recursive bayesian electromagnetic refractivity estimation from...

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Recursive Bayesian electromagnetic refractivity

estimation from radar sea clutter

Sathyanarayanan Vasudevan,1 Richard H. Anderson,2 Shawn Kraut,3 Peter Gerstoft,4

L. Ted Rogers,5 and Jeffrey L. Krolik6

Received 21 November 2005; revised 3 May 2006; accepted 29 November 2006; published 27 April 2007.

[1] Estimation of the range- and height-dependent index of refraction over the sea surfacefacilitates prediction of ducted microwave propagation loss. In this paper, refractivityestimation from radar clutter returns is performed using a Markov state space model formicrowave propagation. Specifically, the parabolic approximation for numerical solutionof the wave equation is used to formulate the refractivity from clutter (RFC) problemwithin a nonlinear recursive Bayesian state estimation framework. RFC under thisnonlinear state space formulation is more efficient than global fitting of refractivityparameters when the total number of range-varying parameters exceeds the number ofbasis functions required to represent the height-dependent field at a given range.Moreover, the range-recursive nature of the estimator can be easily adapted to situationswhere the refractivity modeling changes at discrete ranges, such as at a shoreline. A fastrange-recursive solution for obtaining range-varying refractivity is achieved by usingsequential importance sampling extensions to state estimation techniques, namely, theforward and Viterbi algorithms. Simulation and real data results from radar cluttercollected off Wallops Island, Virginia, are presented which demonstrate the ability of thismethod to produce propagation loss estimates that compare favorably with ground truthrefractivity measurements.

Citation: Vasudevan, S., R. H. Anderson, S. Kraut, P. Gerstoft, L. T. Rogers, and J. L. Krolik (2007), Recursive Bayesian

electromagnetic refractivity estimation from radar sea clutter, Radio Sci., 42, RS2014, doi:10.1029/2005RS003423.

1. Introduction

[2] The determination of microwave propagation con-ditions in the troposphere is important for assessing theperformance of both communications and radar systems.In general, radar coverage over the sea surface isdetermined by the atmospheric refractive index, n, whichalthough very close to unity, depends on meteorologicalconditions. Although in a well-mixed atmosphere,

microwave propagation is typically limited to the lineof sight, in coastal regions where hot, dry air is advectedfrom the land over cool, humid air near the sea surface, itis not uncommon for the refractive index to decreasevery quickly with height. Under such conditions, ductedpropagation can occur, resulting in propagation beyondthe horizon. Figure 1 illustrates the effect of ducting onradar backscatter returns from the sea surface off theVirginia coast. Figure 1 is a plot of power versus rangeand azimuth from a 3 GHz radar pointed at the horizon.The significant clutter returns from ranges well beyondthe approximately 48 km (30 mile) horizon are the resultof surface-based ducting conditions. Prediction of prop-agation loss to points above the sea surface over thecoverage area shown in Figure 1 can be achieved usingan estimate of the profile of index of refraction as afunction of height, range, and azimuth in a computationalelectromagnetic propagation model.[3] Conventional methods for estimating modified

refractivity, M, defined by M = 106 � (n � 1), wheren is the refractive index, can be split into two categories:(1) direct sensing techniques which involve the measure-

RADIO SCIENCE, VOL. 42, RS2014, doi:10.1029/2005RS003423, 2007ClickHere

for

FullArticle

1Sensor Research and Development Corporation, Orono, Maine,USA.

2Applied Signal Technology, Inc., Arlington, Virginia, USA.3Lincoln Laboratory, Massachusetts Institute of Technology,

Lexington, Massachusetts, USA.4Marine Physical Laboratory, Scripps Institution of Oceanography,

University of California, San Diego, La Jolla, California, USA.5Space and Naval Warfare Systems Center, San Diego, California,

USA.6Department of Electrical and Computer Engineering, Duke

University, Durham, North Carolina, USA.

Copyright 2007 by the American Geophysical Union.

0048-6604/07/2005RS003423$11.00

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http://dx.doi.org/10.1029/2005RS003423

ment of atmospheric pressure, temperature and humidityto determine the index of refraction; and (2) remotesensing techniques which infer refractivity more indi-rectly. Instruments for making direct measurementsinclude radiosondes [Rowland and Babin, 1987], micro-wave refractometers, and rocketsondes. A drawback ofthese devices, however, is that they are often expensiveand/or difficult to deploy [Halvey, 1983]. Moreover,these measurements tend to provide estimates of refrac-tivity versus height at a single range while propagationdepends on range-varying profiles at each azimuth.Among remote sensing methods, Doppler spread radarreturns have historically provided very detailed picturesof the dynamics and structure of the turbulent boundarylayer which, in principle, can be used to infer refractivity.In practice, however, the Doppler spread returns areoften contaminated by nonturbulence-related compo-nents [Skolnik, 1980]. Alternatively, lidar can provide ameans of estimating profiles of atmospheric water vaporthat can then be used to infer refractivity. However theperformance of lidar is severely restricted by backgroundnoise levels and high extinction (e.g., cloud) conditions.Ground-based point-to-point microwave propagationmeasurements using multiple transmitter/receiver pairs

[Tabrikian and Krolik, 1999], as well as the usage ofground-based measurements of GPS signals as the sat-ellites rises and sets on the horizon in inferring refrac-tivity [Anderson, 1994] has successfully overcome thesingle time and space line representation of refractivity.The inferred profiles would be characteristic of theintegrated refractive effects along the vertical and hori-zontal paths. In this regard, inferring refractivity fromradar clutter provides an extension of these strategieswhere now the radar itself is used as a remote sensingdevice.[4] The desirability of not having to use additional

equipment to estimate refractivity motivates the estima-tion of range and height varying refractivity from clutter(RFC). Previous work concerning the phenomenology ofsea clutter returns from extended ranges associated withducting conditions has been discussed in the work ofGossard and Strauch [1983]. In addition, it has beenshown that temporal and spatial variations of radarechoes are related to temporal and spatial variations inthe layers of the refractivity profile [Richter, 1969]. Theadvantage of RFC, as discussed in this paper, is that itprovides a synoptic characterization of the duct over thespatial extent of the radar and it overcomes the necessity

Figure 1. Plan position indicator data from SPANDAR radar on Wallops Island, Virginia, at2110 UT on 2 April 1998.

RS2014 VASUDEVAN ET AL.: REFRACTIVITY FROM CLUTTER

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of additional data and/or sensing devices. In addition,RFC has the added advantage of being able to senserange-varying refractivity at a temporal sampling ratethat can track changes in atmospheric conditions.[5] In previous work, RFC estimation was made by

using a maximum a posteriori (MAP) approach to jointlyestimate the refractivity and the spatially varying back-scatter cross section [Tabrikian et al., 1999]. The maindrawback of the method proposed in that paper was thatonly linear variations in refractivity parameters could beobtained. When the algorithm was modified to trackmore complicated range-varying refractivity, the estima-tion was found to be computationally intractable. Sub-sequent methods to improve estimation of morecomplicated range-varying behavior of refractivity wasundertaken by formulating the RFC problem as a non-linear state space problem whose solution was obtainedusing sequential importance sampling technique for statespace estimation [Vasudevan and Krolik, 2001]. Theapproach provided a quick and efficient method ofestimating complicated range-varying refractivity. How-ever, the method failed to update previous estimates ofrefractivity with the arrival of new data and hence theaccuracy of the estimates decreased rapidly with increas-ing range. More recently, a genetic algorithm (GA)approach to estimate refractivity RFC has been proposed[Gerstoft et al., 2003a, 2003b]. In that work, the authorshave presented a method to model the range and heightvarying refractivity of the environment using a total of11 parameters [Gerstoft et al., 2003b] while imposingsome prior constraints [Gerstoft et al., 2003a] in order toincrease the accuracy of results over the coverage area ofthe radar. The parameters are estimated by performing aglobal search using a nonlinear GA optimization ap-proach. In contrast to the need for global optimization, inthis paper the range-recursive nature of the parabolicequation is exploited to yield a potentially more compu-tationally efficient solution. More recently [Barrios,2004], a method has been proposed which utilizes therank correlation between the clutter power observed andthe density of raypaths to estimate refractivity. Theapproach discussed in that paper is designed primarilyfor surface-based ducts because of its utilization of landclutter to estimate refractivity over a coverage area whichincludes a land-sea boundary. The method presented inthis paper has the added advantage that, in principle itcan be used over both sea and land-sea transition byvirtue of the Markovian modeling of the range-varyingrefractivity profile.[6] The remainder of this paper is organized as fol-

lows. In section 2, a simple parameterization of the indexof refraction is described which covers a variety ofdifferent ducting conditions. This is followed by presen-tation of a nonlinear recursive state space model for theelectromagnetic field which is derived from the split-step

Fourier algorithm used to solve the wave equation ininhomogeneous media. Section 3 describes the MonteCarlo recursive Bayesian particle filtering approach usedto estimate range-varying refractivity from sea clutterreturns. Finally, section 4 is a discussion of simulationresults and the demonstration of the method using realradar data collected off the Virginia coast.

2. Statistical State Space Modeling of

Tropospheric Propagation

2.1. Tropospheric Refractivity Model

[7] Although the marine atmospheric boundary layer(MABL) variability which causes microwave ducting isoften complex in nature, for the purposes of propagationstudies, it is often represented at a given range by bilinearand/or trilinear height-dependent refractivity profiles[Rogers, 1996]. In the absence of ducting, standardpropagation conditions are represented by a linear profilewhich appears upward refracting in Earth-flattened coor-dinates [Anderson, 1994; Rogers, 1996]. In the presenceof a surface-based duct, the sudden change in refractivitywhich defines the duct is most commonly modeled bythe MABL structure shown in Figure 2. This simpletrilinear parameterization is quite versatile in its ability toinclude refractivity representations of the standard Earthatmosphere, surface-based ducts, elevated ducts andevaporation ducts. Surface-based ducts are ducts whichare formed at heights within a few hundred meters of thesurface of the Earth while ducts whose heights reach upto 5000 feet are termed as elevated ducts. Evaporationducts are similar to surface-based ducts, but their heightsare restricted to values ranging up to 40 m. Evaporationducting is not readily discernable from a radar’s planposition indicator (PPI), but still results in frequency-dependent extensions of radar range.[8] Instead of determining the refractivity at each point

over height at a given range, it has been shown [Rogers,1996] that the parameterization of the profile can belimited to a few variables without adversely affectingperformance. The refractivity structure for the MABLshown in Figure 2 is parameterized into 3 basicparameters, namely, the base height, thickness and theM deficit. The base height is defined as the height fromthe surface of the sea to the lower boundary of thetrapping layer. The thickness is defined as the distancebetween the lower and the upper boundaries of thetrapping layer. M deficit is the difference in the modifiedrefractivity values at the two boundaries of the trappinglayer. The base height and the M deficit to a large degreeaccount for the magnitude of the ducted field. The baseheight, thickness and M deficit determine the height ofthe duct in the troposphere and the number of rays thatwould be trapped in the duct. In addition, they also affect


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the curvature of rays in the trapping layer. A fourthparameter is the slope of refractivity in the lowest layer.The above trilinear model is versatile in its ability tomodel typical ducting conditions that have been ob-served. For example, when the base height reduces tozero, we end up with a bilinear surface-based duct, whichmodels the scenario where the internal boundary layer(IBL) controls propagation. When the base height rises tovalues above 1000 ft, the trilinear structure models thebehavior of an elevated duct. In practice, the refractivityprofile often changes with range. In this paper, a range-varying extension of bilinear/trilinear profile is achievedhere by modeling the parameters as a Markov process.This modeling, while being physically plausible, has theadded advantage of facilitating computationally efficientestimation of range-varying refractivity. In the abovemodeling, it is assumed that the value of the modifiedrefractivity at the sea level is known to be at 340 M units.The reason for this assumption is that the authors havefound that the most important information is in the slopesof the refractivity rather than their absolute values. As aresult, any commonly observed value in the range of280–360 can be used. In addition, the value of therefractivity in the third layer is assumed to be constantvalue of 0.113 M units/m. This is a valid assumptionsince most electromagnetic fields that enter this regionmakes no contribution to the field being measured at thesurface. Hence, for purposes of modeling, any realisticobserved value that does not make any significant con-

tributions of electromagnetic signals at the surface can beused.

2.2. Nonlinear State Space Formulation of FieldPropagation

2.2.1. State Equation Formulation[9] The most commonly used approach to model wave

propagation in the troposphere is the Fourier split-stepsolution to the parabolic equation [Dockery and Kuttler,1996; Dockery, 1998]. This numerical solution is aforward solver with the ability to handle vertical andhorizontal inhomogeneities in the refractive profile and iscapable of delivering accurate propagation loss estimatesin complicated environments.[10] Let u(rk, z) be the electric field at range rk and

height z. Then, the field at range rk+1 and height z,denoted by u(rk+1, z), is given by the Fourier split-stepsolution to the parabolic wave equation,

u rkþ1; zð Þ ¼ exp jko

2h2 þ 2z

ae� 1

� �dr

� �

� F�1 exp �jp2dx2ko

� �F u rk ; zð Þf g

� �; ð1Þ

where F is the spatial Fourier transform along the height z,dr = rk+1 � rk is the range increment, ae is the radius ofthe Earth, h is the refractive index as a function of heightand range, ko is the wave number and p is the spatialfrequency, or transform variable. It should be noted that

Figure 2. Refractivity modeling of the marine atmospheric boundary layer (MABL).


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the split-step Fourier solution to the parabolic equationcan be interpreted as multiplying the Fourier transform ofthe field at rk by an exponential term that amounts toFresnel diffraction of the wave in free space between thetwo ranges rk and rk+1. The inverse Fourier transform ofthis product, i.e., the field propagated in free space tork+1, is then multiplied by the phase changes broughtabout by aberration due to the inhomogeneous refractivemedium. It should be noted that the phase aberrationterm contains the index of refraction profile between thetwo ranges. Thus, given the field at a range k, the field atrange rk+1 is nonlinearly related to the index of refractionof the medium between the two ranges.[11] As described previously, without loss of general-

ity, let us assume that the index of refraction at range rk isparameterized into L = 4 parameters, namely, the baseheight bk, thickness tk, M deficit dMk and slope ofrefractivity in lower layer _Mk. The values taken by theseparameters are assumed to be a piecewise range-independent approximation to the profile between theranges rk and rk+1. Let gk = [bk, tk, dMk, _Mk]

T be thevector of these refractivity parameters at range rk. Sincethe index of refraction is a function of physical quantitiessuch as pressure, temperature and humidity, it is assumedhere that the parameters of the refractivity do notundergo drastic changes over small range intervals.Moreover, the correlation of the refractive process sug-gests characterizing the relationship between the param-eters from one range step to another by a Markovprocess. Then, the relationship between the refractivityparameters over range can be expressed as

gk ¼ gk�1 þ wk ; ð2Þ

where wk models the uncertainty in the variations ofrefractivity parameters over range. For simplicity, it isassumed to be a zero-mean Gaussian process withcovariance matrix

Pg.

[12] Let uk = [u(rk, z1), u(rk, z2),. . .,u(rk, zM)] be thevector of complex field values at heights z1, z2,. . .,zM atrange rk. The field as given in (1) is a function of all therefractivity parameters up to range rk�1. Thus (1) can bewritten as

uk ¼ f g1; g2; . . . ; gk�1; u0ð Þ; ð3Þ

where the function f(.) represents the repeated applicationof the split-step solution to parabolic equation given in(1) out to range rk and u0 is the field distribution at thestarting range r0. In equation (3), the vector ofrefractivity parameter gk has L unknown parameters ateach range rk. Thus as the range increases, the number ofunknowns increases by L times the number of rangeincrements. Thus, for example, for L = 4, if therefractivity parameters change every kilometer, thenwith this approximation, by the time we reach 20 km,

which is the distance to the horizon for shipboard radarsystems, the total number of parameters to be estimated is80. Normally, estimates are required out to 150–200 km.As a result, the number of parameters to be jointlyestimated is around 600–800. This is clearly aprohibitive number of parameters to estimate given thelimited extent of the radar clutter data and computationalresources.[13] In order to expedite the solution for the g1,

g2,. . .,gk, note that in the split-step solution to the waveequation, the field at range rk can also be modeled as aMarkov process if the complex field at the previousrange step rk�1 is included as part of the state vector. Inparticular, using equation (1), equation (3) can be writtenas

uk ¼ H uk�1; gk�1ð Þ; ð4Þ

where H(.) corresponds to the split-step operation asrepresented by equation (1). Combining equations (1),(2), and (4), we get the new state equation

gkuk

� �¼ gk�1

H uk�1; gk�1ð Þ

� �þ wk

0

� �: ð5Þ

[14] In the above formulation, it can be seen that thesource and refractivity profile up to range rk�1 areincorporated into the field at rk. A significant reductionin the dimensionality of the field vector uk at each rangestep rk is possible when one utilizes the fact that becausetypical M deficits are on an average below 60 M units,only rays that have a grazing angle less than 1 degreetypically propagate within the duct. In light of the above,instead of using the complex field vector over height z ateach range, the complex vertical frequency wave numbercoefficients vk = F{uk} are used instead, where F is thespatial discrete Fourier transform. An upper bound on thenumber of significant vertical wave number coefficientsis given by the spatial frequency aperture-bandwidthproduct. This is analogous to the time-bandwidth productdimensionality of an approximately time and bandwidthlimited process. The maximum vertical wave number is

knmax¼ k sin qmaxð Þ; ð6Þ

where qmax is the maximum angle that actuallypropagates in the medium. The vertical spatial aperturein this case is the maximum expected duct height zmax.Given this maximum vertical wave number, the discreteFourier transform size used in the PE solver of equation(1) is determined using the spatial frequency-apertureproduct given by

zmaxknmax¼ Nf p; ð7Þ


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where zmax is the maximum height to which propagationloss prediction is required and Nf is the Fourier transformsize. For the current problem, the maximum angle thatpropagates in the duct is typically less than 1 degree andassuming the height to which propagation loss predic-tions are required is 200 m, the transform size is 64.Letting K denote the total number of clutter range bins,the advantage of using a Markov model with verticalwave number coefficients in the state vector is thereforethat KL refractivity parameters can be solved recursivelyby updating a state vector of size dim(knmax

zmax) + L KL at each range versus a joint nonlinear optimizationover all KL parameters. This yields a significantly morecomputationally efficient approach.[15] Letting vk denote the vector of vertical wave

numbers coefficients over height at range rk, then inthe spatial frequency domain, equation (4) can be writtenas

vk ¼ F H F�1 vk�1ð Þ; gk�1

� � ¼ T vk�1; gk�1ð Þ; ð8Þ

where F is the discrete spatial Fourier transform, H(.)corresponds to the split-step operation as represented byequation (1) and T(vk�1, gk�1) ¼4 F(H(F�1 (vk�1), gk�1)).Combining (2) and (8) into one equation, we have

gk ¼ gk�1 þ wk

vk ¼ T vk�1; gk�1ð Þ: ð9Þ

[16] It should be noted that no state noise is applied tothe second equation in (9) since this equation corre-sponds to the deterministic split-step PE computation.Finally, defining xk ¼4 [gk, vk] as the state vector at rangerk, the state equations in (9) can written compactly as

xk ¼ h xk�1ð Þ þ mk ð10Þ

where h (�) is the nonlinear relationship between thestate at range rk and rk�1 and mk = [wk, 0]

T is the statenoise which is zero-mean Gaussian with covariance

matrixPm

¼P

g 0

0 0

� �.

2.2.2. Measurement Equation Formulation[17] Having incorporated the propagation model in the

state equation of (10), the measurement equation can beused to model the clutter return. A general expression forthe radar clutter return, y(rk), at slant range bin rk is givenby

y rkð Þ ¼Zr0

Zw0

H rk ; r0;w0ð Þb r0;w0ð Þdr0dw0 þ n rkð Þ; ð11Þ

where b(r0, w0) is the complex coefficient of the seasurface, and the impulse response of the radar, H(rk; r

0,w0), defines the output at rk due to a point source atground range and bearing (r0, w0) on the ocean surface.Note that H(rk; r0, w0) is a function of both radarparameters such as the beam former weights and pulseshape, as well as propagation model parameters such asrefractivity. Because of surface roughness, b(r0, w0) maybe modeled as a complex zero-mean random process.Additive noise at the receiver is denoted by n(rk) and ismodeled as complex zero-mean Gaussian distributedand uncorrelated with the clutter b(r, w). The impulseresponse H(rk; r

0, w0) can be modeled approximately asH(rk; r, w

0) L(rk; g1, g2,. . ., gk)h(r; r0, w0), where L(rk;

g1, g2,. . .,gk) is the two-way propagation loss whichdepends on unknown refractivity profile vector para-meters, g1, g2,. . .,gk, and h(r; r0, w0) is the impulseresponse determined assuming propagation through freespace and using the radar’s beam and pulse width.Thus, for a series of n pulses, the radar return fromequation (11), sampled at range bins (r1, r2, . . .,rK), isgiven by

yn rkð Þ ¼ an rkð ÞL rk ; g1; g2; :::; gkð Þ þ un rkð Þ; ð12Þ

where n = 1, 2,. . .N is the number of pulsestransmitted by the radar. The clutter return at the nthsnapshot (corresponding to the nth pulse transmitted)

an(rk) =R R

Wh(rk; r

0, w0) b(r0, w0)dr0 dw0is modeled as a

complex Gaussian random variable with a variance sa2

which is assumed to be a constant over range. Thevariance of this clutter coefficient is the backscattercross section of the sea surface and depends on the seastate, grazing angle, and the frequency at which theradar operates. Again, L (rk; g1, g2,. . ., gk) is the two-way propagation loss, which is a function of theunknown refractivity parameter vectors g1, g2,. . ., gkwhich we are trying to estimate.[18] Although (12) describes the complex clutter re-

turn, a more common output of radar systems is the planposition indicator (PPI) output, defined as the clutterpower averaged in decibels (dB) over the number ofradar pulses. The PPI is thus defined by

zk ¼10

N

XNn¼1

log10 yn rkð Þj j2: ð13Þ

[19] It can be shown (see Appendix A) that under themodel of (12), zk in (13) is well approximated by a

Gaussian distribution with mean zk =10

log 10ð Þ log (jL(rk; g1,


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g 2 , . . .,g k ) j2 sa2 + su

2 ) � 10glog 10ð Þ and v a r i a n c e

10N log 10ð Þ

�2�2

6þ log 2ð Þ � gð Þ2

h i, where g is Euler’s num-

ber. Given L(rk; g1, g2,. . ., gk), the returns from differentnonoverlapping range bins are uncorrelated since theyilluminate different surface scatterers. The mean of thisprocess is thus a range varying function of propagationloss and depends on the vector of refractivity parametersg1, g2,. . ., gk in a nonlinear manner. In terms of anobservation equation, the PPI output at range rk, can bewritten up to an additive constant in terms of the statevector xk from (10) as

zk ¼10

log 10ð Þ log CHG xkð ÞGH xkð ÞC�� 2s2

a þ s2u

h iþ ek

¼ b xkð Þ þ ek ;

ð14Þ

where G(xk) = F�1 (vk) is the inverse discrete Fouriertransform of the wave number coefficients in the statewhich recreates the field distribution at range rk over height,C= [0, 1, 0, 0, . . .0]T is the vector used to select the intensityof the field near the sea surface from the state vector xk andwhere b(xk) ¼4 10

log 10ð Þlog (jCHG(xk)GH (xk)Cj2 sa2 + su

2) isthe mean of zk, a nonstationary function of range. The fieldsmust be sampled just slightly above the surface because ofthe zero boundary condition at the surface. From (10) and(14), note that the RFC problem has been modeled as anonlinear continuous state space estimation problem. The

objective is now to estimate the parameters of therefractivity gk from the PPI observations.

3. Recursive Bayesian Estimation

[20] The classic approach to state estimation in non-linear state space models is the extended Kalman filter(EKF), which consists of linearizing the state and/ormeasurement equations using Taylor’s series expansions[Gelb, 1974; Anderson and Moore, 1979]. In RFC, anextended Kalman filter approach is problematic becausethe parameters of interest, i.e., refractivity, appear in thecomplex exponential in equation (1) which whenlinearized leads to instability of the EKF and very poorestimates of range-varying refractivity parameters.[21] For discrete-state sequence estimation problems,

the evolution of the state can be described by a latticewhere a column of nodes represents the finite number ofpossible state values at a particular time. Using theMarkov property of the state sequence, a computation-ally efficient method of computing the joint maximum aposteriori (MAP) estimate of the state sequence is theViterbi algorithm [Forney, 1973], based on dynamicprogramming. Even though the number of possiblesequences grows exponentially with the length of thesequence, the computational complexity of the ViterbiMAP estimate grows only linearly. At least formally, aMAP RFC estimate could be achieved by discretizing theelements of the state xk, which contains the refractivityparameters and electromagnetic field, at range, rk, onfixed grid. Let xk,j denote the state vector sampled on a

Figure 3. Discrete state space.


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deterministic set of grid points indexed by j = 1,. . ., J, atrange rk. The resulting lattice for this discrete state spaceis shown in Figure 3. Defining p(x1,. . .,xKjz1,. . ., zK) asthe joint posterior density of the states up to range K,f (zkjxk) as the likelihood of state xk, k = 1,2,. . .K andp(xkjxk�1) is the transition probability associated with atransition from jth state at range k� 1 to ith state at range k,the MAP state sequence estimate is defined by

argmaxx1;x2;...;xK

flog p x1; . . . ; xK jz1; . . . ; zKð Þg

¼ argmaxx1;x2;...;xK

XKk¼1

log f zk jxkð Þ þ log p xk jxk�1ð Þ( )

:

ð15Þ

[22] The Viterbi method finds the optimal path recur-sively by computing

Wi kð Þ ¼maxj Wj k� 1ð Þ þ log pxk jxk�1xk;ijxk�1;j

� � �þ log fzk jxk zk jxk;i

� ; ð16Þ

where Wi(k) is the weight of the optimal path terminatingat xk,i. The sum in equation (15) corresponds tocomputing weights Wi (k) for a particular path throughthe lattice in Figure 3. Note that the total number of statesequences evaluated by equation (16) is JK which growsexponentially in K. However, the computational com-plexity of the Viterbi algorithm grows only linearly withK which gives a dramatic reduction in computation. ForMarkov discrete state sequences, the Viterbi algorithm isguaranteed to find the global MAP estimate.[23] In the case of a continuous state space, as given in

equations (10) and (14), we seek the global maximum tothe continuous posterior density f (XkjZk), where Xk =x1, . . .,xk is a state sequence and Zk = z1, . . ., zk is thesequence of scalar observations out to range rk. Thismaximum could theoretically be estimated using adense discrete grid of sequence values. To uniformlysample L + I parameters (where L is the number ofvertical wave number frequencies propagating and I is thenumber of refractivity parameters) in xk usingN values perparameter, requires J = (L + I)N grid points. If the RFCproblem were solved using a fixed discretization scheme,we would require an exceptionally large number of gridvalues, e.g., N = 3000, that would render the problemcomputationally intractable. In this paper therefore wepursue a particle filtering approach [Godsill et al.,2001], the idea is to construct a grid based onrealizations Xk

(i) drawn from an appropriate probabilitydistribution, where we denote the ith state sequencerealization by Xk

(i). If the support of the density fromwhich these realizations are drawn includes the unknownmaximum, then the maximum over the resulting grid has

been shown to asymptotically approach the globalmaximum in the limit that the number of realizationsdrawn (and thus the number of grid points) approachesinfinity but in practice the solution converges much morequickly [Godsill et al., 2001].[24] Suppose that the number of realizations (a.k.a.

‘‘particles’’) drawn is J. The number of possibletrajectories in the lattice formed by these realizations isthen JK. The posterior density can then be efficientlymaximized over these JK sequences by use of the Viterbialgorithm. An appropriate sampling density must beselected from which to draw the state realizations. In thiswork, for simplicity, we choose the prior distribution asgiven by equation (10), augmenting the sequence of eachrealization Xk�1

i with a new state value xki , which,

because of the Markov property, depends only upon theimmediately preceding state value xk�1

i . An initial priorp(X0) given by

p X0ð Þ ¼ U g0ð Þd v� v0ð Þ; ð17Þ

which is utilized to obtain the initial distribution of thestate space, which comprises the refractivity parametersg0 and the wave number coefficients v0. A uniform prioron the refractivity parameters U(g0) distributed over theparameter space is assumed. The prior on the initial wavenumber spectrum, v0, corresponds to the radar antennapattern. From (17), the priors on g0 and v0 are assumed tobe independent.[25] The state space at any range k is generated from the

state space at range k � 1 by passing the state particlesxk�1,i at range k � 1 through the state equation (10) toobtain the particles at range k. This transition is indicatedby the dotted black lines in Figure 4. This results in agrid, as illustrated in Figure 4, which is the randomlysampled version of continuous state space (as comparedto the fixed which is shown in Figure 3). It should benoted that with fine enough sampling in the fixed statespace model of Figure 3, the lattice encompasses allpossible trajectories whereas in the randomly sampledcontinuous state space shown in Figure 4, the latticeencompasses all possible trajectories only as the numberof samples goes to infinity.[26] Using the randomly sampled grid, the MAP RFC

estimate, by analogy with the discrete state space model,is given by

argmaxx ið Þ1;x ið Þ

2;...;x ið Þ

k8i

nlog p x ið Þ

1; . . . ; x ið Þ

Kjz1; :::; zK

�o¼

argmaxx ið Þ1;x ið Þ

2;...;x ið Þ

k8i

XKk¼1

log f zk jx ið Þk

�þ log p x

ið Þk jx ið Þ

k�1

�h i( );

ð18Þ


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where x1(i), x2

(i), . . .,xk(i) is the state sequence correspond-

ing to the ith sample in the randomly sampled state spacegrid of Figure 4. Again, the Viterbi algorithm is used tofind the optimal path along all the trajectories byrecursively computing

Wið Þ

k ¼ maxj Wjð Þ

k�1 þ log pxk jxk�1x

ið Þk jx jð Þ

k�1

�n oþ log fzk jxk zk jx ið Þ

k

�: ð19Þ

[27] As mentioned previously, as the number of sam-ples increases, we cover all possible trajectories in thecontinuous state space and in the limit that the number ofsamples goes to infinity, the MAP estimate for thecontinuous state space problem converges to the trueMAP estimator [Godsill et al., 2001]. The advantage ofthe particle filtering approach is that in practice only afraction of the number of samples are required whencompared to a fixed gridding method.[28] The algorithm in the MAP RFC method using the

Viterbi approach is summarized as follows.[29] 1. Generation of the randomly sampled distribu-

tion of the continuous state space.

a. For 1 � i � N, x0(i) ¼4 p(x0)

b. For 1 � k � K and 1 � i � N,

xk(i) = h(xk�1

(i) ) + mk(i)

[30] 2. Viterbi algorithm for MAP sequence estimation.

a. For 1 � i � N,

W0(i) = log pxkjxk�1

(x0(i)) + log fzkjxk (zkjx0

(i)),

b. For 1 � k � K and for 1 � i � (L + M),

Wk(i) = max

j{Wk�1

(j) + log pxkjxk�1(xk

(i)jxk�1(j) )}

+ log fzkjxk (zkjxk(i))

ck(i) = argmax

j¼1;2;3;...N{Wk�1

(i) + log pxkjxk�1(xk

(i)jxk � 1(j) )}

c. At the last range bin, i.e., k = K,

i kMAP¼4 argmax

i¼1;2;3;...; LþMð ÞWk

(i)

xkMAP¼4 xk

ikMAP

d. Trace back to get optimal sequence of refractivityparameters across range.

For k = K � 1, K � 2,. . .., 2, 1,

ik = ck+1 (ik+1),

xkMAP= xk

(ik)

[31] Note that the state equation (10) ensures smooth-ness in field values is maintained by constraining thetransition probability density p(xk

(i)jxk�1(j) ). In particular,

Figure 4. Formulation of particle trajectories as state sequences.


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for the case where the PE solver step is deterministic, thetransition probability density takes on the form, (seeAppendix B), given by

p xið Þk jx jð Þ

k�1

�¼ d v ið Þ

k� f v jð Þ

k�1; g

ið Þk

� �jg

ið Þk¼g

jð Þk�1

þwk

� p gið Þk � g

ið Þk�1 þ wk

� �: ð20Þ

[32] If instead of estimating the entire MAP sequenceestimate at each range step, we were interested only inthe estimating xk given data up to range rk, alternativelythe solution for xk can be alternatively computed using afiltering approach which maximizes p(xkjz1, z2, . . ., zk) ateach range rk. Defining, the forward variable, ak

(i) ¼4

p(xk(i), z0, z1, . . ., zk) where ak

(i) is the joint probability ofthe data set z0, z1, . . ., zk and the ith sample of state atrange rk, xk

(i). The forward variable is computed in arecursive manner using the formula [Therrien, 1992]

a ið Þk ¼

XI

j¼1

a jð Þk�1p x

ið Þk jx jð Þ

k�1

�" #p zk jx ið Þ

k

�: ð21Þ

Once all the forward variable have been computed, thefiltered estimates can be easily calculated. In order to doso, calculation of the filtered density function is realized as

a ið Þk ¼4 p x

ið Þk; z0; z1;...; zkð Þ

/ p xið Þk jz0; z1; . . . ; zk

�:

ð22Þ

Thus the filtered distributions are obtained by normalizingthe computed forward variables for each of the particle ateach range,

p xið Þk jDk

�¼ w

ið Þk ¼

a ið ÞkPI

i¼1

a ið Þk

ð23Þ

The forward MAP estimate based on the marginaldistribution of xk at each range step rk is given by

xkMAP¼ max

iw

ið Þk : ð24Þ

In the next section, the Viterbi and forward MAP RFCestimates will be compared in simulations and using realdata.

4. Refractivity From Clutter Results

4.1. Simulation Results

[33] In order to evaluate the RFC methods described inthe previous section, simulations were performed using50 realizations of perfectly trilinear profiles were used to

generate 50 range_varying refractivity profiles bymeans ofa simple Markov process as given in equation (2). Thesimulated range-varying refractivity, along with the initialfield which corresponded to the antenna pattern, were thenused to predict the field distributions by making use of thesplit-step Fourier solution to the parabolic equation (PE)given in (1). In the implementation of the split-step PE, therange increments were 200 m and the Fourier transformsizes were of the order of 512. The field was determined upto a height of 200 m with a Hanning window added forheights beyond 200 m so that any reflections from thecomputational boundary were not significant. The PEsolution was used to compute a simulated clutter data usingequation (14). This simulated clutter data was then used as‘‘data’’ to obtain forward MAP and Viterbi MAP sequenceestimates of range-varying refractivity.[34] For the estimation of refractivity, the initial prior

on each of the refractivity parameters values was assumedto be uniform and independent of one another. In partic-ular, the base height was assumed to be uniform between10 and 150 m, thickness was assumed to take on valuesfrom 5 to 70 m, M deficit was assumed to be uniformbetween 0 and 65 M units, while the slope of the mixedlayer was assumed to be uniform between �0.13 and0.13 m/M units. Five hundred samples of the priordistribution were drawn to represent the set of possiblerefractivity values at the radar. Similarly, 500 copies ofthe radar radiation pattern were used to generate thevertical wave number spectrum used in the state vector ofequation (9). Thus the 500 samples of the refractivity andthe vertical wave number spectrum together represent theset of all possible realizations of the state vector at theradar. Note that this represents a small fraction of the gridpoints that would be required for fixed uniformsampling. Now, the randomly sampled state space shownin Figure 4 was generated using the method presented insection 3. Once the randomly sampled state space isachieved, we use the particle filtering approach to theViterbi MAP sequence estimation detailed in section 3 toproduce estimates of range-varying refractivity.[35] We note that in practice, the transition probability

governing the wave number spectrum was not consid-ered to be a delta function, but rather a more relaxedcondition of a tight uniform distribution was used. Thisallowed for more diversity in the paths through the latticewhich was found to speed convergence. The tight uni-form distribution was defined such that transition jumpsin the field on the order of 5 dB and less were allowedwhile jumps greater than 5 dB were not allowed. Thus(20) was actually implemented as

p xið Þk jx jð Þ

k�1

�¼ p E log v

ið Þk � f v

jð Þk�1; g

ið Þk

� � � �� j

g ið Þk¼g

jð Þk�1

þwk� p g

ið Þk � g

ið Þk�1 þ wk

� �; ð25Þ


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where p(E(log(Fvk(i) � f(Fvk�1

(j) , gk(i))))) � U[�5, 5], E is

the expectation operator and F is the inverse Fouriertransform operator.[36] Example RFC results for the scenario described

are shown in Figure 5. The top plot of Figure 5 is thecomparison of the refractivity estimates with the truerefractivity. The absolute average error between theestimated refractivity and the true simulated refractivitywas found to be 5.4963 M units. The index of refractiongenerated using estimates was then passed through the

split-step PE solution to obtain the field distribution overthe entire area of coverage. The same procedure wasrepeated to obtain the field distribution for the truesimulated refractivity. As the final goal of the work wasthe comparison of the true propagation loss to thepredicted propagation loss, the propagation loss corre-sponding to refractivity estimates and the real simulatedrefractivity were calculated and they are shown as thesecond and third plots of Figure 5. It can be seen thatthere is an excellent match between the two propagation

Figure 5. Matched profile result: Viterbi MAP sequence estimation.


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loss plots. Now, in order to quantify performance, wedefine the absolute average error up to a scaling constantas, e¼4

Pr;h

20 log 10qtrueqestð Þj j, where q is the term whose

performance is being quantified (e.g., clutter, propaga-tion loss or any of the refractivity parameter values). Itshould be noted that, when we quantify performance forclutter, the parameter q is a function of range onlywhereas, when we quantify performance for propagationloss, the parameter q is a function of range and height,the average being computed over the entire domain. Theerror unit is decibels only for the case when we areestimating the error of propagation or clutter loss. Thusthe absolute average error, for the entire area of coverageextending from 0 km to 100 km in range and from 0 m to200 m in height, between the propagation losscorresponding to the refractivity estimates and thepropagation loss corresponding to the true refractivitywas found to be 4.3304 dB. Finally the field valuescorresponding to the refractivity estimates were used togenerate an estimated clutter data using equations (1)and (14), to enable comparison with the true simulatedclutter data generated using the true refractivity profilesas inputs in equations (1) and (14). The bottom plot ofFigure 5 contains the solid blue line which is the plot ofthe data and the estimated clutter data given by thesolid red line. It can be seen that the algorithm is ableto effectively capture the peaks in the data arisingbetween 50 and 65 km and 70 and 80 km, respectively.The average absolute error between the estimatedclutter and the real simulated clutter data was foundto be 3.2811 dB.[37] For over 50 Monte Carlo realizations of range-

varying refractivity, the absolute average error betweenthe propagation loss using the true refractivity and thepropagation loss using the estimated refractivity arecomputed and tabulated. The same was done for theabsolute average error between the true clutter and theclutter estimate obtained using the estimated refractivity.The results are shown in Table 1. In addition to theViterbi MAP sequence estimate, the forward algorithm

results were also tabulated for the 50 Monte Carlorealizations. It can be seen that the Viterbi sequenceestimate performs better than the forward Monte Carloestimate because of the ability of the algorithm toupdate estimates of refractivity at earlier ranges with thearrival of data at later ranges (i.e., step 2d in the algorithm).

4.2. Performance Under Refractivity ModelMismatch Using Real Data

[38] In order to test the robustness of the estimator tomismatches in the perfectly trilinear profile model,simulated clutter data was created using equations (1)and (14) using actual refractivity measurements made bythe Applied Physics Laboratory of Johns Hopkins Uni-versity (APL-JHU) near Wallops Island, Virginia. Ahelicopter-borne instrument package was flown to andfro (multiple times) from the shore to a distance ofapproximately 60 km out to sea measuring the refractiv-ity. Each trip of the instrument package is given a casenumber (10,11,12. . .22) as illustrated in Figure 6. Thesolid black lines in Figure 6 are some of the real profilemeasurements made by the helicopter. Although theground truth was measured out to 60 km, the data wasextended to 100 km by selecting the original measure-ments again by a simple random walk procedure and thisis shown as solid dark lines in Figure 7. The sameassumptions of uniform prior generated in the case ofperfectly matched profiles simulations were used for thisscenario also. Estimates of range-varying refractivitywere then obtained using the Viterbi MAP sequenceestimate from the simulated clutter data. Using theseestimates of refractivity, the propagation loss over thearea of coverage was generated using equation (1). Atypical result for the scenario described is shown inFigure 7. In the top plots of Figure 7, the range-varyingrefractivity estimates (dashed lines) are compared to thetrue range-varying refractivity values (solid lines). Notethat the refractivity estimates match closely to the meas-urements and even capture the range variations of therefractivity quite accurately. The absolute mean value ofthe error between the ground truth refractivity measure-ments and the estimated range-varying refractivity, takenover the entire area of coverage, namely, over range from0 km to 100 km and height from 0 m to 200 m, was3.8673 M units. Similar to Figure 5 of the matchedprofile scenario, the comparison of clutter obtained usingestimates of refractivity in equations (1) and (14) and thatof the of actual clutter, as well as the simulatedpropagation loss and the true propagation loss plots areindicated by the bottom three plots. The absolute averageerror between the propagation loss computed using theestimates of refractivity and those computed using theground truth was found to be 6.2493 dB. Finally,the estimated clutter was generated in the same manner

Table 1. Performance Comparison of Forward MAP Estimate

and Viterbi MAP Sequence Estimate

Number of Trials

Absolute AverageError in ForwardMAP Estimate, dB

Absolute AverageError in ViterbiMAP SequenceEstimate, dB

Clutter Prop Loss Clutter Prop Loss

50 4.5234 6.8976 3.6922 6.608710 6.0145 6.6473 3.7585 6.601312 7.127 7.5123 5.9131 7.0747


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Figure 6. Real data profile fits. Dashed line profiles show RFC Viterbi MAP sequence estimates.Solid line profiles show true helicopter profiles.


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as in the matched profile case and it was compared to thetrue simulated clutter as shown in the fourth plot ofFigure 7. The absolute average error between theestimated clutter and the true clutter for the plot shownis 2.9162 dB. For 10 Monte Carlo realizations mis-matched refractivity, the average absolute error inpropagation loss is tabulated in Table 1. As illustratedin Table 1, even under mismatch, the Viterbi MAPsequence estimate method performs much better than theresults obtained using the forward algorithm estimates.

4.3. Real Data Results

[39] To test RFC with real clutter, data was collectedon 2 April 1998 using the Space and Range radar(SPANDAR) located at Wallops Island, Virginia. Theradar transmitter/receiver is approximately 30 m highand was operating at a frequency of 2.85 GHz with apulse width of 1m sec and a pulse repetition frequency of500 Hz. Real clutter data was measured at azimuthsranging from 0� to 360� at intervals of 0.4�. At eachazimuth, data was collected from 0 km to approximately

Figure 7. Mismatched profile result: Viterbi MAP sequence estimation.


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Figure 8. Errors in propagation loss for real data cases for Viterbi MAP sequence estimates.


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Figure 9. Errors in propagation loss for real data case for forward MAP estimates.


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260 km at increments of 600 m. Figure 6 shows thecomparison between the range-varying refractivity ob-tained using the real clutter data and the ground truthrefractivity measurements made using a helicopter asdescribed earlier. It can be seen from these plots that thefeatures of the duct, especially its range-varying natureare captured in most cases. Figure 8 illustrates the averageabsolute error between the propagation loss estimatesobtained using the Viterbi MAP sequence estimates ofrefractivity and the propagation loss generated by usingthe ground truth refractivity data. For completeness, inFigure 9 we also included the results for the forward MAPmethod which performed less well. This is due to theability of the MAP sequence estimate to update estimatesat earlier ranges with the arrival of new data thus reducingthe propagation of any error in estimates down the range.It can also be seen from Figure 8 that the error is minimalwithin the duct. The majority of the error contributionsarise in areas outside the duct. The average absolute errorin quantities propagation loss and clutter were calculatedfor the 12 real data cases and tabulated in Table 1. Onceagain, it can be seen that the Viterbi MAP sequenceestimate gives better performance than the forward MAPfiltered estimates.

5. Conclusion

[40] This paper has shown how recursive Bayesianestimation when combined with forward and Viterbialgorithms can be used to solve the problem of estimat-ing RFC in a sequential manner. This scheme provides asway of quickly obtaining propagation loss estimatesunder complicated ducting conditions. The simulationand the real data results clearly show that the propagationloss estimates obtained from RFC are comparable tothose obtained from ground truth refractivity measure-ments. Comparison of the Viterbi and forward algorithmresults indicate that it is critical to update the estimate ofrefractivity at all ranges with the arrival of new data inorder to obtain good performance. In the work presented,we have represented a continuous state space by sparsenumber of samples obtained by sampling from a sim-plistic prior distribution. Further work is necessary todetermine more sophisticated methods of sampling thecontinuous state space in order to obtain very accurateresults and faster convergence of the algorithm bysampling from distributions that are data adaptive.

Appendix A

[41] Let us define rn (k) ¼4 ynðkÞj js2f=2, where

s2f ¼4Var yn kð Þð Þ

¼ L xk ; gð Þj j2s2a þ s2

u: ðA1Þ

Then the clutter data can be written as

zk ¼10

N log 10

XNn¼1

log rn kð Þð Þ þ 10

log 10log s2

f=2 �

:

ðA2Þ

The mean of the random variable statistical model of theclutter data zk is given by

E zkð Þ ¼ 10

N log 10

XNn¼1

E log rn kð Þð Þð Þ

þ 10

log 10log s2

f=2 �

: ðA3Þ

[42] Since the clutter returns are independent andidentically distributed (i.i.d.), the expectation does notdepend on the snapshots. Hence the above equationbecomes

E zkð Þ ¼ 10

log 10E log rn kð Þð Þð Þ þ 10

log 10log s2

f=2 �

:

ðA4Þ

The random variable (r.v.) rn(k) is chi-square distributedwith 2 degrees of freedom, as it is the modulus squaredof a complex r.v. The term E(log(rn(k))) can be shown tobe equal to (log(2) � g), where g is Euler’s number.Thus the equation reduces to the form

E zkð Þ ¼ 10

log 10E log 2ð Þ � gð Þ þ 10

log 10log s2

f=2 �

:

ðA5Þ

Substituting (A1) in (A5) and by simple algebraicmanipulation, we get

E zkð Þ ¼ 10

log 10ð Þ log L xk ; gð Þj j2s2a þ s2

u

�� 10glog 10ð Þ :

ðA6Þ

Similarly, the variance of the clutter data can be provento be

Var zkð Þ ¼ 10

N log 10ð Þ

� �2

Var log rn kð Þð Þð Þ

¼ 10

N log 10ð Þ

� �2 p2

6þ log 2ð Þ � gð Þ2

� �: ðA7Þ


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Appendix B

[43] The transition probability can be written as

p xk;ijxk�1;j

� ¼

Zp xk;ijxk�1;j;mk

� p mkð Þdmk ; ðB1Þ

where mk is the state noise given in (10).[44] Now, splitting the terms of the state vector into the

field components and the refractivity parameter compo-nents, we get

p xk;ijxk�1;j

� ¼Z Z

p uk;i; gk;ijuk�1;j; gk�1;j;wg;wu

�� p wg;wu

� dwgdwu; ðB2Þ

where wg and wu are the noise associated with therefractivity parameters and the field in the state equation.[45] Assuming that the joint probability of the noise

can be separated as the product of the marginal proba-bilities, corresponding to noise in refractivity parametersand that in the field, we get, separating the integrals,

p xk;ijxk�1;j

� ¼Z Z

p uk;ijgk;i; uk�1;j; gk�1;j;wg;wu

�� p gk;ijuk�1;j; gk�1;j;wg;wu

�p wg

� dwg p wuð Þdwu:

ðB3Þ

However, the conditional distribution on the refractivityparameter, uk, is simply a delta function since given allthe other terms, it is purely deterministic. Hence theintegral is simplified into

p xk;ijxk�1;j

� ¼ p uk;ijgk;i; uk�1;j; gk�1;j;wg;wu

�p wg

� jgk;i¼gk�1;jþwk

¼ d uk;i � f uk�1;j; gk;i� �

jgk;i¼gk�1;jþwk

� p gk;i � gk�1;i þ wk

� � : ðB4Þ

[46] Acknowledgments. The authors would like toacknowledge the work by Joseph Tabrikian, currently withBen-Gurion University, Ber-Sheva, Israel, in the derivation ofthe clutter model and also in the development of some of thepreliminary ideas associated with the Markov model for fieldpropagation in the troposphere.

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([email protected])

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Lexington, MA 02420, USA. ([email protected])

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Engineering, Duke University, Durham, NC 27708-0291,

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