A sensitivity analysis is performed for the two threshold values of the wet–dry classification, where median(ΔP) is varied from -5 to 0 dB with a step size of 1 dB (because the resolution of the employed data set is 1 dB), and median(ΔPL) is varied from -2 to 0 dB km-1, with a step size of 0.1 dB km-1. Although these threshold values could potentially become positive, this will hardly happen or only small positive values will be obtained. If both threshold values are 0 dB (km-1) this can therefore be considered representative for the situation without wet–dry classification.

The 2-D contour plots in Fig.  display relative mean error (%; black lines) and coefficient of variation (CV) or squared correlation coefficient (ρ2) (colours) in path-averaged 15 min rainfall depths as a function of median(ΔP) and median(ΔPL). The default threshold values are indicated by the black dot. Given the resolution of 1 dB, the default value for median(ΔP), -1.4 dB, implies that median(ΔP) should be lower than -1 dB (hence the black dot is plotted at 1 dB). In order to focus on the performance of the wet–dry classification with respect to the situation without wet–dry classification, the outlier filter has not been applied here. A wet–dry classification generally leads to a clear improvement in terms of ρ2 and CV upon a situation without wet–dry classification.

The contour plots show that application of median(ΔPL) is not necessary and that the best results are obtained for median(ΔP) values around -4 dB, i.e. much lower than the default value. Note that the default values of Aa and α have been used, where an outlier filter was applied. This may be the reason for the variability in relative mean errors. Finding optimal values of Aa and α for every combination of median(ΔP) and median(ΔPL) was considered too computationally demanding and could have compensated for errors in wet–dry classification for each combination of both threshold values.

The nearby link approach with outlier filter and default settings for the parameter values (Table ), gives a CV of the residuals of 3.84, a squared correlation coefficient ρ2 of 0.54, and a relative bias in the mean of 10.5 % compared to the reference: gauge-adjusted, path-averaged radar rainfall depths. Removing step 8 (Appendix ) from the nearby link approach does hardly change these validation results. Therefore, this step can now also be discarded by supplying a function argument. In this step the previous two time intervals and the next time interval are classified as wet if the present time interval is rainy and has more than 2 dB of attenuation for the link under consideration.

Performance of link rainfall estimates clearly deteriorates if the outlier filter is not applied. The CV of the residuals is much larger, 6.03, the ρ2 is much lower, 0.35, and the relative bias in the mean becomes much larger, 24.2 % (compare to previous paragraph). The latter may be related to the fact that Aa and α have been calibrated on a data set where the outlier filter has been applied.

Now the performance for a wide range of threshold values of the outlier filter is investigated for path-averaged 15 min rainfall depths. For the default threshold value of -32.5 dB km-1 h-1, denoted by the grey line, good results are obtained in terms of ρ2 and CV (Fig. ). Results improve for increasing threshold values at the expense of a severe decline in the amount of link data. The best choice for a threshold value is somewhat arbitrary, i.e. a range of values gives good results, while still having many link data. The relative mean error decreases from more than +20 % to less than -15 % from Ft=-300 to 0 dB km -1 h-1. If for each value of Ft the coefficients Aa and α would have been calibrated, the relative mean error would be expected to be more constant. Apparent are the jumps in ρ2 and CV for certain values of Ft. It seems that these are related to some very large “rainfall depths”, which point to malfunctioning links. Note that F is the cumulative difference between a link's specific attenuation and that of the surrounding links. Hence, such jumps for strongly negative values of Ft are likely not caused by rain or (sources of) errors affecting a whole region.

Some of the signal fluctuations that occur in dry weather may also occur during rainy periods. The algorithm does not correct for such errors. Furthermore, nearby links may suffer simultaneously from signal fluctuations not caused by rainfall, resulting in a poor performance of the nearby link approach.

The parameters concerning the wet–dry classification have been optimised using data from another period and are based on received signal level data stored at 0.1 dB resolution . These values have been applied in and in the current manuscript to an independent data set from another brand of links with slightly different antenna covers and a coarser 1 dB power resolution. The sensitivity analysis shows that the parameter values from are suboptimal but give a clear improvement in link-based rainfall estimates compared to the case without wet–dry classification.

The chosen radius depends on the spatial correlation of rainfall and is, hence, not frequency dependent. Because networks are designed in such a way that ΔP will be similar for different microwave frequencies, lower frequencies are utilised for longer link paths and vice versa. Lower frequencies experience less rainfall-induced specific attenuation compared to higher frequencies. Hence, median(ΔP) is nearly independent of frequency.

Ideally, a sensitivity study should be carried out to find optimal threshold values for the wet–dry classification in case of other networks and climates. This will be computationally expensive in case of large data sets.

It is to be expected that the nearby link approach will also work for other temporal resolutions than 15 min. This may require a smaller radius and a higher link density for higher temporal resolutions and may allow a larger radius and lower link density for lower temporal resolutions. For data sampled at very high frequencies, i.e. 1 s, the nearby link approach may become too computationally expensive. This could be circumvented by first averaging signal attenuations over longer durations before applying this approach.

Note that a default 24 h period is considered for determining the reference level and for one step in the nearby link approach. Such a relatively long period is chosen, among other reasons, to increase the probability to obtain at least 2.5 h of dry periods, which helps to determine the reference level more accurately. Note that both periods can be modified.

The filter to remove outliers deals with specific attenuation; i.e. it does not explicitly take into account frequency. It is also suitable for other time interval lengths. Nevertheless, its threshold value is very high, which makes it unlikely that actual rainfall is filtered out accidentally, irrespective of the employed frequency. The outliers are likely caused by malfunctioning links. Perhaps melting precipitation also plays a role. Hence, it makes sense to apply a frequency-independent threshold value. Note that the threshold value may need to be optimised for other networks and climates. The sensitivity analysis on a 12-day data set shows that the chosen default value of Ft is suitable but a range of values shows a similar performance.

The wet antenna attenuation correction Aa will not only compensate for wet antennas. The value of Aa found in the calibration will also be influenced by other errors. Moreover, in case of rain along the link path no, one, or both antennas can be wet, whereas this correction is always applied, and Aa itself has also been optimised on data where the number of wet antennas can vary. In addition, the correction does not depend on rainfall intensity. Hence, applying Aa should be seen as a pragmatic approach towards correcting for wet antennas.

The coefficient α is employed to obtain mean path-averaged rainfall intensities from minimum and maximum path-averaged rainfall intensities and is expected to depend on the time variability of rainfall. Note that the value of α presented here is based on data available at 15 min time intervals. It is expected that this value will be different for disparate time intervals. use the known distribution of rainfall intensities at the point scale to weigh minimum and maximum received signal levels.

The optimised values of Aa and α are representative of summer months in the Netherlands and appear to be relatively close to each other for different frequency classes. Application to data from other months will generally only lead to a small decrease in performance (not shown). Hence, application of the existing values of Aa and α to other rainfall types can still give reasonably good rainfall estimates. Nevertheless, it is advised to recalibrate Aa and α in case of other networks or climates. This may be achieved by comparing link-based rainfall estimates with high-quality (gauge-adjusted) weather radar data, which may provide full coverage over the network, as in . Alternatively, path-averaged rainfall intensities may be estimated from rain gauge data. This may require a dedicated research experiment , preferably with several rain gauges along the link path.

utilise a research link with high temporal resolution for testing the proposed rainfall retrieval algorithm. By applying the rainfall retrieval algorithm, mean path-averaged rainfall intensities can be derived. These can be compared to the corresponding true values obtained from the received signal powers sampled at 10 Hz. Hence, the same instrument, a microwave link, is used to assess the ability of the retrieval algorithm to deal with the sampling strategy. Since the same instrument is used representativeness errors are not present. The probability distribution of minimum and maximum rainfall intensities could be studied in order to obtain more reliable mean rainfall intensities. Moreover, such an experiment may also help to assess attenuation due to wet antennas caused by dew or rain, its dependence on rainfall intensity or antenna cover type, and the time it takes for antennas to dry following a rain event. Preferably a rain gauge or disdrometer should be available near the antenna. A research link is particularly useful when the microwave frequency, path length, and antenna cover are representative of the links in a cellular communication network. Finally, see, for instance, and for other wet antenna attenuation correction methods.

The employed interpolation methodology, ordinary kriging, may not be specifically suited for other regions with different rainfall climatologies. The methodology developed in could be optimised for other climates. This requires long rainfall time series. The assumed stationarity and isotropy will often be violated . However, violation of assumptions does not automatically imply that the interpolation method is not useful. Further, the path-averaged link rainfall intensities are assumed to be point measurements. Hence, it is recommended to improve the interpolation methodology, e.g. by treating the rainfall values as line observations instead of as point observations, which is expected to have the largest impact at local scales for areas with high link densities or at areas with long links. Using data from the same 12 days as employed in , find that link rainfall retrieval errors themselves are the source of error that contributes most to the overall uncertainty in rainfall maps from a commercial link network. Errors due to mapping, i.e. interpolation methodology and link density, play a minor, albeit non-negligible role for the same network as utilised in this study. Hence, despite the limitations of the interpolation methodology its usefulness has been confirmed . Further, study the performance of simulated link rainfall maps as a function of link density. They show that even for low spatial link densities reasonable results can still be obtained.

It may be interesting to apply a tomographic approach in order to obtain link-based rainfall maps . Such an approach can potentially reconstruct the two-dimensional distribution of rainfall from a set of one-dimensional transmission data from many different (nearby or even intersecting) paths . Hence, it would take the line character of the attenuation measurements into account, in contrast to the current approach where a line measurement is assigned to a point at the middle of the link path.

The estimated sill and range will become less accurate by extrapolating to time intervals shorter than 1 h, such as 15 min. use rain gauge data from England to quantify spatial correlation for short time intervals, e.g. 1 and 15 min. These kind of studies can be useful to improve the interpolation of link-based rainfall intensities.

The relationship between path-averaged rainfall intensity and path-averaged specific attenuation is commonly employed in other studies. The provided parameter values of a and b are available for frequencies ranging from 1 to 100 GHz. The value of the exponent b is close to 1 for the frequencies employed in this study, which range from 12.8 to 40.0 GHz. Frequencies between 37.0 and 40.0 GHz are denoted by the grey-shaded area in Fig. , which contains 81 % of the links from the working example, having values of b very close to 1 (right). For other frequencies the corresponding value of b often deviates more from 1 (Fig. , right). High rainfall variability along the link path will lead to overestimation for b<1 and underestimation for b>1 .

Although exponents are often not exactly equal to 1, they are much closer to 1 than the ones typically used in radar reflectivity–rain rate relations. For example, assessed the influence of spatial variability on the link path for these two frequencies. They show that the under- or overestimation will generally be small. In the tropics this problem will be more pronounced because of the high spatial rainfall variability. Particularly for long links operating at low frequencies (e.g. 7 GHz), this may lead to overestimation. Also note that obtain quite accurate rainfall estimates using a 7 GHz microwave link with a length of 29 km in Burkina Faso, having a tropical climate. Previous work has demonstrated that this error is limited for temperate climates such as experienced in the Netherlands . Moreover, 82 % of the links in our network have a length shorter than 5 km, even 97 % of the links are shorter than 10 km. The average link length is only around 3 km.