2018年美賽O獎(jiǎng)?wù)撐?A76082-HF Radio Propagation under Different Terrains

For office use only T1 T2 T3 T4 Team Control Number 76082 Problem Chosen A For office use only F1 F2 F3 F4 2018 MCM ICM Summary Sheet HF Radio Propagation under Different Terrains Summary Even in the satellite era high frequency HF signal communication still plays an important role in everyday communications In order to clearly understand the communication process of HF waves and its influencing factors we first design a mathematical model of signal reflection off the ocean Based on this model we build the ground signal reflection model and compare the two Besides we study the communication process of vessel receivers on a turbulent sea We begin with the establishment of a mathematical model of signal reflection at sea from two aspects On the one hand we study the basic loss of the HF sky wave transmission process On the other hand we investigate the surface properties of the sea Weclassify the ocean surface as a smooth and a rough sea Based on the Fresnel reflection coefficient equation we obtain the reflection intensity of rough and smooth sea surface And their ratio equals to the square of the roughness correction factor We select specific parameters for getting the specific value Then we get the first reflection power of rough sea surface is 0 4378mW and the first reflection power of smooth sea surface is 0 2832mW The first reflection power of rough sea surface is 0 6469 times smooth sea surface As a result using this model we can easily simulate the multi hop path of the signal Taking the selected specific value as the parameter we calculate the maximum number of hops to 8 times if the signal noise ratio threshold is not exceeded Next based on the above models we set up the mathematical model of ground signal reflection Similarly we classify the terrain as a smooth terrain and a mountainous terrain The propagation loss of mountainous terrain is classified as diffraction loss of the mountain and absorption loss of the vegetation We use Epstein Peterson method to study the typical double edged peak diffraction problem Through comparison of the two models we conclude that the ocean surface is more suitable for the transmission of shortwave skywaves than land surfaces What s more we introduce the ship sway model to further establish the communication model of the ship receiver at sea The ship can maintain communication while traveling in the signal coverage We get the longest communication time by calculating the maximum travel time of the ship in the signal coverage area Finally we prepare a synopsis of the results that are suitable as a short note in IEEE Communications Magazine We focus on the transmission process of shortwave skywaves off the ocean The conclusion can help in the communications of maritime transport and fishing industries Keywords Fresnel reflection coefficient equation Sea signal reflection model Transmission loss 更多數(shù)學(xué)建模資料請(qǐng)關(guān)注微店店鋪 數(shù)學(xué)建模學(xué)習(xí)交流 Team 76082Page 1 of 24 Content 1 Introduction 2 1 1Restatement of the problem 2 1 2Notations 3 2 Assumptions 3 3 Skywave basic transmission loss 4 3 1Loss model 4 3 2Calculation ResultsAnalysis 5 4 The Mathematical Model of Ocean signal s Reflection 6 4 1Basic model 6 4 1 1 Sea surface s plural dielectric constant 6 4 1 2 Fresnel reflection coefficient of Sea 7 4 2Comparison of Reflection Intensity between Rough Sea and Smooth Sea 11 4 3Calculation of the maximum number of hops 12 4 3 1Signal to noise ratio calculation of shortwave sky wave communication 12 4 3 2 Calculation results 12 5 The comparison of Ground SignalTransmission and Sea Signal Transmission 13 5 1The mathematical model of ground signal reflection 13 5 1 1 Shortwave sky wave propagation loss in thesmooth terrain 13 5 1 2 Shortwave sky wave in mountainous terraintransmission loss 13 5 2Comparing results 14 6 The communication model of marine ship receiver 15 6 1The model of ship rocks 15 6 2Sea signal transmission model of combining ship swaying 16 6 3The same multi hop path to maintain communication time 18 7 SensitivityAnalysis Time factor 19 8Conclusions 20 8 1Strengths 20 8 2Weaknesses 20 References 24 Team 76082Page 2 of 24 1 Introduction 1 1Restatement of the problem The high frequency HF defined to be 3 30MHz is the portion of the radio frequency spectrum The HF band is a major part of the shortwave frequency band so communication at these frequency is often called shortwave radio Shortwave propagation modes include groundwave and skywaves For frequencies below the maximum usable frequency MUF HF radio waves can travel further through the multiple reflections of the ionosphere and the earth s surface even to the world This method of communication is called skip or skywave Many factors affect the propagation of HF skywaves of which the properties of the reflecting surface are important The properties of the reflecting surface determine the strength of the reflected wave and how far the wave will propagate with the useful signal integrity The most important issue is the reflections on the sea We define the raging sea as a rough sea and relatively speaking we define a calm sea as a smooth sea The problems that we need to solve in this paper are Establish a mathematical model of ocean signalreflection Determine a first reflection intensity of a 100 watt HF constant carrier signal transmitted from a land based source at the turbulent ocean level In this paper we use the size of power on behalf of the strength of the size Compare the above result with the first reflected intensity of the same signal on a calm ocean surface Based on the first issue the remaining reflections of the radio signals occur on a calm sea surface Determining the maximum number of hops the signal can reach before its strength falls below a usable signal to noise ratio SNR threshold of 10 dB Using the results obtained above compared with the results of high frequency radio wave reflections on rugged versus smooth terrain There is a marine vessel that uses high frequencies to communicate and receive weather and traffic reports Transforming the model to accommodate radio transmissions from the ship s receiver on a raging surface Calculate the time that the boat keeps the signal commutating in the same multi hop path Team 76082Page 3 of 24 1 2Notations Let s first define the list of notations used in this article SymbolsDefinition rSea water relative dielectric constant Sea water conductivity Wavelength fThe best available radio frequency RSmooth sea reflection coefficient R Rough sea reflection coefficient Rough correction factor Sea surface dielectric constant RHHorizontalpolarizedwaveFresnelreflection coefficient RvVertical polarization Fresnel reflection coefficient SNR Lb Lbf Li Yp Signal to noise ratio Skywave basic transmission loss Transmission loss of free space Ionospheric absorption loss Extra system loss 2 Assumptions Calm sea is equivalent to a smooth sea so calm sea is the reflection of radio waves is specular reflection The object of our study is the high frequency band of 3 30 MHz If the wave frequency exceeds MUF the electric wave will enter the space through the ionosphere and the ionosphere will change frequently Therefore the best available frequency is usually 0 8 0 9 times we conservatively assume that the best available frequency is f 20MHz When the wavelength and the wave height are comparable or even far less than the wave height the influence of the shadow effect on the radio wave propagation needs to be considered However this dissertation is not applicable Therefore the influence of the shadow effect on the radio wave propagation is ignored Assume that the directional factor ofthe emission and reception antennas is 1 Suppose the emitted electromagnetic wave is a circularly polarized wave Assume that multipath interference is ignored Team 76082Page 4 of 24 3 Skywave basic transmission loss 3 1Loss model In order to clearly describe the spread of the electric wave we give the figure 1 about a simple representation of its jumping process Figure1 Signal transmission diagram A radio wave emitted by the terrestrial source first reaches the ionosphere and reaches the sea level through the reflection of the ionosphere It is its first hopping process After being reflected by the sea it returns to the ionosphere and returns to the sea for the second jump and so on We know that in practice radio waves produce loss of energy when transmitting According to the reason of transmission loss the basic transmission loss of skywave is expressed as 1 Lb Lbf Li Lg Yp 1 Where Lbfis the transmission loss in free space Liis the ionospheric absorption loss Lgis reflection loss on the ground Ypis the additional system loss We mainly discuss the ionospheric absorption loss and space transmission loss Basic transmission loss in freespace Lbf The basic transmission loss in free space is the energy loss the geometric diffusion causes energy loss after the radio wave leaves the transmitting antenna The formula is 2 Lbf 32 44 20lg f 20lgr 2 Where The unit of Lbf is dB f is the working frequency the unit is MHz r is effective Team 76082Page 5 of 24 H path for the propagation of radio waves the unit is km Ionospheric absorption loss Li In the ionosphere there is a region of significant ionization in the atmosphere In accordance with the electronic density changes with height the ionosphere can be divided into D layer E layer F1 layer and F2 layer The F layer is the reflective layer because it is highest and allows the radio waves to travel the furthest distance 3 Therefore we consider the high frequency signals mainly reflect on the F1 layer 150km 200km in thispaper Since the degree of ionospheric absorption relates to many factors it is difficult to make theoretical calculations So we choose semi empirical formula 3 4 L 677 2I seci100 3 bf f f 1 98 10 2 I 1 0 0037R cos 0 881 1 3 i100 arcsin 0 985 cos 4 5 Where fH is the frequency of the magnetic swing at a height of 100 km I is the absorption coefficient and represents the relationship of the ionospheric absorption and solar zenith angle and sunspots R i100is the 100km height of the incident angle is the ray elevation Under certain circumstances these parameters can access data to get a specific value Extra system lossYp The extra system loss is the sum of the losses calculated for other reasons and accurate calculation is difficult Since the additional system loss is basically a function of the local time 1 we can estimate the additional system loss by looking at the data sheet Reflection loss on the groundLg Ground loss occurs from the radio wave through the ground reflection In this model the electric wave is reflected on the sea surface so we do not consider the ground reflection loss 3 2Calculation ResultsAnalysis In order to make the calculation easily we choose the typical numerical value to simulate The parameters we choose are as follows Ray elevation Ionospheric height h 200 km so it is easy to obtain the ffective path of radio wave propagation r 200 2 km Reflection point 123 E 26 N is located in the East China Sea fH 1 24MHz time point is 12 00 on July 1 check the number of sunspots R 110 The sun zenith angle can be calculated by the following formula 4 Team 76082Page 6 of 22 b cos sinasinSx cosacosSxcos Sx 23 44 sin 0 9856 Yn 80 7 180 15 24 t 8 0 t 8 6 Sy 180 15 t 8 8 t 24 Where Sxis the sun s point of latitude Sy is the sun s point of longitude Ynis the number of days from January 1 each year t is Beijing time ais the study point s latitude is the difference between study point s longitude and Sy After simulation we get the transmission lossLf We learn from the formula is 97 4597dB L 10lg Pr Pt 7 Calculate the power Ptof a high frequency constant carrier signal transmitted by a terrestrial source is 100W incident power Pr is 0 43248mW reaching the sea surface after transmission loss 4 The Mathematical Model of Ocean signal s Reflection 4 1Basic model The reflection coefficient of sea waves mainly represents the reflection characteristics of sea waves on the surface of the sea The reflection coefficient is related to the incident angle of the sea waves the size of the waves the electromagnetic parameters of the sea surface and other factors Before studying the reflection characteristics we first study the electromagnetic properties of the sea surface The electromagnetic characteristics of sea surface affect the sea surface reflection intensity of radio waves and it is related to the seawater temperature salinity electromagnetic wave frequency and other factors The complex permittivity of the sea is a parameter describing the electromagnetic properties of the sea surface 4 1 1Sea surface s plural dielectric constant The plural dielectric constant of sea surface is determined by the relative dielectric constant of seawater r sea water conductivity and the wave length the expression is 1 r i60 8 We can calculate constants rand ratio based on the polynomial fit function given by Consultants Committee of the International Radio CCIR Relative dielectric constant of the sea water Team 76082Page 7 of 24 The expression of sea water s relative dielectric constant is 5 432 r ef df cf bf a 1 70 2253 5895 f 2253 5895 f Where f is the radio frequency and its units is MHz a 1 4114535 10 2 b 5 2122497 10 8 c 5 8547829 10 11 d 7 6717423 10 16 e 2 9856318 10 21 Conductivity of the Sea water The expression of sea water s conductivity is 5 wf uf uf 1 sfr 0 5 32 2 tf 1106 207 f 1106 207 f 10 Where f is the radio frequency its units is MHz r 3 8586749 s 9 1253873 10 4 t 1 5309921 10 8 u 2 1179295 10 5 v 6 5727504 10 10 w 1 9647664 10 15 Because we assumefis 20MHz we can get seawater relative permittivity r 70 sea water conductivity 5 0 Sea surface dielectric constant 70 4500i 4 1 2Fresnel reflection coefficient of Sea Fresnel reflection coefficient of smooth sea According to snell s law the Fresnel reflection coefficient of horizontal and vertical polarized waves on a smooth sea surface is 5 RH 11 RV 12 Where is grazing angle of incidence The curve in Figure 2 reflects the relationship between the power reflection coefficient and the grazing incidence angle sin sin cos2 cos2 sin cos2 sin cos2 Team 76082Page 8 of 24 Figure 2 power reflection coefficient and the grazing incidence angle The grazing incidence angle is 45 90 by observing the change of the power reflection coefficient is not obvious For convenience of calculation we set the grazing incidence angle as 45 Fresnel reflection coefficient of rough sea We can easily get a smooth sea reflection coefficient in fact the sea is choppy so we continue to study the reflection coefficient of the rough sea Wave phenomenon is random and non linear so it is difficult to establish an accurate model of waves According to the wave spectrum and the theory of stochastic ocean waves we can regard the actual ocean waves as the result of the superposition of sine waves of different frequencies different initial phases different directions of propagation and different wave heights Figure 3 is our simulation of the waves Figure 3 ocean wave diagram In the raging sea wave height shape and frequency change rapidly and wave Team 76082Page 9 of 24 0 propagation direction may change Tosimplify the model we only consider the effect of frequency and wave height on the roughness of the sea surface At present Pierson Moscowitz spectrum Neutron spectrum NTC spectrum ITTC and two parameter ITTC spectrum are the most widely used marine spectra Among them PM spectrum is the most widely used so we use PM spectrum to describe the frequency of ocean waves The expression of PM spectrum is 8 1 10 3g 2 g4 S i 5 exp 0 74 v 13 Where v is the average wind speed near the height of the sea surface Figure 4 describes the spectrum with the wind speed changes Figure 4Wave spectrum From figure 4 we conclude that wind speed is a major factor affecting the frequency of waves Next we study the influence of wave height on thereflection coefficient According to Phillips 1996 wave model as follows 6 7 h 0 0051v2 14 Where h is the root mean square height of the sea surface v is the wind speed near the height of the sea surface It is obvious that the high wind speed near the sea surface directly affects the root mean square of the sea surface Therefore we believe wind speed is a common major factor affecting frequency and wave height It is obvious that the high wind speed near the sea surface directly affects the root mean square of the sea surface Therefore we believe wind speed is a common major factor affecting frequency and wave height Then we can get rough correction factor expression based on Miller Brown rough surface approximation model 8 9 exp 2 2 g 2 I 2 2 g 2 15 Team 76082Page 10 of 24 Where I0is the first kind of modified Bessel function of order 0 gis the sea surface roughness which is used to describe the magnitude of sea surface fluctuations The formula is g h sin Figure 5 shows the effect of wind speed on the roughness correction factor Figure 5 Relationship between wind speed and power reflection coefficient Figure 5 shows that when the wind speed exceeds a certain level the roughness correction factor decreases rapidly with increasing wind speed Because communication at sea is a communication distance the influence of earth curvature on the correction factor can not be ignored D represents the Earth s curvaturefactor which is calculation formula 16 10 2 1 21 21 sin 2 1 GGR GG D e 16 Where G1is the distance from the RF transmitting end to the specular reflection point and is the distance from the specular reflection point to the RF receiving end which is the radius of the earth the effective earth radius is 6400km Therefore the rough correction factor for earth curvature is taken into account D We use the roughness correction factor to approximate the Fresnel reflection coefficient of the horizontal and vertical polarized waves of a rough sea surface 17 Depending on the relationship of the wavelength and freq