We present a novel method of using two or three collocated microwave link instruments to estimate the three parameters of a gamma raindrop size distribution (DSD) model. This allows path-average DSD measurements over a path length of several kilometers as opposed to the point measurements of conventional disdrometers. Our method is validated in a round-trip manner using simulated DSD fields as well as five laser disdrometers installed along a path. Different potential link combinations of frequency and polarization are investigated. We also present preliminary results from the application of this method to an experimental setup of collocated microwave links measuring at 26 and 38

The use of microwave links to measure rainfall intensity has received significant attention in the last decade. The main driver for this has been the insight that the backhaul links of mobile communication networks are suitable for such measurements and are available in greater numbers than dedicated rainfall measurement stations in many countries

DSD estimates can be used to derive all other bulk rainfall variables and are therefore valuable for a variety of purposes (see, e.g.,

In order to estimate the DSD from a limited number of statistical moments, a parameterization has to be used. The gamma distribution with three parameters provides a good estimate for a wide variety of rainfall types

A handful of different techniques can be identified that have been developed to retrieve DSD parameters from a limited number of polarimetric radar moments: the constrained-gamma method developed by

In this paper we explore the potential of a numerical retrieval from microwave attenuation and/or differential propagation phase shift. We consider two different techniques: the first method uses three measured microwave link variables to derive the three parameters of the gamma distribution. Here, the gamma parameters are weakly constrained (i.e., only a limited range of parameter values is allowed). The other method uses two measured microwave link variables to derive two parameters of the gamma distribution. Here, the third parameter is completely constrained by the other two, similar to the method used by

The paper is organized as follows: in Sect. 2 we present in more detail the datasets used in this paper. In Sect. 3 we present the theory and the methods used to retrieve DSDs from microwave link variables. We also describe the validation methods employed. In Sect. 4 we discuss the results retrieved from the Ardèche dataset. In Sect. 5 we discuss the results of several tests using simulated attenuations from the Wageningen disdrometer dataset. We also consider the efficacy of different frequency combinations and the robustness of the retrieval to measurement uncertainty. In Sect. 6 we apply the developed methods to actual link measurements that were obtained in Wageningen. In Sect. 7 we present our thoughts on the feasibility of the techniques in practice and the choices made in this paper. Finally, in Sect. 8 we come to conclusions and give recommendations for further study.

Our first test case consists of a two-dimensional interpolated DSD field based on polarimetric radar data measured in the Ardèche, France, as part of the HyMeX campaign

We take a transect through this field over the entire length of the field (Fig.

In order to calculate the microwave link variables from the DSD data, we first need to interpolate the data in the diameter dimension from the irregular bins to a regular diameter grid with a resolution of 0.1

Our second test case is a microwave link experimental setup in Wageningen, the Netherlands

In order to use the disdrometer data to represent the DSD of the link path, we take a weighted spatial average over the disdrometer data. As with the Ardèche data, we interpolate the DSD data in the diameter dimension from the irregular bins to regular intervals before calculating the microwave link variables. In order to improve the robustness of the results, we only consider measurement intervals where each one of the five disdrometers counted at least 50 drops (see

To determine the underlying DSD from a limited number of statistical moments, we need an approximation with a limited number of parameters. One of the most widely used approximations for rain DSD is the gamma distribution suggested by

The corresponding dimensionless scattering efficiencies are defined as

Scattering efficiency of raindrops as a function of drop volume-equivalent diameter modeled with the

The variables used as input in the retrieval could be attenuation of horizontally or vertically polarized radiation or phase differences between horizontally and vertically polarized radiation at one or several frequencies. In order to be able to use attenuations and phase differences interchangeably in the retrieval algorithm, we rearrange Eqs. (

Inserting Eq. (

Figure

Several ratios of microwave link observables that can be used as input to the DSD retrieval, as a function of parameters

We use the Powell hybrid method

Even when the algorithm converges, the solution of the system of equations is not necessarily unique. We need an extra set of constraints to make sure we retrieve the parameters that are the most plausible. In order to do so, we use the parameters retrieved by the analytical method of moments of

The mask is then applied to all attenuation/phase-based numerical retrievals to define the range of allowed parameter values. If the estimate is outside this contour, we reset the root-finding algorithm with a new initial guess that is within the contour but slightly perturbed from the previous initial guess as described above. This is continued until we find convergence within the contour or we reach the maximum number of iterations without a solution.

When only two microwave link variables are available, the system of equations of Eq. (

Figure

Ratios of attenuations as a function of the

We test the capability of the methods to accurately retrieve DSDs and their associated statistical moments with two different datasets of drop size distributions: one simulated and one measured. We use Eqs. (

In order to assess the performance of the retrieval methods, we will use a number of statistical measures throughout the Results section. As a measure of the accuracy of the retrieval we use the median of the residuals (MOR). As a measure of the precision of the retrieval we use the median absolute deviation (MAD) of the residuals with respect to the median of the residuals. We chose MAD and MOR over the use of standard deviation and means, because there are a relatively small number of extreme deviations which would otherwise have too much influence and thus would not give much information about the typical precision. The statistical metrics employed here are less influenced by non-normality and outliers. Because the number and severity of the extreme deviations are also an important part of the performance assessment of the retrieval, we also compute the 95th-percentile absolute deviation with respect to the median of the residuals (95AD). Together with the MAD this gives a more complete picture of the distribution of the errors, while still being insensitive to the true outliers. All metrics are normalized with respect to the median of the original quantities; hence they are dimensionless. Furthermore, we also compute the fraction of non-convergent retrievals compared to the total number of retrievals (taking into account the filtering described in Sect. 2.2). This “failure ratio” (also dimensionless) is necessary for a complete picture of the robustness of the method since the other metrics naturally exclude these intervals.

Figure

Figure

Gamma distribution parameters

Looking at the rainfall intensity for both events (Figs.

Gamma distribution parameters

Statistics of rainfall intensity,

We apply both the two-parameter and three-parameter retrieval algorithms to the Wageningen disdrometer dataset. The results are shown in Fig.

Statistics of rainfall intensity,

Gamma distribution parameters

We summarize the differences in accuracy and precision between the two-parameter and the three-parameter method in Table

Statistics of integer statistical moments,

However, the differences in accuracy and precision of the retrieval between the three-parameter and two-parameter retrieval as measured by a range of integer moments is small, and in many cases the two-parameter retrieval proved to be more reliable. Especially the number of sub-millimeter raindrops is severely overestimated by using the three-parameter method, as shown in Fig.

Because these results show that a three-parameter retrieval provides little added value above a two-parameter retrieval and because the two-parameter retrievals are far less computationally intensive than three-parameter retrievals, we will restrict ourselves to two-parameter retrievals in the remainder of this paper.

In order to determine the effect of the carrier frequencies of the links on the accuracy and precision of the retrieval, we perform two-parameter DSD retrievals at many different frequencies and calculate MOR, MAD and 95AD for the third-order moment of the retrieval compared with the third-order moment directly calculated from the measured DSD. We consider both dual-polarization retrievals with frequencies ranging from 10 to 45

MOR, MAD and 95AD of the third-order moment of the DSD estimated using a two-moment dual-polarization retrieval as a function of carrier frequency based on disdrometer data. All statistics are normalized with respect to the median of the moment of the original measured DSD.

MOR, MAD and 95AD of the third-order moment of the DSD estimated using a two-moment dual-frequency retrieval as a function of the two carrier frequencies based on disdrometer data. All statistics are normalized with respect to the median of the moment of the original measured DSD.

Because our retrieval algorithm uses ratios of attenuations as input, it is important that a reliable baseline power level is established from which to calculate the attenuations. To assess the sensitivity of the retrieval technique to inaccuracies in the baseline (dry) power level, we perform the two-moment retrievals based on the simulated attenuations from the disdrometer measurements in Wageningen but with an (equal) offset added to all input attenuations. Figure

DSDs of the two-moment dual-polarization (38

Aside from systematic measurement bias as discussed in Sect. 5.3, there can also be measurement limitations that affect the precision of the attenuation measurements. Because the retrieval method relies on ratios of attenuations, we expect that the retrieval is highly sensitive to such limitations as well. In practice, when processing link attenuation data from operational telecommunication networks, the power quantization error of the analog-to-digital conversion completely overshadows any instrumental error in the analog detector. Therefore, we will focus our analysis here exclusively on such quantization errors. Consequently, we do not have to make any assumptions about the instrumental precision of any particular transceiver model.

We have applied several different magnitudes of rounding to the attenuations calculated from the disdrometer data and performed the retrieval on the rounded attenuations for the complete 9-month dataset. The results are shown in Table

Statistics of rainfall intensity,

Using the double-moment retrieval method, we estimated the DSDs from actual link measurements of the Wageningen setup. The baseline power level of the links showed considerable fluctuations over the course of the measurement period. Therefore, it was not feasible to perform retrievals for the entire 9-month dataset. We selected the event of 27 July 2015 (see Fig.

The resulting DSD is very similar in shape to that obtained in the simulations, with overestimations especially at smaller diameters, but with the general shape of the DSD preserved (not shown). Closer inspection reveals that the bias and scatter compared to the original DSD are actually up to 2 orders of magnitude higher than in the simulations, as can be seen in Table

Gamma distribution parameters

Statistics of accuracy and precision of the retrieval of integer moments of the DSD for the event of 27 July 2015 using actual link data from two different combinations of links. All statistics are normalized with respect to the median of the moments of the disdrometer data.

Constraints on the feasibility of the proposed methods in practice fall into three broad categories: availability of multiple link signals on the same path, quality of the available signals and real-time processing speed.

The use of a three-moment retrieval means that three moments on the same path need to be available. This is rare in commercial networks; therefore this method is most readily applicable to dedicated research networks. There are several different combinations of moments to choose from. However, in our approach we focused on the combination of a horizontal attenuation, vertical attenuation and phase difference at the same carrier frequency. This allows the use of a single set of antennae for all three moments, allowing for the use of a more compact and less expensive device (such as the device that was used in our test setup).

The second concern is with regard to the quality and reliability of the signal. In order to apply the method in practice, it is essential that a baseline (no rain) signal is accurately determined. Because the method relies on the ratios of attenuations, small deviations in the baseline determination can result in large deviations in the retrieval (and even non-convergence). No such problem exists in principle with regard to the phase difference; it is independent of any power baseline. However, phase difference on its own is not sufficient for the retrieval. To have a chance at a successful retrieval, the baseline needs to be as invariant as possible. Where it is not, the variability should be accurately modeled and predicted from auxiliary measurements. In our own preliminary attempts we found our instruments lacking in stability. In particular the clinging of drops to the antenna cover (as described in

The third constraint is only relevant when real-time processing is required. The three-moment method is relatively wasteful with computing cycles because of the repeated re-initialization of the root-finding process. We might expect this to become a bottleneck. However, in its current implementation the processing of 9 months of data from one link requires roughly 3

There are a number of caveats to our methods which could influence the interpretation of the results:
firstly, we use a threshold of 50 drops per disdrometer to filter out low quality measurements before calculating the mask or

Another consideration is the use of the mask itself. The mask is determined on the basis of the measured disdrometer data. We then use this mask in, among others, the retrieval of the DSD from the disdrometer-derived simulated variables. This could potentially lead to a retrieval procedure that is biased towards these particular circumstances and therefore yields more accurate retrievals in this simulation than would be representative for a general application. Nevertheless, these data are never used as input for the root-finding procedure itself. They are only used a posteriori to assess whether the results fall into a plausible range of values. Another potential issue is the use of the predetermined

The third consideration is the underlying assumption that the gamma distribution is an adequate representation of the actual DSD and that the untruncated gamma distribution is applicable throughout the diameter domain. It is on the basis of this assumption that we treat the values of

Using simulated link data, we have shown that a DSD retrieval on the basis of multiple microwave link variables can be successful and accurate, but only when precise high-resolution records of rain-induced attenuation are available. This was confirmed when applied on actual link data, where baseline variations prohibited accurate DSD retrievals. The use of both dual-polarization and dual-frequency retrievals is feasible. However, the use of dual-polarization is less sensitive to systematic inaccuracies in the base power level while being more sensitive to quantization errors than the use of dual-frequency links. Simulated retrievals using a variety of frequencies show that, at least between 10 and 45

In our field experiment we tested a dual-frequency retrieval using 26 and 38

Using phase differences in addition to attenuations is feasible in the simulations. However, in practice these measurements are not accurate enough to yield meaningful solutions. In most instances no convergence was obtained. We have also shown that using three microwave link variables yields no improvements over a retrieval using only two variables, which is also computationally faster and more readily applicable in operational settings. At least in comparable climatologies to those treated here, a predetermined

A follow-up experiment using different microwave links of similar frequencies (preferably commercially available ones) is needed to determine if the base power level of commercial links is sufficiently stable for reliable continuous observations. A tally should also be done on the number of dual-polarized links in cellular communications networks to determine if it is feasible to retrieve spatial DSD information from such networks or whether this technique is only applicable to some individual link paths, either from commercial or research networks.

Another concern is the quantization of data from commercial link networks. As the difference between the attenuation of the horizontally polarized signal and the vertically polarized signal is often a fraction of a decibel and data available from such networks are often rounded to 0.1, 0.5 or 1

The underlying radar and disdrometer data employed to simulate the DSD dataset described in Sect. 2.1 are available through the HyMeX database (

This paper was conceived by TCvL, with some suggestions from RU. TCvL devised the algorithm, wrote the necessary software to perform the retrievals and analyzed the results, with feedback by HL and RU. TCvL wrote the first draft and, after critical revision by all co-authors, wrote the final version.

The authors declare no competing interests.

The Ardèche dataset was kindly provided by Timothy H. Raupach.

This research has been supported by the Netherlands Organization for Scientific Research (NWO-TTW; project number 11944).

This paper was edited by Saverio Mori and reviewed by four anonymous referees.