In situ wind speed measurements taken with cup anemometers or ultrasonic anemometers have a small measurement volume that can be considered a point. Lidar measurements, in contrast, sense wind velocities along an extended stretch of the line of sight of the laser beam. In the case of continuous-wave lidars, the laser beam leaves the lidar optics with a diameter that corresponds to its effective aperture size a0 and is focused onto a focus point. The distance between the lidar optics and the focus point is the focal distance df. The signal of the backscattered radiation originates from anywhere along the illuminated beam, according to a distribution function that has its maximum at the focus point and is proportional to the intensity of the laser light along the beam .

A definite range gate, such as for pulsed lidars, is therefore not applicable to continuous-wave lidars. Instead, the Rayleigh length lR is a measure of the distance between the focus point and the point at which the cross section of the beam has twice the area of the cross section at the focus point. According to , it is given by lR=λdf2πa02, where λ is the laser wavelength and a0 is the effective aperture diameter. The Rayleigh length is quadratically proportional to the focal distance df that increases linearly with the selected measurement height level. The degree of line-of-sight averaging is thus strongly dependent on the measurement height level and is higher for larger heights. The values of lR, a0 and df for the lidar used in our experiments are given in Table .

The intensity of backscattered radiation is a function of the distance s from the focus point along the beam. It is sufficiently well approximated by a Lorentzian function, F(s)=lR/πs2+lR2, where s is the distance from the focus position .

All Doppler spectra that are retrieved during the radial velocity acquisition time are averaged, and the focus point sweeps over a considerable arc of the measurement circle during this time. This arc length lA is lA=DCπN, where N is the number of line-of-sight measurements vr taken during one rotation. In experimental data, the arc averaging effect is contained in the radial velocities. In the models here, we account for this by averaging along the measurement circle.

The Doppler spectra of each line-of-sight measurement resemble the probability density function of the radial wind velocities along the line of sight . But by determining one single velocity value for each line-of-sight measurement, the turbulence information they contain is filtered out.

The additional temporal averaging along the lines of sight is very low, as one measurement takes only 1N s. The effect of line-of-sight averaging is very strong for high wave numbers but has some effect on long turbulent structures as well. The effect of line-of-sight averaging is considered in the numerical models and the discussion in this study. But none of the presented data processing methods can avoid the line-of-sight averaging effect.

As described in Sect.  lidars use all measurement data of at least one full rotation of the prism to reconstruct one wind vector. The resulting system of equations is overdetermined, and in order to find a solution a quadratic best fit is applied. The more line-of-sight velocities that are used to reconstruct a wind vector, the stronger the averaging and thereby the larger the loss of turbulent kinetic energy in the measurement data. The residual of the best fit is a measure of the degree of this form of averaging but is usually not used in the processing.

The diameter DC of the measurement circle is DC=2htan⁡ϕ, with h being the measurement height and ϕ the half-cone opening angle. The spatial separation between the points that one reconstructed wind vector is composed of thus linearly increases with measurement height. The larger the cone diameter, the stronger the circle averaging. Turbulence with a length scale below the diameter of the averaging circle is affected the most.

In addition to the spatial separation of the measurement points along the measurement circle, the acquisition time must be considered. The mean wind motion carries the air while it is probed, which might further increase the separation of measurement points in the mean wind direction. The ZephIR 300 measures one full rotation in 1 s, and the distance the air moves within this time is usually small compared to DC. The effect of temporal averaging is therefore often small compared to the spatial averaging. One example for the path of measurements that is averaged over is given in Fig. a. The circle diameter represents the spatial averaging, and the shift along-wind with the speed U represents the temporal averaging.

Another cause for differences in the shape of turbulence spectra from one-point measurements and their counterparts from VAD scanning lidars is cross-contamination of different velocity components. VAD scanning lidars combine measurements from spatially separated locations where differing velocities may prevail as if they were collected at one point. This leads to a redistribution of turbulent energy among the velocity components u, v and w. Lidar-derived spectra of one of the components can at certain wave numbers show lower energy values than the original wind spectrum of that component but may also show too high values due to a contribution from a different velocity component. To better understand cross-contamination we divide the effect into two different types of separations. First we look into longitudinal separations, i.e., separation along the mean wind direction. Fluctuations at two points separated in line with the wind are highly correlated. If the assumption of frozen turbulence is correct, the coherence would be 1 for all separation lengths and all wave numbers. One example of cross-contamination of correlated fluctuations between two longitudinally separated points is visualized in Fig. . The chosen wavelength of the wind fluctuations equals twice the separation distance. This can be called the first resonance wavelength. The resonance wavelengths are given by λres,n=2DC2n-1. The corresponding resonance wave numbers are kres,n=(2n-1)πDC, where n=1, 2, 3… The resulting values for the first two resonance points are given in Table .

The two beam directions in line with and against the mean wind direction can be used to determine ulidar and wlidar by using the formulas on the right-hand side of the figure. This example looks at these two lines of sight. The v component can be ignored because transverse fluctuations are not detected by the upstream and downstream beams. The example demonstrates a case with isotropic turbulence, i.e., arbitrary but identical amplitudes for fluctuations in all orientations. Averaging along the lines of sight is ignored here for simplicity. The first column of graphs in the figure isolates the u fluctuations u′ and shows the resulting lidar-measured signal for the two radial velocities in the upwind and downwind directions, i.e., vr1 and vr2. When these two signals are combined in the usual way, the reconstructed wind speed components ulidar′ and wlidar′ differ strongly from the real inflow conditions u′ and w′. The lidar is blind to wind speed fluctuations in the u direction and instead attributes the fluctuations to some extent to the estimation of wlidar′. The same is done for w′ in the second column, and the resulting effect is the reverse. The vertical fluctuations w′ are interpreted solely as amplified fluctuations of ulidar′.

The last column combines the two previous cases and shows the resulting distribution of amplitudes that depends on the half-cone opening angle ϕ. When ϕ<45∘ the lidar is more sensitive to vertical variations than to horizontal ones, and the contamination of u′ caused by w′ is more severe than vice versa.

In a more realistic situation, turbulence is non-isotropic and the amplitude of w′ at this first resonance wave number is often considerably lower than the amplitude of u′, which leads to a different distribution of contamination, which can be estimated as follows. We use Eqs. () and () to define the lidar-derived variance in the u direction: σu,lidar2=Δv-2sin⁡ϕ2. In general, the differences of the line-of-sight velocities aligned with the mean wind Δv contain contributions from wind fluctuations in the u and w directions Δvu and Δvw, respectively. Here we look at the resonance case in which Δvu=0 and thus Δv=Δvw. We get σu,lidar,res2=Δvw-2sin⁡ϕ2=2w′cos⁡ϕ-2sin⁡ϕ2=cot⁡2ϕσw,res2≈2.86σw,res2, when ϕ=30.6∘ as for the lidar we used in this study. The subscript “res” indicates that the equation is only valid for inflow fluctuations at resonance, as in the example given before.

In Sect.  we develop a model to predict lidar-derived spectra. This model was used to create the plots shown in Fig. . Figure a shows the modeled spectra of the wind components, uwind and wwind, as solid black and red lines. The parameters of the underlying spectral tensor are given in Table . They were chosen to best represent the wind conditions found during the experiment presented in Sect. . The model was used to estimate the u component of the wind from two lidar beams that point in the upwind and downwind directions. Also here, we did not include line-of-sight averaging to isolate the effect of cross-contamination. The principle of the setup is the same as explained for Fig.  but now we see results for all inflow wave numbers and use anisotropic turbulence. The resulting lidar-derived spectrum ulidar,sum of the u component of the wind is the sum of the lidar's interpretation of the wind components ulidar,u and ulidar,w. We see that the lidar-estimated spectrum of ulidar,sum lies a bit below the target spectrum of uwind for most wave numbers but not at the first and second resonance points that are marked with grey dashed vertical lines. There it exceeds the target spectrum. The reason becomes apparent when we look at the components ulidar,u and ulidar,w that ulidar,sum is composed of. We find that the lidar sees uwind nearly to its full extent for very low wave numbers but when we come close to the resonance points ulidar,u drops to zero. The contribution of the vertical wind ulidar,w shows a mirrored behavior and is amplified according to Eq. () since ϕ<45∘.

When the lines of sight under consideration are not longitudinally but laterally separated, they do not face resonance but instead a second form of cross-contamination. The strength of the contamination depends then on the coherence of the turbulence for the given lateral separation. When the fluctuations at the two selected focus points are very coherent i.e., their correlation is close to unity, we can expect that the lidar-derived wind speed estimates are correct and no cross-contamination occurs. This can be observed at very low wave numbers at which a high degree of coherence is expected. The other extreme is found at the other end of the spectrum at which small fluctuations measured at both focus points are uncorrelated. The lidar-derived spectrum is there a linear combination of the variances of the involved components v and w according to σv,lidar2=Δv-2sin⁡ϕ2=Δvv-2sin⁡ϕ2+Δvw-2sin⁡ϕ2. In the case of fully uncorrelated fluctuations we know that Δvv=-v′sin⁡ϕ and Δvw=w′cos⁡ϕ and the variance σv,lidar,unc2 of the lidar-derived v velocity is σv,lidar,unc2=-v′sin⁡ϕ-2sin⁡ϕ2+w′cos⁡ϕ-2sin⁡ϕ2=12σv,unc2+σw,unc2cot⁡2ϕ≈0.5σv,unc2+1.43σw,unc2 for the lidar with a half-cone opening angle of ϕ=30.6∘. These two situations and all cases in between are shown in Fig. b. The difference from the plots in Fig. a is that the two beams that point into and against the v direction are used here to estimate the v-spectrum vlidar,sum. The target spectrum of the v component of the wind vwind is given as well as the w-spectrum wwind that contaminates the signal. From the vlidar,v and vlidar,w curves it can be seen that at very low wave numbers hardly any contamination occurs but mainly because the w-component wwind itself contains a low energy density at low wave numbers. As it increases for higher wave numbers, the contamination also becomes more severe. In this example wwind dominates the lidar spectrum vlidar,sum for all wave numbers above approximately k1=1.4×10-2 m-1. The result is that the lidar overestimates the v variances for all wave numbers. Such an effect is also reported by . Thus, it is essential for accurate turbulence measurements to minimize spatial separation.

VAD scanning along the whole measurement circle is more complex than using only two beams. Examining the two beams aligned with or perpendicular to the mean wind direction is not sufficient to fully understand the effect of cross-contamination. For circle scans, all three wind speed components are involved in contaminating all the beams that do not point in the four cardinal directions. We refer to the model presented in Sect.  and especially Eqs. (), () and () of the spectral weighting functions therein to better understand which components influence another.

The lidar can also be configured to perform a so-called 3 s scan, in which one measurement cycle is built from data from three full rotations. This limits the cross-contamination but comes at the cost of much stronger averaging along the measurement circle, especially in strong wind cases, and a sampling rate that is 3 times slower. The ability to measure turbulence with this approach is so weak that it is not further investigated in this paper. Instead, the next chapter suggests two methods that can be used to reduce both averaging and cross-contamination.

To begin with, the model predictions of conventional VAD processing and the new SMC method are compared against each other and with regard to the true u target model spectrum acquired from the spectral tensor according to Eq. (). The model prediction of the conventionally processed VAD lidar data shows some attenuation of the spectral energy even for very low wave numbers. This can be partly explained by the infinitely long tails of the line-of-sight averaging function given in Eq. (). That means that even very large eddies are slightly weakened by the underlying Lorentzian function. Averaging along the measurement circle might also have some small additional impact on large-scale turbulence. Both averaging effects become more and more severe for increasing wave numbers until the measured spectral energy reaches values close to zero at roughly k1=10-1 m-1 and above. The tendency of increasing attenuation with regard to the target spectrum is interrupted around the first resonance frequency that is indicated by a vertical grey dashed line at k1=3.4×10-2 m-1. Here the energy density increases and reaches coincidentally roughly the value of the target spectrum. This behavior is as expected an effect of the cross-contamination with energy from both the w spectrum and to a small extent also from v. A resonance effect at the second resonance frequency is hardly pronounced since the energy is nearly fully consumed by the line-of-sight averaging.

The SMC model spectrum predicts a similar shape but without the cross-contamination effect from longitudinal separation. Thus, we find no resonance in the computations. The total variance of the u fluctuations σ2(u′) is lower here since less additional energy from the w component is contained in the uSMC signal. The signal is still contaminated by contributions from other components because the lateral separation cannot be reduced by squeezing. But the averaging along the measurement circle is so strong that for example for wave numbers above around k1=10-2 m-1 less than half of the energy of the target spectrum is expected to be detected by the lidar.

First, when the model is compared with the measurement data, the chosen spectral tensor does not fit the actual wind conditions in the wave number range below k1=10-2 m-1. The extra energy at low wave numbers compared to the spectral tensor model for this site has been observed before and is related to the inhomogeneous landscape at Høvsøre with its sea-to-land transition in the main wind direction and mesoscale effects that overlay the expected spectral gap . Luckily, this does not severely impede the analysis since the most interesting effects are expected at higher wave numbers and tendencies can still be determined from the relative distances between the markers and lines without matching the absolute values. Next, the comparison of data from sonic measurements and VAD as well as SMC-processed lidar data shows in the very low wave number range at k1<3×10-3 m-1 that VAD processing and SMC processing produce similar results with a slight tendency towards lower energy densities in the SMC-measured spectrum that is not found in the model computations. A possible explanation is that the fluctuations of the u and especially the w component in the real wind field are not perfectly correlated, i.e., the frozen turbulence hypothesis that the model assumes is slightly violated. The result is a small contribution of wwind to ulidar that appears to a greater extent in the VAD-processed spectrum. The reason for the difference is that the correlation is closer to unity in the case of SMC processing.

Apart from some exceptions (e.g., at k1=3×10-3 m-1), a relatively increasing averaging effect towards higher wave numbers is found for the lowest wave numbers as expected. In the wave number range k1=10-2 to 6×10-2 m-1 the sonic spectrum and the VAD spectrum follow the corresponding modeled spectra nicely through the first resonance point. That shows that the cross-contamination caused by longitudinal separation is present in the measurements and is properly modeled.

The spectrum derived from SMC-processed data shows a clear tendency towards its modeled spectrum but does not completely reach it. It does not show the resonance effect seen for VAD processing, but the overall energy level is higher than predicted for k1>10-2 m-1. It is not possible to determine what causes this deviation. One possible reason is that the model assumes a perfect delay of the measurement timing. In reality this is not possible due to only discrete acquisition times being available. Also the air packages are in reality not always advected with the exact mean wind speed and direction. Both imperfections justify that the behavior of real SMC processing lies in between the modeled SMC and VAD processing.

For k1>7×10-2 m-1 VAD- and SMC-processed data are nearly identical. As shown in , the assumption of frozen turbulence is not valid for high wave numbers. In this region, fluctuations separated by the distances between the relevant focus points are uncorrelated and the squeezing has no effect. The lack of coherence also explains that the values are higher than predicted because the u spectrum is highly contaminated by w and v fluctuations.

The plotted model spectrum for the conventional two-beam processing method shows a significantly lower averaging effect compared to whole circle processing methods at all wave numbers except in the very low wave number region, where the methods are expected to perform similarly well.

With the two-beam method it is expected that fluctuations with the highest wave numbers analyzed are to some extent included in the spectrum, while they were close to zero when circle processing was applied. The normal two-beam processing in the model is prone to cross-contamination at both resonance points (vertical dashed lines). This situation is explained in detail in Sect. . In contrast, the method of squeezing applied to the two-beam processing shows as expected no cross-contamination in the model calculations.

Overall, spectra calculated from the two-beam processed measurement data show good agreement to the model. It is important to keep in mind that, due to the poor fit of the measured spectra of the horizontal wind components and the modeled spectra at low wave numbers, we can compare the relations between the different methods but not absolute values. At low wave numbers, the measured spectra are on average closer to the target spectrum than in the case of circle processing. The slightly lower energy content of squeezed measurements that we observed and explained for circle processing is found here as well. Also, when it comes to deviations from the modeled behavior, like for example the higher energy density at some wave numbers (e.g., k1=3×10-3 m-1), we find similar tendencies as in circle processing, and the reason is likewise unclear.

The strong cross-contamination at the first resonance frequency is clearly represented in the normal two-beam processing and can be completely avoided by squeezing the two focus points to virtually one point. It is worth mentioning that the squeezing procedure works more like expected when applied to the two-beam method than when applied to the circle processing. This can be explained by the error caused by not having continuous but only discrete delaying times τ available. The relative impact of this error is lower in the case of the two-beam method because then the maximum separation distance DC must be compensated for. In circle processing mode, the shorter separations for which the relative error is larger also contribute to the result.

At k1>7×10-2 m-1 the two processing methods result in nearly identical values again, and we assume the lack of coherence of short eddies to also be the cause here.

The modeled spectra of conventionally VAD-processed lidar measurements predict energy densities that slightly exceed the target spectrum for very long fluctuations with k1<1.3×10-2 m-1. This behavior can be explained by uncorrelated w fluctuations between the eastern and western sides of the measurement circle that contaminate the v signal. This contamination is slightly stronger than averaging that is very weak at low wave numbers.

By contrast, fluctuations shorter than approximately k1=1.3×10-2 m-1 appear dampened in the spectrum, and fluctuations with higher wave numbers k1>10-1 m-1 are not even present in the v spectrum due to the strong averaging. Unlike the u spectrum, the v spectrum does not have characteristic behavior around the first resonance wave number. This is not surprising because the lines of sight that are the most important for the detection of v fluctuations lie, according to Eq. (), orthogonal to the mean wind direction in which turbulence is advected. Thus, no resonance occurs.

When the model spectrum for SMC processing is analyzed, we find a higher variance for all wave numbers above approximately k1=1.3×10-2 m-1. Reduced averaging along the measurement circle is the reason for the higher energy in the SMC spectrum. It is caused by the following: the process of squeezing reduces the longitudinal separation of the focus points ideally to zero while the lateral separation remains unchanged. We know that the lines of site perpendicular to the mean wind direction on both sides of the measurement circle are the most important for the determination of vlidar. Let us assume these are the easterly and westerly beams. The exact east- and westbound beams are not affected by the process of squeezing. But for example the northeast and the southeast beams (respectively the northwest and northeast on the other side) see different turbulent structures in conventional VAD processing. With SMC processing, these two beams see the same structure. In the subsequent calculation of the v component all lines of sight are combined and the pairs of radial velocities that lie in line with the mean wind contribute with the average of their amplitudes. This average of amplitudes is lower than the common amplitude measured by the beam pairs under SMC processing. More simply, there is less averaging along the measurement circle when SMC is applied. As a result, the spectrum of SMC shows higher energy densities for all wave numbers at which uncorrelated fluctuations dominate.

Now we compare the measurements with the model. Unfortunately, similar to the u fluctuations, the target spectrum does not represent the sonic measured values properly, especially for low wave numbers. We will therefore concentrate on the tendencies and proportions between the spectra from different methods. While the model predicts the behavior at the lowest wave numbers more or less satisfactorily, we are faced with two outliers at k1=1.65×10-3 m-1 and k1=2×10-3 m-1 for which both the VAD and SMC processing lead to excessive energy estimations. The reason is unclear and not further investigated. At all other wave numbers, the agreement of model and measurements is very satisfactory. In particular, the differences between the two methods are found in the measurements, as predicted. The good agreement between model spectra and measurement spectra at wave numbers above approximately k1=2×10-2 m-1 might be surprising with regard to the poor agreement of sonic measurements and target spectrum. The reason is that the shape of the lidar v spectra is mainly determined by the cross-contamination from the w component, which, as we describe in Sect. , agrees better with its model representation.

The identity of VAD- and SMC-derived measurement spectra that we saw for u fluctuations for k1>7×10-2 m-1 is found here at k1>10-1 m-1. The reason is obvious when we look at the relevant longitudinal separation distances. They are much shorter when processing v fluctuations than u fluctuations, and the assumption of frozen turbulence is more valid for short separation distances. Therefore squeezing can maintain its effect into a somewhat higher wave number region.

When the two-beam method is applied, i.e., using only the east and west beams to derive the v component of the wind vector, the method of squeezing has no effect. In comparison with the whole circle processing, the two-beam method is characterized by lower energy estimates at low wave numbers and higher energy estimates at higher wave numbers (see Fig. ). One reason for the first is assumed to be the lower coherence of v fluctuations separated by the full distance DC. That implies that two-beam processing gets a somewhat lower contribution of vwind to vlidar. A second reason is that there is not cross-contamination from u on v occurring for the two-beam processing. The higher energy content at high wave numbers results from the absence of averaging along the measurement circle.

The model cannot be compared with measurements because the line-of-sight velocities of the east and west beams were erroneous. The absolute values we measured are unrealistically biased towards nonzero values. This effect has been previously reported . We included the model behavior of two-beam processing for the sake of completeness and to show that the availability of reliable measurement data for the east and west beams would be of hardly any use.

To begin with, we compare the model predictions of conventional VAD processing and the new SMC method against one another and with regards to the w target spectrum. The results of the actual measurements follow. The model prediction of the conventionally processed VAD lidar data shows some attenuation of the spectral energy even for very low wave numbers. The reason is mainly the infinitely long tails of the line-of-sight averaging function and to a lesser extent the averaging along the measurement circle. Both averaging effects become quickly stronger for increasing wave numbers. The spectrum from VAD processed data is expected to drop at the first resonance point marked with a grey dashed vertical line in Fig. . This drop is minor due to the overall low energy level present in the spectrum. The spectrum reaches a value near its final minimum with variance values close to zero already at around k1=5×10-2 m-1 just after crossing the first resonance point. w fluctuations with higher wave numbers are not detectable with conventional VAD processing. According to the model, the SMC processing improves the situation slightly by removing the longitudinal separation that makes lidar blind to w fluctuations at the resonance points with VAD processing. Squeezing the measurements also helps improve the measurements well above and below the resonance wave number. But still, due to the remaining averaging effects, only a minor fraction of the energy in the vertical wind can be detected with both methods at wave numbers above roughly k1=10-2 m-1.

The fit between target spectrum and measurement data in the low wave number region is good for the w component. This was not the case for the u and v components. The results of show that the spectra for vertical fluctuations are not prone to contributions from the mesoscale spectrum. The measurement data overall support these model predictions and show that the process of squeezing functions well over the entire frequency range in this study. In detail, we only find some mismatch for very low wave numbers at which k1<10-3 m-1. The measured spectra lie above the target spectrum here although we expected some attenuation. The discrepancy is caused by the real u-wind spectrum being much higher than the underlying target spectrum; see Fig. . We already found that large-scale u fluctuations are also not perfectly correlated and thus contaminate the measured lidar spectra, which is not considered in the model. At higher wave numbers we find reasonable forecasting of measured w spectra by the model.

The modeled two-beam spectra in Fig. b lie considerably closer to the target w spectrum for all wave numbers. That can be explained by the absence of circle averaging. The strong influence of resonance visible at the two first resonance wave numbers underlines the importance of squeezing when striving for more realistic spectra from lidar measurements.

At low wave numbers with k1<10-2 m-1 the measured spectra contain higher energy densities than modeled spectra. A similar but less pronounced effect was found in circle processing only at the lowest wave numbers. The explanation we gave there must therefore be supplemented by mentioning that the assumed decorrelation is stronger for the maximal separations that are involved in the two-beam method. The further comparison of spectra from experiment and model shows that the process of squeezing also leads to the expected effect in the case of using only two beams to determine the w component of the wind vector. As in the case of u fluctuations, this statement must be limited to wave numbers k1<7×10-2 m-1.