Lidar data quality control and radial velocity measurement error

The lidar data were quality controlled to remove radial velocities corresponding to weak return signals using a simple threshold technique based on the signal-to-noise ratio (SNR). Radial velocities corresponding to large spikes in the SNR field at low elevation angles were also rejected because these are typically associated with hard-target returns, such as terrain features, buildings, trees, power polls, etc. Furthermore, we excluded measurements beyond a range of 4 km for the ASU lidar and 5 km for the ARL lidar. Figure 3 displays quality-controlled radial velocity data for both the ARL and ASU lidars during the assimilation period (1802:25–1805:17 UTC 11 July 2003). During this period, the range of the ASU lidar was limited to approximately 4 km, whereas the ARL lidar provided good-quality radial velocity measurements well beyond 5 km.

The NB method uses estimates of the radial velocity measurement error to account for the range dependence of the data quality. A relationship between radial velocity measurement error and SNR is obtained experimentally, assuming that the dominant source of error in the strong signal regime is random noise that is induced by the mean frequency estimator (Rye and Hardesty 1993; Frehlich and Yadlowsky 1994).

The random noise variance in the radial velocity signal is estimated from an analysis of the fixed-beam data using the autocovariance method as described in Newsom and Banta (2004a) and Mayor et al. (1997). Figure 4a displays average wide-band signal-to-noise ratio (wSNR) and velocity error σ as functions of range for both the ARL and ASU lidars. Here we define the velocity error σ as being the standard deviation of the noise in the radial velocity signal. The wSNR is defined as the ratio of the Doppler spectrum integrated over the width of the signal-to-noise level integrated over the same width. The ARL wSNR and velocity error curves were computed using fixed-beam data that were acquired between 2228:58 and 2258:56 UTC 8 July 2003. The ASU wSNR and velocity error curves were computed using fixed-beam data that were acquired between 2306:46 UTC 13 July and 0646:46 UTC 14 July 2003. For both datasets, the lidars stared nearly horizontally. The ASU wSNR was consistently lower than the ARL wSNR, and this resulted in poorer range performance of the ASU lidar. In the high-wSNR regime, radial velocity error for both lidars is approximately 15 cm s⁻¹. However, the errors increase markedly as the range exceeds 3 km for the ASU lidar and 5 km for the ARL lidar.

The range performance of the lidar can change depending on the concentration and characteristics of particulates in the atmosphere. Thus, a more consistent measure of performance is obtained by relating velocity error to the average wSNR. Figure 4b displays plots of velocity error versus wSNR for both lidar systems. The solid (dashed) curve in Fig. 4b is a fit to the ARL (ASU) data using an analytical expression. The curve fits in Fig. 4b were used to convert wSNR measurements that were acquired during the assimilation period to velocity error.

We should emphasize that Fig. 4 characterizes the performance of the instruments during the JU2003 deployment. Since that deployment, the ASU lidar has undergone tuning. As a result, the performance of the ASU lidar has improved significantly.

Base states

The NB method also requires estimates of vertical profiles of the mean wind and potential temperature as first-guess fields. These profiles are referred to as the base state. The base-state wind and virtual potential temperature profiles are shown in Fig. 5. The virtual potential temperature profile was obtained from a radiosonde that was released at 1800 UTC 11 July 2003. The acquisition of volume scan data was timed to coincide with this radiosonde release. The release site was located inside the CBD, and was 2.6 km west and slightly south of the ARL lidar. This places the release site just slightly outside of the northern limits of the ARL lidar volume scan. Pressure, temperature, and relative humidity data from the sonde were converted to estimates of the virtual potential temperature. The virtual potential temperature profile in Fig. 5 indicates a deep superadiabatic layer from the surface up to about 300 m AGL. The base of the capping inversion layer is located at about 1200 m AGL.

The base-state wind profiles shown in Fig. 5 were computed from the ARL lidar volume scan data using a velocity azimuth display (VAD)-type processing technique (Banta et al. 2002; Chai et al. 2004). The data used in this computation are the same data that are used in the single-Doppler retrieval. Mean wind profiles were also computed using the ASU lidar volume scan data during this same period. The profiles that are derived from the ASU lidar showed good agreement with the ARL-derived profiles. However, the profiles that are derived from the ARL lidar data are less noisy, as a result of the higher SNR.

Volume scan data were acquired during an early afternoon period when conditions at the site were cloud free, hot, humid, and windy. The air temperature at 2 m AGL was about 38°C. The wind profiles shown in Fig. 5 indicate a strong southerly flow that is characteristic for this location at this time of the year. Wind speeds increase from ∼5 m s⁻¹ at the surface to about 9 m s⁻¹ at ∼1000 m AGL, and then decrease to between 6 and 7 m s⁻¹ above that level. Above approximately 1000 m AGL the winds veer, becoming westerly at 2 km AGL. The wind profile in Fig. 5, in fact, shows the remnants of the previous night’s low-level jet. Strong convective mixing during the late morning and early afternoon has eroded the jet structure from below.

Time synchronization and geolocation

For any analysis involving dual-Doppler lidar measurements, it is critical that accurate geolocations are determined for each lidar. It is equally important to ensure that the clocks in the two data systems are synchronized, and that the scanner-pointing angles are referenced to the same coordinate system.

The clocks in both lidar systems were periodically checked throughout the course of the deployment to ensure that they were synchronized to coordinated universal time. Scanner-pointing directions were calibrated in order to establish zero azimuth as true north, and zero elevation as horizontal. Despite this, it is possible that small systematic errors or offsets may exist in the azimuth data from the two lidars. This possibility will be explored in conjunction with the presentation and analysis of the retrieval results in section 4.

Forward model

The forward model simulates dry, shallow incompressible flow under the Boussinesq approximation. The model equations are given by

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The acceleration of gravity is g, Θ_ref is a reference virtual potential temperature, p is the pressure normalized by a constant air density, and S_ij = 0.5(∂u_i/∂x_j + ∂u_j/∂x_i) is the resolved-scale strain-rate tensor. The virtual potential temperature is θ = θ̂ + θ_b, where θ_b is the so-called base-state virtual potential temperature and θ̂ is the departure from the base state. The horizontally averaged virtual potential temperature is 〈θ〉. The coefficient of eddy diffusivity K_m is modeled using an expression that is similar to the one proposed by Troen and Mahrt (1986). This is given by

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where z_max is the height of the computational domain, K_max determines the maximum value of K_m, and α controls the shape of the profile and the height of the maximum. We further assume that the relationship between K_m and the heat diffusivity K_h is given by

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In general, the relationship between K_m and K_h is expected to be a function of stability, such that K_h → 3K_m in the free convection limit, and K_h → K_m in the stable limit (Businger et al. 1971). However, we have found that the retrieval results are not very sensitive to small changes in the proportionality between K_m and K_h (Newsom and Banta 2004b).

The model and its adjoint are integrated using a space-centered forward-in-time scheme on a staggered grid with periodic lateral boundary conditions. At the top and bottom of the domain the vertical velocity vanishes and the potential temperature is set to its base-state value. Horizontal velocities are set to zero at the bottom of the domain. At the top of the domain the horizontal velocities are set the base-state values derived from a VAD-type (Banta et al. 2002; Chai et al. 2004) analysis of the lidar data.

Cost function

Retrievals are performed by finding the initial model state [u(r, t = 0) and θ(r, t = 0)] that minimizes the so-called cost function. The cost function, a quantitative measure of the difference between the measurements and the model output, is given by

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The primary constraint J_obs expresses the difference between the observed radial velocities from the lidar(s) and the radial component of the velocity field generated by the model. A second constraint, or penalty term, J_d is included in order to suppress the divergence of the initial velocity field (Sun et al. 1991; Sun and Crook 1994; Lin et al. 2001; Chai et al. 2004; Newsom and Banta 2004a).

Equation (8) in Newsom and Banta (2004a) can be modified for multiple-Doppler data by simply adding the contributions from each lidar independently. In this case, J_obs takes the following form:

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where

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and

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with analogous expressions for J^ASU_obs and ▵^ASU_m. Summation over the observations from each lidar is denoted by Σ_m, and r̂^ARL_m is the unit vector from the ARL lidar to the mth measurement of the ARL lidar. Likewise, u^ARL_rm refers to the radial velocity measurements from the ARL lidar and the σ^ARL_rms are the corresponding measurement errors. The weighting parameters a^ARL and a^ASU are used to control the contribution of each lidar to J_obs.

The velocity field that is generated by the forward model u is interpolated to the coordinates of each observation. Thus, u_m denotes the model velocity field that is interpolated to the space and time coordinates of the mth observation. The interpolation is essentially a weighted average of u. When interpolating u to the coordinates of the ARL lidar measurements, u_m takes the following form:

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In (10), r is a position vector to a model domain node, thus, Σ_r implies summation over the model domain nodes. The time of the mth observation from the ARL lidar is t^ARL_m, and t_n is the time of the nth time step of the model output. For each observation m, the weighting function W is normalized such that Σ^N−1_n=0 Σ_r W(r − r^ARL_m, t_n − t^ARL_m) = 1. Linear interpolation in space and time is used to transform u to the observational coordinates. Equation (10) is also used to interpolate model output to the coordinates of the ASU lidar data. In this case, we replace t^ARL_m with t^ASU_m and r^ARL_m with r^ASU_m + R, where R is the position vector of the ASU lidar relative to the ARL lidar (see Fig. 1).

The cost function is minimized using a conjugate gradient method that is based on the Polak–Ribiere algorithm (Press et al. 1988). The gradient of the cost function with respect to the initial model state is obtained by solving the adjoint of the forward model.

Newsom and Banta (2004a) derived the adjoint equations for the model described above for the case of single-Doppler input. The gradient of the cost function with respect to u enters as a forcing term in the adjoint equation. Generalizing those results to multiple Doppler lidars is straightforward because the only modification involves the forcing term. In this case, the gradient of J_obs, with respect to the ith component of u, is given by

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where

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with an analogous expression for ∂J^ASU_obs/∂uⁿ_i. Thus, the only change in the adjoint equations presented in Newsom and Banta (2004a) involves the data forcing term. The modification simply involves adding the contribution from each lidar independently.

In the NB method, the gradient of the cost function with respect to the initial model state is computed from a numerical solution of the continuous adjoint equations. Errors are introduced when the cost-function gradient is computed in this fashion, because the discrete representation of the continuous adjoint is not, in general, the same as the adjoint of the discrete forward model. However, these errors are usually fairly small.

Daescu et al. (2003) examined differences between the gradients that are computed using continuous and discrete adjoints of a chemical kinetics model involving several hundred chemical species. In that study, the adjoint method was used to compute the change in the concentration of a single chemical species at the end of the simulation as a result of changes in the concentration of other species at the initial time. Estimates produced by the adjoint method were then compared with a direct finite-difference approximation. The relative errors in the continuous adjoint estimates were, on average, about 0.1%, whereas the discrete adjoint produced relative errors that were, on average, at least one order of magnitude smaller.

To test our adjoint code we compared its estimate of the cost-function gradient with that determined by a finite-difference approximation, as described in Sun et al. (1991) and Newsom and Banta (2004a). For this test we used the data described in section 2. The forward model was integrated over the assimilation period by setting the initial conditions equal to the base state. The adjoint equations were then integrated to compute the cost-function gradient, and that result was compared with the change in the cost function resulting from small random perturbations to the base state. Comparisons were performed in both the single- and dual-Doppler mode. We found that the average relative error in the gradient that was determined from our adjoint was 0.13%, regardless of whether single- or dual-Doppler data were used in the comparison. It is interesting to note that these results are consistent with those of Daescu et al. (2003), despite the vastly different physical models.

The retrieval algorithm was developed in C and executed on a personal computer (PC) with a single 2.4-GHz Pentium-4 processor with 1 Gb of random access memory (RAM). All computations were carried out in double precision. The retrieval algorithm was initiated by setting the first-guess fields to the base-state wind and potential temperature profiles that are shown in Fig. 5. The algorithm was terminated after approximately 160 iterations, or when the fractional change in the cost function, from one iteration to the next, became less than 10⁻⁸. Each retrieval required about 50 h of central processing unit (CPU) time.

Single-Doppler retrieval

Figures 6 and 7 show the horizontal and vertical structure of the wind and temperature fields from the single-Doppler retrieval. Figure 6 shows two-dimensional horizontal cross sections of the retrieved perturbation (u′, υ′) vector field (left panels) and contours of the perturbation virtual potential temperature field (right panels) at two selected vertical levels. Perturbation variables are defined as

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where ξ refers to any prognostic variable, and 〈ξ〉 is the horizontal average of ξ. Figure 7 displays y–z cross sections of the retrieved perturbation (υ′, w′)vector field and contours of the perturbation virtual potential temperature field. Figure 7 is essentially a vertical cross section along the streamwise direction, because the mean flow is southerly within the boundary layer. We note that the x′ and y′ axes shown in Figs. 6 and 7 are coincident with the x and y axes shown in Fig. 1 if it is assumed that there is no systematic azimuth offset in the ARL lidar scanner, as discussed later in this section.

In Fig. 6 it is apparent that the fluctuations in the retrieved fields are confined primarily to the region covered by the scan. This nonphysical effect occurs because the adjoint equations are locally forced by the data (Newsom and Banta 2004b). In regions devoid of data the forcing terms in the adjoint equations are zero and the retrieved fields are influenced only by the surrounding flow through the dynamic and kinematic constraints placed on the system. Away from the data-rich regions, the retrieval remains close to the prescribed base-state wind and temperature profiles. An exception to this behavior occurs as the eddy patterns advect downwind and out of the scan volume. This effect is indicated in Fig. 6.

Several flow features are highlighted in Fig. 7a to facilitate a comparison with the dual-Doppler retrieval. Regions labeled A, B, and D show updrafts, and the regions labeled C, E, and F show downdrafts. Comparing Fig. 7a with Fig. 7b, we observe a strong (positive) correlation between the vertical velocity and the perturbation virtual potential temperature, as expected for the convective boundary layer (CBL).

Profiles of the standard deviations of the retrieved u, υ, w, and θ fields are shown in Fig. 8. These profiles were computed using only those model grid points that occur within the dual-Doppler overlap volume. The averaging was restricted in this way so that differences between single- and dual-Doppler retrievals could be quantified. As indicated, horizontal velocity fluctuations are strongest near the surface and decrease rapidly toward the middle of the mixed layer. Both the u and υ components experience small secondary maxima near 800–900 m AGL. The standard deviation of w increases rapidly just above the surface, and reaches its maximum value of ∼1.4 m s⁻¹ between 100 and 200 m AGL. Above 1000 m AGL the standard deviation of w decreases to about 0.3 m s⁻¹ at ∼1500 m AGL. The shapes of these profiles are roughly consistent with large eddy simulations of shear-driven convective boundary layers (Moeng and Sullivan 1994).

To compare the single-Doppler retrieval with the ASU observations, we linearly interpolate the retrieved velocity field to the space and time coordinates of the ASU radial velocity samples, and then compute the radial component relative to the ASU lidar. The result of this procedure is denoted by u_m · r̂^ASU_m, where r̂^ASU_m is a unit vector from the ASU lidar to its mth radial velocity sample. This comparison is performed using only those samples that fall within the scan volume of the ARL lidar. To establish a baseline, we also compare u_m · r̂^ARL_m with u^ARL_m, where u_m · r̂^ARL_m is the radial component of the retrieved field that is interpolated to the space and time coordinates of the radial velocity data from the ARL lidar u^ARL_rm.

A correlation diagram between u_m · r̂^ASU_m and u^ASU_rm is shown in Fig. 9a. For comparison, Fig. 9c shows the correlation between u_m · r̂^ARL_m and u^ARL_rm. As one would expect, the correlation between u_m · r̂^ARL_m and u^ARL_rm is very good, with relatively little scatter about the oneto-one line. The correlation between u_m · r̂^ASU_m and u^ASU_rm exhibits more scatter but still appears to be reasonably good. We note that in Fig. 9a there is less scatter about the one-to-one line for negative radial velocities. The scatter increases abruptly for positive velocities. An inspection of Figs. 3c and 3d show that negative values of u^ASU_r occur primarily in the stable layer above ∼1200 m AGL, where the flow is relatively uniform. Positive values of u^ASU_r occur primarily in the convective mixed layer where the fluctuations are stronger. In the stable layer the retrieved fields do not deviate significantly from the prescribed base- state. Thus, the strong correlation in this layer is primarily the result of good correlation in the mean vertical structure as observed by each lidar independently.

Distributions of Δ^ASU_m and Δ^ARL_m are shown in Figs. 9b and 9d, respectively. We find that 68% of the ASU radial velocity samples occur within ±1 m s⁻¹ of u_m · r̂^ASU_m. As expected, the distribution of Δ^ARL_m is sharply peaked, with 99% of the ARL radial velocity samples occurring within ±1 m s⁻¹ of u_m · r̂^ARL_m.

The distribution for Δ^ARL_m is very nearly Gaussian with zero mean, while the distribution for Δ^ASU_m exhibits a slight skewness toward positive values, and a peak that occurs at about 0.3 m s⁻¹.

We suspect that the bias in the distribution of Δ^ASU_m is the result of slight systematic offsets in the scanner alignments. The question then becomes: how do we correct for offsets in the azimuth data to eliminate the bias in the distribution of Δ^ASU_m? To address this problem we consider the geometry shown in Fig. 10. Errors in the ARL and ASU azimuth data are denoted by δϕ_ARL and δϕ_ASU, respectively. There are two approaches one could use to evaluate the affects of azimuth errors on the distribution of Δ^ASU_m. The first approach simply involves performing multiple runs of the single-Doppler retrieval with a range of δϕ_ARL values. These retrievals can then be compared with the radial velocities from the ASU lidar using a range of δϕ_ASU values. The second approach involves performing only one single-Doppler retrieval. This retrieval can then be compared with the ASU lidar data by transforming both the position of the ASU lidar and its observations into the frame of reference of the ARL lidar. We opted for the second approach because it involves far less computation.

To implement the second approach we refer once again to the geometry shown in Fig. 10. The position vector from the ARL lidar to mth measurement of the ASU lidar is given by

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We seek to express Eq. (14) in the ARL frame (primed coordinate system) in terms of known quantities. The location of the measurement relative to the ASU lidar r^ASU″_m and the location of the ASU lidar relative to the ARL lidar R are both known. The transformations from the unprimed to the ARL (primed) and ASU (double primed) coordinate systems are given by

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and

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where

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Equations (15) and (16) can be used to express r^ASU′_m in terms of r^ASU″_m, that is,

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In the primed coordinate system, Eq. (14) becomes

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Equation (19) determines the coordinates of a measurement from the ASU lidar given the location of the ASU lidar relative to the ARL lidar R, and the coordinates of a measurement relative to the ASU lidar r^ASU″_m. To compare the single-Doppler retrieval with the ASU observations, we compute the scalar product of the retrieved velocity with r^ASU′_m/|r^ASU_m| at the point determined by Eq. (19).

Using the approach outlined above we evaluated the mean of the distribution of Δ^ASU_m for a range of δϕ_ARL and δϕ_ASU values. Those values of δϕ_ARL and δϕ_ASU that resulted in = 0 are plotted in Fig. 11. This curve is nearly linear and extends well beyond the boundaries of the plot. A unique solution may be determined if we require that δϕ_ARL and δϕ_ASU be as small as possible. This is a reasonable assumption because it is not likely that the lidar operators made significant errors in the scanner alignment. With this assumption we find that the minimum distance from the zero offset point to the curve in Fig. 11 occurs when δϕ_ARL=1.78° and δϕ_ASU=−1.60°.

Figure 12a shows a comparison between u_m · r̂^ASU_m and u^ASU_rm, assuming azimuth offsets of δϕ_ARL=1.78° and δϕ_ASU=−1.60°. A correlation diagram between u_m · r̂^ASU_m and u^ASU_rm is shown in Fig. 12a, and the distribution of Δ^ASU_m is shown in Fig. 12b. The assumption of δf_ARL=1.78° and δf_ASU=−1.60° has two effects. The first effect is that the bias in the distribution of Δ^ASU_m is essentially eliminated. The second effect is that the distribution exhibits slightly less skewness.

Table 2 provides a summary of the results from the single-Doppler retrieval in terms of the coefficients of linear correlation and the root-mean-squared (rms) deviations between u_m · r̂^ASU_m and u^ASU_rm, and between u_m · r̂^ARL_m and u^ARL_rm. Results are shown assuming both zero azimuth offsets (δϕ_ARL=0, δϕ_ASU=0), and nonzero azimuth offsets (δϕ_ARL=1.78°, δϕ_ASU=−1.60°). The coefficient of linear correlation between two variables ξ and η is defined by

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and the rms deviation is given by

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The overbars in Eqs. (14) and (15) denote averages over all samples that occur within the dual-Doppler overlap volume.

Table 2 shows that when the measurements are assumed to contain no azimuth offsets, the coefficient of linear correlation between u^ASU_m · r̂^ASU_m and u^ASU_rm is 0.94, and the rms deviation is 1.18 m s⁻¹. The linear correlation between u^ARL_m · r̂^ARL_m and u^ARL_rm is 0.99, and the rms deviation is 0.24 m s⁻¹. Not surprisingly, the correlation between u^ARL_m · r̂^ARL_m and u^ARL_rm is quite good because the single-Doppler retrieval is derived from the ARL lidar data. Table 2 also indicates that the correlations and rms errors are not very sensitive to small changes in the assumed azimuth errors. This is because these statistics are dominated by the overall trends in the data, as opposed to microscale fluctuations in the velocity field.

Dual-Doppler retrieval

Given the possibility that errors exist in the azimuth data from either lidar, two dual-Doppler retrievals are performed. The two dual-Doppler trials are listed in Table 3. Dual-Doppler trial 1 assumes no azimuth offset, and trial 2 assumes an azimuth offset given by δϕ_ARL=1.78° and δϕ_ASU=−1.60°. The results of these two dual-Doppler trials are compared to the single-Doppler retrieval. When interpreting these results, one must bear in mind that the two dual-Doppler retrievals are performed in the frame of reference of the ARL lidar (the primed coordinates in Fig. 10). As a result, these retrievals are defined in slightly different coordinate systems.

The panels on the left side of Fig. 13 show horizontal cross sections of the perturbation (u′, υ′) vector field from the two dual-Doppler trials. These cross sections are taken at z = 200 m AGL, and can be compared with Fig. 6a. It is clear that the dual-Doppler dataset provides slightly better coverage of the domain. As in the single-Doppler case, fluctuations go to zero in regions that are devoid of data, except where eddies have advected downstream. Artifacts that are caused by the periodic lateral boundary conditions are also evident in Fig. 13. The fluctuations near the left boundary, where there are no observations, occur in response to motion within a data-rich region on the right side of the domain.

Figures 13b and 13d show horizontal cross sections of the difference between the dual- and single-Doppler velocity retrievals. Within the dual-Doppler overlap region there is a general tendency for differences to be larger where ever the ASU lidar dominates the retrieval. In these regions, the difference in the velocity fields can be nearly as large as the velocity fluctuations from either retrieval.

Figure 14 displays vertical cross sections of the retrieved perturbation (υ′, w′) vector field, at x′ = −1 km, for both dual-Doppler trials. These cross sections can be compared with the single-Doppler retrieval shown in Fig. 7a. All retrievals show a relatively weak updraft in region A, but the surrounding flow structure is slightly more complex in the dual-Doppler retrievals. This is expected because of the higher data density in the dual-Doppler dataset. For region B, the single-Doppler result (Fig. 7a) clearly shows an updraft occurring between two counterrotating vortices, whereas dual-Doppler trial 1 (Fig. 14a) shows convergence between an updraft from below and a downdraft from above. By contrast, dual-Doppler trial 2 (Fig. 14b) shows an updraft in this region that is more consistent with the single-Doppler retrieval. All retrievals show a downdraft in region C. Between regions C and D there is a clockwise rotation in both Figs. 14a and 14b that is less obvious in Fig. 7a.

The agreement between the single- and dual-Doppler retrievals tends to improve as the range to the ARL lidar decreases. As the range to the ARL lidar decreases the ARL data density and measurement precision increase, while the ASU measurement precision and data density decrease. Thus, the dual- and single-Doppler retrievals show similar flow structures in regions D, E, and F, because the ARL lidar data dominate the dual-Doppler retrievals in those regions.

Differences between the single- and dual-Doppler retrievals of potential temperature are illustrated by comparing the vertical cross sections shown in Fig. 7b with those in Fig. 15. Because w is strongly coupled to θ, we observe that the differences in θ among the various retrievals are correlated with the differences in vertical motion, as described above.

We caution against making a direct comparison between the dual-Doppler retrievals shown in Figs. 14 and 15. These figures seem to indicate that in some regions there are significant differences between the microscale velocity and temperature fields. These differences, however, are, in part, a result of the fact that Figs. 14a and 15a and Figs. 14b and 15b represent slightly different cross sections through the flow. As discussed above, the dual-Doppler trials were performed in the frame of reference of the ARL lidar. Because trials 1 and 2 assume different azimuth offsets, the coordinate axes for these trials are rotated slightly with respect to one another.

Table 4 lists correlation coefficients and rms deviations between radial velocity components of the two dual-Doppler trials and the observations. The correlation between u · r̂_ARL and u^ARL_r is about the same as in the single-Doppler retrieval. However, the correlation between u · r̂_ASU and u^ASU_r has improved from 0.94 for the single-Doppler retrieval to 0.98 for both dual-Doppler trials. The correlation between u · r̂_ARL and u^ARL_r is slightly better than the correlation between u · r̂_ASU and u^ASU_r, because the ARL lidar data make a larger contribution to the cost function because of the overall better data quality and range performance.

Profiles of the standard deviations of u, υ, w, and θ for dual-Doppler trial 2 are shown in Fig. 16. We note that for dual-Doppler trial 1 these profiles are nearly identical. As in Fig. 8, the averaging has been confined to the dual-Doppler overlap volume so that we may assess the impact of the dual-Doppler data on the retrieval. Comparing Figs. 8 and 16, we see only minor differences in the profiles for both the υ component and θ. In contrast, the rms fluctuations in the u component are significantly stronger in the dual-Doppler retrieval. The w component also indicates somewhat stronger rms fluctuations in the dual-Doppler retrieval.

The differences between Figs. 8 and 16 suggest that the single-Doppler retrieval underestimates the magnitude of velocity fluctuations that are orthogonal to the lidar beam, that is, crossbeam velocities. The addition of the ASU data affects mainly the u component because the ASU lidar is more sensitive to that component, whereas the ARL lidar is more sensitive to the υ component. The w component is less affected because it responds indirectly to the ASU data through the kinematic and dynamic constraints that are placed on the retrieval. The larger fluctuations in the w component are likely the result of its response to the increased u variance through mass continuity.

Figure 17 displays correlation diagrams between prognostic variables, from the single- and dual-Doppler retrievals. The subscripts “s” and “d2” are used to refer to the single-Doppler retrieval and dual-Doppler trial 2, respectively. Again, these comparisons are performed using only those grid points that fall within the dual-Doppler overlap region. The correlation diagrams for u, υ, and θ show distinctive changes in scatter along the one-to-one line. As noted earlier, the scatter about the one-to-one line for these variables is smaller for points within the stably stratified layer, because fluctuations are smaller and both the single and dual-Doppler retrievals remain close to the same prescribed base state in this layer.

In Fig. 17 it is also clear that the w components are less well correlated. Unlike θ and the horizontal velocity components, the horizontal mean of the vertical velocity component is essentially zero. The correlations in θ, u, and υ are heavily influenced by their mean vertical structures, whereas w is not.

The correlation diagram for the u component (Fig. 17a) shows a fairly obvious clustering of points extending above the one-to-one line near u_s = 0. Again, this indicates a tendency of the single-Doppler retrieval to underestimate the magnitude of the crossbeam velocity components. This effect is much more pronounced for the u than for the w component, because the w component is only indirectly affected by the addition of the ASU data.

Table 5 lists the coefficients of linear correlation and the rms differences between prognostic variables from the single-Doppler retrieval and both dual-Doppler trials. The correlations and rms deviations indicate that dual-Doppler trial 2 (δϕ_ARL=1.78°, δϕ_ASU=−1.60°) resulted in slightly better agreement with the single-Doppler retrieval. We also observe that all correlations, with the exception of w, are greater than 0.9. The correlations between the w components are about 0.73. As noted above, the correlations in θ, u, and υ are heavily influenced by their mean vertical structures. The mean profiles of θ, u, and υ from both the single- and dual-Doppler retrievals are determined largely by the same base-state profiles. Thus, the correlations in these variables may be artificially biased toward 1 because all retrievals use the same base-state profiles as the first-guess fields.

To suppress the influence of the mean profile in the comparison between the single- and dual-Doppler retrieval, we define the height-dependent correlation as

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where the perturbation variables (ξ ′ or η′) are defined by Eq. (13), and 〈〈〉〉 implies temporal and horizontal averaging. As before, the averaging is performed over the entire assimilation period for only those points that occur within the dual-Doppler overlap region. Because the horizontal mean is removed from each vertical level, Eq. (22) provides a better statistic for comparing the microscale structure. Figure 18 displays profiles of R_Z for the single-Doppler retrieval and dual-Doppler trial 2. The υ component exhibits the best correlation with values near 0.9 in the middle of mixed layer. The u-component exhibits the poorest overall correlation, with values ranging from 0.4 to 0.8. Once again, these profiles indicate that the retrieval algorithm has difficulty estimating the cross-beam component of the flow from single-Doppler data.

In Fig. 18 the correlations are best in the mixed layer well away from the surface and below the capping inversion. The correlations degrade near the surface and within the stable layer from roughly ∼1100 to ∼1800 m AGL. Above ∼1800 m the correlations improve as the top boundary is approached because both retrievals assume the same boundary conditions at the top of the domain.

The smaller values of R_Z in the stable layer do not contradict our earlier observation that correlations in u, υ, and θ improve in the stable layer (see Fig. 17). The correlations in u, υ, and θ do improve in the stable layer because these variables remain close to their horizontal means. The smaller values of R_Z in the stable layer indicate large uncertainties resulting from weak velocity fluctuations in the stable layer. The data are also generally noisier in this layer as a result of longer-range and reduced aerosol concentrations.

The degraded correlations in the surface layer may be the result of obstructions in the ARL lidar beam. Structures such as trees, power poles, traffic signs, and buildings caused obstructions of the laser beam at low elevation angles in certain azimuths. Regions where the ARL lidar beam was obstructed and where the ASU lidar was not obstructed would contribute to differences between the single- and dual-Doppler retrievals.

Because our data were acquired on a very hot day, it is possible that the effects of refractive turbulence may also be contributing to the degraded correlations in the surface layer. Refractive turbulence causes wave front distortion, which, in turn, decreases the coherent energy of the return signal (Frehlich 2000). The net effect is a decrease in the SNR and an increase in the radial velocity measurement error. Noisier measurements from both lidars would generally degrade the correlation between the single- and dual-Doppler retrievals in this layer.