## Introduction

Measuring both the structure and evolution of turbulent flow through the depth of the atmospheric boundary layer (ABL) remains a difficult and challenging problem. Coherent Doppler lidar is currently one of the most promising technologies for achieving direct measurements of the microscale flow structure. A scanning Doppler lidar is capable of providing radial velocity and aerosol backscatter measurements over an area (sector) or volume of the ABL with resolutions on the order of a few tens of meters. By repeating these scans, it is possible to observe how these quantities evolve in time. However, these measurements form only an incomplete picture of the state of the flow, and current technical constraints restrict the rate at which ABL volumes can be sampled. Four-dimensional variational data assimilation (4DVAR) represents one potentially powerful method that can be used to retrieve three-dimensional, time-varying wind and temperature fields from Doppler lidar observations.

The basic concept behind 4DVAR is to fit the output of a dynamic model to spatially and temporally resolved measurements by variational adjustment of the model’s initial state. The fit is achieved by minimizing a cost function using the adjoint method. The cost function is a measure of the difference between the observed and model-predicted variables, and the adjoint method is used to determine how the cost function changes with respect to changes in the initial model state.

A wind and temperature retrieval technique that is based on 4DVAR was initially demonstrated by Sun et al. (1991), using simulated data. This technique was later applied to a dry gust-front case using real Doppler radar observations (Sun and Crook 1994). More recently, Sun and Crook (1997, 1998) have adapted and applied the technique to study the structure and dynamics of convective storms. Lin et al. (2001) conducted a series of identical twin experiments using a modified version of Sun and Crook’s (1994) original algorithm with synthetic data generated by a large-eddy simulation (LES) to explore the potential of retrieving microscale flow structures from single-Doppler lidar data. Modifications included the introduction of a surface layer parameterization scheme that is based on Monin–Obukhov similarity theory. Chai et al. (2004) then incorporated a variable eddy diffusivity profile and applied this algorithm to single-Doppler lidar data collected under clear-sky, daytime convective conditions during the Cooperative Atmosphere–Surface Exchange Study (CASES-99) field campaign (Poulos et al. 2002).

Newsom and Banta (2004a) recently described a similar 4DVAR retrieval algorithm and also applied it to single-Doppler lidar data collected during the CASES-99 field campaign. This algorithm (hereinafter referred to as the NB method) incorporates a novel data-ingest scheme and accounts for radial velocity measurement uncertainty. Newsom and Banta (2004b) evaluated the sensitivity of the NB method to changes in 1) the prescribed eddy diffusivity profile, 2) the first-guess or base-state virtual potential temperature profile, 3) the phase and duration of the assimilation period, and 4) the grid resolution. Furthermore, that study also examined the characteristics and behavior of turbulence statistics that are derived from the retrieved fields.

The focus of this study is to apply the NB method to single- and dual-Doppler lidar data collected during the recent Joint Urban 2003 (JU2003) field campaign in Oklahoma City, Oklahoma. The purpose of that field campaign was to study the transport and dispersion of contaminants in urban environments. During intensive observational periods (IOPs), sulfur hexafluoride (SF_{6}) was released at several locations within the Oklahoma City central business district (CBD). Numerous meteorological sensors and SF_{6} samplers were installed at various locations in and around the city to provide detailed measurements of key meteorological variables and tracer gas concentrations. Additionally, two coherent Doppler lidars were deployed at two sites outside of the CBD to provide measurements of the urban-scale flow. Thus, the JU2003 field campaign offered a rare and unique opportunity to obtain coordinated dual-Doppler lidar data. The dataset that was acquired enables the first opportunity of which we are aware to validate single-Doppler lidar retrievals of microscale velocity fields.

In this study, we use the dual-Doppler lidar dataset from JU2003 to assess the reliability of single-Doppler retrievals using the NB method. A specific test case is examined in which both lidars performed overlapping volume scans during an afternoon period under clear-sky convective conditions. Single- and dual-Doppler retrievals are performed to answer the following basic questions:

What are the differences between the direct observations of one lidar and the single-Doppler retrieval performed using data from the other lidar?

What are the differences between a retrieval using data from one lidar (single-Doppler retrieval) and a retrieval using data from two lidars (dual-Doppler retrieval)?

To address the first question, a single-Doppler retrieval is performed using data from one lidar. The radial component of the retrieved velocity field relative to the second lidar is then compared directly with the measured radial velocity field from the second lidar. To address the second question, a dual-Doppler retrieval is performed using data from both lidars. That result is then compared with the single-Doppler retrieval.

Another possibility is to compare wind field retrievals performed using 4DVAR with those performed using more traditional dual-Doppler analysis techniques (Armijo 1969; Bousquet and Chong 1998; Dowell and Shapiro 2003; Scialom and Lemaître 1990; Tabary and Scialom 2001). Traditional dual-Doppler analysis methods use the time-independent form of the continuity equation to retrieve three-dimensional wind vectors from dual-Doppler observations. Because no prognostic model is used, one is forced to assume that the observed fields are approximately steady over the volume scan period. For this study, the steady-flow assumption fails because the observed eddy structures evolve rapidly relative to the volume scan period. Retrieval techniques based on 4DVAR account for this evolution in a dynamically consistent manner. This study uses only the 4DVAR-based retrieval method.

This paper is organized as follows. Section 2 describes the experimental setup and observations. This section also presents an analysis of the radial velocity measurement errors that are used by the retrieval algorithm. The retrieval method is briefly reviewed in section 3. Section 4 presents the results of the retrieval experiments, and our conclusions are given in section 5.

## Observations and quality control

JU2003 was conducted in Oklahoma City during the period from 28 June to 31 July 2003. Participants included investigators from government laboratories, universities, and the private sector. The main experimental objective was to provide much-needed high-resolution dispersion (transport and diffusion) data at scales of motion ranging from flows in and around a single city block to the scale of the suburban area covering several kilometers around the CBD. Of primary interest in this study were two coherent Doppler lidars that were used to measure the larger-scale flow regime from locations outside of the CBD.

Figure 1 shows a map of Oklahoma City and the locations of the two lidars. The U.S. Army Research Laboratory (ARL) deployed a Doppler lidar atop a parking garage that was located approximately 1.4 km east-northeast of the CBD. Arizona State University (ASU) deployed a second Doppler lidar that was approximately 4 km to the south-southeast of the ARL lidar. The lidars operated during the IOPs, as well as during non-IOPs. Scanning strategies were coordinated between the two lidars in order to capitalize on the opportunity to acquire dual-Doppler data.

Locations of the ASU and ARL lidars were determined using handheld GPS receivers. That information was then used to compute the location of the ASU lidar relative to the ARL lidar, as defined by the position vector **R** in Fig. 1. We define a Cartesian coordinate system with an origin that is located at the position of the ARL lidar, such that the positive *x* axis is east and the positive *y* axis is north. In this frame of reference, the coordinates of the ASU lidar were determined to be *x* = 1090 m and *y* = −3804 m.

Both the ARL and ASU lidars were manufactured by CLR Photonics, and are nearly identical in their designs. These instruments employ solid-state laser transmitters operating at a wavelength of 2 *μ*m, with a 400-ns (60 m) 2-mJ pulse, at a pulse-repetition frequency (PRF) of 500 Hz. The software interface allows the operator to adjust many of the signal processing parameters. For consistency, both lidars were configured using the same signal processing parameters. Raw signals were processed with 66-m range gates, with 66 m between the centers of each gate. Each range-resolved profile, or beam, was obtained by averaging 100 pulses. As a result, the beam rate, which is the PRF divided by the number of pulses averaged, was 5 Hz.

For this study, wind and temperature retrievals are performed using data from two sequential volume scans that were collected during the early afternoon between 1802:25 and 1805:17 UTC 11 July 2003. The ARL lidar performed a series of vertical raster scans by scanning the beam in elevation from 0° to 45° at 8° s^{−1}, and stepping the azimuth angle from 184.4° to 244.4° in 5° steps. The ASU lidar performed a similar type of scan, with azimuth angles ranging from 275° to 335°. Figure 2 depicts two three-dimensional views of intersecting, nearly simultaneous, vertical scans during the assimilation period. Colors in each scan plane represent radial velocities that are measured relative to the corresponding lidar.

### 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.

## Retrieval method

The NB method fits the output of a model of ABL flow to radial velocity data from a coherent Doppler lidar. The algorithm is similar to that described by Sun et al. (1991), Sun and Crook (1994), Lin et al. (2001), and Chai et al. (2004) but differs primarily in the way that the data are ingested into the model and in the treatment of observational error. The cost function is computed by interpolating model quantities to the measurement points, instead of vice versa. Measurement uncertainty is included in the cost-function evaluation, reducing the impact of regions of low-SNR data.

### Forward model

*g*, Θ

_{ref}is a reference virtual potential temperature,

*p*is the pressure normalized by a constant air density, and

*S*= 0.5(∂

_{ij}*u*/∂

_{i}*x*+ ∂

_{j}*u*/∂

_{j}*x*) is the resolved-scale strain-rate tensor. The virtual potential temperature is

_{i}*θ*=

*θ̂*+

*θ*, where

_{b}*θ*is the so-called base-state virtual potential temperature and

_{b}*θ̂*is the departure from the base state. The horizontally averaged virtual potential temperature is 〈

*θ*〉. The coefficient of eddy diffusivity

*K*is modeled using an expression that is similar to the one proposed by Troen and Mahrt (1986). This is given by

_{m}*z*

_{max}is the height of the computational domain,

*K*

_{max}determines the maximum value of

*K*, and

_{m}*α*controls the shape of the profile and the height of the maximum. We further assume that the relationship between

*K*and the heat diffusivity

_{m}*K*is given by

_{h}*K*and

_{m}*K*is expected to be a function of stability, such that

_{h}*K*→ 3

_{h}*K*in the free convection limit, and

_{m}*K*→

_{h}*K*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

_{m}*K*and

_{m}*K*(Newsom and Banta 2004b).

_{h}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

**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

*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*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).

_{d}*J*

_{obs}takes the following form:

*J*

^{ASU}

_{obs}and ▵

^{ASU}

*. Summation over the observations from each lidar is denoted by Σ*

_{m}_{m}, and

**r̂**

^{ARL}

_{m}is the unit vector from the ARL lidar to the

*m*th measurement of the ARL lidar. Likewise,

*u*

^{ARL}

_{rm}refers to the radial velocity measurements from the ARL lidar and the

*σ*

^{ARL}

_{rm}s 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}.

**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

*m*th observation. The interpolation is essentially a weighted average of

**u**. When interpolating

**u**to the coordinates of the ARL lidar measurements,

**u**takes the following form:

_{m}**r**is a position vector to a model domain node, thus, Σ

_{r}implies summation over the model domain nodes. The time of the

*m*th observation from the ARL lidar is

*t*

^{ARL}

*, and*

_{m}*t*is the time of the

_{n}*n*th 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}

*and*

_{m}**r**

^{ARL}

*with*

_{m}**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.

**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

*i*th component of

**u**, is given by

*J*

^{ASU}

_{obs}/∂

*u*. 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.

^{n}_{i}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.

## Retrieval results

This section presents the results from both single- and dual-Doppler retrievals. Data from the ARL lidar is used to perform the single-Doppler retrieval. The results of that retrieval are compared directly with the ASU data. Dual-Doppler retrievals are then performed using data from both lidar systems, and those results are compared with the single-Doppler retrieval.

Table 1 lists model-specific parameters that were used for both the single- and dual-Doppler retrievals. A model domain was defined with the ARL lidar located at the origin, as shown in Fig. 1. The domain extends from −5 to 0.5 km in *x* (east–west), from −5 to 0.5 km in *y* (north–south), and from 0 to 2.0 km in *z* (vertical). The vertical profile of eddy diffusivity is given by Eq. (4) using *K*_{max} = 20 m^{2} s^{−1}, and *α* = 4. These parameters provide just enough diffusion to keep the model numerically stable. Furthermore, sensitivity studies conducted by Newsom and Banta (2004b) showed that the retrieval results are not very sensitive to small variations in *K*_{max} and *α*.

The number of control parameters listed in Table 1 is the number of individual initial state variables that must be adjusted by the retrieval algorithm to minimize the cost function. The number of adjustable parameters is the same for both single- and dual-Doppler retrievals.

The base-state wind and potential temperature profiles shown in Fig. 5 are used as first-guess fields for both the single- and dual-Doppler retrievals. There were approximately 40 000 radial velocity samples from the ARL lidar and approximately 21 000 samples from the ASU lidar within the model domain during the assimilation period. The weighting coefficients appearing in Eq. (7) were *a*_{ARL} = 1 and *a*_{ASU} = 0 for the single-Doppler retrieval, and *a*_{ARL} = *a*_{ASU} = 1 for the dual-Doppler retrieval. The dual-Doppler retrieval was influenced more strongly by the ARL data because these data accounted for approximately 65% of the total number of samples and exhibited a smaller measurement error.

### Single-Doppler retrieval

*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

*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 _{m} · **r̂**^{ASU}_{m}, where **r̂**^{ASU}_{m} is a unit vector from the ASU lidar to its *m*th 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 _{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 _{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}

*. As one would expect, the correlation between*

_{rm}u

_{m}·

**r̂**

^{ARL}

_{m}and

*u*

^{ARL}

*is very good, with relatively little scatter about the oneto-one line. The correlation between*

_{rm}u

_{m}·

**r̂**

^{ASU}

_{m}and

*u*

^{ASU}

*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*

_{rm}*u*

^{ASU}

*occur primarily in the stable layer above ∼1200 m AGL, where the flow is relatively uniform. Positive values of*

_{r}*u*

^{ASU}

*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.*

_{r}Distributions of Δ^{ASU}* _{m}* and Δ

^{ARL}

*are shown in Figs. 9b and 9d, respectively. We find that 68% of the ASU radial velocity samples occur within ±1 m s*

_{m}^{−1}of

u

_{m}·

**r̂**

^{ASU}

_{m}. As expected, the distribution of Δ

^{ARL}

*is sharply peaked, with 99% of the ARL radial velocity samples occurring within ±1 m s*

_{m}^{−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}

*exhibits a slight skewness toward positive values, and a peak that occurs at about 0.3 m s*

_{m}^{−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}

*? To address this problem we consider the geometry shown in Fig. 10. Errors in the ARL and ASU azimuth data are denoted by*

_{m}*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}

*. The first approach simply involves performing multiple runs of the single-Doppler retrieval with a range of*

_{m}*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.

*m*th measurement of the ASU lidar is given by

**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

**r**

^{ASU′}

_{m}in terms of

**r**

^{ASU″}

_{m}, that is,

**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

*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 _{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}

*is shown in Fig. 12a, and the distribution of Δ*

_{rm}^{ASU}

*is shown in Fig. 12b. The assumption of*

_{m}*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}

*is essentially eliminated. The second effect is that the distribution exhibits slightly less skewness.*

_{m}u

_{m}·

**r̂**

^{ASU}

_{m}and

*u*

^{ASU}

*, and between*

_{rm}u

_{m}·

**r̂**

^{ARL}

_{m}and

*u*

^{ARL}

*. Results are shown assuming both zero azimuth offsets (*

_{rm}*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

Table 2 shows that when the measurements are assumed to contain no azimuth offsets, the coefficient of linear correlation between ^{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}

*is 0.99, and the rms deviation is 0.24 m s*

_{rm}^{−1}. Not surprisingly, the correlation between

u

^{ARL}

_{m}·

**r̂**

^{ARL}

_{m}and

*u*

^{ARL}

*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.*

_{rm}### 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}

*has improved from 0.94 for the single-Doppler retrieval to 0.98 for both dual-Doppler trials. The correlation between*

_{r}**u**·

**r̂**

_{ARL}and

*u*

^{ARL}

*is slightly better than the correlation between*

_{r}**u**·

**r̂**

_{ASU}and

*u*

^{ASU}

*, because the ARL lidar data make a larger contribution to the cost function because of the overall better data quality and range performance.*

_{r}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 “*d*2” 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.

*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*for the single-Doppler retrieval and dual-Doppler trial 2. The

_{Z}*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*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.

_{Z}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.

## Summary

This study uses data that are acquired by two coherent Doppler lidar systems during JU2003 to evaluate single-Doppler retrievals of microscale flow structure in a shear-driven convective boundary layer. The retrieval method is based on a four-dimensional data assimilation technique described by Newsom and Banta (2004a). This paper described modifications of the existing single-Doppler retrieval algorithm for dual-Doppler input.

During the JU2003 field experiment Arizona State University and the U.S. Army Research Laboratory deployed two coherent Doppler lidar systems at locations outside of the central business district of Oklahoma City. The ARL and ASU lidars, which are nearly identical in design, both achieve a minimum radial velocity measurement error of about 15 cm s^{−1} with 66-m range gates and 100-pulse averaging. During the JU2003 deployment, the ARL lidar exhibited better range performance.

A test case was selected from the early afternoon of 11 July 2003. During this period the ARL and ASU lidars executed a series of overlapping vertical raster scans over an area immediately south of the central business district of Oklahoma City. Conditions during this period were cloud free, hot, humid, and windy. The test case coincided with a nearby rawinsonde launch. Data from the rawinsonde were used to establish the base-state virtual potential temperature profile. The basestate wind profiles were determined from a VAD analysis of the ARL lidar data during the assimilation period.

Because of its better range performance, the ARL lidar is used to perform the single-Doppler retrieval. The results of the single-Doppler retrieval are compared directly with the ASU data. Dual-Doppler retrievals are then performed using data from both lidar systems, and those results are compared with the single-Doppler retrieval.

The coefficient of linear correlation and rms deviation between the single-Doppler retrieval and the ASU lidar observations are 0.94 and 1.18 m s^{−1}, respectively. This correlation is quite good, but one should bear in mind that this primarily reflects agreement in the overall trends in the data. The trends in the radial velocity data are determined by the mean flow and the viewing geometry. Thus, the good correlation mainly implies good agreement in the mean flow structure as observed by both lidars.

A comparison between the single- and dual-Doppler retrievals suggests that the single-Doppler retrieval underestimates the magnitude of cross-beam velocity fluctuations. The dual-Doppler retrieval showed significantly stronger variance in the *u* component of velocity and slightly stronger variance in the vertical velocity component. Also, there is a general tendency for differences in velocity to be larger where the ASU lidar dominates the dual-Doppler retrieval. In these regions, the difference in the velocity fields can be nearly as large as the velocity fluctuations from either retrieval.

Linear correlation coefficients between the single- and dual-Doppler retrievals are greater than 0.9 for *u*, *υ*, and *θ*, and about 0.7 for *w*. Correlations for *u*, *υ*, and *θ* are strongly influenced by their mean vertical structure. Because all retrievals use the same base-state profiles as the first-guess fields, the correlations in these variables are artificially biased toward 1.

To compare the microscale eddy structure of the single- and dual-Doppler retrievals, vertical profiles of horizontally averaged correlations are computed for each prognostic variable. This eliminates the influence of the mean vertical structure and more accurately reflects differences in the microscale flow. The height-dependent correlation profiles once again suggest that the retrieval algorithm has difficulty in estimating the cross-beam component of the flow from single-Doppler data. The *υ* component showed the best correlation, with values near 0.9 in the middle of mixed layer, and the *u* component exhibited the poorest overall correlation, with values ranging from 0.4 to 0.8. The correlations are best in the mixed layer and degrade near the surface and within the stable layer. The degraded correlations in the stably stratified layer are likely the result of weak velocity fluctuations and a lower SNR. The degraded correlations in the surface layer may be the result of obstructions in the ARL lidar beam. It is also possible that the effects of refractive turbulence reduce the SNR in this layer.

The results of this study highlight the difficulties associated with the retrieval of microscale flow structures from single-Doppler lidar data. The primary difficulty appears to be in the retrieval of the crossbeam component of the flow.

With further research, algorithms like the one discussed in this study may one day be used to provide very high resolution data for military, emergency response, or aviation safety applications. Further work is required to treat cloudy conditions and complex terrain and to reduce the execution time for near-real-time applications. Improvements in lidar technology are also required in order to improve transverse spatial resolution and reduce the volume scan period.

## Acknowledgments

This research was supported by the DoD Center for Geosciences/Atmospheric Research at Colorado State University under cooperative agreement DAAD19-02-2-0005 with the Army Research Laboratory. Additional support for algorithm development was provided by the National Science Foundation (Grant ATM-9908453). Funding for the JU2003 field campaign was provided by the U.S. Department of Defense, Defense Threat Reduction Agency (DTRA), Army Research Office (ARO), and the U.S. Department of Homeland Security (DHS). We thank Young Yee of ARL for providing rawinsonde data and CLR Photonics for their technical support before, during, and after the field deployment.

## REFERENCES

Armijo, L. 1969. A theory for the determination of wind and precipitation velocities with Doppler radar.

*J. Atmos. Sci.*26:570–573.Banta, R. M., R. K. Newsom, J. K. Lundquist, Y. L. Pichugina, R. L. Coulter, and L. Mahrt. 2002. Nocturnal low-level jet characteristics over Kansas during CASES-99.

*Bound.-Layer Meteor.*105:221–252.Bousquet, O. and M. Chong. 1998. A multiple-Doppler synthesis and continuity adjustment technique (MUSCAT) to recover wind components from Doppler radar measurements.

*J. Atmos. Oceanic Technol.*15:343–359.Businger, J. A., J. C. Wyngaard, Y. Izumi, and E. F. Bradley. 1971. Flux-profile relationships in atmospheric surface layer.

*J. Atmos. Sci.*28:181–189.Daescu, D. N., A. Sandu, and G. R. Carmichael. 2003. Direct and adjoint sensitivity analysis of chemical kinetics systems with KPP: II-Numerical validation and applications.

*Atmos. Environ.*37:5097–5114.Dowell, D. C. and A. Shapiro. 2003. Stability of an iterative dual-Doppler wind synthesis in Cartesian coordinates.

*J. Atmos. Oceanic Technol.*20:1552–1559.Chai, T., C. L. Lin, and R. K. Newsom. 2004. Retrieval of microscale flow structures from high resolution Doppler lidar using an adjoint model.

*J. Atmos. Sci.*61:1500–1520.Frehlich, R. G. 2000. Effects of refractive turbulence on ground-based verification of coherent Doppler lidar performance.

*Appl. Opt.*39:4237–4246.Frehlich, R. G. and M. J. Yadlowsky. 1994. Performance of mean-frequency estimators for Doppler radar and lidar.

*J. Atmos. Oceanic Technol.*11:1217–1230.Lin, C. L., T. Chai, and J. Sun. 2001. Retrieval of flow structures in a convective boundary layer using and adjoint model: Identical twin experiments.

*J. Atmos. Sci.*58:1767–1783.Mayor, S. D., D. H. Lenschow, R. L. Schwiesow, J. Mann, C. L. Frush, and M. K. Simon. 1997. Validation of NCAR 10.6-

*μ*m CO2 Doppler lidar radial velocity measurements and comparison with a 915-MHz profiler.*J. Atmos. Oceanic Technol.*14:1110–1126.Moeng, C-H. and P. P. Sullivan. 1994. A comparison of shear- and buoyancy-driven planetary boundary layer flows.

*J. Atmos. Sci.*51:999–1022.Newsom, R. K. and R. M. Banta. 2004a. Assimilating coherent Doppler lidar measurements into a model of the atmospheric boundary layer. Part I: Algorithm development and sensitivity to measurement error.

*J. Atmos. Oceanic Technol.*21:1328–1345.Newsom, R. K. and R. M. Banta. 2004b. Assimilating coherent Doppler lidar measurements into a model of the atmospheric boundary layer. Part II: Sensitivity analyses.

*J. Atmos. Oceanic Technol.*21:1809–1824.Poulos, G. S. Coauthors 2002. CASES-99: A comprehensive investigation of the stable nocturnal boundary layer.

*Bull. Amer. Meteor. Soc.*83:555–581.Press, W. H., B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. 1988.

*Numerical Recipes in C*. Cambridge University Press, 317 pp.Scialom, G. and Y. Lemaître. 1990. A new analysis for the retrieval of three-dimensional mesoscale wind fields from multiple Doppler radar.

*J. Atmos. Oceanic Technol.*7:640–665.Sun, J. and A. Crook. 1994. Wind and thermodynamic retrieval from single Doppler measurements of a gust front observed during Phoenix II.

*Mon. Wea. Rev.*122:1075–1091.Sun, J. and N. A. Crook. 1997. Dynamical and microphysical retrieval from Doppler radar observations using a cloud model and its adjoint. Part I: Model development and simulated data experiments.

*J. Atmos. Sci.*54:1642–1661.Sun, J. and N. A. Crook. 1998. Dynamical and microphysical retrieval from Doppler radar observations using a cloud model and its adjoint. Part II: Retrieval experiments of an observed Florida convective storm.

*J. Atmos. Sci.*55:835–852.Sun, J., D. W. Flicker, and D. K. Lilly. 1991. Recovery of three-dimensional wind and temperature fields from simulated single-Doppler radar data.

*J. Atmos. Sci.*48:876–890.Rye, B. J. and R. M. Hardesty. 1993. Discrete spectral peak estimation in Doppler lidar. I: Incoherent spectral accumulation and the Cramer-Rao bound.

*IEEE Trans. Geosci. Remote Sens.*31:16–27.Tabary, P. and G. Scialom. 2001. MANDOP analysis over complex orography in the context of the MAP experiment.

*J. Atmos. Oceanic Technol.*18:1293–1314.Troen, I. B. and L. Mahrt. 1986. A simple model of the atmosheric boundary layer; sensitivity to surface evaporation.

*Bound.-Layer Meteor.*37:129–148.

Three-dimensional representations of intersecting scan planes during the assimilation period. (a),(b) The same scans at two different orientations are shown. The positions of the ARL and ASU lidars are indicated. The positive *y* direction is north, and the positive *x* direction is east. Colors represent radial velocities measured relative to the respective lidar.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Three-dimensional representations of intersecting scan planes during the assimilation period. (a),(b) The same scans at two different orientations are shown. The positions of the ARL and ASU lidars are indicated. The positive *y* direction is north, and the positive *x* direction is east. Colors represent radial velocities measured relative to the respective lidar.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Three-dimensional representations of intersecting scan planes during the assimilation period. (a),(b) The same scans at two different orientations are shown. The positions of the ARL and ASU lidars are indicated. The positive *y* direction is north, and the positive *x* direction is east. Colors represent radial velocities measured relative to the respective lidar.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Radial velocity data from individual vertical sweeps near the start and end of the assimilation period: (top) ARL data and (bottom) ASU data. Volume scans were performed by stepping in azimuth and sweeping vertically. The azimuth angle and the mean time of each sweep are indicated above the panels.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Radial velocity data from individual vertical sweeps near the start and end of the assimilation period: (top) ARL data and (bottom) ASU data. Volume scans were performed by stepping in azimuth and sweeping vertically. The azimuth angle and the mean time of each sweep are indicated above the panels.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Radial velocity data from individual vertical sweeps near the start and end of the assimilation period: (top) ARL data and (bottom) ASU data. Volume scans were performed by stepping in azimuth and sweeping vertically. The azimuth angle and the mean time of each sweep are indicated above the panels.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

(a) wSNR and velocity error *σ* as functions of range for the ASU and ARL lidars; (b) *σ* vs wSNR for the ASU and ARL lidars.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

(a) wSNR and velocity error *σ* as functions of range for the ASU and ARL lidars; (b) *σ* vs wSNR for the ASU and ARL lidars.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

(a) wSNR and velocity error *σ* as functions of range for the ASU and ARL lidars; (b) *σ* vs wSNR for the ASU and ARL lidars.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Vertical profiles of the base-state virtual potential temperature *θ _{b}* (solid), and base-state wind speed (dotted) and direction (dashed).

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Vertical profiles of the base-state virtual potential temperature *θ _{b}* (solid), and base-state wind speed (dotted) and direction (dashed).

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Vertical profiles of the base-state virtual potential temperature *θ _{b}* (solid), and base-state wind speed (dotted) and direction (dashed).

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Horizontal cross sections from the single-Doppler retrieval. (left) The perturbation velocity vector fields (*u*′, *υ*′), and (right) the perturbation virtual potential temperature fields (*θ*′) are shown. The cross sections are taken at (a), (b) 200 and (c), (d) 900 m AGL. Shaded areas in (a) and (c) are regions where *w*′ < 0. Shaded areas in (b) and (d) are regions where *θ*′ < 0. These cross sections are taken from the end of the assimilation period, *t* = 172 s.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Horizontal cross sections from the single-Doppler retrieval. (left) The perturbation velocity vector fields (*u*′, *υ*′), and (right) the perturbation virtual potential temperature fields (*θ*′) are shown. The cross sections are taken at (a), (b) 200 and (c), (d) 900 m AGL. Shaded areas in (a) and (c) are regions where *w*′ < 0. Shaded areas in (b) and (d) are regions where *θ*′ < 0. These cross sections are taken from the end of the assimilation period, *t* = 172 s.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Horizontal cross sections from the single-Doppler retrieval. (left) The perturbation velocity vector fields (*u*′, *υ*′), and (right) the perturbation virtual potential temperature fields (*θ*′) are shown. The cross sections are taken at (a), (b) 200 and (c), (d) 900 m AGL. Shaded areas in (a) and (c) are regions where *w*′ < 0. Shaded areas in (b) and (d) are regions where *θ*′ < 0. These cross sections are taken from the end of the assimilation period, *t* = 172 s.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Vertical cross sections along the streamwise direction of the single-Doppler retrieval. (a) The perturbation velocity vector field (*υ*′, *w*′), and (b) the perturbation virtual potential temperature field (*θ*′) are shown. Shaded areas in (b) indicate regions where *θ*′ is negative. The areas labeled A–F in (a) highlight regions of significant upward or downward motion. These cross sections are taken from the end of the assimilation period, *t* = 172 s.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Vertical cross sections along the streamwise direction of the single-Doppler retrieval. (a) The perturbation velocity vector field (*υ*′, *w*′), and (b) the perturbation virtual potential temperature field (*θ*′) are shown. Shaded areas in (b) indicate regions where *θ*′ is negative. The areas labeled A–F in (a) highlight regions of significant upward or downward motion. These cross sections are taken from the end of the assimilation period, *t* = 172 s.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Vertical cross sections along the streamwise direction of the single-Doppler retrieval. (a) The perturbation velocity vector field (*υ*′, *w*′), and (b) the perturbation virtual potential temperature field (*θ*′) are shown. Shaded areas in (b) indicate regions where *θ*′ is negative. The areas labeled A–F in (a) highlight regions of significant upward or downward motion. These cross sections are taken from the end of the assimilation period, *t* = 172 s.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Vertical profiles of horizontally averaged standard deviations of *u*, *υ*, *w*, and *θ* from the single-Doppler retrieval. Averaging is performed over the entire assimilation period, using only those grid points that occur within the overlap between the ARL and ASU lidar scan volumes.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Vertical profiles of horizontally averaged standard deviations of *u*, *υ*, *w*, and *θ* from the single-Doppler retrieval. Averaging is performed over the entire assimilation period, using only those grid points that occur within the overlap between the ARL and ASU lidar scan volumes.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Vertical profiles of horizontally averaged standard deviations of *u*, *υ*, *w*, and *θ* from the single-Doppler retrieval. Averaging is performed over the entire assimilation period, using only those grid points that occur within the overlap between the ARL and ASU lidar scan volumes.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

(a),(c) Correlation diagrams and (b),(d) histograms showing comparisons between the single-Doppler retrieval and the (top) ASU and (bottom) ARL radial velocity observations. For comparison, the solid diagonal lines indicate perfect correlation.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

(a),(c) Correlation diagrams and (b),(d) histograms showing comparisons between the single-Doppler retrieval and the (top) ASU and (bottom) ARL radial velocity observations. For comparison, the solid diagonal lines indicate perfect correlation.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

(a),(c) Correlation diagrams and (b),(d) histograms showing comparisons between the single-Doppler retrieval and the (top) ASU and (bottom) ARL radial velocity observations. For comparison, the solid diagonal lines indicate perfect correlation.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Geometry used to account for possible azimuth offsets in the ASU and ARL lidar data. The primed and double-primed axes define the reference frames of the ARL and ASU lidar, respectively. The azimuth offsets for the ARL and ASU are denoted *δϕ*_{ARL} and *δϕ*_{ASU}, respectively.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Geometry used to account for possible azimuth offsets in the ASU and ARL lidar data. The primed and double-primed axes define the reference frames of the ARL and ASU lidar, respectively. The azimuth offsets for the ARL and ASU are denoted *δϕ*_{ARL} and *δϕ*_{ASU}, respectively.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Geometry used to account for possible azimuth offsets in the ASU and ARL lidar data. The primed and double-primed axes define the reference frames of the ARL and ASU lidar, respectively. The azimuth offsets for the ARL and ASU are denoted *δϕ*_{ARL} and *δϕ*_{ASU}, respectively.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

The solid line indicates the azimuth offset in the ASU and ARL observations required to achieve

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

The solid line indicates the azimuth offset in the ASU and ARL observations required to achieve

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

The solid line indicates the azimuth offset in the ASU and ARL observations required to achieve

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

(a) Correlation diagram and (b) histogram illustrating differences between the single-Doppler retrieval and the ASU radial velocity observations assuming an azimuth offset of *δϕ*_{ARL} = 1.78° and *δϕ*_{ASU} = −1.60°.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

(a) Correlation diagram and (b) histogram illustrating differences between the single-Doppler retrieval and the ASU radial velocity observations assuming an azimuth offset of *δϕ*_{ARL} = 1.78° and *δϕ*_{ASU} = −1.60°.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

(a) Correlation diagram and (b) histogram illustrating differences between the single-Doppler retrieval and the ASU radial velocity observations assuming an azimuth offset of *δϕ*_{ARL} = 1.78° and *δϕ*_{ASU} = −1.60°.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Perturbation horizontal velocity fields (*u*′, *υ*′) from dual-Doppler trials (a) 1 and (c) 2, respectively. (b) The difference between dual-Doppler trial 1 and the single-Doppler retrieval; (d) the difference between dual-Doppler trial 2 and the single-Doppler retrieval. The velocity for the single-Doppler retrieval is denoted **u*** _{s}*, and the velocity for dual-Doppler trial 2 is denote

**u**

*. All cross sections are taken at*

_{d2}*z*= 200 m AGL, and shaded areas are regions where

*w*′ < 0. The scan areas for both lidars are indicated.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Perturbation horizontal velocity fields (*u*′, *υ*′) from dual-Doppler trials (a) 1 and (c) 2, respectively. (b) The difference between dual-Doppler trial 1 and the single-Doppler retrieval; (d) the difference between dual-Doppler trial 2 and the single-Doppler retrieval. The velocity for the single-Doppler retrieval is denoted **u*** _{s}*, and the velocity for dual-Doppler trial 2 is denote

**u**

*. All cross sections are taken at*

_{d2}*z*= 200 m AGL, and shaded areas are regions where

*w*′ < 0. The scan areas for both lidars are indicated.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Perturbation horizontal velocity fields (*u*′, *υ*′) from dual-Doppler trials (a) 1 and (c) 2, respectively. (b) The difference between dual-Doppler trial 1 and the single-Doppler retrieval; (d) the difference between dual-Doppler trial 2 and the single-Doppler retrieval. The velocity for the single-Doppler retrieval is denoted **u*** _{s}*, and the velocity for dual-Doppler trial 2 is denote

**u**

*. All cross sections are taken at*

_{d2}*z*= 200 m AGL, and shaded areas are regions where

*w*′ < 0. The scan areas for both lidars are indicated.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Vertical cross sections along the streamwise direction of the perturbation velocity vector fields (*υ*′, *w*′) from the two dual-Doppler retrievals. The cross sections are taken at *x*′ = −1.0 km. (a) The regions labeled A–F are the same areas labeled in Fig. 7a.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Vertical cross sections along the streamwise direction of the perturbation velocity vector fields (*υ*′, *w*′) from the two dual-Doppler retrievals. The cross sections are taken at *x*′ = −1.0 km. (a) The regions labeled A–F are the same areas labeled in Fig. 7a.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Vertical cross sections along the streamwise direction of the perturbation velocity vector fields (*υ*′, *w*′) from the two dual-Doppler retrievals. The cross sections are taken at *x*′ = −1.0 km. (a) The regions labeled A–F are the same areas labeled in Fig. 7a.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Vertical cross sections along the streamwise direction of the perturbation virtual potential temperature fields (*θ*′) from the two dual-Doppler retrievals: (a) trial 1 and (b) trial 2. The cross sections are taken at *x*′ = −1.0 km. Shaded areas indicate negative values.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Vertical cross sections along the streamwise direction of the perturbation virtual potential temperature fields (*θ*′) from the two dual-Doppler retrievals: (a) trial 1 and (b) trial 2. The cross sections are taken at *x*′ = −1.0 km. Shaded areas indicate negative values.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Vertical cross sections along the streamwise direction of the perturbation virtual potential temperature fields (*θ*′) from the two dual-Doppler retrievals: (a) trial 1 and (b) trial 2. The cross sections are taken at *x*′ = −1.0 km. Shaded areas indicate negative values.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Vertical profiles of horizontally averaged standard deviations of *u*, *υ*, *w*, and *θ* from dual-Doppler trial 2 (*δϕ*_{ARL} = 1.78°, *δϕ*_{ASU} = −1.60°). These profiles are averaged over the entire assimilation period. The averaging is performed only over those grid points that occur within the overlap between the ARL and ASU lidar scans.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Vertical profiles of horizontally averaged standard deviations of *u*, *υ*, *w*, and *θ* from dual-Doppler trial 2 (*δϕ*_{ARL} = 1.78°, *δϕ*_{ASU} = −1.60°). These profiles are averaged over the entire assimilation period. The averaging is performed only over those grid points that occur within the overlap between the ARL and ASU lidar scans.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Vertical profiles of horizontally averaged standard deviations of *u*, *υ*, *w*, and *θ* from dual-Doppler trial 2 (*δϕ*_{ARL} = 1.78°, *δϕ*_{ASU} = −1.60°). These profiles are averaged over the entire assimilation period. The averaging is performed only over those grid points that occur within the overlap between the ARL and ASU lidar scans.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Correlation diagrams between the single- and dual-Doppler retrievals of *u*, *υ*, *w*, and *θ* for dual-Doppler trial 2 (*δ *_{ARL} = 1.78°, *δϕ*_{ASU} = −1.60°). The single-Doppler variables are denoted *u _{s}*,

*υ*,

_{s}*w*, and

_{s}*θ*. The variables from dual-Doppler trial 2 are denoted

_{s}*u*

_{d2},

*υ*

_{d2},

*w*

_{d2}, and

*θ*

_{d2}. These results are taken during the middle of the assimilation period,

*t*= 86 s, using only those grid points that occur within the dual-Doppler overlap volume. For comparison, perfect correlations are indicated by the solid diagonal lines: (a)

*u*, (b)

*υ*, (c)

*w*, and (d)

*θ*.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Correlation diagrams between the single- and dual-Doppler retrievals of *u*, *υ*, *w*, and *θ* for dual-Doppler trial 2 (*δ *_{ARL} = 1.78°, *δϕ*_{ASU} = −1.60°). The single-Doppler variables are denoted *u _{s}*,

*υ*,

_{s}*w*, and

_{s}*θ*. The variables from dual-Doppler trial 2 are denoted

_{s}*u*

_{d2},

*υ*

_{d2},

*w*

_{d2}, and

*θ*

_{d2}. These results are taken during the middle of the assimilation period,

*t*= 86 s, using only those grid points that occur within the dual-Doppler overlap volume. For comparison, perfect correlations are indicated by the solid diagonal lines: (a)

*u*, (b)

*υ*, (c)

*w*, and (d)

*θ*.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Correlation diagrams between the single- and dual-Doppler retrievals of *u*, *υ*, *w*, and *θ* for dual-Doppler trial 2 (*δ *_{ARL} = 1.78°, *δϕ*_{ASU} = −1.60°). The single-Doppler variables are denoted *u _{s}*,

*υ*,

_{s}*w*, and

_{s}*θ*. The variables from dual-Doppler trial 2 are denoted

_{s}*u*

_{d2},

*υ*

_{d2},

*w*

_{d2}, and

*θ*

_{d2}. These results are taken during the middle of the assimilation period,

*t*= 86 s, using only those grid points that occur within the dual-Doppler overlap volume. For comparison, perfect correlations are indicated by the solid diagonal lines: (a)

*u*, (b)

*υ*, (c)

*w*, and (d)

*θ*.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Correlation profiles between the single-Doppler retrieval and dual-Doppler trial 2. The single-Doppler variables are denoted *u _{s}*,

*υ*,

_{s}*w*, and

_{s}*θ*. The variables from dual-Doppler trial 2 are denoted

_{s}*u*

_{d2},

*υ*

_{d2},

*w*

_{d2}, and

*θ*

_{d2}.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Correlation profiles between the single-Doppler retrieval and dual-Doppler trial 2. The single-Doppler variables are denoted *u _{s}*,

*υ*,

_{s}*w*, and

_{s}*θ*. The variables from dual-Doppler trial 2 are denoted

_{s}*u*

_{d2},

*υ*

_{d2},

*w*

_{d2}, and

*θ*

_{d2}.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Correlation profiles between the single-Doppler retrieval and dual-Doppler trial 2. The single-Doppler variables are denoted *u _{s}*,

*υ*,

_{s}*w*, and

_{s}*θ*. The variables from dual-Doppler trial 2 are denoted

_{s}*u*

_{d2},

*υ*

_{d2},

*w*

_{d2}, and

*θ*

_{d2}.

Citation: Journal of Applied Meteorology 44, 9; 10.1175/JAM2280.1

Model parameters used for both single- and dual-Doppler retrievals.

Comparisons between the single-Doppler retrieval and the observations. The radial components of the retrieved field relative to the ASU and ARL lidars are ** u** ·

**r̂**

_{ASU}and

**·**u

**r̂**

_{ARL}, respectively. The observed radial velocities from the ASU and ARL lidars are

*u*

^{ASU}

*and*

_{r}*u*

^{ARL}

*, respectively. The assumed azimuth errors for ARL and ASU lidars are*

_{r}*δϕ*

_{ARL}and

*δϕ*

_{ASU}, respectively. Correlations and rms deviations for the ASU data were computed using only those data that occur within the dual-Doppler overlap volume.

Assumed azimuth offsets for two dual-Doppler trials.

Correlations and rms deviations between the observed radial velocities and the radial component of the retrieved velocity field for the dual-Doppler retrievals. Correlations and rms deviations were computed using only those data that occur within the dual-Doppler overlap volume.

Correlations and rms deviations between prognostic variables from the single- and dual-Doppler retrievals. Variables corresponding to the single-Doppler retrieval are denoted by *ξ _{s}*, where

*ξ*is either

*u*,

*υ*,

*w*, or

*θ*. Variables corresponding to dual-Doppler retrievals 1 and 2 are denoted by

*ξ*

_{d}_{1}, and

*ξ*

_{d}_{2}, respectively. Correlations and rms deviations were computed over the entire assimilation period, using only those grid points that occur within the dual-Doppler overlap volume.