The DOE ARM program maintains a network of coherent Doppler lidar instruments, which are manufactured by Halo Photonics (Brockamin, Leigh, Worcestershire United Kingdom WR6 5LA GB). The Doppler lidar transmits at a wavelength of 1.548 µm, with ∼ 150 ns (22.5 m) pulse width and < 100 µJ pulse energy at a rate of 15 kHz, providing time- and range-resolved measurements of attenuated backscatter and radial velocity (Newsom and Krishnamurthy, 2020; Newsom et al., 2017). When operated in vertical fixed-point mode, the system measures vertical velocity at 1 Hz temporal and 30 m vertical spatial resolution. The lowest acceptable range gate is 105 m. The primary scattering mechanism is atmospheric aerosol. To date, the main application of this instrument has been the observation of turbulence within the boundary layer (Newsom et al., 2015; Williams and Qiu, 2023). When operated in hemispheric scanning mode, the instrument yields 2D wind fields as a function of height. The DOE ARM program collected the lidar data. The instruments are operated by DOE personnel, and data are distributed through publicly accessible archives (Newsom and Krishnamurthy, 2021; Shippert et al., 2022). Data in the archive have undergone a first pass of data processing. The vertical fixed-point data files, which are primarily used here, contain attenuated backscatter coefficients, signal-to-noise ratios, and vertical velocities at approximately 1 s intervals.

The lidar backscatter is attenuated by the two-way transmission through the atmosphere. The true backscatter can be found via (Platt and Collins, 2015) 1βr=βattr1-2LR∫r1r2βattrdr, where β(r) is the true backscatter coefficient, βatt(r) is the measured attenuated backscatter coefficient, LR is the lidar ratio defined in Eq. (2), and the integration is carried out between ranges r1 and r2. In Eq. (1) it is assumed that the lidar ratio does not vary with range. The true backscatter coefficient and attenuated backscatter coefficient are equal in the first range gate. The lidar ratio represents the ratio of the extinction cross-section and 180∘ backscatter cross-section and varies between 5 and ∼ 100 sr. Its value depends on the wavelength, aerosol refractive index, aerosol size distribution, aerosol hygroscopicity, and the presence of absorbing gases in the atmosphere. Most prior studies have focused on systems with wavelengths 355, 532, and 1064 nm. Consequently, there is limited information about values for LR for wavelength > 1 µm. However, the lidar backscatter and lidar ratio can be estimated from the aerosol size distribution, aerosol hygroscopicity, and Mie theory. 2αmie=∫0∞πD24Qextλ,m,DdNdln⁡Ddln⁡D3βmie=∫0∞πD24Qbackλ,m,DdNdln⁡Ddln⁡D Here, αmie is the extinction coefficient derived from Mie theory, βmie is the aerosol lidar backscatter coefficient derived from Mie theory, D is the particle diameter (m), Qext and Qback are the extinction and backscatter efficiencies determined from Mie theory, dN/dlnD is the aerosol size distribution in units of spectral density, λ is the wavelength (of the lidar), and m=n+ki is the complex aerosol refractive index with a real (n) and imaginary (k) component. The integration is performed over the entire size distribution. The Mie solution assumes that the particles are spherical. The modeled lidar ratio is 4LR=αmieβmie. The effects of scattering and extinction by molecules are not considered here.

At elevated relative humidity particles may swell and take up water. The refractive index of the mixed particle can be obtained from the volume-weighted average of the refractive indexes of the dry aerosol and water (Shettle and Fenn, 1979) and the water content estimated using the aerosol hygroscopicity parameter (Petters and Kreidenweis, 2007): 5m=mw+maer-mwkaw1-aw+1-1, where m is the refractive index of the wet particle, mw is the refractive index of pure water, maer is the refractive index of the dry aerosol particle, aw= RH/100 % is water activity neglecting the Kelvin effect, and κ is the hygroscopicity parameter. The refractive indices m=n+ki include a real (n) and imaginary (k) component. Equation (5) is derived by combining Eq. (6) in Shettle and Fenn (1979) and Eq. (1) in Carrico et al. (2010).

Figure 2 illustrates the change in optical properties with RH as modeled via Eqs. (1)–(5) and using the average size distribution measured by the OPC on 2 August 2022. A few notable trends can be summarized as follows. The backscatter coefficient βmie varies ∼±50 % between 0 % and 80 % RH. However, most of the variability is within ±20 %. The function only weakly depends on the assumed hygroscopicity parameter. For real refractive indices n > 1.5, the backscatter coefficient can decrease with increasing RH. Increasing the RH increases this scattering cross-section due to hygroscopic growth. However, the backscatter decreases due to a decrease in the refractive index. For aerosols with a larger refractive index, the latter effect can dominate in the 40 %–90 % RH range. The modeled lidar ratio varies between ∼ 20 and 80 sr. Under dry conditions, the main controlling factor of the lidar ratio is the refractive index. Larger values of n amplify backscatter and thus result in a lower lidar ratio. Larger values of k amplify extinction and thus increase the lidar ratio. The lidar ratio can increase up to a factor of 2 with increasing RH, consistent with previous similar numerical simulations (Ackermann, 1998; Zhang et al., 2022).

The modeled βmie is primarily sensitive to changes in aerosol number concentration and to a lesser extent the shape of the size distribution of particles in the 1 < D < 10 µm diameter range. Figure 3 shows a statistical analysis of the relationship between the aerosol size distribution and optical properties for 2 August 2022. The assumed refractive index is m= 1.55 + 0i. The value was picked arbitrarily due to a lack of knowledge of the refractive index of the aerosol and is used for illustration purposes. The aerosol size distribution shown in Fig. 3a (red fitted line) shows a coarse mode with a mode diameter ∼ 1 µm. Figure 3b shows that particles 2 < D < 6 µm contribute most of the signal to the total backscatter coefficient. Figure 3c shows the correlation of the integrated number concentration for particles D > 3 µm against the total βmie. Linear regression of these data (R2= 0.98) can be used to relate an observed β to aerosol number concentration. The regression yields an intercept value that corresponds to the βmie that is not explained by particles D > 3 µm. As indicated in Fig. 2a, the modeled βmie depends weakly on RH. This implies a dependence of the regression slope on RH. Indeed, the example in Fig. 3d shows a slightly smaller slope for the assumed RH = 80 %. The regression analysis can be applied to arbitrary lower size cuts for the integrated number concentration. One would expect the strength of the correlation to decrease if smaller size particles were included. For example, the correlation for number concentration with D > 0.01 µm and total βmie is likely small, since Aitken- and accumulation-mode particles insignificantly contribute to the backscatter. However, number concentrations within the coarse mode will be strongly autocorrelated if the shape of the coarse-mode distribution is unchanged. In this case, the numbers of D > 1 µm, D > 3 µm, and D > 7 µm will all yield a strong correlation with the total βmie. Figure 3c tests the extent to which the correlation degrades for different lower size thresholds. This example shows R2 > 0.75 even for a lower threshold Dlo= 0.53 µm but essentially no correlation for Dlo < 0.5 µm. The degradation of the correlation is evident as larger scatter in Fig. 3e and f compared to Fig. 3a. Although the details change due to day-to-day variability of the shape of the size distribution, the assumed refractive index, and the assumed hygroscopicity, the overall trends in Fig. 3b–f are repeatable. It is proposed that the empirical regressions shown in Fig. 3d–f, combined with a sensible Dlo, can be used to retrieve aerosol number concentration N(D > Dlo) from measured lidar backscatter coefficients.

Optical models like the one given by Eqs. (1)–(5) have been used successfully to relate OPC size distributions and backscatter for lidar returns from polar stratospheric clouds and cirrus clouds (Cairo et al., 2011; Snels et al., 2021). Here, however, the aerosol refractive index, the aerosol hygroscopicity, the contribution from molecular absorption to extinction, and the particle aspherical shape are unknown. Furthermore, the Doppler lidar backscatter is obtained via factory calibration (Newsom and Krishnamurthy, 2020). A back-of-the-envelope comparison of aerosol optical depth inferred from lidar backscatter against aerosol optical depth from the AErosol RObotic NETwork (AERONET) at a nearby site shows that the lidar backscatter correlates strongly with aerosol optical depth, but its value may be biased high. Combined, these uncertainties are too large to rely on the optical model alone to relate observed inverted backscatter to particle number concentration. Instead, this work relies on empirical correlations between the lidar-observed attenuated backscatter at z= 105 m (the lowest range gate) stratified by relative humidity and surface-based particle number concentration. Note that the two-way attenuation of the backscatter close to the ground is minimal and attenuated backscatter at z = 105 m approximately equals the true backscatter.

Figure 4 summarizes the campaign average correlation between the lidar-observed attenuated backscatter at z= 105 m and particle number concentrations Dlo > 0.53, Dlo > 1.03, and Dlo > 3.25 µm. The height z= 105 m is the lowest range gate to the surface that has complete time coverage above the signal-to-noise ratio. The shown correlations are analogous to those shown from the model in Fig. 3c. For the regression analysis, a lower threshold in number concentration was imposed (N > 2, 0.5, and 0.02 cm-3 for Dlo > 0.53, > 1.03, and Dlo > 3.25 µm, respectively). Below this threshold the correlations are poor, and the regression analysis obscured the intercept value. The correlations show the same pattern as the model. An intercept of the regression line for N= 0 cm-3 indicates the portion of the backscatter that is not explained by particles with D > Dlo. The change in the slope with RH is relatively small and shows a slight decrease in backscatter at 50 % < RH < 65 %, which is qualitatively comparable to the trends in Fig. 2a. The Pearson correlation coefficients are similar for the three shown cutoff sizes. Moving the lower size threshold to the lowest diameter measured by the OPC (Dlo > 0.28 µm) results in poor correlations (R2 < 0.2) for all RHs. This consistency with the model simulations in Fig. 3d is satisfying. The R2 values decrease with increasing RH. At RH > 90 %, the correlation is poor (0.2 < R2 < 0.5). This suggests either increasing interference of other backscattering particles at higher RH or increasing uncertainty due to uncertainty in the RH value itself. Specifically, the comparisons between the interpolated sonde product and the ground-based meteorological station suggest that the precision of the interpolated sonde RH is not better than ±7 % in absolute RH units (supporting information). Furthermore, the R2 values decrease with height, suggesting interference from backscatter attenuation and/or decorrelation of the aerosol at height z with those at the surface. In summary, the results in Fig. 4 suggest that backscatter fluctuations are indicative of particle number concentration if backscatter exceeds the value of the intercept, if a threshold Dlo > 0.53 µm is selected, and if RH < 90 %.

Backscatter flux is obtained using the eddy covariance technique, 6Fβ=<w′β′>, where w′ and β′ are the instantaneous fluctuations of vertical velocity and attenuated backscatter and < > indicates the time average. The measured backscatter coefficients are thresholded at a signal-to-noise ratio of -17 dB, which corresponds to a velocity precision of 20 cm s-1 (Newsom and Krishnamurthy, 2020). The backscatter coefficient data show occasional spikes, possibly due to the transit of larger objects through the beam including birds. These spikes were removed using the following despiking algorithm. A low-pass filter with a cutoff frequency of 0.01 Hz and a fourth-order Butterworth filter function are applied to the backscatter coefficient. Values outside the 0.01 and 0.99 quantiles of the ratio of filtered and measured backscatter are considered spikes and replaced with the value from the filtered data, which approximately corresponds to the average ±100 s of the removed data point. This ensures continuity in the data during the spike event.

The Doppler lidar operates in fixed-point vertical orientation in contiguous blocks ∼ 780 s in length with vertical velocity and backscatter reported at ∼ 1 Hz frequency. This is followed by ∼ 120 s breaks while the instrument plan position indicator scans. The temporal spacing between consecutive timestamps is 1.025 ± 0.15 s. Here each contiguous block is used to derive a particle flux. First, the despiked backscatter data were detrended using a linear fit (Behrendt et al., 2020). Despiking removes spurious peaks from the data, while detrending subtracts both the mean and possible linear trends from the time series. Both are needed to compute accurate fluxes. Figure 5 shows an example of a contiguous block showing detrended and despiked vertical velocity and backscatter data. The corresponding signal variances are σw2= 0.79 m2 s-2 and σβ2= 0.093 Mm-2 sr-2, respectively. The calculated backscatter flux is Fβ= < w′β′>= 1.6 × 10-5 s-1 sr-1.

The lidar vertical velocity and backscatter data contain uncorrelated random noise stemming from a finite number of scatterers in the sampling volume and low photon counting statistics (Lenschow et al., 2000). The variance from uncorrelated noise can be separated from the signal using various methods. One common approach is the autocovariance method (Lenschow et al., 2000; Wulfmeyer et al., 2016). First, the autocovariance function Ax(τ) of the time series is computed via 7Axτ=covxt,xt+τ, where xt is the time series of interest, τ is the lag time, and cov is the covariance function. Ax(0) equals the variance of xt. Next Ax(τ) from the data is fit to a model of the form 8Amodelτ=ν-kτ23, where ν and k are fitted parameters. The fit is obtained for lags up to the first zero crossing of Ax(τ). The model extrapolated to zero lag Amodel(0) equals the noise-free variance. The variance attributed to noise is 9δx2=Ax0-Amodel0.

Finally, the integral timescale, I, is obtained from the fit parameters via 10I=25vk32. Figure 6 shows an example of the autocovariance method applied to the contiguous data block shown in Fig. 5. The autocovariance (Eq. 7, black lines) is largest for lag zero and decreases for larger lag times. The model (Eq. 8, red lines) shows that the decrease with increasing lag is well-modeled using the 2/3 power-law relationship. The derived noise variances, taken from the difference between the calculated covariance and model at lag time zero (Eq. 9), are δ2= 0.161 m2 s-2 and δ2= 0.052 Mm-2 sr-2 for vertical velocity and backscatter data, respectively. The integral timescales derived from Eq. (10) are I= 21 and I= 24 s, for vertical velocity and backscatter data, respectively. The contribution of the noise variance to the total variance, evaluated as δ2/σ2 is ∼ 0.2 and ∼ 0.56 for the vertical velocity and backscatter data, respectively. This indicates that significant noise is present in the data. The derived integral timescales for the two series are similar. For lags > ∼ 50 s, the autocorrelation for both time series is minimal. The example in Fig. 6 illustrates how δ2 and I were determined for each contiguous flux segment.

Spectral analysis is used to estimate the frequency where noise overwhelms the signal. Figure 7 shows the spectral decomposition of the variance of vertical velocity S(w) and backscatter S(β) data. The spectra for S(β) are flat for frequencies larger than 0.035 Hz, which indicates white noise. Integrating ∫S(β)df from 0.035 Hz to the Nyquist frequency equals the noise variance δβ2 derived from the autocovariance analysis. Conversely, the spectra for S(w) do not flatten. Thus, the transition to white noise cannot be used to determine the noise limit. An estimate of the noise variance δw2 can be found by integrating ∫S(w)df from 0.05 Hz to the Nyquist frequency. The visual depiction and the magnitude of these thresholds are similar for other contiguous segments. This suggests that derived backscatter fluxes for frequencies > ∼ 0.035 Hz are not well-resolved due to noise in the lidar signal.

Random noise error. The random noise error in the flux Fβ is approximated for each contiguous segment using Eq. (A22) in Wulfmeyer et al. (2016): 11σnoise≅<β′2>δw2N+<w′2>δβ2N, where σnoise is the uncertainty in the flux due to random noise, and δw2 and δβ2 are noise variances derived from the autocovariance analysis.

Figure 8 demonstrates the application of the various data quality control and uncertainties metrics. Figure 8a shows a 3 d time series of the noise-thresholded and despiked backscatter curtain for broader context. The backscatter shows values more than 100 Mm-1 sr-1 during nighttime (UTC) at z∼ 400 m (pink colors), likely due to the presence of low clouds. Since our analysis focuses on z= 105 only, no additional cloud screening was performed on the dataset. Backscatter signals up to 1500 m are retrieved at approximately noon local time. Figure 8b shows the derived backscatter flux with stable conditions identified via z/L > 0 being grayed out. Superimposed in red is the limit of detection (LOD) derived for each contiguous segment using the lag analysis. The observed fluxes significantly exceed the values from the LOD analysis. Note that the LOD fluxes can be positive or negative. In contrast, the observed Fβ values are all positive (indicating particle emission from the surface). Figure 8c shows that most flux values are from data that pass the stationarity test, i.e., abs(ξ) < 0.3. Figure 8d shows flux-loss-corrected hourly averaged flux values during unstable conditions only. The gray shading comprises the cumulative uncertainty of σnoise, σsample , and σensemble and remains small relative to the absolute value of the retrieved backscatter flux. Taking the combined data quality and uncertainty metrics into account, the data in Fig. 8 show that the backscatter fluxes are statistically significant.

Fluctuations in elastic backscatter correspond to fluctuations in aerosol number concentration (Pal et al., 2010). As demonstrated in Figs. 3 and 4, the Doppler lidar is sensitive to particle number concentration D > ∼ 0.5 µm. The relationship between backscatter and particle number is 15βS=∂β∂NSNS+cS, where S is the saturation ratio (S= RH/100 %), β(S) is the backscatter at saturation ratio S, N(S) is the number concentration of particles larger than a specified threshold diameter, (∂β/∂N)S is the slope, and c(S) is the intercept of the regression lines shown in Fig. 4a–l. In practice, (∂β/∂N)S and c(S) were empirically evaluated in intervals [S; S + 0.05]. The ambient S value is obtained at the lidar height and sample time from the interpolated sonde product. The closest calibrated slopes and intercepts are used to derive N(S) from the observed β(S).

Figure 9 summarizes the retrieved particle number concentration from lidar backscatter via Eq. (15). Here retrievals are limited to conditions wherein RH < 90 %, there is no precipitation, and the observed backscatter exceeds 1.5 times the intercept value in Eq. (15). The latter limit is necessary to avoid retrievals where the backscatter is dominated by signals unrelated to particle number concentration. The cutoff value was selected to filter most noise while maintaining sufficient signal coverage. Increasing the value to > 1.5 does not further improve the accuracy of the comparison between the lidar and OPC shown in Fig. 9. The lidar-retrieved concentrations are visibly noisier, which is likely due to the noise in the backscatter data (Figs. 6 and 7) and the fact that the calibrated slopes were obtained from the campaign average, thus not accounting for variations in refractive index and hygroscopicity in the retrieval. Nevertheless, the retrieval captures the broad trends in particle number concentration, including the transition from lower-concentration to higher-concentration periods on 11 June, 15 July, and 20 July. Pearson correlation coefficients between the OPC and lidar-derived number concentrations are R2= 0.74 for the time series shown in Fig. 9c–e. The R2 values are not identical but very similar, which is explained by the autocorrelation of number concentration within the coarse mode, i.e., because the shape of the coarse mode is approximately constant throughout the campaign. Overall, the results in Fig. 9 confirm that the variability in backscatter is related to changes in coarse-mode particle number concentration.

Based on the preceding paragraph, we assert that high-frequency fluctuations in backscatter are related to high-frequency fluctuations in number concentration. To obtain the number flux from backscatter flux, high-frequency fluctuations in relative humidity also need to be considered (Fairall, 1984). This is because hygroscopic growth increases the particle size, which in turn affects the backscatter. During periods of intense sensible and latent heat flux, there are systematic differences in relative humidity or saturation ratio in updrafts and downdrafts, resulting in a saturation ratio flux. This saturation ratio flux can lead to an apparent particle flux that is not related to turbulent transport. Thus, turbulent flux measurements require correction for false fluxes during periods of high saturation ratio flux (Fairall, 1984; Kowalski, 2001; Vong et al., 2004; Islam et al., 2022).

To derive the influence of saturation ratio fluctuations on the lidar-derived number flux, we use an equation analogous to Eq. (18) in Fairall (1984): 16β′=∂β∂NSN′+∂β∂SNS′. Therefore, the number flux can be written as 17<w′N′>=<w′β′>/∂β∂NS-∂β∂SN/∂β∂NS<w′S′>, where <w′N′> is the eddy covariance flux of particles and <w′S′> is the saturation ratio flux. Evaluation of the terms (∂β/∂S)N, (∂β/∂N)S, and <w′S′> is provided in the Supplement. Note that the saturation ratio flux depends on the latent and sensible heat flux and can be either positive or negative. Furthermore, <w′N′> is a net flux and includes contributions from surface emissions and dry deposition. The emission flux can be obtained by removing the estimated contribution from particle dry deposition (Nilsson et al., 2021): 18Femission=<w′N′>+vd<N>, where vd is the dry deposition velocity, and <N> is the average number concentration. Note that by our convention positive <w′N′> corresponds to an upward flux. In the absence of emissions (Femission= 0), the observed <w′N′> would be negative (downward), reflecting the dry deposition process and using the convention that the deposition velocity is a positive number. Deposition velocity is a function of particle diameter and land use type. Precise values remain poorly constrained within data showing approximately 2 orders of magnitude in scatter (Emerson et al., 2020). Here we consider vd= 1 cm s-1 as an approximate upper bound for particles in the 0.5 to 5 µm size range depositing on a grassland surface (Emerson et al., 2020).

Combining all the correction included in Eqs. (14), (17), and (18), the emission flux can be conceptually decomposed into four terms: 19Femission=F+Fflc+FwS+Fdep, where F= <w′β′>/(∂β/∂N)S is the first-order conversion from backscatter to number flux and Fflc is the additional flux computed from the flux loss correction due to the low-frequency response of the Doppler lidar (Sect. 2.4). If no correction is required Fflc= 0. The sign of Fflc is the same sign as F. The term FwS is the apparent contribution to the flux due to variation in the saturation ratio in updrafts and downdrafts. FwS= -(∂β/∂S)N/(∂β/∂N)S<w′S′> and can be either positive or negative. The term Fdep=vd<N> is always positive.