Brief review on PE method application to propagation channel

Cent. Eur. J. Eng. • 2(1) • 2012 • 19-38 DOI: 10.2478/s13531-011-0049-y

Central European Journal of Engineering Brief review on PE method application to propagation channel modeling in sea environment Review article Irina Sirkova∗ Microwave Physics and Technologies Laboratory, Institute of Electronics, Bulgarian Academy of Sciences Blvd. Tzarigradsko chaussee 72, Sofia 1784, Bulgaria

Received 5 July 2011; accepted 2 September 2011 Abstract: This work provides an introduction to one of the most widely used advanced methods for wave propagation modeling, the Parabolic Equation (PE) method, with emphasis on its application to tropospheric radio propagation in coastal and maritime regions. The assumptions of the derivation, the advantages and drawbacks of the PE, the numerical methods for solving it, and the boundary and initial conditions for its application to the tropospheric propagation problem are briefly discussed. More details are given for the split-step Fourier-transform (SSF) solution of the PE. The environmental input to the PE, the methods for tropospheric refractivity profiling, their accuracy, limitations, and the average refractivity modeling are also summarized. The reported results illustrate the application of finite element (FE) based and SSF-based solutions of the PE for one of the most difficult to treat propagation mechanisms, yet of great significance for the performance of radars and communications links working in coastal and maritime zones – the tropospheric ducting mechanism. Recent achievements, some unresolved issues and ongoing developments related to further improvements of the PE method application to the propagation channel modeling in sea environment are highlighted. Keywords: Radio propagation channel modeling • Parabolic equation method • Tropospheric ducting • Numerical methods

© Versita sp. z o.o.

1.

Introduction

Knowledge of the propagation characteristics of the radio channel is essential for the design and, subsequently, the performance of different wireless communications and radar systems. The propagation conditions in coastal and maritime regions are often complicated by the high variability of the meteorological parameters and sea surface state [1–3], which makes the preliminary assessment of microwave propagation in those regions difficult and sub∗

ject to significant errors [4, 5]. The coastal orography and sea breeze can create sharp gradients in air temperature and water vapor leading to respective changes in the vertical gradient of the tropospheric refractive index n. The microwave propagation in coastal and maritime areas is strongly affected by a specific clear-air propagation mechanism – the formation of tropospheric duct. The tropospheric duct is due to deviation in tropospheric refractivity N (N = (n−1)106 ) from the standard conditions. The appearance of negative vertical gradient of the modified refractivity M (defined as M = N + (z/ae )106 , with z the height above the surface and ae – the Earth’s radius) indicates the presence of tropospheric duct [2, 6]. When ducting occurs, the electromagnetic energy is trapped be-

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• •

the duct) signal by weather radars [10, 13, 14]; is the cause of unexpected interference between satellite and/or terrestrial communications systems [15-18], wrong determination of the angle of arrival in the case of satellite to ship communications [19] and errors in target position estimation by surveillance radars [20]. Brief review on PE method application to propagation channel modeling in sea environment is a major cause for multipath propagation, thus leading to short-term fading, inter-symbol interference and increase in data bit-error-rate in mobile communications [21].

the summer months and during the daytime [8, 22]. On a worldwide basis surface-based ducts are known to appear about 14 % of the time, in certain regions – up to 50 % of the time [22, 23]. As for Europe, the climatology summarized in von Engeln and Teixeira [22] and Lopez [24] shows high ducting probability mostly in the Mediterranean region (see also Bech et al. [13]); some areas in the temperate zone are especially prone to anomalous propagation: the Black sea [25, 26], the British Channel, and East Anglia [27–29]. Even the northern parts of Europe are affected: Figure 1. M(z) profiles for: (a) evaporation duct; (b) surface duct; (c) Fig. 1 M(z) profiles for: (a) evaporation duct; (b) surface duct;thickness. surface-based duct; (d) elevated duct; zd – duct data for duct occurrence and parameters has been reported (c) surface-based duct; (d) elevated duct; zd – duct thickness. for two coastal and one maritime radiosondes station in The evaporation duct is practically always present over the ocean at lower latitudes, its thickness increases Norway [30]; evaporation ducting over Baltic Sea has also ring the summer months and during the daytime [8, 22]. On a worldwide basis surface-based ducts are known to tween the top of the region of negative gradient (the trapbeen reported [31, 32]. Most affected by ducted propagapear about 14 % of the time, in certain regions - up to 50 % of the time [22, 23]. As for Europe, the climatology mmarized in von Engeln andlayer) Teixeira and Lopez [24] at shows highM ducting probability ping and[22] either the level which exceeds the mostly tion in arethethe frequencies above about 1 GHz up to 20 GHz editerranean region (seevalue also Bech et al.top [13]); areas in thelayer temperate zonecase are especially prone to anomalous at the of some the trapping (in this elevated (for higher frequencies other propagation mechanisms may opagation: the Black sea [25, 26], the British Channel, and East Anglia [27-29]. Even the northern parts of Europe duct is formed) or the Earth’s surface (surface-based duct). play predominant role). e affected: data for duct occurrence and parameters has been reported for two coastal and one maritime diosondes station in Norway [30]; evaporation ducting over Balticduct Sea (surface has also been reported 32].above Most statistics and effects due to ducting make the A special case is the evaporation duct) for [31,The fected by ducted propagation are the frequencies above about 1 GHz up to 20 GHz (for higher frequencies other which the negative gradient of M is due to evaporation precise radio channel modeling in coastal and maritime opagation mechanisms may play predominant role). from large of make waterthe[7–10] it is typicalin for The above statistics and effects duebodies to ducting preciseand radiothus channel modeling coastal and maritime regions an important task requiring accounting for sitegions an important task requiring site-specific environmental parameters coastal andaccounting maritimeforregions (the different physical mech-and application specific of environmental parameters and application of sophisticated propagation prediction methods. During the last decades a large range of wave propagation problems anisms of ductsparameters formation not addressed in this paper, phisticated th great variety in geometry and electrical hasare been successfully modeled using the Parabolic Equation propagation prediction methods. During the E) method [33, 34]. This to the wave equation has the advantage being easily forparaxial them approximation the reader is referred to Reddy and Reddy of[3], last solved decades a large range of wave propagation problems merically and provides accurate solutions for complex propagation environments when rays-based [2, 35] or Babin et al. [7], and Mesnard and Sauvageot [10]). The with great variety in geometry and electrical parameters rmal-modes based [35, 36] techniques are unreliable or too resource consuming. The PE uses meteorological types of arecomplex schematically presented Fig. 1.ofAccordhaspatterns. been successfully modeled using the Parabolic Equarameters as input, allowsduct application boundary conditions andon inclusion different antenna ue to these characteristics operability, become a preferred method tion for solving ingand tothe thedemonstrated vertical gradient of the M, PE thehastropospheric condi(PE) method [33, 34]. This paraxial approximation icrowave propagation prediction/assessment problems requiring simultaneous accounting for terrain irregularities tions are classified in four types [1, 2, 6] given in Table 1 to the wave equation has the advantage of being easd clear air propagation mechanisms such as tropospheric ducting [16, 27, 29, 31, 32, 37-52]. The goal of this work is tothe summarize the PE method andare its given application to tropospheric radio numerically and provides accurate solutions for where respective values for theory dN/dz as well. ily solved opagation modeling in coastal and maritime environments. A the PE is given Section 2.1, The values dM/dz = 118 km−1brief andintroduction dN/dz =to−39.2 km−1 in complex propagation environments when rays-based [2, 35] companied with short discussion on the assumptions of its derivation, its advantages and drawbacks, and the the standard troposphere. normal-modes based [35, 36] techniques are unreliable merical methods used correspond for solving it.toSection 2.2 provides more details on the most widely usedornumerical chnique, the split-step Fourier-transform (SSF) method,mechanism to solve the PE, discussing also the boundary or condition at The ducting propagation seriously affects radio too resource consuming. The PE uses meteorologicommunications links and radars’ performance: cal parameters as input, allows application of complex boundary2 conditions and inclusion of different antenna • it is responsible for the formation of radar “holes” patterns. Due to these characteristics and the demon[11, 12]; strated operability, the PE has become a preferred method for solving microwave propagation prediction/assessment • gives rise to difficulties in the rain clutter differenproblems requiring simultaneous accounting for terrain irtiation from the anomalously propagated (trapped regularities and clear air propagation mechanisms such as in the duct) signal by weather radars [10, 13, 14]; tropospheric ducting [16, 27, 29, 31, 32, 37–52]. The goal of this work is to summarize the PE method • is the cause of unexpected interference between theory and its application to tropospheric radio propagasatellite and/or terrestrial communications systems tion modeling in coastal and maritime environments. A [15–18], wrong determination of the angle of arrival brief introduction to the PE is given in Section 2.1, acin the case of satellite to ship communications [19] companied with short discussion on the assumptions of and errors in target position estimation by surveilits derivation, its advantages and drawbacks, and the nulance radars [20]; merical methods used for solving it. Section 2.2 provides more details on the most widely used numerical technique, • is a major cause for multipath propagation, thus the split-step Fourier-transform (SSF) method, to solve leading to short-term fading, inter-symbol interferthe PE, discussing also the boundary condition at the ence and increase in data bit-error-rate in mobile Earth’s surface, the radiation boundary and initial condicommunications [21]. tions of the problem. In Section 2.3 the reader may find The evaporation duct is practically always present over information on the validation of the PE, some PE-based the ocean at lower latitudes, its thickness increases during propagation modeling tools, as well as further discussion

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I. Sirkova

Table 1.

Types of refractive condition.

Refraction types

sub-refraction

normal

super-refraction

ducting

dM/dz, M-Units/km dN/dz, N-Units/km

dM/dz > 157 dN/dz > 0

157 ≥ dM/dz > 78 0 ≥ dN/dz > −79

78 ≥ dM/dz > 0 −79 ≥ dN/dz > −157

dM/dz ≤ 0 dN/dz ≤ −157

on the limitations of the two-dimensional (2D) PE. Section 2.4 considers briefly the environmental input to the PE, especially the methods for tropospheric refractivity profiling, their accuracy, limitations, and the average refractivity modeling. Examples for the PE application are reported in Section 3. The last Section highlights some further developments and unresolved issues related to the PE application in coastal and maritime regions.

2.

Theoretical background

2.1.

Introduction to the PE

The PE is a paraxial approximation to the Helmholtz equation that assumes the electromagnetic field predominantly propagates in some preferred direction, say, the x-axis in a Cartesian coordinate system. This approach, firstly described in Leontovich and Fock [53], with practical application initially in underwater acoustics [54], was for the first time applied to tropospheric propagation in Ko et al. [55]. The most widely used 2D narrow-angle forwardscatter scalar form of the PE (called also standard PE) applied to the tropospheric electromagnetic propagation is given by (1) [33, 34]:


∂U(x, z) i ∂2 U(x, z) ik 2 = m (x, z) − 1 U(x, z), (1) + 2 ∂x 2k ∂z 2 where k is the free-space wave number, m = 1 + M · 10−6 is the modified refractive index of the troposphere, U(x, z) is the reduced or slow-varying along the preferred propagation direction function, x and z stand for range and altitude, and exp(−iωt) time dependence is assumed. Equation (1) holds for both horizontal and vertical electric dipole source fields, the difference between them being contained in the boundary conditions at the Earth’s surface. The modified refractive index m is introduced, besides the immediate indication of the duct formation provided by its negative gradient, for one more reason: it allows the use of the flattened-Earth concept [2] when one is interested in low altitudes above the Earth’s surface. Assuming the problem has azimuthal symmetry, Eq. (1) is obtained from the scalar wave equation in usual spherical coordinates (r, Θ, φ) with origin at the center of the

Earth and source located along Θ = 0. To obtain (1), the following steps are to be carried out: • introduction of variable transformation leading to the rectangular coordinates (x, z); • definition of the reduced function U(x, z) by factoring out the rapid fluctuations (the term exp(ikx)) of a transverse field component Φ (E or H depending on the polarization, see Eq. (2) below); • application of the far-field approximation (i.e., assuming kx  1) to the obtained wave equation in (x, z) coordinates, and; • application of the paraxial approximation (which means assuming that |∂2 U/∂x 2 |  k|∂U/∂x|). For the complete derivation of (1) the reader is referred to Levy [33], and Kuttler and Dockery [34]. The tropospheric propagation problems normally extend to heights and ranges for which x, z  ae . Under this condition the rectangular coordinates are related to the spherical coordinates through the Earth-flattening change of variables, i.e., x = ae Θ, z = r − ae [34]. This allows (x, z) in (1) to be thought of as range and altitude (with z = 0 at the Earth’s surface and x = 0 at the antenna location). The reduced function U(x, z) is related to the transverse electric or magnetic field component as [33]:


1 U(x, z) = exp(−ikx)Eφ (r, Θ) r sin(Θ) 2 ,  1 r sin(Θ) 2 U(x, z) = exp(−ikx)Hφ (r, Θ) . n2

(2)

The far-field assumption is not a restriction to the applicability of the PE to the tropospheric microwave propagation where one is normally interested in far-field calculations. The paraxial approximation limits the accurate representation of the field to propagation directions close to the horizontal. The elimination of the term ∂2 U/∂x 2 implies that the backscattered field is neglected. Also, this approximation requires U(x, z) (and, consequently, n(x, z)) to be a slowly varying function in the direction of preferred propagation. The microwave propagation through the lower troposphere involves electromagnetic field variations over scales much larger than the wavelength and

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Brief review on PE method application to propagation channel modeling in sea environment

(often) small incident angles. Under these conditions the forward-propagated field plays a dominant role [44]. This, with the smooth horizontal variation of the tropospheric refractive index n, assures the applicability of the paraxial approximation and (1) to the tropospheric propagation modeling. Equation (1) has only a first derivative in x. This determines its advantage in comparison to the usual (elliptic) wave equation: (1) is easily solved numerically through marching algorithms, provided the field is known on an initial plane and appropriate boundary conditions on the boundaries in the domain of interest are imposed. The PE given by (1) is very accurate at angles within ±15° of the preferred direction of propagation [56]. Some problems require larger propagation angles to be considered, such an example is the propagation over irregular terrain with steep slopes. The splitting of the far-field approximation to the wave equation into two equations [33] leads to a more general form of the PE for the function U(x, z) that allows wide-angle applications: 

√ ∂ + ik(1 − 1 + Z ) ∂x



 √ ∂ + ik(1 + 1 + Z ) U = 0, ∂x (3)

where Z = A + B, A = (1/k 2 )∂2 /∂z 2 , B = m2 − 1. The first and second terms in (3) describe the forward and backward propagating fields, respectively. The applicability of (3) to larger angular sectors depends on the approximation assumed for the square root operator and the numerical scheme used [33, 57]. If one neglects the backscattered field and assumes the simplest approximation, i.e., (1 + Z )1/2 ≈ 1 + Z /2, (3) transforms into (1). Three main numerical techniques to solve (1) have become popular: the SSF-based algorithms [34, 44–48, 58], finite difference (FD) based algorithms [19, 33, 59, 60] and algorithms using the finite element (FE) method [41–43, 61–65]. The choice of the PE solution method depends strongly on the specific propagation scenario. The FD- and FEbased numerical schemes permit more flexibility in the implementation of various boundary conditions. The SSF method has proven to be more stable numerically, thus allowing larger step sizes in comparison to the two previous techniques and shorter computational time. The last has made the SSF a preferred method for solving long-range tropospheric radiowave propagation problems. The FE-

and FD-based solutions to (1) are sketched below; more details are provided in Section 2.2 on the SSF solution to the PE. In all following Sections, although the discussion is essentially based on the SSF solution to the PE, literature references and examples related to specific issues concerning the FD- and FE-based solutions are also provided. For FE-based solution one usually makes use of the standard FE method as described in Jin [66]. This solution encompasses the following steps: • the one-dimensional cross-section of (1) is divided into line segments (elements); • in each element the unknown solution U is expressed through suitably chosen (i.e., leading to sparse matrices) basis functions; • the weak Galërkin approximation is applied to (1) and the expansion for U is substituted into the weighted residual for each element; • a summation of the residuals over the elements is performed to obtain the system of equations over the entire cross-section; • the solution at range x +∆x is obtained from that in range x applying the Crank-Nicholson algorithm to the matrix form of the entire cross-section system of equations. In Sirkova and Hernandez-Figueroa [61] this procedure has been applied to the microwave propagation modeling in surface duct formed over smooth perfectly conducting earth surface. More complex boundary conditions have been introduced in Apaydin and Sevgi [42] where a FEbased solution to (1) is developed to study surface wave propagation along mixed sea-land-sea paths. In Arshad et al. [64] wide-angle FE-based PE is applied to the propagation modeling over irregular terrain and on urban streets. To construct a FD-based solution of (1) the first step is to cover the domain of interest with a grid of dimensions ∆x, ∆z and indexes j and i in x and z directions, respectively. In order to obtain the solution at range xj+1 from that at range xj , one considers initially the solution at a midpoint (xj + ∆x/2, zi ) while approximating the partial derivatives in (1) with finite-difference expressions:

∂U(xj + ∆x/2, zi ) U(xj+1 , zi ) − U(xj , zi ) ≈ ∂x ∆x ∂2 U(xj + ∆x/2, zi ) U(xj + ∆x/2, zi+1 ) + U(xj + ∆x/2, zi−1 ) − 2U(xj + ∆x/2, zi ) ≈ , ∂z 2 ∆z 2

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(4) (5)

I. Sirkova

where all the involved points are inside the integration domain; to complete the system of equations, expressions for the boundary conditions at the top and bottom of the computational domain should be added. The next step is to approximate U at the midpoint range xj + ∆x/2 by averaging the values at the neighboring ranges xj and xj+1 : U(xj + ∆x/2, zi ) ≈

U(xj+1 , zi ) + U(xj , zi ) 2

based solution, is of Crank-Nicholson type. For details on FD-based implementation of the PE the reader is referred to Levy [33] and Marcus [59].

2.2.

The SSF technique makes use of the fact that the Fourier transform of Eq. (1) has a simple solution at x+∆x in terms of the solution at x, provided ∆x is sufficiently small so that the refractive index may be considered as a constant in this interval. The inverse Fourier transform is then applied to the solution of the transformed equation and the desired U(x + ∆x, z) is obtained as (8):

(6)

Finally, one is able to express the values of U at range xj+1 as a function of values at range xj in the form of a linear system of equations which, similarly to the FE-

Z e p) = ={U(x, z)} ≡ U(x, −6 ∆x

U(x + ∆x, z) = eikM(z)10

where (7) gives the appropriate Fourier transform, p is the transform variable (referred to as the vertical wave number), the first exponent in (8) denotes the environment propagator and the second, the free space propagator (i.e., the diffraction effects are decoupled from the refractive effects). Note that (8) is an exact solution to (1) for M = constant; the error introduced in (8) for M = M(x, z) and the related limitations on ∆x are discussed in Kuttler and Dockery [34]. To obtain U(x, z) from (8) at every range step, both forward and inverse Fourier transforms are needed. Also, (7) requires knowledge of the field above and below z = 0. For smooth, perfectly conducting earth/sea surface, the boundary conditions for horizontal polarization and vertical polarization are satisfied if U(x, z) is odd or even, respectively, about z = 0; in Dockery [67] this symmetry has been extended to impedance-type boundary condition for smooth finitely conducting surfaces. The symme-

−6 ∆x

U(x + ∆x, z) = eikM(z)10

SSF solution to the PE

∞

U(x, z)e−ipz dz

−∞

=−1

n

o 2 e−i(p /2k)∆x ={U(x, z)} ,

(8)

try means that only the field for z ≥ 0 is to be considered in the real calculations. Actually, at any range the transverse field is obtained by moving back and forth between z and p domains via the fast Fourier transform (FFT) and inverse FFT that calculate the real sine and cosine transform instead of the full Fourier transform; this ensures the effectiveness of the SSF method. The application of the discrete Fourier transform imposes limits to the transform variable p (p = k ·sin(Θ)), with Θ the local angle from the horizontal: −pmax < p < pmax ; pmax determines the maximum angle to be used in the calculations and is related to the mesh spacing by the Nyquist criterion [45]. The approximate splitting of the square root operator in the forward propagating part of Eq. (3) as (1 + Z )1/2 = (1 + A + B)1/2 ≈ [(1 + A)1/2 + (1 + B)1/2 − 1] leads to a wideangle forward-propagating PE for which the application of the SSF procedure gives the solution (9), where the second exponent denotes the wide-angle propagator [33]:

  √   −i k− k 2 −p2 ∆x =−1 e ={U(x, z)}

For a discussion on the conditions and assumptions needed to decouple the diffraction and refraction effects in (9) the reader is referred to Levy [33] and Kuttler [68], where numerical and analytical examinations and com-

(7)

(9)

parisons of the narrow and wide-angle propagators (for n2 = 1) are done. The solution to (1) must satisfy the boundary condition at the earth surface as well as the radiation condition at

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Brief review on PE method application to propagation channel modeling in sea environment

infinity. Equation (10) provides the impedance boundary condition for the transverse field components [34]: ∂Φ + γh,v Φ|z=0 = 0 ∂z z=0 γh,v = ik sin(α)

1 − Reff , 1 + Reff

Reff = RFh,v Rrfac ,

(10)

(11)

where γh,v is a parameter related to the surface impedance with subscripts h, v referring to horizontal and vertical polarization, α is the local grazing angle, Φ is the respective field component depending on the polarization, Reff stands for the “effective” reflection coefficient [33], RF is the smooth-surface Fresnel reflection coefficient, Rrfac is a roughness reduction factor (RRF), see Eqs. (14), (15) and (16). Relation (11) provides direct relation of γh,v to the surface reflection coefficient, thus allowing the introduction of fine-scale roughness effects through the substitution of the smooth-surface Fresnel reflection coefficient by the “effective” reflection coefficient Reff . This approach is suitable when sea surface roughness is to be considered. The SSF solution of (1), combined with the impedance boundary condition (10), (11), was firstly derived in Kuttler and Dockery [34] (referred to as Mixed Fourier Transform or MFT) and further improved in Kuttler and Dockery [69] and Kuttler and Janaswamy [70] by proposing a more effective method in the sense of numerical stability and number of transforms DMFT (Discrete Mixed Fourier Transform) algorithm. The application of boundary conditions (10) and (11) requires knowledge of the grazing angle α at the next step in order to advance the solution to that step. Note that the PE solutions do not provide the grazing angles in a straightforward manner. Common approaches to the local grazing angles problem are: 1) application of the spectral estimation (SE) [71] technique at each range

step to determine the dominant grazing angle; or 2) application of an internal ray-tracing (RT) code to estimate the grazing angles. The RT is used in the case of evaporation duct, whereas SE is better suited for terrain applications; see the discussion on the properties of those methods in Kuttler and Dockery [69], Kuttler and Janaswamy [70], and Dockery et al. [72]. The littoral zone includes both variable terrain and ducting and obtaining α for this scenario requires combination of both SE and RT [72, 73]. In Barrios [45] the accounting for larger scale roughness (i.e., the terrain irregularities) with the PE method is done (for horizontal polarization and boundary condition for perfectly reflecting surface) by applying a general coordinate transformation [74] that flattens the surface elevations and introduces an additional term, responsible for the terrain effects, in the modified refractive index. This transformation retains the PE for the reduced function redefined in the new coordinates in the form of Eq. (1). Although this is a narrow angle equation, in Barrios [45] a solution with wide-angle propagator as in (9) has been conjectured and successfully applied. In Donohue and Kuttler [57] a terrain flattening transformation (referred to as “shift map”) is developed, which extends Beilis and Tappert [74] to obtain a proper wide-angle forward PE for the transformed in the new coordinates field function Ψ: s ∂Ψ ∂2 e z)Ψ = i k 2 + 2 Ψ + ik m(x, ∂x ∂z

e z) accounts not only for the Earth’s curvature where m(x, but also for the terrain elevation and x, z refer now to the new height and range variables. The SSF developed for flat boundaries is applicable to (12) and provides the solution with wide-angle propagator in the form of (13) [57]:

√ n  o 2 2 e 1 ,z)∆x −1 e 1 ,z)∆x Ψ(x + ∆x, z) = ei(k/2)m(x = ei∆x k +p = ei(k/2)m(x Ψ(x, z) ,

where x1 is in the middle of the interval ∆x. Note that the type of the Fourier transform in (13) is a function of the boundary conditions; the application of the “shift map” technique for finitely conducting boundary with elevations approximated by piecewise linear segments is explained and validated in Donohue and Kuttler [57]. It is to be noted also that this technique is valid only if the slope of the terrain irregularity is lower than 15 degrees. For higher slope values, another terrain model, called “staircase approximation”, is applicable: the terrain profile is modeled as a series of flat surfaces located at different

24

(12)

(13)

heights [33]. Above each surface the SSF is applied depending on the boundary condition. The field inside the obstacle is set to zero and the origin of the vertical field in the cross-section is shifted according to the height variations of the terrain. The drawback of this model is that the reflections on the obstacles are not properly taken into account. The fine-scale surface roughness, characteristic for the sea surface, is normally accounted for through the incorporation of the Miller-Brown [75] RM−B , or Ament [76] RA , RRFs in the boundary conditions (10) and (11):

I. Sirkova

    RM−B = exp −2k 2 σξ2 sin2 (α) I0 2k 2 σξ2 sin2 (α)   RA = exp −2k 2 σξ2 sin2 (α)

(14) (15)

where σξ is the standard deviation of the surface height ξ, I0 in (14) is the modified Bessel function of the first kind of order zero, k and α have the same meanings as aforementioned. The comparisons of the propagation prediction results obtained with these two RRFs combined with different propagation models to the data from propagation measurements do not allow concluding which of them is more accurate; see the discussions in [77–79]. Note that both the above correcting coefficients affect only the magnitude of the complex Fresnel reflection coefficient and do not account for the shadowing. In order to explain some discrepancies between the propagation modeling and the observed experimental results, theoretical efforts have been directed towards the improvement of the accuracy in accounting for the effects of ocean roughness on radio wave propagation by introducing the shadowing into the RRF Rrfac . Under grazing incidence angles (typical for ducted propagation) the shadowing may have significant influence on the propagation factor PF [79]. (The PF is defined as the square of the ratio of the electric field amplitude E, received at a given point under specific conditions to the amplitude of the electric field E0 , received under free-space conditions, with the beam of the transmit antenna directed toward this given point, where E participates with its polarization component that coincides with the polarization of E0 [2].) On the basis of a newly derived Probability Density Function (PDF) only for the illuminated points of a one-dimensional rough surface with Gaussian statistics of heights and slopes and for forward propagation, in [80] the following RRF that accounts for the shadowing has been obtained:

Q 2 σeξ2 eξ − Rrf = exp −iQ m 2

! ,

Q = 2k sin(α)

(16)

e ξ , σeξ are the mean value and standard deviation where m of the illuminated surface heights. The RRF (16) is obtained using the same statistics as the one assumed for the derivation of RA given by (15). Expression (16) introduces a phase to the Ament’s correction coefficient RA ; it also corrects its amplitude by the introduction of σeξ instead of σξ . It should be noted that RM−B (14) is derived using PDF obtained for sea surface formed by random variable, whose amplitude has Gaussian distribution with zero mean, and whose phase is uniformly distributed in the interval [−π/2, π/2]. This statistics differs from the one for

e ξ and σeξ in (16) are obtained, i.e., their application which m to the Miller-Brown RRF will not be correct. In Freund et al. [79] an expression for the Rrfac with shadowing is proposed which improves the “unshadowed” RRFs (14), (15) while maintaining their functional form: " Rfit

2kσξ sin(α) = exp − δ


2 # eiχ ,

(17)

where the parameter δ and the function χ are determined via numerical curve fitting, see Freund et al. [79] for details. The numerical solution of (1) requires truncation of the natural infinite propagation domain (the troposphere) to the limited size of the computational window. To avoid any parasitic reflection from this artificially introduced boundary, a transformation of the zero-boundary condition at infinity to the boundary condition at the computational window upper boundary is needed. Two main groups of solutions to this problem are known. The first one (most commonly used with the SSF solution of the PE) consists in the insertion of an artificial absorbing layer adjacent to the concerned boundary. Within this layer, the discretized field is “filtered” applying a suitable attenuation function [42, 45, 58] or the reflection is removed by adding a small imaginary part to the refractive index near the extended computational window boundary [37]. A variation of the absorbing layer approach, also often used with the SSF PE solution, is the Perfectly Matched Layer (PML) technique, firstly proposed for electromagnetic waves application in Berenger [81]. Within the PML, the field is attenuated by means of fictitious dielectric and permeability constants. If properly built, the artificial absorption procedure is very accurate. Its drawback is the need to adjust the absorber parameters to each change in the problem parameters. The absorbing layer increases additionally the integration domain; in the tropospheric case this domain is inherently large. This incited the development of analytical proper transparent boundary conditions (TBCs) that would minimize the magnitude of the reflected field inside the computational window – the TBCs form the second group of solutions to the radiation boundary condition problem. Due to the ability of FD- and FE-based numerical schemes to model different kind of boundaries, the TBCs have been developed for PE solutions based on those schemes. The rigorous treatment of the radiation boundary condition problem leads to non-local TBCs (NTBCs) [33, 59, 82]. The NTBC for narrow-angle PE has been proposed and successfully applied to the microwave tropospheric propagation under ducting conditions in Levy [83] using a FD-based numerical scheme; for a wide-angle NTBC the reader is referred to Zebic-Le Hyaric [84]. Although the NTBCs provide mathematically correct solu-

25

Brief review on PE method application to propagation channel modeling in sea environment

tion and optimize the computational window vertical size zb to the domain of interest, their application is limited by the arduous implementation. To provide operational solution, some compromises in accuracy could be made applying simpler or more complex local TBCs (LTBCs) [85]. The simpler LTBC [85], firstly proposed and used in integrated optics, is validated for FE-based solution of (1) and tropospheric surface duct case in Sirkova [61] and later compared to the NTBC [83] for elevated ducts, evaporation ducts and cascades in Sirkova [62] and Sirkova [86]. The start of the computation procedure for the PE requires knowledge of the initial field U(0, z). It is obtained using the relation between the far-field antenna pattern F(p), which is normally a known function, and the aperture field distribution A(z): F (p) and A(z) form a Fourier transform pair [33, 45, 67] (see also [38] and [87] for the 3D case). The application of the boundary condition for perfectly conducting surface and image theory allow expressing U(0, z) through A(z −za ) and A∗ (z +za ), where za is the antenna height and the asterisk denotes the complex conjugate. By Fourier-transforming this representation of e p) expressed U(0, z) one obtains the reduced function U(0, via F (p): e p) = F (p) exp(−ipza ) − F ∗ (−p) exp(ipza ), U(0, e p) = F (p) exp(−ipza ) + F ∗ (−p) exp(ipza ) U(0,

(18)

where the first of equations (18) refers to horizontal polarization, the second refers to vertical polarization. The required U(0, z) is obtained applying inverse Fourier transform to (18). In (18) the antenna height is included using the Fourier shift theorem; similarly, the antenna elevation angle Θs may be accounted for by placing F (p−ps ), where ps = k · sin(Θs ), instead of F (p). Thus, besides for diffraction, refraction, reflection and scattering mechanisms, the PE accounts also for the antenna pattern which is introduced in a natural way into the SSF numerical scheme. For the FD and FE solutions of PE that are not based on the Fourier transform, the initial field A(z) may be obtained by Fourier transforming the F (p) function; see the near field/far field transformation procedure in Levy [33]. Up to now the forward propagating PE has been considered. To account for the backward scattered field, which may be important in the case of propagation over irregular terrain under anomalous tropospheric conditions, Ozgun [88] proposes a two-way SSF algorithm to solve (1). The algorithm starts from an initial field at x = 0 that is marched out in the x > 0 direction in the usual manner, see (8), until an obstacle (with staircase approximation) is reached. Then the reduced function is split into two components, Uf and Ub , propagating in forward and backward

26

directions, respectively. In the region x < x0 , where x0 is the location of the obstacle, the field is given by:

Φ(x, z) = exp(ikx)Uf (x, z) + exp(−ikx)Ub (x, z)

(19)

The initial field Ub (x0 , z) of the backward-wave is obtained by imposing the boundary conditions for Φ(x, z) at the obstacle. Then the backward-wave is marched out in −x direction by solving (1) through (8), where the signs of k and ∆x are reversed. Each time the field reaches the edge of an obstacle it is split into two waves propagating in opposite directions. At each range step the total field is obtained as a sum of the backward and forward fields. This procedure continues until a given threshold criterion for the difference between the total fields at the nth and (n − 1)th steps is reached.

2.3. PE

Validation of the PE and limitations of 2D

During the last two decades the numerical implementation of the PE solutions has been repeatedly verified through comparisons to other propagation models [33, 34, 45, 65, 67, 69]. The adequacy of the PE method in modeling the radio wave propagation has been also extensively validated by means of comparisons to measured data [27, 31, 32, 89–93]. Both verification and validation refer to different inhomogeneous environments and a broad range of frequencies and propagation scenarios. A quite good agreement between the simulation results obtained with wide-angle SSF PE and GTD (Geometrical Theory of Diffraction) for propagation in standard troposphere over irregular terrain is reported in Barrios [45]; an excellent agreement is found in Dockery [67] comparing the PE and the rigorous normal-mode method for surface duct conditions; a favorable comparison for surface duct over a rough surface (with Miller-Brown RRF) between the SSF PE solution and normal-mode based solution is reported in Dockery and Kuttler [69]. A recent work, Apaydin and Sevgi [65], reports canonical test cases and systematic comparisons between the SSF-based and FE-based solutions to PE (1) as well as calibration of both solutions against analytically exact solutions. The validation of the PE through measurements for the case of maritime propagation requires organization of trials that involve, among other equipment, ships moving on outbound and inbound courses and/or buoys with sensors for meteorological parameters located along the propagation path. Such an experimental investigation on anomalous microwave propagation in a tropical maritime environment is described in Kulessa et al. [91]. A general agreement between the measured and predicted field strength

I. Sirkova

amplitude values, obtained by solving the PE, are observed. Comparisons of the PF calculated through the PE method and obtained via measurements in real conditions over the Baltic Sea (accompanied with environmental parameters collection using sensors mounted on buoys, on a research vessel, and radio probes on balloons) have shown that the PE model gives a good estimation for the PF over a wide range of frequencies (X, Ka and W band) and a variety of atmospheric conditions [31]; some discrepancies are reported concerning the fading: the computed fading was not always as deep as the measured one [32]. Another experiment, the Rough Evaporation Duct (RED) experiment [92, 93], specifically organized to investigate the effects of ocean waves on the meteorological parameters and, hence, on the evaporation duct, has provided also data for the validation of meteorological and propagation models designed for the marine surface layer. Measurements of microwave propagation loss under evaporation duct conditions were compared to prediction estimates based on the PE given by (1). The surface roughness was modeled with Ament RRF (15) where σξ = 0.0051u2 , with u the wind speed in m/s. For λ = 3 cm and a 25.7km over-sea path (for details on the experiment see [93]) the mean and standard deviation of the difference between the observed and modeled propagation loss was −0.2 and 4.0 dB for a transmitter with za = 13 m and 4.3, and 3.4 dB for a transmitter with za = 5 m (receiver at 4.7 m). Similar results for the comparisons are reported for λ = 1.5 and 10 cm as well. Those differences of a few decibels between the observations and model were considered satisfactory. As it has been mentioned, the sophisticated ray methods (GTD, UTD (Uniform Theory of Diffraction)) and normalmodes methods (or hybrid combinations of them), which overcome the difficulties experienced by the simple ray theory in complex propagation environments (caustics, shadow zones, ducted propagation), are not easy to include in routine calculations. The great flexibility demonstrated by the PE has turned its simplest form (1) into the preferred technique for solving tropospheric propagation problems in a number of thoroughly validated and practically applied radiowave propagation assessment tools as: VTRPE (Variable Terrain Radio Parabolic Equation, solution of (1) based on SSF) [94], PCPEM (PC Parabolic Equation Model, SSF based) [37], TPEM (Terrain Parabolic Equation Model, uses SSF) [45], TERPEM (TERrain Parabolic Equation Model, successor of PCPEM, combines PE and ray-trace techniques) [95], TEMPER (Tropospheric Electromagnetic Parabolic Equation Routine, based on DMFT) [69], [72], APM (Advanced Propagation Model, hybrid model that uses ray optics and DMFT-based PE) [73], PREDEM (PREdiction of ElectroMagnetic Detection) [96], AREPS (Advanced Refractive Effects Prediction System) [97]. Some of the limitations of

this widely spread modeling of the 3D environment with 2D PE are discussed below. The 2D PE (1) loses validity when problems involving propagation over terrain with steep transverse gradients leading to substantial lateral (cross-range) effects (diffraction, scattering, depolarization) are to be solved [38, 56]. Given the rigorous 3D modeling of those scenarios with accounting for an inhomogeneous troposphere is extremely difficult (including their treatment with Integral Equation methods as well, see the discussion in [98]), an alternative may be searched for in pseudo-3D PE applications. The full treatment of those problems requires a 3D vector PE [87, 99–101]. In Awadallah [38] (see also Awadallah et al. [87]), a pseudo-3D version of the 2D PE (1) implementation in TEMPER [69] is proposed. The 2D TEMPER code is run on multiple azimuths, with 1D terrain elevations along each azimuth, and then the azimuths are combined over the 3D domain of interest to approximate propagation in a 3D environment. The pseudo-3D TEMPER model has been used to examine the lateral diffraction and scattering effects in two scenarios: an urban environment and propagation over terrain variations with steeper transverse gradients. Then the results for the one-way PF have been compared to a forward-marching 3D Vector Parabolic Equation (VPE) model. As reported by Awadallah [38, 87], the lateral propagation effects are much more pronounced in the urban environment than over the irregular terrain: laterally diffracted and scattered field strengths by vertical building edges can enhance the field strength by 10–20 dB over 2D model predictions whereas in the second scenario the agreement between the VPE model and the pseudo-3D TEMPER model was within 5 dB at high power levels. For spherical coordinates and under the assumption that the atmosphere varies only in the radial direction, in Nunes da Silva et al. [101] the lateral effects due to irregular terrain are studied by expressing the vector electric and magnetic fields in terms of scalar electric and magnetic Hertz potentials. This formulation guarantees divergent-free electric and magnetic fields in the region of interest. Further, the terrain height function is introduced through a mapping that transforms the irregular terrain into a spherical surface, the reduced functions are expressed through the potentials and the parabolic approximation is applied to the modified Helmholtz equation for the Hertz potentials. The potentials, and consequently the field components, are coupled through a vector impedance boundary condition at the irregular surface. Finally, the sparse system of equations, formed by the finite difference version of the above equations resulting from the Crank-Nicholson scheme, is solved by specialized software. The results are similar to those reported by Awadallah et al. [87]. The 3D formulations in the aforementioned works disregard the backscattered and multiple

27

Brief review on PE method application to propagation channel modeling in sea environment

scattered fields, but they clearly illustrate the importance of considering the lateral propagation effects in scenarios with steep transverse terrain gradients between the transmitter and the receiver.

2.4.

Modeling of the environment

As it is clear from the previous sections, the appropriate form of the parabolic approximation and numerical solution to it, chosen according to the specificity of the problem, determines in a great degree the reliability of the acquired results. Not less important for the fidelity of the results is the accuracy in the environmental modeling that serves as input to the PE. This modeling encompasses the availability of a suitable digital dataset for the terrain representation and a reliable model for the non-linear changes of the tropospheric refractivity. Normally the resolution of standard digital terrain databases is sufficient for the terrain representation. More complicated is the matter of the reconstruction of the refractivity profiles; this especially refers to the evaporation duct profiles. The importance of this problem has led, in addition to the well known methods for in situ collection of the meteorological parameters, to the development of indirect techniques to derive the refractivity, such as the use of ground-based GPS (Global Positioning System) measurements [102], Raman lidar system based measurements [103], or RFC (Refractivity From Clutter) algorithms that extract the refractivity from the radar clutter return [104, 105] (in the latter two works PE (1) with SSF solution is used to create the radar replica signals). The remote sensing techniques have the advantage of providing profiles characteristic for the integrated refractive effects along the entire propagation path. Despite the variability of the marine boundary layer, for practical purposes one usually assumes lateral homogeneity for the refractivity and applies a single profile, approximated to account for the average behavior of M(z), along the entire propagation path. As shown in Goldhirsh and Dockery [106], for long distances this assumption can result in significant errors in the PF, especially when paths close to the shore are considered. The solution is to apply a range-dependent approach with multiple profiles taken along the path. Very often the averaged M(z) profile is approximated by piecewise linear curve. This curve is frequently reduced to bi-linear or tri-linear profile to model surface ducts or surface-based and elevated ducts, respectively. This simplification allows pointing out the influence of the basic duct parameters: duct height zd and M-deficit ∆M, ∆M = M(zd ) − M0 , M0 = M(z = 0), in the case of surface duct, and base height, M-deficit and thickness of the trapping layer, for surface-based and elevated ducts. In addition, those parameters are easily extracted from the available duct statistics [26, 107]. The bi- and tri-linear

28

models set the slope above and below the M inversion to 0.118 M-units/m (which corresponds to the standard troposphere). Due to its frequent occurrence and particular importance for ship borne radars and communications, special attention is paid to the evaporation duct modelling. Most often the evaporation duct is modelled by log-linear height profile of the modified refractivity M, with governing parameter the duct thickness zd [108, 109]:    z + z0 M(z) = M0 + 0.13 z − zd ln z0

(20)

where z0 is the aerodynamic roughness parameter usually taken to be 1.5 × 10−4 m [33, 108, 110]. The physics behind this profile, based on the Monin–Obukhov similarity theory, is explained elsewhere [7, 108, 109]. As indicated by Paulus and Anderson [111], (20) is a good practical approximation to the average M(z) profile, and has been widely used as input to the PE propagation model applied under evaporation duct conditions [27, 33, 40, 73, 97, 111– 115]. Equation (20), however, has been obtained assuming thermally neutral troposphere stratification and does not account for the tropospheric stability effects on the M profile [7, 29, 33]. Attempts at improving (20) have been made by changing the slope in different parts of the log-linear curve [116, 117], thus making it to better fit the experimental profiles and, hence, include the influence of tropospheric stability. Recent efforts of the US Naval Postgraduate School to develop a state-of-the-art bulk evaporation duct model (referred to as NPS model) based on measured (or modeled) values of wind speed, air and sea temperature, relative humidity and atmospheric pressure have led to restoration of high-fidelity profiles and evaporation duct heights [118]. However, the NPS model still leaves room for improvements: the model is very accurate under unstable troposphere (when the air above the sea is colder than the sea surface), whereas in highly stable conditions (namely, combination of large positive air-sea temperature difference and low wind and humidity) the NPS model becomes untrustworthy [29]. For further discussion on (20), the evaporation duct models, and comparison between them, the reader is referred to [7, 9, 29, 110, 119]. The above discussed evaporation duct model applied to an open ocean. To account for the spatio-temporal development of the ducting layers in the littoral zones [3, 39, 120], mesoscale numerical weather prediction (NWP) modeling [121, 122] has been used to generate 4D refractivity fields. The generated refractivity profiles may be used as input to the PE model [12, 39, 122]. It should be noted that the current state-of-the-art mesoscale NWP models, as applied to the refractivity field modeling, still provide rather qualitative predictions of duct height and strength; intensive

I. Sirkova

efforts are endorsed towards the quantitative prediction of those parameters [122]. The varied topics of Section 2, together with the most appropriate literature references to each topic, are summarized in Table 2.

3.

Results and discussion

This Section presents results that illustrate some of the above discussed topics, namely: the implementation of the radiation and initial boundary conditions; the ability of PE to incorporate different input data and provide quantitative assessment of essential propagation channel parameters; and the combination of some recent achievements in the modeling of sea surface roughness with the PE method. The reported results are obtained with FEbased and SSF-based solutions to (1) applied to ducting conditions over the sea and refer to the PF and the major parameter used for radar coverage and communication links budget analysis – the path loss (PL). Further discussion on the over-sea propagation modeling is presented as well. The relation between the PL and the PF (both expressed in dB) is given in Eq. (21), where the expression of the PF in terms of the reduced PE field U(x, z) is reported as well [33]:



 PL = 20 log

(a)

4πr λ

− PF , (21)

(b)

PF = 20 log |U(x, z)| + 10 log(r) + 10 log(λ) Figure 2. where λ is the free-space wavelength, r is the distance between the corresponding points, and the first term in the right-hand side of the expression for the PL is the free-space loss. The initial field, required to start the PEbased calculations, is often provided by a source with normalized Gaussian beam pattern (22) leading to an aperture function of the form (23): 



 ln(0.707)(θ − θs )  F (θ) = exp    2 2

θ0 2

A(z) = e−B(z−za ) , 2

B=

−k 2 θ02 4 ln(0.707)

(22)

(23)

where θ0 and θs are the half power beamwidth and the antenna elevation angle. The Gaussian beam antenna is used to provide the initial field for Fig. 2 and Fig. 3, where the FE-based PE solution (used for Fig. 2) applies (23), whereas the SSF-based PE solution (used for Fig. 3)

Path loss in dB for a cascade of two M inversions over a smooth sea surface with LTBC (a) and NTBC (b) applied at zb = 500 m.

applies (22) with (18). Figures 4 and 5, for which the SSF-based PE solution is used, are obtained for initial field provided by an omni directional antenna (i.e., in (18) F (p) = 1 for all angles θ). Figures 2(a) and 2(b) present PL for piecewise linear M profile (with parameters given in Table 3) with radiation boundary condition assured through LTBC and NTBC application to the upper computational window boundary (zb = 500 m), respectively. The profile from Table 3 gives a cascade of two inversions, the first one forms a surface duct with zd = 50 m, the second one is responsible for an elevated duct with zd = 226 m situated between 74 m and 300 m (trapping layer between 200 m and 300 m). The other parameters are: θ0 = 10 , θs = 00 , za = 20 m, transmitting frequency F = 10 GHz, horizontal polarization, smooth perfectly conducting sea surface. The com-

29

Path loss, dB

Path loss, dB

Division Multiple Access-Frequency Division Duplex) downlink band, f=fDLmin=2112.4 MHz, za=30 m, θ0 =50, θs =00, horizontal polarization, perfectly conducting smooth sea surface, and fixed height for the receiver zr=20 m. The computation is based on the APM [73]. Figure 3 (a) shows comparison between PL variations obtained for Mreview on PEparameters. method application to propagation channel sea statistics, environment profiles based on monthly mean values of theBrief surface-duct It should be noted that,modeling for the in used zr is always “submerged” within the duct; the same is true for za except for January, March and May – this implies noticeable influence of ducting on the PL, especially for long range paths. For distances greater than 6 km, PL under ducting differs significantly from month to month and may exceed or be lower than the standard troposphere case, reaching a difference morecorrelation than 30 dBbetween for September and r=7.8 km. In this study, September is characterized by oration duct). In this case there is muchofbetter measured Table 2. Summary of the solutions and environmental input to the PE. moderate zd, but also by the strongest ∆M among the other months. Due to the bi-linear profile of M, months with solutions References demonstrate similar Figure similar ratio ∆M/zdusing rsus range for fixed height obtained as PE input to the PL PE behavior. surface-duct M 3 (b) depicts PL variations for seasonal mean surfaceduct parameters. For summer and autumn months PL variations compared to 70, standard troposphere case exceed 15 narrow-angle [33, 57, 58, 69, 88] a two year statistical study 34, 44–48, near model, and data for zd and ∆M derived fromSSF wide-angle [33, 45, 57, 68, 84] dB for some ranges (r=9.8 km, r=12 km, respectively). Comparisons between PL obtained for M-profiles based on ul. These statistical data are based on radiosonde measurements recorded at a [41–43, 61–65] [52]. For more detailed investigation, a longdaytime nighttime mayisbechosen found in Sirkova e Bosporus strait [26]. Theand parameters formean Fig.2Dduct 3 FE are:parameters the frequency FD [19, 33,duct 59, 60] term Mobile duct statistical study differentiating between different types and accounting for range-dependent duct Telecommunications System Wideband Code MA-FDD (Universal Validation [27, 31–34, 45, 65, 67, 69, 89–93] 0 this study shows: a) the PL differences of scores of dB due thickness and strength is needed. Even though tentative, MHz, zpropagation =5 , [37, θs 45, 69, 72, 73, 94–97] Division Duplex) downlink band, f=fDLmin=2112.4PE-based a=30 m, θ0tools surface ducting will affect the link budget and increase the requirements to the WCDMA power control range; b) conducting smoothtosea surface, and fixed height 3D for the receiver zr=20 m. The [38, 56, 87, 99–101] depending on the specific climatic characteristics, the Muse of annually averaged (and even seasonally averaged) 3]. Figure 3 (a) shows comparison betweenEnvironmental PL variations obtained for input to the PE [7, 9, 29, 92, 108–111, 116–119, 121, 122] values for the duct parameters, is athe common practice, may be not enough for accurate PL prediction. of the surface-duct parameters. It should be notedwhich that, for used statistics, uct; the same is true for za except for January, March and May – this implies Jan, black PL, especially for long range paths. For distances greater than 6 km, PL under Feb, blue Table 3. Parameters for piecewise linear modified efractivity profile. th to month and may 150 exceed or be lower than the standard troposphere case, 150 StanTrop StanTrop Mar, cyan (b) dB for September and r=7.8(a)km. In this study, September is characterized by Height, m 0 50 200 300 1300 Apr, magenta 140 140 ∆M among the other months. Due to the bi-linear profile of M, May, months yellowwith M, M-units 5 0 17.7 2.7 120.7 Jun, purple r PL behavior. Figure130 3 (b) depicts PL variations for seasonal mean surface- 130 umn months PL variations 120 compared to standard troposphere case exceed 15 120 km, respectively). Comparisons between PL obtained for M-profiles based on Jul, red 110 Winter, black 6 and 15 therein). Aug, orange meters may be found 110 in Sirkova [52]. For more detailed investigation, a long- seen in Gunashekar et al. [27] (see Figs. Spring, red Sep, green 100 ing between different100duct types and accounting for range-dependent duct A comparison is made on the measured and PE-predicted Summer, green Oct, gray SSF PE solution and AREPS)Autumn, signalblue strength for though tentative, this 90 study shows: a) the PL differences of scores of dB due (using 90 Nov, pink budget and increase the requirements to the WCDMA power control GHz radio waves propagation over long-range sea path Dec, range; black b) 2 80 haracteristics, the use 80 of 0annually averaged in 0the British Channel First in [27]) 2 4 6 Islands. 8 10 (Fig. 12 6 14 16 the 2 4 6 (and 8 even 10 seasonally 12 14averaged) 16 s a common practice, may be not enough Range, for accurate Range, comparison is made only usingkmdata for evaporation duct km PL prediction.

Path loss, dB

(with M(z) according to (20)) as input to the PE – there Fig. 3 (a) Comparison(a)between path losses obtained for is M-profiles based on monthly mean values of duct Jan, black little correlation between measurements and prediction Feb, blue parameters; (b) Comparison between path losses obtained for M-profiles for seasonal mean duct parameters. 150 and no coincidence between them. Later (Fig. 15 in [27]) StanTrop Mar, cyan (b) the comparison is made including in the input to the PE, Apr, magenta 140 Searching to improve the propagation modeling and increase the agreement between predicted and measured May, yellow besides the evaporation duct, also higher-level refractivity 130 a number of authors have combined the PE (1) with different RRFs and applied it to simulate over-sea Jun, purple fields,

(that is, some similarly as in proposed Fig. 2, applying one or more propagation in various scenarios [112-115]. As discussed in data Section 2.2, recently RRFs account for 120 M-inversions above the evaporation duct). In this case the shadowing caused by the sea surface roughness on electromagnetic waves propagating under grazing incidence Jul, red 110 there(16), is much better correlation between on measured and black through RRF introduced on the PF and PL is illustrated Fig. 4 and Aug, orange angles. The influence of the shadowing, Winter, Spring, red predicted signals. Sep, green Fig.100 5. Figures 4 (a) and (b) show comparisons between PFs obtained with original Ament (15) and phase corrected Summer, green Oct, gray Ament (16) RRF, and original Ament and Miller-Brown (14) respectively, forPLevaporation duct Fig.RRF, 3 reports results for versus range for conditions. fixed height 90 Autumn, blue Nov, pink = 14 m, z = 5 m, F = 5 GHz, range from the source r =M5 km, The other parameters are: M(z) given by (20) for z d a obtained using as input to the PE surface-duct proDec, black 80 ~ ~ u = 7 m/s (u is the wind speed at 10 meters above the sea surface; about the relation of u to , from mξ model, σ ξ and (16) 0 210 4 6 8 10 12 14 16 12 14 16 10 files reconstructed through the 10 bi-linear data Range,with km surface impedance characteristics see [80]), vertical polarization calculated as a afunction frequency in for zd and ∆M derived from two yearofstatistical study accordance with [123]. The(b)computation is based on the APM Close to formation the sea surface, Fig. 4 (b), the statistiAment on [73]. surface ducts over Istanbul. These ath losses obtainedand for Miller-Brown M-profiles based on monthly mean values of duct RRFs lead to almost identical results forcal thedata PF are for based the used scenario; the difference between on radiosonde measurements recorded en path losses obtained for M-profiles for seasonal mean duct parameters. Figure 3.

(a) Comparison between path losses obtained for Mprofiles based on monthly mean values of duct parameters; agreement (b) Comparison between path losses obtained for Mand increase the between predicted and measured profilesRRFs for seasonal mean ductitparameters. (1) with different and applied to simulate over-sea

ation modeling mbined the PE 115]. As discussed in Section 2.2, some recently proposed RRFs account for ce roughness on electromagnetic waves propagating under grazing incidence g, introduced through RRF (16), on the PF and PL is illustrated on Fig. 4 and putation is carried out using the (15) FE-based solution to parisons between PFs obtained with original Ament and phase corrected [61],RRF, with respectively, computationalfor steps chosen afterwards [62]. and Miller-Brown(1)(14) evaporation duct conditions. relatively to thoser given m, LTBC za = 5 presents m, F = 5results GHz, range fromclose the source = 5 km, by (20) for zd = 14The ~ , σ~ see by the NTBC (for the details andofmore [86]). meters above the sea surface; about relation u10 toexamples from (16) m ξ ξ

seen is the trapping as of energy in both and in surface impedanceClearly characteristics calculated a function of surface frequency elevated ducts. ion is based on the APM [73]. Close to the sea surface, Fig. 4 (b), the Ament st identical resultsThe for difficulties the PF forinthe used scenario; difference between predicting the field the strength for maritime links, and the importance of higher M-inversions may be

30

13

at a meteorological station situated near the Bosporus strait [26]. The parameters for Fig. 3 are: the frequency is chosen at the lower end of the UMTS WCDMA-FDD 13 (Universal Mobile Telecommunications System Wideband Code Division Multiple Access-Frequency Division Duplex) downlink band, f = fDLmin = 2112.4 MHz, za = 30 m, θ0 = 50 , θs = 00 , horizontal polarization, perfectly conducting smooth sea surface, and fixed height for the receiver zr = 20 m. The computation is based on the APM [73]. Fig. 3(a) shows comparison between PL variations obtained for M-profiles based on monthly mean values of the surface-duct parameters. It should be noted that, for the used statistics, zr is always “submerged” within the duct; the same is true for za except for January, March

the accounting for the shadowing will increase the destructive effect of the sea roughness on the duct structure trapping capability, especially as far as the long-range ducted propagation is concerned. Further comparisons between (14), (15), and (16) and their effects on the PF and PL for the case of ducted propagation over the sea are I. Sirkova reported in [114, 115].

ights above 100 m. The introduction of the shadowing in the Ament RRF, tween PLin obtained for M-profiles based on daytime and interference minima and maxima of the PF (due to the phase correction nighttime mean and minima magnitudes. Figure 5 compares the PL for smooth and rough duct parameters may be found in Sirkova For more detailed investigation, a long-term duct staface duct conditions. For this Figure, the duct is modeled by a[52]. bi-linear tistical study differentiating between different duct types za =accounting 5 m, for range-dependent duct thickness and units and duct thickness zd = 50 m. The other parameters are: and face, vertical polarization. Figure 5 (a) refers to smooth sea surface, Figs.is5 needed. Even though tentative, this study strength m/s and ce introduced through phase-corrected Ament RRF with u10 = 7shows: a) the PL differences of scores of dB due to surducting will affect the link budget and increase the sea surface roughness destroys the trapping property of the ductface structure 16) reduces to the greatest degree (in comparison to (14) and requirements (15)) the to the WCDMA power control range; b) depending on the specific climatic characteristics, the use of ence between the direct and reflected fields. Thus it may be expected that annually averaged (and even seasonally averaged) values ncrease the destructive effect of the sea roughness on the duct structure for the duct parameters, which is a common practice, may the long-range ducted propagation is concerned. Further comparisons be not enough for accurate PL prediction. ects on the PF and PL for the case of ducted propagation over the sea areto improve the propagation modeling and inSearching

crease the agreement between predicted and measured fields, a number of authors have combined the PE (1) with different RRFs corrected and appliedAment it to simulate Fig. 4 Comparisons between PFs obtained with original Ament and phase RRF,over-sea (a), and propagation in various scenarios [112–115]. Ament and Miller-Brown RRF, (b), for evaporation duct conditions. As discussed in Section 2.2, some recently proposed RRFs account for the shadowing caused by the sea surface roughness on electromagnetic waves propagating under grazing incidence angles. The influence of the shadowing, introduced through RRF (16), on the PF and PL is illustrated on Fig. 4 and Fig. 5. Figures 4(a) and 4(b) show comparisons between PF s obtained with original Ament (15) and phase corrected Ament (16) RRF, and original Ament and Miller-Brown (14) RRF, respectively, for evaporation duct conditions. The other parameters are: M(z) given by (20) for zd = 14 m, za = 5 m, F = 5 GHz, range from the source r = 5 km, u10 = 7 m/s (u10 is the wind speed at 10 meters above the sea surface; about the relation of e ξ , σeξ from (16) see [80]), vertical polarization with u10 to m (b) surface impedance characteristics calculated as a function ed with original Ament and phase corrected Ament RRF, (a), and oforiginal frequency in accordance with [123]. The computation is Figure 4. Comparisons between PFs obtained with original Ament based on the APM [73]. Close to the sea surface, Fig. 4(b), -Brown RRF, (b), for evaporation duct conditions. and phase corrected Ament RRF, (a), and original Ament and Miller-Brown RRF, (b), for evaporation duct conditions. the Ament and Miller-Brown RRFs lead to almost identical results for the PF for the used scenario; the difference between these two RRFs reflects on the PF for heights above 100 m. The introduction of the shadowing in the and May – this implies noticeable influence of ducting on Ament RRF, Fig. 4(a), leads to clearly seen shift of the inthe PL, especially for long range paths. For distances terference minima and maxima of the PF (due to the phase greater than 6 km, PL under ducting differs significantly correction in (16)) as well as to changes in the maxima and from month to month and may exceed or be lower than the minima magnitudes. Fig. 5 compares the PL for smooth and rough sea (with shadowing included) under surface standard troposphere case, reaching a difference of more duct conditions. For this Figure, the duct is modeled by than 30 dB for September and r = 7.8 km. In this study, September is characterized by moderate zd , but also by a bi-linear height profile with M-deficit of 10 M-units the strongest ∆M among the other months. Due to the and duct thickness zd = 50 m. The other parameters are: za = 5 m, F = 10 GHz, perfectly conducting sea surbi-linear profile of M, months with similar ratio ∆M/zd demonstrate similar PL behavior. Fig. 3(b) depicts PL face, vertical polarization. Fig. 5(a) refers to smooth sea surface, Figs. 5(b) and 5(c) show PL for rough sea survariations for seasonal mean surface-duct parameters. For face introduced through phase-corrected Ament RRF with summer and autumn months PL variations compared to standard troposphere case exceed 15 dB for some ranges u10 = 7 m/s and u10 = 10 m/s, respectively. In general, the sea surface roughness destroys the trapping property (r = 9.8 km, r = 12 km, respectively). Comparisons be(a)

original

14

31

Brief review on PE method application to propagation channel modeling in sea environment

of the duct structure and changes the PL pattern. The RRF (16) reduces to the greatest degree (in comparison to (14) and (15)) the specular scattering and, hence, the coherence between the direct and reflected fields. Thus it may be expected that the accounting for the shadowing will increase the destructive effect of the sea roughness on the duct structure trapping capability, especially as far as the long-range ducted propagation is concerned. Further comparisons between (14), (15), and (16) and their effects on the PF and PL for the case of ducted propagation over the sea are reported in [114, 115]. (a) Benhmammouch et al. [112] studied the influence of the wind direction on electromagnetic waves propagation over rough sea surface by hybridizing two approaches to the sea surface representation: (i) the large sea surface roughness is generated on the basis of a sea spectrum multiplied by an angular spreading function, thus introducing the angle difference between the wind direction and electromagnetic wave propagation direction; (ii) the small roughness is introduced by replacing the standard deviation σξ in (14) with the standard deviation of capillarity waves. The simulations are based on the SSF solution to (1) and refer to evaporation duct conditions given by (20). In comparison to the PL obtained for the original (b) Miller-Brown RRF, the results presented show greatest PL pattern destruction and energy leakage for Fig. wind5diPath loss under surface duct conditions: (a) smooth sea surface; (b) phaseFig. 5 Path loss under surface duct conditions: (a) smo rection coinciding with the direction of radiocorrected propagation. Ament RRF with u10=7 m/s; (c) phase-corrected Ament RRF with u10=10 m corrected Ament RRF with u10=7 m/s; (c) phase-corrected As suggested by Hristov et al. [78], the deviation of predicted from measured field strengths in over-sea propagaBenhmammouch et al. [112] studied the influence of the wind direction on electromagnetic wa et al. [112] influence of the wind direl tion may be due, other than over to therough non-accounting for the sea surface by Benhmammouch hybridizing two approaches to studied the sea the surface representation: (i) the rough by hybridizing twoanapproaches to thefunction, sea surf shadowing, also to the nonlinear interactions betweenover the the roughness is generated on basis sea of asurface sea spectrum multiplied by angular spreading roughness generated on and the basis of a sea spectrum multiplieddirectio by an sea waves, which can causethe the angle sea surface statistics to difference between the iswind direction electromagnetic wave propagation difference between the wind differ from Gaussian. In Levadnyi et al.is[113] these the twobyangle (14) withand the electromagnetic standard deviati roughness introduced replacing the standard deviation σξ indirection mechanisms have been compared SSF-based roughness PE is introduced by replacing deviation waves.using The simulations are based on the SSF solution to (1) andthe referstandard to evaporation duct σcondition ξ in (14 solution to obtain the PF , In andcomparison the results have to thedemonPL obtained for the original Miller-Brown RRF, the results presented show waves. The simulations are based on the SSF solution to (1) and refergrt strated that the shadowing destruction has prevailing effect in comand energy leakage for windtodirection coinciding with direction of radio propagation In comparison the PL obtained thethe original Miller-Brown RRF, th (c) for parison to the sea surface’s non-Gaussianity (especially, As suggested by destruction Hristov et al. [78], the deviation of predicted from measured field the stren and energy leakage for wind direction coinciding with di under evaporation duct conditions (20) and may for centimeterpropagation be due, other than to the non-accounting for the shadowing, also to the nonli Figure 5. Path loss under surface duct conditions: (a) smooth sea As suggested by Hristov et al. [78], the deviation of predicted Fig. 5 Path loss under surface duct conditions: (a) smooth sea surface; (b) phasesized waves range). (b)surface phase- corrected Ament RRF with u10 = between the sea waves,propagation which can cause the sea statistics to differ from Gaussian. Leva maysurface; be (c) due, other than the non-accounting forInthe sh m/s; (c) phase-corrected Ament RRF withtouRRF =10 m/s. corrected Ament RRF withinuthe 10=7 10 7 m/s; phase-corrected Ament with u 10 = 10 m/s. It should be noted that the these resultstwo reported above mechanisms have been compared using SSF-based PE solution to obtain the PF, and between the sea waves, which can cause the sea surface statistics to di discussed papers should notdemonstrated be extrapolatedthat to different the shadowing has prevailing effect in comparison to the sea surface’s Benhmammouch et al. [112]scenarios; studied the influence of the wind direction electromagnetic waves SSF-based propagation PE solu these two mechanisms haveonbeen compared using frequency ranges and propagation rough (especially, underthe evaporation duct conditions (20) and for centimeter-sized waves range). over rough sea surface by hybridizing two approaches to the sea surface representation: (i) the large sea demonstrated that the shadowing has prevailing effect feasibilitypapers of (more or surface sea surface scattering effects stillIt need additional invesshould be noted that theapplied results AREPS reportedtoininvestigate the abovethe discussed should notinbecompar extrapo roughness is generated on the basis of a sea spectrum multiplied by an angular spreading function, thus introducing (especially, under evaporation duct conditions (20) and for less stable) beyond-the-horizon radio communitigations, both theoretical and experimental. frequency ranges and propagation scenarios; the roughover-sea sea surface scattering centimeter-s effects still the difference between directiontoand electromagnetic wave propagation (ii)above the small It should be noted the results reported in the discussed p cations links basedthat on the capability ofdirection; the evaporation As aangle reliable method, the PE has the beenwind recently investigations, both used theoretical and experimental. in (14) with the standard deviation of capillarity roughness is introduced by replacing the standard deviation σ ξ tohas ensure increased propagation range. accurate frequency and propagation scenarios; the rough sea surfa provide accurate PL prediction for coastal and marAsthe a reliable method, theduct PEranges been recently used to provide PL prediction for waves. Theinsimulations based on the Watson SSF solution to (1) andboth refertheoretical to evaporation duct conditions given by (20). itime zones a number of are new applications: and investigations, and experimental. maritime zones in a number of new applications: Watson and Coleman [124] studied the feasib In comparison to thethe PLfeasibility obtained offora novel the original Miller-Brown RRF, the results show recently greatest PL pattern Coleman [124] studied technique As a reliable method, the presented PEonhas to provide technique for measuring atmospheric refractivity based thebeen reception andused processing of d destruction and energy leakage for wind direction coinciding with the direction of radio propagation. for measuring atmospheric refractivity on the receptelevisionbased signals; Tang et al. [125] applied PE in cognitive radios to cope with the multipath fadi maritime zones in a number of new applications: Watson and Colem As processing suggestedofby Hristov al. [78], the deviation of predicted from measured field strengths in over-sea tion and digital radio et and television links; Ruxton et al.signals; [49], Woods et al. [50], and Iqbal and Jeoti [51] have applied AREPS to technique forformeasuring atmospheric refractivity based on the rece propagation mayapplied be due, other than to the non-accounting the shadowing, alsothe to theory the nonlinear interactions Tang et al. [125] PE in cognitive This work has briefly described and current feasibility of radios (more to orcope less stable) beyond-the-horizon over-sea radio communications links based signals;toTang al. [125] applied PE in cognitive radios to c between the sea waves, which can cause seatelevision surfacestatus statistics differetfrom Gaussian. In Levadnyi et al. [113] of the propagation PE-based propagation channel prediction with the multipath fading in links;the Ruxton ofover the water evaporation ductlinks; toetensure increased range.et al. Ruxton et al. [49], Woods [50], and Iqbal and these two mechanisms been and compared PE emphasis solution on to its obtain the PF,inand the and results haveJeoti [5 al. [49], Woods et al. [50],have and Iqbal Jeoti [51]using have SSF-based method with applications coastal

4.

32

Concluding Comments

feasibility or less to stable) beyond-the-horizon over-sea radio c demonstrated that the shadowing has prevailing effect ofin(more comparison the sea surface’s non-Gaussianity 4. Concluding Comments of the duct to ensure increased propagation range. (especially, under evaporation duct conditions (20) andevaporation for centimeter-sized waves range). It should be noted that the results reported in the above discussed papers should not be extrapolated to different frequency ranges and propagation scenarios;4.theConcluding rough sea Comments surface scattering effects still need additional investigations, both theoretical and experimental. As a reliable method, the PE has been recently used to provide accurate PL prediction for the coastal and

I. Sirkova

maritime regions. During the last two decades, the strength of the PE-based modeling in accounting for diffraction, refraction, reflection and scattering propagation mechanisms has been demonstrated in a number of studies, and the method has won recognition for its high reliability. Nevertheless, to make correct use and take full advantage of the capabilities of the PE method when applying it to the planning and operation of different radio systems, one should take into consideration both the approximations of the method [126] and possible sources of errors related to the accuracy (and some specificities of their collection) of the used input data [127]. The complicated environmental conditions in coastal and maritime areas require further sophistication of the PEbased propagation model (followed by experiments for conclusive validation) as well as extensive measurements to provide more reliable and refined local climatology statistics. Ongoing theoretical efforts are mainly directed to: (i) the development of 3D PE models and improving their use friendliness for inclusion in routine calculations; (ii) improvements in the sea surface roughness representation; and (iii) progress in the evaporation duct modeling, particularly with respect to more accurate determination of the height of this type of duct. Further development may be also expected in a PE-based time-dependent propagation modeling as explored in Dockery et al. [72]. To improve the environmental input to the PE model, advanced climate analysis techniques are applied; in spite of that, the supply of refractivity data with necessary quality to meet the real-time operational requirements of microwave systems operated, especially, in littoral environments, is still an open issue.

Acknowledgments This work has been performed in the framework and with the support of Action IC0802 “Propagation tools and data for integrated Telecommunication, Navigation and Earth Observation systems” from the European COST program.

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