V F Kravchenko,V I Lutsenko,I V Lutsenko

(1. Kotelnikov Institute of Radio Engineering and Electronics of Russian Academy of Sciences,Moscow 125009,Russia; 2. Usikov Institute of Radiophysics and Electronics of National Academy of Sciences of Ukraine,Kharkov 61085,Ukraine)

0 Introduction

The intensity of backscatter by the sea depends on many factors,the most important of which are the wind speed,the grazing angle and the azimuthal angle of surface radiation relative to the wind direction. The wind speed of developed waves fully determines the state of the sea surface,and the degree of roughness,influencing greatly on specific radar cross-section (SRCS) reflected at low grazing angles. The grazing angle determines that radiation of the reflective surface area occurs in the line of sight or in the region of interference when the radiation area of the surface field is formed not only by the direct signal,but also by reflections from the surface,and the azimuth of surface radiation is relative to the direction of the main travelling wave which is in turn matches for the developed wind wave with the wind direction.

In Ref.[1] it was proposed to represent the sea specific RCS (SRCS) in the form of three factors for the first time: one of which taking into account the influence of wind speed,the second one allowing for the grazing angle,and the third factor taking the azimuth of surface radiation into consideration. Later models[2-4]also used such presentation of the seaSRCS,but in Ref.[2,5] the effect of high refraction of the driving layer of troposphere was considered and the supplement ofSRCSwas introduced due to scattering by breaking waves. As shown in experimental studies[6,7],the contribution to the seaSRCSby breaking waves in the short millimeter wave (wavelength is 2 mm) is very noticeable. In the developing process of these models,we assume that the saturation ofSRCSoccurs when the wind speed is approximately 5-7 m/s. However,the experimental studies show that the saturation ofSRCSat shortening wavelength of the radiation field occurs at lower wind speed. When describing the azimuthal dependence ofSRCS,the results obtained by Moore were not considered. In this paper we propose a phenomenological model of the seaSRCSat the grazing angle of radiation surface of millimeter and centimeter waves.

In high resolution of range and angular coordinates,there are significant deviations of the distribution law of fluctuations reflected by underlying surfaces,and “clear” sky signal from standard is observed[8,9]. It is caused by sequential and separate observation of surface regions with different statistical properties of irregularities which lead to the unsteadiness and non-Gaussianity of scattered signal. For the sea surface it is areas in which there are crests of sea waves with high background reflectance. In this paper the possibility of using finite atomic functions[21-24]to descript statistics of signal reflected by the sea has been discussed for the first time.

1 Model of SRCS

1.1 Dependence on wind speed

The results of experimental study of characteristics of the backscatter by the sea surface are used as the input data of the model. These results are obtained by the coherent-pulse systems with wavelengths of 3 cm,8 mm and 4 mm[8,9]. Height of the antenna of measurement systems (about 11 m) allows conducting a study at grazing angles from 2° to 0.1°. Measurements are carried out near the Sevastopol city in the water area of the gulf of Kalamitsky. Restricted fetch areas (up to 100 km),considerable depths (greater than 100 m) at a distance from the coast more than 500 m,and steady wind allow measuring in fully developed waves and extrapolating the results of open sea in most cases.

Fig.1 shows the dependence of the SeaSRCSon the wind speed,which is obtained at wavelengths of 8mm and 4 mm and at grazing angles greater than the critical,that is to say,when reflections from the surface do not affect the level of radiation field,and at the same time its approximation is got by the least squares method of the following function

(1)

whereU0is the wind speed after which there is a rapid saturation ofSRCS. Choosing an approximating function based on the fact that the wind speed is low (U≤U0),theSRCSincreases in proportion,soσ0≈U4; while the wind speed is high (U≥U0),it quickly saturates,soσ0≈σ0=const. Dependence of choosing the quartic is due to the fact thatSRCSincreases in proportion to the variance of the surface roughness height. In this case,the RMS height of sea waves increases in proportion toUnandn=2 or 3[10]. Earlierly,the dependence ofσ0on wind speed was described by the expression[11]

σ0～Uγ,γ≈3,

whereγ≈3 is its empirical coefficient.

In Table 1 there are obtained values of coefficients of Eq.(1). Fig.1 shows the dependence of theSRCSof sea on the sind speedUw.

Fig.1 Dependence of the SRCS of sea on the wind speed Uw

Table 1 Values of approximate coefficients depending on sea SRCS with wind speed

As seen in Table 1 the wind speedU0at which the saturation of the seaSRCSis starting depends on the wavelength of the radiation field. The parameterσ0=algb,which characterizes the maximum of attainable value ofSRCS,is about the same for both wavelengths. In the models[2,5-7]it was assumed that the wind speedU0which determined saturation of the seaSRCS,was about the same for all wavelengths,at which saturation of the spectral density of resonant scattering ripple appears. In another word,there is a formation of equilibrium Phillips range of spectrum[12]and its value is 5-7 mps. In the case,characteristic values of the wind speed,which determine the beginning ofSRCSsaturation,increase with wavelengths of the radiation field. This is due to the fact that the saturation of the seaSRCSwill be determined not only by saturation of the spectral density of resonant scattering ripple,but also by saturation of values of the surface roughness,the value of which will be affected by wavelengths more obviously than the resonant ripple[13],which the saturation occurs at larger values of the wind speed than the Phillips spectrum. Assuming that the dependenceU0on the wavelengthλis showed by the form

U0≈U0λ0.

(2)

We can confirm thatU0≈2.76 mps andn≈0.70.

It should be noted that the speed value approximately corresponds to the wind speed,and the air stream over surface is laminar (about 3 mps)[12]. Enough extensive experimental materials were used by the authors[1]for the phenomenological model of scattering by the sea surface that the frequency ranged from 1 GHz to 100 GHz. According to the empirical relation in the millimeter wavelength range,for a weak roughnessσ0～λ-(0.5-1)and a strong wave emotion,the dependence isσ0～λ0or evenσ0～λ+0.5for vertical transmitting and receiving. The results agree with these data whenn≈0.7.

1.2 Dependence on grazing angle

For empirical dependence ofSRCSon the grazing angle there is data obtained from approximately the same states of the sea surface. It's presented in Fig.2.

Fig.2 Dependence of SRCS on the grazing angle ψ

These figures show the approximation of experimental data using the following form of dependence

(3)

whereψ0is the critical grazing angle that characterizes the transition from the region of interference forσ0≈ψ4,and a strong influence is exerted on the irradiating element of the field surface reflected just from the surface to the plateau region forσ0≈const,and the reflection from the surface practically has no effect on the level of its radiation. The approximate coefficients obtained of current (○) and averageSRCS(◆) are shown in Table 2.

Table 2 Approximation coefficients of the sea SRCS depending on the grazing angle

Shortening the wavelength leads to increasing of theSRCSunder identical experimental conditions,and decreasing in values of critical angle. The seaSRCSincreases with wind speed. The critical angle is the grazing angle,at which the surface radiation level is determined not only by the direct signal field but also by a signal reflected from the surface,that is the radiation occurring in the areas of interference of these fields. Michel suggested[11]an empirical relation to determine the angle,that is

(4)

whereHis the average height of waves.

Dependence of the heightHof roughness on wind speedUis shown in Fig.3,that is quoted from the Ref.[10].

1-average height of 1/10 of the highest waves; 2-indicative height; 3-average heightFig.3 Dependence of sea wave height on wind speed

For the coefficientPi,i∈[1,4],its empirical relation is showed in Table 3. And it satisfy the following relation:

H=p1*U3+p2*U2+p3*U1+p4.

(5)

Table 3 Approximation coefficients of sea wave height

Using the Eq.(5) and data in Table 3 we can estimate the average height of waves which should be used to estimate the critical angle by Eq.(4) and calculate the distant dependence ofSRCSby Eq.(3). In general whenSRCSgot by Eq.(1) allows evaluating a maximum at a given speed ofSRCSvalue and then Eq.(3) is used to estimateSRCSat a given grazing angle.

1.3 Effect of increased refraction on the sea SRCS

Refraction of troposphere in the marine surface layer leads to changes in the local angle radiation surface and itsSRCS. The refractive additive to the geometric grazing angle[14]is defined by the relation as follow

(6)

wheregnis a gradient of the refractive index;Ris a distance to the reflecting surface element.

(7)

1.4 Dependence on the surface radiation azimuth

To account the azimuthally dependence ofSRCSwe can use the Eq.(8) proposed by Moore[15]

σ0=a0+a1cosΔθ+a2cos2Δθ,

(8)

where Δθ=θ-θ0is the azimuth angle of surface radiation relative to the wind directionθ0.

Knowing the relations ofSRCSat radiation angles of 0°,90° and 180°,we can estimate the coefficientsσiby the Eq.(8)

(9)

Fig.4 shows relations ofSRCSand grazing angle,andSRCSis obtained experimentally by Ref.[10,16] with centimeter waves at radiation of the sea surface against the wind. In Fig.5 there are variations ofSRCSat changing surface radiation azimuth obtained by authors with the millimeter wavelengths.

Uw=5 mps,λ=3.2 cm,-○- HP,-×- VP; λ=10 cm,---VPFig.4 Relation of sea SRCS against the wind and grazing angle

λ=8 mm,□ HP,? VP; λ=4.1 mm,▲ VPFig.5 Influence of surface radiation azimuth on the sea SRCS at millimeter wavelengths

1.5 Additive to SRCS due to reflections from breaking waves and splashes

To assess the effect of reflections from breaking waves and splashes,the key issue is to determine the water content (the number of drops when collapsing of a breaking wave). Poor knowledge of the issue led to the fact that the authors had to account the water content of splashing purely on the assumption of speculative considerations to create a model of scattering by breaking waves. In the Ref.[13,15] there was an attempt to account the water content using empirical data on the height distribution of drops in the surface layer[15]. However,this approach is productive to assess the splash fractions stably existing in the layer and it is poorly suited to assess the water content of splashes formed at collapsing breaking waves. The collapses take place when the breaking waves reach a stability limit. In this case,the following wave with smaller amplitude (energy) is formed,but a part of the mass is transformed into splashes. Using the diagram characterizing changes of the wave height at the collapse[17]and assuming that the wave form at the collapse does not change,it is easy to show that the water mass is proportional to its height so the wave mass of height change characterizes transformes to splashes.

1.6 Polarization features of sea SRCS

Reflections by vertical polarization of radiation and reception dominate at decimeter and longer wavelengths,thereby coinciding with the two-scale model of scattering[13]. In a centimeter wavelength at high wind speed,the reflection by horizontal polarization begins to dominate. To explain the effect we use the model of microwave scattered by sharp crests of waves-scattering by a wedge. At millimeter wavelengths and only at low wind speed (up to 4-5 mps) there is scattering mainly by vertical polarization,as it is shown in the two-scale model. At high wind speed signal reflected by sea in the horizontal and vertical polarization is about the same as shown in Fig.6.

Fig.6 Influence of radiation polarization on the sea SRCS of the 8 mm wave at wind speed more than 5 mps

2 Statistical description of reflections from sea

A statistical description of the reflections signal by sea is based on the two-component imbedded stochastic processes {S(t),θ(t)},in which one component is continuousS(t) and the otherθ(t)=υiis the discrete andtis a generalized coordinate[18-20]. At any time the process is in one ofkpossible phase statesHi∈υ1,υ2,…,υk,and it is assumed that the initial state is knownθ0=υiat timet=0,and the one-step transition probabilityπij,wherei,j=1,2,…,K. Let’s associate each nonzero array elementsπijof the transition probability with a random variableTijwith the distribution densityfij(t),which will be called the time-out in the stateυibefore passing to the stateυi. If valuesTijare distributed by the exponential law then the process is Markovian. In practice,in many cases such assumption is unsatisfied in particular for reflections from the sea,and then the process in which the state transition is described by the Markov chain and the distribution density of existence time in each of them differs from the exponential one and belongs to a class of semi-Markov,wherei=0 is a pause andi=1 is a spike in the reflected signal due to reflection from the sea wave crest. Inside each of the statesυi∈(0,1) is assumed to be that the process is quasi-stationary,and being described by its statistical scattering matrix. That is

Density distribution of values is

And spectrum is

This means that the statistical scattering matrix of the process ‖S(t)‖ as well as the density matrix of distribution of values ‖P(s)‖ and spectrums ‖S(ω)‖ to account for polarization of scattering is block vector-each elementiof the vector is a square matrix2×2. In some cases the standard models of Gaussian processes can be used to describe the process in the phase state. However,intensity of the signal scattered by sea is determined by the large wind wave steepness which cannot be infinitely large because a certain steepness hydrodynamic waves lose stability and collapse. It is a prerequisite that there is an amplitude limitation of the signal scattered by sea. So it is logical to assume that statistics of the scattered signal in each of the phase states should be appropriately described by finite functions[21-24].

3 Technique of data processing and results

The radar reflections from the sea surface were used to process photographic images. The photos were obtained from screen of the pulse-coherent radar station with a wavelength of 2 cm and a radiation pulse duration of 0.4 ms (range resolution of 60 m),which operated in the sector scan mode of Fig.7. The antenna was located at a height of about 11 m. During experiment the sea state was about 6 points (average wind speed exceeded 13 mps). The range of reflection from the sea was from 0.5 km to about 10 km and grazing angles ranged from 1.5° to 0.05°. Photographs were got at different distances such as 0.5-5 km or 5.5-10 km. There is a periodicity of radar reflections associated with a sea wave period as shown in Fig.7. Changing the irradiation direction will lead to change the spatial period of radar images. It is minimum at sea surface irradiation towards the wave and maximum at surface irradiation along waves. At the distance of up to 2-2.5 km the sea surface reflections from rough sea has a clearly defined spatial periodic structure with a period determined by the projection of spatial period of sea waves in the direction of irradiation. At irradiation towards waves the spatial period of reflections is about 120-130 m. Long-range reflections are sporadic and only crests of the highest waves are visible. The density histogram of radar images of the sea surface for a zone is bimodal as shown in Fig.7.

Fig.7 Radar images of the sea surface of the wave of 2 cm at a strong breeze with the sea disturbance number: 6,density of value distribution and their approximation by Kravchenko and Gaussian functions: a,b) radar images,scanning speed of 4 (°)/s; c,d) approximation by Kravchenko functions; e,f) approximation by Polly-Gaussian distribution; a,c,e) 1 scan,sector scan of 120°,range of 0.5-5 km; b,d,f) 2 scans,range of 0.5-5 km

The first maximum of the histogram corresponds to the zone of weak reflections; the second maximum corresponds to the zone of strong reflections (crests of waves). At long ranges the difference of maximum densities of distribution for levels of “black” and “white” is so big that there is only one maximum-the “black” level. Densities approximation of distribution obtained experimentally can be got by

(10)

wherepkis the final probability of each ofkphase states; and in generalk∈(1,2,3),wherek=1 is the black level,k=2 is the grey level,k=3 is the white level.

The Gaussian densities of distribution are used as approximation functional,that is

(11)

and Kravchenko[21-24]finite functions is

a=2.

(12)

Calculation of the distribution density by atomic functionup(x) can be made by the inverse Fourier transform of a characteristic function obtained as a product of characteristic functions of rectangular pulses,or by calculating the density of distribution of a series formed by the sum of a series of random numbers with uniform distribution laws[21-24]. The latter approach is used for obtaining the basic mother functionup(x). The derived random quantity is scaled by coefficientsbkand is shifted perIk. In this way the valueIwith the density of distributionφk(I) used for experimental data approximation is formed. The key is to obtain a minimum error variance. Parameters (pk,Ik,bk) are determined for atomic functions and (pk,Ik,σk) are for Gaussian functions,which can minimize the error variance. It can be seen that both approximations have similar results. However,in about 1/3 cases for density and in a half of cases for distribution function of radar images of the sea surface,the finite atomic functions can get better results than the standard Gaussian function.

4 Conclusion

The empirical relation for calculation of the seaSRCSat millimeter and centimeter wavelengths taking into account the effect of wind speed,the angle of incidence and polarization of the radiation field has been obtained. The simulation model of the signal scattered by the sea has been proposed on the basis of semi-Markov nested processes. It is shown that the reflected signal at spikes and pauses can be locally described by the Gaussian model and the finite Kravchenko[21-24]functions.

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