Our experiment is conducted using two commercial microwave links (CMLs), a microwave scintillometer (MWS) and an eddy-covariance system (EC) at the Ruisdael Observatory at Cabauw, the Netherlands (Fig. ). The links and scintillometer transmit along an 856 m path between 51.9743° N, 4.9235° E and 51.9676° N, 4.9296° E. On both sides, the CMLs and MWS are mounted on a 10 m high vibration-free mast as designed for a project of. The site is located in a European marine west-coast climate (Cfb in the Köppen classification). The water table is managed so that the soil water content in the root zone is kept as near as possible to field capacity e.g.. The surrounding terrain consists mostly of grass fields regularly separated by open-water ditches (see Fig. a) and some small villages. Under the prevailing southwesterly wind conditions, the scintillometer footprint does not contain any major obstacles within more than 2 km, except the 213 m flux tower. Elevation differences in the area are within a few metres for distances of up to 20 km .

For this study, we use data from two collocated CMLs and from an MWS. Both CMLs were formerly part of a commercial mobile phone network operated by T-Mobile Netherlands (currently, Odido Netherlands). These are a Nokia Flexihopper, mounted at 10 m above the surface, transmitting at 38.1745 GHz with a bandwidth of 0.9 MHz, and an Ericsson MiniLink RAU2, mounted at 9 m above the surface, transmitting at 38.1605 GHz with a bandwidth of 7 MHz. The diameters of the antennas of both links are 0.3 m. Both links are bidirectional and transmit in the opposite direction at approximately 39.4 GHz. For this study, we only use the 38 GHz data the 39 GHz data can be found in. Both devices transmit and receive only horizontally polarized radiation.

Similar to , all signal intensities are sampled with a Campbell Scientific CR1000 data logger at a 20 Hz sampling frequency. To sample the signal intensity, we direct the analogue detector signal used for automatic gain control to the data logger. To convert the measured voltages to received signal intensities, we use the calibration curve provided by for the Nokia Flexihopper: 5I=-34.228V+22.433, in which V is the voltage measured [V] by the data logger, and I is the intensity [dB].

For the Ericsson MiniLink, the following standard equation is used (Sander Gombert, employee Alfatech, personal communication, 4 June 2024): 6I=-40V+120. The Nokia Flexihopper was installed on 11 September 2023 and the Ericsson MiniLink on 4 October 2023. We perform our analysis based on 30 min time intervals, a typical time interval for the determination of turbulent heat fluxes e.g., until 18 October 2023. After this date (towards winter), the turbulent heat fluxes reduce, so they are less clearly reflected in the Cnn estimates. For the Nokia Flexihopper, the transmitting 38 GHz antenna was accidentally moved on 25 September, slightly reducing the received signal intensity. In order to account for this, we exclude this day from our analysis and treat our data as two separate subsets, i.e. before and after this day.

As a reference, we use a Radiometer Physics RPG-MWSC-160 microwave scintillometer, transmitting at 160.8 GHz and sampled at 1 kHz using the internal data logger of the MWS. The aperture of the MWS is 0.3 m. Data from the MWS are available during the entire period, with only minor data gaps, 1 h d⁻¹ at most. The MWS directly provides an analogue-to-digital converter level ranging between 0 and 65 536 that is proportional to signal intensity, which can be used in the subsequent analysis. The MWS is specifically designed to measure the full spectral range of the signal intensity fluctuations caused by the scintillation effect and to link these fluctuations to the turbulent heat fluxes.

To compare the Cnn estimates obtained with the CMLs with the estimates from the MWS, we assume that the Cnn for 38 and 160 GHz scintillation measurements are the same, as suggested by the calculation proposed by . Other studies suggest that these values might slightly differ, although insignificantly in comparison to other uncertainties in our study. For example, using the analysis of for a sensible heat flux of 100 W m⁻² and a latent heat flux of 200 W m⁻² (and an air density of 1.2 kg m⁻³, friction velocity of 0.2 m s⁻¹, relative humidity of 50 % and a temperature of 293 K), the Cnn for 38 GHz is 6.384×10-12 m^−2∕3, and the Cnn for 160 GHz is 6.392×10-12 m^−2∕3, a difference ≪ 1 % based on the parameters of.

To allow for a comparison of the power spectra of the CMLs with the MWS, we convert the scintillation measurements of the 160 GHz MWS to equivalent 38 GHz scintillation data. To do so, we need to transform the variance on the y axis and the scintillation frequency of the MWS, i.e. the frequency on the x axis in the power spectrum, into fMWS,160GHz , i.e. a coordinate transformation that conserves variance. The variance can be transformed through Eq. (). Following based on Eq. (), the scintillation frequency is transformed as follows: 7fMWS,38GHz=fnorm,38GHzfnorm,160GHz×fMWS,160GHz, in which fMWS,38GHz is the transformed frequency axis for the equivalent 38 GHz MWS data [Hz], and fnorm [Hz] is commonly used to normalize the frequency axis e.g.. The value of fnorm depends on the transmitting frequency; hence, the values for 38 GHz (i.e. fnorm,38GHz) and 160 GHz (i.e. fnorm,160GHz) differ. To compute fnorm, the following equation is used: 8fnorm=u⟂2πλL-1/2, which reduces the fraction in Eq. () to 38/160 = 0.4873. Hereafter, when referring to the MWS data, we refer to the equivalent 38 GHz MWS data.

Additionally, we smooth the power spectra similar to in . To do so, the power spectrum is smoothed by averaging the power at each frequency with those at the neighbouring frequencies in a specified window. We set the width of the window to 20 % of the specific frequency (to account for the increase in number of values towards higher frequencies in the power spectra). The weighting of these points within the window is assumed to be bell-shaped so that the adjacent points have more influence on the smoothing than the points at the far end of the window do.

After studying the Ericsson link time series and variances, we decided to exclude this link from this scintillometry analysis. The 0.5 dB power quantization of the device prevents us from obtaining representative variances. Graphs of the time series and variances of the Ericsson link are available in Appendix . For the influence of power quantization on σln⁡(I)2 of the Nokia link, see Appendix .

For our analysis, we do not consider nighttime intervals (i.e. incoming shortwave radiation below 20 W m⁻²), intervals during which it rained or those that follow within 1 h of a rain event (to exclude wet-antenna attenuation in our analysis), and intervals with horizontal wind speeds above 8 m s⁻¹ independent of the wind direction. The latter is applied because the Nokia CML vibrates above this wind speed, and we observed an increase in variances above this limit in our data (not shown). We divide all time intervals that do not meet the previously described conditions randomly over a calibration and a validation set. We use 80 % of the data for calibration and 20 % for validation. Additionally, for our corrected Cnn estimates (Sect. ), we remove all time intervals with Cnn estimates larger than 6.49×10-12 m^−2∕3, which we expect to be the maximum value for our dataset. Using Eqs. () and (), this value is based on the assumption that 80 % of the maximum incoming shortwave radiation (i.e. approximately 800 W m⁻² for this dataset) is used for the turbulent heat fluxes, with a minimum Bowen ratio (which results in a maximum Cnn) of 0.2 (and an air density of 1.2 kg m⁻³, friction velocity of 0.2 m s⁻¹, a temperature of 293 K and a specific humidity of 0.015 kg kg⁻¹).

EC measurements are used to compute additional independent Cnn estimates. The EC system consists of a sonic anemometer (Gill-R50) and an open-path H2O/CO2 sensor (LICOR-7500) and is installed at 3 m above the ground . The measurement frequency of the system is 10 Hz.

To estimate Cnn with EC measurements, we compute CTT, Cqq and CTq from the raw temperature and humidity measurements, defined as e.g. 9Cyy≡y(x)-y(x+r)2‾r2/3=y(t)-y(t+Δt)2‾u‾Δt2/3, in which y(x) is either T or q at location x, and r [m] is a separation distance. To estimate structure parameters from time series, Taylor's frozen turbulence hypothesis has to be assumed, so y(x) is replaced by y(t), and r is replaced by the mean horizontal wind speed u‾ multiplied by Δt. Additionally, we have to correct for the height difference between the EC measurements (i.e. 3 m) and the links (i.e. 10 m), as the structure parameters are not constant with height. To do so, we use Eqs. () to () in Appendix .

It should be noted that the 10 Hz temperature and wind speed components for the EC show unexpected behaviour because some temperatures and wind speeds occur much more frequently than other temperatures and wind speeds that are approximately the same (see Fig. S1 in the Supplement for a histogram of the wind speed, temperature and humidity measurements during a full day, i.e. 11 September 2023, to illustrate this unexpected behaviour). However, the overall behaviour of these components does not show any abnormalities. Therefore, we expect this only has a minor influence on the CTT, Cqq, u* and H calculations, the latter two being required in Eq. ().

For our analysis, we also make use of other meteorological measurements at Cabauw available from the, such as air temperature, humidity, wind speed, precipitation and radiation. The majority of these measurements are needed for the conversion of the MWS data to 38 GHz (e.g. u⟂) and to correct the Nokia CML variances (Sect. ).

In this study, we compare Cnn estimates of the various instruments. For all comparisons, we use the relative mean bias error (RMBE), the 10–90 inter-quantile range (IQR) and Pearson's correlation coefficient (r). For all metrics, we use the logarithmic values of the Cnn estimates since Cnn typically exhibits a log-normal distribution throughout the day e.g.. We define the RMBE in comparison to our reference instruments and calculate it as 10RMBE=log⁡(y)-log⁡(x)‾, in which log indicates the decimal logarithm; y are the Cnn estimates of the instrument on the y axis and x are the Cnn estimates of the instrument on the x axis, i.e. the reference instrument. Intuitively, the RMBE represents the order of magnitude that the values on the y axis are larger (or smaller) than the reference values on the x axis due to the use of logarithmic values. The IQR is calculated as follows: 11IQR=P90-P10, in which P90 and P10 are the 90th and 10th percentiles of the difference between the logarithmic Cnn estimates of the instrument on the y axis and the logarithmic Cnn estimates of the instrument on the x axis of a scatterplot. The IQR can be interpreted as how many orders of magnitude the 90th percentile of the residuals is larger than the 10th percentile of the residuals. For r, we use the logarithmic values of the Cnn estimates so that this value visually corresponds to the correlation on a log–log plot.

Our first method assumes that there is a constant noise floor with (scintillation) frequency and over all time intervals present in the Nokia CML signal, probably as a consequence of the noise floor design in the receiving antenna. Under this assumption, we can write the variance in the CML as 12σCML2=σabsorption2+σscintillations2+σnoise2. The method consists of estimating the contribution of the noise floor to σln⁡(I)2 by subtracting a low quantile of all Nokia-CML-derived values of σln⁡(I)2 (or Cnn) from itself based on the calibration part of the dataset. All values below this percentile are removed since these would become negative after correction. Step 1.

Noise estimation (only in the calibration part of the dataset). a.

Absorption filter. For each time interval, we apply a high-pass filter at 0.015 Hz by subtracting the moving average with a window size of 1/0.015 = 66.7 s from the signal intensity time series. We have selected this high-pass filter value as it retains 95 % of the variance due to scintillation for the CML at crosswind speeds of 0.5 m s⁻¹ in our setup. For higher crosswind speeds, the spectrum shifts towards higher frequencies so that an even larger fraction of the variance is retained.

In this method, we make use of the MWS to determine the noise contribution to the Nokia CML signal. Also, we take into account the crosswind condition, as the scintillation spectrum shifts to higher frequencies with higher crosswind speeds. We therefore select, depending on the crosswind, those parts of the spectrum where the Nokia CML and the MWS data behave similarly. For example, in Fig. a between approximately 0.1 and 1 Hz, the Nokia CML and the MWS show similar behaviour, although with an offset for the Nokia CML. After computing the (partial) variance in the selected parts of the spectrum, we correct for the fraction of σln⁡(I)2 omitted based on the theoretical spectra (Eq. ). For operational CMLs this method is typically not possible, but it shows the potential for using CMLs as scintillometers. Step 1.

Noise estimation (only in the calibration part of the dataset). a.

Absorption filter. Similar to step 1a in Sect. , we apply a high-pass filter at 0.015 Hz for each time interval by subtracting the moving average with a window size of 1/0.015 = 66.7 s from the signal intensity time series (Fig. a).

Time series of a sunny day versus a cloudy day (Fig. a and b) show that both methods capture the daily cycle typically found in Cnn estimates but with some more outliers compared to the reference instruments. Similar to those of the reference instruments, the Cnn estimates of our corrections are generally higher on the sunny day than on the cloudy day. Both methods show more outliers on the cloudy day than on the sunny day, especially the spectral-noise method. For other cloudy days (and occasionally the start and end of the day), similar, more noisy behaviour is observed for both methods. We attribute this to the relatively low Cnn during these days (and moments), which makes it more complex to extract the scintillation signal from the noise-dominated Nokia signal.

For our entire dataset, both proposed methods show a huge improvement (Fig. ) in comparison to the unprocessed Nokia CML Cnn estimates (Fig. ). The RMBE related to both the MWS and EC has reduced from 1.2 to at least 0.3, which is a major improvement in comparison to the RMBE of the comparison between the reference instruments (i.e. 0.01), indicating that the proposed methods overestimate Cnn at most by a factor of 2 (i.e. 100.3), which is also visible in Fig. a, where both methods are seen to overestimate the references during the entire day. Also, both our proposed methods increase the correlation coefficient, especially the spectral-noise method. The IQR increases slightly after our correction methods (partly a consequence of taking the logarithmic values of Cnn). Between the two correction methods, the spectral-noise method has a lower IQR, which is a consequence of the nature of our corrections, as the constant-noise method only subtracts a constant value, while the spectral-noise method removes the part of the spectrum in which the influence of the noise on σln⁡(I)2 is relatively high (i.e. f > 1 Hz). Moreover, the performance of both methods does not seem to show any large dependence on weather conditions, such as temperature, crosswind, humidity and incoming shortwave radiation (Fig. )

As a proof of concept, our results show that under certain conditions, CMLs could be used to estimate Cnn, although with a larger uncertainty and bias with respect to both reference instruments, an MWS and an EC, than the comparison between the reference instruments among each other. Our two proposed methods to correct the Nokia CML scintillation spectra and to obtain Cnn estimates show comparable behaviour, although the spectral-noise method performs slightly better than the constant-noise method, especially regarding the spread. An advantage of the constant-noise method is that it is a relatively simple correction method that does not require the use of an MWS. Overall, this shows that considering crosswind conditions, which cause a shift in the power spectrum along the frequency axis, and selecting the frequency ranges over which scintillations are best resolved also improve the Cnn estimation. However, it also requires a more elaborate study of the power spectrum of the CML and the MWS, which might not always be possible.

As the spectral-noise method requires the presence of an MWS to determine the contribution of noise to Cnn, the ability to transfer our methods to other datasets is limited. When an MWS is available to install next to a CML, both our methods can be used to estimate Cnn using CMLs, under the condition that the noise in the CML is of a similar nature to the noise in the Nokia CML. This even holds for different experimental conditions, such as other path lengths or installation heights, since these are indirectly accounted for in our methods. The only difficulty might arise when the contribution of noise to the signal intensity fluctuations is relatively large in comparison to the scintillation fluctuations. Moreover, for the spectral-noise method, when assuming that the noise is caused by a stationary white-noise floor in the receiving antenna e.g., installing an MWS next to the CML would not be required for a full experimental period, but it would be sufficient to perform a one-time determination of the noise floor, possibly even for a single type of CML.

However, usually an MWS is not available to install next to a CML, let alone an entire network of CMLs. Therefore, the constant-noise method is the most promising for application in other experimental conditions. However, for other CML types, most often having a collocated MWS will be required in order to determine the nature of the signal, including the noise. Having the full information on the introduction of noise into the receiving antenna of CMLs in advance would allow for a more precise correction of the noise, possibly not even requiring the use of an MWS. For example, this could shed light on the dependency of a noise floor on the signal intensity, temperature or the possible presence of any frequency-domain filter. However, usually this information is not available or is not shared publicly, complicating the Cnn estimation.

Previous scintillometer studies confirm the correspondence obtained between microwave scintillometer Cnn estimates and in situ EC measurements. , who built on the work of , reported Cnn estimates with a 30 GHz scintillometer at 60 m above the surface that showed similar behaviour to Cnn estimates obtained with high-frequency temperature, humidity and wind measurements. Similarly, showed that Cnn measurements performed by a 173 GHz scintillometer only slightly underestimated Cnn estimates obtained with EC high-frequency meteorological measurements. and reported CTT, Cqq and CTq estimates from an EC system, which were comparable to measurements from a dual-beam scintillometer setup (optical and microwave). Hence, compared to previous studies, our Cnn estimates from CMLs exhibit a relatively large uncertainty.

Even though other studies outperform our Cnn estimates, these all require high-quality meteorological input data, which are not often available, whereas Cnn estimates obtained from CML signal intensities would be a more direct method to obtain Cnn, do not require any additional measurements and are available from a potentially larger number of devices with a nearly continental coverage. used single-level observations to obtain the energy balance and used the Penman–Monteith equation to estimate Cnn. A comparison of their simulated Cnn estimates with EC-based Cnn estimates over grassland seems to outperform our comparison between CML and EC estimates, although their method shows a large dependence on the quality of the meteorological input data. Similarly, estimated Cnn using two-level meteorological observations of wind speed, temperature and humidity. Also, provided Cnn estimates over snow and ice using meteorological observations and emphasized the strong dependence of the estimates on the nonlinear relation between the fluxes and Cnn and on the assumed Bowen ratio.

Several aspects of CML networks could prevent us from obtaining similar Cnn estimates, as CMLs are not designed to measure the scintillations. Firstly, power quantization affects the measured variances in signal intensity. From the devices used, a Nokia and an Ericsson CML, we rejected the Ericsson CML estimation of Cnn using scintillation theory because of the 0.5 dB power quantization (i.e. the discretization of the signal intensity). Power quantization is a method commonly applied to CML networks, typically ranging between 0.1 and 1 dB . We tested the impact of power quantization on our data and expect that for the smallest quantization steps, Cnn estimates could still be feasible, although quantization would be an additional source of uncertainty (Fig. a and b).

Secondly, the CMLs have not been designed with the aim of measuring scintillations, which is also reflected by the presence of noise in the signal intensity of the Nokia CML. To correct for this inability to capture the scintillations, we determined our noise levels with the MWS, which usually is not possible for a CML. In order to determine how antennas modify the received signal intensity, e.g. as a consequence of different internal hardware design choices, a comparison with an MWS would be required for each specific type of CML antenna before being able to estimate Cnn or before having full information on the noise. Alternatively, the noise sources for each component of a CML device could be characterized if high-frequency data are available. Moreover, the mounting mechanism of the CMLs is not designed to be vibration-free, as the Nokia CML started to vibrate above wind speeds of 8 m s⁻¹, even though the mast itself remained free of vibrations. In addition, the masts used in CML networks might also not be vibration-free. Note that for long paths, saturation of the scintillation signal could also influence the obtained Cnn estimates e.g. seefor the saturation limit for microwave frequencies.

Thirdly, typical temporal sampling strategies applied to CML networks are on a coarser temporal resolution than our 20 Hz sampling. Typically, CML signal intensities are stored in the network management system every 15 min, with minimum and maximum values of the signal intensity (and occasionally also with a mean intensity included). Another sampling strategy developed by allows us to select an instantaneous sampling strategy with time intervals as small as 1 s, the variances of which might approach actual signal variances (Fig. c). Our selected 20 Hz sampling strategy mimics the typical instantaneous sampling strategy on which the coarser sampling strategies are based. However, it could be that adding the variance to the operationally reported signal intensities is relatively easy, as calculating the variance is only one additional computation compared to calculating the mean value per time interval.

This study focused on obtaining Cnn estimates, while to compute the turbulent heat fluxes, additional information, and thus uncertainty, related to the distribution between temperature and humidity fluctuations is required. For scintillometer setups, an optical link is usually collocated next to the MWS. The optical link is mostly sensitive to temperature fluctuations (and can also be used to solely determine the sensible heat flux), so the structure parameter of humidity can be extracted from the Cnn estimates by the MWS. For (the vast majority of) CMLs, no in situ measurements are available, complicating the required separation between the temperature and humidity structure parameters. To do so, we would have to use global meteorological data, such as satellite measurements or model data, but the accuracy and usefulness for the eventual retrieval of the turbulent heat fluxes are questionable. Either way, the required assumptions in this computation step introduce additional uncertainty, possibly making the overall uncertainty in the turbulent heat fluxes relatively large. In a follow-up study, we will focus on obtaining the turbulent heat fluxes from the presented methods to estimate Cnn. As a potential solution to reduce the relatively large uncertainties, we will look into the influence of upscaling the 30 min estimates to daily estimates. Additionally, we aim to use a more extensive dataset (around a full year) instead of 37 d in September and October to identify potential influences of other weather circumstances on the Cnn estimates.