the Best Constant U, V experiment, while the RMSE in the runs with the smaller values of K were less than the RMSE in the Best Constant U, V runs. The analyzed υ _{ r }, AIR υ _{ r }, and errorinυ _{ r } fields in the K = 0.002 m^{2} s^{−4} run are presented in Fig. 7. The corresponding U, V fields are shown in Fig. 8. The dominant northward translation evident in Fig. 8 is consistent with the motion seen in animations of the reflectivity field and our perception of cloud motions during the field deployment. The errorinυ _{ r } field (lower panel of Fig. 7) appears to contain both amplitude and phase errors. A peak error of almost 3.7 m s^{−1} is found near the point x = −1.0 km, y = −2.5 km, in a lobe of relatively weak radial velocities.
Experiments using AIR data from a band of heavy convective rainfall on 4 Sep 2018. Radial velocity data from the 5.5° elevation angle are supplied at ~2 min intervals (t _{ i } = 2202:53 UTC, t _{ m } = 2204:58 UTC, t _{ f } = 2207:03 UTC). The analyzed radial velocity is compared to the AIR radial velocity at t _{ υ } = 2203:55 UTC. An asterisk indicates a “K threshold” experiment in which the procedure did not converge but subsequently increasing K by 0.001 m^{2} s^{−4} did yield convergence. The Best Constant U, V run is the Specified Constant U, V run with the lowest RMSE (see Fig. 6).
Results from the 27 May case are summarized in Table 2. As in the tests with 4 September data, the best of the 27 May advectioncorrection experiments had lower RMSE and higher CC than any of the nonadvectioncorrection experiments. Again, the RMSE decreased with decreasing K, and was lowest in the K threshold run (K = 0.002 m^{2} s^{−4}). Statistics from the Specified Constant U, V runs and the LargeK run are shown in Fig. 9. As in the 4 September case, errors in the LargeK run were similar to but somewhat larger than in the Best Constant U, V run, while the RMSE in the runs with the smaller values of K were less than the RMSE in the Best Constant U, V run. The analyzed υ _{ r }, AIR υ _{ r }, and errorinυ _{ r } fields in the K = 0.005 m^{2} s^{−4} run are shown in Fig. 10. The corresponding U, V fields are shown in Fig. 11. Although the errors in υ _{ r } tend to be somewhat larger near the tornado than in many other locations, the largest error (≈8.7 m s^{−1}) is far from the tornado, near a point (x = −17.5 km, y = 4.5 km) on the northern flank of a lobe of negative radial velocities (blue blob). At this location, the AIR υ _{ r } is negative, with a magnitude of a few meters per second, while the analyzed υ _{ r } is positive and of similar magnitude. This error might be a phase error enhanced by evolution effects not properly accounted for by the procedure (size of blue blob is too small in the analysis).
Experiments using AIR data from a decaying tornado and its parent storm on 27 May 2015. Radial velocity data from the 4.0° elevation angle are supplied at ~2 min intervals (t _{ i } = 2203:52 UTC, t _{ m } = 2206:01 UTC, t _{ f } = 2207:59 UTC). The analyzed radial velocity is compared to the AIR radial velocity at t _{ υ } = 2204:56 UTC. An asterisk indicates a “K threshold” experiment in which the procedure did not converge but subsequently increasing K by 0.001 m^{2} s^{−4} did yield convergence. The Best Constant U, V run is the Specified Constant U, V run with the lowest RMSE (see Fig. 9).
Tables 1 and 2 show that the value of K used in the 27 May LargeK run is 10 times larger than in the 4 September LargeK run. We settled on these K values because they produced similar (small) spatial changes to U and V in both datasets, with peak domainwide variations of ≈0.05 m s^{−1}. Additionally, the threshold for K at and beneath which the procedure did not converge was about twice as large in the 27 May case than in the 4 September case. Despite their failure to converge, these threshold tests produced the lowest RMSE and largest CC for both datasets. The relatively larger values of K in the LargeK and threshold runs in the 27 May case compared to the 4 September case are qualitatively consistent with the amplitude principle for K discussed in section 5. In additional tests in which K was beneath its threshold value and further decreased, the amplitude of the U, V–iteration cycle increased, and the procedure eventually became unstable.
The analytical and real data tests described herein were obtained with parameter settings that had not been optimized for computational efficiency. Additional tests using AIR data (not shown) showed that a significant speedup could be achieved simply by increasing the tolerance (TolOUT) for convergence of U and V in the outer loop. However, to prevent the procedure from converging prematurely in that loop—which could substantially increase analysis errors—the blending parameter wgt in (5.4) should be increased in tandem with the increase in TolOUT. When the original 27 May analysis using K = 0.005 m^{2} s^{−4} was rerun with TolOUT increased a hundredfold to 0.1 m s^{−1} and wgt increased tenfold to 0.25, the execution time decreased from 112 to 3 min. Similarly, when the original 4 September analysis using K = 0.002 m^{2} s^{−4} was rerun using these same increased parameters, the execution time decreased from 87 to 5 min. In both spedup runs, the increase in RMSE (ΔRMSE) over the RMSE in the corresponding original run was negligible (ΔRMSE ≈ 0.007 m s^{−1} for the 27 May analysis and ≈0.002 m s^{−1} for the 4 September analysis). Similar speedups were achieved by setting TolOUT to 0.1 m s^{−1} without changing wgt, but at the expense of a much larger ΔRMSE (ΔRMSE ≈ 0.174 m s^{−1} for the 27 May analysis and ≈0.324 m s^{−1} for the 4 September analysis).
8. Conclusions
A procedure to advection correct (analyze) the Doppler radial velocity field using spatially variable patterntranslation components (U and V) is developed, with encouraging results obtained in proofofconcept tests using data from the Atmospheric Imaging Radar (AIR). The advectioncorrection procedure is based on the minimization of a cost function that accumulates errors in a secondderivative frozenturbulence constraint on the radial velocity field, while penalizing spatial gradients in U and V. The radial velocity field and associated U, V fields are analyzed together as the iterative solution of the Euler–Lagrange equations arising from the minimization of the cost function. Radial velocities from three data input times are incorporated into the analysis. As in many advectioncorrection techniques that use the reflectivity field as a tracer, the new radialvelocitybased technique may be subject to solution nonuniqueness associated with temporal aliasing. Solution nonuniqueness was demonstrated in an analytical test in which the first guesses for U and V were set close to values associated with a spurious solution, but did not appear to be an issue in the AIR data tests.
After performing verification tests using analytical data, we conducted experiments using AIR data from a convective rainband on 4 September 2018 and a decaying tornado and its parent storm on 27 May 2015. In these AIR data tests, the procedure yielded radial velocities with lower RMSE and higher CC than radial velocities analyzed in any of the control runs (Forward Persistence, Backward Persistence, Linear Time Interpolation, Specified Constant U, V, and Large K).
The procedure is sensitive to the value of a smoothness parameter (K). In AIR data tests spanning three orders of magnitude of K, the best results were obtained with relatively small values of K, though for values not much smaller than these, the procedure became unstable. For relatively large K, the retrieved U, V fields were nearly constant, and the errors in the analyzed radial velocity field were similar to (though slightly worse than) those from the Best Constant U, V experiment. As the procedure lacks an objective method to determine appropriate values of K, we recommend that such values be explored on a casebycase basis, using trial and error, perhaps guided by visual estimates of physically reasonable U, V fields from animations of the data.
The stability and execution time of the procedure are sensitive to computational parameters such as the convergence thresholds (tolerances) for U and V on the inner and outer iteration loops and an underrelaxation (blending) parameter used to weight the updated and previous iterates on the outer loop. Experimentation may also help determine appropriate values of these parameters on a casebycase basis.
The primary utility of the new procedure is to the analysis of nonsimultaneously collected radial velocity data from two or more radars to common analysis times for use in multipleDoppler wind syntheses of severe convective phenomena. Since some experimentation is required to establish appropriate smoothing and computational parameter values or to explore the threat of solution nonuniqueness, the procedure is currently more suitable for nonrealtime research applications than use in an operational analysis or data assimilation setting. Future work will focus on determining procedures to estimate some of these parameters objectively.
Acknowledgments
This research was supported by the National Science Foundation (NSF) under Grant AGS1623626. Data collection and analysis of the 27 May 2015 case were supported by NSF AGS1303685 and AGS1823478; Jim Kurdzo led the deployment, Javier Lujan assisted with the data collection, and Casey Griffin dealiased and quality controlled the data. The AIR is maintained and operated by the Advanced Radar Research Center (ARRC) of the University of Oklahoma. We are grateful to the ARRC engineering staff for their support of our radar experiments. Shawn Riley provided computing assistance. The authors thank Lou Wicker, Qin Xu, Nikola Petrov, and the anonymous referees for their helpful comments.
Data availability statement
The AIR data and Fortran code used to generate the analyses reported on herein are available from the corresponding author upon request.
APPENDIX A
Analysis Using Two Time Levels of Data
If υ _{ r } data are only available at two data input times, then (3.11) and the equivalent system (3.12)–(3.14) apply throughout the analysis window, and (3.18a) and (3.18b) are replaced by the single equation Θ = (A _{1} t + B _{1})E _{1}(t). In view of (3.10a), (3.10b), and (3.12), A _{1} = B _{1} = 0, so D ^{2}Γ/Dt ^{2} is identically zero. Equations (3.12) and (3.13) then show that Θ and Ψ are identically zero, and (4.8) and (4.9) reduce to Laplace’s equations for U and V, the solutions of which [subject to the zeronormalgradient conditions (4.2a), (4.2b) and (4.10a), (4.10b)] are arbitrary constants, with no links to the υ _{ r } field. Such an analysis has no practical value.
APPENDIX B
Analysis Using an Arbitrary Number of Time Levels of Data
To incorporate data from an arbitrary number n of input times, the analysis window should include n − 2 intermediate data input times and n − 1 analysis subwindows. The solution of (3.11) [analog of (3.20a) and (3.20b)] then contains 4(n − 1) integration constants, which satisfy 4(n − 1) linear algebraic equations: 1 equation from imposing data at the initial time, 1 equation from imposing data at the end time, 2(n − 2) equations from imposing data at the intermediate data input times (as in the n = 3 case, data at each intermediate data input time are used twice), 2 equations from (3.10a), (3.10b), and 2(n − 2) equations expressing continuity of DΓ/Dt and D ^{2}Γ/Dt ^{2} at the intermediate data input times. The integration constants can thus be obtained by solving 4(n − 1) linear algebraic equations. U and V then follow as solutions of (4.8) and (4.9) subject to boundary conditions (4.2a), (4.2b) and (4.10a), (4.10b).
APPENDIX C
Classification of (4.8) and (4.9) for U and V
The secondorder quasilinear operator (C.1) is classified using the same procedure as for linear operators (Garabedian 1998). Since κ, α, t _{ f } − t _{ i }, and (Γ_{ x })^{2} are nonnegative, (C.3) shows that γ is nonnegative. Equations (C.2) then show that A, C, and AC − B ^{2} are strictly positive for all U and V, so (C.1) is an elliptic operator. Similarly, it can be shown that the operator analogous to (C.1) arising from (4.9) is elliptic.
APPENDIX D
Comment on the Forcing Terms in (4.8) and (4.9)
As there are no restrictions on Q ^{ N } that would make its integral vanish, (D.4) will generally be violated, and no solution for U ^{ N+1} exists.
As the discriminant c ^{2} − bg in the quadratic formula solution of (D.15) for U _{ d } as a function of V _{ d } is negative (c ^{2} ≤ bg), U _{ d } is imaginary if V _{ d } is real. However, since U _{ d } and V _{ d } should be real, we must have U _{ d } = V _{ d } = 0 (U _{2} = U _{1}, V _{2} = V _{1}), and the solution is unique.
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We use the strong and weakconstraint terminology of Sasaki (1970). A constraint that is imposed exactly is a strong constraint. A constraint that is imposed approximately (i.e., in a least squares error sense) is a weak constraint.
At least three time levels of input data were also required in the υ _{ r }based procedure of GalChen (1982), in which a ∂^{2} υ _{ r }/∂t ^{2} term was discretized. As shown in appendix A, minimizing (2.3) with υ _{ r } data incorporated at only two time levels yields meaningless results.
In the numerical solution procedure (section 5), trajectories are launched forward and backward in time from every analysis point at every analysis/computational time.
Treated as field variables, U, V, and divU are in a steady state. However, due to their spatial variations, these same variables vary with time along trajectories. We obtain U, V, and divU along trajectories by interpolating their values from neighboring grid points to the location of the virtual particle tracing out the trajectory.
To ensure continuity of Γ at time t _{ m }, the input data are used twice: for Γ_{1} at the end of subwindow 1, and for Γ_{2} at the beginning of subwindow 2.
Second derivative terms in U and V which appear in the expansion of Ψ must be taken into account in the classification of (4.8) and (4.9) (appendix C).
Unless stated otherwise, the first guesses for U and V in our experiments (i.e., at the first iteration) are 0 m s^{−1}.
We refer to “large” or “small” values of K in a relative sense, that is, with respect to a given dataset. A value of K large enough to produce nearly constant U and V in one dataset may be too small to produce nearly constant U and V in another dataset, even if the two datasets have the same spatial and temporal resolution (as in the two AIR datasets considered in section 7).