## 1. Introduction

Accurate prediction of weather, climate, and air quality change requires knowledge about the turbulent processes in the atmospheric boundary layers (ABLs). With field measurements acquired using in situ sensors on aircraft, towers, and tethered balloons, or by radar, sonar (SODAR), and lidar remote sensors, the understanding of the ABL has been greatly improved. Nevertheless, a genuine picture of the dynamic processes in the ABL is far from complete. With advances in computer technology, computational fluid dynamics has played a significant role in studying ABL flows. Although direct numerical simulation of atmospheric flow is still impossible with today's computational power, application of large eddy simulation (LES) to model the ABL has significantly contributed to the current knowledge of turbulent processes in the ABL. For instance, vortical structures and low-speed streaks have been observed in the neutrally stratified ABLs (Moeng and Sullivan 1994; Lin et al. 1996). Convective rolls are seen in the slightly convective ABL (Glendening 1996; Khanna and Brasseur 1998). The strongly convective ABL is now known to be characterized by elongated updrafts (Khanna and Brasseur 1998; Lin 2000). Although LES models are able to generate detailed flow structures in the ABL, they are often limited by idealized boundary conditions, which decouple the interaction between mesoscale (2– 2000 km) and microscale (2 mm–2 km) atmospheric structures (Jacobson 1999). Near the surface, LES relies on complicated subgrid-scale (SGS) models for physical realization of eddy structures. Some argue that organized coherent structures derived from LES are quite sensitive to SGS models and grid resolution effects.

For field measurements, Doppler radars have been widely used for diagnostic studies of convective systems, severe weather detection, and short-term forecasting. They can provide full volume scan data every 3 to 10 min, with 1 km^{2} horizontal resolution, 500-m vertical resolution, and velocity accuracy of 0.5 to 1.0 m s^{−1}. Doppler radars with higher spatial resolution are also available (see Thomson and List 1999; Wurman and Gill 2000). Over the past 20 yr, different lidar systems have been developed. Lidars are different from radars in that they use optical rather than radio frequencies. They normally have better range resolutions with narrower beams than radars. For instance, the 248-nm water Raman lidar (Eichinger et al. 1994) has a 3-m range resolution, which is better than the grid resolutions used in most ABL flow simulations. This lidar has detected Rayleigh–Bernard-like “dumbbell shaped” coherent structures of 10 to 20 m in diameter in the lower surface of the equatorial marine boundary layer (Cooper et al. 1997). The high-resolution Doppler lidar (HRDL), designed and developed at the National Oceanic and Atmospheric Administration (NOAA) Environment Technology Laboratory (ETL), is a solid-state coherent Doppler lidar that operates at a wavelength of 2 *μ*m, with 30-m range resolution and is capable of 10 cm s^{−1} velocity measurement precision (Grund et al. 2001; Wulfmeyer et al. 2000). HRDL has demonstrated its effectiveness at detecting coherent structures in the ABL in numerous field campaigns since its first successful deployment during the Lasers in Flat Terrain (LIFT) experiment in 1996. Weckwerth et al. (1997) presented the first observation of linearly organized coherent structures in the ABL surface layer using HRDL data. Linear streaks with horizontal spacing of about 200 m are clearly recognizable from HRDL radial velocity contour plots.

Nevertheless, the analysis of lidar data is quite limited at present. In published works, radial velocity and reflectivity fields are often presented to show the presence of coherent structures or other phenomena without considering cross-beam velocity components. Although some structures are distinguishable from displays of plan position indicator (PPI) or range–height indicator (RHI), they are incomplete and might be misleading. Deploying multiple lidars could clear up the ambiguity, but it is impractical due to the operational cost.

The objective of the work is to apply the four-dimensional variational data assimilation (4DVAR) method to HRDL radial velocity data to recover complete wind and temperature fields for the study of microscale atmospheric structures in the ABL. The 4DVAR technique used was developed for mesoscale Doppler radar observation (Sun and Crook 1998) and later modified to deal with microscale ABL flows (Lin et al. 2001; Lin and Chai 2002; Chai et al. 2002). Before applying the 4DVAR to HRDL data, identical twin experiments (ITEs) are conducted to address issues concerning boundary conditions and retrieved data quality. In these ITEs, the scan volume of synthetic radial velocity data is constructed to mimic a realistic scan volume, which has available data only within certain ranges of elevation and azimuth angles. The idealized boundary condition employed in the ITEs of Lin et al. (2001) becomes impractical in HRDL data retrieval. The ITEs in this work aim to assess the effects of Dirichlet and inflow/outflow boundary conditions in HRDL data retrieval. Buffer regions enclosing the scan volume work effectively with the foregoing two boundary conditions. Finally, HRDL observations are assimilated using the 4DVAR that also treats eddy viscosity and thermal diffusivity profiles as control variables (Chai and Lin 2002). A flow structure resembling a dry microburst is identified from the retrieved wind field, and its characteristics are discussed.

The paper is organized as follows. Section 2 briefly describes the numerical formulations of the present 4DVAR, including the prediction model, the adjoint equations, and the optimization method. Section 3 presents the results of the ITEs to assess boundary condition and buffer zone effects. Section 4 discusses HRDL radial velocity data retrieval. Section 5 summarizes results and draws conclusions.

## 2. Numerical method

The current 4DVAR system (Lin et al. 2001; Lin and Chai 2002; Chai and Lin 2002) has several features for microscale turbulent structure retrieval. In addition to the initial wind and temperature fields, eddy viscosity and thermal diffusivity model parameters are treated as control variables. All control variables are updated iteratively to minimize the cost function, which measures the difference between model predictions and field observations. A set of adjoint equations constructed via variational analysis calculates the cost function gradients with respect to control variables.

### a. Prediction model

**U**has three components

*U*

_{1},

*U*

_{2}, and

*U*

_{3}(

*U,*

*V,*and

*W*) in

*x*

_{1},

*x*

_{2}, and

*x*

_{3}(

*x,*

*y,*and

*z*) directions, respectively. Spatial variables

*x*

_{1}and

*x*

_{2}correspond to the two horizontal directions, and

*x*

_{3}is the vertical direction;

**G**= (0, 0, −

*g*) and

*g*is the gravitational acceleration;

*ρ*

_{o}is the reference density;

*θ,*Θ, and Θ

_{o}are fluctuating, background (function of

*z*only), and reference virtual potential temperature, respectively. Lowercase variables denote fluctuating parts of variables. Eddy viscosity

*ν*(

*z*) and thermal diffusivity

*κ*(

*z*) are assumed functions of height only. Dependent variables are arranged on a staggered, orthogonal grid. A second-order finite difference method is applied for spatial differencing. The second-order Adam–Bashforth method is used for time advancement of dependent variables. Mass conservation is enforced by solving a pressure Poisson equation derived from the continuity equation (1) and the momentum equations (2).

To apply this model for flow structure retrieval in a convective boundary layer (CBL), a surface momentum and temperature flux model is implemented to enforce satisfaction of the Monin–Obukhov similarity theory at the first vertical grid level immediately above the surface. The gradient-free boundary condition is imposed at the domain top for *U,* *V,* and *θ,* whereas the Dirichlet boundary condition *W* = 0 is used for *W.* Two lateral boundary conditions are evaluated in section 3.

### b. Optimization method

*U*

_{rad}and

*U*

^{obs}

_{rad}

_{t}and Σ

_{x,y,z}denote summations over time and space, respectively;

*α*is a validity coefficient indicative of the quality of the observational data. In the current study, it is taken as unity for good observations and zero for bad ones. The second term, which is a penalty term, represents nondivergent constraint for the initial velocity field. Given that this penalty term does not dominate the cost function, the value for coefficient

*β*should be as large as possible to obtain a divergence-free initial velocity vector field. We chose

*β*= 100, as suggested in Lin et al. (2001). The spatial smoothness constraint, shown by the last term, is no more than 20% of the total cost function by adjusting

*ζ*

_{F}

The adjoint method is used to calculate the gradients of the cost function with respect to all control variables. These gradients are then used by the limited memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) method (Liu and Norcedal 1989) to find the optimal initial guess and parameters *ν*(*z*) and *κ*(*z*) for the prediction model.

## 3. HRDL observational data

The HRDL observational data were obtained during the Cooperative Atmospheric Surface Exchange Study—1999 (CASES-99) field experiment near Leon, Kansas. The dataset was collected from 1923:57 to 1953:19 UTC on 24 October 1999. It was in a typical convective afternoon, when the local central daylight time (CDT) was from 1423:57 to 1453:19 CDT. The platform of the lidar location was at latitude 37.6360° and longitude −96.7339°. The observational data were processed with 90-m range gates and 100 pulse averages, with the pulse repetition frequency as 200 Hz. Due to 2 times oversampling in time and 3 times oversampling in range, the data indicate a 0.25-s time resolution and 30-m range gates.

The observational data include 19 scan volumes, which are made up of a sequence of PPI scans. There are 10 PPI sectors in each scan volume, with elevation angles ranging from 1° to 19°. The PPI sectors are spanned from azimuth angle −30° to 30°, centered to the north. In each sector, light beams are about 2° apart, with 198 range gates in each beam. It is unlikely to get the same azimuth angles in different sector scans. During operation, actual elevation and azimuth angles, along with other parameters, such as the observational time associated with light beams, were recorded in the original radial velocity data. Radial velocities corresponding to target returns and low wideband signal-to-noise ratio (wSNR) are tagged and not used in the 4DVAR analysis. The wSNR threshold is based on the height of the noise floor in the far range gates. Hereafter, the tagged radial velocities are referred as “bad” observations. The radial velocity contours in scan planes with elevation angles of 1° and 19° are shown in Fig. 1. Although the biggest range for the scan is about 6 km, only the ranges less than or equal to 3 km are shown here since there are almost no valid data beyond 2.5 km range. Blank spots in the scan planes represent bad observations. The displayed volume is the 15th in the 30-min period, denoted as VOL15 hereafter. In this volume data, a high radial velocity region, outlined by isosurface *U*^{obs}_{rad}^{−1} in Fig. 1, can be identified. Figure 2 displays a time sequence of slices at 5° elevation angle, from scan volumes VOL13 to VOL17, suggesting that a high-speed structure moved through the scanned region. Lack of the cross-beam velocity components makes it difficult to infer additional information without further analysis. HRDL takes about 90 s to sweep one volume of data. The time variation between any two beams in a single volume must be taken into account in the 4DVAR.

## 4. Boundary conditions and buffer zones

In this section, we study the effects of boundary conditions and buffer zones on retrieved results by performing ITEs. To emulate lidar scan features, range constraints on elevation and azimuth angles are imposed on the synthetic observational data. Buffer zones are defined as the regions between the domain boundaries and the lidar scan volume that contain no data.

### a. Synthetic observational data

A computational domain of 5 km × 5 km × 1.875 km, with a uniform grid of *NX* × *NY* × *NZ* = 48 × 48 × 45 is used for most cases. A smaller domain of 3.333 km × 3.333 km × 1.875 km, with a grid size of 32 × 32 × 45, is applied to study the effects of computational domain. A geostrophic wind of 10 m s^{−1} and a surface temperature flux of 0.24 K m s^{−1} are imposed to drive the simulated CBL flow. At *z* = 980 m, a capping inversion layer is imposed. The roughness height *z*_{o} is 0.16 m. Coriolis parameter *f* is 10^{−4} s^{−1}. Stability parameter −*z*_{i}/*L* is about 15, where *z*_{i} is the averaged CBL height and *L* is the Monin–Obukhov length.

The prediction model Eqs. (1)–(3) are integrated forward in time to generate synthetic observational data. Initial conditions for velocity and temperature fields, as well as profiles of eddy viscosity and thermal diffusivity are obtained by running the National Center for Atmospheric Research (NCAR) large eddy simulation (LES) code (Moeng 1984; Sullivan et al. 1994). Simulated turbulence flow develops and evolves over a physical time period of 1.56 h. The mean and variance profiles of the simulated velocity fields are displayed in Figs. 3a and 3b. The CBL is capped by an inversion layer, where temperature increases with height. The results show that the simulated data agree reasonably well with field data (Lenschow et al. 1980).

Recorded 3D flow fields permit calculation of radial velocities. In Fig. 4, the lidar is assumed at (0, 0, 0) m. For the current HRDL dataset, each volume scan spans about 90 s. To emulate the scan pattern in a simple manner, several *x*–*y* planes of data are assumed to be acquired every 25 s. A total of three volumes of data are available in a period of 300 s. Readers are advised to read Lin et al. (2001) for the detail of the sampling scheme in constructing the volume data. The positive *x* direction is chosen as the zero azimuth angle. The azimuth angle of scan volumes varies from 15° to 75°. The maximum elevation angle is set at 20°. There is no restriction on the range in the radial direction because the radial range of the HRDL is large enough to cover the computational domain. The data outside the constraints are assumed void.

*σ*

_{F}

^{′}

_{o}

*x*–

*y*plane. In calculating correlation coefficient

*σ*

_{F}

*ϵ*

_{F}

*x*–

*y*plane, only the scanned region is considered. The

*ϵ*

_{F}

*σ*

_{F}

*z*< 400 m only. Results at the middle of the assimilation time window are more accurate than those at the beginning or end (Lin et al. 2001). The results presented subsequently are taken from the middle of the assimilation time window.

### b. Boundary conditions

We consider two lateral boundary conditions: Dirichlet boundary conditions and inflow/outflow boundary conditions. For Dirichlet boundary conditions, mean vertical profiles of *U,* *V,* and *θ* are estimated and assigned to lateral boundary nodes. For inflow/outflow boundary conditions, velocity solutions from the previous iteration are used to determine inflow/outflow conditions. If the velocity vector points into the computational domain, that is, the inflow boundary, dependent variables are specified with available mean profiles. Otherwise, if the velocity points outside, that is, the outflow boundary, gradient-free Neumann conditions are applied to the *U,* *V,* and *θ* variables. For all ITEs, *W* values are set to zero at the boundary nodes. Global mass conservation is enforced at each iteration by uniformly distributing a mass residual on lateral boundary nodes.

The ITEs conducted are described in Table 1. Correlation coefficients and rms errors are summarized in Table 2. Using the sectorlike scan volume as a reference, there are various ways of selecting the computational domain for the 4DVAR analysis. Figure 4 illustrates two choices. One domain is made big enough to cover all volume data to utilize all available information. The other domain is smaller, at 3.333 km × 3.333 km × 1.875 km with a grid of 32 × 32 × 45, based on the assumption that a high ratio of data to grid points may improve retrieval quality. The aforementioned boundary conditions are applied to the two domains for comparison.

Table 1 shows that only cases IT3 and IT4 use the small computational domain. With Dirichlet boundary conditions applied, case IT1, having a big domain, produces better results than case IT3 (Table 2). The rms error of velocity components for case IT1 is about 0.1 m s^{−1} smaller than for case IT3. The rms temperature error for case IT1 is 0.061 K smaller than that for case IT3. The correlation coefficients for case IT1 are all greater than those for case IT3. Comparison between cases IT2 and IT4, which implement inflow/outflow boundary conditions, shows consistently that the big domain case IT2 yields significantly better retrieval than the small domain case IT4. It is noted that the correlation coefficients and rms errors are averaged over their own computational domains for all the cases. If only the common domain is considered, the improvement of the big domain over the small domain is even more significant except for *σ*_{θ}. For instance, the correlation coefficients calculated over the common domain for case IT2 are 0.899, 0.915, 0.896, and 0.724, for variables *U,* *V,* *W,* and *θ,* respectively. The rms errors averaged in the common area are also smaller, with *ϵ*_{U}, *ϵ*_{V}, *ϵ*_{W}, and *ϵ*_{θ} as 0.452, 0.453, 0.494 m s^{−1}, and 0.193 K.

These ITEs suggest that lateral boundary condition treatment has only minor effect on retrieval, as indicated by both correlation coefficients and rms errors. Inflow/ outflow boundary conditions are slightly better than Dirichlet boundary conditions. Implementation of Dirichlet boundary conditions, however, is much easier. Although additional observations by using a big domain may result in better retrieval, buffer zones are speculated to be the main cause for the improvement of retrieval and will be examined next.

### c. Buffer zones

The buffer zone concept is found in oceanic data assimilation literature. Gavart et al. (1999) implemented a special boundary treatment in the assimilation of satellite altimeter data into a primitive equation ocean model. A surrounding recirculating area was added to separate the interior. The added surrounding regions were referred to as buffer zones. Ezer and Meller (2000) applied the same concept in their sensitivity studies of the Princeton Ocean Model, with temperature and salinity fields relaxed in buffer zones near the boundaries. The application of buffer zones to the 4DVAR is the focus of this section.

For big domain cases IT1 and IT3, only 26% of the region increased over the small domain contains radial velocity data. The extra data, however, may not effectively improve retrieval quality. It is speculated that the added void region, which contains no data, acts as a buffer which eases the transition from the externally enforced boundary condition to the central data-rich region. To clarify this conjecture, case IT5 is conducted. This case is almost the same as IT1 except that constraints on elevation and azimuth angles are removed so that the amount of data nearly doubles. With more observational data, case IT5 should have yielded better results than IT1 if the buffer zone had insignificant effect on retrieval quality. The results, however, show that case IT5 yields worse retrieval, indicated by rms errors and correlation coefficients for variables *U,* *V,* and *θ.* Although the rms error of the retrieved *W* velocity for case IT5 is 0.008 m s^{−1} smaller than that of IT1, it is insignificant compared with the *U* and *V* components.

*N*is the number of control variables. NTER is the number of iterations, and

*ξ*

_{i}is the cost function gradient with respect to the

*i*th control variable. Figure 5 illustrates the effect of buffer zone size on retrieval. The number of iterations required to meet the termination criterion, the normalized cost function, the rms errors, and the correlation coefficients of these cases are plotted versus buffer zone size. The results show that solutions converge faster with buffer zones introduced. An optimal buffer zone size for retrieval is indicated by variations of the cost function, the rms errors, and the correlation coefficients. Buffer zones with two vertical grid planes on each side are most effective in retrieval of the “centered column” volume data. For a sector-shaped lidar scan volume, it is difficult to determine optimal buffer size because volume scan shape depends on elevation and azimuth angles. Different lidar scan schemes yield different constraints on these angles and the range. The above experiments recommend use of a big model domain covering sector-shaped HRDL data for real lidar data retrieval.

### d. Observational errors

In contrast with the radial velocity provided for the previous ITEs, real lidar observations always contain errors. Lin et al. (2001) conducted sensitivity tests on observational errors of various amplitude and spatial correlation. With introduction of buffer zones, this issue must be reexamined. As quality control is routinely applied in postprocessing, errors are not severe in HRDL observational data. For conditions typically encountered in the ABL, the radial velocity measurement precision is often specified as 0.1 m s^{−1} (Grund et al. 2001; Blumen et al. 2001). To simulate actual lidar observational errors, uniformly distributed random errors are added to observations used in cases IT1 and IT2, which apply different boundary conditions. The error amplitudes added to case IT1 are chosen as 20% of radial velocities. The case is listed as IT6 in Tables 1 and 2. Case IT7 is modified from case IT2 by adding random errors in the range of −0.1 to 0.1 m s^{−1}. Comparison of the results between cases IT6 and IT1 shows that the retrieved flow fields are only slightly affected. Increases of the rms errors *ϵ*_{U} and *ϵ*_{V} by 0.018 and 0.010 m s^{−1}, respectively, are observed after introducing the random errors. The difference is negligible. Comparison between cases IT7 and IT2 shows consistently that observational errors only slightly affect the solution if not negligible. These results show that the 4DVAR is insensitive to the presence of small random observational errors.

### e. Retrieved flow structures

In this section, a detailed inspection of retrieved flow structures is presented. In Table 2, case IT2 produces the best retrieval. Vertical distributions of the correlation coefficients and the rms errors of this case are displayed in Fig. 6. Since the number of radial velocity data decreases with height, the retrieval quality deteriorates with height too, having larger rms errors and smaller correlations. Figure 7 displays the retrieved instantaneous flow field at *t* = 150 s in the horizontal plane *z* = 229 m and the corresponding flow field used to generate observations. The mean value calculated at that level has been deducted from the total in order to show the flow pattern clearly. There are discrepancies in the small-scale flow structures, but the larger-scale flow structures agree well. The magnitudes of the retrieved fluctuating components are in general smaller than those of the original flow field used for synthetic observation generation, especially for the cross-beam components. This may be attributed to the smoothness constraint that removes the effect of high-spatial-frequency observational errors. In the next section, buffer zones and inflow/outflow boundary conditions are applied to the 4DVAR analysis of HRDL observations.

## 5. HRDL data retrieval

### a. Computational grid

A computational domain of 2.4 km × 2.4 km × 1.25 km is employed in the 4DVAR analysis of HRDL observations. Except in one case for grid sensitivity test, all other cases use a grid of 48 × 48 × 25. Axes *x,* *y,* and *z* direct to the north, west, and upward, respectively. The coordinates of the HRDL location is (0, 0, 25) m. Figure 1 shows that the chosen computational domain covering most of the scan volume. Buffer zones are located between the lateral boundaries and the region containing data.

*d*

^{(i)}is the distance between vertex

*i*and point c. Bad observational data as determined by the lidar quality control algorithm are excluded from averaging. If all eight vertices are bad,

*U*

^{obs(c)}

_{rad}

### b. Input data and mean profiles

*z*= 0 m) to 1914:33 UTC (

*z*= 1250 m) on the same day is provided at CASES-99 Web site (http://www.joss.ucar.edu/cases99/). The virtual potential temperature profile can then be calculated (Jacobson 1999). The resulting profile is shown in Fig. 8. The height of the mixing layer is about 700 m. Mean velocity profiles, used in inflow/outflow boundary conditions, are approximated using a modified velocity– azimuth display (VAD) technique (Banta et al. 2002), where the scan volumes are uniformly divided into bins along the vertical. Data in each bin are processed independently to obtain mean velocities of

*U*

_{b}and

*V*

_{b}at the given height. Assuming zero mean vertical velocity,

*U*

_{b}and

*V*

_{b}can be obtained through an optimization procedure. The cost function of each bin is defined aswhere

*γ*is the elevation angle,

*ϕ*is the azimuth angle, and

*U*

_{rad}is the measured radial velocity. The running index

*j*represents each of the observations in a given bin. Minimization of the cost function (9) by solving ∂

*L*/∂

*U*

_{b}= 0 and ∂

*L*/∂

*V*

_{b}= 0 gives the values of

*U*

_{b}and

*V*

_{b}for a bin. Figure 9 shows the wind profiles obtained with two division heights; that is,

*dz*= 10 m and

*dz*= 50 m, using all 19 scan volume datasets. The results are consistent although the profile using a small

*dz*= 10 m is not smooth due to decreasing data density in each bin. These “bin” mean profiles vary slightly, but are found to have little effect on retrieval if buffer zones are applied. For the following analyses, the mean velocity profile approximated with

*dz*= 50 m using VOL13–VOL17 data is applied to specify inflow/outflow boundary conditions. The mean velocities above a capping inversion layer (

*z*≥ 750 m), where data are scarce, are specified as

*U*

_{b}= 6.7 m s

^{−1}and

*V*

_{b}= 0.0 m s

^{−1}. The surface temperature 298.0 K, the net radiation temperature flux 0.25 K m s

^{−1}, and the friction velocity 0.35 m s

^{−1}, from the CASES-99 Web site, are provided to the 4DVAR model for specifying surface boundary conditions.

As shown in Fig. 2, a flow structure characterized by high radial velocity moved through the scanned region during VOL13–VOL17. Four 4DVAR experiments are conducted based on the five volumes of data to study this structure. The simulation time window for each case is 180 s, which covers two volume scans. Lin et al. (2001) showed that two volumes of data are needed for the 4DVAR using Doppler lidar observations. Each case thus assimilates two volumes of data. For instance, case AC1 uses observations VOL13 and VOL14 as inputs, and case AC2 uses VOL14 and VOL15, etc., as shown in Table 3.

### c. Radial velocity misfit

*U*

_{rad}denotes would-be radial velocity, and

*U*

^{obs}

_{rad}

^{−1}in the first 10 iterations. The convergence speed slows down eventually. From iteration 100 to 200, Δ only reduces from 0.178 to 0.131 m s

^{−1}. The termination criterion shown below is set for all the cases in this section:

^{−1}

*U*

^{obs}

_{rad}

^{−1}, and are quantified below.

Figure 12 shows the probability density function (PDF) of radial velocity misfit for case AC2. The PDF is constructed using an interval of 0.01 m s^{−1} with 33 422 observations. Distribution is nearly symmetrical with respect to *U*_{rad} − *U*^{obs}_{rad}^{−5} m s^{−1}. The standard deviation of misfit is 0.2 m s^{−1}, consistent with the specified termination criterion. A normal distribution function based on the same mean and standard deviation is plotted for comparison. The actual distribution has a higher probability of occurrence at a smaller misfit than the normal distribution. This implies good compatibility of retrieved wind data with actual lidar observations. Such an unbiased distribution of the misfit indicates that the current prediction model could reasonably describe the dynamics associated with the observations. Figures 13 and 14 show the comparable values for the instantaneous velocity fields. However, the model is far from being perfect. Model errors are always present due to lack of resolution, inaccuracy of physical parameters, and inexact boundary conditions. It should be noted that the magnitude of the misfit is not equivalent to the accuracy of the retrieval, which depends on both observational and model errors.

### d. Grid resolution

Use of a fine grid is expected to generate more detailed information, but it is prohibitive in practice due to intensive computation and memory demands. The previous HRDL data assimilation would require 4 GB of computer memory if the spatial resolution were doubled in each dimension. ABL flow usually varies more in the vertical direction than in the horizontal directions, so case AC5 with *NZ* = 50 that doubles the vertical grid resolution of previous cases is performed to evaluate the retrieval sensitivity to grid resolution.

Figure 15 shows eddy viscosity profiles retrieved from cases AC3 and AC5. The two profiles have similar shapes. The one with lower resolution, case AC3 with *NZ* = 25, results in larger eddy viscosities. Since subgrid Reynolds stresses are approximated by eddy viscosities, a low-resolution case containing more subgrid turbulence energy is expected to generate large eddy viscosities. This explains the difference in the two profiles. Local maxima of the two profiles suggest that turbulent motions are the strongest near the ground (*z* ≈ 50 m). There exists another strong turbulent region close to the capping layer, where *z* ≈ 600 m.

Retrieved velocity fluctuations of cases AC3 and AC5 are displayed in Fig. 16. Most large-scale flow features captured are the same in both cases. There are some differences in small-scale flow structures. To identify the source of the differences, the 3D velocity vectors of case AC3 are subtracted from those of case AC5. This requires interpolation of AC3 data into the fine mesh of AC5. Absolute differences in *U,* *V,* and *W* components averaged throughout the domain are 0.29, 0.52, and 0.65 m s^{−1}, respectively. Figure 17 shows difference vectors in horizontal and vertical planes. The main differences lie in the cross-beam direction, normal to the radial-beam direction of lidar, due to the lack of tangential direction data. To quantify the differences, rms values of vectors in Figs. 17a and 17b in radial and tangential directions are computed. They are 0.16 and 0.83 m s^{−1} in the radial and tangential directions in Fig. 17a, and are 0.23 and 0.69 m s^{−1} in Fig. 17b.

Figure 18 displays the PDF distribution of observations and rms magnitudes of the above difference vectors as a function of radial distance (range) from the lidar. The PDF distribution of observations indicates data frequency, showing HRDL data becoming scarce at very small and large distances, as expected. Maximum PDF occurs at around the range of 1400 m. Data frequency obviously correlates with the rms of differences between fine and coarse grid solutions, as shown in the figure. Large differences occur when data are lacking. Since the central region of the model domain is where most data are located, the retrieval in the region is expected to be the most accurate.

### e. Observational errors

It is often a concern that data assimilation results might be sensitive to observational errors. Therefore, it is important to conduct error sensitivity tests with real observational data. We expect that the 4DVAR analysis is not sensitive to perturbation within the measurement precision ranges. Otherwise, the 4DVAR results are not trustworthy.

Depending on different aerosol backscatter conditions and lidar operating parameters, HRDL radial velocity precisions achieved during field experiments were documented from 0.06 to 0.3 m s^{−1} (Grund et al. 2001). Three sets of uniformly distributed random errors are added to the observational data used for case AC2. As listed in Table 4, the maximum magnitudes of added random errors are 0.1, 0.2, and 0.4 m s^{−1}, respectively. Thus, three pseudo-observational datasets are generated. 4DVAR experiments similar to AC2 are repeated accordingly. Comparing the retrieval results with that of AC2, we calculate rms differences between the results at the middle of the assimilation time window for variables *U,* *V,* *W,* and *θ.* They are listed in Table 4. The rms differences seem close to the magnitudes of perturbations added in the pseudo-observations. The retrieved flow structures from four observational datasets, that is, the original HRDL dataset and the three pseudo-observational datasets, at *z* = 175 m, are shown in Figs. 13 and 14. When the maximum magnitude of added random errors is 0.1 or 0.2 m s^{−1}, it is difficult to identify the effect of the added perturbation to the observations. Even if the perturbation is much larger than the typical HRDL precision, the retrieval results still do not change much. These results suggest that the current data assimilation is not sensitive to the observational errors of HRDL data and the minimization solution is unique.

### f. Instantaneous flow structures

HRDL scan volume is composed of radial velocity along light beams in the radial direction at different instants. The 4DVAR fills in variables that the HRDL cannot sense, throughout time and space, and integrates different variables consistently to develop missing information. Recovered 3D instantaneous velocity vectors make it possible to study microscale atmospheric structures, which are difficult to model and observe.

Figures 19 and 20 display horizontal and vertical slices of 3D flow fields. These are taken from cases AC1– AC4 retrieved results (Table 3) at the middle of the assimilation time window. In Fig. 19, velocity fluctuations in a horizontal plane at *z* = 175 m are displayed. Radial velocity *U*_{rad} = 4.0 m s^{−1} contours denoted by bold dotted–dashed lines are superimposed on vector fields to identify high-speed structures. A high-speed structure is seen coming from the left (south) in Fig. 19a. Another is found near the upper-right corner. Recovered velocity fluctuations are consistent with these contour lines. In subsequent frames (Figs. 19b–19d), the structure on the left progresses toward the north and veers slightly to the west, finally merging with the structure on the right, becoming a big structure. Tracing a region marked by a solid circle in Figs. 19a–19c, the front of the high-speed structure moves at about 6 m s^{−1}.

Figure 20 displays fluctuating velocity vectors in a vertical plane at *y* = 0 m, corresponding to the event identified in Fig. 19. This reveals horizontally aligned vortical (roll) structures, denoted by solid lines with arrows, progressing from the left to the right (Figs. 20a– 20d). Pressure fluctuation contours are superimposed to illustrate the downdraft and to locate the outflow front in the vertical planes. Comparison with the locations of vortices suggests that vortices are formed ahead of and above the moving front. These vortices cause strong updrafts and undulation aloft, indicated by the dashed lines in Fig. 20. In theory, the downdraft should be physically associated with low-temperature heavy air. The recovered temperature field thus is examined in Fig. 21. The low-temperature region coincides with the downdraft, which pushes the high-temperature region to the right (north). The highest temperature fluctuations occur around *z* = 500 m and correlate with the strong updrafts, which may entrain warm air from the capping layer into the boundary layer.

During field experiment CASES-99, there were a lot of other instruments deployed. Figure 22 shows a 60-m main tower located at the center of our computational domain, downwind of the large radial velocity blob seen in the lidar data. Vertical velocity components measured by eight sonic anemometers along the tower are plotted in Fig. 23. The plot covers the period when volumes VOL13 to VOL16 were scanned. It shows a relatively strong updraft occurring around 1942:50 UTC. The downdraft occurring 90 s later is only apparent near the top of the tower. It is certainly possible that the downdraft is even stronger above the tower top, but it cannot be said with certainty. Note that case AC1 covers from 1942:28 to 1945:28 UTC in time, while AC2 covers from 1944:00 to 1947:00. The general trend in the tower vertical velocity measurements seems to correlate with the retrievals shown in Fig. 20. That is, there is an updraft advecting away from the lidar, passing through the tower. It is then followed by a downdraft. Quantitative comparison bewteen the magnitudes of retrieved and measured vertical velocity at 50 m is difficult. The reason is as follows. The spatial resolution in the vertical direction for cases AC1 and AC2 is only 50 m, which matches well with that of HRDL observations. At the tower location, the lidar scans with elevation angles of 1° and 3° correspond to *z* = 25 m and *z* = 76 m. While the sonic anemometers measure winds at their exact locations, the retrieved vertical velocity represents the “resolved” component (a volume-average value from the finite-volume point of view). The total vertical velocity at any given point should consist of resolved and SGS components. The latter is nontrivial to calculate.

The downdraft observed in Fig. 20 resembles a microburst (Lester 1995), which is depicted as a strong downburst with a small length scale. Table 5 lists the features of a typical microburst and those of the retrieved structure described previously. The length scale of the retrieved downdraft is 1000–2000 m, estimated from the region of high pressure fluctuations in Fig. 20. While the same downdraft is found in the retrieved flow fields for all four cases using observations VOL13– VOL17, it is not observed in the data retrieved from VOL12 or VOL18. This suggests that the downdraft lifetime is at least about 8 min, spanning from VOL13 to VOL17. The peak outflow velocity of the retrieved structure is about 6 m s^{−1}, smaller than 10–12 m s^{−1} of a typical microburst. It appears that the retrieved downdraft has similar length scale and lifetime as a typical microburst, but with smaller strength in peak outflow velocity. The typical microburst is induced by thunderstorm (wet) or virga (dry), but there were no such events present at the time of the current retrieved downdraft. Thus, the retrieved downdraft is not a microburst event. Here, a microburst is simply used for comparison with the retrieved downdraft flow structure.

After data assimilation, many analyses are possible with the retrieved flow field information. For instance, 3D vortical structures can be revealed by displaying the pressure distribution (Jeong and Hussain 1995). Figure 24 shows the isosurface of negative pressure fluctuation *p*/*ρ*_{o} = −5.0 m^{2} s^{−2}, with pressure contours and fluctuation velocity vectors shown in the two vertical planes at *x* = 1575 m and *y* = −25 m. The high pressure region indicates the downdraft and the velocity divergence can also be clearly identified near the ground in Fig. 24. The data are retrieved from case AC5 (the fine-grid case) at *t* = 90 s. A big spanwise vortical structure is seen at the center of the computational domain close to *z* = 500 m. Another vortex at (*x,* *y*) = (1500, −500) m touches the ground. Fluctuating velocity vectors in the two vertical planes at *y* = 175 m and *y* = −575 m are shown in Fig. 25, confirming that these low pressure regions indicated by the arrows are indeed vortical motions. It is noteworthy that the locations of the vortices occur at the heights where retrieved eddy viscosities, shown in Fig. 15, are local maxima.

The above observations are illustrated in Fig. 26. A downdraft creates strong high-speed outflow motion. The motion subsequently generates vortex A, which resembles the vortex ring in a typical microburst; vortex B, which precedes the outflow front; and wavelike motion between the vortices. Upward motion of vortex A could entrain the warm air from the capping layer into the boundary layer.

## 6. Conclusions and discussion

In this paper, a 4DVAR method is applied to high-resolution Doppler lidar radial velocity data. Before assimilating actual lidar data, ITEs are used to evaluate the effects of buffer zones and boundary conditions. Results show that buffer zones placed outside lidar volume data improve retrieval quality. It is speculated that the data void eliminates the discontinuity between the observations and externally enforced inaccurate boundary conditions. With buffer zones employed, the 4DVAR can yield good results with a simplified lateral boundary condition. The current 4DVAR treats the initial fields and profiles of eddy viscosity and thermal diffusivity as control variables. The model is then applied to assimilate HRDL observations. Radial velocities reconstructed from the retrieved flow fields agree well with the original data. Credibility of retrieved data is first assessed by a grid sensitivity test. It is found that the major source of errors in the 4DVAR lies in the cross-beam velocity component when data are scarce. Then sensitivity tests are conducted on observational errors. By adding random errors with the magnitude up to 0.4 m s^{−1} to the original HRDL observations, we find that the retrieval is not sensitive to the observational errors. A number of retrieval experiments are performed to generate a time sequence of 3D wind and temperature fields in a convective boundary layer. A downdraft resembling a dry microburst is identified. The flow structure generates vortices and wavelike motions.

During the month of CASES-99 field experiments, several interesting phenomena were observed. For instance, Newsom and Banta (2003) and Blumen et al. (2001) analyzed a Kelvin–Helmholtz billow event in the nocturnal boundary layer. The huge amount of measurement data from CASES-99 offers a great opportunity to study atmospheric structures. It also poses a big challenge to extract useful information from measurements. The current application of the 4DVAR to HRDL observations reveals microscale atmospheric flow structures, which are otherwise not apparent or visible by merely displaying radial velocity observations. The ability to generate coherent flow fields through the combination of observations and prediction model demonstrates that 4DVAR can be a useful tool for the study of microscale atmospheric flow structures.

In our application of the 4DVAR to real data, the physical parameters, that is, the vertical profiles of eddy viscosity and thermal diffusivity, are optimized along with the initial conditions. The flexibility that it brings to the prediction model helps to reduce the strong constraint of a perfect model approach, adopted in the current 4DVAR application. However, model errors still exist. The weak constraint approach, for example, Zupanski et al (2002), may be used to properly account for model deficiency.

## Acknowledgments

This work was supported by the National Science Foundation (Grant ATM-9874925), monitored by Dr. Roddy R. Rogers. The National Center for Atmospheric Research, the National Partnership for Advanced Computational Infrastructure, and the National Center for Super-computing Applications, sponsored by the National Science Foundation, are acknowledged for the computing time used in this research. Funding for field measurements was provided by the Army Research Office, the Center for Geosciences/Atmospheric Research at Colorado State University, and the National Science Foundation (Grant ATM-9908453). The authors are indebted to the organizers of the CASES-99 field program and to the members of the Optical Remote Sensing Division at NOAA/ETL. The authors would like to thank three anonymous reviewers for their helpful suggestions and comments.

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ITE descriptions: BC is boundary conditions; restrictions, restriction on ranges of elevation, and azimuth angles for observations

Correlation coefficients (*σ*) and rms errors (*ϵ*) of ITEs at 50 iterations, averaged in the scan volume throughout the computational domain below *z* = 400 m

Descriptions and results of cases using HRDL data

Rms differences between retrieval results of case AC2 and the sensitivity tests. Random errors are added to the original HRDL observational data for the sensitivity tests

Comparison between typical microbursts (Lester 1995) and the retrieved downburst