Accumulated Mixed Precipitation Estimation Using Measurements from Multiple Microwave Links

Recently, microwave communication networks have been shown to be valuable tools for rainfall monitoring, based on the wellknown Power-Law which relates rain-rate to attenuation in microwave frequencies. However, once precipitation other than pure rain exists (e.g., snow), the Power-Law relation is no longer accurate. In this paper we propose a model which relates the induced attenuation to rain, snow, and sleet. Based on this model we propose estimating the total accumulated precipitation, regardless of the precipitation type, using measurements from multiple microwave links. Our technique takes advantage of the commercial communication networks, need for redundancy, which dictates the use of multiple microwave links at the same area. We show that by using measurements from at least three microwave links better estimation of the total accumulated precipitation fall can be provided, when rain, snow, sleet, or a mixture of them coexists. To demonstrate the proposed approach, it has been applied on actual microwave links attenuationmeasurements, which were provided by a cellular carrier.The estimation results were compared with Rain-Gauges and disdrometer measurements and show very good agreement and improved accuracy.


Introduction
Since microwave communication networks (MCNs) usually make use of the 10 GHz-40 GHz frequencies range, which is known to be affected by precipitation, it was only natural to assume that rainfall should have a measurable effect on these networks.And indeed, it has been shown back in 2006 [1], that it is possible to monitor rainfall accurately using only the standard attenuation measurements which are being recorded by the MCNs operators (i.e., the cellular operators).Soon after, numerous studies presented different methods for precipitation monitoring, using the widespread MCNs as tools.Among these methods, there are methods for detection and separation of wet and dry periods [2], estimation of the rainfall intensity [3,4], and analysis of the rain-rate estimation errors [5,6].
These studies, however, have been focused on rainfall.Once snow and sleet (i.e., a mixture of snow and rain) particles are considered as well, it was suggested [7] to use a specific predetermined dual-frequency Microwave Links (MLs) setup, which recorded the phase as well as the Received Signal Level (RSL) data, in order to estimate the precipitation rate.These techniques, unfortunately, cannot be implemented with current MCNs protocols, since in a commercial setup the MLs frequencies are predetermined by the operators, and the data which is regularly logged is the RSL attenuation alone (thus, the phase information is unavailable).
In this study we suggest a new approach which can be used in order to achieve reliable estimation of the total accumulated amount of the fallen precipitation, even when multiple types of precipitation exist, using MCNs.This new estimation procedure uses the fact that commercial MCNs, especially in developed and highly populated areas, are generally built from a large number of different MLs.Thus, multimeasurements of the same phenomenon are available, and multiparameter estimation can be performed.
The rest of the paper is organised as follows.In Section 2 we describe the precipitation induced Microwave Link (ML) attenuation, as well as the Proposed Estimation Procedure.Section 3 describes a real setup which was used to

Materials and Methods
First, it is imperative to understand how rain, snow, and sleet affect the ML attenuation.This is going to be inquired in the following sections.
(1) Rainfall Induced Attenuation.The relationship between the instantaneous rain-rate and the ML attenuation is given by the well-known Power-Law [8]: where   is the rain induced attenuation in (dB),   is the ML-path averaged rain-rate in (mm/h),  is the ML length in (km), and  and  are parameters which are determined by the ML radiation frequency, polarisation, the surrounding temperature, and the Drop Size Distribution (DSD).Thus, the  and  parameters are specific for different environments and conditions, and their typical values can be found in literature [9,10].
(2) Snowfall Induced Attenuation.Similar to the rainfall Power-Law, an equivalent "Snowfall-Law, " which describes the ML attenuation due to ice particles such as snow, has been developed (The following equation has been developed considering microwave frequencies, while assuming a surrounding temperature of ≈0 ∘ C. For different frequencies and lower temperatures, the coefficients differ.)[11,12]: where   is the attenuation due to snow in (dB),   is the MLpath averaged (liquid equivalent) snowfall-rate in (mm/h),  is the ML length in (km), and  is the ML radiation specific wavelength in (cm).
Previous studies have shown that dry-snow particles (i.e., a mixture of pure ice and air) induce a much smaller effect on the ML attenuation, compared to the effect induced by rainfall [10,13,14].This assumption, however, is no longer met once a mixture of liquid and solid water particles, such as sleet [15,16], is considered.Since the sleet induced attenuation model is lacking, we will next suggest a Sleet-ML Attenuation relationship.
(3) Sleet Induced Attenuation.A sleet particle can be described as an ice core, covered by a thin layer of liquid water.Sleet particles usually form near the freezing point of water, where both rain and snow particles coexist [12,17].
Since the sleet particles' outer layers are made of liquid water, it is not surprising that the sleet induced attenuation has been shown to resemble the one induced by rain particles [12].Thus, an approximation of the sleet induced attenuation is proposed, taking the existing rainfall Power-Law (1) as a basis [18]: where  Sl is the sleet induced attenuation in (dB),  Sl is the average (liquid equivalent) sleet fall-rate in (mm/h) throughout the ML-path,  is the ML length in (km), and ã and b are analogue to the Power-Law (1)  and  parameters.Now, the following approximations are assumed.
(i) Since sleet particles are made either from rain particles which pass through the freezing layer and/or from snow particles which pass through the melting layer near the freezing point of water [17,19], it can be assumed that during sleet-fall, liquid precipitation particles exist.Thus, we propose to model the instantaneous sleet-rate to be proportional by an unknown ratio to the instantaneous liquid particles fall-rate (i.e., the rain-rate): where   is a unit-less ratio between  Sl and   .Thus,   ⋅   is the effective instantaneous sleet fall-rate.
(ii) Since the Power-Law (1) dependence on the DSD has been shown to be of a second order for typical MLs frequencies [20], and noting that for those frequencies the Power-Law (1)  parameter is close to one ( ≈ 1) [9,10], the sleet Power-Law equivalent ã and b (3) parameter can be approximated to the rain-rate Power-Law  and  (1) parameters.Further discussion regarding this assumption is presented in Section 5.
Combining these two assumptions, the instantaneous sleet induced attenuation (3) can be written as where and  and  are the Power-Law (1) known parameters.Thus,   ⋅   is the approximated instantaneous sleet fall-rate.
(4) Total Precipitation Induced Attenuation.Putting (1), (2), and (5) together, the total instantaneous ML attenuation induced by precipitation,  pr (  ), can be formalised.Written in (dB), the instantaneous ML induced attenuation due to rain, snow, and sleet particles can be presented as where  pr (  ),   (  ),   (  ), and  Sl (  ) are the total precipitation, the rainfall, the snowfall, and the sleet fall induced attenuation, respectively, sampled at time   .Using (1), (2), and (5), an explicit form of (7) can be formalised:  9), the measurements model can be formulated as follows: where (  ) is the ML attenuation in (dB) at time index   and (  ) is the model parameters vector at   (8).

Paramater Estimation.
It is clear that in order to estimate the parameters vector (10c), a minimum of three different ML attenuation measurements are needed, in which (  ) is identical.Fortunately, due to the high Quality of Service (QoS) desired by the cellular operators, MCNs are built to be redundant by using multiple MLs within the same path.Thus, the following equation can be rewritten as where   (  ) is the total ML attenuation in the th ML in (dB), ℎ  ((  )) equals  pr (  ) (8) for the th ML (and thus it is a function of each ML time-indepenenet specific properties), and    (  ) is the additive noise portion in the th ML measurement.
Noting that ( 11) is a nonlinear parameter estimation problem with (  ) as the parameters vector, the estimation can be performed via various parameter estimation methods, such as Least Squares [24].
Denoting the resulting estimate for time index   by θ : where R (  ), R (  ), and ĉ (  ) are the estimated variables, from which the total fallen precipitation rate Prec(  ) can be calculated in (mm/h) (1), (2), and (5) as follows: 2.1.3.Accumulated Precipitation Calculation.Since we are interested in the accumulated precipitation fall, rather than the precipitation rate, the following calculation is performed: where Ptot () is the total accumulated precipitation, ℎ is the sampling interval, and  is the number of samples within a given weather event.So,  ⋅ ℎ is the observation interval.

Application Using Actual Data
In order to demonstrate the feasibility of the proposed procedure, a test case was designed and executed during 2010-2013.The site chosen for this test is kibbutz Ortal, located in northern Israel.This specific site was chosen since it holds a major microwave communication network (MCN) node, which acts as a base-station for numerous fixed terrestrial MLs, employed for transmission purposes by the Israeli cellular operator Cellcom, as well as the fact that only northern Israel has a probable chance to experience snow and sleet weather events.Since the aim of this test is to present the potential of the proposed procedure, a reference for the local precipitation fall was needed.For this purpose, readings from three Rain-Gauges (RGs) operated by the Israeli Meteorological Service (IMS) were gathered.
In order to classify the fallen precipitation, an OTT Parsivel Disdrometer Weather Sensor [25-27] was installed near Ortal's MCN node.The disdrometer, as well as Cellcom's MCN tower can be seen in Figure 1(a).An example of the MRSL/mRSL data series recorded by a given ML during a storm is presented in Figure 2(a).

Rain-Gauges.
In order to capture the actual precipitation fall, three RGs located in kibbutz Ortal, Merom-Golan, and El-Rom were monitored.These three kibbutzes (and hence, RGs) are distributed roughly evenly and in close proximity (≤2.2 km) to the MLs path.The RGs are operated by the IMS and record the daily fallen accumulated precipitation (The RGs data can be accessed online under: http://www.ims.gov.il/).The RGs locations can be seen in Figure 1   (iii) mixture of both liquid and solid precipitation types: light and moderate rain drizzle and snow, strong rain drizzle and snow, and freezing rain; precipitation in this group was treated as "sleet".
An example of the raw data which was reported by the Parsivel disdrometer is shown in Figure 2

Analyzed Weather Events.
During December 2010 and January 2013, two major weather events (which include rain, snow, and sleet particles) have been analyzed.Since an ML monitors the path-averaged precipitation fall, for each event, the accumulated precipitation values reported by the three available RGs (as presented in Figure 1(b)) were monitored and averaged.Furthermore, measurements from the Parsivel Disdrometer Weather Sensor were used to classify the types of the fallen precipitation.In addition, two periods of pure rainfall (with no other precipitation types detected) were analyzed in order to examine the proposed procedure performance during rain-only events.Details regarding the four weather events are presented in Table 2, where the accumulated precipitation column presents the averaged total accumulated precipitation as reported by the RGs, and the type via disdrometer column presents the percentage of the different precipitation types throughout each event, as reported by the disdrometer.
It is worth noting, that for the available weather events (of a duration of ≥48 h) and location (northern Israel), the variation in the total accumulated fallen precipitation along the MLs path and the corresponding RGs' locations (which are ≤2.2 km away from the MLs path) were shown to be negligible [28].

Data Preparation.
We will now go into details regarding the data preparation needed for the estimation processes.

ML Attenuation Data Series.
Eventhough each ML contributes both the MRSL and the mRSL data series, previous study has found that for moderate and heavy storms, the contribution of the MRSL data series for the estimation process in this region is negligible [29].Thus, in this demonstration, only the mRSL data series were used.
For each mRSL data series, the value of  dc (as defined in ( 9)) was determined by taking the mRSL most common value during the dry period prior to each storm.Further discussion regarding the ZL is presented in Section 5.

3.3.2.
and , , and  Values.As mentioned in Section 2, the values of  and  (1) parameters are well-studied and can be found in the literature [9,10].However, those values are applicable for instantaneous attenuation measurements.In our case, we have access only to the mRSL/MRSL measurements.In [18,30] we show that by replacing  by an appropriate   > , the overestimation of the rain estimates due to the usage of the mRSL only can be compensated.We have calibrated   empirically for this region to the value of 0.39 (dB/km)(mm/h) − [18], which is used in the sequel.
The value of  was taken as 1.12, which corresponds to the 18-19 GHz frequencies range [9,10].Since the expected difference in the values of  and  due to the MLs different polarisations and frequencies within the 18-19 GHz range is small [9], the same  and  set was used for all MLs.
The value of  (2) was taken as 1.6 cm, corresponding to a frequency of 18.74 GHz, which is roughly the average frequency used by the four MLs (Table 1).It is worth noting, that each ML uses a slightly different frequency.However, since the difference between those frequencies is small, the changes to  are negligible.
Lastly, the value of  (1), (2) was taken as 12.08 km, which is the MLs path's length.
Note, that  and , , and  parameters are timeindependent, and thus their specific values were taken as constants during the entire experiment duration.

Accumulated Precipitation Estimation.
For each weather event (as presented in Table 2), two different estimation processes have been performed.First, the proposed model based estimation was executed (11), and the estimated precipitation rate (13) was calculated.Second, the known Power-Law (1)   estimation was executed (by using the same estimation algorithm (11), with   = 0 and   = 0).This was done in order to establish the current widely used estimation process [1,3], which treats the snow and sleet effects as negligible, as a baseline for comparison.These two estimations were performed for every sampling interval.
Lastly, by using ( 14), the total accumulated precipitation during each storm was calculated, both for the proposed procedure and for the Power-Law model based estimation.

Results
From December 2010 until January 2013, data from two extreme weather events in Israel have been collected and analysed.These events include different types of precipitation, including rain, snow, and sleet particles.In addition, two rain-only events were analysed, in order to test the proposed model mismatch during rain-only scenarios.
In Figures 3 and 4, the proposed model based estimation results for every event, Ptot () of ( 14), is plotted, as well as the average of the measurements of the three RGs.For comparison, the Power-Law (1) estimates are plotted as well.In addition, Ptot () specific values throughout the events can be seen in Table 3, where PM represents the proposed model estimates, PL represents the comparison Power-Law estimates, and RG represents the RGs measurements average.

Results
Analysis.Looking at Figure 3 and Table 3, it is easy to conclude that once other than pure rain precipitation types are involved, the proposed model based estimation outperforms the Power-Law estimation by a significant margin.The estimation based on the proposed model achieved a total  accumulated precipitation values which are much closer to the RGs' observed ones.Furthermore, by inspecting Table 3, it is clear that the proposed model estimates achieved better performance throughout the events progression, compared with the Power-Law based estimation model.
While Figure 3 emphasises the robustness properties of the proposed precipitation estimation procedure versus the specificity of the Power-Law based estimation model, the pure rainfall events (Figure 4) have put to a test the possible drawbacks of such robustness.And indeed, during the rain-only events, the Power-Law estimation systematically achieved more accurate results than the proposed pocedure estimation.However, once the total accumulated precipitation fall detected in these events (18 mm-23 mm) are considered, it can be seen that the absolute errors of the proposed procedure are rather small (≈11 mm in 96 h for the 16 January 2012 event and ≈4 mm in 48 h for the 28 January 2013 event), suggesting that the proposed model mismatch during rain-only events damages the estimation accuracy only mildly.
Nonetheless, it is worth mentioning, that during periods of strong precipitation rate, both estimates tend to underestimate the accumulated precipitation fall.This underestimation can be attributed to the finite dynamic range of the mRSL.During extremely intense precipitation fall, the mRSL may drop to its absolute minimum, and thus stronger precipitation rate will not be distinguished.Example of such an intense precipitation rate can be seen in Figure 3(b), starting from 07 January 2013.

Conclusion and Discussion
In this paper we established a new approach for monitoring other than pure rain precipitation, by using MCNs.This approach takes advantage of the fact that MCNs often contain multiple MLs at the same location, which gave us the possibility to estimate more unknown parameters and therefore to employ an expanded model, from which the total accumulated fallen precipitation, regardless of the precipitation type, can be estimated.And indeed, by using actual MCN data, we have demonstrated that during weather events with mixed precipitation this proposed model and the resulting estimation exhibited better performance compared to the standard Power-Law based estimation.In addition, despite the robustness nature of this new method, it exhibited only mild absolute errors during estimation of pure rain weather events.Thus, this study suggests a new tool, which can be used for precipitation estimation, especially at locations where mixed types of precipitation are common.
This research presents promising results.However, in order to deal with specific physical phenomena, as well as with challenges which arose due to the usage of actual MCNs measurements, we made a number of assumptions and approximations which may have introduced some errors into the estimation process.Since these sources of errors are either out of the scope of this paper or are yet to be fully understood, their effects on the estimation accuracy should be further studied.We will now list the main sources of errors which should be addressed further.

Sources of Errors due to Physical Phenomena.
In this research we have developed a procedure which is capable of treating multiple types of precipitation.This development is based on some assumptions and approximations regarding the sleet, which may introduce some errors.Additionally, other physical phenomena may also introduce errors into the estimation process [31,32].

The Sleet Induced Attenuation Approximations.
In Section 2, we presented a new precipitation induced attenuation model, which generalises the well-known individual models of rain and snow induced attenuation, while adding a new term which accounts for the sleet induced attenuation.This model is based on a number of approximations.The sleet induced attenuation term ( 5) is approximated based on the assumption that the sleet induced attenuation can be modelled via the rain induced attenuation term.While the assumption that sleet affects the microwave radiation similar to the effects caused by rain has been proven [11,12], once considering sleet particles, the specific  and  parameters are yet to be determined.Due to the lack of better knowledge, we considered the fact that the  and  set for sleet should be similar to the set already found for rain [20], and thus the same set of  and  was taken both for rain and for sleet.However, since the sleet DSD may be different than the DSD of rain particles, better understanding of the sleet specific  and  parameters should be considered.In addition, the proposed model is based on the assumption that the sleet fallrate is proportional to the rain-rate alone.Eventhough snow induced attenuation is negligible compared to the attenuation induced by rain or sleet [12,13], refining the sleet relationship further in relation to both rain and snow particles may yield better overall accuracy.

Zero-Level, Wet Antenna, and Estimation
Bias.Establishing the ZL has been a topic of past research [2,21,23], and it is out of the scope of this paper.In general, we assume a known ZL, meaning that it needs to be preestimated.During our experimental study, for simplicity reasons, and because we concentrated on demonstrating the feasibility of the proposed procedure, the ZL was taken as a constant value, which was determined by the dry period prior to the storm.However, in recent studies, it has been suggested that the ZL may fluctuate during the storm, as well as be affected by a bias caused by the Wet Antenna (WA) effect [33,34].While these errors' influence during moderate and heavy storms is small [29,30], during lighter storms the ZL and the WA induced errors may indeed be significant.Thus, in order to achieve a more accurate estimation, adjustments for the ZL and the WA should be considered.Lastly, due to the high variations of the precipitation fall (as can be seen in Figure 2(b)), the use of more advanced estimation algorithms may be considered as well.

Hail Effects.
In this research, we demonstrated the proposed procedure abilities by using storms data available to us.These storms included mixtures of rain, snow, and sleet particles.Eventhough that in theory hail induced ML attenuation should be similar to the attenuation induced by sleet [12,17], we have yet to deal with hail, so further study regarding the effects of hail is suggested.

Sources of Errors due to Actual MCNs Usage.
Actual MCN measurements suffer from limitations from our point of view, since they aim into communication purposes, rather than precipitation monitoring.These limitations are additional sources of errors.

Quantization Error.
As mentioned throughout this paper, the mRSL/MRSL data series often pass a quantizer which introduces quantization errors.In our specific experimental setup, the mRSL/MRSL data series were affected by a quantization error of 1 (dB).While the quantization errors are relatively small during heavy and long storms (due to the large dynamic range of the mRSL/MRSL compared to the 1 (dB) quantization error, as well as the fact that the quantization error bias is zero), once lighter or shorter storms are estimated, the weight of the quantization errors may rise and possibly influence the estimation accuracy.Since the quantization error is nonlinear and nonadditive, treating this kind of error is not straight-forward and should be addressed in the future.

General MCN Architecture.
The approach proposed in this research takes advantage of multiple MLs which share the same path.While actual MCN setup is inherently redundant, having four MLs sharing the same path is not unusual; the question of generalising our results to MLs at spatial diversity is an open one.Indeed, in a limited area, precipitation fields are coherent [28], and this coherency can be extracted to achieve better performance.Preliminary tests have shown that using the spatial locations of the different MLs, and creating a covariance matrix as part of an alignment preprocess, may achieve promising results.However, this subject is yet to be fully examined.

3. 1 .
Available Data.During the test case, both MLs Attenuation Data and Meteorological Data were obtained.3.1.1.Available MLs Attenuation Data.From the dozen commercial MLs transmitting from and/or to Ortal node, data from four MLs which share the same path has been collected.These four MLs are clustered in a single path of 12.08 km, between kibbutz Ortal and kibbutz Odem (Figure1(b)).It is worth noting, that by using only MLs which share the

Figure 1 :
Figure 1: The installed OTT Parsivel Disdrometer Weather Sensor near Ortal's MCN base-station tower (a), and the test area map showing the ML's 12.08 km path, as well as the three available RGs: RG-Ortal, RG-Merom-Golan and RG-El-Rom (b).(Image (a) was taken by Y. Dagan.Image (b) was created via Google Earth).

Figure 2 :
Figure 2: An example of mRSL and MRSL data series, sampled at 15 min intervals, with 1 dB quantization error (a), and the actual measured fallen precipitation divided into rain, snow, and sleet, as detected by the OTT Parsivel Disdrometer Weather Sensor (b). (b).

3. 1 . 3 .
Parsivel Disdrometer Weather Sensor.Since RGs cannot distinguish between different types of precipitation, the OTT Parsivel Disdrometer Weather Sensor was monitored in order to classify the different fallen precipitation types.During the (b).The full list of precipitation types which are supported by the Parsivel disdrometer can be found in[25,26].

Figure 3 :
Figure 3: Accumulated precipitation between 10 December 2010 and 14 December 2010 (a) and the accumulated precipitation between 04 January 2013 and 10 January 2013 (b), as observed by the Rain-Gauges, as estimated by the proposed model, and as estimated by the Power-Law model.The estimates errors throughout the storms are presented: It can be seen, that the proposed model achieved better estimates throughout the storms.

Figure 4 :
Figure 4: Accumulated precipitation between 16 January 2012 and 20-Jan-2012 (a) and the accumulated precipitation between 28-Jan-2013 and 30-Jan-2013 (b), as observed by the Rain-Gauges, as estimated by the proposed model, and as estimated by the Power-Law model.The estimates errors throughout the storms are presented: It can be seen that in general, the Power Law Model achieved better estimates throughout the storms.However, the proposed model estimation error is contained.

Table 1 :
Available microwave links.

Table 1 .
Due to the rough and nonlinear preprocessing made by the cellular operator, each ML contributes only the maximum Received Signal Level (MRSL) and the minimum Received Signal Level (mRSL) for every 15 minutes quantised in 1 dB.

Table 3 :
Results throughout the events.