Vertical Precipitation Estimation Using Microwave Links in Conjunction with Weather Radar

When monitoring rain rates by weather radar in semi-arid regions and when measuring precipitation at an arid region; precipitation particles, rain, or snowflakes may evaporate before reaching the ground. This evaporation is regarded as the Virga phenomenon and may cause a false representation of the precipitation amount that actually reaches the ground. The Virga occurs naturally when the air below the cloud is relatively dry, and continues until humidity below the base of the cloud is high enough to decrease the evaporation. This paper suggests a method of combining near ground Commercial Microwave Links (CMLs) attenuation measurements, in conjunction with data from several weather radar beams, observing different heights, in order to produce estimates of the vertical profile of the rain-rate values and of the Cloud Base level (ClB). We propose an estimation method and demonstrate it using real-data measurements of two major storm events in the dead-sea area. We verify the validity of the estimation near ground by comparing the results with Rain Gauges’ (RGs) actual measurements in addition to comparing the estimated ClB with real ClB observations of a nearby weather station. While the storm events selected indeed show great evaporation, the suggested method provides excellent results, with a correlation of up to 0.9615, when correlated with real measurements of RGs of two storms from 2014 to 2016.


Introduction
The attenuation recorded at Commercial Microwave Links (CMLs), due to precipitation, produced many studies for over more than a decade now, while continuing to be the source of new research in the field of precipitation estimation, and rain estimation in particular, based on Received Signal Level (RSL) measurements.Since the CMLs are widely spread by Commercial Microwave Networks (CMNs) around the world, this presented an opportunity to use them as precipitation sensors in addition to the commonly used methods for precipitation measuring, as it has been shown that there is a direct link between the presence of water droplets in the path of the CML and the signal attenuation.Principally, the rain-induced-attenuation of the transmission signal over CMNs links, is calculated as the Transmitted Signal Level (TSL) at the first base station of the link after deducting the RSL in the opposite side of the CML, and after removal of a base zero-level; it was shown to be directly connected to the amount of rain per unit of time in the CML's path by the Power-Law relation [1].Based on this Power-Law relation, the ongoing study has produced many methods, specifically for rainfall monitoring, such as but not limited to, rainfall estimation [2][3][4], rain detection [5,6], and rainfall mapping [7,8].The CML precipitation monitoring methods grant us better ground coverage, compared with currently used technologies, such as Rain Gauges (RGs) and weather radar, and improves the overall accuracy of the precipitation measurements [9,10].Nonetheless, the aforementioned research's focus was presented toward near-ground level rainfall monitoring.The performance of these studies is often compared with radar; however, such a comparison is only reasonable, even though radar measures rain much above ground, only if there is not much variance with height above ground (i.e., when the relative humidity is approaching 100%), but, it is not the case when dealing with arid and even semi-arid regions.
In previous research [11], it was shown that precipitation particles, rain, or snow flakes, may evaporate before reaching the ground when measuring precipitation in an arid region.This evaporation is followed by the creation of visible precipitation shafts below the cloud base level.The precipitation shafts are observed at height by the weather radar beams, but are not actually measurable at the ground level.This evaporation is regarded as the Virga phenomenon [12] and occurs naturally when the air below the cloud is dry, and continues until humidity below the base of the cloud is high enough to decrease the evaporation; the precipitations then reach the ground [13].
The Virga is a frequent phenomenon.Despite being a commonly seen event, the actual amount of evaporation that occurs between the time the raindrops are released at the cloud base and the time that the rainfall reaches the earth's surface is not well understood.This evaporation is particularly important in regions with high cloud bases where, in some instances, the rainfall evaporates completely before reaching the ground.In particular, for semi-arid areas, this evaporation may lead to considerable over-estimation of rainfall by the weather radar, while the ground might remain nearly dry.Current ground monitoring devices such as RGs are usually sparse and represent an area of less than 1 square meter [14].Moreover, the Virga phenomenon is much more likely to occur in arid and semi-arid regions such as found, for example, in the east and south of Israel.As CMLs are widespread with high density, and cover long areas mainly across lengths of 1-30 km, in this paper we suggest a method of profiling vertical rain-rates in arid areas using a combination of several radar beams, observing at different heights, along with cellular links, in order to achieve more accurate above-ground level rain-rates where the Virga may affect the radar measurements of rain, while verifying the suggested method on two cases of storms in the semi-arid zone of Israel where the Virga phenomenon, i.e., the evaporation of rain was of significance.
The rest of the paper is organized as follows: Section 2 provides the necessary background and the methodology used.Section 3 details the experimental setup and the results.Lastly, Section 4 concludes this paper.

Theory
The power law equation [1,15], which connects the CMLs' power attenuation A(dB) caused by the effect of rainfall at a specific rate P(mm/hr), allows us to use CMLs for the sake of rain rate monitoring: where a, b are parameters that are affected by the CML's frequency, polarization and the Drop Size Distribution (DSD), and L(km) is the CML's length.The a and b parameters have been calculated for different DSDs [1], but, as the Power-Law became a main tool for planning communication systems exposed to rainfall, global recommendation based on experimental studies have been suggested by bodies like the International Telecommunication Union [15] and the Federal Communication Commission [16].
Next, the rate of rainfall (in (mm/hr)), at distance Z below Cloud Base level (Cl B ), can be written as [17]: where P(0) is the rain rate at the height of Cl B, C and α are constants that are affected by the DSD, and Ψ(Z) can be presented as: Environments 2018, 5, 74 and K 1 , K 2 are affected by the surrounding environment's parameters, and are calculated by: when ρ l is the density of liquid water, k is the thermal conductivity of air, L the latent heat of condensation, R υ is the gas constant for water vapor, T the temperature, e s (T) the saturation vapor pressure and D w the diffusivity of water vapor in air.Assuming constant relative humidity along the Z axis, Equation (2) could be written as: where (C * C/(K 1 + K 2 )).Our aim is to use rain rate measurements P(Z) at different heights in order to find the unknown parameters in Equation ( 5), so the vertical profile of the rain from the cloud base to the ground level can be determined.That is, using the available measurements we need to estimate the parameters vector Θ = P(0) α C * Cl B t , in effort of achieving an equation that will define the vertical profile of the rain rate below the Cl B and downwards.Exactly how we suggest doing so will be explained in the following subsection.

Methods
For the sake of estimating our parameters vector Θ, we suggest using CML data in conjunction with different radar beams, and later validate our results with RGs nearby.First, we need to present our measurements vector y, which is provided by using various rain rate monitoring tools: weather radar sampling rain rate at different heights, and CMLs sampling at the antenna height.Second, we show how we estimate Θ from the measurements.Finally, we show how the RGs' rain rate could be calculated from Θ to validate the results.In addition, the relative humidity S 0 is considered and is provided by weather stations in the vicinity.
Given a CML at a height of H CML above ground level, the CMLs' rain rate measurements P (in (mm/hr)), when following Equation (5) and Equation ( 1) can be presented as: Next, we need to introduce a method to present the weather radar rain rate.Since the radar beams have an opening of a certain degree, when monitoring at a distance, this translates to integration and averaging of the precipitation P(Z 1 → Z 2 ) by the radar, along the vertical axis, where the beam is observing from height Z 1 to height Z 2 above ground level.The integration of a specific radar beam can be written as:

(7)
A diagram of the different measurement sources along the vertical axis Z, which actually goes from the CLB downwards, is depicted in Figure 1.To summarize the above, the samples vector y can be represented, as: where n is the added measurement noise, but as one can see, it is problematic to differentiate between Θ 2 , Θ 3 , Θ 4 ; this is where additional prior knowledge is required.We suggest using the values of α and C that were derived according to the research done by Schlesinger et al. [17], and calculate C * by using the surrounding environment parameters K 1 , K 2 .Next, we can present a new, and of lesser dimension, parameters vector: and the samples vector is: This problem's solution is known and can easily be solved using a Least Squares Estimation (LSE) to extract Θn .After doing so, our estimated values for P(0) and Cl B could be calculated by: Finally, for the sake of validating the results, we can calculate, from the estimated parameters, the rain rate at the RG location and height, and compare it with the actual RG measurements.Given a RG height above ground level is less than a few meters and is roughly at ground level, the rain measurements in (mm/hr), when following Equation ( 5) and using the estimated parameters from Equation ( 9), can be represented as: where S 0 is taken from the measurements at the specific weather station where the RG is located.
In the next section, the experiment and measurements selected in order to solve our estimation problem will be presented, as well as the corresponding results.
and the samples vector is: This problem's solution is known and can easily be solved using a Least Squares Estimation (LSE) to extract Θ .After doing so, our estimated values for (0) and Cl could be calculated by: Finally, for the sake of validating the results, we can calculate, from the estimated parameters, the rain rate at the RG location and height, and compare it with the actual RG measurements.Given a RG height above ground level is less than a few meters and is roughly at ground level, the rain measurements in (mm/hr), when following Equation ( 5) and using the estimated parameters from Equation ( 9), can be represented as: where  is taken from the measurements at the specific weather station where the RG is located.
In the next section, the experiment and measurements selected in order to solve our estimation problem will be presented, as well as the corresponding results.

The Experimental Setup
An experiment was performed in order to verify the validity of the suggested method for estimating P (0) and Ĉl B , thus effectively deriving the vertical profile of the rain-rates, as described in the previous section.The location of southern Israel was selected for our experiment site, as this region is of semi-arid nature, and thus can enable the occurrence of the Virga phenomenon.This location is at the vicinity of the city of Arad and of the site of Shani, and has a CML of 16 km length, deployed between Arad and Shani.The CML is situated at about 40 meters above the ground level, while recording maximum attenuations once every 15 min.As suggested in the previous section, we included measurements of the weather radar of the Israeli Meteorological Service (IMS); the IMS weather radar performs a complete scan, at different elevations, every five minutes.Two different radar elevation levels were relevant for the experiment: The first one observes at heights of between 355 and 1940 m, and the second one observes between 1275 and 2862 m above ground level for the selected experiment site.Once we were able to estimate the Cl B and P(0) using the CML attenuation measurements together with the weather radar, as suggested in the previous section, we were able to later derive the estimated values of rain rate at the RGs locations P(RG), with the corresponding humidity measurements for each RG location, and compare them with actual RG measurements.The experimental site holds two RGs for the sake of this validation, one near Arad, and the other near Shani, which samples the accumulated rain every 10 min.The experimental site, the mentioned CML, the weather radar, and the two RGs are shown in Figure 2.

Humidity (%)
Estimated P(0) The Humidity -So (%) The storms of 7-8 May 2014 and of 25-27 March 2016 were selected to be analyzed, since the months of March to May are generally dry in Israel in this region and the humidity could drop dramatically, thus creating good conditions for significant evaporation of rain drops, as discussed in the introduction.
In addition, since the CML records only the maximum attenuation every 15 min, and not instantaneous measurements, when deriving rain rates from CML samples, we had to use the power law parameters: a, b as was suggested recently [18].In a recent study [18], a power law parameters calibration was performed to use maximum attenuation values of a CML, at the same test site location.The zero-level attenuation, when there is no actual rain, and is composed of the CML free path attenuation due to dry air, water vapor etc. [7], was measured prior to the onset of the selected storm events, and was deducted from the CML measurements during the storms.
Furthermore, when analyzing the weather radar measurements together with the CML derived rain rates, we had to address the different area spans and different timestamps of the samples.As a result, we used an averaging of 124 weather radar area cells above the CML location, which is called the Averaged Radar Cells Over Microwave Link (ARCOML) [19,20].The weather radar cells that were selected for the ARCOML were of an area of about 125 m over 1 km, as the resolution of the radar is 125 m by 1 • , and located above the CML path.In addition, when combining the samples of different timestamps and resolutions in time, we simply took the smallest common denominator, which is 30 min samples, when taking the resolution of the RGs into account.
To summarize, there are three different sources of measurement, the CML in addition to the weather radar beams at two elevations, while there two parameters we wish to estimate.Thus, as suggested in the previous section, we can now estimate P(0) and the Cl B by using LSE, while assuming the values of α and C as they were calculated previously [17] in addition to using the average of the relative humidity samples from the weather stations at Arad and Shani, which are located near the two ends of the CML used in this work.The estimation for P(0) and the Cl B was done every 30 min, and based on these estimation results, the near ground rain rates were estimated at the location of the two RGs every 30 min, using the relative humidity values for that RG location, via Equation (10).

Results
As described in the previous subsection, we conducted an experiment to calculate rain rates using multimodal measurements, in addition to using our suggested method, to interpolate measurements near the ground, at the Shani and Arad RGs locations.For each of the storms, we first estimated average rain rates at the Cl B and the Cl B itself for every 30-min interval.The estimation results for the storm of May 2014 is depicted in Figure 3 The estimated values for P(0) for the duration of the storm, above the CML, ranged between 0 and 18.2 mm/hr while the Cl B was varying up to 6 km and averaged at 3.18 km above ground level.The Cl B estimated values' average is compared to actual Cl B observances at a somewhat nearby weather station (about 30 km away).However, while the estimated values for P(0) were physically within reason for this specific area and for a storm at this season, without access to real ground truth measurements for P(0), we decided to validate the results by calculating the rain rate near ground at the RGs-P(RG)-and correlate it with actual RGs measurements.In Table 1, the correlation between the different measurement sources of rain rate (CML, RGs, ARCOML at two different elevations, and the suggested method estimated rain rate near ground-P(RG)) is shown for the duration of 25 h on 7-8 May 2014 in southern Israel.The suggested method produced the highest correlation values, except for correlation with Arad RG, where the CML was slightly better correlated but by a very negligible margin: 0.9627 vs. 0.9615.When comparing the results of our method vs. Shani RG, we got a correlation of 0.5569, more than 10% over the second-best correlation value.
Figure 4 depicts the time series plot of all measurement sources for the May 2014 storm in addition to our suggested method.In comparison of our method with the CML, the ARCOML, or average of the two RGs, a higher correlation is found between our method and the CML than with the RGs.This could be attributed to the fact that the suggested method uses CML data, but also due to the pin-pointed location of the RGs, which is not representing its surrounding rain rate well, and the fact that both the CML and our method show results at near ground while the ARCOML does not.In Figure 4, it is also noticeable that when comparing the near ground methods to the ARCOML at the heights between 355 and 1940 m, as long as the relative humidity is around 50%, the radar beam overestimated rain rates as and nearly matches the other methods in the second half of the storm when the relative humidity rises to near 100%.This is because the Virga effect diminishes significantly with 100% relative humidity.
Figure 2. The experiment site area: Southern Israel, the weather radar of the Israeli Meteorological Service (IMS) (white target), Arad and Shani RGs (white triangles), and the commercial microwave link marked with two black antenna markers and the actual microwave link between them (purple).
Figure 3. May 2014 event, top subplot: estimated rain rates at the Cl (brown) and the average humidity (green circles).Bottom subplot: estimated Cl levels (brown stars), the average of the estimated Cl (brown), the average of a real Cl measurements nearby (blue), and the average humidity (green circles).The temporal rain rate according to the average of Arad and Shani RGs (in blue), the rain rate according to the microwave link (in black), the lower weather radar beam (ARCOML 1, in red), as well as according to the suggested method (in green) in correspondence with the event's relative humidity averaged from measurements at the weather station in Arad and Shani, over 25 h on 7-8 May at southern Israel.
In addition to the storm event detailed above, we decided to include an additional event to test our suggested method.For the event of March 2016, the estimated values of Cl B are compared to actual (more accurate than for May 2014 storm) Cl B observances at a somewhat nearby weather station (about 30 km away) and are depicted in Figure 5; our estimated Cl B matches closely with those of the station.In Table 2, again, as we have no ground truth for P(0), we calculated rain rates at ground level near the two rain gauges and calculated the correlations.Once again, our method provided the highest correlations, except when the rain gauges were compared to their own average (of course, this will show high correlation as for this storm, the correlation between the two rain gauges was high in the first place).In Figure 6, we show the temporal rain rate of the different sources for the March 2016 event, versus the humidity values, over 20 h on 26-27 March at southern Israel.Again, we see a good matching of our suggested method to the CML rain rates, and although it may seem that the values produced by our method are high, we need to remember that this storm produced very low rain rates, so the absolute difference between the ground sources (suggested method, rain gauges and ML) are low.Table 2.The correlation of the ground truth rain rate measurements of rain gauges (RGs) in Arad, Shani and their average, with the different measurement sources for rain rate (RGs, commercial microwave link (CML), averaged radar cells over microwave link (ARCOML) at two different elevations, and the suggested method of estimating rain rates near ground by combining radar and microwave link measurements) over the duration of 20 h on 26-27 May 2016 at the experiment site between the cities of Arad and Shani in southern Israel.Values marked in green show best values, values marked at yellow show second best values, and third best marked in purple for a specific column.Table 2.The correlation of the ground truth rain rate measurements of rain gauges (RGs) in Arad, Shani and their average, with the different measurement sources for rain rate (RGs, commercial microwave link (CML), averaged radar cells over microwave link (ARCOML) at two different elevations, and the suggested method of estimating rain rates near ground by combining radar and microwave link measurements) over the duration of 20 h on 26-27 May 2016 at the experiment site between the cities of Arad and Shani in southern Israel.Values marked in green show best values, values marked at yellow show second best values, and third best marked in purple for a specific column.

Figure 1 .
Figure 1.A diagram of the different measurement sources in respect to the vertical axis Z.The picture depicts two radar beams at different elevation levels and gives an example of the heights the elevation of 0.44° observes:  ,  in respect to the Z-axis, which actually goes from the cloud base (ClB) downwards to the ground.On the ground level, we can see a rain gauge (RG) and cellular antennas of the CML.

Figure 1 .
Figure 1.A diagram of the different measurement sources in respect to the vertical axis Z.The picture depicts two radar beams at different elevation levels and gives an example of the heights the elevation of 0.44 • observes: Z 1 , Z 2 , in respect to the Z-axis, which actually goes from the cloud base (Cl B ) downwards to the ground.On the ground level, we can see a rain gauge (RG) and cellular antennas of the CML.

Figure 2 .
Figure 2. The experiment site area: Southern Israel, the weather radar of the Israeli Meteorological Service (IMS) (white target), Arad and Shani RGs (white triangles), and the commercial microwave link marked with two black antenna markers and the actual microwave link between them (purple).

Figure 2 .
Figure 2. The experiment site area: Southern Israel, the weather radar of the Israeli Meteorological Service (IMS) (white target), Arad and Shani RGs (white triangles), and the commercial microwave link marked with two black antenna markers and the actual microwave link between them (purple).

Figure 3 .
Figure 3.May 2014 event, top subplot: estimated rain rates at the Cl B (brown) and the average humidity (green circles).Bottom subplot: estimated Cl B levels (brown stars), the average of the estimated Cl B (brown), the average of a real Cl B measurements nearby (blue), and the average humidity (green circles).

Figure 4 .
Figure 4.The temporal rain rate according to the average of Arad and Shani RGs (in blue), the rain rate according to the microwave link (in black), the lower weather radar beam (ARCOML 1, in red), as well as according to the suggested method (in green) in correspondence with the event's relative humidity averaged from measurements at the weather station in Arad and Shani, over 25 h on 7-8 May at southern Israel.

Figure 5 .
Figure 5. March 2016 event, estimated Cl levels (blue circles and dashes) and the ground truth of observed Cl at nearby weather station (black).

Figure 5 .
Figure 5. March 2016 event, estimated Cl B levels (blue circles and dashes) and the ground truth of observed Cl B at nearby weather station (black).

Table 1 .
The correlation of the ground truth rain rate measurements of Rain Gauges (RGs) in Arad, Shani and their average, with the different measurement sources for rain rate (RGs, Commercial Microwave Link (CML), Averaged Radar Cells Over Microwave Link (ARCOML) at two different elevations, and the suggested method of estimating rain rates near ground by combining radar and microwave link measurements) over the duration of 25 h on 7-8 May 2014 at the experiment site between the cities of Arad and Shani in southern Israel.Values marked in green show best values, values marked in yellow show second-best values for a specific column.