## Introduction

**Humidity remains in the atmosphere even on bright days**. Water of all three states can be found naturally in the atmosphere: liquid (rain, fog, and clouds), solid (snowflakes, ice crystals), and gas (water vapour). Water in any state is an obstacle in the link of the electromagnetic wave. When the wave passes through the water particles, a part of its energy is absorbed and a part is scattered. Therefore the electromagnetic wave is attenuated. Prediction of the influence of these factors is very important in radio system design. Attenuation due to rain, fog, and clouds can lead to the perturbations of the wireless, mobile, satellite and other communications. Another problem is the refractive index of the atmosphere, which affects the curvature of the electromagnetic wave path and gives some insight into the fading phenomenon. The anomalous electromagnetic wave propagation can cause disturbances to radar work, because variation of the refractive index of the atmosphere can induce loss of radar coverage. Accurate prediction of losses due to these factors can ensure a reliability of the radio system, decrease an equipment cost, furthermore, the radio systems can become less injurious to health of people.

**When there are no possibilities to gather** data for calculations of the specific attenuation due to rain, clouds and fog, and atmospheric refractive index, the values recommended by the International Communication Union’s Radiocommunication sector (ITU-R) can be used. But the recommended values are not always exact. In design of the radio links, the most desirable operating frequencies are below 10 GHz, because in such cases atmospheric absorption and rainfall loss may generally be neglected (Freeman, 2007). However, in most countries, the frequency-band below 10 GHz is highly congested. In addition, high frequencies provide larger bandwidth, narrower beam width, good resolution and smaller component size (Bhattacharyya et al., 2000). Therefore, the operating frequencies of 10 GHz and above are often used in design of radio systems. The higher the operating frequency, the greater attenuation due to hydrometeors (rain, cloud, fog, snow, and etc.) is observed (Tamosiunaite et al., 2010a).

**In (Ishimaru, 1978),** it was mentioned that the electromagnetic wave attenuation due to snow is less than attenuation due to rain, and that the attenuation due to dry snow may be neglected in microwave band. However, the attenuation due to wet snow is higher. Some results of attenuation due to hail are presented in (Ishimaru, 1978). In this topic, our attention would be concentrated on the attenuation due to rain, clouds, and fog. The variation of the radio refractivity will be the object of our investigation presented there as well.

## Attenuation due to rain

**The electromagnetic wave attenuation due to rain** (the rain attenuation) is one of the most noticeable components of excess losses, especially at frequencies of 10 GHz and above (Freeman, 2007). The methods of prediction of the rain attenuation can be grouped into two groups: the physical (exact) models and the empirical models. The physical models attempt to reproduce the physical behaviour involved in the attenuation processes while the empirical methodologies are based on measurement databases from stations in different climatic zones within a given region. The empirical methods are used widely and frequently with the best success (Emiliani et al., 2004). Two main causes of attenuation are scattering and absorption. When the wavelength is large compared to the size of raindrop, scattering is predominant. Conversely, when the wavelength is small compared to the raindrop’s size, attenuation due to absorption is predominant (Ivanovs & Serdega 2006). Water molecules are dipoles. The raindrop’s dipoles have the same time variation as the electromagnetic waves and therefore act as an antenna, which re-radiates the electromagnetic wave energy. Hence, a raindrop becomes an "antenna" with low directivity. Consequently, some energy is reradiated in arbitrary directions giving a net loss of energy in the direction towards the receiver (Ivanovs & Serdega 2006). Water is a loss-making dielectric medium. The relative dielectric constant of water is high, compared to the dielectric constant of the surrounding air. It depends on temperature and the operating frequency of the radio system. The specific heat of the water is high. Therefore, water absorbs a large amount of warmth, while warms itself. The surface tension of water is high. This is the reason why the molecules of water are holding together. One of the problems in prediction of electromagnetic wave power losses is description of shape of the raindrop. It depends on the size of droplet. It is known, that only very small droplets are like spheres. Such droplets form in clouds, as water vapour condenses on the nuclei of condensation. Further, these droplets grow by coalescence. Shape of the raindrops, that are larger than 1 mm in diameter, is no more spherical. They are not tear-shaped, as it commonly presented in pictures. The shape of falling large raindrops is more like a hamburger shape. Therefore, horizontally polarized waves suffer greater attenuation than vertically polarized waves (Freeman, 2007).

**As mentioned above**, the water molecules are polar ones. Those molecules rotate in such way that positive part of one molecule would be as near as possible to the negative part of another molecule. Therefore, molecules are rotating, hammering one on another and heating (Tamosiunaite et al., 2010a). The water molecule also rotates when a negative charge is brought near to it. The fields of electromagnetic wave vary up as time goes and force the water molecules to rotate respectively to the variation of fields.

### Specific rain attenuation

One of the most widely used rain attenuation prediction methods is an empirical relationship between the specific rain attenuation a [dB -km-1] and the rain rate R [mmr h-1] (Freeman, 2007, Rec. ITU-R P.838-3, 2005):

where a and b are functions of operating frequency f and rain temperature t; the value of R [mm'h-1] is for an exceedance of 0.01% of the time for point rainfall rates with an integration time of one minute. The coefficients a and b (coefficients ah and bh to be used for horizontal polarized waves; coefficients av and bv to be used for vertical polarized waves) are presented in (Freeman, 2007; Recommendation ITU-R P. 838-3, 2005).

**Rain rate**

**In determination of the rain attenuation,** the main parameter is rain rate R, which is expressed in [mmh-1]. Gauges at the surface measure the accumulation of rain-water (flux) in a known time interval and report the result as a rain rate (accumulation per unit time) averaged over some measurement or aggregation interval (Crane, 1996). The rain rate can be described as the thickness of the precipitation layer, which felled down over the time period of one hour in the case when the precipitation is not evaporated, not soaked into the soil, and is not blown away by the wind (Tamosiunaite et al., 2010a). The evaluation of Rvalue is the first step in the rain attenuation prediction. The rain attenuation depends on the meteorological conditions in the considered localities. This is the reason to analyze the rain attenuation in particular locations (eg. country, city, climatic region).

First attempts to predict the rain attenuation under Baltic region climate conditions are described in (Tamosiunas et al., 2005, 2006; Ivanovs & Serdega, 2006; Zilinskas et al., 2006, 2008). It was mentioned in (Ivanovs & Serdega, 2006), that rain events produce unavailability of microwave link, which sometimes lead operators to economical losses or even license loosing.

**The significant differences in annual,** seasonal, monthly, and daily amounts of rainfall are observed in localities of Lithuania. The noticeable local differences of rainfall amounts are characteristic of Lithuania as well. The precipitation amount is probably the most changeable meteorological index on Lithuania’s territory. It varies from 901 mm in Silale district to 520 mm in Pakruojis district (Bukantis, 2001). No month of a year could be described as "an average month" in Lithuania. This is the reason to revise the suitability of the models that derived under climatic conditions other than Lithuanian ones. The models using only annual amount of rainfall was analyzed in (Tamosiunas et al., 2005). Considering the peculiarities of Lithuania’s climate, the change in (Chebil et al., 1999) model was made. This new model for the electromagnetic wave attenuation due to rain medium in atmosphere for the first time has been presented in (Tamosiunas et al., 2006). Calculation of radio wave attenuation due to rain using annual precipitation and heavy rainfall data is described in (Zilinskas et al., 2006). The heavy rainfall events and showers with thunderstorms occur during the warm season (from May to September) in Lithuania.

**Integration time**

**As was mentioned above**, the R-values are expressed in [mm'h-1]. However, time intervals between the readings of rainfall amount in many cases must be much shorter. Those intervals are called the integration time t. In (Ivanovs & Serdega, 2006; Tamosiunas et al, 2007; Tamosiunaite et al., 2010a) it was mentioned, that the period of time between the readings of the rainfall amount values is a very important parameter, because it can significantly change the R-value. High R-values "hides" when t is long.

**Consider an example.** There were raining. The duration of the rain was 5 minutes. The total amount of the precipitation was 5 mm. It did not rain during remaining 55 minutes of one hour. Thereby, if we would count the average R-value for that hour (r = 60 min.), it would be equal to 5 mm h-1. But if we would count the R-value for every minute of that hour, we would find that R-values are much higher. Consider that in every of those 5 rainy minutes the amount of the precipitation was 1 mm. Consequently, for each of those 5 minutes the Rvalue would be 60 mm’h-1. That is why the average R-values are unreliable. In Lithuania, the t values must be as small as possible (Tamosiunaite et al., 2010a).

**"One-minute" rain rate**

**Almost all rain attenuation methods require** "one-minute" rain rate value. The "one- minute" rain rate value R(1 min.) is expressed in [mmrh"1]. R(1 min.)-value can be defined as the R-value for 0.01% of time of the year, obtained using the rainfall amount value, which was measured in r = 1 min and multiplied by 60 (Karasawa & Matsudo, 1991).

However, in many instances data collection is oriented toward agricultural and hydrological purposes, for which annual, monthly, daily, and less commonly, 3- and 6-hourly totals are collected. Therefore the models for conversion of R(r min.)-values into R(1 min.)-values are used.

A review of models for estimation of 1 min rainfall rates for microwave attenuation calculations are presented in (Tattelman & Grantham, 1985).

**One of such conversion models was presented in (Moupfouma &Martin, 1995):**

where R^ min) is the "one-minute" rain rate value, R(r min) is the rain rate value measured in r minutes (r> 1 min.).

In (Zilinskas et al, 2008) another model (4) for calculation of the R^ min) -value was presented. That model was derived on the basis of model presented in (Rice & Holmberg, 1973) in accordance with the peculiarities of Lithuanian climate.

where MV_IX is amount of rainfall which precipitated in May-September, t is the number of hours in a year when the value of rain rate could be equal or exceed the R(1 min.) -value. According to data that was collected in Lithuanian weather stations and (4) formula, the average R^ min)-value for Lithuanian territory was calculated. That value is 60.23 mm’h-1.

This value is double the value, which is suggested by ITU-R (Tamosiunaite et al., 2010a). According to (1) formula, the values of coefficients a and b (presented in Freeman, 2007), and the value of R^ min) = 60.23 mmr h-1, the dependency of the average specific electromagnetic wave attenuation due to rain, a, on the operating frequency f was estimated. The results are shown in Fig. 1.

**Fig. 1. The dependency of the average specific electromagnetic wave attenuation due to rain a on the operating frequency f, in Lithuania.**

**Worst month statistics**

**The "Worst-month" model was proposed by ITU-R in (Rec. ITU-R P.481-4, 2005).** This model is a supplement of the "One-minute" models, which were explained above. In "One-minute" models a lot of precipitation data must be collected and calculated. Furthermore, majority of those models are appropriate only in cases when the reliability of the radio wave system must be equal 99.99%. The main advantage of the "Worst-month" model is that only the worst-month statistics must be collected. Furthermore, the "Worst-month" model is appropriate in cases when the required reliability of the radio system is other than 99.99%. The worst-month is the month (or 30 days period) from a year (or twelve consecutive calendar months), during which the threshold is exceeded for the longest time. This month is not necessarily the same month in different year. The fraction of time when the threshold value of rain rate (so, and rain attenuation value) was exceeded is identical to probability that the threshold value of rain rate would be exceeded (Crane, 1996). The average annual worst-month time percentage of excess, pm is proportional to the average annual time percentage of excess, p, in such relation:

where Q is the conversion factor; pm [%] and p [%] must refer to the same threshold levels (the same rain rate value).

**The conversion factor Q is a two-parameters (Qi, j0) function of p. In most cases a high reliability of the radio system is required (p < 3 %). Then Q can be expressed as (Rec. ITU-R P.481-4, 2005):**

For global planning purposes the following values of the parameters Q1 and ft may be used: Q1 = 2.85 and P = 0.13 (Rec. ITU-R P.481-4, 2005).

**For global rain rate applications**, the following values for the parameters Q1 and ft should be used: Q1 = 2.82 and P = 0.15, for tropical, subtropical and temperate climate regions with frequent rain; Q1 = 4.48 and P = 0.11, for dry temperate, polar and desert regions. Yet ITU-R recommends that more precise values of Q1 and ft should be used where possible. Since

and (6), consequently:

According to (2), (3) and annual data, the relation between p and R(1 min) can be found.

This relation could be compared to the relation calculated according to (8) and ITU-R suggested Q1 and P values. According to Lithuanian climate, the values Q1 = 2.82 and P = 0.15 should be appropriate.

For example, we evaluated the "Worst-month" model in Vilnius, the capital of Lithuania. The results are shown in Fig. 2. As can be seen, the values Q1 = 2.82 and P = 0.15 are appropriate only in cases when R(1 min) > 38 mm’h-1.

**Fig. 2. The correlation between the real, calculated and corrected values of p (in Vilnius).**

When R(1 min) < 38 mm^h-1, the calculated values are apparently distant from the real values. Therefore, the values of Q1 and P must be corrected. The best correlation is when in (6) there are q = 0.5 and % = 1.03 . Consequently, the corrected Q1 and P values should be Q1 = 2 and P = 0.03 . But still, as can be seen in Fig. 2, the corrected values are only correct when R(1 min) < 30 mmr h-1. Furthermore, when R(1 min) > 34 mm’h-1, the values Q1 = 2.82 and P = 0.15 are more proper than Q1 = 2 and P = 0.03 . As a result, in cases when R(1 min) < 34, the values Q1 = 2 and P = 0.03 should be used, and in cases when R(1 min) > 34 mm’h-1, the ITU-R suggested values Q1 = 2.82 and P = 0.15 may be used.

## Attenuation due to clouds

**The effect of rain attenuation is greater than** that of clouds in many cases, but clouds occur more often than rain. In clouds, water droplets are generally less than 0.01 cm in diameter (Freeman, 2007). In (Altshuler & Mart, 1989), it was mentioned that cloud attenuation was primarily due to absorption by the cloud droplets, and scattering losses were secondary. With increase in operating frequency the attenuation due to clouds also increases, but as the temperature of the clouds decreases the attenuation value increases (Sarkar et. al., 2005). On average, the clouds cover more than 50% of the territory of Lithuania. According to the data of its weather stations, November and December are the cloudiest months. The clearest sky is in May and June. There are about 100 overcast days in the year.

### Liquid water content

**The liquid water content M is one of the most important** parameters of the clouds. M describes the mass of water drops in the volume units of the cloud. It has been mentioned in (Freeman, 2007) that the specific cloud attenuation aC [dB/km] is a function of the liquid water content M [g/m3], the frequency f, and the temperature within the cloud T. The measurements of M at a point in space or averaged over a radio wave path are very complicated. Direct methods for measuring M consists of extracting a known volume through a cotton pad or of rotating cups in an impeller apparatus, both to be weighed; also, resistance changes can be measured with a hot wire probe attached to an aircraft flying through clouds (Liebe et al., 1989). The liquid water content in the cloud varies in a wide range. In most of the cloud attenuation models, it is required to know the value of M. The climate conditions (humidity, temperature, etc.) and cloud morphology are different over various localities of several regions; accordingly, the liquid water contents differ within the clouds as well. This factor must be considered when analyzing rain attenuation and cloud attenuation. Our first attempt to determine the specific cloud attenuation under the Lithuanian climatic conditions is presented in (Tamosiunaite et al., 2008; Zilinskas et al., 2008). The humid weather predominates over the year in Lithuania.