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full-protect. potentially proprietary. Cmarston1929 (talk) 12:54, 10 July 2008 (UTC)

(1.1) STATISTICAL NARROW BAND MODELS (Molisch, page 119)

 * Modeling of small-scale fading: Within a sufficiently limited physical area, multipath amplitude variations admit to a random variable modeling of the associated multipath phase ensemble which is characterized by a Doppler spatial correlation and a Rayleigh distribution of the resulting absolute amplitude.
 * Modeling of large -scale fading : small-scale Rayleigh-distributed amplitudes, if averaged over larger areas, are manifested as a log-normal distribution of the resulting field strength and the concomitant normal distribution of the associated log of field strength (path loss).
 * Waveguide Model: Appropriate for low antennas in the mobile channel(which requires field distribution analysis ),the Waveguide Model imposes a Poisson distribution on an ensemble of fields in an urban environment modeled as a 3D wave guide to calculate path loss.
 * Boltzman Model: Adapted from fluid flow models, modeling of urban channel as dynamics of particles on  a 3D lattice, the Boltzman model is well adapted to complex boundary conditions and provides good agreement with path-loss measurements.

OUTDOOR MODELS (Molisch, page 119)

 * Okumura Hata- valid only for base station (transmitter) above roof tops, path loss is modeled with three parametrically adjustable coefficients in the equation $$ L = A + B \log(d) + C $$.
 * Hata Model- Empirical realization of Okumura-Hata Model with parametric adaptability to suburban and rural areas as well as urban areas, but not reliable for smaller cell communications less than one kilometer in radius.
 * COST-231 Walfisch-Ikegami- Path loss is modeled as the sum of a multi-screen diffraction due to multiple buildings assumed to be of uniform spacing and height, a roof-to street loss due to diffraction from the mobile station horizon (final building of the multi- screen diffracter) and a reflection loss involving a  surface provided by the  building immediately adjacent to the mobile antenna in the direction of propagation.
 * Dual slope model- based on a two ray model in which the slope of the loss as a function of the log of the range coordinate is varied from an initial to a final vale at  the onset of a range-correlated  critical optical configuration associated  with sufficient penetration of the first Fresnel zone.

INDOOR MODELS ( Sarkar, et. al. page 57)

 * Conventional Indoor Models
 * The Motley-Keenan Model: site-specific indoor model which requires explicit floor plans and antenna position coordinates and includes explicit wall attenuation contribution, but in its neglect of indirect but often less-attenuated paths, is considered unreliable.
 * Computationally Efficient models
 * Improved Ray-Tracing: Adaptation of results from the combination of several ray tracing methods with UTD in a 2D environment are used in a novel 3D  propagation prediction model which is more accurate  than conventional 2D models and more efficient than  full 3D ray tracing models.
 * Selective Use of FDTD-localized implementation to model  effect of walls on indoor propagation channel. The inadequacy of ray-tracing methods to properly  model the effects of the inner structure of walls and of the assumption of specular reflection may be addressed through the use of Floquet constructs in an FDTD simulation to describe periodic structures inherent in wall fabrication materials as required to properly model non-specular reflections and diffusive scattering effects.

SOLUTIONS BASED ON INTEGRAL EQUATIONS

 * Method of Moments
 * Conventional MOM: Formally exact solution for fields as an integral over unknown sources (weighted by Green functions) appropriate for selected dimensionality in which a discretized representation of know field values is expressed as the product of an impedance matrix and a vector of unknown currents. The unknown currents may then be solved through a formal inversion of the impedance matrix or other standard methods of coupled linear algebraic equations.
 * Hybrid variations: Ray Tracing and Moment Method
 * Periodic Moment Method: Initial use of Moment Method adapted to periodic structure to prepare initiation rays which are then propagated with ray methods
 * Fast Far-Field Approximation Method: Introduction of Green function feature accelerates traditional Integral Equation (IE) methods and has been applied effectively to undulating and hilly terrain and terrain with urban features; technical details involving terrain profile truncation and small-scale-roughness have been effectively addressed as discussed in the linked article: [[Media:FFFAM.pdf]].

SOLUTIONS BASED ON DIFFERENTIAL EQUATIONS

 * Finite Element Methods: General numerical method in which partial differential equations for fields are reformulated as ordinary differential equations by first expressing solutions in a discretzied space in terms of localized basis functions to thus provide a bilinear set of coupled equations with implicit boundary conditions which may be solved using standard techniques of linear algebra including matrix inversion. For problems in the frequency domain,the bi-lnear matrix equations reduced to an eignevlue equation which may be solved by matrix diagonalization to provide resonant frequencies (eigenvalues)and an (orthogonal) set of basis functions for general field solutions (eigenvectors).
 * FDTD : coupled time-dependent equations for electric and magnetic fields are solved simultaneously on a temporal-spatial grid in which time derivatives of a given field are computed as spatial derivatives of the conjugate field as required by the Maxwell equations.
 * Reduced dimensionality formulations
 * Hybrid formulation of FDTD with geometrical optics
 * (Vector) Parabolic Equation: Vector variation of scalar versions permits modeling of 3D scattering; especially effective for single building or groups of buildings at microwave frequencies.
 * Novel Unitary (Time-Dependent) Propagator expanded in Chebychev basis

CONVENTIONAL RAY METHODS
The Maxwell Equations admit to an exact solution expressed as a summation in powers of the wavelength which in the high frequency limit reduces to the first term which defines the (ray) elements of geometrical optics. Geometrical optics (GO) provides a suitable approximation for propagation involving  distances  and environmental features with  linear dimensions large  with respect to the wavelength. If generalized with the elements of the Unified Theory of Diffraction (UTD), the resulting methodology provides high frequency approximations of LOS propagation, reflection, refraction and diffraction. diffraction) and are inadequate in moderately complex environments.
 * Image Method: Secondary and higher order sources are located as images of the transmitter with respect to all potential reflecting surfaces and all such rays along rays paths  to the receiver are superimposed to construct the received field. The methods are systematic, complete and practical in an optically simple environment, but require an establishment of an upper bound of the number of rays to be considered in each mode of propagation (reflection and
 * Ray Launching (single source to omni-directional targets) (Molisch, Section 7.5.1)
 * free space loss
 * reflection
 * diffraction
 * diffuse scattering
 * Ray tracing (point-to-point raypath ) (Molisch, Section 7.5.2, pages 132-133)
 * Longley-Rice Model (ITM): valid in the range 40 MHz-100 GHz, the model includes (1) an area mode prediction option to predict path parameters which does not require a terrain profile and (2) a point-to-point ray tracing option which requires a terrain profile as input, modeling terrain-induced diffractions according to the Fresnel-Kirchoff knife edge model.

ENHANCEMENTS TO IMPROVE EFFICIENCY OF RAY METHODS

 * Ray Splitting- Acknowledging a fundamental assumption of Geometrical optics that a given ray represents only a finite solid angle into which it is directed, ray splitting introduces a companionate ray at a critical distance associated with exceeding an upper bound of solid angle for valid single ray representation. (Molisch, page 131)
 * Reduced dimensionality methodology
 * Restriction to the vertical plane if transmitter is high and loss is dominated by diffraction from the mobile station's horizon to the receiver and by reflection from the closest building in the direction of propagation.
 * Restriction to horizontal (slant) plane if both antennas are below roof-top level
 * (2.5D)for larger distance urban channels : sum of contributions from vertical plane for horizontal diffracting edges and slant(horizontal) plane to negotiate vertical diffracting edges.
 * Geographical Databases: The accuracy of deterministic methods is ultimately limited by the accuracy of relevant geographical and morphological features of the environment. (Molisch, page 134)
 * Building plans- available in digital form are of particular relevance to the indoor propagation channel.
 * Geographical and morphological data bases (land use), variably accessible according to nationality, are of particular relevance to propagation models selected for rural areas.
 * Urban propagation data base optimization: vector data providing critical coordinates of building structural features and pixel data area coverage providing land use descriptions which are directly complementary to coexisting structural features.

(1.3 ) HYBRID DETERMINISTIC AND STOCHASTIC MODELS
While retaining a deterministic treatment of path loss (of a specified fade margin), inaccessible causal parameters are replaced by a statistical modeling of multi-component phase contributors to thus statistically model impulse responses

(1.4 ) ARTIFICIAL NEURAL NETWORKS MODELS
Addressing accuracy deficiencies of statistical models and efficiency limitations of site specific models, Artificial Neural Network models provide accurate field strength estimates of noisy data while presenting the additional benefit of intrinsic parallelism. Developmental calculations have explored exhaustive data bases of physical surroundings and while requiring a lengthy learning phase, the model was fast in the final field level prediction phase and underscored the significance of the quality of the data base in determining the accuracy of the final result. Similar investigative studies introducing topological and morphological data  have attempted to  facilitate the learning process with supplementary heuristic and deterministic equations. The slow convergence and unpredictability of field-strength predictions  has been addressed through the introduction of radial basis functions which, though linear in the fundamental input parameters, provide a best non-linear-approxmation capability for the model.

( 2 )WIDE-BAND MODELS (Molisch, Section 7.3)

 * Tapped delay
 * Models for power delay profile
 * Arrival times of rays and clusters
 * Standardized channel model

(3) DIRECTIONAL MODELS (Molisch, Section 7.4)

 * General model structure and factorization
 * Angular dispersion at base station
 * Angular dispersion at base station
 * Polarization
 * Model Implementation
 * Standardized directional models
 * MIMO models

REPRESENTATIVE UNIFIED-PARAMETER VALIDATION DATA FOR COLORADO PLAINS
The following data is typical of that for the Colorado Plains in which the receiver site at a fixed location is varied from one to 13 meters with the expected decrease in excess loss which in the example below recovers from 15.3 dB to a gain of 6.8 dB. Unlike the data at NTIA, all relevant parameters including geographic coordinates of both antennas, propagation parameters and structural heights required for a given measurement are provided in a single record.

TRANSMITTER          RECEIVER TX SITE    CODE  LAT    LON         LAT      LON       FMHZ       H_TX        H_RX   XS LOSS(DB) RANGE (KM) R1-0_5-T1  O   40.0944 -105.1189   40.0939 -105.1255  230.0000    6.6000    1.0000   15.3000    0.5700 R1-0_5-T1  O   40.0944 -105.1189   40.0939 -105.1255  230.0000    6.6000    2.0000    9.8000    0.5700 R1-0_5-T1  O   40.0944 -105.1189   40.0939 -105.1255  230.0000    6.6000    3.0000    6.1000    0.5700 R1-0_5-T1  O   40.0944 -105.1189   40.0939 -105.1255  230.0000    6.6000    4.0000    4.1000    0.5700 R1-0_5-T1  O   40.0944 -105.1189   40.0939 -105.1255  230.0000    6.6000    5.0000    2.2000    0.5700 R1-0_5-T1  O   40.0944 -105.1189   40.0939 -105.1255  230.0000    6.6000    6.0000    0.8000    0.5700 R1-0_5-T1  O   40.0944 -105.1189   40.0939 -105.1255  230.0000    6.6000    7.0000   -0.9000    0.5700 R1-0_5-T1  O   40.0944 -105.1189   40.0939 -105.1255  230.0000    6.6000    8.0000   -2.5000    0.5700 R1-0_5-T1  O   40.0944 -105.1189   40.0939 -105.1255  230.0000    6.6000    9.0000   -3.7000    0.5700 R1-0_5-T1  O   40.0944 -105.1189   40.0939 -105.1255  230.0000    6.6000   10.0000   -4.9000    0.5700 R1-0_5-T1  O   40.0944 -105.1189   40.0939 -105.1255  230.0000    6.6000   11.0000   -5.6000    0.5700 R1-0_5-T1  O   40.0944 -105.1189   40.0939 -105.1255  230.0000    6.6000   12.0000   -6.3000    0.5700 R1-0_5-T1  O   40.0944 -105.1189   40.0939 -105.1255  230.0000    6.6000   13.0000   -6.8000    0.5700

$$ cf= { \over{\pi}} $$

Consider construction of a text reference in the text 1. The following data is typical of that for the Colorado Plains in which the receiver site at a fixed location is varied from one to 13 meters with the expected decrease in excess loss which in the example below recovers from 15.3 dB to a gain of 6.8 dB. Unlike the data at NTIA, all relevant parameters including geographic coordinates of both antennas, propagation parameters and structural heights required for a given measurement are provided in a single record.

ALTERNATE MECHANISMS FOR ENFORCING IMPEDANCE BOUNDARY CONDITIONS USING THE SPLIT-STEP FOURIER APPROACH TO SOLVING PARABOLIC EQUATIONS
In his original modern implementation of the Split-Step Fourier Operator in solving the parabolic approximation of the Helmholtz equation, Ryan [2] was able to satisfy the impedance boundary conditions by introducing a modified index of refraction in the coordinate space portion of the SSFPE unitary propagator, a technique which was also employed by Barrios [4] who noted the essential difference of her work and that of Dockery who has since [3] introduced the mixed Fourier (MFT)  transform formalism for satisfying the impedance boundary conditions, a technique which has also since  been employed by Frank Ryan [5]. Dockery however reports [6] that the MFT was unstable for high frequencies and that it introduced a factor of two in the computational time, difficulties which have since been removed by Dockery and Kuttler. However in a recent review article, [8] [[Media:Dockery_2007.pdf]], Dockery reported that the method remained problematic for small scale roughness at certain frequencies but that these problems have been effectively solved by Kuttler and Janaswami [7]. Both schools (NOSC) and (JHU-APL) are in accord that for homogeneous smooth earth, the boundary conditions can be solved for vertical and horizontal polarization by expressing the solution as linear combination of even(V) or odd functions(H) which can be respectively propagated with (fast ) Fourier cosine or sin transforms and that for realistic electrical ground constants, a linear combination of these fundamental solutions  may be analytically employed to solve the more general problem. As an exercise in the enforcement of impedance surface boundary conditions, work at the PSU ARL will first develop computational proficiency in the elementary Fourier projection and propagation techniques by reproducing the work in Dockery's paper[3] for propagation over the smooth earth.

PREPARATION OF INITIAL FIELD DISTRIBUTION FUNCTIONS
Since the most general impedance boundary conditions are enforced by expressing the field as linear combinations of even and odd functions (with respect to vertical coordinate z), it becomes desirable to derive the symmetry components from an arbitrary function such as an initial field distribution which may be constructed from an antenna amplitude field pattern confined to the permissible values of the k-space coordinate. Given a suitable antenna radiation pattern in p space such as f(p)= sin( ap)/ap, the associated even and odd functions are constructed as

ce = [f(p) + f (-p)]/2

co= [f(p) - f(-p) ]/2

The original field pattern, f_{+}(p) and its mirror image (about z=0) f_{-} (p)  may be  recovered from the even and odd projections as

f_{+} (p) =     ce + co

f_{-} (p) =     ce - co

With suitably constructed wave forms in momentum space which exhibit the desired even and odd symmetry characteristics, the corresponding z-space representations of the even and odd functions are recovered as the cosine and sine transforms of the corresponding momentum space wave functions. Adjustments for the desired maximal momentum coordinate amplitude (such as the selection of beam directivity according to the relation p_0 = k_0 sin (theta_0) ) assures that the momentum values p_0 so constructed will correspond to that value in p-space with the largest amplitude in the p-space representation. Adjustments for antenna height z0  are also made in momentum space by introducing the phase factors e^((+/-) i p z0)  to the corresponding terms in Eqs(1).

ce = [f(p)e^(-i p z0) + f (-p)e^(i p z0)]/2

co= [f(p)e^(-i p z0)  - f(-p)e^( i p z0)]/2

The z-space representations of the even and odd components are obtained as cosine and sine inversions of the corresponding even and odd p-space functional forms.

psi_e( z) =      F_{C}^{-1} c_e (p)

psi_o( z) =      F_{S}^{-1} c_o (p)

ANALYTIC INITIAL FIELDS FOR SPLIT-STEP FOURIER PROPAGATOR
Using the methods described above for an initial Gaussian antenna radiation pattern, the general form for the initial wave function in p-space is given by $$ \psi_{+}(x=0; p) = e^{ -\alpha{(p-p_{0})}^2} e^{ -i p (z_{0})} $$ where the quantity p0 is related as stated above to the angle of inclination of the main beam above the horizontal and where the phase factor in $$z_{0}$$ is the p-space manifestation of antenna heights above the surface of the (smooth) earth. Given this analytic form for the positive argument p-function,the corresponding negative argument function follows from the transformation $$ p\mapsto -p $$, to yield $$ \psi_{-}(x=0; p) = e^{ -\alpha{(-p-p_{0})}^2} e^{ i p (z_{0})}  $$ The even (odd) function ( with respect to p=0) is obtained from \psi_{+} and $$\psi_{-} $$ as their sum (difference) to yield

$$ \psi_{e}(x=0; p) = \psi_{+}+\psi_{-} = e^{ -\alpha{(p^2+p_{0}^2) } } \cosh( 2\alpha p p_{0} + i p z_{0} )$$

and $$ \psi_{o}(x=0; p) = \psi_{+}-\psi_{-} = e^{ -\alpha{(p^2+p_{0}^2) } } \sinh( 2\alpha p p_{0} + i p z_{0} )$$

,

The associated even(odd) functions in z-space are obtained as the Fourier cosine(sine)inversions of the corresponding  even(odd) functions in p-space and the total solution in z -space may either be constructed from these transformed components or obtained directly as the general Fourier inversion of a composite complex p-space wave function constructed from the even and odd functions and extended to the negative half space as appropriate for the respective understood symmetry. A typical initial function n p-space modeled as a displaced Gaussian (centered at p0, black) and its associated even(red) and odd(blue) components and the cooresponding representations in coordinates space ( as a shifted Gaussian centered at z0, same color scheme) is shown at right.

$$ \psi_{e}(x=0;z) = {\sum_{j=1}^{N}} {\cos(\pi j k/N) \psi_{e}(0,p)} ={(1/2 \alpha)}^{1/2}{\cosh( z z_{0}/2 a -i p_{0} z) e^{(-(z^2+z_{0}^2)/(4 a) ) } e^{i p_{0} z_{0}} } $$

$$ \psi_{o}(x=0;z) = {\sum_{j=1}^{N}} {\sin(\pi j k/N) \psi_{o}(0,p)} = {(1/2 \alpha)}^{1/2}{\sinh( z z_{0}/2 a -i p_{0} z) e^{(-(z^2+z_{0}^2)/(4 a) ) } e^{i p_{0} z_{0}} } $$

$$ \psi_{+}(x=0;z) = {\sum_{j=1}^{2N}} {e^{(i 2\pi j k/(2 N))} \psi_{+}(0,p,-p)} = {(1/2 \alpha)}^{1/2}e^{- { (z-z_{ 0 } )^{ 2 }/(4 \alpha  ) } } e^{ -i p_{0} (z-z_{0}  )  } $$

The even function $$ \psi_{e}(x=0;z) $$ can may be rewritten as

$$ \psi_{e}(x=0;z)= {(1/2 \alpha)}^{1/2}{ \big( \cosh(z z_{0}/2 a)\cos( p_{0} z)- i \sinh(z z_{0}/2 a)\sin(p_{0} z) \big) e^{(-(z^2+z_{0}^2)/(4 a) ) } e^{i p_{0} z_{0}} } $$

which can be more transparently written as

$$

\psi_{e}(x=0;z)= {(1/2 \alpha)}^{1/2} \big(e^{-{ (z+z_{0} ) }^2/(4 a) } e^{ i  p_{0} (z + z_{0}  )   } +e^{-{ (z-z_{0}  ) }^2/(4 a) } e^{-i  p_{0} (z - z_{0}  )   }\big) $$

which is the sum of two Gaussians,  centered at $$ \pm z_{0}  $$ and partitioned between real an imaginary sinusoidal factors  also centered at $$\pm z_{0}$$, whereas the odd function is of a similar form but wth a center of inversion through the origin

$$ \psi_{o}(x=0;z)= {(1/2 \alpha)}^{1/2} \big(-e^{-{ (z+z_{0} ) }^2/(4 a) } e^{ i  p_{0} (z + z_{0}  )   } +e^{-{ (z-z_{0}  ) }^2/(4 a) } e^{-i  p_{0} (z - z_{0}  )   }\big) $$

A quadratic coefficient of $$\alpha=128$$ was observed numerically to provide conjugate functions well localized within their respective discrete representations and therefore permits a calculation  of the associated beam width from the relation $$ \alpha (p-p_{0})^2 = \big((\log(2)/2)/{{p_{1/2}}^2 \big)(p-p_{0})^2} => p_{1/2} = {(\log(2)/(2\alpha))}^{1/2} =.0520347. $$ Analytic expressions for the second derivatives (in z) of these initial wave functions are of interest for validation of a single action of the unitary propagator as described in the next section and are given below:

INFINITESIMAL VALIDATION OF SPLIT-STEP FOURIER PROPAGATOR
The Split-Step Fourier Propagator approach to solving the parabolic equation resulting from an application of the paraxial approximation upon the (scalar) Helmholtz equation relevant to a transverse electromagnetic field  normal to the propagation plane defining the dimensionally reduced 2-D problem, involves iterative  application of a unitary propagator in which the coordinates-space representation of the field  is first transformed to   k-space where the action of the square root operator is excuted through a multiplicative process to produce an intermediate operand which is then inverted to coordinate space.

$$ \mathbf{F}^{-1}(p\mapsto z) \big( e^{ - i (\delta x)\mathbf {H}}\big) \mathbf{F}(z\mapsto p)|\psi(x,z)> $$
 * \psi(x+\delta x, z)>=

For strictly vertical or horizontal polarization, the resulting boundary conditions are enforced by first extracting even or odd symmetry representations of the wave fnction and replacing the general Fourier transforms $$ \mathbf{F} $$ by the Fourier cosine $$ (\mathbf{C})$$ or sine transform $$(\mathbf{S})$$ thus:

$$ \mathbf{C}^{-1}(p\mapsto z) \big( e^{ - i (\delta x)\mathbf {H}}\big) \mathbf{C}(z\mapsto p)|\psi_{e}(x,z)> $$
 * \psi_{e}(x+\delta x, z)>=

$$ \mathbf{S}^{-1}(p\mapsto z) \big( e^{ - i (\delta x)\mathbf {H}}\big) \mathbf{S}(z\mapsto p)|\psi_{o}(x,z)> $$
 * \psi_{o}(x+\delta x, z)>=

(General boundary conditions for finite ground constants involve linear combinations of the above symmetry forms where the general Fourier transformation is obtained as the sum of the two half-transforms acting upon their respective symmetry-specific operands.)

$$ \mathbf{F}    = \mathbf{C}      + i \mathbf{S}$$ $$\mathbf{F}^{-1} = \mathbf{C}^{-1} - i \mathbf{S}^{-1} $$

The unitary propagator may be expanded for small arguments to produce expressions which may readily evaluated numerically to replicate the effective action of the operator for a single iteration, which for propagation in a vacuum reduces to the expression $$ e^{ - i (\delta x)\mathbf {H}} \approx  \mathbf{1}   - i (\delta x)\hat{\mathbf{P}}^{2}/ (2 k_{0}) $$ which in the coordinate representation assumes the form $$ e^{ - i (\delta x)\mathbf {H}}\approx \mathbf{1}  - i (\delta x)\boldsymbol{ \hat{\nabla}_{z} }^{2}/ (2 k_{0}) $$ $$  e^{ - i (\delta x)\mathbf {H}}\approx  \mathbf{1}   - i (\delta x) \big(\partial^{2}/{\partial{ z^2}\big)/(2 k_{0}}) $$ $$  e^{ - i (\delta x)\mathbf {H}}-  \mathbf{1}\approx   - i (\delta x) \big(\partial^{2}/{\partial{ z^2}\big)/(2 k_{0}}) $$ which indicates that the single application of the unitary propagator for small $$\delta x$$ can be approximated by the sum of the unit matrix and a (diagonal) correction which is proportional to the second derivative with respect to the elevation coordinate, z. The exact expressions for the (second derivatives in z ) of the initial wave functions can presumably be very closely approximated by subtracting the unit matrix from the action of the propagator upon the wave function.

$$ \partial^{2}/\partial{z^2} \psi_{e}(z)=e^{-(1/4 \alpha)(z^2+z_{0}^2) + i p_{0}z_{0}}\big(\big( 2 a_{v} + b_{u}^2 + 4 {(a_{v} z)}^{2}\big)\cosh(b_{u} z)+ \big(4 a_{v} b_{u} z \sinh(b_{u} z\big) \big) $$  $$  \partial^{2}/\partial{z^2} \psi_{o}(z)=e^{-(1/4 \alpha)(z^2+z_{0}^2)+ i p_{0}z_{0}}\big(\big( 2 a_{v} + b_{u}^2 + 4 {(a_{v} z)}^{2}\big)\sinh(b_{u} z)+ \big(4 a_{v} b_{u} z \cosh(b_{u} z\big) \big) $$

where the constants $$a_{v}$$ and $$ b_{u} $$ are given by

$$ a_{v} =-(1/4 \alpha) $$

$$ b_{u} =(z_{0}/2\alpha) -  i p_{0} $$

$$ e^{ - i (\delta x)\mathbf {H}}-  \mathbf{1}\approx   - i (\delta x) \big(\partial^{2}/{\partial{ z^2}\big)/(2 k_{0}}) $$

which exhibits a diagonal form with matrix elements given by

$$ \delta_{ij}\big(e^{ - i (\delta x)\mathbf {H_{ij}}}-  1\big)\approx   - i (\delta x) \big(\partial^{2}/{\partial{ z^2}\big)/(2 k_{0}}) $$ An effort will be made to reproduce the work of Dockery for propagation over the smooth earth at a frequency of 3000 MHz with a transmitter height of 30 m and a receiver at fixed ranges of 40 km and 80 km  and of a variable height between 0 and 500 m, which has been replicated below using  an application of Norton's adaptation of Sommerfeld's methods for propagation over the smooth earth.

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